Paper Title: Slow decay rate of correlations induced by long-range extended Dzyaloshinskii-Moriya interactions.





Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute,  Chhatnag Road, Jhunsi, Prayagraj - 211019, India


§ ABSTRACT 
    We examine the impact of long-range  Dzyaloshinskii-Moriya (DM) interaction in the extended XY model on the phase diagram as well as the static and dynamical properties of quantum and classical correlation functions. It is known that in the nearest-neighbor XY model with DM interaction, the transition from the gapless chiral phase to a gapped one occurs when the strengths of the DM interaction and anisotropy coincide. We exhibit that the critical line gets modified with the range of interactions which decay according to power-law.   Specifically, instead of being gapless in the presence of a strong DM interaction,  a gapped region emerges which grows with the increase of the moderate fall-off rate (quasi-long range regime) in the presence of a transverse magnetic field. The gapless chiral phase can also be separated from a gapped one by the decay patterns of quantum mutual information and classical correlation with distant sites of the ground state which are independent of the fall-off rate in the gapless zone. We observe that the corresponding critical lines that depend on the fall-off rate can also be determined from the effective central charge involved in the scaling of entanglement entropy.  We illustrate that in a non-equilibrium setting, the relaxation dynamics of classical correlation,  the decay rate of total correlation, and the growth rate of entanglement entropy can be employed to uncover whether the evolving Hamiltonian and the Hamiltonian corresponding to the initial state are gapped or gapless. 

 Slow decay rate of correlations induced by long-range extended Dzyaloshinskii-Moriya interactions
    Tanoy Kanti Konar, Leela Ganesh Chandra Lakkaraju, Aditi Sen(De)
    
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§ INTRODUCTION


Long-range order across quantum systems is desirable for a variety of quantum technological tasks, as it facilitates the sharing of correlation among distant parts of the system. Historically, such long-range order has been observed in many-body systems at quantum criticality or when the spectrum becomes gapless. In the case of the nearest-neighbor (NN) XY model with nonvanishing anisotropy parameter, it has been demonstrated that the classical correlations decay with the distance between the spins  exponentially (polynomially) when it is away from (at) the criticality.
Beyond NN models, long-range interacting (LR) systems with power-law decay in the range of interactions have been shown to exhibit polynomial decay of correlations, even when the spectrum is not gapless. These systems become an intense topic of research in the last few years, especially in the context of building analog quantum simulator<cit.> and due to their natural occurrence in several physical platforms such as the Rydberg atom arrays <cit.>, dipolar systems <cit.>, polar molecules <cit.>, trapped-ion setups <cit.>, and cold atoms in cavities <cit.>. Moreover, several established results that hold for nearest-neighbor systems came into the light of investigation in these LR systems in terms of Lieb-Robinson bound <cit.>, area law <cit.>, novel phases of matter  <cit.>,  dynamical phases <cit.> and from the perspectives of usefulness in quantum technologies like  quantum metrology <cit.> and quantum computation <cit.>. 








The nearest-neighbor Dzyaloshinskii-Moriya (DM) interaction, on the other hand,  involves an asymmetric exchange of spins caused by spin-orbit coupling <cit.>.

It has been widely explored in a variety of solid-state compounds, including Cu(C_6D_5COO)_2. 3D_2O <cit.>, Yb_4As_3<cit.>, Crl_3 <cit.> etc, exhibiting fascinating magnetic properties, like gapless chiral phase when the DM interaction strength is stronger than the anisotropy parameter in the transverse Ising <cit.>,  XY <cit.> and the gamma model <cit.>. It was also shown that the DM interaction can host a ground state in the gapless phase that contains logarithmic entanglement behavior, similar to the LR interacting systems which can enhance the fidelity in quantum teleportation <cit.>, thermal entanglement <cit.> and performance in quantum engines <cit.>.  



In the non-equilibrium domain, the DM interaction has also been examined in the context of non-equilibrium thermodynamics <cit.>, dynamical quantum phase transitions <cit.>, topological phase transition <cit.> and quantum speed limit <cit.>. 



 In this paper, we explore the long-range extended XY model in the presence of long-range extended DM interactions, which decay according to a power-law.  Specifically, we aim to analyze the responses of both long-range and DM interactions in equilibrium and non-equilibrium physics. In the former scenario, we present the phase-classification utilizing conventional order parameters, the law for the decreasing slopes of classical and quantum correlations, and the scaling of entanglement entropy (EE) with the corresponding central charge. In contrast to the NN model <cit.>, we show that given a fixed fall-off rates, there exists a gapped region in which the DM interaction is stronger than the anisotropy parameter. However, we establish that whenever the system is gapless, it is chiral. 
 Notably, in the static case, when the ground state belongs to the chiral gapless region, the classical correlation and quantum mutual information decay with constant decreasing exponent while this is not the case in the gapped regime.   In this model, we observe that the effective central charge associated with EE exhibits interesting behaviors: in the presence of LR and DM interactions, the central charge acts nonlinearly with multiple kinks that are absent from the extended Ising model without DM interactions.

  In comparison to the equilibrium scenario, the dynamical phases could not be efficiently characterized as they depend on the  initial state, evolving operator,  time and system parameters. 
   In order to distinguish between the gapless and gapped phases, we resort to a recently proposed relaxation dynamics of two-point classical correlations <cit.> and entanglement entropy of the evolved state in the transient regimes, as well as mutual information and classical correlation in the steady state domains
  (cf. <cit.> for other dynamical quantities). In our case, the initial state is  prepared by tuning the parameters of the Hamiltonian which have either gapless or gapped energy spectra while  evolution happens when the range of interactions is suddenly quenched, resulting in a gapless or gapped Hamiltonian. Specifically, in the transient  regime,  we show that the long-range DM interactions modify the exponent in the relaxation time of the two-point classical correlation between modes  and establish a generic  scaling law for the growth of entanglement entropy over time. Further, we  observe that  the behavior of mutual information and classical correlation between two arbitrary sites in the steady state differs depending on whether the Hamiltonians corresponding to the initial and dynamical states are in the same (gapped or gapless) phase or in different phases. 

  
  

  
  

 













 

The paper is organized as follows. In Sec. <ref>, we introduce the long-range extended XY model with an additional long-range DM term and find the energy spectra and eigenvectors. While 

we identify the conventional phases in Sec. <ref>, we indicate the decay of classical and quantum correlations in Sec. <ref>. The behavior of  entanglement entropy and the corresponding central charge with respect to system parameters are studied in Sec. <ref>. We then move on to the investigation of features in dynamical states, namely the relaxation dynamics of classical correlations, and the rate of entropy growth in the transient regime in Secs. <ref> and  <ref> respectively while we examine the decay of mutual information with distance between the spins  in the steady state  in Sec. <ref>. The results are summarized  in Sec. <ref>.



§ EMERGENT GAPPED PHASE WITH STRONG DM INTERACTIONS: PHASE DIAGRAM



The  critical lines  for the long-range interacting XY spin models in the presence of Coloumb-like interaction, described by a parameter α and for the nearest-neighbor transverse XY model with DM interaction  are known in the literature <cit.>. However,  the effects of long-range DM interactions has never been addressed before. In this situation, a possible query can be  – can the introduction of long-range DM interactions change the critical lines and phases of the LR Ising models? We answer this question affirmatively by first presenting the analytical prescription of this model for obtaining energy spectra and the eigenvectors, thereby leading to the modified phase-portrait of the extended long-range XY model <cit.> having long-range extended Dzyaloshinskii-Moriya interaction both in the thermodynamic limit and for finite systems.  In particular, we report the gapless chiral phase and its dependence on the fall-off rate, α. In addition, we identify appropriate order parameters that can reveal α-dependence in the modified magentic phase diagram. From the decay patterns of classical correlation including Landau parameters and quantum correlations in terms of quantum mutual information and block entanglement entropy, we again separate the gapless region from the gapped one in this model. 



 §.§ Energy spectrum of the long-range XY and DM interactions


The Hamiltonian, describing long-range extended XY model having long-range extended DM interaction in the presence of transverse magnetic field, reads as


    H   =   ∑_j=1^N∑_r=1^N/2 -J_r^'/𝒩 [1+γ/4σ_j^xℤ_r^zσ_j+r^x+1-γ/4σ_j^yℤ_r^zσ_j+r^y
          +D'/4(σ_j^xℤ_r^zσ_j+r^y-σ_j^yℤ_r^zσ_j+r^x) ]-h'/2∑_j=1^Nσ_j^z,

where ℤ_r^z = ∏_l=j+1^j+r-1σ_l^z, with ℤ_1^z=𝕀
and σ^k(k=1,2,3) is the Pauli matrix, J_r'=J/r^α with α being the strength of power-law decay of the model, γ is the anisotropy parameter, D is the strength of the DM interaction, 𝒩=∑_r=1^N/21/r^α is called the Kac-scaling factor <cit.> which ensures extensivity of the energy in the case of finite-size systems α<1 in the finite-size limit and h' is the strength of the external magnetic field. To make the analysis dimensionless, we redefine the magnetic field as h=h'/J and DM interaction strength as  D=D'/J. We consider the periodic boundary condition (PBC), i.e., σ_N+1≡σ_1.  This model can be solved analytically by mapping spins into free fermions under the following Jordan-Wigner transformation <cit.>:

    σ^x_n    =  ( c_n + c_n^†)
     ∏_m<n(1-2 c^†_m c_m) 
    σ^y_n    =i( c_n - c_n^†)
     ∏_m<n(1-2 c^†_m c_m) 
    and  σ^z_n   =1-2 c^†_n  c_n,

where c_m^†(c_m) is creation (annihilation) operator of spinless fermions and they follow fermionic commutator algebra. The corresponding free fermionic version of the Hamiltonian takes the form 

    H   =   ∑_n∑_rJ_r/2 ((1+iD)c_n^† c_n+r+γ c_n^† c_n+r^† + h.c )
          +h(c_n^† c_n-1/2),

where J_r=1/r^α. One can observe that the Hamiltonian which is quadratic in fermionic operators have long-range interaction both in hopping and superconducting terms. In addition, an extra hoping term emerges due to the DM interaction which also induces a complex phase. It is worthwhile to mention here that in the absence of DM interaction for γ=0, the model is 𝕌(1) symmetric while it is ℤ_2 symmetric with γ 0, D=0. The introduction of  DM interaction leads to a competition between these two symmetries.   In the short-range (SR) models, it is known that the conformal sector changes due to the shift of the symmetry sector <cit.>, responsible for a non-trivial phenomenon. 


Due to the presence of translational invariance in the system, momentum is a good quantum number. Hence, we perform a Fourier transform of the form, c_n=1/√(N)∑ e^-iϕ_kc_k where ϕ_k=2π k/N ∀ k∈[-N/2,N/2] and the corresponding Hamiltonian in the momentum basis can be written as H=k⊕ H_k with 

    H_k    =   ∑_k>0- [∑_r(Re(J_k^α)+DIm(J_k^α))c_k^† c_k
           +(Re(J_k^α)-DIm(J_k^α))c_-k^† c_-k
          -iγIm(J_k^α) (c_k^† c_-k^†- c_-k c_k)]
           +h(c_k^† c_k+c_-k^† c_-k)-h,

and J_k^α = ∑_r = 1^N/2J_r e^iϕ_kr. This Hamiltonian can be diagonalized by applying the Bogoliubov transformation,

    [    c_k; c_-k^† ]=[    u_k -v_k^*;    v_k  u_k^* ][    τ_k; τ_-k^† ],

where u_k=cosθ_k, v_k=isinθ_k and θ_k is called Bogoliubov angle, given by cosθ_k=-h+Re(J_k^α)-λ/√(2(λ^2-(Re(J_k^α)-h)^2)) and λ=√((h-Re(J_k^α))^2+(γ Im(J_k^α))^2). The eigenvalue corresponding to each momentum, k, is given as

    ϵ_± k=-D Im(J_k^α)  ±λ.

Note first that for D 0, although the Bogoliubov basis, (u_k, v_k)^T, are independent of DM interactions,  ϵ_kϵ_-k,  making the analysis non-trivial.

Further, when D=0, this model shows criticality at h=1 and at h=-1+2^1-α for α>1  corresponding to the both ends of the Brillouin zone, i.e., ϕ_k=0 and ϕ_k=π respectively.  In the case of α<1, the energy of the system diverges in the thermodynamic limit, hence, we consider large but finite system size <cit.>. With the introduction of DM, new gapless region emerges, which changes with α for D>γ. Such a region is known for the  nearest-neighbour XY model when D>γ <cit.>  and  gapless to gapped transition occurs at D= γ. However, we now establish that even when D>γ, long-range interactions can induce a gapped phase instead of a gapless one, which implies the modification of the critical lines in presence of LR interactions.






















 §.§ The gapless chiral phase depending on LR interactions


Apart from quantum phase transitions at zero temperature in which the energy gap vanishes, the LR Ising models possess another three distinctive regimes with respect to power-law fall-off rate α, namely, non-local (0 < α < 1), quasi-local (1 < α < 2) and local (α > 2) according to different scaling law of correlation length <cit.>. It is important to note here that in the absence of DM interaction, gap-closing never occurs by tuning the parameter α for h>0 except h_c=1. Further, we know that α≫ 2, the system with NN interactions becomes gapless for D>γ depending on h and the chiral phase emerges along with paramagnetic and ferromagnetic phases. Due to the long-range DM interactions, a competition between the range of interactions, α and the strength of DM interactions surfaces which becomes responsible for the change in condition for the gapless  to gapped transition. More specifically, for moderate values of α(<2), D>γ does not guarantee chiral gapless phase. We find the following in case of LR model. 




    For a given value of the magnetic field h (both in h≥ 0 and h<0), a gapped to gapless transition occurs when  D>γ depending upon the value of α instead of D=γ, known for the NN model.



    The proof is done by analyzing the dispersion relation in Eq. (<ref>). A system is said to be

gapped when the expression, Δ <cit.>, defined as

    Δ=max{min{ϵ_k},0}>0;  ∀ϕ_k∈ [-π, π],
 
otherwise, the spectrum is gapless (with Δ = 0). The exact point, where the gap-closing takes place, depends on the parameters of the Hamiltonian, and the corresponding momentum, k_F, called the Fermi point, can be obtained as a solution of ϵ_kk=0. However, due to the presence of long-range order, it is cumbersome to find the Fermi point analytically. By differentiating numerically, we observe that the gapless phase never occurs when D<γ while for D>γ, the gapless phase occurs although it is not ubiquitous, and changes with the variation of h,D,γ and α. In particular, when h<0, with the decrease of α, the system is gapped even when D>γ and two distinctive gapped regions emerge when 1≤α<2 and 0<α<1 which depend on the strength of D, thereby inducing (destroying) gapped (gapless) regions. However, for α>2, D > γ guarantees gapless phases and the gapless to gapped transition occurs at D=γ. On the other hand, we find, that when 1≤α<2, strong DM interactions are required to obtain the gapless phase and the desired asymmetric interaction strength for obtaining the gapless phase increases with the decrease of α (see Fig. <ref>). Note, further, that the maximum D necessary for the gapless phase does not happen exactly at α=1, but at a nearby point which can be due to finite size analysis.  



It is now evident that the interplay of α and D along with γ and h can change the critical lines of the phase diagram. Therefore, we now examine the phases in the (h,γ)- and (α, D)-planes.






α-dependent chiral phase. In the nearest-neighbor  XY model with DM interactions, the gapless phase gives rise to a chiral order that measures local current flow between the nearest neighbor sites. It can be established, by computing <cit.> the order parameter,

    𝒞ℋ= 1/4N∑_i=1^N σ^x_iσ^y_i+1-σ^y_iσ^x_i+1

between any two neighboring sites, i and i+1 in the ground state which describes the helical alignment of spins or chiral order in the z-direction. 


In the nearest-neighbor case, i.e., for the NN XY model with D > γ and suitable h values, it was shown that 𝒞ℋ > 0 when Δ = 0. 

In the LR model, we find that for a fixed h, α <2 and D > γ, whenever Δ = 0, i.e., the system is gapless, it possesses chiral order with order parameter 𝒞ℋ > 0 (as shown in Fig. <ref>). Interestingly, however, there exists region in which for D> γ, Δ>0 (as shown in the above proposition), and the corresponding chiral order parameter vanishes, i.e., 𝒞ℋ =0. 








Ferromagnetic and paramagnetic phases. We now calculate the long-range magnetic order parameter, C^xx_N/2 =  σ^x_1σ^x_N/2. When C^xx_N/2 0, the system is said to be in the ferromagnetic-x phase (FM_x) while paramagnetic when it vanishes. On one hand, we show that the FM_x phase occurs when the system is gapped and 0<h<h_c^1=1, 1<α<2, and D < γ (see Fig. <ref>). On the other hand, when h is negative, the system becomes ferromagnetic, i.e., C^xx_N/2 > 0  when h > h_c^2=-1+2^1-α with 1<α<2. When the parameters  are neither chiral nor ferromagnetic, the system is in a paramagnetic phase.




  













 §.§ Constant decaying exponent of correlations with DM interactions



From the above analysis, it is evident that the transition from the chiral gapless phase to the other magnetic gapped phase heavily depends on the fall-off rate α. Let us now investigate how the scaling of classical correlations and quantum mutual information between two spins (a measure of total correlations containing both quantum and classical correlations components) <cit.> can recognize the α-dependent phases discussed above. 

The classical correlation (CC) between two sites, separated by a distance R,  can be represented as C^xx_i,i+R = ⟨σ_i^x σ_i+R^x ⟩ while the quantum mutual information is defined as 
ℐ_i:i+R=S(ρ_i)+S(ρ_i+R)-S(ρ_i, i+R), where S(σ) = -(σlog_2σ) indicates the von-Neumann entropy of the state σ, ρ_i and ρ_i+R are the reduced density matrices of the joint state ρ_i,i+R. As we impose periodic boundary condition, the site index i can be ignored and both the classical and total correlation can be studied as C^xx_R and ℐ_R, where R = {1,2,…N/2}. In typical one-dimensional (1D) quantum spin models, the CC decays exponentially when it is away from criticality, whereas, at the criticality, the polynomial decay with R is observed. However, the extended Ising model deviates from this norm by showing different scaling laws near the critical points and away from it <cit.>. Moreover, in the long-range Kitaev chain (see Eq. (<ref>) with D = 0), quantum mutual information (QMI) persists between two distant segments of the chain, provided the system is in the non-local regime, i.e., α < 1, thereby  predicting the correlation between two distant regions of the system  <cit.>.

In our study, we expect to have a notable scaling law due to the presence of long-range and additional DM interactions which is indeed the case. Firstly, we observe that the trends of C_R^xx and ℐ_R with R is qualitatively different when the system is in the chiral phase and when it is not. Secondly, a counter-intuitive observation is that if one tunes the parameters 

 D, γ, h and α in such a way that 𝒞ℋ 0, the decay exponents of mutual information and classical correlation become constant as depicted in Fig. <ref>. Specifically, we observe that irrespective of the details of the power-law exponent and the strength of the magnetic field, the decay exponent of CC and QMI falls off as 

    ℐ_R    ∝ R^-1  and
    
        C^xx_R    ∝ R^-0.45,

which are independent of α provided the ground state belongs to the gapless chiral region (see Fig. <ref> and Table <ref>). This behavior of ℐ_R and C^xx_R highlights a more complex and highly correlated patterns between α and D within the gapless region. It indicates that although the range of interactions can control the critical lines in the (D,γ)-plane, differentiating gapless and gapped phases, the decay rate of CC and QMI in the gapless phase remains unaltered  and the chiral phase favors the sharing of both quantum and classical correlations between distant sites. More importantly, such α-dependent scaling behavior holds both for quantum and classical correlations (see Fig. <ref>).  

On the other hand,  when the system is in the non-chiral (which we refer to as achiral) phase, the ground state displays the decay exponent of both classical and quantum correlations which vary with α, i.e., depending on the internal descriptions of the model (see Fig. <ref> and Table <ref>). 

Through these observations, we elucidate the nuanced role of long-range DM interactions in modulating correlation properties across both chiral and achiral regions leading to the following proposition.


    In the chiral region of the LR XY model with DM interactions, the scaling exponents of CC and QMI are constant (independent of the fall-off rate, α) while their exponents depend on α in the achiral region.











 §.§ Amendment in scaling of entanglement entropy and central charge owing to long-range DM interactions



In a many-body system, an insight of universality classes can be obtained from the generic features of the ground state. A prominent physical quantity that serves the purpose is the block entanglement entropy  of block-size l <cit.>. In particular, after partitioning the N-party ground state into two blocks, containing l and N-l sites (with l ≪ N), we compute the block entanglement of the ground state as S_l = S(ρ_l) with ρ_l being the reduced density matrix of the N-party ground state.  The l-block entropy can be calculated from the two-point correlation functions between modes, m and n, given as

    C_mn=⟨Ψ|c_m^† c_n | Ψ⟩,    F_mn=⟨Ψ|c_m^† c_n^† |Ψ⟩,

where |Ψ⟩ is the ground state and the corresponding correlation matrix, consisting of correlation functions, can be written as

    ℂ_l=[ 𝕀-C   F; F^†   C ].

If λ_ls are the eigenvalues of ℂ_l, the von-Neumann entropy of |Ψ⟩ in the bipartition l:N-l is given as S_l=-∑_l=1^2Nλ_llog_2λ_l.

If a system is gapped which typically happens away from the criticality and when the range of interactions is relatively local, i.e., the norm of the Hamiltonian does not increase as the system size increases, S_l follows the area law, which implies S_l^NC∝ l^d-1, where d is the spatial dimension of the system and NC stands for “not at criticality" while it deviates from the area law at criticality <cit.>. Hence from the scaling behavior of S_l, one can detect the transition from a gapped to a gapless phase which is interesting since a simple scalar quantity can describe the essential properties of the Hamiltonian instead of a complete microscopic description. 

On the other hand, when the system becomes gapless, representing the critical point, e.g., in the transverse NN XY model,  S_l scales logarithmically, i.e., for a translationally invariant spin chain, at critical points, S_l^C= c/3log_2[N/πsinπ l/N]+a <cit.>, where c is the conformal or effective central charge in the conformal field theory and a is a non-universal constant. Further, it was shown that the central charge carries the signatures of an underlying symmetry. For example, c takes value 1/2 for the transverse-field Ising-like Hamiltonian at the critical point for which the Hamiltonian adheres to the ℤ_2-symmetry, while with the addition of asymmetric nearest-neighbor DM interaction, the Hamiltonian becomes  gapless region having c=1 and gapped with c=1/2. Hence EE turns out to be a powerful tool to separate a gapped phase from a gapless one.


It was also recently shown that if one considers long-range interactions instead of short-range ones <cit.>, where magnetic criticality depends on the value of α, EE depends upon α and the free-fermionic version of long-range Ising spin shows fractal entanglement apart from the volume and area law <cit.> and the scaling of entropy at critical points may not follow S_l^C. In this case, we can define an effective conformal charge, c_eff, to describe the universal properties of the system.

We are interested in finding how the scaling of S_l in the LR Ising Hamiltonian gets altered in the presence of DM interactions. Specifically, we will show the revision of a central charge c_eff, which occurs in S_l^C. Before presenting the effects of DM interaction on the central charge, we observe that at criticality, i.e., h_c^1 = 1 of the extended Ising model, c_eff monotonically increases when 1 < α < 2 and saturates to 1/2 at α≥ 2, (which is the scaling known for the NN model) as shown in Fig. <ref>(a). When α < 2, c_eff varies non-linearly with α which points out that due to the presence of long-range interaction, conformal symmetry breaks down. 

On the other hand, the introduction of long-range  DM interaction makes more modifications in c_eff –  (i) the saturation value changes to 1 when α→α_c = 2, with α_c being the point which mimics the NN transverse Ising model with NN DM interactions; (ii) for α<α_c, the conformal symmetry breaks down whereas when α > α_c,  c_eff→ 1 (see Fig. <ref>(b)). 
Our study reveals that even with the introduction of long-range asymmetric interaction, the conformal symmetry is not restored for all the values of α although c_eff indicating the conformal invariance begins for smaller values of α_c<2 which is found in the absence of DM interactions. This behavior can again be attributed to the fact that when the DM interaction is present, the system becomes gapless upon varying the power-law decaying factor α.











§ DECAY OF DYNAMICAL CORRELATION: UNCOVERING CHIRAL AND ACHIRAL PHASES



Let us now study the trends of physical quantities including quantum correlations in the time-evolved state in the transient and the steady state regimes, starting from the product state or the ground state which evolves according to the LR Hamiltonian with DM interactions. Our aim is to demonstrate that the scaling of correlations with time or with distant sites can indicate the transition from the chiral phase to the achiral ones. 



 §.§ Relaxation behavior of dynamical correlations






Let us first present the method to capture the relaxation of dynamical correlation <cit.> which is shown to comprehend distinct phases in equilibrium. The initial state and the evolution of the system in the momentum basis can be represented by the Bogoliubov–de Gennes (BdG) equation, given as


    ψ_k(0)=[u_k(0), v_k(0)]^T  and  ψ_k(t)=e^-iH_ktψ_k(0).

The corresponding dynamical correlation (DC) is defined as C_mn(t)=c_m^† c_nψ(t) which is a fermionic correlation between two modes. The relaxation of this correlation can be measured by taking the difference between correlation in arbitrary time and time in which the system reaches the steady state, given by δ C_mn(t)=C_mn(t)-C_mn(∞), describing the decay of the correlation function with time. Let the Bogoliubov angle of the initial and quenched Hamiltonian be η_k and η̃_k respectively and the difference between the angels is given as α_k=η_k-η̃_k. The relaxation of DC takes the form <cit.>

    δ C_mn(t) = 
       k∈gapped∫sin 2η̃_ksin 2α_k cos(ϵ_k+ϵ_-k)tcos[k(n-m)]= 
        
       k∉gapless∫sin 2η̃_ksin 2α_k cos(ϵ_k+ϵ_-k)tcos[k(n-m)],

where the initial state belongs to the gapped  and gapless regions in Eqs. (<ref>) and (<ref>) respectively. In our case, we have chosen m-n=1, but our result holds for any arbitrary m  and n.

When δ C_mn(t) ∼ t^-χ, χ indicates the dynamical relaxation exponent characterized by the distinctive values of the system parameters.

In the case of the nearest-neighbor XY model, it was shown that the exponent depends on the phase being commensurate or incommensurate. Specifically, in the commensurate phase, the derivative of the dispersion relation with respect to the momentum vanishes at the extreme ends of the Brillouin zone, while in the incommensurate phase, the derivative vanishes inside the Brillouin zone. If the initial state is a product state, i.e., the ground state of the model when the magnetic field goes to infinity and is then quenched with the Hamiltonian corresponding to the commensurate phase, we find χ = 3/2 while χ = 1/2 when quenched with the incommensurate phase <cit.>. 

In the NN XY model with DM interaction, if the initial state belongs to the gapless phase, the exponent changes to χ = 1 if the quenching Hamiltonian is in the commensurate phase, which indicates the presence of DM interaction. In the case of quenching by a Hamiltonian in the incommensurate phase, the exponent is either χ = 1/2 or χ = 1 depending on the parameters of the initial state <cit.>. On the other hand, considering the LR Kitaev chain, it was found that the scaling law behaves similarly to the SR ones when the initial and post-quench Hamiltonian are non-critical although the exponent changes with the parameters chosen from an equilibrium phase transition <cit.>. 

The model considered in this work possesses both LR DM interactions and LR interactions in the xy-plane and hence the interplay between α and D as mentioned in the static scenario can influence the scaling of δ C_mn(t), resulting to different χ. For investigations, possible three situations arise in the non-local regime (α < 1) - (ni) both pre- and post-quench Hamiltonian are in the chiral phase; (nii) only pre-quench Hamiltonian belongs to the chiral phase; (niii) only post-quench Hamiltonian is chosen from the chiral phase. All three cases can also be considered when α is chosen from a quasi-local regime, i.e., when α > 1 which we refer to as (qi) - (qiii). Eg., we find that χ∼ 1.5 for the case of (niii) and when (qii) χ∼ 0.3 (see Fig. <ref>). 

This shows that the exponent in DC can distinguish between the gapped and the gapless phases  of the post- and pre-quench Hamiltonians  (see Fig. <ref>). In addition, the robustness of correlation can be defined as a quantity 1/χ such that if the value of χ is small, the system does not reach the steady state correlation value for a significantly long time. This reinforces the above result that the state from the gapless phase contains a strong correlation both in terms of space and time. 



 §.§ Slow decay of total correlation and two-point correlation with LR DM interaction



We now focus on six scenarios ((ni) - (niii) and (qi) - (qiii)) to study the decay pattern of total correlation and classical correlation with R in a steady state limit, represented by ℐ_R(∞) and 𝒞^xx_R(∞) respectively. Even though the evolution is unitary, as we are studying the properties of a subsystem, a steady state can be reached in the dynamics. When the quenching Hamiltonian is in the gapped phase ((nii)), the steady state maintains both classical and quantum correlations between distant spins, and  persists even at large distance R (see Figs. <ref>(b), (c), (f) and (g)). Conversely, a gapless quenching Hamiltonian ((ni) and (niii)) results in a steady state where classical correlations are present only between nearby spins as depicted in Figs. <ref> (a), (d), (e) and (h). Notice that the decaying natures of ℐ_ℛ(∞) and C^xx_R(∞) are independent of the initial state. Remarkably, the behavior of total and classical correlations remains similar when the distance between the spins is small. 
















 §.§ Effects of long-range in dynamics of EE



In a final attempt to distinguish the gapless and gapped phases under dynamics, we investigate the
growth of entropy with time. The initial states are prepared as the ground states of gapped as well as from the gapless regions and EE, S_l(ρ(t)) ≡ S_t for a fixed l is computed after quenching the system to the gapped or gapless regions by changing the parameters α for a given γ, h and D. Let us denote the initial and final fall-off rates be α_i and α_q respectively. Firstly, we notice that the rate of growth is shown to be linear in time in the case of the XY model with NN <cit.> and long-range interacting model <cit.>, with the initial state being a product state. The trends of S_t with time depend on both α_i and α_q, thereby indicating its dependence on pre- and post-quench Hamiltonian. 

  Secondly, the rate of increment of EE with time is faster if the initial state is in a gapless region and the EE value gets saturated around time t ∼ l/2, as observed in the case of the nearest-neighbor Ising model quenched from a non-critical regime to a critical one <cit.>. However, if the initial and final states are chosen from a gapless region, the increment rate is slow and the saturation is reached faster around t=l/2 compared to a gapped case (see Fig. <ref>(d) and (e)). Also, the scaling of entropy increment until it reaches saturation can be approximated as

    S_t^±∝ a_1t/α_i ±α_q+a_2,

where a_1, and a_2 are constants (see Fig. <ref>).

When both the initial and the final Hamiltonian are gapped or one of the pre- or post-quench Hamiltonian is gapped, the saturation of EE requires much larger time than that obtained with  the Hamiltonians corresponding to the initial and final states being gapless. 








§ CONCLUSION



Quantum long-range systems are intriguing due to their non-local properties, often displaying counter-intuitive phenomena compared to a short-range Hamiltonian. These models are pervasive in physically realizable systems such as trapped ions and atomic, molecular, and optical setups.   Therefore, comprehending their universal features is essential for the advancement of quantum technologies and condensed matter physics.





We found that the long-range extended Dzyaloshinskii-Moriya (DM) interactions with the extended XY model have unique consequences on the critical behavior of the system with a variation of the fall-off rate, α, in which the range of interactions between the sites decays. Specifically,  even with a comparable or more DM interactions than the anisotropy parameter, we proved that the system remains gapped depending on α, which  varies with the direction of the transverse magnetic field although the gapless phase still possesses chiral order.  
By analyzing classical and total correlations between two arbitrary sites of the ground state in the presence of DM interactions, we observed that both  of them decay with the distance between the sites and the decay exponent is independent of α in the gapless region while it depends on the system parameters in the gapped regime and  decays faster in this case than the gapless zone.  Additionally, we exhibited   the departures from the universal trend in the block entanglement entropy in presence of long-range DM interactions which could successfully capture the interplay between long-range and DM interactions.





This study investigated how dynamical  correlations change in time when the energy spectra of the initial and the evolving Hamiltonian are gapped and gapless, driven by  α for a fixed strengths of DM interactions, and magnetic fields. 

The scaling of both classical correlations and mutual information in the  steady states reveals that the sharing of correlations is favorable when the evolving Hamiltonian is gapped.  Moreover, we found that the growth of entanglement entropy with time  is faster in the gapless regime than in the gapped phase and it saturates quickly when both the pre- and post-quench Hamiltonians are gapless.   On the one hand, these findings contribute to the understanding of fundamental aspects of quantum phase transitions, while on the other, sharability of correlations and entanglement  of the ground state as well as their dynamical patterns can be vital for establishing quantum links amongst quantum computers.



We acknowledge the support from the Interdisciplinary Cyber Physical Systems (ICPS) program of the Department of Science and Technology (DST), India, Grant No.: DST/ICPS/QuST/Theme- 1/2019/23. We  acknowledge the use of https://github.com/titaschanda/QIClibQIClib – a modern C++ library for general purpose quantum information processing and quantum computing (<https://titaschanda.github.io/QIClib>), and the cluster computing facility at the Harish-Chandra Research Institute. This research was supported in part by the INFOSYS scholarship for senior students.







 

§ CORRELATION FUNCTIONS




The two-point correlation functions are the fundamental constituents of our study. In the context of the ground state or the thermal state, the two-site correlation function between sites separated by a distance of R can usually be expressed as a determinant of an R × R Toeplitz matrix <cit.>.
However, as the Hamiltonian includes asymmetric interaction, the bipartite reduced state obtained from the N-party ground  state after tracing out N-2 parties includes the correlators like C^xy≡⟨σ^x_i σ^y_i+R⟩ = tr (σ^x ⊗σ^y ρ_R) and C^yx along with  C^ii (i=x,y,z).  Thus for the corresponding density matrix of the two distant parties of the spin chain and the time-evolved state, we need to deal directly with the Pfaffians in order to evaluate the correlations as a function of time as well as distance since there is a formation of chiral phase due to DM interaction. 

Using the Pfaffian formalism, we first write the spin correlation functions as 








    
    .
    [ C^lm_i,i+R = ⟨σ_i^l σ_i+R^m ⟩= c(l,m)pf  |  . I^lm_1,2                                                      …                                             I^lm_1,R-1                                                 J^lm_1                                                 F^lm_1                                               G^lm_1,2                                                    c .                                                      …                                               G^lm_1,r;                                                                                                             …                                                      …                                                      …                                                      …                                                      …                                                    c .                                                      …                                                      …;                                                                                                                                                         I^lm_R-2,R-1                                               J^lm_R-2                                               F^lm_R-2                                             G^lm_R-2,2                                                    c .                                                      …                                             G^lm_R-2,R;                                                                                                                                                                                                                    J^lm_R-1                                               F^lm_R-1                                             G^lm_R-1,2                                                    c .                                                      …                                             G^lm_R-1,R;                                                                                                                                                                                                                                                                               E^lm                                                 D^lm_2                                                    c .                                                      …                                                 D^lm_R;                                                                                                                                                                                                                                                                                                                                    K^lm_2                                                    c .                                                      …                                                 K^lm_R;                                                                                                                                                                                                                                                                                                                                                                                         H^lm_2,3                                                      …                                               H^lm_2,R;                                                                                                                                                                                                                                                                                                                                                                                                                                                       …                                                      …;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H^lm_R-1,R ] |,






where c(x,x)=c(y,y)=(-1)^R(R+1)/2, 

    I^xx_μ,ν      =  ⟨ A_l+μ(t) A_l+ν(t)⟩, 
    
    J^xx_μ,        =I^xx_μ,R, 
     
    H^xx_μ,ν      = ⟨ B_l+μ-1(t) B_l+ν-1(t)⟩, 
    
    K^xx_ν      = H^xx_1,ν, 
     
    G^xx_μ,ν      = ⟨ A_l+μ(t) B_l+ν-1(t)⟩, 
     
    F^xx_μ      =G^xx_μ,1,  
    
    E^xx      =G^xx_R,1,  
    
    D^xx_ν      =G^xx_R,ν,



    I^yy_μ,ν      =  ⟨ A_l+μ-1(t) A_l+ν-1(t)⟩, 
    
    J^yy_μ      =I^yy_μ,R, 
     
    H^yy_μ,ν      = ⟨ B_l+μ(t) B_l+ν(t)⟩, 
    
    K^yy_ν      = H^yy_1,ν, 
     
    G^yy_μ,ν      = ⟨ A_l+μ-1(t) B_l+ν(t)⟩, 
     
    F^yy_μ      =G^yy_μ,1,  
    
    E^yy      =G^yy_R,1,  
    
    D^yy_ν      =G^yy_R,ν,

with s(x,y)=s(y,x)= -i (-1)^R(R-1)/2, we have

    I^xy_μ,ν      =  ⟨ A_l+μ(t) A_l+ν(t)⟩, 
    
    G^xy_μ,ν      = ⟨ A_l+μ(t) B_l+ν(t)⟩, 
    
    J^xy_μ      = G^xy_μ,0, 
     
    F^xy_μ      =G^xy_μ,1,  
    
    H^xy_μ,ν      = ⟨ B_l+μ(t) B_l+ν(t)⟩, 
    
    E^xy      =H^xy_0,1,  
    
    D^xy_ν      = H^xy_0,ν, 
      
    K^xy_ν      =H^xy_1,ν,

and

    I^yx_μ,ν      =  ⟨ A_l+μ-1(t) A_l+ν-1(t)⟩, 
    
    G^yx_μ,ν      = ⟨ A_l+μ-1(t) B_l+ν-1(t)⟩, 
    
    J^yx_μ      = I^yx_μ,R, 
     
    F^yx_μ      =I^yx_μ,R+1,  
    
    E^yx      =I^yx_r,r+1,  
    
    D^yx_ν      = G^yx_R,ν,  
    
    K^yx_ν      =G^yx_R+1,ν, 
     
    H^yx_μ,ν      = ⟨ B_l+μ-1(t) B_l+ν-1(t)⟩.


Each of the elements in the Pfaffian can be constructed from the expectation values of one of the operators, A_l A_l+R, B_l B_l+R and A_l B_l+R with

A_i=c_i^†+c_i,   B_i=c_i^†-c_i.





When recasted in the same Fourier basis as taken while diagonalization, the operators for the k^th momentum have the matrix form

    (A_l A_l+R)_k =  [[          cos(kR)        -isin(kR)                0                0;        -isin(kR)          cos(kR)                0                0;                0                0 cos(kR)-isin(kR)                0;                0                0                0 cos(kR)+isin(kR) ]],


    (B_l B_l+R)_k =   [[          -cos(kR)           sin(kR)                 0                 0;          -sin(kR)          -cos(kR)                 0                 0;                 0                 0 -cos(kR)+isin(kR)                 0;                 0                 0                 0 -cos(kR)-isin(kR) ]],



    (A_l B_l+R)_k =   [[   cos(kR)  isin(kR)         0         0; -isin(kR)  -cos(kR)         0         0;         0         0         0         0;         0         0         0         0 ]].
 
The single-site transverse magnetization in the same Fourier basis is given by

    σ_z^k = [[ -1  0  0  0;  0  1  0  0;  0  0  0  0;  0  0  0  0 ]].






The corresponding expectation value of the operators for each value of k, say O^k, with respect to the time-evolved  state is computed as ⟨ O ⟩ = ∑_k=1^N/2Tr(ρ_β^k(t) O^k).

Once we find all the single-site magnetizations m_j^α ∀ α={x,y,z} at j=i and j=i+R and all possible two-site correlation functions C_i,i+R^l,m ∀  l,m={x,y,z}  from the Pfaffians described above, we can construct the two-site reduced density matrix between sites i and I+R as we know any two party density matrix for the given Hamiltonian is given as


    ρ_ij=1/4 (𝕀_4 + m^z(σ^z_j+σ^z_i) + ∑_k,l C^kl_ijσ^k_i ⊗σ^l_j),

where k,l∈{x,y}.








 Paper Title: Generalised BBGKY hierarchy for near-integrable dynamics.





Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France


Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland


Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland


 Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France

§ ABSTRACT 
  We consider quantum or classical many-body Hamiltonian systems, whose dynamics is given by an integrable, contact interactions, plus another, possibly long-range, generic two-body potential. We show how the dynamics of local observables is given in terms of a generalised version of Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, which we denote as gBBGKY, which is built for the densities, and their correlations, of the quasiparticles of the underlying integrable model. Unlike the usual cases of perturbation theory from free gases, the presence of local interactions in the integrable model “lifts" the so-called kinetic blocking, and the second layer of the hierarchy reproduces the dynamics at all time-scales. The latter consists of a fast pre-equilibration to a non-thermal steady state, and its subsequent thermalisation to a Gibbs ensemble. We show how the final relaxation is encoded into a Boltzmann scattering integral involving three or higher body-scatterings, and which, remarkably, is entirely determined by the diffusion constants of the underlying integrable model. We check our results with exact molecular dynamics simulations, finding perfect agreement. Our results show how gBBGKY can be successfully employed in quantum systems to compute scattering integrals and Fermi's golden rule transition rates.


Generalised BBGKY hierarchy for near-integrable dynamics
    Jacopo De Nardis
    Submitted August 5, 2024
========================================================





  
Introduction –


The field of non-equilibrium many-body physics continually astounds us with its profound complexity, capable of driving the emergence of entirely new behaviors. While classical theories of chaos, turbulence, and active dynamics remain vibrant areas of research, recent years have seen quantum many-body systems enter the non-equilibrium domain <cit.>. 
In low spatial dimensions, a significant class of dynamical systems is represented by integrable systems, which are characterized by stable quasiparticles. These quasiparticles play essential roles in transport, entanglement spreading <cit.>, anomalous scrambling <cit.>, long coherence times <cit.>, as well as exotic hydrodynamics <cit.>. Typically, interacting integrable systems involve elementary constituents, bosons, spins, or fermions, with two-body contact (or very short-range) interactions. Remarkable examples include the Lieb-Liniger <cit.> gas and the Fermi-Hubbard chain <cit.>. This prompts the question: what happens to quasiparticles and the overall dynamics when additional, possibly longer-range, interactions are incorporated into the Hamiltonian? This problem of integrability breaking has garnered significant attention over recent years <cit.>, largely driven by cold atomic experimental settings <cit.> where anomalously long thermalization times have been observed for nearly two decades. While progress has been made in systems where the underlying integrable theory is free, much less has been accomplished for interacting integrable systems. In free theories, the matrix elements of local operators are typically known, enabling the use of Boltzmann scattering via the Fermi's golden rule. In contrast, interacting integrable models feature matrix elements that are complex functions of both the state and the model. Despite some recent advances in specific settings <cit.>, there remains a need for a general and practical framework to compute the dynamics of any integrable model with generic additional weak two- (or more) body potentials.




In this paper, we introduce a novel approach to the problem: we generalise the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy <cit.>, to the cases where the unperturbed theory  (classical or quantum) involves strong integrable contact interactions. This can be envisioned as a hard-core repulsion V^ int_a(r) with radius a on top of a long-range field φ(r), giving the two-body potential,  see also Fig. <ref>,

    U(r) =  V^ int_a(r) + V(r),      V(r) = (V_0/ξ)  φ(r / ξ  )  ,

with V(r) (repulsive V_0 >0 or  attractive V_0 < 0) acting on the characteristic distance ξ and it is well-known for example to appear in atomic dipolar interactions <cit.>. We demonstrate two main distinctions from the known free case: (a) the n-point correlation functions must be divided into spatially local contributions, namely δ-like terms from the contact integrable potential, and regular contributions induced by the extra potential; (b) the free propagator for all correlations is dressed by integrable scatterings, incorporating diffusive and higher-order derivative terms. This feature is crucial for the thermalization dynamics, resulting in a thermalization time (starting from any stationary state of the unperturbed theory) of order E_ Kin^3/2/(V_0  a)^2, with E_ Kin the mean energy of the unperturbed Hamiltonian. This contrasts with the the standard one-dimensional BBGKY for perturbed free gases, i.e. with a = 0, where the thermalisation time scales,  due to the so-called kinetic blocking, as  E_ Kin^7/2/V_0^4,  <cit.>.

 Our approach gives a fully predictive theory for generic dynamics close to integrability, encompassing classical systems but also quantum bosonic, fermionic, and spin chains.   








  
The gBBGKY hierarchy –
We consider the following Hamiltonian with the two-body potential of eq. (<ref>) with strength V_0

    Ĥ = Ĥ^ int_a + V̂,   V̂ = 1/2∫xx'V(x-x')q̂_0(x)q̂_0(x'),

where q̂_0(x) denotes the local particle density operator (which is a functional in the classical case) and Ĥ^ int_a is given by the sum of standard kinetic terms and an integrable interaction V̂^ int_a,  typically short ranged, and with interaction strength denoted by a, (for example in the Lieb-Liniger gas V^ int_a(x-x') ∼ a δ(x-x')) with a=0 corresponding to the free case. More generally, q̂_i(x) is the density of conserved charge Q̂_i, which commutes with the integrable Hamiltonian [Ĥ^ int_a,Q̂_i]=0. The first standard conserved quantities are particle number Q̂_0 = N̂, momentum Q̂_1 = P̂ and kinetic (unperturbed) energy Q̂_2 = Ĥ^ int_a.  Using relations for commutators between local charges,  the quantum generalisation of the classical Liouville's equation reads as  <cit.>, 

    ∂_t q̂_i(x)  +∂_x ( ĵ_i(x)  + ĵ_0,i(x) ∫ d x'  V(x-x') q̂_0(x') )=
       -1/2∫ dx' V'(x-x') [ q̂_0(x')ĵ_i,0(x)+ĵ_i,0(x)q̂_0(x') ] ,

where the (generalized) currents ĵ_k,i satisfy i[Q̂_k,q̂_i]+∂_x ĵ_k,i=0. For a system on the continuum and a potential coupling to the density as in eq. (<ref>) we have ĵ_0,i=0 due to Galilean invariance (the case of coupling to generic density will be considered in <cit.>). The equation (<ref>) is written for operators, and it is the action of taking expectation values that generates the hierarchy of dynamical equations for multipoint operators involving arbitrary products of charge densities. For all such equations, we take the average in a generic (Generalised) Gibbs ensemble, (G)GE, ⟨·⟩_β =  Tr(e^- ∫ dx β^i(x) q̂_i(x))/Z and isolate only their connected parts. The two-point correlations we denote S_o_1,o_2(x,y)= ⟨ô_1(x) ô_2(y) ⟩ _β - ⟨ô_1(x) ⟩_β⟨ô_2(y) ⟩_β and analogously for higher point correlations.  










In the same spirit as standard BBGKY theory, we move from the dynamics of the charges to the dynamics of the (quasi-)particles of the Hamiltonian Ĥ^ int_a. We introduce the particle density  ρ̂(θ, x) = ∑_j δ(θ - θ_j) δ(x-x_j), with average ⟨ρ̂(θ,x) ⟩ = ρ(θ,x),  (we use the generic labelling θ for each quasi-particle, denoted as momenta). Local charges (as well as other generic local operators) and their currents can be easily expressed in terms of the densities,  ⟨q̂_i ⟩ = ∫θh_i(θ)ρ(θ), with h_n(θ) = θ^n/n! the single-particle eigenvalue of Q̂_n. We can proceed analogously to introduce generic multi-particle correlations, obtaining this way the hierarchy involving ρ(θ,x) and its multi-point correlations as S_θ_1,θ_2(x_1,x_2), S_θ_1,θ_2,θ_3(x_1,x_2,x_3), and so on. 


It is crucial to stress the structure of the correlators such as S_θ_1,θ_2(x_1,x_2). In the unperturbed theory V_0=0 these objects carry non-trivial correlations induced by the local, integrable interactions. In particular, in any equilibrium state given by a generic density of quasiparticles ρ(θ), they read as 
    S_θ_1,θ_2(x_1,x_2)|_V_0=0≡δ_x_1,x_2(δ_θ_1,θ_2 g^(1,2)_(1,2)   +g^(1,2)_(1|2)),

 where we denoted the x dependence as upscript and the θ as subscript (see also eq. (<ref>)). 
Each correlator therefore consists of two main types of terms: one singular in space and one singular both in space and momenta. When interaction is added, V_0 ≠ 0,  the correlations also acquire an additional, smooth long-range component. Such a structure generalizes to higher point-functions and in order to have a well-defined and self-consistent hierarchy, the singular parts have to be extracted and the latter must be written only for the regular factors of the correlations.  Namely, we impose the following generic structure for the correlations S_θ_1,…,θ_s: 

    S_θ_1,…,θ_s(x_1,…,x_s) = ∑_P ∈𝒫_sδ^P(𝐱) ∑_R ∈ℛ_Pδ^R(θ) g^P_R.

Here 𝒫_s is the set of all partitions of a set of s elements. 
R is a refinement of P, that is a further fragmentation of the partition and by ℛ_P we denote a set of all refinements of P. With δ^P(𝐱) we denote a product of Dirac δ-functions that enforces the pairing of the coordinates according to P. For example, δ^(1,2|3,4)(𝐱) = δ(x_1 - x_2) δ(x_3 - x_4) with a standard notation for partitions in which vertical bar separates different subsets. In the momenta space we use analogous notation. Finally, g^P_R denotes contribution to the correlation functions of singular structure specified by the two partitions, see eq. (<ref>). We note that (<ref>) contains also regular terms corresponding to the full partition with cardinality |P|=s. Moreover, we stress that the correlators S_θ_1,…,θ_s(x_1,…,x_s) are symmetric under simultaneous exchange of position and momenta (we shall use the notation [· ]_(1,2) for symmetrization with respect to both), reducing therefore the complexity of (<ref>). The resulting hierarchy is written for the density of particles ρ and the functions g_R^P. We find the following first two lines of the gBBGKY hierarchy 


    𝒟^(1)_(1)ρ_(1)^(1)= ∫x_2θ_2𝒜^(1|2)_(1|2)( ρ_(1)^(1)ρ_(2)^(2) + g^(1|2)_(1|2)),





    𝒟^(1|2)_(1|2) g^(1|2)_(1|2)   = [𝒜^(1|2)_(1|2)ρ_(1)^(1)g^(2,3)_(2,3)+𝒜^(1|2)_(1|2) g^(1|2,3)_(1|2,3)+𝒜^(1|2)_(1|2)∫θ_3ρ_(1)^(1)g^(2,3)_(2|3)+𝒜^(1|2)_(1|2)∫θ_3g^(1|2,3)_(1|2|3)+
       +∫x_3θ_3𝒜^(1|3)_(1|3)ρ_(1)^(1)g^(2|3)_(2|3)+∫x_3θ_3𝒜^(1|3)_(1|3)g^(1|2|3)_(1|2|3)]_(1,2).


Here, the local 1-point correlations are denoted as densities of particles g_(1)^(1)≡ρ_(1)^(1) = ρ(θ_1, x_1), and we have introduced respectively the generalised acceleration induced by the perturbing potential 𝒜^(i|j)_(i|j)= V'(x_i-x_j)(∂_θ_i-∂_θ_j). 

The gBBGKY eqs. (<ref>) and (<ref>) read very similar to the known BBGKY equations for perturbed free particles, but with some major, important differences. First,  the left-hand side kinetic term 𝒟^(i|j)_(i|j), which describes the evolution of the multipoint correlations under the Hamiltonian Ĥ^ int_a depends on the interactions a: for free particles a=0 it reduces to the constant velocities spreading of the multipoint correlations 𝒟^(1|2)_(1|2) = ∂_t + ∑_j=1^2  θ_j ∂_x_j.  For finite interactions it is instead given (in a first order in V_0) in terms of the flux Jacobian <cit.> A_i^k = ∂⟨ĵ_i ⟩ /∂⟨q̂_k ⟩ (which is not diagonal in the interacting case) and by the diffusion matrix 𝔇_i^k <cit.>). For the case of two particles for example this reads 

    D^(1|2)_(1|2)= ∂_t + ∑_j=1^2∂_x_j A_θ_j^·  +  1/2∑_j=1^2∂_x_j 𝔇_θ_j^·∂_x_j  ,

where the operators A_θ_j^· and 𝔇_θ_j^·  acts on the momenta on their right. The flux Jacobian gives an effective ballistic spreading similarly to the free case, but where the velocities depends on the local density ρ(θ,x). Moreover, there is also a diffusive spreading of their trajectories, due to random scatterings with single particles. As at least two-body scattering are necessary, diffusion typically scales as 𝔇∼ (a ρ̅)^2, with ρ̅ characteristic density. Finally, we remark that higher order derivative terms in the hydrodynamic expansion of eq. (<ref>) can also be included, giving higher order corrections in a/ξ. 

The second major difference is that gBBGKY contain also correlations between particles at the same points, and different momenta, see eq. (<ref>). These correlations are not present in free gases, and they are a feature of Ĥ^ int_a being a short-range interacting Hamiltonian, which therefore can correlate particles at some microscopic scales, much smaller than the scale ξ at which the two-body perturbation acts.  

To truncate the hierarchy, one needs to estimate the order of the different contributions in (<ref>). Given that the system is characterized by the two interaction parameters a and V_0, the truncation of the hierarchy follows from scaling of correlation functions with these parameters, which reads g^P_R ∼ a^|R| - |P| V_0^|P|-1. We notice that the three-point function is of order g^(1|2|3)_(1|2|3) = O((V_0)^2) and enters the dynamics of ρ_(1)^(1) with order (V_0)^4. When interested in the dynamics at order (V_0)^2 we can therefore neglect it, together with the time evolution of g_(1,2)^(1,2), g_(1|2)^(1,2)  and of  g_(1|2|3)^(1|2,3), also of order (V_0)^2. 


We can recognize some of the terms in the RHS of Eqs. (<ref>) and  (<ref>). The O(V_0) term of (<ref>) is the  generalised Vlasov equation, which neglects two-particle correlations. It corresponds to a fully mean-field coupling of two space-dependent densities, and it gives no dynamics when the system is homogenous, namely when the density ρ(θ,x) does not depend on x. Similarly, the O(V_0) terms in (<ref>), the first on the right-hand side, are the driving terms of the generalised Landau equation, when the hierarchy is integrated into a single, effective kinetic equation. In one-dimensional free gas, setting a=0, the diffusive kinetic term in eq. (<ref>) vanishes and the  Landau equation gives only a short-time dynamics, and it cannot describe the thermalisation of the system at later times. This phenomenon is known as kinetic blocking in one-dimensional systems, and as we shall demonstrate later, the picture changes drastically when local integrable interactions are added.




  
The example of hard rods  –



We apply gBBGKY hierarchy truncated to its second layer,  eq. (<ref>), (<ref>) truncated to (V_0)^2 order in (<ref>), to a gas of classical impenetrable hard spheres with diameter a <cit.>, interacting with the long-range potential of eq. (<ref>). Hard spheres (or rods) gas in one dimension is integrable in the absence of extra interactions. The expressions for the  flux Jacobian and diffusion matrix are  well known <cit.> and they share the same hydrodynamics as quantum integrable systems, modulus changing the statistical factor <cit.>. We focus  on a homogenous system with constant density and we compare gBBGKY prediction with exact microscopic simulations.  We initialize the time-evolution starting from two boosted finite temperature states ρ_±; β; θ_0≡∑_σ = ± e^-β (θ- σθ_0)^2 /Z, and with g^(1|2)_(1|2)(t=0)=0. The state is stationary under the unperturbed Hamiltonian Ĥ^ int_a but it evolves non-trivially as long as V_0 is finite. As a probe of thermalisation dynamics, we compute the kurtosis of the (normalised) quasiparticle distribution and the two-point correlations. The kurtosis is expected to be zero in the thermal state, where quasiparticle distribution is Gaussian. In Fig. <ref> we show how gBBGKY reproduces the full molecular dynamics even at large values of V_0, modulo finite-time effects that involves higher order in the hierarchy. We can moreover observe two regimes in the evolution of the quasiparticle kinetic energy E_ Kin=v̅^2/2 =⟨Ĥ^ int_a⟩, Fig. <ref> (a): a fast evolution at short times, instated by the change in the interaction energy due to the build-up of the density-density correlations in the system (see. Fig. <ref> (b)), and the kinetic regime where it is mostly constant and where the dynamics is well described by the Boltzmann equation, which we describe in the coming section.
 

 







  
The interacting Landau equation and Boltzmann scattering kernel –
A common approach to handling the perturbed dynamics described by eq. (<ref>) is to use the Boltzmann scattering integral, which is a functional of the quasiparticle distribution at time t, ℐ_θ[ρ(·,t)]. This integral is derived by summing over all intermediate excitations induced by the operator V̂, as described by the celebrated Fermi's golden rule <cit.>. The matrix elements between the state |ρ⟩ at time t and its excitations, commonly referred to as form factors, define a typical timescale for each intermediate state | ex⟩, proportional to the square of its matrix element τ_ ex^-1∼ |⟨ρ | V̂ |  ex⟩ |^2 (in classical physics, a similar approach can be used, where cross-sections replace matrix elements). The Boltzmann scattering integral defines the Boltzmann equation as follows <cit.>:

    ∂_t ρ(θ,t) = ℐ_θ[ρ(· ,t)],

which, given that ℐ_θ∼ (V_0)^2, describes the late-time thermalization dynamics on timescales of order (V_0)^-2. Although efforts have been made to derive ℐ_θ for interacting integrable models, the general expression has remained elusive due to the complexity of the matrix elements involved.

Using algebraic manipulation (see <cit.>), we demonstrate that by truncating (<ref>) to terms of order V_0, and performing a time integration, we can precisely derive the Boltzmann scattering integral. By introducing the super-operators acting on two-point correlations 𝐀≡ (A ⊗1 - 1⊗ A), the diffusion matrix 𝐃 = (𝔇⊗1 + 1⊗𝔇), and 𝐂'_θ,θ'≡∫γ ( ∂_θρ_θC_γ,θ' - ∂_θ'ρ_θ'C_θ,γ ) (with 1 being the identity matrix in momentum space), we obtain the compact expression:

    ℐ_θ = ∂_θ∫_-∞^+∞k/2π k (V(k) )^2 [(k 𝐃 - i 𝐀)^-1]_θ,θ'^γ,γ' 1_θ'𝐂'_γ,γ'

where the repeated momentum indices are integrated over and V(k) denotes the Fourier transform of the potential (<ref>).  In <cit.>, we show that it possesses all the properties required for a scattering integral: it leaves thermal states steady, conserves the three canonical equations of motion, energy density, and momentum.

First, we observe that ℐ_θ∼ (V_0)^2, as expected. The first non-trivial terms in the expansion of the operator (k 𝐃 - i 𝐀)^-1 are of order k 𝐃∼ a^2, as the zeroth order in k integrates to zero. By expanding the inverse of the operator in (<ref>), we can see that ℐ_θ includes three or higher particle scatterings, where two particles scatter due to V̂ and all the others via V̂_a, see Fig. <ref> (b). Eq. (<ref>) provides a compact general form of the timescale for thermalization, even in a generic system given by Ĥ^ int_a with local interactions and a perturbing long-range potential V̂. It remarkably shows the role of the diffusion constants of Ĥ^ int_a in quantifying thermalization timescales: denoting D̅ the typical diffusion constant, we can extract the timescale τ^-1∼ (D̅/v̅^3)  ∫ dx (V'(x))^2 ∼ (a ρ̅ V_0)^2 /(ξ^3 E_ Kin^3/2).

The Boltzmann equation with the scattering integral given by eq. (<ref>) can also be understood as a generalization of the Landau equation for interacting integrable models. The latter clearly vanishes in the limit a → 0 and this can be either understood by the kinetic blocking or equivalently by the fact that density-density matrix elements of free systems cannot generate more than two particle scatterings <cit.>. Therefore, in this case 3-point correlations needs to be added to eq. (<ref>), which, after time integration, yield a higher-order scattering integral of order (V_0)^4 ρ̅^2/(ξ^3 E_ Kin^7/2)  <cit.>, which is also characterised by a dramatically different scaling with the kinetic energy.

 



 

  
Conclusions   –
 
In this work, we have generalized the BBGKY hierarchy (gBBGKY) for systems with integrable contact interactions by introducing ultra-local correlations and integrating kinetic theory with the underlying hydrodynamics of integrable models. We have demonstrated that the resulting kinetic equation overcomes the kinetic blocking, and the combination of short- and long-range interactions results in thermalization rates that are quadratic in the strength of both interactions. Kinetic blocking physics however still manifests itself from the fact that the thermalisation rate is O( (V_0 a)^2) which remains quartic in interaction strength. 

Our method is broadly applicable, with its main ingredient being generalized hydrodynamics, which offers a universal framework for many classical and quantum integrable models. Specifically, unlike previous approaches based on the Fermi's golden rule and matrix elements, the gBBGKY allows to directly access the Boltzmann scattering integrals and the time evolution of multi-point correlations  in generic perturbed integrable models.

While the partition separation of interactions between an integrable and a non-integrable part can be approached in various ways, we focused here on an experimentally relevant scenario of a short-range integrable model perturbed by long-range, dipole-like interactions. Nonetheless, many natural extensions, including dynamics from inhomogeneous initial states, perturbing operator coupling to other conserved charges, or systems on a lattice such as long-range spin chains or fermionic systems <cit.>, are now directly conceivable.




  
Acknowledgments–
MŁ and MP acknowledge support by the National Science Centre (NCN), Poland via project 2022/47/B/ST2/03334, J.D.N. and L.B. are founded by the ERC Starting Grant 101042293 (HEPIQ) and the ANR-22-CPJ1-0021-01. 




 























Supplementary Material 
 

Generalised BBGKY hierarchy for near-integrable dynamics






§ GBBGKY HIERARCHY IN THE HOMOGENOUS SETTING
 

 

We here specialize equations (<ref>) and (<ref>) from the main text to a homogeneous setup. In this case, the 1-point correlation does not depend on the spatial coordinate, while the 2-point correlation function depends only on the relative distance z = x_1 - x_2. This allows us to simplify the notation and use ρ(θ) and g^(2)_θ, θ'(z) for the two correlators respectively. We shall also make use of the filling function n(θ). The latter are the normal modes of the hydrodynamics,  defined as the modes that diagonalise the flux Jacobian of the model Ĥ^ int_a via the rotation matrix R

    A_θ,θ' = [R    diag(v^ eff) R^-1 ]_θ,θ',   δ n(θ)/δρ(θ') = R_θ,θ',    R_θ,θ'≡δ_θ,θ'-T_θ,θ'n_θ' ,

with T_θ,θ' the scattering shift of the integrable model. We also define v^ eff=(ε')^ dr/(p')^ dr with ε(θ) and p(θ) energy and momentum as function of momenta and with the dressing operator f^ dr≡ R^-1f.

The dynamics of the 1-point correlator is given by

    ∂_t ρ_θ=∫zθ'V'(z) ∂_θg^(2)_θ,θ'(z),

with the Vlasov term disappearing for a homogeneous system. In a similar way, the dynamics for the 2-point correlator is














    (D_(1|2)^(1|2) g^(2))_θ, θ'(z) =
       ∫ dz' γ V'(z-z') (∂_θρ( θ)g^(2)_γ,θ'(z')-∂_θ'ρ( θ')g^(2)_θ,γ(z'))
        +V'(z) (∂_θρ( θ)f(θ')ρ(θ')-∂_θ'ρ( θ')f(θ)ρ(θ))+ 
        
       
        + V'(z) (∂_θ' f(θ)g^(2)_θ,θ'(z)- ∂_θ f(θ')g^(2)_θ,θ'(z))
        +V'(z) ∫γ(∂_θρ( θ)g^(2)_{γ,θ'}-∂_θ'ρ( θ')g^(2)_{θ,γ})+ 
        
       
        + V'(z) ∫γ(∂_θ' g^(3)_{θ,γ},θ'(z)- ∂_θ g^(3)_θ,{γ,θ'}(z)) .

In writing this expression we introduced g^(2)_{γ,θ'} which corresponds to g_(1|2)^(1,2) of the main text and in a homogeneous system is space independent. Similarly, we write g^(3)_θ,{γ,θ'}(z) for g_(1|2|3)^(1|2,3) which depends only on the relative distance between particle at x_1 and group of particles (2,3) at x_2. The action of the kinetic term is 

    (D_(1|2)^(1|2) g^(2))_θ, θ'(z) = ∂_t g^(2)_θ,θ'(z) + ∂_z( A_θ^γ g^(2)_γ,θ'(z)- A_θ'^γ  g^(2)_θ,γ(z)) - ∂_z (𝔇_θ^γ∂_z  g^(2)_γ,θ'(z)+𝔇_θ'^γ∂_z  g^(2)_θ,γ(z) ).

Finally, f is the statistical factor depending on the statistics of the quasi-particles (fermions f= 1- n, bosons f=n, classical f=1).

The equation for g^(2)_θ,θ' depends on two more correlators namely the reduced 2-point  correlator g^(2)_{θ,θ'} and the reduced 3-point correlation g^(3)_{θ_1, θ_2 }, θ_3(z). They obey dynamic equations that are obtained following the procedure described in the main text. The kinetic part follows from the integrable dynamics, while the right hand side follows from the BBGKY hierarchy refined by the singularity structure of the correlations. In this way, the reduced 2-point correlator obeys

    ∂_t g^(2)_{θ,θ'} =  ∫zγ    V'(z)  (∂_θg^(3)_{θ,θ'},γ(z)-∂_θ'g^(3)_γ,{θ,θ'}(z)),

while the equation for the 3-point correlator, here written only in the leading term in V_0, is









    (D_(1,2|3)^(1|2|3) g^(3))_{θ_1, θ_2}, θ_3(z) = V'_1,3[ ∂_θ_1,θ_3g^(2)_{θ_1,θ_2}ρ_θ_3+  ∂_θ_1,θ_3∫θ_4 g^(3)_{θ_1,θ_2,θ_4}ρ_θ_3]_(1,2)
      +𝒪((V_0)^2),

where 

    (D_(1,2|3)^(1|2|3) g^(3))_{θ_1, θ_2}, θ_3(z) = ∂_t g^(3)_{θ_1,θ_2},θ_3 + 
       + 1/2∂_x_1( A_θ_1^γ g^(3)_{γ, θ_2},θ_3+H_θ_1^α,βg^(2)_{α ,θ_2}g^(2)_β,θ_3 +A_θ_2^γ g^(3)_{γ, θ_1},θ_3+H_θ_2^α,βg^(2)_{α ,θ_1}g^(2)_β,θ_3) + ∂_x_3( A_θ_3^γ g^(3)_{θ_1, θ_2},γ) +𝒪((V_0)^2).

In particular, here we defined

    H_θ^α,β≡δ  A_θ^α/δρ(β)= ∫γσ(R^t)^-1_θ ,γn_γ/ρ_γT^ dr_γ,σΔ v^ eff_σ,γ(R^t_σ, kR^t_γ,α+R^t_jkR^t_σ,β)  ,  Δ v^ eff_σ,γ≡ v^ eff_σ - v^ eff_γ .


Note that the kinetic term depends also on the lower order correlation functions. 
There are also higher orders contributions to the 3-point function that ultimately lead to higher order corrections to the dynamics of ρ_θ. However, the orders that we keep are enough to capture both short time dynamics and long-time thermalization. 

The dynamic equations has to be supplemented with the initial data. For the reduced 2-point and 3-point functions these are (see section <ref> for an explicit derivation)

    g^(2)_{θ,θ'}|_t =0 
       = [C]^ reg_θ,θ'   g^(3)_{θ,θ'}, θ” (z)
      |_t=0  = [(R^t)^-1_θ,ξ(R^t)^-1_θ',ξ
       f_ξg_ξ,θ”(z)]^ reg .


Results of the numerical solution to the set of equations (<ref>), (<ref>), (<ref>) and (<ref>) are presented in the main text.  

However, to capture the leading non-trivial dynamics the hierarchy can be further truncated to an equation for the density and

    (D_(1|2)^(1|2) g^(2))_θ, θ'(z)
        = V'(z) ∫γ(∂_θρ( θ)C_γ,θ'-C_θ,γ∂_θ'ρ( θ')) + 𝒪(V_0^2),

where in the equation for g^(2) we kept only the leading term of order V_0 and neglected all the terms that are of higher order. Whereas this set of equations is a further approximation to the dynamics, it has a benefit of being tractable analytically. In that regard we present two results. First, we will compute the correlations in the stationary state. Second, in the next Section, we will use it to derive a kinetic equation, the interacting Landau equation. 

To this end it is useful to introduce compact notation to express operations in the momenta space. The 2-point function depends on 2 momenta and the dynamic equations involve various contractions (integrations) of these momenta as actions of linear operators. The two momenta can be then thought as belonging to two different vector spaces. The operators acting on g^(2) naturally act in their tensor space. With this point of view we write 

    (𝐀g^(2)(z))_θ,θ' = 𝐀_θ,θ'^γ,γ'g^(2)_γ,γ'(z) = A_θ^γ g^(2)_γ,γ'(z) - A_θ'^γ g^(2)_γ,γ'(z),

where 𝐀 = A ⊗1  - 1⊗ A, with 1 being the identity operator which in the momenta space is represented by the δ-function. In a similar way, for the diffusion operator we have 𝐃 = 𝔇⊗1  + 1⊗𝔇. With this notation the kinetic term (<ref>) is given by

    (D_(1|2)^(1|2) g^(2)(z) )_θ, θ'= ((∂_t + 𝐀∂_z - 𝐃∂_z^2 ) g^(2)(z) )_θ,θ'.

Introducing as in the main text the matrix 

    𝐂_θ,θ' = ρ'_θ∫γC_γ,θ'-∫γC_θ,γρ'_θ',

the equation for g^(2) reads

    ((∂_t + 𝐀∂_z - 𝐃∂_z^2 ) g^(2)(z))_θ,θ' = V'(z) 𝐂_θ,θ'.

In the following we will also need that C_θ,θ'=(R^t)^-1fρ R^-1.

We conclude the discussion by computing the 2-point correlation in the stationary state. 












In order to find a stationary solution to this equation we suppose the following expression for correlations g^(2), Eq_θ,θ'(z)=V(x)F[ρ]_θ,θ' . We also assume that 𝔇_θ^γF[ρ]_γ,θ'=0.
The stationarity condition leads to the equation 𝐀 F[ρ] = 𝐂. Inverting this relation we find

    F[ρ]_θ, θ' = (𝐀^-1𝐂)_θ, θ' .

















We note that 𝐀^-1 operator has a simple form

    𝐀^-1= ((R^t)^-1⊗(R^t)^-1)1/v^ eff⊗1 -1⊗ v^ eff((R^t)⊗(R^t)) .

The expression for the stationary 2-point correlation is supplemented with an equation for the stationary 1-point function ρ. To simplify the discussion, we assume that the stationary state of the system is thermal. In that case we can evaluate F[ρ] explicitly as follows.

For a thermal state ρ_θ^ th we have the following expression

    ∂_θρ^ th_θ=δρ^ th_θ/δ n^ th_γ∂_γn^ th_γ=-β(R^t)^-1_θ,γρ_γf_γv^ eff_γ 1^ dr_γ ,

which follows from the general structure of thermal states in integrable models as described by the Thermodynamic Bethe Ansatz. With this expression it is a matter of simple manipulation to evaluate F[ρ^ th]. In the process we find the following formula

    𝐂^ th = -β((R^t)^-1⊗ (R^t)^-1)( v ⊗ 1 - 1 ⊗ v ) (ρ_ tot f ρ^ th⊗ρ_ tot f ρ^ th),

which implies that 

    (𝐀^-1𝐂^ th)_θ, θ' = [(R^t)^-1ρ_ tot fρ^ th]_θ  [(R^t)^-1ρ_ tot fρ^ th  ]_θ'.

The 2-point correlator in the stationary and thermal state is then

    g^(2),λ, th_θ,θ'(z)= - β V(z)[(R^t)^-1ρ_ tot fρ^ th]_θ  [(R^t)^-1ρ_ tot fρ^ th  ]_θ' + 𝒪(V_0^2)  .

We observe that the spatial profile follows the shape of the long-range potential, while correlations in the momenta simply factorize.

We can also verify that the diffusion operator acting on the stationary correlations is zero. This follows from the fact that 𝐃𝐀^-1𝐂^ th =0. To prove this relation we observe that diffusion operator 𝐃 acts independently on the θ and θ' variables and then use 𝔇=(R^t)^-11^ dr𝔇̃(1^ dr)^-1R^t together with 𝔇̃ρ f=0. This confirms that indeed g^(2),λ, th_θ,θ'(z) is the stationary correlation also in the presence of the diffusion.
 


 §.§ Derivation of three point correlations at equilibrium, eq. (SA.9)


We derive here an analitic expression for g^(3)_{θ,θ'}, θ”(z) in a GGE. In particular, reintroducing x and y, where z=x-y, we write

    g^(3)_{θ,θ'}, θ”(x,y) = ⟨δρ(ξ,x)δρ(ξ',x)δρ(θ”,y) ⟩ = δρ_θ/δ n_ξδρ_θ'/δ n_ξ'⟨δ n(θ,x)δ n(θ',x)δρ(θ”,y) ⟩
         =
       (R^t)^-1_θ,ξ(R^t)^-1_θ',ξ
       (ρ^ tot_ξ(x))^2
       ⟨δ[n(ξ,x)/ρ^ tot_ξ(x)] δρ(θ”,y)⟩=
       
       
       = (R^t)^-1_θ,ξ(R^t)^-1_θ',ξ(ρ^ tot_ξ(x))^2δ/δρ_α(x)[n(ξ,x)/ρ^ tot_ξ(x)] 
       ⟨δρ(α,x)δρ(θ”,y)⟩
       
       = (R^t)^-1_θ,ξ(R^t)^-1_θ',ξ(ρ^ tot_ξ(x))^21/(ρ^ tot_ξ(x))^2[R^t_ξ,α+n_ξT_ξ,α] 
       ⟨δρ(α,x)δρ(θ”,y)⟩
       
       = (R^t)^-1_θ,ξ(R^t)^-1_θ',ξ⟨δρ(ξ,x)δρ(θ”,y)⟩ .

where the repeated indices are integrated over. We also used the following relations

    δ n(x,θ) δ n(x,θ') =   δ_θ,θ'δ[ n(x,θ)/ ρ^ tot(x) ]  ,  δρ^ tot_α/δρ_β = T_α,β  ,  δρ_α =(R^t)^-1 _α,βρ^ tot_βδ n_β .
 
 
 


§ DERIVATION OF THE BOLTZMANN SCATTERING INTEGRAL


As discussed on the previous section, the BBGKY hierarchy for near-integrable dynamics, in the leading order in the perturbing potential V_0 yields the following set of equations 

    ∂_t ρ_θ =∫zθ'V'(z) ∂_θg^(2)_θ,θ'(z), 
       ((∂_t + 𝐀∂_z - 𝐃∂_z^2 )
        g^(2)(z))_θ,θ' = V'(z) 𝐂_θ,θ',








In the following we will derive the kinetic equation for ρ_θ which supersedes the Landau equation for interacting models.

The equation for the correlations is a linear equation and has the shape of a forced convective-diffusive equation. It can be solved by the Green's function technique. Its Green's function obeys

    ((∂_t + 𝐀∂_z - 𝐃∂_z^2 ) 𝐆(z, t) ) = δ(τ) δ(z) ( 1 ⊗ 1 ).

This equation can be solved by usual techniques and 

    𝐆(z,τ)=θ_H(τ)/2π∫ke^ikz[e^ik𝐀τ-k^2𝐃τ].

Note that the operators 𝐀 and 𝐃 depends also on τ through ρ_θ and C_θ, θ'.
At time t the correlations read

    g^(2)_θ,θ'(z,t)=∫_0^tτ∫z' V'(z') (𝐆(z-z', t - τ) 𝐂)_θ, θ'.




Inserting this result into the equation for the density we obtain

    ∂_t ρ_θ = ∫_0^tτ∫zz' V'(z) V'(z') ∫θ'∂_θ(𝐆(z-z', t - τ) 𝐂)_θ, θ'.

At this stage it is customary to perform two approximations. First, we assume that all operators 𝐀, 𝐂 and 𝐃 depend on time t and not on t - τ as expected from the formulas. The difference between the two evaluations is of order V_0 and hence does not contribute to the leading term. Second, we assume that the lower integration limit over τ can be extended from 0 to the minus infinity. This is motivated by assuming that the particles' distributions and correlations evolve at different timescale. The former being much shorter than the latter implies that the correlations relax to the temporary stationary state before the distribution itself changes. This permits to extend the temporal integral. With these two assumptions we obtain

    ∂_t ρ_θ = ∫_0^∞τ∫zz' V'(z) V'(z') ∫θ'∂_θ(𝐆(z-z',τ) 𝐂)_θ, θ',








where τ appears only explicitly in the expression for the Green's function and 𝐀 and 𝐃 are computed at time t. Hence, upon inserting the expression for 𝐆, the integral over τ can be performed. At the same time we express V'(x) through the Fourier transform of the potential,

    V(x) = ∫k/2π e^i k x Ṽ(k),

to obtain

    ∂_t ρ_θ= ∂_θ∫k/2π∫θ'
        k Ṽ^2(k)[(k𝐃-i𝐀)^-1𝐂]_θ,θ'.

This is the kinetic equation generalizing the Landau equation. To show that the RHS of Eq.(<ref>) is real we expand the kernel

    (k𝐃-i𝐀)^-1 = -i ∑_m≥ 0 (ik)^m (𝐀^-1𝐃) ^m 𝐀^-1 .

finding 

    ∂_t ρ_θ= -∂_θ∫θ'∑_m≥ 0∫k/2π (ik)^m+1Ṽ^2(k)[ (𝐀^-1𝐃) ^m 𝐀^-1𝐂]_θ,θ'.

Finally, we observe that each complex term of the sum with even m is vanishing because of the symmetry of the potential Ṽ(k)=Ṽ(-k).



























 §.§ Stability of the thermal state


We will show, that the thermal state is a stationary state of the interacting Landau equation. This guarantees the stability of the thermal equilibrium. The prove is simple. From the expansion (<ref>) we see that each term contains factor 𝐃𝐀^-1𝐂. However as derived in Appendix <ref>, this expression, when specialized to a thermal state, vanishes, 𝐃𝐀^-1𝐂^ th = 0. Therefore, every thermal state is a stationary state. 




 §.§ Conservation of density, energy and momentum


The interacting Landau equation respects the usual 3 conservation laws, that is conservation of particle number, momentum and the energy. In the standard Landau equation, the conserved form of the energy is the kinetic energy. Here, the integrable dynamics, which governs the left hand sides of the dynamic equations, takes into account also short-range interactions. Hence, the resulting kinetic equation conserves the sum of the kinetic energy and the potential energy of the short-range interactions. Equivalently, we interpret it as the kinetic energy of the quasiparticles. 

To prove the conservation laws it is convenient to rewrite the equation in a form of the diffusion equation

    ∂_t ρ_θ= 
        ∂_θ(𝔇^ LR_θ,γ∂_γρ_γ),

where we introduced

    𝔇^ LR_θ,γ=∫θ'∫k/2π  kṼ^2(k)(𝐌_θ,θ'^γ,γ'-𝐌_θ,θ'^γ',γ)∫αC_γ',α  ,  𝐌_θ,θ'^γ,γ'= Re[ (k 𝐃 - i 𝐀)^-1]_θ,θ'^γ,γ',

and used eq. (<ref>) for 𝐂. We observe that the second operator 𝐌 is partially transposed (only in γ momenta) compared with the first one. Such partial transposition is not a symmetry of the operator, unlike the full transposition, 𝐌^T = 𝐌. 

The particle number Q_0 is given by an integral over ρ_θ. Its conservation is immediate, (where L the volume of the system)

    L^-1∂_t Q_0=  ∂_t ∫θρ_θ= ∂_t ∫θ∂_θ(𝔇^ LR_θ,γ∂_γρ_γ) = 0 .

The total momentum Q_1 is given by

    L^-1 Q_1 =  ∫θ p_θρ_θ.

In a Galilean invariant theory p_θ = θ and 

    L^-1∂_t Q_1 = ∫θp_θ∂_tρ_θ   =- ∫θ(
        𝔇^ LR_θ,γ∂_γρ_γ).

The integral over 𝔇^ LR_θ,γ is 

    ∫θ𝔇^ LR_θ,γ   =∫θθ'∫k/2π kṼ^2(k)(𝐌_θ,θ'^γ,γ'-𝐌_θ,θ'^γ',γ)∫αC_γ',α=0  ,

upon exchanging in the second term the dummy integration variables θ and θ' and using the symmetry of 𝐌.

Finally we show that kinetic energy is also conserved 

    L^-1∂ Q_2 = ∫θϵ_θ∂ρ_θ.

 For this proof it is convenient to go back to the original formulation (<ref>) of the kinetic equation,

    L^-1∂_t Q_2    = ∫θε_θ∂_tρ_θ = -∫k/2π k Ṽ^2(k)∫θθ'ε'_θ[(k𝐃-i𝐀)^-1𝐂]_θ,θ'
       
        = -1/2∫k/2π k Ṽ^2(k)∫θθ'(ε'⊗1-1⊗ε')
        [(k𝐃-i𝐀)^-1𝐂]_θ,θ'
       
        = -1/2∫k/2π k Ṽ^2(k)∫θθ'𝐀[(k𝐃-i𝐀)^-1𝐂]_θ,θ' =0

In going to the last line we used that for any function f_θ of the momenta, it holds

    ∫θε'_θf(θ)=∫θγξε'_θ(R^t)^-1_θ,γ1^ dr_γ/1^ dr_γR^t_γ,ξf_ξ=∫θγξ(R^t)^-1_θ,γv^ eff_γR^t_γ,ξf_ξ=∫θγA_θ,γf(γ) .

In the last we used that upon expanding the inverse operator we get ∫θ𝔇_θ,γ which is indeed zero. This concludes our proof. 












§ NUMERICAL SIMULATIONS


In this section we explain in details how we performed the numerical simulations used to obtain the reults presented in the main text.

Firstly, we detail the method used to integrate the set of gBBGKY equations (<ref>), (<ref>), (<ref>) and (<ref>). This is a set of coupled non-linear integro-differential equations and its solution represents a highly expensive numerical task in terms of computational time and memory consumption. 
In order to overcome this complexity we perform the following crucial approximation

    g^(2)_{θ,θ'} (t)
       ≃ [C]^ reg_θ,θ'(t)+𝒪(V_0)    g^(3)_{θ,θ'}, θ” (z,t)  ≃ [(R^t)^-1_θ,ξ(R^t)^-1_θ',ξ
       f_ξg_ξ,θ”(z)]^ reg(t)+𝒪(V_0) .

that permits to reduce the initial set of equations to the set of equations (<ref>) and (<ref>). This simplification neglects only subleading corrections at order ∼ aV_0^2 to the evolution of g^2_θ,θ'(z,t).

Then, we discretize space and momentum and we solve the two coupled equations (<ref>) and (<ref>) using the 4^ th-order Runge Kutta method with time interval Δ t=0.02. More precisely we used the following grid of points: θ∈[-θ_ max,θ_ max] with θ_ max=7 and Δθ=14/100; z∈[-L,L] with L=200 and Δ z=2. 
Additionally, in order to avoid spurious space reflections at the borders during the space propagation of g^(2)_θ,θ'(z), we forced it to be vanishing at z=± L.
The choice of Δ t,Δθ, Δ z, L and θ_ max was made such that it guarantees the convergence of the numerical solution.



The numerical simulations for the evolution of the hard rods gas with long range interaction was made using the same algorithm presented in  Appendix A of <cit.>. In particular we used the symplectic integrator 
implemented in the  function of the  library, i.e. an explicit Runge-Kutta method of order 8 <cit.>.
We used a time step Δ t_ HR=0.0005 that guarantees an error in the total energy of the order of 10^-10.

More precisely, we simulated a system on a ring of total length L_ HR=500 with particle density ρ̅=0.2, namely with ∼ 100 particles per realisation. We performed the ensemble average over a total of 2000 independent initial states for each set of parameters. 















































































































































































 Paper Title: Properties of Krylov state complexity in qubit dynamics.



APS/123-QED






 Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India




§ ABSTRACT 
We analyze the properties of Krylov state complexity in qubit dynamics, considering a single qubit and a qubit pair. A geometrical picture of the Krylov complexity is discussed for the single-qubit case, whereas it becomes non-trivial for the two-qubit case. Considering the particular case of interacting Rydberg atoms, we show that the Krylov basis obtained using an effective Hamiltonian minimizes the complexity compared to that obtained from the original Hamiltonian. We further generalize the latter property to an arbitrary Hamiltonian in which the entire Hilbert space comprises two subspaces with a weak coupling between them.    



Properties of Krylov state complexity in qubit dynamics
    Rejish Nath
    August 5, 2024
=======================================================







§ INTRODUCTION
 
The concept of complexity has found various implications in physics. For instance, in computation, it can be the number of resources one requires, like that of bits/qubits, operations, or the depth of the circuits, etc. <cit.>. For implementing a multi-qubit unitary operation, it can be the minimum number of predefined elementary unitary operations <cit.>. It can also be interpreted as the minimum geodesic distance between the identity and a target unitary operator in an operator space with an appropriately chosen metric, as in the case of  Nielsen complexity <cit.>. Similarly, one can define the complexity of a quantum state, which can be the minimum number of unitary operations from a predefined set of operators required to synthesize the given target state from a reference state <cit.>. The latter also led to studies for finding the minimal geodesics using various metrics <cit.>. The concept of complexity also finds applications in black holes, where correspondence between the growth of an interior volume of a black hole and growth of the complexity of its dual quantum state on the boundary is conjectured, termed as the holographic complexity <cit.>. Despite all these developments, the above complexity measures suffer from ambiguities, for instance, the choice of metric in the case of the Nielsen complexity, in choosing the set of elementary unitary operations for achieving a target unitary operation from the
identity operator or a target quantum state from an initial state. 

Subsequently, a new measure of complexity was put forward for states <cit.>, motivated by a similar measure for operators <cit.>, called the Krylov complexity <cit.>. It is quantified as the depth or spread of an initial state in its Krylov basis <cit.> over its time evolution. The advantages of the Krylov complexity are twofold. First, given an initial state and a Hamiltonian, the complexity is unambiguously defined via the Krylov basis with a calculation using the Lanczos algorithm <cit.>. Second, the Krylov basis has been shown to minimize the spread of an initial state among all choices of ordered bases <cit.>. The Krylov complexity further gained attention in studying quantum chaos and integrability <cit.> and found applications in quantum phase transitions <cit.>, black holes <cit.>, neutrino oscillations <cit.>, modular Hamiltonians <cit.>, quenched systems <cit.>, non-Hermitian systems <cit.>, periodically-driven systems <cit.>, and others <cit.>. Also, there are efforts to establish a connection between Krylov complexity and other complexity measures. For instance, it is shown that the time average of the Krylov complexity can be used to set an upper bound for the Nielsen complexity of the corresponding unitary operator using a specific metric that is defined using the Krylov basis <cit.>. A unified framework for describing the operator and state complexity was discussed in <cit.>, which has generalized this measure to mixed states and reduced density matrices <cit.>. There are efforts to interpret the Krylov complexity geometrically <cit.>, and a correspondence between the Krylov complexity and the wormhole length is found <cit.>. 
In this work, we study the Krylov complexity of a single qubit and a two-qubit system in search of fundamental insights into the Krylov complexity. While it has been shown in <cit.> that the Krylov complexity or its square root cannot act as a measure of distance between states in general, we find that the square root of the complexity does, indeed, relate to the distance between time-evolved states for a single qubit. However, once we consider two non-interacting qubits, which could have been described independently, we find that such a geometrical interpretation is no longer possible precisely because the overall Krylov complexity is more than just a sum of the individual complexities. We further consider a pair of two-level Rydberg atoms <cit.> which exhibit correlated dynamics under Rydberg blockade  <cit.>. The doubly excited states are inhibited in the Rydberg blockade dynamics and an effective two-level picture emerges. In that case, one expects the dynamics of Krylov complexity to be that of a single two-level system, i.e., it periodically oscillates between zero and one. However, we observe that the Krylov basis obtained using the original Hamiltonian exhibits a complexity having an amplitude larger than one. In contrast, we find an ordered basis, obtained as the Krylov basis of an effective Hamiltonian, that minimizes the complexity. We further generalize this aspect for an arbitrary Hamiltonian for which the entire Hilbert space comprises of two subspaces with a weak coupling between them. 

The paper is organized as follows: in section <ref>, we review the Krylov state complexity and some of its properties. In section <ref>, we study the single qubit and provide a geometrical interpretation of the Krylov complexity. We extend the study to two qubits in section <ref>, considering both uncoupled and coupled qubits. The particular case of a pair of Rydberg atoms is discussed in Sec. <ref>. We generalize the minimization of the Krylov complexity in Sec. <ref>. Finally,  we summarize our results in Sec. <ref>. 





§ KRYLOV STATE COMPLEXITY

For a closed system described by a Hamiltonian Ĥ, the time evolution of an initial state |ψ_0⟩ is 

    |ψ (t) ⟩ = e^-iĤt |ψ_0 ⟩ = ∑_n=0^∞(-it)^nn! |ψ_n ⟩,

where |ψ_n ⟩ = Ĥ^n |ψ_0⟩. Employing the Gram-Schmidt process to the set of states {ψ_0, Ĥψ_0, Ĥ^2ψ_0, ...} generates an ordered, orthonormal basis, 𝒦={K_1, K_2, ..., K_n} for the part of the Hilbert space that the time-evolution explores. The new basis 𝒦 is called the Krylov basis <cit.>. The zeroth element in the Krylov basis is the initial state itself, i.e., |K_0⟩=|ψ_0⟩. The other Krylov basis states (n>1) are constructed recursively via the Lanczos algorithm <cit.> from the initial state and the Hamiltonian, i.e., via |K_n+1⟩ =b_n+1^-1|A_n+1⟩, where 

    |A_n+1⟩ = (Ĥ - a_n) |K_n⟩ - b_n |K_n-1⟩

with a_n=⟨ K_n|Ĥ|K_n⟩ and b_n= ⟨ A_n|A_n⟩^1/2 are the Lanczos coefficients. Note that b_0=0. By construction, the dimension of the Krylov space may be smaller than the full Hilbert space. 

Given an initial state, we can define a cost function on all ordered bases, ℬ = {|B_0⟩, |B_1⟩, , |B_D-1⟩}, of the Hilbert space, given by

    𝒞_ℬ (t) = ∑_n=0^D-1 c_n | ⟨ψ (t)|B_n⟩|^2 = ∑_n=0^D-1 c_n P_ℬ (n,t)

where c_n is a monotonically increasing function of n with c_n ≥ 0, P_ℬ (n,t) = | ⟨ψ (t)|B_n⟩|^2, and D is the dimension of the Hilbert space. The state or spread complexity is obtained when c_n = n and is minimized for the Krylov basis, atleast in the vicinity of t=0 <cit.>. The latter is called the Krylov complexity and is given by <cit.>, 

    𝒞_𝒦 (t) = ∑_n = 0 n |⟨ψ (t)|K_n⟩|^2.

By construction, 𝒞_𝒦 (t=0)=0 and in general, 𝒞_𝒦 (t=0)≥ 0. 

Interestingly, a given initial state only evolves through states with the same Lanczos coefficients. To show that, consider two different initial states, |ψ_0⟩=|K_0⟩ and |ψ_0^'⟩ = e^-iĤt |ψ_0 ⟩, connected by a unitary time evolution. We can see that a_0^' = ⟨ψ_0^'|Ĥ|ψ_0^'⟩ = ⟨ K_0|e^iĤtĤ e^-iĤt|K_0⟩ = a_0 and b_1^' = ⟨ A_1^' | A_1^'⟩^1/2 = b_1 since |A_1^'⟩ = e^-iĤt (Ĥ - a_0) |K_0⟩ = e^-iĤt |A_1 ⟩. Similarly, we can show that a_1^' = a_1, b_2^' = b_2, |K_1^'⟩ = e^-iĤt |K_1⟩, and thereby, |A_2^'⟩ = e^-iĤt |A_2⟩. Following the same, we get |A_n+1^'⟩ = e^-iĤt|A_n⟩ assuming that |A_j^'⟩ = e^-iĤt |A_j⟩, a_j^' = a_j and b_j^' = b_j for all j ≤ n. Thus, |ψ_0^'⟩ has the same set of Lanczos coefficients as |ψ_0⟩ and hence, the Krylov complexity <cit.>.

In the Krylov basis, the Hamiltonian satisfies,

    Ĥ |K_n⟩ = a_n |K_n⟩ + b_n |K_n-1⟩ + b_n+1 |K_n+1⟩,

and forms a tridiagonal matrix, which resembles the Hamiltonian of a tight-binding model on a chain with a site-dependent hopping amplitude b_n between n and n-1 sites. a_n acts as a local energy offset or a chemical potential. Now, the Krylov complexity 𝒞_𝒦 (t) can be interpreted as the expectation value of the position on this lattice, where
the origin is at the first site [n=0 in Eq. (<ref>)]. In other words, the Krylov complexity measures how far the state has evolved away from the initial site in the above tight-binding model.




§ SINGLE QUBIT


In this section, we discuss the properties of the Krylov complexity in the dynamics of a single qubit. Let the eigenstates of the Hamiltonian be |±⟩ with eigenenergies ±ω/2 so that the Hamiltonian in the energy eigenbasis takes the form Ĥ = (ω/2) σ̂_z (where we have set ħ = 1). The initial state |ψ_0⟩ = α |+⟩ + β |-⟩ forms the first Krylov basis state |K_0⟩. The second one is orthogonal to the initial state and is |K_1⟩ = β^* |+⟩ - α^* |-⟩, where α and β are the probability amplitudes. |K_1⟩ is defined up to a global phase factor. The Hamiltonian of a qubit in the Krylov basis can be written as (after neglecting constants), 

    Ĥ=(a_0-a_1)/2σ̂_z+b_1σ̂_x,

where a_0= (|α|^2 - |β|^2) ω/2, a_1=-a_0, b_1=|α||β|ω and σ̂_x, y are the Pauli spin-1/2 matrices. 

A general qubit state in Krylov basis can be written as |ψ(t)⟩=c_0(t)|K_0⟩+c_1(t)|K_1⟩, where c_0,1 is the probability amplitude of finding the qubit in the corresponding Krylov basis states. The Krylov complexity of a qubit for the initial state |ψ_0⟩ is obtained as <cit.>

    𝒞_𝒦 (t) = |c_1(t)|^2=4 |α|^2 |β|^2 sin^2 ( ω t2),

 satisfying 0 ≤𝒞_𝒦 (t) ≤ 1 with its maximum value of 4 |α|^2 |β|^2 and a time-averaged value of 2|α|^2 |β|^2. The maximum value of one is attained when the initial state is an equal superposition of energy eigenstates, i.e., when |α|^2=|β|^2. In terms of Lanczos coefficients, the Krylov complexity becomes,

    𝒞_𝒦 (t) = 4b_1^24b_1^2 + (a_0 - a_1)^2sin^2 ( t√(4b_1^2 + (a_0 - a_1)^2)2),

which is symmetric in interchange between a_0 and a_1. Thus, an initial state and the corresponding orthogonal state of a qubit exhibit same complexity dynamics. If the initial state is stationary or when the two levels are degenerate, 𝒞_𝒦 (t) remains zero, and those cases are disregarded from further discussion. Once the energy eigenvalues of the Hamiltonian are kept the same, the initial state determines the amplitude of the oscillation of 𝒞_𝒦 (t), i.e., how far the state is evolved away from the initial state. 





 §.§ Shannon entropy and inverse participation ratio


Now we look at the Shannon entropy S_ Sh(t) and the inverse participation ratio Π(t) computed using the Krylov basis <cit.> and provides the relation with the Krylov complexity. The Shannon entropy determines how a given quantum state is spread over the Hilbert space spanned by the Krylov basis <cit.>, whereas Π(t) is typically used to analyze the localization/delocalization properties of quantum dynamics <cit.>, expecting  a simple relation with the Krylov complexity. They are defined as follows,

    S_ Sh=-P_nlog_2 P_n 
    Π=1/∑_n P_n^2-1,

where P_n(t)=|⟨ψ(t)|K_n⟩|^2. In terms of Krylov complexity, we get,

    S_Sh   =   -𝒞_𝒦log_2 𝒞_𝒦 - (1 - 𝒞_𝒦) log_2 (1 - 𝒞_𝒦) 
    Π   =   12𝒞_𝒦^2 - 2 𝒞_𝒦 + 1-1.

 The dependence of S_Sh and Π on 𝒞_K is shown in Fig. <ref>. When 𝒞_K=0 or 1, the system is in a Krylov basis state and both S_Sh and Π vanish, indicating the localization in the Krylov space. When 𝒞_𝒦=0.5, they attain the maximum value of unity, indicating the delocalization. Above relations indicate a non-trivial functional dependence of S_ Sh(t) or Π(t) on 𝒞_𝒦.








 §.§ Geometrical interpretation


Here, we establish a geometrical interpretation of the Krylov state complexity for a single qubit. The Krylov complexity itself does not act as a measure of distance between states <cit.>. In particular, it is shown that 𝒞_K and some functions of the Krylov complexity do not satisfy the triangle inequality, i.e. 𝒞_𝒦 (t_3 - t_1) ⪇𝒞_𝒦 (t_3 - t_2) + 𝒞_𝒦 (t_2 - t_1). Now, we show that in the case of a single qubit, the square root of the Krylov complexity does act as a measure of distance between the states connected by unitary time-evolution and satisfies the triangle inequality. 




The state vector |ψ⟩ is mapped to a point, α⃗_|ψ⟩, in a three-dimensional parameter space via the density matrix written in the energy eigenbasis as

    ρ̂(t) = |ψ (t)⟩⟨ψ (t)| = ℐ2 + α⃗_|ψ⟩·σ̂,

where ℐ is the identity matrix. Henceforth, we drop the subscript |ψ⟩ in α⃗_|ψ⟩ for convenience. For an initial state |ψ_0⟩ = cos (θ_0/2) |+⟩ + sin (θ_0/2) e^iϕ_0 |-⟩ at t=0 and under the unitary evolution given by the Hamiltonian, Ĥ = (ω/2) σ̂_z, the components of the vector α⃗ (t) are obtained as,

    α_x    = sin(θ_0)2cos (ϕ_0 + ω t),
    α_y    = sin(θ_0)2sin (ϕ_0 + ω t), 
    α_z    = cos(θ_0)2.

Thus, the time-evolution of an initial state |ψ_0⟩ represents a circular motion in the α_x-α_y plane, keeping θ(t)=θ_0, as shown in Fig. <ref>. At this point, we define the distance between two states as the Euclidean distance between the corresponding points in the α⃗- space and obtain,

    |α⃗ (t_2) - α⃗ (t_1)|^2 = sin^2 θ_0 sin^2 ( ω (t_2 - t_1)2) = 𝒞_𝒦 (t_2 - t_1),

where the Krylov complexity is independent of ϕ_0. Strikingly, the square root of the Krylov complexity can be identified as the magnitude of displacement of the state vector from the initial state.

Triangular inequality. We now prove that this does satisfy the triangle inequality. In the following, we set t_ij = t_i - t_j for brevity. The proof of the triangle inequality then follows from:


    √(𝒞_𝒦 (t_31))   = √(𝒞_𝒦 (t_32 + t_21))
       = sinθ_0 |sin(ω (t_32 +  t_21)2)| 
       = sinθ_0 |sin(ω t_322) cos(ω t_212) + sin(ω t_212) cos(ω t_322) | 
       ≤sinθ_0 ( |sin(ω t_322) | | cos(ω t_212) | 
        + |sin(ω t_212) | | cos(ω t_322) |  ) 
       ≤sinθ_0 ( |sin(ω t_322) | + |sin(ω t_212) | ) 
       = √(𝒞_𝒦 (t_32)) + √(𝒞_𝒦 (t_21)).

Note that this is still consistent with the results shown in <cit.>, where it is argued that √(𝒞_𝒦) violates the triangle inequality for three states separated by small time intervals. This is shown in <cit.> by Taylor expanding √(𝒞_𝒦 (t)) as,


    √(𝒞_𝒦 (t))≈ b_1 t - [ (a_0 - a_1)^2 + 2 (2b_1^2 - b_2^2 ) ]24 t^3 + 𝒪 (t^5).

Depending on the initial state and its Lanczos coefficients, the coefficient of t^3 may be positive, in which case √(𝒞_𝒦 (t_1+t_2))≰√(𝒞_𝒦 (t_1)) + √(𝒞_𝒦 (t_2)), i.e., not sub-additive. However, this coefficient is negative for all initial states for the single qubit, as b_2 = 0. This necessitates looking at higher-order terms in the series expansion, or the function itself, which we have explicitly shown above does satisfy the triangle inequality. 

Finally, we remark that while our mapping to the α-parameter space appears to be dependent on our initial choice of basis (the energy eigenbasis), we show below that a change in basis only results in mapping to different points in the parameter space and does not affect the displacement. We consider a unitary transformation, |ψ^'⟩ = 𝒰 |ψ⟩, under which the density matrix transforms as


    ρ^' (t) = 𝒰ρ (t) 𝒰^†   =12·ℐ + α⃗^' (t) ·σ̂

Now we show that dα⃗^' 2 (t)=dα⃗^2(t)=𝒞_𝒦 (t). Considering the unitary transformation in its general form,

    𝒰 = cos (a) ·ℐ + i sin (a) (n⃗·σ̂)

where n⃗·n⃗ = 1 and 2a is an angle of rotation about an axis along n⃗. By a straightforward, albeit slightly tedious calculation using the properties of Pauli matrices, it can be shown that 𝒰 (α⃗·σ̂) 𝒰^† = α⃗^'·σ̂ where

    α⃗^' = cos (2a) α - sin (2a) (n×α) + 2 (sin^2 (a) ) (n·α) n

For brevity, we set α (t_1=0) = α_1 and α (t_2 = t) = α_2 and α_21 = α_2 - α_1, so that (dα)^2 (t) = α_21^2. Then, we have 

    (dα⃗^')^2 (t)    = [ (cos 2a) α⃗_21 - (sin 2a) (n⃗×α⃗_21) + 2 (sin^2 a) (n⃗·α⃗_21) n⃗]^2 
       = (cos^2 2a) α_21^2 + (sin^2 2a) [α_21^2 - ( n⃗·α⃗_21)^2 ] 
        + 4 sin^4 a ( n⃗·α⃗_21)^2 + 4 sin^2 a cos 2a ( n⃗·α⃗_21)^2=(dα⃗)^2 (t),

i.e., our geometric interpretation is basis-independent.   



 §.§ A two level atom


At this point, we consider a two-level atom in which the ground state |g⟩ is coupled to the excited state |e⟩ by a light field of Rabi frequency Ω and with a detuning Δ, described by the Hamiltonian (ħ=1),

    Ĥ = - Δσ̂_ee + Ω2σ̂_x,

where the operator σ̂_ab=|a⟩⟨ b| with a, b∈{g, e} and σ̂_x=σ̂_eg+σ̂_ge. The energy eigenvalues of Ĥ are E_± = ±√(Δ^2 + Ω^2)/2. In terms of the energy eigenstates |±⟩, the bare states are,
    |e⟩ = sin (θ/2) |+⟩ + cos (θ/2) |-⟩ and |g⟩ = cos (θ/2) |+⟩ - sin (θ/2) |-⟩,
where cos (θ/2) = [(√(Δ^2 + Ω^2) + Δ)/(2√(Δ^2 + Ω^2))]^1/2. If we take either |g⟩ or |e⟩ as the initial state, the Krylov complexity is the same and is

    𝒞_𝒦(t) = Ω^2Δ^2 + Ω^2sin^2 ( √(Δ^2 + Ω^2)t2),

which characterizes the well known Rabi oscillations. Using 𝒞_𝒦(t), we can compute the Shannon entropy and the inverse participation ratio using the Eqs. (<ref>) and (<ref>) as well.




§ TWO QUBITS

Now, we extend the calculations to two qubits. In particular, we consider both uncoupled and coupled qubits. For the latter, we take the case of a pair of interacting two-level Rydberg atoms.


 §.§ Non-interacting qubits



In the following, we discuss the Krylov complexity of a pair of non-interacting qubits governed by the Hamiltonian, Ĥ=Ĥ_1 + Ĥ_2, where Ĥ_1 (Ĥ_2) is the Hamiltonian of the first (second) qubit. Let the energy eigenstates of H_i be |±⟩_i and eigenvalues E_±^(i)=ω_i/2. Considering a general initial product state for the two qubits: (α_1 |+⟩_1 + β_1 |-⟩_1) ⊗ (α_2 |+⟩_2+β_2 |-⟩_2), we obtain the Krylov complexity as,

    𝒞_𝒦 (t) = 𝒞_𝒦^(1) (t) + 𝒞_𝒦^(2) (t) + ℱ(t)

where 𝒞_𝒦^(i) (t) is the Krylov complexity of the individual qubit, given in Eq. (<ref>) and  


    
    ℱ (t)    = (ω_1 cosθ_1 - ω_2 cosθ_2)^2ω_1^2 sin^2 θ_1 + ω_2^2 sin^2 θ_2𝒞_𝒦^(1) (t) 𝒞_𝒦^(2) (t) 
        + [ sin^2 θ_1 sin^2 θ_2 [(ω_1 cosθ_1 - ω_2 cosθ_2)^2 + 2 (ω_1^2 sin^2 θ_1 + ω_2^2 sin^2 θ_2)](ω_1^2 + ω_2^2 - 2ω_1 ω_2 cosθ_1 cosθ_2)(ω_1^2 sin^2 θ_1 + ω_2^2 sin^2 θ_2)][ω_-2sin(ω_+t2) - ω_+2sin(ω_-t2) ]^2,


where ω_±=ω_1±ω_2, sinθ_i = 2 |α_i| |β_i|, and cosθ_i = |α_i|^2 - |β_i|^2. See Fig. <ref> for the Bloch sphere representation of the product state of two independent qubits as they evolve independently on the Bloch sphere with two circular trajectories. As we find, Krylov complexity is the sum of Krylov complexity of each qubit and an additional term, ℱ, which is positive-valued at any instant t. If ω_1 = ω_2, the function ℱ simplifies to

    ℱ (t)= [(cosθ_1 - cosθ_2)^2sin^2 θ_1 + sin^2 θ_2] 𝒞_𝒦^(1) (t) 𝒞_𝒦^(2) (t),

which exhibits periodic oscillations and vanishes when θ_1=θ_2 as shown in Fig. <ref>(a). The amplitude or the maximum of ℱ (t) as a function of θ_1 and θ_2 for ω_1=ω_2=ω is shown in Fig. <ref>(b), and is independent of ω. The maximum amplitude is 0.5 when θ_1=π/4 and θ_2=3π/4 or vice versa. For ω_1≠ω_2, ℱ (t) is still periodic but exhibits more complex behaviour as shown in Fig. <ref> and the maximum of the peak value of ℱ (t) is found when θ_1=θ_2=π/2.

For vanishing ℱ (t), the geometric interpretation for the single qubit can be straightforwardly extended to the case of two non-interacting qubits. As the Krylov complexity, in this case, is simply the sum of the complexities of the individual qubits, the total complexity can be viewed simply as a restatement of the Pythagoras theorem. The individual complexities can represent squared distances in orthogonal directions, and the total complexity can represent the hypotenuse squared. However, when ℱ (t)≠ 0, the square root of the Krylov complexity generally does not satisfy the triangle inequality. 












Two-level atoms. For a pair of two non-interacting two level atoms coupled by light fields, the Hamiltonian is

    Ĥ = ∑_i=1^2 [ - Δ_i σ̂^i_ee + Ω_i2σ̂_x^i ].

The two atom bare states are |gg⟩, |ge⟩, |eg⟩ and |ee⟩. For a global driving, i.e., when Δ_1 = Δ_2 = Δ and Ω_1 = Ω_2 = Ω, the states |ge⟩ and |eg⟩ are degenerate, and only the symmetric state |+⟩=(|ge⟩ + |eg⟩)/√(2) is relevant to the dynamics and the antisymmetric state |-⟩=(|ge⟩ - |eg⟩)/√(2) can be disregarded. In the case of global driving, we obtain the Krylov complexity 𝒞_K^α (t) for an initial product state |α⟩ as, 

    𝒞_𝒦^gg (t) = 2Ω^2Δ^2 + Ω^2   sin^2 (√(Δ^2 + Ω^2)t2), 
    𝒞_𝒦^ge (t) = 2Ω^2Δ^2 + Ω^2   sin^2 ( √(Δ^2 + Ω^2) t2) 
       + 2Δ^2 Ω^2(Δ^2 + Ω^2)^2sin^4 ( √(Δ^2+Ω^2)t2),
    
𝒞_𝒦^ee (t)=𝒞_𝒦^gg(t) and 𝒞_𝒦^eg (t)=𝒞_𝒦^ge(t). For the symmetric state, we get 

    𝒞_𝒦^+ (t) = Ω^2Δ^2 + Ω^2sin^2 (√(Δ^2 + Ω^2)t ) + 
    8Ω^2 Δ^2(Δ^2 + Ω^2)^2sin^4 (√(Δ^2 + Ω^2) t2),

which cannot be expressed in terms of 𝒞_𝒦^eg (t) and 𝒞_𝒦^ge (t). Although in a single qubit, the Krylov complexity is identical for initial states |g⟩ and |e⟩, in a pair of qubits, despite being entirely uncoupled, 𝒞_𝒦 is found to depend on the initial state, whether being in |gg⟩ or |ge⟩. The latter can be attributed to the function ℱ(t) in Eq. (<ref>). When Δ=0, all states are degenerate, ℱ(t) vanishes and the complexity simplifies to 𝒞_𝒦^gg = 𝒞_𝒦^ge = 2 sin^2 (Ω t/2) and  𝒞_𝒦^+ = sin^2 (Ω t).




 §.§ Interacting qubits: a Rydberg atom pair


At this point, we consider the coupling between the qubits, and in particular, we consider the case of a pair of interacting two-level Rydberg atoms. The Rydberg atoms are two-level atoms in which the ground state |g⟩ is coupled to the Rydberg state |e⟩. When both qubits are in the Rydberg state, they interact repulsively, and the governing Hamiltonian in the interaction picture is,

    Ĥ = ∑_i=1^2 [ - Δ_i σ̂^i_ee + Ω_i2σ̂_x^i ] + V_0 σ̂^1_eeσ̂^2_ee

where V_0 = C_6/R^6 is the interaction strength, and R is the distance between the two Rydberg atoms.    





  §.§.§ Rydberg blockade


For the global driving and V_0≫Ω,
the system exhibits coherent Rabi oscillations between the states |gg⟩ and |+⟩ with an enhanced Rabi frequency of √(2)Ω, resulting in an effective two-level atom or called the superatom <cit.>. The doubly excited state |ee⟩ is completely inhibited in the dynamics and is called the Rybderg blockade. We would, therefore, expect the Krylov state complexity of the two initial states |gg⟩ and |+⟩ to be identical and resemble that of a two-level system, with the complexity peaking at one and exhibiting oscillations at the enhanced Rabi frequency. However, as we show in Fig. <ref>(a), the dynamics of 𝒞_𝒦 (t) is different for these two initial states, and in particular, the amplitude of oscillation of 𝒞_𝒦 (t) for initial state |+⟩ exceeds one, although the population dynamics is identical for both initial states as shown in Figs. <ref>(b) and <ref>(c). As shown below, it is different since the Krylov basis is different for the two initial states. 



For the initial state |gg⟩, apart from |gg⟩, the Krylov basis consists of,

    |K_1⟩^gg   = 1√(2)(|ge⟩ + |eg⟩) = |+⟩
    
        |K_2⟩^gg   = |ee⟩
    
        |K_3⟩^gg   = 0,

with Lanczos coefficients a_0 = a_1 = 0, a_2 = V_0, and b_1 = b_2 = Ω/√(2). Note that only the first two Krylov basis states (|K_0⟩^gg and |K_1⟩^gg) are relevant in the blockade regime. In that case, we can use Eq. (<ref>) and obtain the Krylov complexity as 𝒞_𝒦 (t) = sin^2 ( Ω^' t2) with Ω^' = √(2)Ω, the enhanced Rabi frequency. 

In contrast, for the initial state |+⟩, the Krylov basis comprises of |K_0⟩^+=|+⟩ and,

    |K_1⟩^+   = 1√(2)(|gg⟩ + |ee⟩) 
    
        |K_2⟩^+   = 1√(2)(-|gg⟩ + |ee⟩) 
    
        |K_3⟩^+   = 0.

 with Lanczos coefficients a_0 = 0, a_1 = a_2 = V_0/2, b_1 = Ω, and b_2 = V_0/2. Here, the first three Krylov basis states are needed to describe the blockade dynamics, indicating that 𝒞_𝒦(t) can go beyond the value of one, as shown by a dashed line in Fig. <ref>(a). It is surprising given that we observe coherent Rabi oscillations between |gg⟩ and |+⟩, as can be seen in Fig. <ref>(b) and <ref>(c), we expect to see identical Lanczos coefficients and Krylov complexity for both the states, as discussed in section <ref>. Using an ordered basis consisting of  {|+⟩, |gg⟩,|ee⟩, |-⟩}, the spread complexity dynamics starting from |+⟩ is found to be identical to the Krylov complexity obtained for the initial state |gg⟩ [solid line in Fig. <ref>(a)]. It implies that for the initial state |+⟩, another ordered basis exists for which the spread complexity becomes minimal, but not the Krylov basis constructed using the original Hamiltonian. Below, we show that the new ordered basis for which the spread complexity is minimum can be obtained as the Krylov basis obtained using a zeroth order effective Hamiltonian.   
 
The Krylov complexity attains even a larger amplitude (∼ 3) under Rydberg blockade [see the dashed line in Fig. <ref>(a)] if the initial state is |eg⟩ or |ge⟩. As we see below, all four Krylov states become relevant to characterize the dynamics and give an amplitude of three. The population dynamics shown in Fig. <ref>(b) is characterized by the periodic oscillations between |ge⟩ and |eg⟩ via |gg⟩. It implies that we can choose an ordered basis {|ge⟩, |gg⟩, |eg⟩, |ee⟩} such that the spread complexity amplitude is two [see the solid line in Fig. <ref>(a)], making it less than that of the Krylov complexity. For the initial state, |K_0⟩ = |ge⟩, we get

    |K_1⟩^ge   = 1√(2)(|gg⟩ + |ee⟩) 
    
        |K_2⟩^ge   = 1√(2)Ω_V(-V_0 |gg⟩ + 2Ω |eg⟩ + V_0 |ee⟩) 
    
        |K_3 ⟩^ge   = 1Ω_V(-Ω |gg⟩ - V_0 |eg⟩ + Ω |ee⟩,

 where Ω_V = √(2Ω^2 + V_0^2). The corresponding Lanczos coefficients are a_0 = 0, a_1 = V_0/2, a_2 = V_0^3/(2Ω_V^2), b_1 = Ω/√(2), b_2 = Ω_V/2 and b_3 = V_0^2Ω/(√(2)Ω_V^2). All four Krylov basis vectors have significant projections to |gg⟩, |eg⟩, or |ge⟩, making them all relevant for the blockade dynamics and resulting in a larger Krylov complexity. 

 The above results indicate that if the exact dynamics occur only in a subset of the full Hilbert space, the spread complexity is generally not minimal for the Krylov basis. This ambiguity can be removed by truncating the original Hilbert space and using an effective Hamiltonian. For instance, in the perfect blockade scenario, where the population in |ee⟩ is zero, we can arrive at an effective Hamiltonian (zeroth order) for V_0≫Ω <cit.>, 
 
    Ĥ_ eff = Ω2(σ̂_gg^1σ̂_x^2+σ̂_x^1σ̂_gg^2).

 Using the truncated Hilbert space and the effective Hamiltonian, we now compute the new Krylov basis for a given initial state and find that the modified Krylov complexity is minimal. For instance, for the initial state |ge⟩, using the effective Hamiltonian, we obtain the Krylov basis as |ge⟩, |gg⟩, |eg⟩, |ee⟩} and the Krylov complexity as 𝒞_𝒦 (t) = 2 sin^2 (Ω t/2√(2)), which has an amplitude of two not three. This aspect of reduced Hilbert space and an effective Hamiltonian to minimize the Krylov complexity is generalized in Sec. <ref>.





  §.§.§ Rydberg-biased freezing


Now, we examine the dynamics of the Krylov state complexity 
under the phenomenon of Rydberg-biased freezing, which arises from a combined effect of Rybderg blockade and an offset in the Rabi couplings of the two Rydberg atoms <cit.>. Again, we keep Δ_1 = Δ_2 = 0. When Ω_2>>Ω_1 and V_0 >> Ω_2, starting from |gg⟩, the first atom is essentially frozen in the ground state while the second atom undergoes Rabi oscillations between |g⟩ and |e⟩. In other words, Rabi-oscillations occur between |gg⟩ and |ge⟩ states as shown in Fig. <ref>(a), and |eg⟩ and |ee⟩ states do not contribute to the dynamics. 



For the initial state |K_0⟩^gg = |gg⟩, the Krylov basis states are,

    |K_1⟩^gg   = 1Ω̅(Ω_2 |ge⟩ + Ω_1 |eg⟩) 
    
        |K_2⟩^gg   = |ee⟩
    
        |K_3⟩^gg   = 1Ω̅( -Ω_1 |ge⟩ + Ω_2 |eg⟩) .

where Ω̅= √(Ω_1^2 + Ω_2^2). Even though, three Krylov basis states (|K_0⟩^gg, |K_1⟩^gg, and |K_3⟩^gg) look relevant to the dynamics, in the limit  Ω_2≫Ω_1, |K_2⟩^gg and |K_3⟩^gg≈ |eg⟩ can be disregarded. The corresponding Lanczos coefficients are a_0 = a_1 = a_3 = 0, a_2 = V_0, b_1 = Ω̅/2, b_2 = (Ω_1 Ω_2)/Ω̅≈Ω_1, and b_3 = (Ω_2^2 - Ω_1^2)/Ω̅≈Ω_2 and the Krylov complexity is obtained as 𝒞_𝒦 (t) ≈sin^2 (Ω_2 t/2), which oscillates between zero and one, as expected, in agreement with the numerical results shown in Fig. <ref>(b).

If the initial state is |ge⟩, then the Krylov basis states are given by 

    |K_1⟩^ge   = 1Ω̅(Ω_2 |gg⟩ + Ω_1 |ee⟩)  
    
        |K_2 ⟩^ge   = 1Ω_V Ω̅(-V_0 Ω_1 |gg⟩ + Ω̃^2 |eg⟩ + V_0 Ω_2 |ee⟩) 
    
        |K_3⟩^ge   = 1Ω_V ( -Ω_1 |gg⟩ - V_0 |eg⟩ +  Ω_2 |ee⟩ )

where Ω_V = √(Ω_1^2 + Ω_2^2 + V_0^2) here. The Lanczos coefficients are obtained as a_0 = 0, a_1 = V_0 Ω_1^2/Ω̅^2, a_2 = V_0(Ω_2^2 Ω_V^2 - Ω_1^2 Ω̅^2)/(Ω_V^2 Ω̅^2), a_3 = V_0Ω_1^2/Ω_V^2, b_1 = Ω̅/2, b_2 = Ω_1 Ω_2 Ω_V/Ω̅^2, and b_3 = √(Ω̅^22 + Ω_1^2 Ω_2^2 Ω_V^2Ω̅^2 - Ω_1^2 Ω̅^2(Ω_2^2 + V^2)Ω_V^4 - Ω_1 Ω_2 Ω_VΩ̅).
In the limit V_0 ≫Ω_2 ≫Ω_1, the Krylov basis states become |K_1⟩^ge→ |gg⟩, |K_2⟩^ge→ |ee⟩ and |K_3⟩^ge→ |eg⟩. Thus, the Krylov complexity oscillates between zero and one and is identical to what we obtained for initial state |gg⟩. 



§ MINIMIZATION OF THE STATE COMPLEXITY


It is proved that among the available ordered bases, the cost function in Eq. (<ref>) is minimized in the Krylov basis obtained from a given initial state and the original Hamiltonian, such that for 0 ≤ t ≤τ for some τ > 0, 𝒞_𝒦 (t) ≤𝒞_ℬ (t) <cit.>, where ℬ is any ordered basis other than the Krylov basis. However, in Sec. <ref>, we have shown that the Krylov basis computed using the zeroth order effective Hamiltonian minimized the state complexity rather than that obtained using the original Hamiltonian under the Rydberg blockade. Here, we discuss this in a more general context. We consider the Hamiltonian, Ĥ = Ĥ_A + Ĥ_B + Ĥ_AB with 

    Ĥ_A    = ∑_i,j = 1^N_A c_ij^A |a_i⟩⟨ a_j| 
    Ĥ_B    = ∑_i,j = 1^N_B c_ij^B |b_i⟩⟨ b_j| 
    Ĥ_AB   = ∑_i=1^N_A∑_j=1^N_B( d_ij |a_i⟩⟨ b_j| +  h.c. )

where A and B are orthogonal subspaces of the Hilbert space, spanned by {|a_i⟩| 0 ≤ i ≤ N_A } and {|b_j⟩| 0 ≤ j ≤ N_B } respectively. The coefficients c_ij^X (X = A, B)  represent the matrix elements of the Hamiltonian in the subspace X, while d_ij  provide the coupling between the two subspaces. We assume that the subspaces A and B are energetically well-separated and are only weakly coupled between them, i.e.,

    |d_ij| ≪min({ℰ_B}) - max({ℰ_A}).

Our results then suggest that whenever the dynamics of an initial state that lies completely in one of the subspaces, say A without any loss of generality, can be described by an effective Hamiltonian on A, the state complexity is minimized by the Krylov basis obtained using the effective Hamiltonian rather than the complete Hamiltonian. This effective Hamiltonian is Ĥ_A at zeroth order (as was the case in Sec. <ref>), or could be computed by including effects of Ĥ_AB at higher orders (see <cit.> and references therein, for instance). We discuss below the case when Ĥ_eff = Ĥ_A.  

Let 𝒦_A and 𝒦 represent the Krylov basis obtained using the effective Hamiltonian, Ĥ_A, and the complete Hamiltonian, Ĥ, respectively. The first basis state in both 𝒦 and 𝒦_A is the initial state |ψ_0⟩. We denote the basis states in 𝒦_A as |K_A,j⟩ and the Lanczos coefficients obtained using Ĥ_A as a_A,j and b_A,j. We further assume that the first M states in 𝒦 and 𝒦_A are identical, i.e. |K_A,j⟩ = |K_j⟩ for 0 ≤ j ≤ M-1. It also implies that the first M Krylov basis states in 𝒦 reside solely in the subspace A. We further assume that M is smaller than the size of 𝒦_A; if not, both 𝒦_A and 𝒦 result in identical Krylov complexity. 

Following the Lanczos algorithm, we obtain

    b_M |K_M⟩ = b_A,M |K_A,M⟩ + Ĥ_AB |K_A,M-1⟩,

where we have used Ĥ_B |K_A,M-1⟩ = 0, Ĥ_A |K_A,M-1⟩ = a_A,M-1|K_A,M-1⟩ + b_A,M |K_A,M⟩ + b_A,M-1 |K_A,M-2⟩, which is obtained using Eq. <ref>, and |K_A,j⟩ = |K_j⟩ for 0 ≤ j ≤ M-1. The latter implies we have identical Lanczos coefficients for j ≤ M-1. Eq. (<ref>) can be rewritten as |K_M⟩ = α |K_A,M⟩ + (1 - α)e^iλ |b⟩, up to a global phase with 0 < α≤ 1, where the state |b⟩ contains states from B-subspace, which is populated by Ĥ_AB. For α≠ 1, |K_A,M⟩ is not exactly identical to any state in 𝒦 and must therefore be a superposition of two or more states in 𝒦. 

Given that the dynamics is taking place only in A in this weakly-coupled limit, when the M+1'th basis state, i.e., |K_A, M⟩ becomes relevant to the dynamics (after the first M basis states, which contribute equally to the spread in both 𝒦 and 𝒦_A), 𝒦_A is more efficient than 𝒦 in minimizing the spread of the state, as the overlap of the state with |K_A, M⟩ will necessarily involve at least one extra state in 𝒦. Thus, we find that at least up to this point, 𝒞_𝒦_A (t) ≤𝒞_𝒦 (t).

In a larger Hilbert space, the initial state in A may couple to other states in A through virtual processes involving the subspace B, that may otherwise not feature in the Krylov basis obtained from Ĥ_A. By following the same line of reasoning above, we expect that the Krylov basis obtained using the effective Hamiltonian instead, which takes into account such higher-order couplings to B, will further minimize the spread complexity compared to 𝒦_A. 

This discrepancy arises because the contribution from different subspaces on a Krylov basis may not always reflect their importance in the dynamics. This is because the construction of the Krylov basis only involves the couplings between different states and does not directly feature the energy separation between them, which plays a vital role in the dynamics. Returning to our example of the Rydberg blockade, the two subspaces can be identified as A = span({|gg⟩, |ge⟩, |eg⟩}) and B = {|ee⟩}. Then we have


    Ĥ_A   = Ω2(σ̂_gg^1σ̂_x^2+σ̂_x^1σ̂_gg^2) 
    Ĥ_B   = V_0 σ̂_ee^1 σ̂_ee^2 
    Ĥ_AB   = Ω2(σ̂_ee^1σ̂_x^2+σ̂_x^1σ̂_ee^2)

The c_ij^X's (X = A, B) can be read off from above, while the non-vanishing d_ij's are equal to Ω/2. Diagonalizing Ĥ_A and Ĥ_B, we obtain ℰ_A = {0, ±Ω/√(2)} and ℰ_B = {V_0}. Thus, |d_ij|/( min( ℰ_B) - min( ℰ_A) ) ∼𝒪(Ω/V_0), which was equal to 0.01 for our choice of values. Note, however, that d_ij is identical to the non-vanishing off-diagonal matrix elements in Ĥ_A, as a result of which the Krylov basis states from both subspaces appear on an equal footing in the Krylov basis of |+⟩ and |ge⟩, even though |ee⟩ does not take part in the dynamics in the blockade regime. 

As we conclude this section, we note that while we only considered two isolated subspaces A and B in the limit of very weak coupling between them, we expect the discussion above to be equally applicable for more than two subspaces as well, where the coupling between two distinct subspaces is much smaller than the minimum energy gap between them to restrict the dynamics to only one subspace effectively.



§ SUMMARY


We analyzed the Krylov state complexity in the quantum dynamics of a single qubit and a pair of qubits. In the single qubit case, we demonstrated that the square root of the Krylov complexity measures the distance between time-evolved states by explicitly constructing an associated parameter space on which the states in the Hilbert space are mapped. In the case of two non-interacting qubits, we found that the total Krylov complexity was not simply the sum of the individual complexities of the two isolated qubits as one may have expected but consisted of an additional term that only vanishes uniformly for certain initial states. We further noted that this extra term in the complexity breaks the subadditivity of the square root of the complexity, rendering it impossible to treat the Krylov complexity as a measure of distance between states in general for this system.

We further considered the case of two interacting Rydberg qubits in the blockade regime, where two atoms simultaneously occupying the excited Rydberg state is prohibited. Hence, there is a redundancy in the Hilbert space. It has an important consequence in the dynamics of the Krylov complexity. We found that the Krylov basis obtained using the original Hamiltonian does not necessarily minimize the complexity, but the same obtained using an effective Hamiltonian describing the reduced Hilbert space minimizes it. 





We wish to acknowledge J. Bharathi Kannan for useful discussions. S.S. acknowledges funding support from the Junior Research Fellowship (JRF) awarded by the University Grants Commission (UGC), India. We further acknowledge DST-SERB for the Swarnajayanti fellowship (File No. SB/SJF/2020-21/19), MATRICS Grant No. MTR/2022/000454 from SERB, Government of India, National Supercomputing Mission for providing computing resources of “PARAM Brahma” at IISER Pune, which is implemented by C-DAC and supported by the Ministry of Electronics and Information Technology and Department of Science and Technology (DST), Government of India, and acknowledge National Mission on Interdisciplinary Cyber-Physical Systems of the Department of Science and Technology, Government of India, through the I-HUB Quantum Technology Foundation, Pune, India.












 Paper Title: Low energy excitations in bosonic quantum quasicrystals.






alejandro.mendoza@ufsc.br
Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, Brazil
Max Planck Institute for the Physics of Complex Systems, Nothnitzerstr. 38, 01187 Dresden, Germany


Max Planck Institute for the Physics of Complex Systems, Nothnitzerstr. 38, 01187 Dresden, Germany


Max Planck Institute for the Physics of Complex Systems, Nothnitzerstr. 38, 01187 Dresden, Germany
Theoretical Physics III, Center for Electronic Correlations and Magnetism,
Institute of Physics, University of Augsburg, 86135 Augsburg, Germany

§ ABSTRACT 
   We present the first principles construction of the low-energy effective action for bosonic self-organized quantum quasicrystals. Our generalized elasticity approach retains the appropriate number of phase- and corresponding conjugate density- degrees-of-freedom required for a proper description of the Goldstone modes. For the dodecagonal and decagonal quasicrystal structures we obtain collective longitudinal and transversal excitations with an isotropic speed of sound. Meanwhile, for the octagonal structure, the coupling between phononic and phasonic degrees of freedom leads in turn to hybridization of the latter with the condensate sound mode, producing collective excitations with a longitudinal and transversal component, and an anisotropic speed of sound. Finally, we discuss the fate of each excitation mode at the low and high density phase transitions limiting the quantum quasicrystal phase.  
      
    
    
     

    


Low energy excitations in bosonic quantum quasicrystals
    F. Piazza
    
=======================================================



Introduction.–Quasicrystals (QCs) are an exotic state of matter where crystalline and disordered behaviour meet. They retain long range orientational order but lack any discrete translational invariance <cit.>.  Although they were initially discovered in metallic alloys <cit.>, it is known today that this kind of exotic state can be stabilized in many different contexts <cit.>, as long as an appropriate effective competition between two or more characteristic length scales is present in the system <cit.>. One of the most intriguing versions of quasi crystalline systems  are the so-called quantum quasicrystals <cit.>, which appear as ground states of quantum many-body Hamiltonians. Such is the case of QCs produced by spin-orbit interactions in dipolar bosons <cit.>, quasicrystalline phases induced in ultracold bosonic gases by quasiperiodic optical traps <cit.> and more recently, self-organized quasicrystals in Bose-Einstein condensates produced by cavity mediated interactions <cit.>.
In this Letter, we present the first principles construction of the low energy elastic theory of bosonic quantum quasicrystals, having supersolids as a particular case. Our calculations maintain full connection with the microscopic properties of the system allowing us to make qualitative and quantitative predictions regarding the hybridization of the low momentum gapless energy modes, the corresponding propagation speed and the behavior of these excitations at the phase boundaries of the  quasicrystal phase. As particular examples we consider the three most relevant quasiperiodic structures in two dimensions, the octagonal, the decagonal and the dodecagonal quasicrystals. To our knowledge, this is the first microscopic elastic theory of bosonic quantum quasicrystals able to corroborate the symmetry arguments regarding the existence of five gapless excitation modes for these systems. Our results show that the hybridization process between phonons, phasons and the longitudinal condensate sound mode qualitatively depends on the kind of quasicrystal structure.


 Model and Method.–We consider a 2D bosonic gas at zero temperature, interacting through an isotropic non-local two-body interaction of the form U v(r), where U represents a dimensionless parameter characterizing the intensity of the non-local interaction with respect to the kinetic energy. The nature of the two-body interaction between particles is such that the system hosts a quasicrystalline pattern at high enough particle densities in the ground state. As discussed in the literature <cit.>, the key ingredient for the stabilization of such phases is the presence of competing minima in the interaction potential v̂(k). In dimensionless units, the path-integral formulation of the grand canonical partition function of the system 𝒵=∫ Dϕ Dϕ^*e^-S[ϕ,ϕ^*] <cit.> is characterized by the action: 

    S[ϕ,ϕ^*]      =∫_0^∞ dτ∫ d^2x  ϕ(x,τ)^*(∂_τ-1/2∇^2-μ.
          .+U/2∫ d^2x'v(x-x')|ϕ(x',τ)|^2 )ϕ(x,τ).


To access the low-energy properties of the system we should allow long-wavelength fluctuations of the global phase, as well as of the phase of the modulated pattern of the quasicrystal ground-state wave-function which can be expanded in a Fourier basis as ϕ_0(x)=√(ρ)∑_nc_nexp(iG_n·x). Here ρ stand for the average particle density of the system and c_n and G_n corresponds to the Fourier amplitudes and wave vectors of the pattern respectively. We thus propose a perturbed ground state wave function of the form:  

    ϕ(x,τ)=√(ρ)c_0√(1+δ n_0(x,τ)/ρ c_0^2)exp(iθ(x,τ))
       +   ∑_n≠0√(ρ)c_n√(1+G_n·Π(x,τ)/Z+G_n,⊥·Π_⊥(x,τ)/Z_⊥)
          exp(i G_n·x+i G_n·u(x,τ)
        + i G_n,⊥·w(x,τ).
       +   .iθ(x,τ)),

where the fluctuation fields u(x,τ) and w(x,τ) represent the phonon and phason fields introduced as in the classical case <cit.>. Moreover, due to the quantum nature of the system we also have to admit fluctuations in the canonical-conjugate density fields associated to these phase variables, analogously to what was proposed by Pretko et al. <cit.> for supersolids. Accordingly, we have included the fields Π(x,τ) and Π_⊥(x,τ) as conjugate to u(x,τ) and w(x,τ), respectively. Finally, we added the overall phase fluctuation field θ(x,τ) of the wave function and its canonically conjugate zero momentum density fluctuation δ n_0(x,τ). It is understood that all fluctuation fields are slowly varying functions in space and imaginary time, with zero mean over the size of the system and, as expected, in the absence of fluctuations our ansatz for ϕ(x,τ) reduces to the ground state wave function ϕ_0(x). Furthermore, the set of vectors {G_n,⊥} is constructed from a permutation of the set {G_n}, such that the extended basis G̃_n=G_n⊕G_n,⊥ is orthogonal in the sense ∑_nG̃_n,αG̃_n,β∝δ_α,β <cit.>.

It is simple to show that the proposed ansatz satisfies the normalization condition of the boson field  ∫_A d^2x<|ϕ(x,τ)|^2>=N=ρ A, as long as ∑_nc_n^2=1. Such normalization constraint allows the calculation of the chemical potential μ for a system with a given particle density ρ. Lastly, the free parameters Z and Z_⊥ in Eq. (<ref>) are chosen to enforce the expected relation between canonically conjugate fields in the low energy effective action of the system <cit.>. By replacing the above ansatz for ϕ(x,τ) in Eq. (<ref>) and expanding up to quadratic order in the fluctuation fields we obtain the following long-wavelength effective action:
 
    δ S   =∫ dτ d^2x (i δ n_0(x,τ)∂_τθ(x,τ)+i Π(x,τ)·∂_τu(x,τ).
       +.i Π_⊥(x,τ)·∂_τw(x,τ))+ ∫ dτ d^2x 1/2[ρ(∇θ)^2.
       +γ_0δ n_0(x,τ)^2+γΠ(x,τ)^2+γ_⊥Π_⊥(x,τ)^2
       +2Π·∇θ+.E^T C̃ E],

where E stands for the extended phonon-phason strain field defined as E={∂_xu_x,∂_yu_y,(∂_xu_y+∂_yu_x)/2,(∂_yu_x+∂_xu_y)/2,∂_xw_x,∂_yw_y,∂_yw_x,∂_xw_y} and C̃ represents the phonon-phason elastic tensor. Closed formulae for the elements of this tensor as well as for the γ's couplings in terms of the ground state wave function are provided in the Supp. Matt. <cit.>. The expression in Eq. (<ref>) for the low-energy effective action of a bosonic quasicrystal is one of the main results of this work. It allows qualitative and quantitative predictions since all couplings are known in terms of the ground state wave function of the system. Moreover, it generalizes the well-known density-phase action of a homogeneous condensate <cit.>, not only to the present case of quantum quasicrystals, but also to supersolids if we eliminate the phasonic degrees of freedom <cit.>.



Elementary excitations for the dodecagonal QC– The effective action in Eq. (<ref>) can be recast as δ S=1/2∫dω/(2π)d^2q/(2π)^2η̂(q,ω)^†M(q,ω)η̂(q,ω), where η̂(q,ω)=(δn̂_0(q,ω),
θ̂(q,ω),
  Π̂_x(q,ω),
  û_x(q,ω),
  Π̂_y(q,ω),
  û_y(q,ω), Π̂_x,⊥(q,ω),
  ŵ_x(q,ω),
  Π̂_y,⊥(q,ω), ŵ_y(q,ω)), the energy-momentum dispersion relations ϵ(q)=ω(q) can be computed by solving the equation [M(q,iω(q))]=0 <cit.>. The dodecagonal quasicrystal (Fig. <ref>(a)) is the simplest case due to the absence of the coupling between phonons and phasons. The latter are deformations of the density pattern which are absent in ordinary crystals. We obtain analytical expressions for the dispersion relation of the five gapless modes <cit.>. In this case there are three longitudinal modes and two transversal modes, as schematically represented in Fig. <ref>(b). Two of the longitudinal modes result from the hybridization of the condensate sound mode (corresponding to a modulation of the overall phase θ) with the longitudinal phonon mode, while the transversal phonon as well as the phason mode remain decoupled. 
  
 
  


To deepen our understanding of the excitation modes we calculate the eigenvectors η_j(q) using the charateristic equation M(q,i ω_j(q))η̂_j(q)=0 for each excitation mode. Each eigenvector corresponds to a fluctuation field in real space and time given by η_j(x,t)=Re[η̂_j(q)exp(i(q·x-ω_j(q)t))]. By expanding Eq. (<ref>) to linear order in the fluctuation fields we obtain a general expression for the correction to the ground state wave function: δψ_j,q=[u_j,q(r)exp(i(q·r-ω_j(q)t))-v_j,q(r)^*exp(-i(q·r-ω_j(q)t))], allowing us to identify the Bogoliubov modes u_j,q(r) and v_j,q(r) <cit.>. 
The local phase and density fluctuations can be then written as Δρ_j,q=| u_j,q(r)-v_j,q(r)|^2 and Δφ_j,q=| u_j,q(r)+v_j,q(r)|^2 <cit.>. In Fig. <ref>(c-l) we show the typical behavior of these quantities. Each column corresponds to a mode in Fig. <ref>(b) in increasing order of excitations energy. In agreement with previous observations for supersolids <cit.>, the hybridized  longitudinal modes present significant (O(1)) phase and density fluctuations, while the transversal phonon mode presents significant density fluctuations and  O(q) phase fluctuations. On the other hand, while showing the same scaling (O(1) vs O(q)) of the transversal phonon modes, phasons feature density and phase fluctuations patterns which presents a different behavior around the center of the quasicrystal structure [The center of the quasicrystal here is being understood as the special point where all reflections symmetry planes of the quasicrystal pattern intersects. Given the form of our anzats for ϕ_0(x) it coincides with the origin of our coordinates system.] compare e.g. panel (e) and (i) of Fig.<ref>.  Mathematically, this is ultimately a consequence of the phason's property of producing a redistribution of the QC structure destroying the center of the quasicrystal without altering the average properties of the system <cit.>. Finally, we would like to remark that differently from the case of supersolids, here all density and phase fluctuations patterns are aperiodic. 


Excitations in decagonal and octagonal QCs.–In the case of ten- and eight-fold quasicrystals, the presence of the phonon-phason coupling leads to a characteristic equation [M(q,iω(q))]=0 with convoluted analytical solutions. However, some general properties can be obtained from the numerical solution of this problem. In the case of the decagonal quasicrystal, the longitudinal and transversal modes separately hybridize among themselves, leading to two groups of three and two modes, respectively, all with isotropic dispersion relations. In contrast, for the octagonal quasicrystal, the structure of the elasticity action characterized by an anisotropic phason contribution and a finite phonon-phason coupling produces anisotropic dispersion relations ϵ(q) with propagation speed c_s that depend on the orientation of the momentum q when q→0.  Additionally, unless the momentum of the excitation q is oriented along one of the symmetry axes of the quasicrystal, every mode has a longitudinal and a transversal component. As an example, we show in Fig. <ref>(a) the anisotropic behavior of the propagation speed and hybribization of modes for an octagonal QC. The numerical evaluation was carried out considering values of the couplings for which the system displays strong anisotropy and  hybridization effects <cit.>. The red to blue color gradient indicates the longitudinal and transversal character of each mode, using the angular orientation of the field u as a measure of this property. Interestingly, the propagation of excitations along non-equivalent symmetry axes of the octagonal quasicrystal produces an exchange of the longitudinal and transversal character for some modes. Moreover, the evaluation of the local density and phase fluctuations profile in this case shows that the scaling of the fluctuations when the propagation direction of the excitation do not coincide with a symmetry axes of the quasicrystal are all O(1). Meanwhile, in the opposite case, longitudinal modes have density and phase fluctuations of amplitude O(1) and transversal modes with density fluctuations O(1) and phase fluctuations O(q). The aperiodic patterns of phase and density fluctuations for each scenario discussed can be found in <cit.>.


Instabilities of the quantum quasicrystal phase.– A first analysis of the instabilities can be based on the behaviour of the propagation speed of the gapless modes. With this aim we consider the case of the decagonal QC structure and pursue the calculation of the couplings in the action (<ref>), considering small amplitudes of the QC modulation <cit.>. 
In order to make a numerical evaluation possible without defining a particular form of the interaction potential, we assume that the QC amplitude is a concave increasing  function of ρ U [Mathematically this means ∂ c_1(Uρ)/∂ (Uρ)>0 and ∂^2 c_1(Uρ)/∂ (Uρ)^2<0, where c_1(Uρ) stand for the modulation amplitude of ϕ_0.]. Although this is the expected behavior of this quantity, such ansatz does not intend to capture the actual quantitative behavior of a certain model <cit.>. The results for the propagation speed of each mode are shown in Fig. <ref>(b) as a functon of ρ U. One recognizes a discontinuous behavior of the propagation speed at the homogeneous-superfluid-to-quantum-quasicrystal phase transition, analogous to previous observations in supersolids <cit.>, and a continuous behavior at the quantum-to-classical quasicrystal transition at high enough densities where the propagation speed of the lowest mode vanishes <cit.>,[Although Mean Field treatments of our action consistent with the Gross-Pitaevskki description are not able to reach a strictly zero propagation speed for the lowest mode, beyond Mean Field descriptions have succeeded in observing rigorously the supersolid to normal solid transition <cit.>.]. As we shall see next, this transition corresponds to the breaking of quantum phase coherence. 


To further explore the nature of the instabilities we compute the second-order correlations of the fluctuation fields. At the lower density boundary, when the QC amplitude is small and the ground state wave function is dominated by the main Fourier amplitude of the pattern (c_1), we find that Z, Z_⊥ and all elastic couplings in C̃ approaches zero as c_1^2, while γ and γ_⊥ diverges as c_1^-2. This behavior lead us to the conclusion that in this regime ⟨δ n_0^2⟩ and ⟨θ^2⟩ remain finite, while ⟨Π^2⟩∝ c_1^2, ⟨Π_⊥^2⟩∝ c_1^2, ⟨u^2⟩∝ c_1^-2  and ⟨w^2⟩∝ c_1^-2, as c_1→0, with c_1≫ c_n>1. In this scenario the average wave function can be roughly approximated as ⟨ϕ(x,τ)⟩≈√(ρ)∑_nc_nexp(iG_n·x)×exp(-G_n^2⟨u^2⟩/4-G_n,⊥^2⟨w^2⟩/4-⟨θ^2⟩/2). This result confirms that close enough to the superfluid-quantum-quasicrystal stability boundary, where we have large phonon and phason fluctuations, there is a strong non-perturbative “renormalization" of the Fourier amplitudes of the quasicrystal profile [The strong renormalization process is signaled by the non-analytic behavior of ⟨ϕ⟩ in the limit c_1→ 0 upon inclusion of phonon and phason fluctuations ]. This phenomenology suggests that in cases in which the mean field superfluid to quantum quasicrystal phase transition is weak enough, the nature of the transition could be modified to a continuous one, even with the presence of intermediate hexatic-like phases. 

At the upper boundary of the quantum quasicrystal phase (γρ→ 1^+) all couplings of the effective action in Eq. (<ref>) remain finite, however in this limit the propagation speed of the lower hybridized mode approaches zero, as observed  before. This behavior  signals the disappearance of this mode, in the same way in which the lower energy excitation mode disappears at the supersolid to normal solid phase transition <cit.>. Calculating the correlations of the fluctuation fields in this regime, we obtain to leading order that ⟨δ n_0^2⟩ ⟨u^2⟩, ⟨w^2⟩ and ⟨Π_⊥^2⟩ remain finite as the stability boundary is approached, while ⟨θ^2⟩∝(γρ-1)^-1/2 and ⟨Π^2⟩∝(γρ-1)^-1/2. This result is to be expected since in the high ρ U regime the localization of particles should eventually lead the system to the loss of quantum phase coherence. Interestingly, due to the effective Π·∇θ interaction the de-stabilization of the θ(x,τ) field also produces the destabilization of the Π(x,τ) field, indicating the natural breakdown of our theory at this phase boundary. All evidence available in this case indicates a second order phase transition in this limit, as observed in supersolid systems <cit.>.  






Conclusions.–We have developed the elastic theory of a self-organized quantum quasicrystal with 8, 10 or 12-fold rotational symmetry. We obtained the effective action of the system described in terms of the global wave function phase, the two-dimensional phonon and phason fields and their respective conjugate density fields. From here, we were able to prove from first principles the existence of five gapless extended low energy excitation modes [It is worth to mention, that due to the lack of periodicity quantum quasicrystals can also host particle-hole localized excitations with an energy arbitrarily close to zero. See Refs. <cit.>.] for the considered two-dimensional QC structures  <cit.>. We showed that the presence of the Π·∇θ coupling in our effective theory produces the hybridization of the condensate sound mode with phonon and phason modes, leading to a rich phenomenology depending on the QC structure. A particularly interesting finding was that, due to the phonon-phason coupling, the octagonal QC structure develops an anisotropic speed of sound that depends on direction of the propagation, as well as that all modes show a mixed longitudinal-transversal character. Finally, our results shed light on the behavior of the low energy fluctuation fields at the thermodynamic boundaries of the quantum quasicrystal phase, elucidating possible scenarios for phase transitions in two dimensions.            


Acknowledgments: A.M.C. acknowledge MPIPKS for financial support and hospitality.



 
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Supplemental Materials: Low energy excitations in super quasicrystals: a generalized elasticity approach 














§ I. CONSTRUCTION OF THE EFFECTIVE ACTION

In this section we present details about the  construction of the effective action for the low energy behavior of the octagonal, decagonal and dodecagonal cluster quasicrystalline structures. We assume that the pair interaction potential v(x) is such that the model self-stabilizes one of these phases as its ground state in the quantum regime of the system <cit.>. By quantum regime, we refer to the region of parameters in which particle delocalization is strong enough to produce a finite superfluid fraction in the system. As a consequence, we consider that the stationary configuration that minimizes the action in Eq. (1) can be expressed as a Fourier expansion of the form

    ϕ_0(x)=√(ρ)∑_n c_nexp(i G_n·x),

where c_n and G_n represent the amplitudes and wave vectors of the Fourier expansion of ϕ_0. Moreover, for the kind of QCs considered in this work we have G_n=k(n_1e_1+n_2e_2+n_3e_3+n_4e_4), where k stands for the characteristic wave vector of the quasicrystal and the basis vectors e_i's are specific for each kind of pattern. The sum in the Fourier expansion is performed over a four dimensional integer vector n whose components n_i's are all integer numbers. A possible definition of the vectors e_i's for each of the cases considered is presented in Table <ref>. 


The low energy excitation properties can be accessed considering the effects of long wave-length fluctuations in the Fourier amplitudes and in its corresponding conjugate phases, in the same spirit as presented in ref <cit.>. In our case we propose a general ansatz for the perturbed wave function field of the form 

    ϕ(x,τ)=√(ρ)c_0√(1+δ n_0(x,τ)/ρ c_0^2)exp(iθ(x,τ))
        +∑_n≠0√(ρ)c_n√(1+G_n·Π(x,τ)/Z+G_n,⊥·Π_⊥(x,τ)/Z_⊥)
          exp(i G_n·x+i G_n·u(x,τ)
        + i G_n,⊥·w(x,τ)
        +iθ(x,τ)),

where the set {G_n,⊥} is constructed as a permutation of the set {G_n} such that the extended four dimensional basis G̃_n=G_n⊕G_n,⊥, is orthogonal in the sense that ∑_nG̃_n,αG̃_n,β∝δ_α,β <cit.>. For the purpose of calculations we can write G_n,⊥=k(n_1e_1,⊥+n_2e_2,⊥+n_3e_3,⊥+n_4e_4,⊥), where the basis {e_i,⊥} corresponding to our particular choice of {e_i} is chosen as shown in Table <ref>. Moreover is important to notice that the index vector {n_i} for two corresponding G_n and G_n,⊥ coincide. 


Since we are interested in the low energy behavior of the system we can work in the hydrodynamic regime, in which the fluctuation fields introduced vary slowly in space and imaginary time. Mathematically, this argument allows us to neglect all components of the effective theory containing the product of slowly varying terms with spatially oscillatory contributions.

Now we can proceed with the evaluation of each term of the original action (Eq. (<ref>)) as a function of the fluctuation fields, keeping only up to quadratic contributions. The first contribution can be recast in the symmetric form 

    δ S_1=∫ dτ d^2x1/2(ϕ(x,τ)^*∂_τϕ(x,τ)-ϕ(x,τ)∂_τϕ(x,τ)^*),

which up to quadratic order in the fluctuation fields yields

    δ S_1=∫ dτ d^2x (i δ n_0(x,τ)∂_τθ(x,τ)+i (ρ∑_nc_n^2G_n^2/2Z)Π(x,τ)·∂_τu(x,τ)+i (ρ∑_nc_n^2G_n,⊥^2/2Z_⊥)Π_⊥(x,τ)·∂_τw(x,τ)).

In this way, by choosing Z=ρ∑_nc_n^2G_n^2/2 and Z_⊥=ρ∑_nc_n^2G_n,⊥^2/2 we obtain for δ S_1, the expected structure combining the phases and the corresponding conjugate density fluctuation fields. 

The second contribution of our action can be recast as

    δ S_2=∫ dτ d^2x1/2∇ϕ(x,τ)^*·∇ϕ(x,τ).

In this case, the use of our ansatz leads us to the expression 

    δ S_2   =   δ S_2^0+∫ dτ d^2x{1/8ρ c_0^2(∇δ n_0)^2+1/8Z((∇Π_x)^2+(∇Π_y)^2)+1/8Z_⊥((∇Π_x,⊥)^2+(∇Π_y,⊥)^2).
       +   .ρ c_0^2/2(∇θ)^2+ρ∑_n≠0c_n^2/2(∇θ)^2+Z/2((∇ u_x)^2+(∇ u_y)^2)+Z_⊥/2((∇ w_x)^2+(∇w_y)^2)+Π·∇θ},

where δ S_2^0, represents a fluctuation-independent contribution. The linear order contributions in the fluctuation fields are not presented in this and the rest of the terms of the effective action because they have a zero total contribution, once we are considering fluctuations around the global minimum of our action. At this point it is worth noticing that the particular form considered for our ansatz in Eq. (<ref>) as well as the appropriate selection of Z and Z_⊥ are responsible for producing a form of δ S_1 and δ S_2 preserving the expected relation between the couplings associated to conjugate density and phase fields. Other choices for this ansatz, for instance including  a dependence of Z and Z_⊥ with |G_n|, even when preserves the symmetries of the original expression will not be able to reproduce this property.  

The third contribution of the original microscopic action, δ S_3=∫ dτ d^2xμϕ(x,τ)^*ϕ(x,τ), does not provide any quadratic contribution to the total effective action in the long wave limit. Finally, we can proceed with the contribution resulting from the non-local interaction term 

    δ S_4=∫ dτ d^2xd^2x'1/2U v(x-x')|ϕ(x,τ)|^2|ϕ(x',τ)|^2.

This last term produces two kinds of contributions, one including only density-density interactions defined as δ S_4,1 and a second one corresponding to the phonons and phasons deformation fields defined as δ S_4,2. Terms of density-phase interactions cancel out exactly due to the reflection and rotational symmetries of the modulated patterns considered. Taking into account that in the determination of δ S_4,1 and δ S_4,2 we have to deal with a convolution with a non-local interaction kernel, it is easier to perform the evaluation of these terms in momentum space. After a lengthy but strightforward calculation we obtained that δ S_4,1 in the long wave limit is given as

    δ S_4,1   =   ∫ dτd^2q/(2π)^21/2(γ_0δn̂_0(q,τ)δn̂_0(-q,τ)+γΠ̂(q,τ)·Π̂(-q,τ)+γ_⊥Π̂_⊥(q,τ)·Π̂_⊥(-q,τ)
    ),

where 

    γ_0   =   -1/2∑_n1≠0,n3,n4Uv̂(G_n1)c_n1c_n3c_n4/c_0^3δ(G_n1+G_n3-G_n4,0)+∑_nUv̂(G_n)c_n^2/c_0^2,
    γ   =   -1/4∑_n1,n2,n3,n4Uv̂(G_n1-G_n2)ρ^2Π_i=1^4 c_ni/Z^2(G_n1-G_n2)_x^2δ(G_n1-G_n2+G_n3-G_n4,0),
    γ_⊥   =   -1/4∑_n1,n2,n3,n4Uv̂(G_n1-G_n2)ρ^2Π_i=1^4 c_ni/Z_⊥^2(G_n1,⊥-G_n2,⊥)_x^2δ(G_n1-G_n2+G_n3-G_n4,0).

The obtained expression for each γ reduces to the one well known for the homogeneous scenario (c_0=1 and c_n≠0=0), γ_0=Uv̂(0), γ=0 and γ_⊥=0. It is also important to remark that since V̂(k) has a negative part responsible for exciting the main Fourier modes of the modulated pattern, we will have γ_0, γ and γ_⊥ as positive parameters, which guarantees the stability of our effective theory. We should note that in the long wave limit the terms proportional to the gradient of the density fluctuation fields in δ S_2 are subleading respect to the contribution in δ S_4,1, which is why these terms do not appear in Eq. (<ref>) of the main text.     

The last contribution in δ S_4 corresponds to the phonon and phason interaction. This term can be written in general as 

    δ S_4,2   =   ∫ dτd^2q/(2π)^2U ρ^2/4∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)
       ×   (2v̂(q-(G_n1-G_n2))-2v̂(G_n1-G_n2))((G_n1-G_n2)·û(q,τ)+(G_n1,⊥-G_n2,⊥)·ŵ(q,τ) )
          ((G_n1-G_n2)·û(-q,τ)+(G_n1,⊥-G_n2,⊥)·ŵ(-q,τ) ).

It is possible to convince ourselves that unlike all previous terms in our effective action this contribution has a different form for each quasicrystal. As expected in the long wave limit δ S_4,2 reproduces the structure of the well-known classical elastic Hamiltonian for each kind of quasicrystal structure <cit.> yielding 

    δ S_4,2   =   1/2∫ dτd^2q/(2π)^2[((2μ+λ)q_x^2+μ q_y^2)û_x(q,τ)û_x(-q,τ)+((2μ+λ)q_y^2+μ q_x^2)û_y(q,τ)û_y(-q,τ).
       +   . (2μ+2λ)q_xq_yû_x(q,τ)û_y(-q,τ)+(K_1q_x^2+(K_1+K_2+K_3) q_y^2)ŵ_x(q,τ)ŵ_x(-q,τ).
       +   . (K_1q_y^2+(K_1+K_2+K_3) q_x^2)ŵ_y(q,τ)ŵ_y(-q,τ)+2(K_2+K_3)q_xq_yŵ_x(q,τ)ŵ_y(-q,τ).
       +   .(R q_x^2-R q_y^2)(û_x(q,τ)ŵ_x(-q,τ)+û_y(q,τ)ŵ_y(-q,τ))+2Rq_xq_y(û_x(q,τ)ŵ_y(-q,τ)-û_y(q,τ)ŵ_x(-q,τ))
    ].

The expression presented corresponds to the most general scenario, consistent with the octagonal quasi-crystalline pattern. In the case of the decagonal quasi-crystal we will have K_2=-K_3, which makes zero the off-diagonal phason-phason interaction proportional to ŵ_x(q,τ)ŵ_y(-q,τ) and finally, in the case of the dodecagonal quasicrystal we will have R=0, which makes zero all elastic phonon-phason interactions. It is important to mention that the particular form obtained for our elastic phonon-phason action is a direct consequence of the particular choice we made for G_n,⊥. Different but equivalent forms can be obtained for δ S_4,2 if we consider other permutations of G_n in the definition of G_n,⊥. Furthermore, from the comparison between Eqs. <ref> and <ref> it is possible to conclude that


    (2μ+λ)   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1-G_n2)_x^2∂_qx^2v̂(q-(G_n1-G_n2))|_q=0
    μ   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1-G_n2)_x^2∂_qy^2v̂(q-(G_n1-G_n2))|_q=0
    
    K_1   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1,⊥-G_n2,⊥)_x^2∂_qx^2v̂(q-(G_n1-G_n2))|_q=0
    
    K_T   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1,⊥-G_n2,⊥)_x^2∂_qy^2v̂(q-(G_n1-G_n2))|_q=0
    
    R   =U ρ^2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1,⊥-G_n2,⊥)_x(G_n1,⊥-G_n2,⊥)_x
    ×   ∂_qx^2v̂(q-(G_n1-G_n2))|_q=0,

where K_T=K_1+K_2+K_3. These expressions complete the determination of all couplings in our effective theory from the knowledge of the ground state wave function. Finally, we can notice that the gradient terms of the phonon and phason fields in Eq. (<ref>) can be absorbed in δ S_4,2 modifying the elastic coefficients as follows (2μ̃+λ̃)=(2μ+λ)+Z, μ̃=μ+Z, K̃_1=K_1+Z_⊥ and K̃_T=K+Z_⊥. This allows us to write our total effective action in the following more compact way 

    δ S   =∫ dτ d^2x (i δ n_0(x,τ)∂_τθ(x,τ)+i Π(x,τ)·∂_τu(x,τ)+i Π_⊥(x,τ)·∂_τw(x,τ))+ ∫ dτ d^2x 1/2[ρ(∇θ)^2.
       .+γ_0δn̂_0(x,τ)^2+γΠ(x,τ)^2+γ_⊥Π_⊥(x,τ)^2+2Π·∇θ+E^T C̃ E],

where E stands for the extended phonon-phason strain field defined as E={∂_xu_x,∂_yu_y,(∂_xu_y+∂_yu_x)/2,(∂_yu_x+∂_xu_y)/2,∂_xw_x,∂_yw_y,∂_yw_x,∂_xw_y} and C̃ represents the phonon-phason elastic tensor <cit.> given by


    C̃=
    [ λ̃ +2μ̃      λ̃       0       0       R       R       0       0;      λ̃  λ̃+2μ̃       0       0      -R      -R       0       0;       0       0      μ̃      μ̃       0       0      -R       R;       0       0      μ̃      μ̃       0       0      -R       R;       R      -R       0       0  K̃_̃1̃     K_2       0       0;       R      -R       0       0     K_2  K̃_̃1̃       0       0;       0       0      -R      -R       0       0    K̃_T     K_3;       0       0       R       R       0       0     K_3    K̃_T ].







§ II. EXCITATION PROPERTIES OF THE DODECAGONAL QUASICRYSTAL

As we can notice, the effective action in Eq. (<ref>) can be written in the general form

    δ S=1/2∫dω/(2π)d^2q/(2π)^2η̂(q,ω)^†M(q,ω)η̂(q⃗,ω),

where the following definition of the Fourier transform of f(x,τ) and its inverse have been used

    f̂(q,ω)   =   ∫d^2q dω/(2π)^3e^i(q·x-ωτ)f(x,τ)
    
    f(x,τ)   =   ∫ d^2x dτ e^-i(q·x-ωτ)f̂(q,ω).

Moreover, η̂(q,ω)={δn̂_0(q,ω),θ̂(q,ω),Π̂_x(q,ω),û_x(q,ω),Π̂_y(q,ω),û_y(q,ω),Π̂_x,⊥(q,ω),ŵ_x(q,ω),Π̂_y,⊥(q,ω),ŵ_y(q,ω)} and the interaction matrix M(q,ω) is written as

    M(q,ω)=
    [                   γ_0                    -ω                     0                     0                     0                     0                     0                     0                     0                     0;                     ω                 ρ q^2                 -iq_x                     0                 -iq_y                     0                     0                     0                     0                     0;                     0                  iq_x                     γ                    -ω                     0                     0                     0                     0                     0;                     0                     0                     ω (2μ̃+λ̃)q_x^2+μ̃q_y^2                     0         (μ̃+λ̃)q_xq_y                     0      R/2(q_x^2-q_y^2)                     0               Rq_xq_y;                     0                 i q_y                     0                     0                     γ                    -ω                     0                     0                     0                     0;                     0                     0                     0         (μ̃+λ̃)q_xq_y                     ω (2μ̃+λ̃)q_y^2+μ̃q_x^2                     0              -Rq_xq_y                     0      R/2(q_x^2-q_y^2);                     0                     0                     0                     0                     0                     0                   γ_⊥                    -ω                     0                     0;                     0                     0                     0      R/2(q_x^2-q_y^2)                     0              -Rq_xq_y                     ω   K̃_1q_x^2+K̃_Tq_y^2                     0       (K_2+K_3)q_xq_y;                     0                     0                     0                     0                     0                     0                     0                     0                   γ_⊥                    -ω;                     0                     0                     0               Rq_xq_y                     0      R/2(q_x^2-q_y^2)                     0      (K_2+K_3)q_x q_y                     ω  K̃_T q_x^2+K̃_1q_y^2 ].



As we can observe the interaction matrix 𝐌(q,ω) separates in two independent blocks of 6×6 and 4×4 when R=0, i.e. in the case of the dodecagonal quasicrystal pattern. This case corresponds to the simplest scenario, in which phasons and phonons fields do not interact with each other.

Now we can proceed with our study of the low energy behavior of the system. The dispersion relation ω(q) corresponding to each excitation mode can be obtained by solving the equation [M(q,iω(q))]=0. The solution of this equation leads to two groups of solutions corresponding to the two different blocks of the matrix M(q,ω) in this case. The solutions arising from the 6×6 block of M(q,ω) are given by

    ω_1(q)   =   √(A+√(A^2+B))q
    ω_2(q)   =   √(A-√(A^2+B))q
    ω_3(q)   =   √(γμ̃)q,

where A=(ργ_0+γ(2μ̃+λ̃))/2 and B=(2μ̃+λ̃)γ_0(1-γρ). The form of the solutions indicates a hybridization process of the modes with dispersion relation ω_1(q) and ω_2(q), the analysis of eigenvectors associated to these eigenvalues allow us to conclude that these two excitation modes correspond to a hybridization of the longitudinal Bogoliubov mode with the longitudinal phonon mode, while ω_3(q) corresponds to the transversal phonon mode. A similar procedure for the 4×4 block of M(q,ω) yields the dispersion relations 

    ω_4(q)   =   √(γ_⊥K̃_1)q
    ω_5(q)   =   √(γ_⊥K̃_T)q,

corresponding to the longitudinal and transversal phason modes, respectively. 



§ III. FLUCTUATION EIGENSTATES OF THE DODECAGONAL QUASICRYSTAL

The longitudinal or transversal character of the excitation modes in the dodecagonal quasicrystal refers to the way in which the vectorial fields Π(x,τ), u(x,τ), Π_⊥(x,τ) and w(x,τ) oscillate respect to the momentum of the excitation q. For a better understanding of the nature of these modes we calculated the corresponding eigenvectors η̂_j(q) in a situation in which q is oriented along the "x"-axis, yielding 


    η̂_1(q_x)   =   { (c_1^2-γ(2μ̃+λ̃))c_1 q_x, -iγ_0(c_1^2-γ(2μ̃+λ̃)), γ_0(2μ̃+λ̃)q_x, -iγ_0c_1,0,0,0,0,0,0}
    η̂_2(q_x)   =   { -(c_2^2-γ_0ρ)c_2 q_x, iγ_0(c_2^2-γ_0ρ), γ_0(2μ̃+λ̃)q_x, -iγ_0c_2,0,0,0,0,0,0}
    η̂_3(q_x)   =   {0, 0, 0, 0, c_3 q_x, -iγ, 0, 0, 0, 0}
    η̂_4(q_x)   =   {0, 0, 0, 0, 0, 0, c_4 q_x, -i γ_⊥, 0, 0}
    η̂_5(q_x)   =   { 0, 0, 0, 0, 0, 0, 0, 0, c_5 q_x, -iγ_⊥},

where c_i=∂_q ϵ_i(q), refers to the sound velocity of the corresponding excitation mode. From here, the fluctuation fields in real space and time writes as η_j(x,t)=Re[η̂_j(q)exp(i(q·x-ω_jt))] and the correction to the ground state wave function as 

    δψ   =   √(ρ)(δ n_0(x,t)/2ρ c_0^2+ic_0θ(x,t)+1/2∑_n≠0c_n(G_n·Π(x,t)/Z+G_n,⊥·Π_⊥(x,t)/Z).
          .×exp(iG_n·x)+i∑_n≠0c_nexp(iG_n·x)× (θ(x,t)+G_n·u(x,t))+G_n,⊥·w(x,t)).).

Comparing this expression with the usual ansatz for the solution of the linearized Gross-Pitaevskii equation <cit.>, δψ_j,q=u_j,q(r)exp(i(q·r-ω_j(q)t))-v_j,q(r)^*exp(-i(q·r-ω_j(q)t)), we identify in each case the  functions u_j,q(r) and v_j,q(r). This allows us to compute the phase and density fluctuations for the excitation modes as Δρ_j,q=| u_j,q(r)-v_j,q(r)|^2 and Δφ_j,q=| u_j,q(r)+v_j,q(r)|^2 <cit.>. In Fig.<ref>(c-l) we show the behavior of these quantities considering typical parameters for the ground state wave function and for the pair interaction potential. For the numerical evaluation, the effective couplings of the theory can be roughly estimated using a single mode expansion for the wave function of the dodecagonal QC. In this case, we can write ϕ(x)=√(ρ)(c_0+c_1∑_j=1^12exp(i ke_j·x)), where {e_i} is a set of unitary vectors forming regular dodecagon. In the small c_1 limit we can reach to 

    Z   ≈   Z_⊥≈6ρ k^2c_1^2
    γ   ≈   γ_⊥≈ Uv̂(k)/(6c_1^2k^2)
    γ_0   ≈   Uv̂(0)
    
    (2μ̃+λ̃)   ≈   K̃_T≈6ρ k^2c_1^2+3Uρ^2c_1^2(kv̂'(k)+3k^2v̂”(k))
    μ̃   ≈   K̃_1≈6ρ k^2c_1^2+3Uρ^2c_1^2(3 kv̂'(k)+k^2v̂”(k)).

Considering U=1, ρ=0.4, k=1,  v̂(0)=2, v̂(k)=-1, v̂'(k)=0, v̂”(k)=1.5, c_0=1, c_1=0.15 and a momentum of the excitation q=0.1(1,0) we obtain the density and phase fluctuations patterns shown in Fig.1 (c-l).





§ IV. NUMERICAL CONSIDERATIONS FOR FIG. 2

For the numerical evaluation associated to Fig.2 (a) we considered the following set of arbitrary parameters ρ=4, γ_0=4, γ=1 γ_⊥=1, 2μ̃+λ̃=52, μ̃=20, K̃_1=24, K̃_T=56 and R=32.  

Moreover, as explained in the main text for Fig.2 (b) we need more realistic estimates of the behavior of the couplings of the effective theory in Eq.(3), these can be obtained from a single mode expansion, which for the decagonal quasicrystal, yields 

    Z   ≈   Z_⊥≈5ρ k^2c_1^2
    γ   ≈   γ_⊥≈ Uv̂(k)/(5c_1^2k^2)
    γ_0   ≈   Uv̂(0)
    
    (2μ̃+λ̃)   ≈   5ρ k^2c_1^2+5/2Uρ^2c_1^2(kv̂'(k)+3k^2v̂”(k))
    μ̃   ≈   5ρ k^2c_1^2+5/2Uρ^2c_1^2(3 kv̂'(k)+k^2v̂”(k))
    K̃_1=K̃_T   ≈   5ρ k^2c_1^2+5Uρ^2c_1^2(kv̂'(k)+k^2v̂”(k))
    
    R   ≈   5/2Uρ^2c_1^2(-kv̂(k)+k^2v̂”(k)).

Additionally, for numerical evaluations we use the anzats  c_1 (Uρ)=0.035+0.51(Uρ-0.2)/(Uρ)^0.65. This ansatz is only intended to capture the two main features of the dominant Fourier amplitude of the QC pattern, i.e. c_1(Uρ) is a strictly growing function defined from the QC phase boundary (Uρ=0.2). Secondly, c_1(Uρ) should be concave reflecting the tendency of the system to avoid collapse as Uρ is increased. Furthermore, for the pair interaction parameters we consider v̂(k)=-1, v̂'(k)=0, v̂”(k)=1 and v̂(0)=1.     




§ BEHAVIOR OF THE DENSITY AND PHASE LOCAL FLUCTUATIONS IN THE OCTAGONAL QUASICRYSTAL PATTERN


In this section we present the typical behavior of the local density and phase fluctuations for the octagonal quasicrystal for different orientations α of the propagation direction of the excitation respect to one of the main directions of the quasicrystal. For numerical calculations all the couplings of our effective theory takes the same values considered for the construction of Fig.(<ref>). 

The first case presented corresponds to the most general scenario for which all low energy modes are hybridized. As a consequence the scaling of the density and phase fluctuations is O(1) for all modes. For the numerical evaluation in this case we select α=π/16, the results are presented in Fig.(<ref>). Next we consider separately the cases of α=0 in Fig.(<ref>) and α=π/8 in Fig.(<ref>). For these special directions, as they correspond to symmetry directions of the octagonal quasicrystal structure, modes are purely longitudinal or transversal and the hybridization takes place only between modes in each of these separate groups. Because of this feature the scaling of the density and phase fluctuations for the longitudinal and transversal modes remains different in these cases.  


  









 Paper Title: Ultra-high-amplitude Peregrine solitons induced by helicoidal spin-orbit coupling.





Research Group of Nonlinear Optical Science and Technology, Research Center of Nonlinear Science,
 School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan 430200, China

qinzhou@whu.edu.cn
Research Group of Nonlinear Optical Science and Technology, Research Center of Nonlinear Science,
 School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan 430200, China
State Key Laboratory of New Textile Materials and Advanced Processing Technologies, Wuhan Textile University, Wuhan 430200, China

Department of Physical Electronics, School of Electrical Engineering,
	Faculty of Engineering, and the Center for Light-Matter University, Tel Aviv
	University, Tel Aviv, Israel
Instituto de Alta Investigación, Universidad de Tarapacá,
Casilla 7D, Arica, Chile



§ ABSTRACT 
In the framework of the model of a spatially non-uniform Bose-Einstein
condensate with helicoidal spin-orbit (SO) coupling, we find abnormal
Peregrine solitons (PSs) on top of flat and periodic backgrounds, with
ultra-high amplitudes. We explore the roles of the SO coupling strength and
helicity pitch in the creation of these anomalously tall PSs and find that
their amplitude, normalized to the background height, attains indefinitely
large values. The investigation of the modulation instability (MI) in the
same system demonstrates that these PSs exist in a range of relatively weak
MI, maintaining the feasibility of their experimental observation.


Ultra-high-amplitude Peregrine solitons induced by helicoidal
spin-orbit coupling
    B. A. Malomed
    
=================================================================================





Introduction. Rogue waves (RWs), first discovered as extreme events
in the ocean <cit.>, have been
widely studied, due to their unique properties and potential applications,
in nonlinear optics <cit.>,
plasmas <cit.>, Bose-Einstein condensates (BECs) <cit.>, magnetics <cit.>,
financial markets <cit.>, and various other settings <cit.>. A widely
recognized RW prototype is provided by the exact Peregrine-soliton (PS)
solution of the nonlinear Schrödinger equation (NLSE) <cit.>, whose characteristic features are the threefold peak
amplitude and spatiotemporal localization on top of the background field
<cit.>. Several landmark experiments have directly demonstrated
this remarkable phenomenon and its ramifications <cit.>.

Spin-orbit (SO) coupling in BECs have drawn much interest since its
experimental implementation <cit.>, as it
offers the realization of the SO-coupling phenomenology in the uniquely
clean form <cit.> and make it possible to create artificial
vector gauge potentials <cit.>. Recently, models
of BECs with non-uniform SO coupling have been introduced, as they provide
high tunability of this effect, and enhance the role of the intrinsic
nonlinearity in the SO-coupled BECs <cit.>.
In this context, soliton dynamics in the BEC with non-uniform landscapes of
the SO coupling has been investigated <cit.>, where, in particular, the
helicoidal gauge potential may originate from the light propagation in a
helical waveguide array <cit.>. The propagation of matter-wave
solitons in a BEC with a random SO coupling was addressed too <cit.>.

SO-coupled BECs are modeled by systems of two (or several) coupled
Gross-Pitaevskii equations (GPEs). In this connection, it is relevant to
stress that PSs exist in multi-component NLSE models, such as the famous
Manakov system, but, due to the energy transfer between different
components, the PS amplitude is no longer fixed, although it still does not
exceed the triple background height <cit.>. Nevertheless, recent
studies have shown that, under the action of self-steepening effects, the
amplitude of fundamental PSs can exceed the threefold limit, reaching up to
fivefold the background height <cit.>. In particular, exceptional
PSs, which feature ultra-high peak amplitudes, have been reported too in the
vector derivative NLSEs including the self-steepening effect <cit.>.

In this work, we focus on the following questions: can the fundamental PS
with an ultra-high peak amplitude be excited in other ways, besides using
higher-order effects, such as self-steepening, and to what extent is it
possible to increase the PS amplitude? To answer these questions, we first
consider a BEC model with non-uniform helicoidal SO coupling (cf. Ref. <cit.>), which offers experimental feasibility. We construct its
exact fundamental PS solutions on top of flat, alias continuous-wave (CW),
and periodic backgrounds. Through the analysis of the PS amplitude, we find
that PS with ultra-high peak amplitude, reaching indefinitely large values
(as normalized to the background height), can be created with the help of
the helicoidal SO coupling.

To explore the PS dynamics under the action of spatially non-uniform gauge
potentials, we consider the GPE for the spinor wave function Ψ
=(Ψ _1,Ψ _2)^T of an effectively one-dimensional two-component
BEC, including the helicoidal SO coupling. In the scaled form (with M=ħ
=1, where M is the atomic mass), the GPE is  <cit.>

    i∂Ψ/∂ t=1/2Q^2(x)Ψ
    -(Ψ^†Ψ)Ψ,

where the helicoidally molded SO coupling is represented by the generalized
momentum operator,

    Q(x)=-i∂ _x+ασ·n(x).

Here α is the SO-coupling strength, which is tunable in the
experiment <cit.>, σ
=(σ _x,σ _y,σ _z) is the vector of the Pauli matrices,
and the spatial modulation is represented by vector

    n(x)=(cos (2κ x),sin (2κ x),0),

with κ <0 and κ >0 corresponding to the left- and right-handed
helicity, respectively <cit.>. As usual,
it is assumed that the inter- and intra-species attractive interactions have
equal strengths. Special forms of Eq. (<ref>) include the
uniform Rashba-Dresselhaus SO coupling <cit.> when κ =0,
and the canonical Manakov system <cit.> when α =0.

Fundamental PS solutions. Eq. (<ref>) is made
gauge-equivalent to the integrable Manakov system,

    i𝐮_t+1/2𝐮_xx+(𝐮^†𝐮)
    𝐮=0,   𝐮=(u_1,u_2)^T,

by means of the transformation <cit.>

    Ψ=
    [ ν _+e^-i(k_m+κ )x  ν _-e^i(k_m-κ )x; ν _-e^-i(k_m-κ )x -ν _+e^i(k_m+κ )x ]𝐮,

where k_m=√(α ^2+κ ^2) is the effective
momentum of the lowest-energy states, and

    ν _+=   sgn(α )√(( k_m- κ)
    /( 2k_m) )
    ν _-=   √(( k_m+ κ) /( 2k_m) )

Below, k_m plays a crucial role determining properties of PSs,
especially as concerns the amplification of their amplitudes.

The Manakov system (<ref>) possesses the Lax pair <cit.> and admits the solution by means of the Darboux dressing method 
<cit.>. To begin with, we take the CW seed solution of Manakov
system (<ref>), with components

    u_j0=aexp [-i(k_jx-ω _jt)],  j=1,2,

which is determined by the amplitude (a), wavenumbers (k_j), and
frequencies

    ω _j=2a^2-k_j^2/2.

Making use of the Manakov system invariance with respect to the rotation of
the set of the two components, we choose them in Eq. (<ref>) with equal
amplitudes a. Subsequent results demonstrate that the helicoidal SO
coupling makes PS heights different in the two components Ψ _1,2 for
the same background amplitudes a, see Eqs. (<ref>) and
(<ref>) below.

Utilizing the known PS solutions for Manakov system (<ref>)
derived by means of the Darboux transform <cit.>, and substitution (
<ref>), we obtain the following exact fundamental PS solutions of the
underlying Eq. (<ref>):

    Ψ _1   =   ae^-iκ x[ ν _+( 1-ℛ_1/𝒩_1) e^iθ _1+ν _-( 1-ℛ
    _2/𝒩_2) e^iθ _2] ,  
    Ψ _2   =   ae^iκ x[ ν _-( 1-ℛ_1/𝒩_1) e^iθ _1-ν _+( 1-ℛ
    _2/𝒩_2) e^iθ _2] ,  
    θ _1   =   -(k_m+k_1)x+ω _1t,  θ _2=(k_m-k_2)x+ω _2t,

where we define

    𝒩_j   =   [ (θ +μ t)^2+ζ ^2t^2+4/ζ
    ^2] {[δ +(-1)^jμ ]^2+ζ ^2},  
    ℛ_j   =   8i{ζ ^2t-[μ +(-1)^jδ ](θ +μ
    t)}+16,  
    μ   =   ±√(2)/2[ √(δ ^2(8a^2+δ ^2))
    -4a^2+δ ^2] ^1/2,

in the case of |δ |≥ a, with δ≡ k_1-k_2, or

    𝒩_j   =   [ θ ^2+(ζ +μ ^')^2t^2+4/
    (ζ +μ ^')^2] (2a^2+ζμ ^'),  
    ℛ_1   =   4i(4a^2-δ ^2+2ζμ ^')t-4i(-1)^jδθ +8,  
    μ ^'   =   ±1/√(2)[ 4a^2-δ ^2-√(δ
    ^2(8a^2+δ ^2))] ^1/2,

in the case of |δ |<a. In either case, we set θ≡
2x+(k_1+k_2)t and ζ≡( 1/√(2)) [ √(δ ^2(8a^2+δ ^2))+4a^2-δ ^2] ^1/2. Using the
translational symmetry, we shift the above solutions to the origin, to
produce compact expressions for them. Note that these PS solutions are
non-singular ones in the entire parameter range.



In addition to the same features which are demonstrated by the conventional
PSs, that exist in some multi-component systems, such as PSs of the
bright-dark type, PS doublets, etc., the helicoidal SO coupling can
generate more intricate PS structures, among which the most salient aspect
is, as shown below, the possibility of having PSs with uniquely large heights.

The consideration of the exact solution (<ref>) reveals that the
PS is generally located on top of a periodic background formed by the
superposition of two different CWs. The exact solution for the periodic
background is

    |Ψ _1^bg|    =   a√(1+α/k_mcos[
    (δ +2k_m)x+k_1^2-k_2^2/2t] ),  
    
    |Ψ _2^bg|    =   a√(1-α/k_mcos[
    (δ +2k_m)x+k_1^2-k_2^2/2t] ).

It is moving with speed v=( k_2^2-k_1^2) /[
2(δ +2k_m)], where k_1,2 are the same wavenumbers
as in Eq. (<ref>).

Note that, if wavenumbers k_1,2 and the momentum minimum k_m
satisfy the following relationship,

    k_1=-k_2=-k_m,

the cos terms vanish in Eq. (<ref>), i.e., the
periodic background degenerates into a flat CW. Due to the presence of the
helicoidal SO coupling, the constraint (<ref>) is different
from similar ones which provide for the flat background in the
coupled-NLSE system <cit.> and the multi-component
long-wave-short-wave resonance model <cit.>.

PS on the CW background.  To reveal the amplification effect of
the helicoidal SO coupling on the PS amplitude, we first address the PS
solution on top of the flat CW background, subject to constraint (<ref>). The respective background amplitude (<ref>) amounts to

    |Ψ _1^cw|=a√(k_m+α/k_m)
    , |Ψ _2^cw|=a√(k_m-α/k_m).

Under the action of the SO coupling with strength α, the components
of the CW background (<ref>) have different heights.

Taking into regard that the center of the PS solution (<ref>) is
pinned to the origin, enhancement factor |F_j| of component Ψ _j
is defined as the peak-to-background ratio:

    F_1   =   Ψ _1(0,0)/|Ψ _1^cw|=a(ν
    _+f_u_1+ν _-f_u_2)/|Ψ _1^cw|,  
    
    F_2   =   Ψ _2(0,0)/|Ψ _2^cw|=a(ν
    _-f_u_1-ν _+f_u_2)/|Ψ _2^cw|,

where |Ψ _1,2^cw| are given in Eq. (<ref>),
coefficients ν _± are same as in Eq. (<ref>), and factors f_u_1 and f_u_2 are defined, for |δ |≥ a, as

    f_u_1=1-4ζ ^2/ζ ^2+(δ -μ )^2,  f_u_2=1-
    4ζ ^2/ζ ^2+(δ +μ )^2,

and, for |δ |<a, as

    f_u_1=f_u_2=1-2(ζ +μ ^')^2/2a^2+ζμ
    ^'.




Characteristic examples of the PSs featuring large enhancement factors are
presented in Fig. <ref>, which includes a PS with nearly fivefold peak
amplitude for the component Ψ _1, with α =-1/2 and κ
=2/5, in Fig. <ref>(a_1). For comparison, two special cases are
presented too, viz., for α =0 [Figs. <ref>(a_2) and
(b_2)] and κ =0 [Figs. <ref>(a_3) and (b_3)], which
correspond to the Manakov system limit and the uniform SO coupling,
respectively. It is observed that, with α =0, the PS amplitudes are
only twice as large as those of the background (in fact, for the Manakov
system the peak amplitude cannot exceed three times the background value
<cit.>). In addition, for κ =0 the PS with zero background
in component Ψ _1 or Ψ _2 is produced by Eq. (<ref>), depending on the sign of α. Thus, the helicoidal
SO coupling makes it possible to elevate the amplitude of one component to
an exceptional level, while suppressing the other component.



To further unveil the specific role of the helicoidal SO coupling in
generating PSs with exceptionally high amplitudes, we display the dependence
of enhancement factors F_j on the SO-coupling strength α and
rotation frequency κ in Figs. <ref> and <ref>,
respectively. They exhibit an indefinitely large (diverging) enhancement
factor for component Ψ _1 or Ψ _2 at |α |→∞ or κ→ 0. In particular, the insets to these
figures feature the enhancement factor |F_1| with values close to 10
at α =-1 and κ =0.4, and |F_2| close to 5 at α =0.6 and κ =-0.6. A caveat is that the enhancement factor is diverging
when the background amplitude |Ψ _j^CW| is vanishing, as
shown by the cyan dashed curves in Figs. <ref> and <ref>. The
absolute values of the PS peak amplitude may be increased by taking values
of amplitude a>1 in Eq. (<ref>) (recall it is currently fixed as a≡ 1, by means of scaling).



PSs on top of the periodic background. If the constraint (<ref>) does not hold, the above solution (<ref>)
produces the PS built on top of the periodic background. Similar to the case
of the flat CW background considered above, we define the enhancement factor
to analyze the effect of the helicoidal SO coupling on the PSs. In Fig. <ref> we demonstrate a characteristic example exceeding the threefold
contrast between the peak amplitude and periodic background in component Ψ _2, for k_1=-k_m,k_2=k_m-3/2.



Next, we address the modulation instability (MI) of the CW field, Ψ
_j0=a_je^iμ t with μ =a_1^2+a_2^2, where a_1 and a_2 are the uniform amplitudes and μ is the chemical potential, in
the presence of the helicoidal SO-coupled BECs. To this end, we add small
perturbations to the CW fields, viz., Ψ _j=Ψ
_j0{1+p_jexp [-i(β x-Ω t)]+q_j^∗exp [i(β
x-Ω ^∗t)]}, where β and Ω are, respectively, the
real and complex parameters, p_j and q_j being small complex
amplitudes. Linearizing the corresponding Eq. (<ref>) with
respect to p_j and q_j, we derive a quartic equation for the
perturbation eigenfrequency Ω, which determines the MI gain as γ _h=|Im(Ω )|_max. In Fig. <ref>
, we display heatmaps for the so found value of γ _h in the (β
,α ) and (β ,κ ) parameter planes. The plots reveal that
the MI-gain spectra are symmetrically distributed in broad regions of α and κ, which can give rise to the anomalous PS behavior in
a broad range of parameters, in comparison to the usual situation underlain
by the baseband-MI analysis <cit.>. For instance,
Figs. <ref>(a_2) and ( b_2) demonstrate,
respectively, that the gain maximum, γ _h≈ 1.91 at α =-1 and κ =0.4, corresponds to the enhancement factor |F_1|≈ 10 in Fig. <ref>, and the maximum γ _h≈ 1.65, at α =0.6 and κ =-0.6, corresponds to |F_2|≈ 5 in Fig. <ref>. Such relatively small values of the MI
gain, corresponding to the ultra-high PS peak amplitudes, suggest that these
large amplitude values may be relatively easy to attain in the experiment,
as the background will not be vulnerable to the quick destruction by of the
MI-driven blowup, hence these PSs are rather robust modes.





To test the expected robustness of the PSs in the present setting, in Fig. 
<ref> we display results of the numerically simulated evolution of the
PSs from Figs. <ref>(a_1) and (b_1) under the action of 2%
random disturbances. It is observed that the PSs with the ultra-high
amplitudes indeed demonstrate robust propagation.

Conclusion. We have reported the occurrence of abnormal fundamental
PSs (Peregrine solitons) with ultra-high peak amplitudes in the integrable
system of GPEs (Gross-Pitaevskii equations) including the helicoidally
modulated SO (spin-orbit) coupling, which is a gauge isomer of the Manakov
system. The PS solutions are found on top of both the flat and periodic-wave
backgrounds. The results demonstrate the existence of RWs (rogue waves with
the ultra-high amplitude in the context of matter waves (BEC)), while
previously this was reported in models of nonlinear optics <cit.>. The helicoidal SO coupling is crucially important for
generating this abnormal PSs, and the controllable nature of the SO coupling
makes the predicted phenomenology experimentally feasible. The MI
(modulational instability) is also studied in the system, demonstrating that
the high-amplitude PSs readily coexist with moderate MI, thus preventing a
strong background instability and improving chances for the experimental
creation of the predicted tall rogue waves.



Acknowledgments This work was supported by the National Natural Science
Foundation of China (Grant No. 11975172) and the Natural Science Foundation of Hubei Province of China (Grant No. 2023AFB222). The work of B.A.M. is
supported, in part, by the Israel Science Foundation (Grant No. 1695/22).




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 Paper Title: Radio-Frequency Spectroscopy and the Dimensional Crossover in Interacting Spin-Polarized Fermi Gases.






Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Università di Trento, I-38123 Trento, Italy

Department of Physics and CQIQC, University of Toronto, Toronto, Ontario M5S 1A7, Canada

Department of Physics and CQIQC, University of Toronto, Toronto, Ontario M5S 1A7, Canada

Department of Physics and Hong Kong Institute of Quantum Science & Technology, 

The University of Hong Kong, Hong Kong, China


§ ABSTRACT  
Low-dimensional ultracold gases are created in the laboratory by confining three-dimensional (3D) gases inside highly anisotropic trapping potentials. Such trap geometries not only provide access to simulating one-dimensional (1D) and two-dimensional (2D)  physics, but also can be used to study how the system crosses over towards a 3D system in the limit of weak confinement. In this work, we study the signature in radio-frequency (RF) spectroscopy for both the 1D-to-3D and the 2D-to-3D crossovers, in spin-polarized Fermi gases. We solve the two-body scattering T-matrix in the presence of strong harmonic confinement and use it to evaluate the two-body bound state and the RF spectroscopy transfer rate in the high frequency limit, covering both the quasi-low-dimensional and 3D limits. We find that in order to understand the dimensional crossover for spin-polarized Fermi gases with p-wave interactions, one needs to take into account an emergent s-wave interaction. 


Radio-Frequency Spectroscopy and the Dimensional Crossover in Interacting Spin-Polarized Fermi Gases
    Shizhong Zhang
    August 5, 2024
====================================================================================================





§ INTRODUCTION


In recent years there has been an increased interest in understanding the physics of spin-polarized Fermi gases in low dimensions <cit.>. These systems are special as they do not interact with the standard s-wave contact interaction, but rather a momentum dependent short-ranged p-wave interaction. In three-dimensional space (3D) such systems are plagued by severe three-body losses <cit.>, making the strongly interacting limit inaccessible. However, in low dimensions, it has been predicted that these systems are more stable than their 3D counterparts <cit.>. Moreover, strongly interacting low-dimensional spin-polarized Fermi gases offer interesting physics in their own right, such as topological superconductivity in two dimensions (2D) <cit.>, as well as integrable dynamics <cit.> and the Bose-Fermi duality <cit.> in one-dimensional space (1D). 

These low-dimensional spin-polarized Fermi gases can be readily accessed in the laboratory by tightly confining a 3D gas in an anisotropic external trapping potential. If the many-body energy scales, like temperature or chemical potential, are less than the energy gap to the first excited state of the trapping potential, all the atoms will reside in the the lowest lying state of the trapping potential; the system is effectively low-dimensional in this regime. One can then tune the effective one-dimensional interactions using the 3D p-wave Feshbach resonance, resulting in the confinement-induced resonances <cit.>.

On the other hand, when the energy scales of the many-body state become comparable to the energy spacing of the states of the trapping potential, the low-dimensional approximation begins to break down. The atoms can then begin to occupy higher states of the trapping potential. Once enough of these scattering continua are occupied, the system becomes fully 3D. Hence confined quantum gases are an excellent platform for studying not only low-dimensional physics, but also the dimensional crossover.

However, even when the low-dimensional energy hierarchy is intact, namely, when the relevant many-body energy scales are much smaller than the energy spacing of the trapping potential, 3D physics can still be revealed by probing the system at short distances. At scales much smaller than the transverse oscillator length, the confining potential is increasingly irrelevant. Hence tuning the length scale at which one probes the system can reveal  another type of dimensional crossover. 

There are several observables that may reveal this form of the dimensional crossover. For example, consider the particle distribution at high momentum: 
it is well known that the momentum distribution exhibits power-law tails for large momentum: k ≫ k_F, where k is the magnitude of the momentum and k_F is the Fermi momentum <cit.>. For p-wave gases, the high momentum tail changes continuously from k_⊥^-2 in the q2D limit (k_⊥ a_⊥≪ 1) to k^-2=(k_⊥^2 + k_∥^2)^-1 in the 3D limit (k_⊥ a_⊥≫ 1), where k_⊥ (∥) are the momentum components perpendicular (parallel) to the unconfined two-dimensional space <cit.>. The constants of proportionality for the high-momentum tails in the quasi-low- and high-dimensional limits are related to the d-dimensional thermodynamic contacts, which depend on the many-body details of the system and how many states of the confining potential are occupied.

In order to study the dimensional crossover using the momentum distribution, one needs the ability to measure the momentum distribution over many decades ranging from k_⊥ a_⊥≪ 1 to k_⊥ a_⊥≫ 1. 
Although this is possible using time-of-flight expansion,
one needs to optimize the free-expansion time to maintain a good signal-to-noise ratio over many orders of magnitude of momenta. Even in experiments well adapted to such a measurement, spurious k^-4 tails have been observed due to spin impurities <cit.>. Moreover, the continuous nature of the momentum distribution makes it difficult to distinguish definitive signatures of the dimensional crossover.

A more commonly used technique to study the high-momentum distribution is radio-frequency (RF) spectroscopy <cit.>. In RF spectroscopy, a radio-frequency pulse performs a spin flip from some initial interacting hyperfine state |a⟩ to a final non-interacting hyperfine state | b ⟩ at a fixed detuning ω. For large detunings (compared to the energy spacing between the hyperfine states) the RF transfer rate has power-law tails, and the constants of proportionality are again related to the thermodynamic contacts. The benefit of the RF spectroscopy transfer rate is that the dimensional crossover can be studied by simply adjusting the detuning, independent of whether the initial state is in the low-dimensional regime or not. 

In this work we examine the 1D-to-3D and 2D-to-3D crossovers in p-wave spin-polarized  Fermi gases using the RF spectroscopy transfer rate. We obtain analytical results for the two-body scattering in both quasi-one-dimensional (q1D) and quasi-two-dimensional geometries (q2D). This allows us to evaluate the two-body bound states, as well as the RF transfer rate at the two-body level for the whole dimensional crossover. The results for the RF transfer rate are made more rigorous using the operator product expansion (OPE) in the low-dimensional and three-dimensional limits, which can connect the RF tails to the corresponding thermodynamic contacts. Our results for the two-body bound states and the RF transfer rate indicate that there are two effective low-dimensional interactions, one that is p-wave and one that is s-wave. Both interactions are necessary for understanding the dimensional crossover, although only the low-dimensional p-wave interaction is needed for understanding the low-dimensional limit. 

Our study was motivated by a recent experiment <cit.> 
that found evidence of q1D s-wave scattering in spin-polarized Fermi gas and a first signature of the dimensional crossover in RF spectroscopy. Here we provide a complete theoretical description of both the 1D-to-3D and the 2D-to-3D crossovers for the RF transfer rate and the dimer state. This work is also accompanied by an experimental work <cit.> examining the RF spectroscopy and binding energies in q2D Fermi gases.

The remainder of this article is organized as follows. In Sec. <ref> we discuss how three-dimensional p-wave scattering in the presence of harmonic confinement can produce both low-dimensional p-wave and s-wave interactions. In Sec. <ref>, we then explicitly solve the Lippmann-Schwinger equations for two-body p-wave scattering in q1D systems. We then extend this formalism to q2D systems in Sec. <ref> and calculate the two-body bound states along the dimensional crossover in Sec. <ref>.
In Sec. <ref> we determine the RF spectroscopy transfer rate for the whole dimensional crossover. The conclusions are presented in Sec. <ref>.



§ P-WAVE SCATTERING IN HARMONIC CONFINEMENT



Consider two non-interacting identical fermions in an anisotropic external harmonic trap:

    U_ext.( r)=1/2m(ω_⊥^2  r_⊥^2+ω_∥^2 r_∥^2) ,

where r_⊥ is the position vector in the confined directions (the transverse directions), while r_∥ is for the weakly confined directions (the axial directions). To be specific, in the q1D case:  r_⊥ =(x,y)  and r_∥ = z, while for the q2D trap: r_⊥ = z and r_∥ =(x,y). In both cases, we consider high anisotropy and require the transverse oscillator frequency ω_⊥ to be much larger than that of the axial directions ω_⊥≫ω_∥. Furthermore, we shall set that the axial oscillator frequency ω_∥=0 and assume that any effects with a non-zero ω_∥ in actual experiment can be adequately taken care of using the local density approximation. 

At the two-body level, the center of mass motion can naturally be separated from the relative motion. While in general the center of mass motion of the pairs cannot be ignored, for the discussion of the leading high-frequency tails of the RF transfer rate, the inclusion of the center of mass is irrelevant. The relative motion is described by the simple Hamiltonian:

    H =  p_∥^2/2μ +  p_⊥^2/2μ  + 1/2μω_⊥^2  r_⊥^2 + v(r) ,

where μ is the reduced mass, p_∥ is the axial momentum along the unconfined direction, and p_⊥ is the momentum in the transverse directions. The interaction potential v(r = | r|) is short ranged; the range r_0 is much smaller than the typical size of the trapping potential. In particular r_0≪ a_⊥ where a_⊥=√(1/μω_⊥) is the transverse oscillator length. 

To understand the effects of the short-range potential v(r) on the scattering of two identical fermions, it is beneficial to examine its matrix element between two non-interacting states of the relative motion of the two fermions ϕ( r) and χ( r):

    ⟨ϕ | v(r) | χ⟩=∫ d^3 rϕ^*( r)v(r)χ( r) .

Because of the fermionic statistics, the relative wavefunctions must be odd under exchange, or equivalently parity: ϕ( r)=-ϕ(- r) and χ( r)=-χ(- r).
Assuming that the scale over which ϕ( r) and χ( r) changes are much larger than r_0, i.e. if we look at states with energies much smaller than r_0^-2, we can then expand the integrand around r=0 and obtain:

    ∫ d^3 r (ϕ^*( 0)+r_i∇_iϕ^*( 0))v(r)(χ( 0)+r_j∇_jχ( 0)) ,
    
    =   ∫ d^3 r r_iv(r)r_j∇_iϕ^*( 0)∇_jχ( 0) ,
    
    =   [∫ d^3 r x^2v(r)]∇ϕ^*( 0)·∇χ( 0)≡ g∇ϕ^*( 0)·∇χ( 0) .

where we have used the obvious result that χ( 0)=ϕ( 0)=0 in the second line. In the last line, we also used the fact that integration such as ∫ d^3 r xyv(r) are zero due to rotational symmetry. From the final equality, the p-wave matrix element at leading order is equivalent to that of a phenomenological p-wave pseudo-potential:

    v(r) = g ∇_i δ( r)  ∇_i  ,

where the arrows indicate the direction to which the gradient is applied. This allows us to identify the bare coupling constant in terms of the microscopic two-body potential as: g≡∫ d^3 r x^2v(r). In the particular case when both the initial state and final state are described by plane waves with wave vector k and k', then clearly

    ⟨ k  | v(r) |  k'⟩=g/L^3 k · k ' ,

where L^3 is the volume of the system.

Once the form of the pseudo-potential Eq.(<ref>) is chosen, it is clear that the requirement for antisymmetrizing the non-interacting wave function ϕ and χ is no longer necessary since the even parity part is automatically zero in the transition matrix element. One can improve the above approximation by taking into account additional derivatives such as ∫ d^3 r x^2y^2v(r)∇^2_xϕ^*( 0)∇^2_yχ( 0) that are generally suppressed at low energy by the additional derivatives. 

For 3D p-wave scattering in the absence of the harmonic trap, the p-wave pseudo-potential in Eq. (<ref>) suggests that the low-energy scattering T-matrix, T̂_p, can be written in the following form at low-energy <cit.>:

    ⟨ k  | T |  k'⟩   = 2 k · k' T_3D,p(E)   ,
    
    T_3D,p^-1(E)    = 2μ/24π[ 1/V_p + 2μ E/R_p  +(-2μ E -i δ)^3/2] ,

where | k^(')⟩ describes a plane wave state with momentum k^(') and | k|  = | k'|. The factor of 2 in the matrix element accounts for the indistinguishable nature of the fermions.  In Eq. (<ref>), the p-wave scattering volume, V_p, and the 3D p-wave effective range, R_p, depend on the details of the potential v(r) <cit.>, but such knowledge is not needed for our purposes. 

In the presence of the harmonic confinement, Eq. (<ref>) is still valid since it only refers to the short-range form of the wave functions which is insensitive to external confinement. However, now the non-interacting states are plane waves with momentum k in the axial directions, while in the transverse direction they are harmonic oscillator states. The pseudo-potential can now act on either the axial or transverse part of the two-body wavefunction. If the gradients act on the axial directions, the matrix elements of the T-matrix will have the form of a low-dimensional p-wave interaction: ⟨ k | v|  k' ⟩∝ g k · k'. This occurs when the axial two-body wavefunctions are odd functions under parity (or equivalently particle exchange), while the transverse two-body wavefunctions are even function under parity. This is the only possibility when all the atoms reside in the lowest state of the transverse trapping potential.

If instead the gradients act on the transverse directions one obtains an interaction reminiscent of a low-dimensional s-wave interaction: ⟨ k | v|  k' ⟩∝ g. In order to have a non-zero matrix element, the two-body wavefunction has to have even parity along the axial direction and odd parity along the transverse direction. This can only be accomplished when the atoms occupy different single particle transverse states which allows for an odd parity state. 

In the following sections, we will specifically consider scattering in q1D and q2D geometries and show how starting from the fully 3D scattering, one reproduces the results of low-dimensional s- and p-wave scattering, before we investigate the crossover of radio-frequency spectral from its low-dimensional feature to that corresponds to full 3D as one increases the probing rf-frequency. 



§ P-WAVE SCATTERING IN Q1D



We now turn to a specific calculation of the two-body scattering properties inside a q1D harmonic trapping potential. The non-interacting eigenstates and energies (in the center of mass frame) are given by :

    ψ_k,n,m(r_∥, r_⊥)    = e^i k r_∥√(n!/π a_⊥^2(n+|m|)!)
       e^-r_⊥^2/2 a_⊥^2 e^i m ϕ(r_⊥/a_⊥)^|m|ℒ^|m|_n (r_⊥^2/a_⊥^2)  ,   
    
    E_n,m(k)    = k^2/2μ + (2n + |m|+1)ω_⊥ .

In Eqs. (<ref>-<ref>) k is the axial momentum, n is the transverse principle quantum number, and m is the angular momentum in the transverse direction. We have also written r_⊥  = (r_⊥, ϕ) in polar coordinates. In this representation the transverse eigenstates have definite parity: P = (-1)^m.

The matrix elements of the pseudopotential Eq. (<ref>) in this basis are found to be:

    ⟨ k n m |    v | k' n' m ⟩ = 
       2g/π a_⊥^2( k k' δ_m,0 + 2/a_⊥^2√(n+1)√(n'+1)δ_|m|,1) .

When m=0, the interactions are of 1D p-wave character with the required form factor kk', while for m = ± 1 the interactions resemble a 1D s-wave interaction. This follows from the parity considerations discussed in the previous section.

We now solve the Lippmann-Schwinger equation for the two-body T-matrix inside the harmonic confinement. Since the angular momentum m is a conserved quantity, the different angular momentum sectors can be evaluated independently. The full details of the calculation are presented in Appendix <ref>. In the case when m=0 which corresponds to p-wave scattering, the matrix element of T can be written as

    ⟨ k,n,0| T | k',n',0 ⟩= 2 k k' T_p,1D(E) ,

with T_p,1D(E) given explicitly by

    T_p,1D^-1(E)= 
    2μ/2[a_⊥^2/6(1/V_p + 2μ E/R_p)  - 2/a_⊥ζ(-1/2,-ℰ_0 - i δ)] .

In Eq. (<ref>), ℰ_m  = (E-(|m|+1)ω_⊥) / 2ω_⊥ = (k a_⊥/2)^2 is the ratio of the kinetic energy in the axial direction for the mth channel to the transverse level spacing 2ω_⊥, and ζ(s,x) is the Hurwitz zeta function. For the m = ± 1 sector, we have similarly

    ⟨ k,n,± 1 |    T | k',n', ± 1 ⟩ = 2√(n+1)√(n'+1) T_s,1D(E) , 
    
    T_s,1D^-1(E)   =
    2μ/2[a_⊥^4/12(1/V_p + 2μ E/R_p)  . 
    
    + a_⊥/2   .[ ζ(-1/2,-ℰ_1 - i δ) + E/2ω_⊥ζ(1/2,-ℰ_1 - i δ)]] .

Eqs. (<ref>-<ref>) are the exact solutions to the Lippman-Schwinger equation and are valid for arbitrary energies. 

The low-dimensional limit is defined when the kinetic energy is much less than 2ω_⊥, i.e. ℰ_m ≪ 1. First consider m=0. In the 1D or low-energy limit, ℰ_0 ≪ 1, Eq. (<ref>) reduces to:

    T^-1_p,1D(E)    =2μ/2[1/a_p,1D + k^2 r_p,1D +i k ] , 
    1/a_p,1D   = a_⊥^2/6( 1/V_p + 2/R_p a_⊥^2) - 2/a_⊥ζ(-1/2) , 
    
    r_p,1D   = a_⊥^2/6R_p + a_⊥/4ζ(1/2) ,
 

where ζ(s) = ζ(s,1) is the Riemann zeta function. The low-energy form of the T-matrix is the exact result one would obtain from a true 1D calculation with p-wave interactions in 1D. Here a_p,1D and r_p,1D are the 1D p-wave scattering volume and effective range, respectively, and are written in terms of the 3D scattering parameters, V_p and R_p. Eqs. (<ref>-<ref>) have been reported previously <cit.>. 

Similarly for small kinetic energies in the even-wave sector, ℰ_1 ≪ 1, the m=± 1 T-matrix reduces to:


    T_s,1D^-1(E) =   2μ/2[-a_s,1D + k^2 r_s,1D + i/k] , 
    
    a_s,1D =    -a_⊥/2[ζ(-1/2) + ζ(1/2) . 
       . + a_⊥^3/6(1/V_p + 4/a_⊥^2 R_p)]  ,
    
    r_s,1D =   a_⊥^3/16[ζ(1/2) + ζ(3/2)  + 4a_⊥/3 R_p] ,


which is equivalent to a true 1D calculation with even-wave interactions. The only difference between Eq. (<ref>) and a true 1D calculation is that the matrix element for the low-dimensional s-wave scattering has an extra factor of two from the indistinguishability of the spin-polarized fermions. The parameters a_s,1D and r_s,1D in Eqs. (<ref>-<ref>) are the 1D s-wave scattering length and effective range, respectively. To the best of our knowledge, Eqs. (<ref>-<ref>) have only been reported in Ref. <cit.>.



§ P-WAVE SCATTERING IN Q2D



Next, let us consider p-wave scattering in a q2D geometry. In this case, the eigenstates and energies for the Schrödinger equation in the relative coordinates are:

    ψ_k,n,m( r_∥,r_⊥)    = e^i  k · r_∥/π^1/4√(a_z)e^-r_⊥^2/2a_z^2 H_2n+|m|(r_⊥/a_⊥)/√(2^2n+|m| (2n+|m|)!) , 
    
    E_n,m(k)    = k^2/2μ + (2n + m+ 1/2) ω_⊥ .

In Eqs. (<ref>-<ref>), k is the momentum in the axial direction, while n is the harmonic oscillator quantum number in the transverse directions. We have also defined the quantum number m, which controls the parity of the transverse wavefunction P = (-1)^m and which takes on values of m = 0,1.

The matrix elements of the interaction are:

    ⟨   k n m |v | k' n' m⟩ = 
       2g/√(π) a_⊥(f_s(n)f_s(n')   k · k'δ_m,0 + 2/a_⊥^2 f_p(n) f_p(n') δ_m,1) ,

where 

    f_s(n)    = (-1/2)^n √((2n)!)/n! , 
    
    f_p(n)    = (-1/2)^n √((2n+1)!)/n! .


As in the q1D case, when m=0, the interaction matrix element is 1D p-wave in character, while for m=1 it is s-wave in character. The parity in the transverse direction acts equivalently to the angular momentum in the q1D geometry. For low-energies, m=0 occurs when both atoms are in the ground band, while m=1 is possible only when one of the atoms is in a higher transverse state.

The calculation for the q2D T-matrices is similar to the case of q1D scattering and provides the following expressions for m = 0:

    ⟨ k n 0 |   T |k' n' 0 ⟩ = 2 f_p(n)f_p(n')  k · k' T_p,2D(E) , 
    
        T_p,2D^-1   =2μ/8π[2√(π) a_⊥/3(1/V_p + 2μ E/R_p) + 4/a_⊥^2w_0(ℰ_0)] ,

and similarly for m=1:

    ⟨ k n 1 |    T |k' n' 1 ⟩ = 2 f_s(n)f_s(n') T_s,2D(E) , 
    
        T_s,2D^-1   =2μ/4π[√(π) a_⊥^3/6(1/V_p + 2μ E/R_p) - w_1(ℰ_1)] .

We also define ℰ_m = (E-(m+1/2)ω_⊥)/(2ω_⊥) for the q2D case, and the functions w_0,1(ℰ_0) are defined as:

    w_0(ℰ_0)    = ∑_n=0^N(2n-1)!!/(2n)!!(n-ℰ_0) ln(n- ℰ_0 - i δ) -  ,
    w_1(ℰ_1)    = ∑_n=0^N(2n+1)!!/(2n)!!ln(n- ℰ_0 - i δ) -  ,

where N is a high-energy cutoff which we send to infinity, and “norm.” represents the normalization terms. The normalization of Eqs. (<ref>-<ref>) are discussed further in Appendix <ref>.

In the low-energy limit, ℰ_m ≪ 1, these two-expressions collapse onto the desired expressions for 2D s- and p-wave T-matrices. Namely for m=0 we find the T-matrix for 2D p-wave interactions:

    T_p,2D^-1(E) = 2μ/8π[1/a_p,2D + 2μ E ln(1/-2μ E r_p,2D^2-i δ) ] ,

with the 2D scattering area a_p,2D and effective range r_p,2D given by

    a_⊥^2/a_p,2D   = 2 √(π)	a_⊥^3/3(1/V_p+ 1/a_⊥^2 R_p) - α_a  , 
    ln(a_⊥/r_p,2D)    = √(π) a_⊥/3 R_P + α_r ,

where α_a ≈ 0.2173 and α_r ≈ 0.4689. Eqs. (<ref>-<ref>) were derived previously in Ref. <cit.>. 

Similarly for m=1 we obtain the T-matrix for 2D s-wave interactions:

    T_s,2D^-1 (E) = 2μ/4π[ln(1/-2μ E a_s,2D^2 - i δ) + 2μ E r_s,2D] ,


    ln(a_⊥^2/a_s,2D^2)    = √(π)a_⊥^3/6(1/V_p + 3/a_⊥^2 R_p) + ln(2B/π) , 
    r_s,2D/a_⊥^2   = √(π)a_⊥/6 R_P + γ/4 ,

with B ≈0.85 and γ≈ 1.41.



§ TWO-BODY BOUND STATES



Since the T-matrices discussed in Secs. <ref>-<ref> are exact solutions to the two-body scattering, we can calculate exactly the two-body bound state and examine how it evolves from the low-dimensional to the 3D limits. This can be readily done by examining the pole of the two-body T-matrices in q1D and q2D.

First consider the bound-state equation in 3D. In practice there are three two-body bound states corresponding to corresponding to m = -1,0,1, two of which m = ± 1 are degenerate in experiments <cit.> due to the anisotropy of the dipole-dipole interaction. For our purposes we assume the interaction is isotropic so the three bound states are degenerate. In this case, the three-dimensional binding energy E=-E_B,3D for the two-body bound state is the solution of

    E_B,3D = R_p/2μ V_p + R_p/2μ(2μ E_B,3D)^3/2

for V_p > 0. In the shallow-dimer limit (R_p^3/2 |V|^-1/2≪ 1), the leading solution is . Also, for V_p < 0, a positive-energy scattering resonance exists at exactly -E_B,3D^0.  



 §.§ q1D Bound State


For the case of two-body bound states in the presence of harmonic confinement, it is convenient to introduce the bound state energies as: 

    E_B^(m) = (|m|+1) ω_⊥ - κ_m^2/2μ ,

where κ_m^2/2μ is the associated binding energy. The equations determining the bound state energies can then be written as:

    -E_B,3D^0 =E_B^(0)
    +6 R_p/a_⊥ω_⊥ζ(-1/2, (κ_0 a_⊥/2)^2 )  ,

for odd-wave interactions and:

    -E_B,3D^0    = E_B^(1)+3 R_p/a_⊥ω_⊥[ζ(-1/2, (κ_1 a_⊥/2)^2 ) . 
       . + (1 - (κ_1 a_⊥/2)^2) ζ(1/2, (κ_1 a_⊥/2)^2 )] ,

for even-wave interactions. The right hand side of Eqs. (<ref>-<ref>) increases monotonically as a function of the binding energy, κ_0,1^2/2μ. Hence there is always a bound state for V_p<0, which corresponds to a three-dimensional quasi-bound state. This is especially evident when one considers the limit: κ_0,1^2/2μ≫1. In this case one can show that the Hurwitz zeta functions for both p-wave and s-wave scattering yield a term that scales as κ_0,1^3. This is exactly the last term in Eq. (<ref>) (as κ^3 ∼ E_B^3/2), thus recovering the 3D equation for the two-body bound states.



On the other hand, in the limit of shallow bound states, i.e. when κ_0,1^2/2μ≪ 2ω_⊥, Eqs. (<ref>-<ref>) reduce to the expected results for a 1D system with p- and s-wave interactions. In the p-wave case, we find that the equation determining κ_0 and the p-wave binding energy is:

    1/a_p,1D- κ_0 - κ_0^2 r_p,1D=0 .

Similarly, for s-wave case, we have

    -a_s,1D + 1/κ_1 - κ_1^2 r_s,1D=0 .




Thus Eqs. (<ref>-<ref>) describe the behavior of the two-body bound state over the whole dimensional crossover. This is exemplified in Fig. <ref> which presents the results for the full dimensional crossover for the binding energy, i.e. Eqs. (<ref>-<ref>), alongside the limiting low-dimensional and 3D behaviours. As one can see, the full solution smoothly interpolates between both limiting behaviors as a function of the 3D scattering volume or the respective 1D scattering parameter.



 §.§ q2D Bound State

In q2D, the two-body bound state energies are defined as:

    E_B^(m) = (m+1/2)ω_⊥ -κ_m^2/2μ


it is beneficial to solve the T-matrix in a different approach. Eqs. (<ref>-<ref>) are the result of integrating over  all the virtual states in the unconfined two-dimensional plane, describing the scattering in q2D as a discrete set of contributions which come from the discrete nature of the states in the confined direction. 

To study the behavior of the bound state across the dimensional crossover, however, it is more practical to do the reverse, and obtain an expression for the q2D T-matrices which have integrated out the discrete transverse excitations, and leaving an expression in terms of the virtual q2D plane wave modes <cit.>. This is an alternative but identical representation of the q2D T-matrix. The details are discussed in Appendix <ref>. Here we present the equations for the binding energies:


    -E_B,3D^0/ω_⊥ = E_B^(0)/ω_⊥ + 3/4π^1/2R_p/a_⊥∫_0^∞ du     [ u e^-ℰ_0 u/λ u + 4 λ^21/2(1-λ)^2 e^-u + ℰ_0 ((1+λ)^2 - e^-u(1-λ)^2)/((1+λ)^2 - e^-u(1-λ)^2)^3/2. 
        - . 4/(u+ 4 λ)^5/2 - 8/31/4-ℰ_0/(u+4 λ)^3/2] ,




    -E_B,3D^0/ω_⊥ = E_B^(1)/ω_⊥ + 3/π^1/2R_p/a_⊥∫_0^∞ du [ e^-ℰ_1 u/u + 4 λ1/[(1+λ)^2 - e^-u(1-λ)^2]^3/2 - 1/(u+ 4 λ)^5/2 - 2/33/4 -ℰ_1/(u+4 λ)^3/2] ,


where  is the dimensionless binding energies. We also define λ=2(Λ a_⊥)^-2 with Λ as a cutoff parameter which in our model is given by R_p^-1. The left hand side of each equation is the leading low-energy binding energy in the 3D limit, while the first two terms on the right hand side describe the quasi-low-dimensional bound state relative to the zero-point energy. The integral on the left hand side contains two pieces: first is the contribution due to virtual excitations for a quasi-low-dimensional systems, and the second are terms represent the renormalization of the 3D interaction.

In the limit of small binding energies ℰ_0,1≪ 1, we expect the dominant contribution to the integrals in Eqs. (<ref>-<ref>) come from the virtual excitation term. An analysis of the integral in this q2D limit, described in Appendix <ref>, yields the expected low-dimensional expressions for the binding energies:

    κ_0^2 r_p,2D^2 ln(κ_0^2 r_p,2D^2)    = - r_p,2D^2/a_p,2D ,     m=0 ,
    1/κ_1^2 a_s,2D^2ln(1/κ_1^2 a_s,2D^2)    =  r_s,2D/a_s,2D^2 ,     m=1 .

Above we have used the definitions for the two-dimensional scattering parameters, Eqs. (<ref>-<ref>) and Eqs  (<ref>-<ref>). 
These equations can be solved using Lambert's W functions:

    κ_0^2    = -1/a_p,2D W_0^-1(-r_p,2D^2/a_p,2D), 
    κ_1^2    = 1/r_s,2D W_0(r_s,2D/a^2_s,2D).




In the opposite limit of deep dimers ℰ_s,p≫ 1, we expect that both the virtual excitation term and renormalization terms are important. As shown in Appendix <ref>, a similar analysis in this limit gives results consistent with the 3D prediction, Eq. (<ref>).

We present the numerical solution of Eqs. (<ref>-<ref>) in Fig. <ref>, alongside the predictions for the 3D and low-energy 2D theories. In these solutions the effective range was chosen to match the parameterization of the commonly used p-wave resonance in ^40K <cit.>. We find that the both the p-wave and s-wave q2D bound states smoothly cross over from the low- to high-dimensional results with the intermediate condition being ℰ_s,p∼ 1, or κ_s,pa_⊥∼ 1. Ref. <cit.> presents experimental evidence for both the p-wave and s-wave dimers in q2D ^40K, whose energies are as predicted by Eqs. (<ref>-<ref>). 



§ RF SPECTROSCOPY OF SPIN-POLARIZED FERMI GASES



We now apply the results for the two-body scattering to the calculation of the RF spectroscopy transfer rate in the high frequency limit. A RF pulse induces a hyperfine-Zeeman spin flip from state |a ⟩ to state |b ⟩ at a fixed detuning, ω. The transfer rate for this process is defined as <cit.>:

    Γ(ω) = Ω^2/4 Im i ∫_0^∞ dt ∫ d^3  R∫ d^3  r e^i(ω+i δ)t
       ×⟨ T_t  ψ_a^†ψ_b( R + 1/2 r, t)ψ_b^†ψ_a( R - 1/2 r, 0) ⟩ ,

where Ω is the Rabi frequency, and ψ_a^†( r) creates one fermions in hyperfine-Zeeman state a.



 §.§ Two-Body Solution

For large detuning, the RF pulse probes the short-distance physics of the many-body wave function which is dominated by two-body correlations. Thus we proceed to evaluate the RF transfer rate at the two-body level. For both q1D and q2D geometries, the RF transfer rate has a contribution from each value of m:

    Γ(ω) = ∑_m Γ^(m)(ω) ,

which can be evaluated independently. The corresponding RF transfer rate  at the two-body level is determined by the diagrams shown in Fig. <ref>. The two-body calculation is presented in  Appendix <ref>. Here we present the results for q1D:

    Γ^(0)(ω)    ≈Ω^2A_0(E_F,T)/8π(2ω_⊥)^3/2/ω^2
       ×∑_n=0^∞√(ω/2ω_⊥-n) θ(ω/2ω_⊥-n) 
     , 
    Γ^(± 1)(ω)    ≈Ω^2A_± 1(E_F,T)/16π(2ω_⊥)^3/2/ω^2
       ×∑_n=0^∞ (n+1)(ω/2ω_⊥-n)^-1/2 θ(ω/2ω_⊥-n) ,

where the constants A_m(E_F,T) are the sole fitting parameters which describe the many body physics, i.e. dependent on the Fermi energy, E_F, and the temperature, T. We have also assumed that all the atoms scatter in the lowest p-wave and s-wave scattering continua. Similarly in q2D one finds:

    Γ^(0)(ω)    ≈Ω^2A_0(E_F,T)/16√(π)(2ω_⊥)^3/2/ω^2
       ∑_n=0^∞(2n-1)!!/(2n)!!(ω/2 ω_⊥-n)θ(ω/2ω_⊥-n) ,
        
    Γ^(1)(ω)    ≈Ω^2 A_1(E_F,T)/16√(π)(2ω_⊥)^3/2/ω^2
       ∑_n=0^∞(2n+1)!!/(2n)!!θ(ω/2ω_⊥-n) .

We note that similar results were obtained for the case of q2D Fermi gases with s-wave interactions <cit.>.



Eqs. (<ref>-<ref>) are shown in Figs. (<ref>-<ref>) for a wide range of detunings. 
In the limit ω≪ 2ω_⊥, only the initial low-dimensional scattering continuum contributes to the RF transfer rate. In this limit one finds that the RF transfer rate is consistent with the true low dimensional calculation, as shown in Appendix <ref>, as shown by the blue lines in Figs. (<ref>-<ref>). This scaling appears to be robust all the way up to ω = 2ω_⊥, when a new low-dimensional scattering continuum becomes energetically available. The availability of a new scattering continuum leads to a non-monotonic behavior in the RF transfer rate, as is shown in Fig. <ref> for q1D, and Fig. <ref> for q2D. The scaling behavior of this discontinuity only depends on the low-energy behavior of the s- and p-wave  scattering, and the geometry of the system, as can be illustrated by Fermi's golden rule. 

Consider an RF spin flip from an initial state |i ⟩ = |k, n_i,m⟩ to a final state, |f⟩ = |k,n_f,m⟩. The matrix element for the RF spin flip operator, V_RF is related to the Franck-Condon factor, i.e. the overlap between the initial and final scattering states: |⟨ f | V_RF | i ⟩|^2 ∝ k^2(1-|m|). Given that the axial density of states scales like 1/k for 1D and k^0 for 2D, Fermi's golden rule states that the RF transfer rate is: ∝ k for p-wave scattering and ∝ k^-1 for s-wave scattering in q1D. Similarly the RF transfer rate is proportional to k^2 for p-wave scattering and k^0 for s-wave scattering in q2D. In the RF process, the excess kinetic energy is related to k^2/2μ =δω = ω - 2n_f ω_⊥. The universal discontinuity in the RF transfer rate then has the the form √(δω) for the p-wave sector and 1/√(δω) for s-wave interactions in q1D. In q2D these discontinuities are of the following form: δω for p-wave scattering and (δω)^0 for s-wave scattering. This discontinuity is directly captured in the two-body solutions, and can be clearly seen in Figs. (<ref>-<ref>).

In the 3D limit many effective low-dimensional scattering continua become open. In this limit we can replace the sum over transverse states with a continuous integral. In both q1D and q2D the RF transfer rates then converge on the three-dimensional result:

    Γ^(0)(ω)    = Ω^2 A_0(E_F,T)1/12π√(ω) ,  
    Γ^(± 1)(ω)    = Ω^2 A_±1(E_F,T) 1/12π√(ω) .

Eq. (<ref>) is consistent with the leading frequency behavior one would acquire from a 3D calculation with p-wave interactions, as detailed in Appendix <ref>, and is shown in Fig. <ref> by the dashed red dashed line.









 §.§ Operator Product Expansion


Up to this point we have only considered two-body correlation effects and their role on the RF transfer rate. To make this argument more rigorous, we can use the operator product expansion (OPE) to obtain the leading high-frequency behavior and the sub-leading tail. In the OPE we expand the time-ordered correlation function in a series of local operators. The local operators carry all the information of the many-body sate, while the series coefficients contain the information on the detuning ω <cit.>: 

    Γ(ω) = ∑_a W_a(ω) ⟨ O_a ⟩ ,

where ⟨ O_a⟩ is the thermodynamic average of a local operator, O_a, and W_a(ω) is the corresponding Wilson coefficient. This process is most easily done in the low-dimensional and 3D limits. 

The operators relevant to our discussion are the contact operators in  D dimensions, which describe how the energy changes with respect to the low-energy scattering parameters. The d-dimensional s- and p-wave contact operators are summarized in Appendix <ref>. In 3D, there are two contact operators:

    C_V_p   = -∂ H/∂ V_p^-1 ,     C_R_p   = -∂ H/∂ R_p^-1 .

As discussed in Appendix <ref>, one can evaluate the expectation values of the contact operators and compare to the two-body solutions of the RF-transfer rate, Eq. (<ref>-<ref>) to extract the Wilson coefficients. In the 3D limit, it is straight forward to show that both for q1D and q2D:

    .Γ(ω)|_3D≈2Ω^2/√(ω)[⟨ C_V_p⟩ + 3/2ω⟨ C_R_p⟩] ,

which is the correct 3D result. In the low-dimensional limit, one obtains:

    .Γ^(0)_p(ω)|_1D   ≈Ω^2/2ω^3/26/a_⊥^2[ ⟨ C_V_p^(0)⟩. 
       . + 1/2ω( ⟨ C_R_p^(0)⟩ - 2/a_⊥^2⟨ C_V_p^(0)⟩) ] , 
    .Γ^(± 1)(ω)|_1D   ≈Ω^2/4ω^5/224/a_⊥^4⟨ C_V_p^(± 1)⟩ ,

for q1D systems, and similarly for q2D systems:

    .Γ^(0)(ω)|_2D   = πΩ^2/2ω3/2 √(π) a_⊥[ ⟨ C_V_p^(0)⟩. 
       .+ 1/ω( ⟨ C_R_p^(0)⟩ - 1/a_⊥^2⟨ C_V_p^(0)⟩) ] , 
    . Γ^(1)(ω)|_2D   = πΩ^2/4ω^212/√(π)a_⊥^3⟨ C_V_p^(1)⟩ .


Eqs. (<ref>-<ref>) describe the low-dimensional RF transfer rate in terms of the 3D contacts with definite angular momentum (parity) m: C_V_p^(m) and C_R_p^(m). The 3D contacts can then be connected to the low-dimensional ones using the definitions in Eqs. (<ref>-<ref>) for the q1D scattering parameters and Eqs. (<ref>-<ref>) for the q2D scattering parameters. The definitions for the low-dimensional contacts are presented in Appendix <ref>. Here we simply quote the results for q1D systems:

    C_V_p^(0)   = a_⊥^2/6 C_a_p,1D ,
    
    C_R_p^(0)   = 1/3 C_a_p,1D + a_⊥^2/6 C_r_p,1D ,
    
    C_V_p^(± 1)   = a_⊥^4/12 C_a_s,1D , 
    
    C_R_p^(± 1)   = a_⊥^2/3 C_a_s,1D + a_⊥^4/12C_r_s,1D ,

and for q2D systems:

    C_V_p^(0)   = 2 √(π)a_⊥/3 C_a_p,2D ,
    
    C_R,p^(0)   = 2√(π)/3a_⊥ C_a_p,2D + 2 √(π)a_⊥/3 C_r_p,2D ,
    
    C_V_p^(± 1)   = √(π)a_⊥^3/6 C_a_s,2D , 
    
    C_R_p^(± 1)   = √(π)a_⊥^2/2 C_a_s,2D + √(π)a_⊥^3/6C_r_s,2D .

Given the above equations, Eqns. (<ref>-<ref>), one can show that the RF transfer rate collapses onto the desired result of a true low-dimensional system, see Appendix <ref>:

    Γ_p,1D(ω)    = Ω^2/2ω^3/2[⟨ C_a_p,1D⟩ + 1/2ω⟨ C_r_p,1D⟩] , 
    Γ_p,2D(ω)    = πΩ^2/2ω[⟨ C_a_p,2D⟩ + 1/ω⟨ C_r_p,2D⟩] , 
    Γ_s,1D(ω)    = 2Ω^2/4ω^5/2⟨ C_a_s,1D⟩ , 
    Γ_s,2D(ω)    = 2πΩ^2/4ω^2⟨ C_a_s,2D⟩ .

The only difference is that the RF transfer rate for the even-wave interactions is twice that of a true s-wave system, which comes from the indistinguishable nature of the spin-polarized fermions.



§ CONCLUSIONS



In this work we have analyzed the p-wave scattering in the presence of strong harmonic confinement potentials, typically used to create q1D or q2D systems. We found that there were two effective low-dimensional interactions, one that is p-wave and the other that is s-wave in character, and we solved the corresponding two-body problems. Both interactions are necessary in understanding the dimensional crossover in RF spectroscopy experiments. At the two-body level we found a complete description of the dimensional crossover, which is shown in Fig. <ref> for the 1D-to-3D crossover, and in Fig. <ref> for the 2D-to-3D crossover. To make these results more rigorous we used the OPE to relate the leading and sub-leading high-frequency power-law tails to the thermodynamic contacts in the low-dimensional and 3D limits.

These results can be tested experimentally. Already in Ref. <cit.> a clear signal was observed in the RF spectroscopy indicating the opening of a new s-wave scattering channel for a spin-polarized Fermi gas of ^40K atoms in a q1D trapping geometry, matching Eq. (<ref>). The experimental observation of q2D s-wave channel is reported in Ref. <cit.>. 
The non-monotonic behavior of the RF transfer function is quite general and depends only on the form of the interaction and the geometry of the system. Similar results were obtained for the RF spectroscopy of a q2D Fermi gas with s-wave interactions <cit.>. 

Our analysis is quite general and can be also applied to a number of different scenarios. In particular understanding the dimensional crossover from spectroscopic probes should be possible in, for example, the structure factor at zero momentum, or the closed-channel fraction. This calculation can also be readily extended to study other dimensional crossovers with other higher partial wave interactions.


A particularly interesting result is the presence of different types of low-dimensional scattering, i.e. scattering with different parity or angular momentum, at higher energies. This opens up a rich new area of physics where atomic gases can occupy a finite number of transverse harmonic oscillator states. In such systems, the different low-dimensional interactions can compete, and can result in strong correlations, as observed in Ref. <cit.>.




  
*Acknowledgements– The authors would like to thank Kevin G. S. Xie for useful discussions. This work is supported by HK GRF Grants No. 17306024, CRF Grants No. C6009-20G and No. C7012-21G, and a RGC Fellowship Award No. HKU RFS2223-7S03. JM is supported in part by the Provincia di Autonoma di Trento.




equationsection






§ SOLUTION OF LIPPMANN-SCHWINGER EQUATION



In this appendix, we solve the Lippmann-Schwinger equation in the presence of the external trapping potential that is described by the Hamiltonian given by Eq. (<ref>). The inter-atomic potential v(r) is given by the p-wave pseudo-potential, Eq.(<ref>). The derivation presented here follows closely that of Ref. <cit.>.

We begin with the definition of the T-matrix in free space:

    T_3D(E) = v + v G_3D^0(E) T_3D(E) ,

where G_3D^0(E) =(E-H_0+iδ)^-1 is the bare propagator in three-spatial dimensions without confinement. Here H_0 is simply the kinetic energy. Given the low-energy T-matrix, one can solve for the bare interaction:

    v^-1=T_3D(E)^-1+G_3D^0 .

The bare interaction can similarly be solved in terms of the T-matrix in the presence of the confinement T(E):

    v^-1=T(E)^-1+G^0(E) ,

where now G^0(E) is the bare propagator in the pressence of the confinement potential H_t: G^0(E) = (E-H_0-H_t+iδ)^-1. Eliminating v^-1 which is identical in both free space and confined scattering, one obtains T_3D(E)^-1+G_3D^0=T(E)^-1+G^0(E). Re-arranging, for T(E) one can relate the scattering from some interaction v inside a trapping potential to scattering to the same scattering in free space: 

    T(E) = T_3D(E) + T_3D(E)(G^0(E) - G_3D^0(E) )T(E) .

This result was first derived in Ref. <cit.>.

For p-wave scattering it is possible to simplify Eq. (<ref>) since both the 3D free space T-matrix and the T-matrix inside the harmonic trapping potential are separable:

    T_(3D)(E) = ∇·T̃_(3D)(E) ·∇ ,

where the (3D) T-matrix has been written in position space, and the arrows denote whether the gradient is applied to the bra or the ket. Eq. (<ref>) is consistent with the matrix elements of the pseudo-potential. Inspired by the separable form of the 3D T-matrix, we propose the following ansatz that solves Eq. (<ref>):

    T(E)    = ∇·T̃(E) ·∇ , 
    T̃(E)^-1   = T̃_3D^-1
       - lim_ r,  r'→ 0∇_ r·∇_ r'[G^0(E, r, r')-G^0_3D(E, r, r')] .

From here it is necessary to consider the specific structure of the trapped and free-space propagator. First consider q1D scattering. The non-interacting propagator for a q1D system:

    G^0(E; r_∥, r_⊥; r_∥',  r_⊥')    =  
    - 2μ a_⊥/4∑_m=-∞^∞∑_n=0^∞ϕ_n,m( r_⊥) ϕ_n,m^*( r_⊥')/√(n - ℰ_m- i δ)
       × e^-2/a_⊥√(n - ℰ_m- i δ)|r_∥-r_∥'| ,

where ℰ_m = (E-(|m|+1)ω_⊥)/(2ω_⊥), and ϕ_n,m( r_⊥) is the wavefunction of a 2D harmonic oscillator, see Eq. (<ref>). To make a direct connection to the trapped propagator, the free-space propagator must also be expanded in terms of modes with definite angular momentum, m:

    G_0(E; r_∥, r_⊥; r_∥',  r_⊥') =-2μ/4L∑_m= -∞^∞∑_n=0^∞k_n,m/√(k_n,m^2 - 2μ E- i δ)
       × e^-√(k_n,m^2 - 2μ E - i δ)|r_∥-r_∥'|
       × J_|m|(k_n,mr_⊥)J_|m|(k_n,mr_⊥)e^i m (ϕ-ϕ') ,

where J_m(x) is the mth Bessel function, and k_nm(2n+1 + |m| +1/2) π/2L, L is the length of the system, and r_⊥ = (r_⊥, ϕ). 

From Eqs. (<ref>, <ref>) one can then show in the limit r,  r' → 0:

    ∇_ r·   ∇_ r'[G^0(E, r, r')-G^0_3D(E, r, r')] = 
       2μ/π a_⊥^3δ_m,0[∑_n=0^N √(n- ℰ_0 - i δ) -2/3N^3/2] 
    
    -   2μ/2 π a_⊥^3δ_|m|,1∑_n=0^N [ √(n-ℰ_1- i δ) - 2/3N^3/2. 
       . + E/2ω_⊥( 1/√(n-ℰ_1-i δ) - 2N^1/2) ] ,

where N is a UV cutoff which should be taken to infinity. The sums in Eq. (<ref>) are regularized and are equivalent to the Hurwitz zeta functions, ζ(s,x) <cit.>. This gives the final result, Eqs. (<ref>-<ref>), for the T-matrix inside the harmonic trapping potential.

The calculation for a q2D system is identical to the q1D case, except that one now uses the q2D basis functions, Eq. (<ref>) to construct the propagator in the presence of the trap. The free-space propagator regularizes the leading divergent behavior of the q2D T-matrix, producing Eqs. (<ref>-<ref>). However, we find that for low-dimensional p-wave interactions, there is still a divergence present that is not regularized by the 3D interaction. This is due to our approximation for the low-dimensional T-matrix following Eq. (<ref>). This additional divergence is identical to the one expected for a true low-dimensional p-wave interaction and can then be regularized by examining the effective low-dimensional scattering at low energies. This additional divergence does not appear if one examines the dimensional crossover using the two-body wavefunction <cit.>. However we note that both the T-matrix and wavefunction approaches give consistent results once we fixed this spurious divergence.



§ ALTERNATIVE SOLUTION TO THE T-MATRIX FOR Q2D





 §.§ Deriving the Equations for the Binding Energy


In this section, we consider an identical way of solving the two-body T-matrix following Ref. <cit.>. In this method we use a modified pseudo-potential which in 3D has the form:

    ⟨ k | v |  k'⟩=2g/L^2e^-k^2+k'^2/Λ^2 k · k' ,

with Λ acting as a soft UV cutoff, compared to Eq. (<ref>) which uses a hard UV cutoff. Solving for the two-body T-matrix in 3D the bare coupling constant g can be related to the 3D scattering parameters:

    1/g = 2μ/24π[ (1/V_p - Λ^3/4√(2π)) + (1/R_p - Λ/√(2π)) 2μ E] ,

where V_p and R_p are the 3D p-wave scattering volume and effective range, respectively. Once these parameters are set, the physics should be identical to the zero-ranged pseudo-potential used in the main text.

From this modified pseudo-potential, the q2D geometry can be readily evaluated:

    ⟨ k, n,m| v |  k',n',m'⟩ = 2g/L^2f_n,mf_n',m' ( k · k')^1-mδ_m,m' ,

where m=0,1. The constants f_n,m are related to the transverse wavefunction and its derivative evaluated at the origin ψ_n,m(0):

    f_n,m= ∑_k_⊥e^-k_⊥^2/Λ^2k_⊥^mψ_n,m(k_⊥) .


The benefit of this representation is that f_n,m can be evaluated exactly:

    f_n,m   = (-1)^n (-i)^m 2^m/2/π^1/4a_⊥^(2m+1)/2√((2n+m)!/2^2n(n!)^2)
       1/(1+λ)^(2m+1)/2(1-λ/1+λ)^n ,

with λ = 2/(Λ^2 a^2_⊥). From here the two-body scattering T-matrix following Eq. (<ref>) can be written as:

    T_m^-1 = 1/g   - 1/2∑_n=0^∞∫d^2p/(2π)^2(p^2/2)^1-m
       |f_n,m|^2 e^-2p^2/Λ^2/E-p^2/2μ - (2n+|m|+1/2)ω_⊥ .

In Eq. (<ref>) g is defined in Eq. (<ref>), μ is the reduced mass and a_⊥^2 = (μω_⊥)^-1. When investigating the two-body bound state:  with binding energy κ_m^2/(2μ). In this case, the denominator of Eq. (<ref>) is positive semidefinite, and allows for an analytical summation over the transverse quantum states. Namely, by denoting: u = p^2/(2μ)/(κ^2_m/(2μ)+2nω_⊥), ℰ_m = κ_m^2/(4μω_⊥) and using the relations:

    ∑_n=0^∞(2n)!/(n!)^2x^n    = 1/√(1-4x) ,
    ∑_n=0^∞(2n)!/(n!)^2 n x^n    = 2x/(1-4x)^3/2 , 
    ∑_n=0^∞(2n+1)!/(n!)^2x^n    = 1/(1-4x)^3/2 ,

one can obtain Eqs. (<ref>-<ref>).



 §.§ Low-Dimensional Limits of the Binding Energies


Next we consider the limiting behaviours of Eqs. (<ref>-<ref>). In order to discuss the limits, we will focus on the behaviour of the virtual excitation term at small and large binding energies, corresponding to the low- and high-dimensional limits. To that end let us define:

    J_0    = ∫_0^∞du/λue^-ℰ_0 u/4λ + u1/η^3/2[ℰ_0 η + 1/2e^-u(1-λ)^2] , 
    
        J_1    = ∫_0^∞ du e^-u/4λ+u1/η^3/2 ,

where η =(1+λ)^2-e^-u (1-λ)^2.

In the low-dimensional limit, ℰ_0,1≪ 1, we expect that the regularization terms are not important as they regulate the high energy, or 3D, physics. In this case we can simplify the integral by noting that the virtual excitation term for small binding energies is weighted at large u, and as such: η∼ 1. Restricting the region of integration to u>4λ allows one to evaluate the integral analytically. Then taking the limit λ→ 0 we find:

    J_0    ≈1/λ -4ℰ_0ln(e^-γ_E/4λℰ_0) , 
    
        J_1    ≈ln(e^-γ_E/4λℰ_1) .

We note that the q2D p-wave term J_0 has an additional divergence of λ^-1. As discussed previously, this is exactly the divergence one would expect for a true 2D p-wave system, and is an artifact of our solution to the T-matrix. As such, this term is required to regularize the q2D T-matrix, and can be safely removed. The resulting terms are the correct energy dependencies for true 2D p-wave and s-wave systems respectively. Substituting these expressions into Eqs. (<ref>-<ref>) yield the correct low-dimensional equations for the binding energies. 

In the opposite limit ℰ_0,1≫ 1 we expect to recover the 3D equation for the binding energy. In this case, we expect the virtual excitation term to produces divergences that will cancel the regularization terms in Eqs. (<ref>-<ref>), i.e. terms that scale as λ^-3/2 and λ^-1/2, while the leading regular term scales as ℰ_0,1^3/2.  In this limit the dominant contribution in Eqs. (<ref>-<ref>) is at small u. For the s-wave case, one can approximate η≈ u to find 

    J_1 ≈1/121/λ^3/2 - ℰ_1/λ^1/2 + 4√(π)/3ℰ_1^3/2

which gives the correct 3D result when substituted into Eq. (<ref>). An analogous analytic expression for the limiting case of J_0 was not found, however the full numerical solutions of Eqs. (<ref>-<ref>) shown in Fig. <ref> confirm that the bound states do indeed approach the 3D solution of Eq. (<ref>). 



§ TWO-BODY RF AND OPERATOR PRODUCT EXPANSION


Consider an initial state |Q, N, M, k,n,m⟩ with Q,N,M being the center of mass momentum, principal transverse quantum number, and the transverse angular momentum (or parity), respectively. Similarly k,n,m correspond to the quantum numbers for the relative momentum, principal transverse quantum number, and the transverse angular momentum (or parity). The main contribution to the RF spectroscopy at the two-body level comes from Fig. <ref>, where the black circles are the two-body T-matrices in Eq. (<ref>-<ref>), and the open circles are a) the operator insertions from Eq. (<ref>) or b) the contact operators. This is equally valid for both q1D and q2D. For simplicity of the presentation, we will ignore the center of mass coordinate and provide explicit calculations for the case of q1D. The extension to q2D is straightforward with minimal change. 

The first step in our OPE calculation is to evaluate Eq. (<ref>) using our two-body state. For our chosen initial state and q1D scattering, the diagram in Fig. <ref>(a) has the form:

    I_m(ω)    ≈ -∑_n'∫dℓ/2π[f_n^(m)(k) T^(m)(E)]^2 (f_n'^(m)(ℓ))^2 
       ×1/E+ω - E_n',m(k) + i δ1/(E- E_n',m(k) + i δ)^2 ,

where we have performed the internal frequency integral, and we have also defined the form factors: f_n^(0)(k) = k √(1 /(π a_⊥^2)) and f_n^(± 1) = √(2 /(π a_⊥^4)). Here we also  use T^(m) are the solutions to the Lipmann-Schwinger equation with definite angular momentum (parity) m.

The high-frequency tails of the RF transfer rate arise from the  non-analytic terms in Eq. (<ref>). From the non-analytic terms in Eq. (<ref>), one finds the two-body solution for the RF transfer rate:

    Γ^(0)   ≈[f_n^(0)(k) T^(0)(E)]^2  
       ×∑_n 1/8π(2ω_⊥)^3/2/ω^2√(ℰ_0-n) θ(ℰ_0 -n) ,

for p-wave interactions (m=0), and for s-wave interactions (m=± 1):

    Γ^(±1)   ≈[f_n^(±1)(k) T^(±1)(E)]^2  
       ×∑_n n+1/16π(2ω_⊥)^3/2/ω^21/√(ℰ_1-n)θ(ℰ_1 -n) ,

where again  ℰ_m = (E - (|m|+1) ω_⊥)/(2ω_⊥).

Eqs. (<ref>-<ref>) are the same as Eqs. (<ref>-<ref>), where we have replaced the factor that depends on the initial state, [f_n^(m)(k) T^(m)(E)]^2, with a constant that depends on the many-body physics, A_m(E_F,T).

The second step of the OPE is to connect this two-body calculation to the expectation value of local operators. The operators of interest to us are the 3D contact operators discussed in Appendix <ref>. To facilitate our calculation, we can express the contact operators in the basis of two-body states in the presence of transverse confinement. It is straightforward to show the contact operators have the form:

    C_V_p =∑_m  C_V_p^(m)   =∑_m ∑_Q,N,M∑_k,n∑_k',n'
       g^2/12πf_n^(m)(k) f_n'^(m)(k')(Φ_Q,N,M^k,n,m)^†Φ_Q,N,M^k',n',m , 
    
    C_R_p =∑_m  C_R_p^(m)   =∑_m ∑_Q,N,M∑_k,n∑_k',n'
       (E-Q^2/4-(2N+|M|+1)ω_⊥)
       g^2/12πf_n^(m)(k) f_n'^(m)(k')(Φ_Q,N,M^k,n,m)^†Φ_Q,N,M^k',n',m ,

where g is the p-wave coupling strength, f_n^(0)(k) = √(1/π a_⊥^2)k and f_n^(± 1) = √(2(n+1)/π a_⊥^4) are the q1D form factors. The operator  Φ_Q,N,M^k,n,m is the annihilation operator for a two-body state with energy E, center of mass quantum numbers Q,N,M, and relative quantum numbers k,n,m.

At the two-body level, the calculation of the expectation value of the contact operators are related to the diagrams in Fig. <ref>)(b). The result is:

    ⟨ C_V_p⟩   = ∑_m 2/12π[f_n^(m)(k) T^(m)(E)]^2 , 
    ⟨ C_R_p⟩   = ∑_m 2/12π[f_n^(m)(k) T^(m)(E)]^2 E .


Eq. (<ref>) matches the leading and subleading energy dependence of Eqs. (<ref>-<ref>). From there it is possible to extract the Wilson coefficients, W(ω), for the scattering volume and effective range contacts. This is most readily done in the 1D limit, ω≪ 2ω_⊥, and in the 3D limit, ω_⊥≪ω. In the 1D limit the Wilson coefficients depend on the angular momentum, m:

    C_V_p^(0):     W(ω)    =  Ω^2/2ω^3/26/a_⊥^2(1- 1/a_⊥^2ω) ,  
       C_V_p^(± 1):      W(ω)    = Ω^2/4ω^5/224/a_⊥^4 , 
       C_R_p^(0):     W(ω)    = Ω^2/2ω^5/23/a_⊥^2 ,

Conversely, in the 3D limit, the Wilson coefficients are isotropic:

    C_V_p^(m):     W(ω)   = Ω^2/2ω^1/2 ,
       C_R_p^(m):     W(ω)   = 3Ω^2/4ω^3/2 ,

as expected in the true 3D limit. The Wilson coefficients in Eqs. (<ref>-<ref>) produce the 1D and 3D results.

The calculation described above can be readily extended to the case of q2D scattering. There one finds the following Wilson coefficients in the 2D limit (ω≪ 2ω_⊥):

    C_V_p^(0):     W(ω)    =  πΩ^2/2ω3/2√(π)a_⊥(1-1/a_⊥^2ω)  , 
       C_V_p^(1):      W(ω)    =  πΩ^2/4ω^212 /√(π) a_⊥^3 , 
       C_R_p^(0):     W(ω)    = πΩ^2/2ω^23/2 √(π) a_⊥ .

In the 3D limit (ω≫ω_⊥), the Wilson coefficients are the same as those in Eq. (<ref>), recovering the true 3D result.



§ REVIEW OF D-DIMENSIONAL CONTACTS AND RF SPECTROSCOPY


In this section we review the two-body scattering for d-dimensional gases. We also quote the relevant definitions for the contact operators, and the RF transfer function. For a d-dimensional gas with s- and p-wave interactions we consider the following Hamiltonians, H_s and H_p, respectively:

    H_s    = ∫ d^d rψ_σ^†( r) (-1/2∇_ r^2 ) ψ_σ( r) 
       + g ∫ d^d rψ_↑^†( r)ψ_↓^†( r) ψ_↓( r)ψ_↑( r) , 
    
    H_p    = ∫ d^d rψ^†( r) (-1/2∇_ r^2 ) ψ( r) 
       - g/2∫ d^d rψ^†( r)∇_ rψ^†( r) ψ( r)∇_ rψ( r) .

In defining H_s and H_p we let ψ_(σ) is the annihilation operator for the (spin-1/2) spin-polarized fermions, g is the (s) p-wave interaction strength, and we have defined: ∇_ r = (∇_ r-∇_ r)/2. We have also set ħ = m = 1. These Hamiltonians are consistent with the pseudo-potential shown in Eq. (<ref>).

The low-dimensional T-matrices are defined in Eqs. (<ref>, <ref>, <ref>, <ref>), which depends on the low-dimensional scattering parameters: a_s,dD the s-wave scattering length and r_s,dD the s-wave effective range, a_dD,p, the p-wave scattering volume, and r_dD,p the p-wave effective range.




 §.§ Thermodynamic Contacts

The low-dimensional contacts are defined in terms of the change of energy with respect to the low-dimensional scattering parameters. For s-wave interactions the contacts are:

    d   = 1     C_a_s   = ∂ H_s/∂ a_s    C_r_s   = -∂ H_s/∂ r_s
    
        d   = 2     C_a_s   = -∂ H_s/∂ln(a_s^-2)    C_r_s   = -∂ H_s/∂ r_s

while for p-wave interactions <cit.>:

    d   = 1     C_a_p   = -∂ H_p/∂ a_p^-1    C_r_p   = -∂ H_p/∂ r_p
    
        d   = 2     C_a_p   = -∂ H_p/a_p^-1    C_r_p   = -∂ H_p/∂ln(r_p^-2)
    
        d   = 3     C_a_p   = -∂ H_p/∂ V_p^-1    C_r_p   = -∂ H_s/∂ R_p^-1

In Eqs. (<ref>-<ref>) we have muted the dimensional index on the scattering parameters for transparency. Microscopically, the contacts in d dimensions can be succinctly written as:

    C_a_s   = g^2/𝒜_s∫ d^d rψ_↑^†( r)ψ_↓^†( r) ψ_↓( r)ψ_↑( r) , 
    
    C_r_s   = g^2/𝒜_s∫ d^d rψ_↑^†( r)ψ_↓^†( r)[i ∂_t - H_cm]ψ_↓( r)ψ_↑( r) , 
    
    C_a_p   = -g^2/𝒜_p∫ d^d r1/2ψ^†( r)∇_ rψ^†( r) ψ( r)∇_ rψ( r) , 
    
    C_r_p   =  -g^2/𝒜_p∫ d^d r1/2ψ^†( r)∇_ rψ^†( r)[i ∂_t - H_cm]ψ( r)∇_ rψ( r) ,

where H_cm is the center of mass momentum. The explicit definitions of the contacts and the values of 𝒜_s and 𝒜_p are tabulated in Tab. <ref>. 





 §.§ RF Spectroscopy

The RF transfer rate is microscopically defined in Eq. (<ref>). The large detuning tails of the RF transfer rate can be obtained using the operator product expansion (OPE) technique <cit.>. Here we simply quote the relevant results for s-wave interactions <cit.>:

    d   = 1    Γ_s(ω)    = Ω^2/4ω^5/2⟨ C_a_s⟩
    
        d   = 2    Γ_s(ω)    = πΩ^2/4ω^2⟨ C_a_s⟩

and for p-wave interactions <cit.>:

    d   = 1    Γ_p(ω)    = Ω^2/2ω^3/2[⟨ C_a_p⟩ + 1/2ω⟨ C_r_p⟩] 
    
        d   = 2    Γ_p(ω)    = πΩ^2/2ω[⟨ C_a_p⟩ + 1/ω⟨ C_r_p⟩] 
    
        d   = 3    Γ_p(ω)    = Ω^2/2ω^1/2[⟨ C_a_p⟩ + 3/2ω⟨ C_r_p⟩]



 



 Paper Title: Nonlinearity-induced dynamical self-organized twisted-bilayer lattices in Bose-Einstein condensates.





























#15pt#1

    GThese authors contribute equally to this work.Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaThese authors contribute equally to this work.Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, Chinalishuai0999@stu.xjtu.edu.cnMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, Chinaliubophy@gmail.comMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, China§ ABSTRACT 
        Creating crystal bilayers twisted with respect to each other would lead to large periodic supercell structures, which can support a wide range of novel electron correlated phenomena, where the full understanding is still under debate.
        Here, we propose a new scheme to realize a nonlinearity-induced dynamical self-organized twisted-bilayer lattice in an atomic Bose-Einstein condensate (BEC).
        The key idea here is to utilize the nonlinear effect from the intrinsic atomic interactions to couple different layers and induce a dynamical self-organized supercell structure, dramatically distinct from the conventional wisdom to achieve the static twisted-bilayer lattices. To illustrate that, we study the dynamics of a two-component BEC and show that the nonlinear interaction effect naturally emerged in the Gross-Pitaevskii equation of interacting bosonic ultracold atoms can dynamically induce both periodic (commensurable) and aperiodic (incommensurable) moiré structures. One of the interesting moiré phenomena, i.e., the flat-band physics, is shown through investigating the dynamics of the wave packet of BEC. Our proposal can be implemented using available state-of-the-art experimental techniques and reveal a profound connection between the nonlinearity and twistronics in cold atom quantum simulators.
        Nonlinearity-induced dynamical self-organized twisted-bilayer lattices in Bose-Einstein condensates
            Bo Liu
            August 5, 2024
        ===================================================================================================
        
        
        
        § INTRODUCTION
        
        Superimposed crystal bilayers twisted with respect to each other provide a new knob to unveil a wide
        range of novel electron correlated phenomena <cit.>.
        In particular, moiré patterns associated  with small twist angles result in various novel physical properties <cit.>, including the flat band <cit.>, fractional Chern insulators <cit.>, quantum Hall effect <cit.>, moiré ferroelectricity <cit.>, exotic magnetism <cit.>, moiré excitons <cit.>, non-Abelian gauge potentials <cit.>, and unconventional superconductivity <cit.>.
        In parallel to these developments in solid state materials, examples of twisted-bilayer are also studied in other artificial quantum systems, such as in optics <cit.>, acoustics <cit.> and cold atoms  <cit.>.
        In photonic lattices, the moiré pattern and its induced phenomena, such as the localization of light and flat band physics, have been studied in the photorefractive medium <cit.> and nanostructured photonic moiré superlattices <cit.>.
        
        Ultracold atoms in optical lattices are another one of the most promising platform to
        investigate the remarkable physical properties of twisted artificial structures <cit.>.
        It thus has stimulated tremendous amount of efforts,  including theoretically predicting moiré physics in ultracold fermions <cit.> and bosons  <cit.>. In particular,  experimental breakthrough observation of the spatial moiré pattern and momentum diffraction of a Bose-Einstein condensate (BEC) in twisted bilayer optical lattices has recently been achieved <cit.>.
        However, the conventional wisdom in past studies focuses on achieving the static twisted-bilayer lattices <cit.>.
        How to realize such twisted lattice structures in a dynamical way has so far eluded efforts.
        
        
        
        In this work, we propose a new scheme to dynamically achieve twisted-bilayer lattices that builds on the concept of nonlinearity. Our scheme relies on the  nonlinear interaction effect, which naturally emerges in the Gross-Pitaevskii equations of interacting bosonic ultracold atoms. We find that such nonlinear interaction effect amazingly results in a dynamical self-organized twisted-bilayer lattice. This new scheme frees out the usual requirement of adding external single-particle type coupling between different layers and only relies on the nonlinear effect from the intrinsic atomic interactions,  dramatically distinguished from previous schemes  <cit.>. We show that both dynamical self-organized periodic (commensurable) and aperiodic (incommensurable) twisted-bilayer lattice structures can be realized through our proposed scheme. In particular, for the aperiodic case, we find the localization of the Gaussian BEC wave packet, resulting from the emergence of the flat bands in the nonlinearity-induced self-organized twisted-bilayer lattices. It indicates that our proposed nonlinear interaction effect can dynamically produce the tunable moiré lattices in BEC.
        
        
        
        § EFFECTIVE MODEL
        
        The system considered here is schematically illustrated in Fig.<ref>. A two-component BEC, for instance ^87Rb atoms, is used,  where a spin-dependent harmonic potential is applied <cit.>. Atoms in component 1 are trapped in a tight harmonic trap and the atoms in component 2 are loaded in a square optical lattice confined by a very shallow harmonic trapping potential. After the system reaches equilibrium, the tight harmonic trap for component 1 atoms is tuned to be very shallow and a square optical lattice is added. At the same time, the inter-species interaction is turned on. Such a system can be described by the following model Hamiltonian
        
            𝐇   =   ∫dxdy ∑_α=1,2ψ_α^†[ 𝐤^2/2m+V_trap,α(x,y)+V_α(x,y) ] ψ_α
               +   ∫dxdy(g_11n̂_1^2+g_22n̂_2^2+2g_12n̂_1n̂_2),
        
        where ψ_α(α=1,2) denote the Bose field operators for two-component bosons, respectively. The corresponding density operators  are defined as n̂_α=ψ^†_αψ_α. V_trap,α denotes the harmonic trap. The spin-dependent optical  lattices are  V_1(x,y)=V_1[sin^2(cosθ/2k_Lx-sinθ/2k_Ly)+sin^2(sinθ/2k_Lx+cosθ/2k_Ly)] and  V_2(x,y)=V_2[sin^2(cosθ/2k_Lx+sinθ/2k_Ly)+sin^2(-sinθ/2k_Lx+cosθ/2k_Ly)], respectively, with θ being the twist angle. k_L=2π/λ is the wavevector
        of the laser field with the corresponding lattice constant defined as a_L=π/k_L, and λ is the wavelength of the laser. V_α captures the corresponding lattice depth.
        g_11 and g_22 characterize the intra-species interaction strength for component 1 and  component 2 atoms, respectively. The inter-species interaction strength is captured by g_12.
        
        From Eq. (1), the following
        dimensionless energy functional can be constructed under the Gross-Pitaevskii (GP) mean-field theory
        
            ε   =   ∫dx'dy' ∑_α=1,2ψ_α^* [ -∇^2/2+V'_trap,α(x',y')+V'_α(x',y') ] 
               ×   ψ_α +∫dx'dy'(g'_11|ψ_1|^4+g'_22|ψ_2|^4+2g'_12|ψ_1|^2|ψ_2|^2),
        
        where dimensionless spatial coordinates are defined as x'=x/λ and y'=y/λ, respectively. V'_trap,α=V_trap,α/(ħ^2/mλ^2) represents the harmonic trapping potential. V'_α=V_α/(ħ^2/mλ^2) labels the dimensionless lattice depth. g'_11(12,22)=g_11(12,22)/(ħ^2/m) are the corresponding dimensionless interaction strength.
        The wavefunction is normalized as ∬ |ψ_α|^2dx'dy'=N_α with N_α being the particle number of the two-component atoms, respectively.
        Based on Eq. (2), the dynamics of the system can be well described by the following coupled GP equations
        
            i∂ψ_1/∂ t' =    [ -1/2∇^2+V'_trap,1(x',y')+V'_1(x',y')+g'_11|ψ_1|^2 
               +g'_12|ψ_2|^2 ] ψ_1  
            
            			i∂ψ_2/∂ t' =   [ -1/2∇^2+V'_trap,2(x',y')+V'_2(x',y')+g'_12|ψ_1|^2 
               +g'_22|ψ_2|^2 ] ψ_2,
        
        where t'=t/(mλ^2/ħ) denotes the  dimensionless evolution time.
        
        
        § MOIRÉ LOCALIZATION INDUCED BY THE  NONLINEAR INTERACTION EFFECT
        
        In the following, we will demonstrate that distinguished from previous studies <cit.>, even without
        external single-particle type interlayer coupling,  the existence of nonlinear interaction term in GP equation, i.e., w=g'_11|ψ_1|^2
        +g'_12|ψ_2|^2, can dynamically result in the twisted-bilayer lattices and  lead to one of the interesting moiré phenomena.
        To show that, we monitor the dynamics of BEC for different twist angles θ. Here the initial state for component 1 atoms is the ground state of the harmonic trap, which is chosen as a Gaussian wave packet (left part in Fig. <ref>(a)). While for component 2 atoms, the initial state is chosen as the ground state (right part in Fig.<ref>(a)) determined by the effective Hamiltonian in the right hand-side of Eq. (<ref>) with g'_12=0.
        
        Interestingly, as shown in Fig. <ref>, when tuning the twist angle θ, in the presence of nonlinear term w, even without external single-particle type coupling between two layers, one of the interesting moiré physics, i.e., both the localized and delocalized behavior of component 1 atoms, can be achieved.
        For instance, as shown in Fig. <ref>(b) and (c), when θ is chosen as a Pythagorean angle, for instance, θ= arctan3/4 (Fig. <ref>(b)), the Gaussion wave packet for component 1 atoms displays the delocalized behavior.
        While considering θ being the non-Pythagorean twisted angle, for instance, θ = π/6 (Fig. <ref>(c)), the wave packet for component 1 atoms turns out to be localized.
        Note that here we consider the case with large number imbalance between two components, i.e., N_1 ≪ N_2 and lager intra-species interaction, i.e., g'_22 > g'_12. The effect of inter-species interaction on the dynamics of component 2 atoms is highly suppressed and the density profile of component 2 atoms almost does not change during the evolution, as shown in the right column of Fig. <ref>.
        
        
        
        To quantitatively distinguish between the  delocalized and localized behavior of the wave packet of component 1 atoms, we analyze the following mean square displacement (MSD) <cit.>
            σ(t)=[∬dxdy (x^2+y^2)|ψ_1(t)|^2/∬dxdy|ψ_1(t)|^2]^1/2,
        
        which can characterize the width of the wave packet of component 1
        atoms. The different dynamical behaviors of the wave packet can be
        quantified as σ(t)∝ t^κ with distinct dynamical index κ.  As shown in Fig. <ref>,  when the twist angle θ is chosen as a Pythagorean angle, for instance, θ= arctan3/4, σ(t) tends to linearly increase with respect to the evolution time, i.e., σ(t) ∝ t.  The dynamical index is 1,  indicating
        its delocalized behavior. While θ being a non-Pythagorean angle, for instance, θ = π/6, σ(t) will reach a stable value in a fairly short time, indicating the dynamical index κ being 0. It thus characterizes the localized behavior of the wave packet.  Therefore, through tuning the twist angle θ, one of the moiré physics, i.e., both the localized and delocalized behavior of the wave packet of component 1 atoms, can be achieved.
        
        
        
        § NONLINEARITY-INDUCED DYNAMICAL SELF-ORGANIZED TWISTED-BILAYER LATTICES
        
        In the following, we will explain the reason for the appearance of the moiré physics in our proposed system. Let us focus on Eq. (<ref>) in the GP equation describing the two-component atomic BEC. In the presence of the nonlinear term w,  component 1 atoms suffer from a  dynamical self-organized effective potential V_eff≡ V'_trap,1+V'_1+w=V'_trap,1+V'_1+g'_11|ψ_1|^2+g'_12|ψ_2|^2.
        As shown in Fig. <ref>(a), when the twist angle θ is chosen as a Pythagorean
        angle, for instance, θ =arctan 3/4, the effective potential V_eff possess a spatially periodic structure. While considering θ being a non-Pythagorean twist angle, for instance,  θ=π/6, as shown in Fig. <ref>(b), V_eff behaves as a twisted-bilayer lattice with a spatial aperiodic structure. Therefore, distinguished from previous studies <cit.>, even without external single-particle type interlayer coupling, the nonlinear effect resulting from the intrinsic atomic interactions provides a new way to dynamically induce the twisted-bilayer lattices. Through tuning the twist angle θ, we find that both periodic (commensurable) and aperiodic (incommensurable) moiré structures can be created in such a dynamical twisted-bilayer lattice.
        
        
        
        
        To further understand the distinct dynamical behaviors of the wave packet of component 1 atoms, we rely on the analysis of the appearance of the flat band physics in our proposed nonlinearity-induced self-organized twisted-bilayer lattices. To do that, we calculate the band structure of V_eff for both periodic (commensurable) and aperiodic (incommensurable) moiré structures.
        In particular, when considering the aperiodic moiré lattice, the band structure can be calculated through approximating the non-Pythagorean twist angle by a Pythagorean one. For instance, here we use θ=arctan 120/209 to approximate θ= π/6.
        The band structure of V_eff can thus be obtained by the plane-wave expansion method through introducing the Bloch basis ψ_nk⃗=∑_G⃗ u_nk⃗,G⃗|k⃗+G⃗⟩ with the Bloch vector k⃗ and the reciprocal lattice vector G⃗. Here n labels the band index. Then, the energy spectra can be obtained through solving the eigen-problem via the following relation
        
            (k⃗+G⃗)^2λ^2/2u_nk⃗,G⃗ + ∑_G⃗'⃗⟨k⃗+G⃗ | V_eff| k⃗+G⃗'⃗⟩ u_nk⃗,G⃗'⃗   
            
            		= E'_nk⃗ u_nk⃗,G⃗,
        
        where E'_nk⃗=E_nk⃗/(ħ^2/mλ^2) is the dimensionless energy spectra.
        
        
        
        
        As shown in Fig. <ref>, the band structures of the corresponding
        twisted-bilayer lattices V_eff  demonstrated in Fig. <ref> are obtained. We find that when changing the twist angle from a Pythagorean angle to a non-Pythagorean angle, more dispersive lower bands become extremely flat, accompanying with the lattice structure changing from the periodic (commensurable) case to the aperiodic (incommensurable) one. In particular, the appearance of the lowest flat band will lead to the localization of the wave packet of component 1 atoms, since such a BEC wave packet will mainly concentrate at the lowest band after a long enough time evolution. Furthermore, it is known that the flat band can support quasi-nondiffracting localized modes. Therefore, the wave packet of component 1 atoms loaded in such a  nonlinearity induced self-organized aperiodic (incommensurable) twisted-bilayer lattice will tend to be localized.
        
        
        
        § CONCLUSION
        
        In summary, we have demonstrated a new approach to
        dynamically achieve twisted-bilayer lattices in an atomic BEC.
        To emphasize a remarkable difference, our scheme
        does not require external single-particle type
        interlayer coupling. Instead, we only require the nonlinear effect
        arising from the intrinsic atomic interactions.
        We further show that both periodic and aperiodic  moire lattices, i.e.,  nonlinearity-induced dynamical self-organized twisted-bilayer lattices, can be designed through tuning the twist angle. One of the interesting phenomena in twisted-bilayer lattices, i.e., the flat-band physics, is also shown through investigating the dynamics of the wave packet of BEC. Such nonlinearity-induced approach is rather generic to other quantum systems, such as in photonics and electrical circuits, than restricted to the setup considered in this work, since the nonlinear effects, such as  Kerr nonlinearity, is common in these systems. Our results reveal a profound connection between
        the nonlinearity and twistronics.
        
        
        § ACKNOWLEDGMENT
        
        We acknowledge helpful discussions with L. Chen. This work is supported by the National Key R&D Program of China (2021YFA1401700), NSFC (Grants No. 12074305, 12147137, 11774282), the Fundamental Research Funds for the Central Universities (Grant No. xtr052023002), the Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 23JSZ003) and  Xiaomi Young Scholar Program. We also thank the HPC platform of Xi'an Jiaotong University, where our numerical calculations was performed.
        
        
        apsrev


 Paper Title: The impact of quantum noise on limit cycles and entrainment in an atom-cavity system.







National Institute of Physics, University of the Philippines, Diliman, Quezon City 1101, Philippines



National Institute of Physics, University of the Philippines, Diliman, Quezon City 1101, Philippines



§ ABSTRACT 
Quantum noise is inherent in any open quantum system as it affects not only the statistical properties of the initial state but also the time evolution of the system. A transversely pumped atom-cavity setup is a prototypical example of such an open system, wherein limit cycles (LCs) have been observed and identified as continuous time crystals (CTCs). Using truncated Wigner approximation (TWA), we show that the inherent quantum noise pushes the system to exhibit signatures of LCs for interaction strengths lower than the critical value predicted by mean-field theory, suggesting a noise-induced enhancement of temporal ordering. By comparing the oscillation frequencies of the LCs for the one-dimensional (1D) and two-dimensional (2D) regimes, we find that the LC frequencies have larger shot-to-shot fluctuations in 1D than in 2D. We demonstrate that this has an important consequence in the effectiveness of entrainment of LCs for a periodically driven pump intensity or light-matter coupling strength.


The impact of quantum noise on limit cycles and entrainment in an atom-cavity system
    Jayson G. Cosme
    August 5, 2024
====================================================================================





§ INTRODUCTION

Typically, noise is expected to inhibit well-defined oscillatory response in complex systems. Nevertheless, it has been suggested that noise could induce order, instead of the expected chaotic behavior <cit.>.
Indeed, phase synchronization in systems with common noise <cit.> and uncorrelated noise <cit.> has been observed.
Moreover, applied noisy drive has been shown to produce nontrivial ordered phases in interacting quantum systems <cit.> and, in other cases, can have the opposite effect of destabilizing an existing LC or CTC <cit.>. In Ref. <cit.>, fluctuations corresponding to the intrinsic quantum property and coupling to an external environment, and their role on the stability of discrete time crystals have been explored. In this work, we focus on the role of inherent quantum noise in the emergence and stability of LCs, particularly in the context of CTCs and their dynamical control <cit.>. These LCs are characterized by oscillating observables at a well-defined frequency even though the underlying equations governing the system's time evolution is manifestly time-independent.

The emergence of LCs in various many-body quantum systems is widely studied, especially in the context of time crystals <cit.>. In particular, for an atom-cavity system <cit.>, artificial noise applied to the light-matter interaction strength is found to destabilize the LC, which then begs the question if the inherent fluctuation associated with the photon dissipation in the cavity <cit.> could inhibit the appearance of LCs. This is important in understanding the quantum nature of LCs in open systems <cit.>.
Exploring the precise role of quantum noise can be experimentally challenging as the inherent noise cannot be turned off naturally in experiments. However, in a theoretical setting, an approach such as the truncated Wigner approximation (TWA) <cit.> allows for the possibility of choosing which noise to be included in the theoretical description of a given system. In atom-cavity systems, beyond mean-field approximation of the quantum dynamics using TWA has been shown to capture the correct qualitative behaviour of time crystals observed in experiments <cit.>.


 



In this paper, we investigate the effects of quantum noise on the robustness and dynamics of the LC phase in the 1D and 2D atom-cavity systems. To this end, we employ both mean-field simulations and TWA, wherein we sample the quantum noise of the initial state and the stochastic noise in time. In doing so, we can compare the dynamics for cases with and without quantum noise. This allows us to infer the effects of quantum noise to the emergence of LCs. Moreover, we investigate the entrainment of LCs <cit.> in both 1D and 2D by periodically driving the pump intensity. Consequently, we are able to observe a hallmark of entrainment, which is frequency locking, and we determine which of the atom-cavity systems will exhibit a more robust entrainment.


This paper is structured as follows. We first discuss the atom-cavity system in Sec. <ref>. Next, we present the mean-field diagrams for the 1D model in Sec. <ref>. We also show the spectrogram for different values of the pump intensity. In  Sec. <ref>, we analyze the average response frequencies together with the corresponding standard deviation to infer the qualitative and quantitative effects of quantum noise. We compare the entrainment of LCs for the 1D and 2D atom-cavity systems in Sec. <ref>. Lastly, we conclude the paper in Sec. <ref>.





§ ATOM-CAVITY MODEL

We focus on a model for an atom-cavity system of ^87Rb-atoms forming a Bose-Einstein condensate (BEC) with particle number N_a = 6.5× 10^4 trapped inside a Fabry-Pérot resonator as shown in Fig. <ref>(a). A pump beam with intensity ε = ϵ E_rec, where E_rec = ħω_rec is the recoil energy, is applied in the direction perpendicular to the cavity axis. The pump has a wavelength λ_p = 792.55 nm and the recoil frequency is ω_rec = 2π^2ħ/mλ_p, where m is the mass of a ^87Rb atom.
Photons leaking out of the cavity is characterised by the cavity photon decay rate κ= 2π× 3.6 kHz. We use realistic parameters based on experimental realizations in the good cavity regime κ∼ω_rec <cit.>.

The Hamiltonian for the atom-cavity system consists of  cavity, atom, and atom-cavity interaction terms

    Ĥ = Ĥ_C+Ĥ_A+Ĥ_AC.

Here, we neglect the contact interaction between the atoms, which corresponds to a mean-field breaking term in this model <cit.>.
The Hamiltonian for the cavity with a mode function cos(kz) is given by

    Ĥ_C = -ħδ_Cα̂^†α̂,

where δ_C is the pump-cavity detuning and α̂^† (α̂) is the creation (annihilation) operator for the cavity photon. Note that δ_C = δ_eff +N_a U_0/2, where U_0 is the lattice depth induced by a photon in the cavity. The Hamiltonian for the atoms is

    Ĥ_A = ∫Ψ̂^†(y,z) [-ħ^2/2m∇^2+εcos^2(ky)]Ψ̂(y,z)  dy dz

where Ψ̂(y, z) is the bosonic field operator associated with the BEC.
Lastly, we consider the atom-cavity interaction

    Ĥ_AC =∫Ψ̂^†(y,z)  ħ U_0  [cos^2(kz)α̂^†α̂
    
    	+√(ε/ħ|U_0|)cos(kz)cos(ky)(α̂^† +α̂)]Ψ̂(y,z)  dy dz.

In the 1D limit, the dynamics along the pump direction is frozen Ψ̂(y,z) →Ψ̂(z), such that Eq. <ref> reduces to just the kinetic energy along the cavity direction. Moreover, the atom-cavity interaction Eq. <ref> simplifies to only contain z-dependent terms, which can be obtained by taking cos(ky) → 1. The dynamics of the system is given by the Heisenberg-Langevin equations

    ∂/∂ tΨ̂   = i/ħ[Ĥ, Ψ̂] 
    ∂/∂ tα̂   = i/ħ[Ĥ, α̂]-κα̂+ξ,

where ξ is the stochastic noise due to the cavity dissipation with ⟨ξ^*(t)ξ(t')⟩ = κδ(t-t') <cit.>. 


In our numerical simulations, we represent the quantum operators as c-numbers using TWA. This enables us to approximate the dynamics using an ensemble of stochastic trajectories. The TWA includes the leading order quantum corrections to the mean-field theory, which is valid for N_a→∞ <cit.>. In open quantum systems, perturbations due to quantum noise in the initial states and stochastic noise ξ associated with the photon loss are present. In theory, we can choose which source of quantum noise we include in our TWA simulations.

To numerically solve the relevant stochastic differential equations, we utilize the predictor-corrector method, otherwise known as Heun's method. We consider a time step of 10^-8 s and record the values every 20 steps. For 1D simulations, we use a real-space basis for Ψ̂(z) <cit.>, while for 2D simulations, we employ a momentum-space basis for Ψ̂(y,z) <cit.>.







§ MEAN-FIELD RESULTS

In this section, we first focus on mean-field simulations to serve as a benchmark for TWA simulations, which will be shown in the next section.


To introduce the phases found in this configuration of an atom-cavity system, we implement the two-ramp pump protocol shown in Fig. <ref>(b). In Fig. <ref>(c), we show the dynamics of the intracavity photon number, |α|^2, used to distinguish between different phases in the system. The normal phase (NP) is identified by a zero intracavity photon number. By increasing the pump intensity, it will transition to the superradiant phase (SR) when ε≥ε_SR, where ε_SR is the critical value for the NP-SR transition.
The SR phase is characterized by a non-zero constant intracavity photon number <cit.> as highlighted in Fig. <ref>(c). Hence, its long-time trajectory as shown in Fig.<ref>(d) is a steady state in the phase space spanned by the real and imaginary parts of the cavity mode α. Increasing the pump intensity further leads to a transition into the LC phase. This dynamical phase is characterized by an oscillating |α|^2 at a well-defined limit cycle frequency ω_LC as shown in Fig. <ref>(c). Because it oscillates at a certain frequency, the long-time trajectory in phase space forms a closed loop as presented in Fig. <ref>(e).

In Fig. <ref>, we present the phase diagram in the δ_eff-ϵ space for a 1D system. We note that the existence of LCs in the 1D regime for blue and red atom-pump detunings was first predicted in Refs. <cit.> and <cit.>, respectively. 
To construct Fig. <ref>, we linearly increase the pump intensity until ϵ for t∈ [15, 20] ms. Then, we fix ϵ to a constant value for 15 ms. We consider the time frame of the last 5 ms to obtain the long-time average, maximum value, and the standard deviation of the rescaled peak heights of the cavity occupation. In classifying the dynamics, we apply the following threshold values. 
 
A cavity occupation dynamics is classified to be that of a LC if the standard deviations is lower than 0.025, and the difference between the maximum and average |α|^2 is greater than or equal to 10% of the average. The SR phase, on the other hand, is identified by having a standard deviation less than 10% of the average cavity occupation. A response with an average photon occupation less than or equal to 10^-3 is considered as a NP. Lastly, dynamics with standard deviation greater than or equal to 0.025 is determined as a chaotic response (CH). We obtain the LC frequency ω_LC from the dominant frequency in the power spectrum dynamics of the intracavity photon number |α|^2 for the same time window. Then, ω_LC is obtained as the frequency with the highest peak in the power spectrum. 
In Fig. <ref>, both the blue-detuned and red-detuned cases have LCs. 

In Fig. <ref>, we further investigate the dynamics of |α|^2 by including a gallery of spectrograms for different ε/ε_LC, MF, where ε_LC, MF is the intensity required to transition from SR to LC according to MF theory. This allows us to take note of how the dynamics change for different intensities in the absence of stochastic noise. Accordingly, we expect that the system is in the SR phase when ε/ε_LC, MF<1.0, and it is in the LC phase when ε/ε_LC, MF≥1.0. 
For the pump protocol, we linearly increase the pump intensity until the desired ε for 5 ms, then kept it constant for 395 ms. Afterwards, we obtain the spectrogram of |α|^2 using a moving time window of ≈ 3.3 ms. The intensity in units of dB corresponds to 
10log_10(P) where P is the power spectral density. Hence, the dominant oscillation frequencies of |α|^2 are characterized by positive intensity shown as brighter shades in the color bars. Thus, a dominant frequency seen as streak of positive intensity at a constant ω signifies the appearance of a LC. On the other hand, the SR phase is identified by the lack of nonzero frequency peak in the long-time limit due to the absence of photon oscillations. 

We find that in 1D, as depicted in Fig. <ref>(a), transient oscillations appear but they eventually vanish for long times as ε approaches ε_LC, MF. 
We note for the discussions in the next section that the transient oscillations in Fig. <ref> persist for t<100 ms.
To inspect these dynamics, we show in Fig. <ref>(c-f), the adjusted intracavity dynamics |α|^2_ctr = |α|^2-|α|^2_mean where |α|^2_mean is the mean value of |α|^2 for t∈[0.1, 0.4] s, and its corresponding spectrogram for ε/ε_LC, MF<1.0. We find that the amplitude of the |α|^2_ctr oscillation decreases and eventually vanishes. 
 

This is also reflected in the features of the spectrogram in Fig. <ref>(d). It is also striking that a consistent dominant frequency is present, and that it is larger than the expected ω_LC. On the other hand, the 2D system shows a different trend in Fig. <ref>(b). While we find that the transient oscillations vanish after some time similar to the 1D case, we find that the photon dynamics approach what appears to be plateaus that increase in value as portrayed in Fig. <ref>(e). Moreover, between these plateaus, we observe short-lived small amplitude oscillations that become more frequent the closer ε is to ε_LC, MF. The sharp increase in the photon number appears as streaks at constant times in the spectrogram in Figs. <ref>(b) and <ref>(f). 

Overall, the mean-field limit provides us with a baseline of the expected behavior of the system. To summarize, we found that transient oscillations are present in the absence of stochastic noise. Hence, one may mistakenly identify these transient oscillatory response as LC if the simulation time is not sufficiently long. We will thus classify such a behaviour with an oscillatory response faster than the expected LC frequency as a SR phase. 



§ TWA SIMULATIONS



 §.§ Noise and limit cycles

In the previous section, we discussed the mean-field dynamics of |α|^2 for different pump intensities. Next, we explore the effects of quantum noise in the initial state and stochastic noise associated with the cavity dissipation. We will assume that the stochastic noise in time can be freely switched off within TWA as done in the supplementary materials of Ref. <cit.>. By comparing the results with and without stochastic noise, we can pinpoint the fundamental effect of stochastic noise.


In Fig. <ref>, we present the results for the same pump protocol used in Fig. <ref> albeit now in the presence of stochastic noise. Note that we use exactly the same initial state for a fair comparison. As expected when temporal noise is included in the dynamics, we find that stochastic noise broadens the power spectrum, which results to a decrease in the intensities of the dominant frequencies. Interestingly, the transient high-frequency oscillations in the SR phase seen in Fig. <ref> have vanished in the presence of stochastic noise. This means that the inherent quantum noise due to dissipation overwhelms the transient oscillations making them unlikely to be observed in actual experiments. Moreover, it also pushes the system to oscillate at a well-defined frequency such that ω_peak∼ω_LC, MF as seen from the bright streaks close to the expected LC frequency in Fig. <ref>. This will become more obvious when we discuss Fig. <ref> later. The spectrograms for the SR phases close to the SR-LC transition point already show signs of a LC by having a faint bright streak at the expected LC frequency in Fig. <ref>(b), for example. In the remainder of this section, we will further explore this apparent noise-induced enhancement of the LC phase.






To further investigate the noise-induced temporal ordering, we now use 100 trajectories in our TWA simulations for varying pump intensities as depicted in Figs. <ref> and <ref> for the 1D and 2D systems, respectively. For the pump protocol, we linearly increased the intensity for 5 ms and then kept it constant for 15 ms. We then analyze the Fourier transform of the intracavity photon number for t∈[15, 20] ms, F{|α|^2(t)}= |α̃|^2(ω).
The dominant frequency ω_peak is determined as the frequency corresponding to the maximum value of the normalized Fourier spectrum given by the crystalline fraction defined as Ξ = max{|α̃|^2(ω)/∑_ω |α̃|^2(ω)} <cit.>. 

In Figs. <ref> and <ref>, we present two sets of results depending on which type of quantum noise is included in the TWA. In both cases, quantum noise in the initial state is included assuming a coherent state for the BEC and a vacuum state for the cavity mode. However, in one case, the stochastic noise is absent meaning the dynamics is fully deterministic, while in the other, it is present resulting to a stochastic dynamics of the system. Furthermore, Fig. <ref> depicts the results for the 1D case while Fig. <ref> corresponds to the 2D case.
In Figs. <ref>(a) and <ref>(a), the average values of ω_peak for 100 trajectories are indicated as circles and diamonds for those with and without stochastic noise, respectively. The error bars denote the standard deviation for the recorded peak frequencies. In Figs. <ref>(b)-(g) and <ref>(b)-(g), we present the histograms of the crystalline fraction and peak frequency for exemplary choices of the pump intensity. Here, the intensity of the color bar represents the number of trajectories following the binning procedure for the histograms.

The SR phase manifests as a zero-frequency oscillation in MF as shown in Fig. <ref>(b), and low-frequency oscillation due to stochastic noise in Fig. <ref>(e). 
Trajectories with significantly high frequency without stochastic noise are attributed to SR phase instead of the LC phase as discussed in Sec. <ref>. These low amplitude high-frequency oscillations emerges the closer the pump intensity is to ε_LC, MF. In some trajectories, these high frequencies have a well-defined oscillation as noted by a high value of Ξ in Fig. <ref>(b) and <ref>(c). We emphasize that these fast oscillations are transient as discussed in the previous section, which means that all trajectories in Fig. <ref>(b), for example, correspond to a SR phase. Interestingly, Figs. <ref>(c), <ref>(f), <ref>(c), and <ref>(f) exhibit trajectories with ω_peak∼ω_LC, MF even though ε<ε_LC, MF. 
More importantly, comparing Figs. <ref>(c) and <ref>(f), we find that the presence of stochastic noise in time has eliminated the presence of trajectories in the SR phase oscillating at high frequencies. This phenomenon is even more pronounced in 2D as seen in Figs. <ref>(c) and <ref>(f). Hence, the suppression of the high-frequency oscillations in the SR phase by stochastic noise reveals an early signature of the onset of LCs. This noise-induced enhancement of LC is emphasized in Fig. <ref>(f), in which most trajectories possess ω_peak∼ω_LC, MF. Nevertheless, we point out that not all trajectories can be considered as LC due to some of them having a relatively low Ξ.
Furthermore, the presence of noise in the initial state leads to a shift of the phase diagram because of the slight variations of N_aU_0 for constant U_0. Thus, trajectories would either be an SR or LC near the SR-LC transition since the quantum fluctuations in the initial state effectively changes the δ_eff. 




 

To investigate the shot-to-shot fluctuations found deep in the LC regime, we employ TWA for 500 trajectories as shown in Fig. <ref>. We use the same pump protocol as in the mean-field simulations and kept it constant for 20 ms and 35 ms for 1D and 2D, respectively. We calculate the Fourier transform of the cavity occupation dynamics for each trajectory over the last 5 ms, which means that our Fourier transform can only detect frequency changes larger than 200 Hz. This is important to note when we consider multiple trajectories within TWA as they could in principle have different ω_LC. From the Fourier spectrum, we record its real and imaginary parts at the LC frequency predicted by MF, N_P(ω_LC,MF), for fixed  δ_eff and ϵ. Furthermore, we rescale the spectrum by the maximum value recorded for all trajectories. Notice that some of the trajectories do not have the same LC frequency as the mean-field prediction, which is evident from the points scattered near the origin of Figs. <ref>(a) and <ref>(b). Specifically, we note that some of the trajectories have frequency around ω_peak = ω_LC,MF± 2π× 200 Hz. This can be explained by the smooth variation of the LC frequency in the mean-field phase diagram. Quantum fluctuations in the initial state lead to shot-to-shot variations of the total particle number N_a, which means that TWA can be considered as a sampling of MF results from slightly different choices of δ_eff in the mean-field phase diagram. This then leads to different LC frequency for each trajectory, the variation of which depends on how fast the LC frequency changes as a function of δ_eff. In comparison to Fig. <ref>(a), we find that there are less fluctuations of the LC frequency in the 2D system as shown in  Fig. <ref>(b), wherein most trajectories are found along the unit circle. 






 §.§ Entrainment

We now investigate the consequence of fluctuating LC frequency on the phenomenon of entrainment when the pump intensity is periodically modulated <cit.>. Entrainment, otherwise known as injection locking in optics, is observed by periodically driving a system with a limit cycle state <cit.>. This results to the “locking" of the limit cycle frequency by driving the system to a ratio of ω_LC <cit.>. In the atom-cavity system, this was proposed to create a more robust time crystal or limit cycle phase <cit.> and was later reported experimentally <cit.>. In both of these papers, they focused on subharmonic injection locking, wherein the pump intensity is periodically driven at a frequency ω_dr= 2ω_LC.

In the following, we first choose a set of parameters deep in the LC regime with frequency ω_LC, MF. Then, we periodically modulate the pump intensity once we have a stable LC by driving it at twice the LC frequency ϵ=ϵ_0[1+f_0cos(2ω_LC, MFt)]. A successful entrainment is marked by |α|^2 oscillating at ω_LC, MF, thereby breaking the discrete time translation symmetry imposed by the periodic drive. This constitutes a phase transition from continuous to discrete time crystals <cit.>. 

 

In Fig. <ref>, we present the response frequencies for 100 trajectories for different driving strengths f_0. The mean value of the response frequency ω_peak is indicated as a circle and its standard deviation is represented by the error bar. The dashed lines refer to the LC frequency according to MF, ω_LC, MF. We observe in Fig. <ref>(a) that deviations in the response frequencies for all f_0 is present. Hence, the entrainment in a 1D system is not as robust as the one observed in Ref. <cit.>.
The weak entrainment can be attributed to the strong shot-to-shot variations of the LC frequency in 1D as discussed in the previous subsection. While ω_LC,MF is fluctuating for each trajectory, the driving frequency is fixed to just one value for all those trajectories. That is, not all trajectories are resonantly driven, leading to deviations of the response frequency even for smaller values of f_0 in 1D. In contrast, the 2D case exhibits a more robust entrainment with zero standard deviation for f_0∈ [0.1, 0.35]. This demonstrates that, while a 2D system also experiences shot-to-shot fluctuations of the LC frequency (see Fig. <ref>(b)), albeit less likely and weaker than the 1D case, it can still result to a successful entrainment as seen in Fig. <ref>(b). This is further corroborated by a comparison of the 1D case in Fig. <ref>(a) and the zoomed-in view of the results for 2D in the inset of Fig. <ref>(b), wherein the standard deviations are considerably higher in 1D than in 2D.





§ CONCLUSIONS

In summary, we have explored the role of quantum noise in the emergence and entrainment of LCs in an atom-cavity system. Specifically, we compare the photon dynamics obtained using MF and TWA in the 1D and 2D regimes. The TWA allows us to freely switch on and off the stochastic noise in time associated with the cavity dissipation. We found that the inclusion of stochastic noise results to an earlier onset of LCs for smaller pump intensities or, equivalently, light-matter interaction strengths, compared to the corresponding mean-field prediction. In particular, stochastic noise suppresses the high-frequency oscillations in the SR phase, which in turn reveals an earlier signal for the onset of LCs through weak yet detectable oscillations at the expected LC frequency. Furthermore, we have demonstrated that the shot-to-shot fluctuations of the LC frequency are larger in the 1D limit compared to the 2D limit of the atom-cavity system. This then results to less robust entrainment in 1D as some of the trajectories oscillate at frequencies that are non-resonant with respect to the applied periodic drive of the pump intensity. 

Our work highlights the unique role that quantum noise plays in the emergence and stability of dynamical phases, such as limit cycles and time crystals. In particular, we have shown that the fluctuations corresponding to the photon dissipation, a necessary ingredient for beyond mean-field approaches in open quantum systems to ensure that the bosonic commutation relations for the photons are preserved, may reveal a temporal order, or some hints of it, that is otherwise hidden or completely absent in a standard mean-field theory. Resonance phenomena involving dynamical phases, exemplified in this work by entrainment of LCs, can be adversely affected by quantum fluctuations of the initial state depending on the rigidity of the characteristic frequency of the dynamical phase. It would be interesting to explore in future works these phenomena in the full quantum regime, which may aid in the quest for understanding the quantum origins of various behaviours in nonlinear dynamics, such as the appearance of LCs and bifurcations of stable states <cit.>.




§ ACKNOWLEDGEMENTS

R.J.L.T. and J.G.C. acknowledge support from the DOST-ASTI’s COARE high-performance computing facility. This work was funded by the UP System Balik PhD Program (OVPAA-BPhD-2021-04).




















 Paper Title: 2D Skyrmion topological charge of spin textures with arbitrary boundary conditions: a two-component spinorial BEC as a case study.





romero@fisica.unam.mx
Instituto de Física, Universidad
Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, Mexico. 


§ ABSTRACT 

We derive the most general expression for the Skyrmion topological charge for a two-dimensional spin texture, valid for any type of boundary conditions or for any arbitrary spatial region within the texture. It reduces to the usual one Q = 1/4π∬f⃗·(∂_x f⃗×∂_y f⃗) for the appropriate boundary conditions, with f⃗ the spin texture. The general expression resembles the Gauss-Bonet theorem for the Euler-Poincaré characteristic of a 2D surface, but it has definite differences, responsible for the assignment of the proper signs of the Skyrmion charges. Additionally, we show that the charge of a single Skyrmion is the product of the value of the  normal component of the spin texture at the singularity times the Index or winding number of the transverse texture, a generalization of a Poincaré theorem. We illustrate our general results analyzing in detail a two-component spinor Bose-Einstein condensate in 2D in the presence of an external magnetic field, via the Gross-Pitaevskii equation. The condensate spin textures present Skyrmions singularities at the spatial locations where the transverse magnetic field vanishes. We show that the ensuing superfluid vortices and Skyrmions have the same value for their corresponding topological charges, in turn due to the structure of the magnetic field.

2D Skyrmion topological charge of spin textures with arbitrary boundary conditions: a two-component spinorial BEC as a case study
    S. Sánchez-Reséndiz, E. Neri, S. González-Hernández, and V. Romero-Rochín
    
=================================================================================================================================

Keywords: Skyrmion charge, Vortices, BEC, Ultracold Quantum Matter





§ INTRODUCTION


Skyrmions are topological defects or excitations that emerge in continuous fields and spin textures due to intrinsic chiral interactions  or to the presence of external inhomogeneous magnetic or gauge fields. These excitations arise in diverse materials or fields as in nuclear physics <cit.>, particle physics <cit.>, notably in magnetic systems in condensed matter, such as in 2D electron gas <cit.>, magnetic metals <cit.>, transition metal films <cit.>, frustrated magnets <cit.> and in a host of other magnetic situations <cit.>, liquid crystals <cit.> and, the context of the present paper, in Bose-Einstein condensates, both 2D <cit.> and 3D  <cit.>. Ever since Skyrme seminal work <cit.>, its analysis has been pursued in field theoretical models such as the non-linear σ-model in high energy physics <cit.>, or in terms of the Dzyaloshinskii-Moriya theory in magnetic systems <cit.>, or within the Gross-Pitaevskii (GP) equation in Bose-Einstein condensates, as in essentially all of the theoretical works already referenced. In this paper we deal with Skyrmions generated by an external gauge field in a two-component spinorial Bose-Einstein condensate in two spatial dimensions, described by two coupled GP equations. This stylized model allows for a simple and clear description of  spin textures that can show a rich variety of Skyrmions, with the allowance for the determination of all its  inherent and topological properties. In most of the experimental Skyrmion realizations, BEC studies have been performed in real alkaline weakly interacting Bose clouds with spin F=1  <cit.> and  F = 2 <cit.>, however, the topological properties of a given spin structure should be independent of their physical origin.




An outstanding feature of Skyrmions is their topological character ingrained and described by its topological charge. In most of the studies on 2D Skyrmions  the topological charge has been considered to be given by the expression <cit.>

    Q = 1/4π∬f⃗·( ∂f⃗/∂ x×∂f⃗/∂ y) dxdy ,

where f⃗(x,y) is the 3D vector spin texture or continuous field of the problem at hand, obeying f⃗·f⃗ = 1 everywhere in the XY-plane. It is expected that the charge Q is an integer. This expression, however, requires particular boundary conditions. Skyrme <cit.> and others afterwards <cit.>  considered that f⃗ = constant as |r⃗| →∞ and, in such a case, Q is a positive or negative multiple integer, with the interpretation that Q is the number of times the mapping of f⃗ covers the unit sphere. Additionally, if the spin texture lies on the plane at the boundary, then Q = ± 1/2, a defect called a half Skyrmion <cit.>. These boundary conditions, however, need not appear or exist in a given physical problem <cit.> or in a multiple Skyrmion configuration, where one may need to evaluate the local charge of the Skyrmions present. In these cases expression (<ref>) cannot account for the calculation of the whole charge that, from topological consideration, must be an integer. As an example, in figure <ref> we present a spin texture arising in the 2D spin 1/2 spinor Bose-Einstein condensate (BEC) here studied, the details of such a solution given further below. We observe a configuration of three Skyrmions, one pointing on the positive z-direction and the other two in the negative one. Each defect should have its own topological charge, properly defined, and the total one of the full configuration should be the sum of the individual ones. The main purpose of this article is to show that a topological charge Q_S of the whole or part of the spin texture configuration can be defined within an arbitrary closed curve, which may or may not be the true boundary of the spin texture. Parametrizing the curve by r⃗_c(t), with t: 0 → T and  r⃗_c(0)= r⃗_c(T), we will find that such a topological charge is given by  

    2π Q_S    =   ∬_A f⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy + ∮_r⃗_cf⃗·( d f⃗/dt×d^2f⃗/dt^2) /|d f⃗/dt|^2 dt  
           + ∑_i=1^s sin^-1f⃗_i ·( . d f⃗/dt|_i^ +×. d f⃗/dt|_i^ -)/| d f⃗/dt|_i^ +| d f⃗/dt|_i^ -,

with Q_S ∈ℤ, a positive or negative integer. The first term is the usual one, except for a factor of 2, with A the area enclosed  by the curve r⃗_c. The second is a boundary term with the integral along the curve r⃗_c, and the third one takes into account the possible fact that the vector t⃗ = (d f⃗/dt)/|d f⃗/dt|, tangent to the parametric 3D curve f⃗(r⃗_c(t)), 
may be discontinuous at a finite number of s points along the curve. All these aspects should become clearer in the derivation given below. The above expression indeed reduces to the usual one Q_S = 2 Q for the boundary conditions described in the previous paragraph. Figure <ref> shows the same spin texture as in figure <ref> but with several boundary curves r⃗_c. The small circular ones each enclose the local Skyrmion singularities, yielding Q_S = +1 for the  two of them pointing down and Q_S = -1 for the one pointing up, using (<ref>). The large circle and the square curve enclose the three Skyrmions and yield Q_S = +1, which is the sum of the local ones. We remark that, for the square curve, the last term of (<ref>) must be taken into account since it gives rise to discontinuities on the tangent vector mentioned above.




While we claim that the above expression is a general one for any spin texture, since the derivation that we provide makes no reference to the physical origin of the spin texture, for purposes of illustrations and for a better understanding of the Skyrmion charge, we will use the spin textures arising from  a spinorial BEC in the presence of an external magnetic field, as a case study. In addition, an important and interesting point, is that such a system shows another well known type of topological defects, namely, quantum vortices <cit.>. We will explicitly show the profound relationship between the quantum vortices and the Skyrmions, namely, that their corresponding local charges have exactly the same values. The present stylized model allows for finding that, for the present problem, the external magnetic or gauge field ultimately defines the topological structure of the vortices and the Skyrmions.


The article is organized as follows. In Sections II and III we present a detailed analysis, both in a simplified analytic and in a full numerical fashion, of stationary solutions to the Gross-Pitaevskii equations of a 2D 1/2 spinorial Bose gas in the presence of an external inhomogeneous magnetic field. We show how the quantum vortices and Skyrmions can be predicted from the knowledge of the magnetic field. In Section IV, the main contribution of this article, we provide a general derivation of the topological charge of Skyrmions present in any given spin texture, for arbitrary boundary curves. Section V is devoted to a discussion between the similarities and discrepancies between the Skyrmions topological charge and the Euler-Poincaré characteristic of a surface in terms of the Gauss-Bonet theorem <cit.>. 
In an Appendix we illustrate our main result with a variety of numerical solutions of different spin textures generated by different magnetic fields, and for different types of arbitrary boundary curves. Final remarks are given in Section VI.



§ 1/2 SPINOR BEC IN THE PRESENCE OF AN EXTERNAL MAGNETIC FIELD


We consider stationary states of a two-component spinorial BEC in 2D, with a Zeeman coupling to an external magnetic field, as described by the coupled Gross-Pitaevskii (GP) equations

    μψ_α(r⃗) = [-ħ^2/2m∇^2 + 1/2 mω^2 r⃗^ 2
    + g (|ψ_+|^2 + |ψ_-|^2) ]ψ_α(r⃗) - B⃗(r⃗)·μ⃗^αβψ_β(r⃗) ,

where r⃗ = (x,y) and α = +,- denotes the two components of the spinor. Summing over repeated indices is assumed. μ is the chemical potential and the magnetic dipole operator is given by

    μ⃗^αβ = μ_0 σ⃗^αβ,

where σ⃗= x̂σ_x + ŷσ_y + ẑσ_z  are Pauli matrices and  μ_0 > 0 the magnitude of the magnetic dipole moment. The magnetic field is 

    B⃗(r⃗) = x̂ B_x(r⃗) + ŷ B_y(r⃗) + ẑ B_z .

We denote B⃗_T(r⃗) = B_x(r⃗) x̂ + B_y(r⃗) ŷ as the transverse magnetic field and, for simplicity, we will consider B_z(r⃗) = constant . We choose  B⃗_T(r⃗)  such that it vanishes at certain points r⃗_1, r⃗_2, …, r⃗_s. We shall see that at those points there appear quantum vortices and the origins of Skyrmions <cit.>.  The normalization condition is

    ∫ d^2 r  (|ψ_1|^2 + |ψ_2|^2) = 1 .


Writing the solution as ψ_α = √(ρ_α) e^i Θ_α, the superfluid density ρ_α and  the superfluid velocity v⃗_α, in each component, are given by

    ρ_α   =    |ψ_α|^2 ,
    v⃗_α    =    ħ/m∇Θ_α.


The circulation C of the velocity field in an arbitrary closed circuit C is defined by

    C =  ∮_ Cv⃗_α· dr⃗.

As we will explicitly calculate, if the circuit does not enclose a quantum vortex the circulation vanishes, but when a vortex or vortices are enclosed, the circulation yields the vortex topological charge Q_V within the circuit <cit.>

    C =  2 πħ/m Q_V

with Q_V ∈ℤ, a positive or negative integer. 


The condensate spin texture f⃗(x,y) is an observable, independent of the spin basis, and defined at any spatial point within the condensate,

    f⃗ = 1/ρ_+ + ρ_-(ψ_+^* ψ_-^*) [x̂σ_x + ŷσ_y + ẑσ_z]
    (
    [ ψ_+; ψ_- ]) .

It can be verified that the spin texture has magnitude one f⃗·f⃗ = 1 everywhere.  


Writing f⃗ = x̂ f_x + ŷ f_y + ẑ f_z, the components explicitly are

    f_x    =   ψ_+^* ψ_- + ψ_-^* ψ_+/ρ_+ + ρ_-
       =    2 √(ρ_+ρ_-)/ρ_+ + ρ_-cos(Θ_- - Θ_+) 
    
    f_y    =   - iψ_+^* ψ_- - ψ_-^* ψ_+ /ρ_+ + ρ_-
       =    2√(ρ_+ρ_-)/ρ_+ + ρ_-sin(Θ_- - Θ_+)
    
    f_z    =   ρ_+ - ρ_- /ρ_+ + ρ_-.




§ VORTICES AND SKYRMIONS IN THE VICINITY OF POINTS OF VANISHING TRANSVERSE MAGNETIC FIELD


While we will consider a variety of magnetic fields, all have in common that they have spatial points r⃗_1, r⃗_2, …, r⃗_s where the transverse magnetic field vanishes, B⃗_T (r⃗_i) = 0. This property imprints both the appearance of vortices and of Skyrmions at those points. To understand how this occurs, with no loss of generality, we consider a transverse magnetic field that vanishes at the origin r⃗ = 0 and analyze the behavior of the spinor wavefunctions in its vicinity. For this we explicitly write both GP equations for the two spinor components, assuming that B_z = 0,

    μψ_+    =   [-ħ^2/2m∇^2 + 1/2 mω^2 r^2
    + g ρ]ψ_+ - μ_0 (B_x - i B_y) ψ_- 
    μψ_-    =   [-ħ^2/2m∇^2 + 1/2 mω^2 r^2
    + g ρ]ψ_- - μ_0 (B_x + i B_y) ψ_+ ,

where ρ = |ψ_+|^2 +  |ψ_+|^2 and r= √(x^2+y^2). In the vicinity of the origin the magnetic field generically vanishes as <cit.>

    B_x - i B_y ≈ B_0  r^n  e^i n φ

where B_0 is the field strength, not necessarily real, tanφ = y/x and n a positive integer.  We note that if B_0 is real, the resulting field obeys ∇·B⃗ = 0 and ∇×B⃗ = 0, namely, it would be a true magnetic field. However, one can consider arbitrary “artificial” gauge fields <cit.> that do not necessarily satisfy Maxwell equations but that allow for different sorts of fields and which in turn give rise to different spin textures, as we show in the rest of the paper. In this way, generically, there are four different types of fields in the vicinity of a vanishing location, with B_0 real, positive or negative,

    I)B_x - i B_y    ≈    B_0  r^n  e^i n φ II) B_x - i B_y ≈ i  B_0  r^n  e^i n φ
     III) B_x - i B_y    ≈    B_0  r^n  e^-i n φ IV) B_x - i B_y ≈ - i  B_0  r^n  e^-i n φ.

The first two obey Maxwell equations, the last two do not; as a matter of fact, case (I) with n=1 is commonly produced in a Ioffe-Pritchard trap, and n=2 is a quadrupole trap <cit.>.  Figure (<ref>) shows the structure of the magnetic fields for the case n = 1. Case (III) is the hedgehog configuration usually considered in Skyrmion discussions <cit.>. 

As we will see, the local structure of the transverse magnetic field at its vanishing points plays a fundamental role in the final signs of the topological charges of the vortices and Skyrmions. As discussed in Section V below, such a structure can be associated to a topological quantity, namely, the Index of the field at the vanishing point. <cit.>


Let us consider case (I) for the moment. We will summarize all cases below.  For this case there are two possible solutions,

    i)ψ_+(r⃗)    =    F_+(r) e^-inφ andψ_-(r⃗) = F_-(r)  
     ii)ψ_+(r⃗)    =    F_ -(r)   andψ_-(r⃗) = F_+(r) e^inφ.

Substitution of these solution into GP equations (<ref>) shows that, very near r = 0, one obtains

    F_+(r)    ≈    A  r^n 
    
    F_-(r)    ≈   μ/g,

where A is a positive constant that can be numerically determined. The important features are that F_+(r) vanishes as r^n while  F_-(r) yields a constant as r → 0. Therefore, case (i) in (<ref>) is a vortex in the ψ_+ component and a density “spike” in the ψ_- one (see below), and the opposite in (ii), namely a spike in ψ_+ and a vortex in ψ_-. As we see now, in the former case the vortex charge is Q_V = - n, while in the latter Q_V = + n. This can be concluded right away from the velocity fields,

    i)v⃗_+(r⃗)    =    - n ħ/mφ̂/r andv⃗_-(r⃗) = 0  
     ii)v⃗_+(r⃗)    =    0   andv⃗_-(r⃗) = + n ħ/mφ̂/r,

where φ̂= - x̂sinφ + ŷcosφ. Indeed, the velocity fields circulate in the clockwise (negative) direction in the ψ_+ component for case (i), and in the anticlockwise (positive) direction for the ψ_- one, in case (ii). The circulations are, using a circular contour of radius r and dr⃗ = φ̂  r dφ,

    C_±   =   ∮_ Cv⃗_±· dr⃗
        =    ∓  n ħ/m∮ dφ
        =    ∓ 2π nħ/m,

finding the values of the topological charges already quoted. Clearly, if the vortex is not enclosed by the contour, the circulation vanishes, C_± = 0. Figure <ref> shows the solution to the GP equations (<ref>), as well as the velocity fields, corresponding to figures (<ref>) and (<ref>), including the magnetic field that generates such a configuration; see the Appendix for details of the calculation. One can observe that, in the vicinity of the vortices, the top one is of type (II) while the lower ones are of type (I) in (<ref>). In panels (A) and (C) of <ref> we can also see what we mean by a “density spike”: while the vortex in one component occurs, accompanied by a density depletion, in the other component there is a rapid increase of the density at the same spatial location, namely a spike, identified in the figure by a bright spot in the density profile. This increment of the density in one spinor component almost cancels the depletion at the vortex in the other component <cit.>.  
From expressions (<ref>) of the fields, the vortex structure of case (II) is the same as case (I) just discussed, however in cases (III) and (IV) the solution is as in (I) given by (<ref>), but with the velocity circulation reversed and, therefore, the charges are now Q_V = +n for solution (i) and Q_V = -n for (ii). As we will see below, the Skyrmion charges will have exactly the same values as Q_V, with their own spin texture structure.




We now turn our attention to the spin texture and, for the sake of argument, we also study case (I). Again, in the vicinity of zero field, one finds the spin texture from (<ref>), for both solutions (i) and (ii) in (<ref>)

    f_x    =   F_+ F_-/F_+^2 + F_-^2cos n φ
    
    f_y    =    -  F_+ F_-/F_+^2 + F_-^2sin n φ
    
    f_z     =    ±F_+^2 - F_-^2/F_+^2 + F_-^2.

where the ± signs in front of f_z refer to solutions (i) and (ii), respectively, see (<ref>). First, we note that the transverse spin texture f⃗_T = f_x x̂ + f_y ŷ  is parallel and in the same direction to the transverse magnetic field B⃗_T, as physically expected, see (I) in (<ref>), thus showing that both have the same spatial pattern. This is true in general and it is exemplified in the gallery of figures in the Appendix.
Evidently, as r → 0, the components f_x and f_y vanish since either F_+ or F_- vanish as well, and the spin texture becomes parallel to the ẑ vector, pointing upwards or downwards, f⃗ = ±ẑ. This latter direction is determined by whether the solution is (i) or (ii) as above in (<ref>): for solution (i) |ψ_+| = F_+(0) = 0, |ψ_-| = F_-(0)  0 and, therefore, the spin texture points downwards, f⃗(0) = - ẑ; for solution (ii) |ψ_-| = F_+(0) = 0, |ψ_+| = F_-(0)  0 and the spin texture now points upwards, f⃗(0) = + ẑ. For the other fields (II), (III) and (IV) the same rule applies for the peaks: if the vortex appears in the ψ_+ component, the texture points downwards, but if the vortex is in ψ_- then it points upwards. The ensuing structure of the spin texture is what we call the Skyrmion and, for the sake of describing it, we shall call the location r⃗_i where f⃗(r⃗_i) = ±ẑ, as the origin of the Skyrmion. 





§ THE TOPOLOGICAL CHARGE OF THE SKYRMIONS


As mentioned in the Introduction, the topological charge of the Skyrmions is usually given by (<ref>) but, as also argued, such an expression requires particular boundary conditions on the spin texture f⃗. As also already mentioned, in general those boundary conditions need not occur, or a spin texture can have a multiplicity of coexisting local Skyrmions, whose boundaries among them can also be quite arbitrary. Moreover, in all cases one should be able to asses the individual charges of the local Skyrmions in order to validate that the charges add as any topological index. The spin textures here presented are clear examples of the just two described situations. There is another fundamental aspect to be considered as well. In descriptions of common spin textures and their charge given by (<ref>), it is affirmed that the topological charge is the number of times that the unit sphere is covered by the spin texture. This follows, on the one hand,  by assuming that f⃗(x,y), being a unit vector, is by itself a parametrization or mapping of the unit sphere and, on the other, again, by the mentioned usual boundary conditions. While such a parametrization can indeed be assumed, there is subtle aspect that we believe has not been addressed before, namely, that different spin textures give rise to mappings to the unit sphere but with different orientations. That is, the normal vector to the sphere, generated by the parametrization f⃗(x,y), can point outwards or inwards to the sphere, depending on the given spin texture. And, for configurations with several Skyrmions, this can occur within the same spin texture. This is a very interesting and important point to consider in the elucidation of a general expression for the identification of the Skyrmion topological charge. In this section we present a derivation of the Skyrmion topological charge, in an as possible straightforward fashion, by a procedure similar to the derivation of the Gauss-Bonet (GB) theorem <cit.>. We will  leave the discussion of the mentioned delicate issues to the following section, where we also argue about the similarities or, rather, dissimilarities of the Skyrmion charge with the Euler-Poincaré characteristic χ, arising from the GB theorem.


Let us assume that a spin texture f⃗(x,y) is given in all or part of the XY-plane, such that it has a single location r⃗_0 where either f⃗(r⃗_0) = +z or f⃗(r⃗_0) = -z. We call this point the origin of the Skyrmion. Consider now an area A that encloses this point. The area 
has a boundary ∂ A described by a parametrized closed curve

    r⃗_c(t) = x̂ x_c(t) + ŷ y_c(t) t : 0 → T r⃗_c(0) = r⃗_c(T) .

 Notice that although the curve is continuous, its tangent may be discontinuous at a finite number of points r⃗_c(t_1), r⃗_c(t_2), …, r⃗_c(t_s). Further, and very important,  the curve is assumed, one, to run or be oriented in the positive counterclockwise direction and, two,  the curve does not cross itself, say the red curve in figure (<ref>).
 


Evidently, since the texture f⃗(r⃗) is a unit vector, it can be considered to be the outward normal vector r̂(θ,ϕ)  to the unit sphere S^2, with (θ,ϕ) the usual azimuthal and polar sphere angles, (please note that we use ϕ for the polar angle in 3D and φ in 2D)

    cosθ   =    f_z 
    tanϕ    =    f_y/f_x.

At the same time one could assume that f⃗(r⃗) is by itself a parametrization of the unit sphere. As discussed in detail below why, we will not consider this point of view. As we show below, perhaps surprisingly at  first sight, the two mentioned assumptions are not necessarily consistent with each other. Nevertheless, any value of the spin texture f⃗(r⃗) can be identified with a single point (θ,ϕ) on the sphere, although the opposite is not true in general.  That is, depending on the order of the Skyrmion, parts of the sphere can be covered more than once. Figure (<ref>) shows a simple example of these observations, where we can see how the spin texture within and on the square are mapped to unique points on the sphere. And, with the identification of the unit normal and the spherical angles, the usual orthogonal sphere unit vectors (r̂, θ̂, ϕ̂) along the directions (r,θ,ϕ), can also be expressed in terms of the spin texture,

    r̂    =    x̂ f_x + ŷ f_y + ẑ f_z 
    θ̂    =    x̂f_z f_x/√(f_x^2+f_y^2)+ ŷf_z f_y/√(f_x^2+f_y^2) - ẑ√(f_x^2+f_y^2)
    ϕ̂    =     -x̂f_y/√(f_x^2+f_y^2)+ ŷf_x/√(f_x^2+f_y^2).

 

In an analogous way, we can observe that the curve r⃗_c(t) on the plane gets mapped to a 3D curve f⃗(r⃗_c(t)), denoted as C, that lies on the surface of the unit sphere. The arclength of such a curve can be found from

    ds = |df⃗(r⃗_c(t))/dt| dt ,

and the tangent t⃗(s) to the curve is given by, see again figure (<ref>) as an example, 

    t⃗(s) = df⃗(s)/ds.

This is a unit vector, t⃗·t⃗ = 1. Since it is also tangent to the sphere, then t⃗·r̂ = 0 and, therefore, t⃗ lies on the θ̂-ϕ̂ plane. We define  α as the positive angle that t̂ makes with θ̂ and, thus, we can write

    t⃗ = θ̂cosα + ϕ̂sinα.

α can take any positive value, see figure <ref>. Note again that for every texture vector f⃗ on the curve C there is an angle α = α(f⃗), but the inverse is not necessarily true.


With the above preliminaries we can now start the derivation of the expression for the topological charge Q_S of a single Skyrmion. Consider the following integral along the curve C,

    I = ∮_ Cf⃗(s) ·(t⃗(s) ×dt⃗(s)/ds) ds.

Before we proceed with the evaluation of the integral, we note two points. First, while the integrand may seem to be the geodesic curvature of C on the sphere, we will see that it is not in general, differing by a minus sign when it is not. And second, although the curve r⃗_c(t) on the plane has been considered to run in the positive counterclockwise direction in the XY-plane, the curve C on the sphere, in general, is not necessarily oriented in the same direction. These apparent discrepancies, to be discussed in the Section V, have their origin in the previous observation that f⃗ is not being formally considered to be a parametrization of the sphere.


 Substituting the tangent vector (<ref>) into the integral (<ref>) yields,

    ∮_ Cf⃗(s) ·(t⃗(s) ×dt⃗(s)/ds) ds = ∮_ C dα + ∮_ Ccosθ∇ϕ· ds⃗.

Let us analyze the first integral over the α angle. Here it is important to notice that, because of the assumptions on the boundary curve r⃗_c(t), the tangent t⃗ to the curve C is also discontinuous at the points (f⃗(r⃗_1), f⃗(r⃗_2), ⋯, f⃗(r⃗_s)). This, in turn, implies that the angle α is discontinuous at those  points: Let t⃗_i^ - be the incoming tangent of the curve at the point f⃗(r⃗_i) and  t⃗_i^ + the outgoing one. We call the corresponding angles α_i^- and  α_i^+. Hence, 

    ∮_ C dα   =   ∫_α(0)^α_1^- dα + ∫_α_1^+^α_2^- dα + ⋯ + ∫_α_s^+^α(T) dα
        =    α(T) - α(0) - ∑_i=1^s ε_i,

where α(0) = α(f⃗(0)) and α(T) = α(f⃗(T)) with f⃗(0) = f⃗(T). We have assumed that such a point is not on any discontinuity. ε_i = (α_i^+ - α_i^-) is the angle between the tangents t⃗_i^ + and t⃗_i^ - at the discontinuity, given by

    sinε_i = f⃗_i ·(t⃗_i^ -×t⃗_i^ +) ,

see figure <ref>. 
Below we return to the analysis of the values of α(T) - α(0). 


The second integral in (<ref>), over the angle ϕ, can be rewritten in a way that allows for the determination of another contribution of the Skyrmion charge, using Green theorem. For this, we first recall that the origin r⃗_0 of the Skyrmion occurs where the spin texture becomes f⃗(r⃗_0) = ±ẑ, that is, where the azimuthal angle equals θ = 0 or π, respectively. Since the contour C encloses either of those points, the integrand of the second integral in (<ref>) is not analytic at θ = 0 or at π, preventing the direct use of Green theorem. However, adding and subtracting an appropriate quantity one obtains 

    ∮_ Ccosθ∇ϕ· ds⃗ = - ∮_ C(± 1 - cosθ) ∇ϕ· ds⃗±∮_ C∇ϕ· ds⃗,

where the + sign is used when the singularity is at θ = 0, namely f⃗(r⃗_0) = + ẑ, and the - sign when θ = π or f⃗(r⃗_0) = - ẑ. In this way, the integrand of the first integral on the right side of (<ref>) is analytic at the respective points and the integral can be performed using Green Theorem. This yields, up to a sign, the solid angle subtended by the contour C around the North or the South Pole,

    ∮_ C(± 1 - cosθ) ∇ϕ· ds⃗  = ±∬_ C dΩ.

The sign on the right side depends on the positive or negative orientation of the contour C, which in turn depends on the structure of the Skyrmion in the vicinity of the singularity. The above term can also be seen as a Berry phase type contribution <cit.>. However, the previous integral can be written in a manner that is more appropriate for our purposes, namely, by rewriting it in terms of the spin texture itself:

    ∮_ C(± 1 - cosθ) ∇ϕ· ds⃗    =    ∮_r⃗_c(± 1 - f_z )  ∇(arctanf_y/f_x) · d r⃗
        =    ∬_A{∂/∂ x[ f_z ∂arctanf_y/fx/∂ y] -
    ∂/∂ y[ f_z ∂arctanf_y/fx/∂ x] } dxdy 
        =    ∬_A f⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy .

The first equality follows by writing the angles θ and ϕ in terms of the f⃗ components, see (<ref>), hence allowing for expressing the line integral along the original contour r⃗_c(t). The second equality is obtained by means of Green Theorem, the surface integral being performed over the whole area A enclosed by the curve r⃗_c(t). The derivation of the last equality is a laborious exercise that requires the use of f⃗·f⃗ = 1. Note that in the expression of the last line one does not need to worry about the sign of the integral, as this will emerge naturally from the current spin texture.  The important issue of these signs will be further addressed below. 


Collecting the previous results (<ref>), (<ref>) and (<ref>) into (<ref>), we partially obtain

    α(T) - α(0) ±∮_ C∇ϕ· ds⃗   =   ∬_A f⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy + ∮_ Cf⃗(s) ·(t⃗(s) ×dt⃗(s)/ds)ds 
           + ∑_i=1^N arcsinf⃗_i ·(t⃗_i^ +×t⃗_i^ -)  .

First of all, it is simple to conclude that the left-hand-side of this expression must be 

    2π Q_S = α(T) - α(0) +  ∮_ C∇ϕ· ds⃗

 with Q_S ∈ℤ. This integer will be carefully  identified below as the topological charge of the Skyrmion enclosed by the curve r⃗_c. Quite generally, it must be an integer since, first, f⃗(0) = f⃗(T), then sinα(f⃗(0)) = sinα(f⃗(T)) and, hence, α(T) - α(0) = 2 π N with N an integer. Similarly, since

    ±∮_ C∇ϕ· ds⃗   =   ±∮_ C d ϕ
        =     2 π M,

with M a positive or negative integer, since the contour C encloses either the North or the South Pole of the unit sphere. Therefore, Q_S =  N +  M is an integer. However, it remains to understand how this integer number relates to the given spin texture f⃗(x,y) and to its behavior in the neighborhood of the singularity f⃗(r⃗_0) = ±ẑ. 



Let us address the role of the angles α and ϕ in the determination of the integer value of the Skyrmion charge Q_S in (<ref>). First, one can conclude that α(0) = α(T), namely N = 0. 
This follows from expression (<ref>) for α and from the geometrical facts, one, that 
θ̂ always points in the direction of increasing θ, two, that the tangent vector t⃗ has a definite orientation, counter or clockwise for any given curve C and, three, that either the North or South Pole are enclosed by C.
These three facts imply that the angle α retraces itself around C without completing a single turn. Thus, this leaves the value of Q_S to depend solely on the integral of ϕ in (<ref>) and (<ref>). This has an intrinsic topological solution. For this, we write the integral over ϕ in terms of an integral of the spin texture over the original contour r⃗_c(t),

    ±∮_ C_i dϕ = ±∫_r⃗_c^(i)∇(arctanf_y/f_x) · d r⃗_c^(i).

The interesting and relevant observation if that the integral on the right-hand-side is precisely the Index I(f⃗_T(r⃗_0)) of the transverse spin texture field f⃗_T(r⃗), times 2π, around r⃗_0 <cit.>. Such an Index is I = ± n with n a positive integer that counts the number of turns the angle ϕ does around the corresponding Pole. 
Based on our findings that the  transverse spin texture f⃗_T(r⃗) follows the transverse magnetic field B⃗_T(r⃗), we can extrapolate and assert that any spin texture, in the vicinity of the singularities f⃗_T(r⃗_0) = 0, behaves just as the magnetic field given in (<ref>). Hence, for cases (I) and (II) the index is I = -n while for (III) and (IV) one obtains I = +n. Therefore, in a very general way, the topological charge of a single Skyrmion can be expressed as,

    Q_S = f_z(r⃗_0)  I(f⃗_T(r⃗_0))  ,

demonstrating that the topological charge is a positive or a negative integer, its sign being determined by both, the value of f_z and the Index of f⃗_T, at the singularity. We believe that this last identification of the Skyrmion charge has not been pointed out before. Figure <ref> shows two cases of  Skyrmions with singularities at f⃗ = +ẑ and f⃗ = +ẑ, produced by the same transverse magnetic field but with opposite B_z component; all the thermodynamic properties are the same, except that the Skymion charges have opposite values.




Returning to the identification of  Q_S in (<ref>), and rewriting the integrals in terms of the spin texture f⃗ and the original boundary curve r⃗_c, one arrives at the sought for expression for the topological Skyrmion charge given in (<ref>) in the Introduction, but for a single Skyrmion. The extension to the total topological charge of a spin texture that holds an arbitrary number N of Skyrmions, whose origins are at r⃗_1, r⃗_2, …, r⃗_N, is straightforward. This is achieved with the 
reasonable assumption that, while in the vicinity of the origins the spin texture behaves as in the case of a single Skyrmion, the “local” textures can be  joined smoothly in a continuous and differentiable fashion. This assumption is verified by the numerical examples shown in the Appendix.
Very importantly, the boundaries between those local cases can be quite arbitrary and complicated, such as the boundary of our example of the three Skyrmions in figure (<ref>). This corroborates the importance of having obtained the local charge of a Skyrmion for an arbitrary boundary curve r⃗_c. Figure <ref> shows a 
sketch of the union of local boundary curves r⃗_c^(i) of several Skyrmions that yields a total curve r⃗_c, inside of which all Skyrmions are enclosed. Writing the different contributions of a local Skyrmion charge as Q_S^(i) = Q_2^(i) + Q_1^(i) +Q_0^(i), where

    Q_2^(i)   =   ∬_A_if⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy 
    
    Q_1^(i)    =    ∮_r⃗_c^(i)f⃗·( d f⃗/dt×d^2f⃗/dt^2) /|d f⃗/dt|^2 dt  
    
    Q_0^(i)    =    ∑_j=1^s^(i)sin^-1f⃗_j ·( . d f⃗/dt|_j^ -×. d f⃗/dt|_j^ +)/| d f⃗/dt|_j^ -| d f⃗/dt|_j^ +,

it is evident, since all boundary curves r⃗_c^(i) run with same positive orientation, that 

    Q_S = ∑_i=1^N Q_S^(i)

with an analogous expression for all contributions Q_2, Q_1 and Q_0. The obtained expression for the total Skyrmion charge Q_S = Q_2 + Q_1 + Q_0 is given by (<ref>) of the Introduction, with r⃗_c = ⋃_i=1^N r⃗_c^(i) the external boundary curve, A =  ⋃_i=1^N A_i, the total area, and the sum over the angles is only on those on the external boundary r⃗_c since all the internal angles cancel out, see figure <ref>. It is also of interest to conclude that the total Skyrmion charge can be written as,

    Q_S = ∑_i=1^N  f_z(r⃗_i)  I(f⃗_T(r⃗_i))  .

As we will argue in the following section, this last expression appears as a generalization of Poincaré theorem <cit.>. 



In the Appendix we illustrate the results of this section with explicit numerical calculations of several spin textures obtained by solving GP equations (<ref>) for our case study of a spinor BEC in the presence of an external field. 




§ DISSIMILARITIES BETWEEN THE SKYRMION CHARGE Q_S AND THE EULER-POINCARÉ CHARACTERISTIC Χ



The expression (<ref>) for the Skyrmion topological charge Q_S has a deceivingly similar appearance to the Euler-Poincaré characteristic χ of the Gauss-Bonet theorem of a given 2D surface embedded in 3D <cit.>. As a matter of fact, the derivation here presented for Q_S is similar, but not equal, to the derivation of the Gauss-Bonet theorem presented by do Carmo <cit.>. However, as we have been alluding to in the text, there are notable differences between them: A technical difference being that the spin texture vector f⃗ cannot always be both the normal vector and the parametrization to the sphere at the same time. A fundamental difference being that in the Gauss-Bonet theorem the main role is played by the surface itself, while in the Skyrmion case the protagonist is the spin texture f⃗. 


To be precise, given an orientable surface S, the GB theorem states that the Euler-Poincaré characteristic χ of a region R of the surface S, with a given closed boundary C and with M possible exterior angles θ_i is, <cit.>

    2πχ = ∬_R K  dA + ∮_C k_g ds + ∑_i=1^M θ_i

where K is the gaussian curvature at a point on the surface, k_g the geodesic curvature at a point of the curve C, the boundary of R. The curve C must be oriented in the positive direction. These are intrinsic properties of the surface and of curves drawn on it. If the region R is a simple region of the surface S, then χ = +1 always.  If the surface is compact with no boundaries, then χ = 2 - 2g where g is the genus or number of handles of the surface. And there are surfaces with χ = -1 such as the pseudo sphere and the so-called pair of pants. The above expression for χ is written in an invariant form, independent of any parametrization of the surface. For sure, one can write it in terms of a parametrization, but this one must be consistent with an assumed orientation of the surface and with the positive orientation of the curve C. To see this, let X⃗(σ_1,σ_2) be such a parametrization, then, the normal to the surface is

    n̂ = ∂_1 X⃗×∂_2 X⃗/|∂_1 X⃗×∂_2 X⃗|,

where ∂_α is the derivative with respect to σ_α. This normal must have the same orientation on the whole surface. For instance, if the surface is a sphere, then n̂ must point outwards or inwards over the whole sphere. The gaussian curvature can be expressed as,

    K = n̂·(∂_1 n̂×∂_2 n̂) .

Along the same lines, the tangent to the curve C is

    t⃗ = ∂X⃗(s)/∂ s,

and, hence, the geodesic curvature of C is

    k_g = n̂·(t⃗×dt⃗/ds) .

The exterior angles θ_i are, in an obvious notation,

    sinθ_i = n̂·( t⃗_i^-×t⃗_i^+) .


Therefore, if we compare χ, from (<ref>)-(<ref>), with Q_S from (<ref>), they look quite similar if we identify the parametrization X⃗ with f⃗, the surface thus being a sphere, and the normal n̂ with f⃗. However, these two identifications are incompatible in general. This can be readily seen if we calculate the normal to the sphere from the parametrization X⃗ = f⃗, 

    n̂ = ∂_x f⃗×∂_y f⃗/|∂_x f⃗×∂_y f⃗|.

One finds that, depending on the spin texture, n̂ = f⃗ or n̂ = -f⃗. And this can occur within the same spin texture, such as in the example of figure (<ref>). Therefore, if f⃗ is a parametrization, then f⃗ cannot be the normal always, and viceversa. This would result in severe problems since the normal would discontinuously change on the sphere. Hence, we have opted to consider 
that f⃗ is the outward normal to the sphere, at the expense of abandoning f⃗  as a parametrization. It is then not a surprise that we do not obtain GB theorem and, therefore, Q_S is not χ, since the sign of the former depends on the spin texture f⃗ and the latter is a topological intrinsic property of the given surface. Indeed, the fact that we can find positive or negative Skyrmion charges, simply by inverting the value of the origin of the Skyrmion f⃗(r⃗_0) is a consequence of not having considered f⃗ as a parametrization of the sphere. In this way, the integrands of Q_2 and Q_1 are not necessarily the gaussian or the geodesic curvatures of the sphere or of the curves drawn on it.
It is also of interest to point out that there is a theorem, due to Poincaré <cit.>, that states that χ = ∑_i I_i with I_i the Indices of a differentiable vector field on a compact surface with isolated singular points. That is, given the surface, the singularities must exist and their Indices add to χ. In the present case, see (<ref>), the singularities can be quite arbitrarily given by the the spin texture and, as a consequence, their topological charges depend on an additional sign given by f_z(r⃗_0) at the singularity. 


The previous observations are not definite, however, as we are aware that topology can yield results that may be difficult to visualize geometrically. That is, although we have pointed out evident differences between the Skyrmion topological charge and the Euler-Poincaré characteristic, we also concede that their respective expressions appear strikingly similar. Hence, we leave open the question of whether they may truly be the same topological object or not, perhaps for not so obvious or evident surfaces such as the unit sphere. This should be the subject of further scrutiny. Along the same lines, an approach in terms of homotopy theory <cit.> should complement our approach, basically rooted in differential geometry.




§ FINAL REMARKS


In this article we have provided a derivation of the Skyrmion topological charge Q_S of an arbitrary spatial region A ⊂ℝ^2 of a spin texture f⃗(r⃗), given by (<ref>). If the spin texture is continuous and differentiable everywhere, except perhaps at the origins of the Skyrmions, where f⃗ = ±ẑ, one can find the local topological charges of the present Skyrmions and, simply by joining the local regions, one can calculate the topological charge of several Skyrmion regions and, eventually, of the whole area where the spin texture is defined. While we can calculate the topological charge as the sum of the three main contributions Q_S = Q_2 + Q_1 + Q_0, one can also demonstrate that the local Skyrmion charge is the product Q_S = f_z(r⃗_0) I(f⃗_T(r⃗_0)), with I the Index of the transverse spin texture at the singularity r⃗_0. The total Skyrmion charge of a region, or of the whole spin texture, is the addition of the local Skyrmion charges. It may be of formal interest to highlight that the 2D contribution Q_2, the 1D Q_1 and the 0D Q_0 are all expressed in terms of a triple scalar product involving the spin texture f⃗ times the relevant cross product of the contribution at hand. We also find that, depending on the value of the spin texture at the boundary curve or curves, one recovers the usual expression for the Skyrmion charge, given solely in terms of the contribution Q_2. For instance, if  the texture completely covers the sphere, f⃗ = ±ẑ, or f_z = 0 at the boundary, the last two terms in (<ref>) or (<ref>) vanish. Notice also that we have defined the Skyrmion charge with a factor 1/2π in (<ref>), instead of 1/4π as in (<ref>), because of the similarities  with the Gauss-Bonet representation of the Euler-Poincare characteristic. Thus, while usually the Skyrmion charge for a closed sphere is Q = ± 1, here it is Q_S = ± 2, and as the charge of a half Skyrmion is considered Q = ± 1/2, here it is a Q_S = ± 1. As we indicated in the Introduction, the derived expression is quite general and independent of the origin of the spin texture.


In order to produce spin texture within a physically sensible system, we have studied a 1/2 spinor BE condensate in the presence of an external magnetic field. This system, in addition to generate interesting and non trivial spin textures, with which we can make a detailed and systematic verification of our main result, presents quantum vortices, defects of topological nature as well. As we have indicated, one can trace the appearance of the vortices and of the Skyrmions to the behavior of the external magnetic field in the vicinity of the locations r⃗_0 where the transverse magnetic field vanishes. Indeed, by looking a the emergence of quantum vortices, say, in (<ref>), one can conclude that the vortex local topological charge is Q_V = ± I(B⃗_T(r⃗_0)), where the sign ± is if the vortex appears in the ψ_+ or ψ_- spinor component, respectively, and I(B⃗_T(r⃗_0)) is the Index of the transverse magnetic field in the vicinity of its vanishing location r⃗_0. However, since the transverse spin texture f⃗_T follows exactly the same pattern as the transverse magnetic field B⃗_T, see Appendix, then the local quantum vortex and Skyrmion topological charges are exactly the same. That is, for the present case study, the external magnetic field determines the topology of all ensuing structures. These results can definitely be tested with the currently achieved experimental capabilities in the study of spinor Bose-Einstein condensates. For other physical spin textures one can perhaps always find the underlying field or interaction responsible for the emergence of the corresponding topological structures. 


While we have left out several additional observations of this problem, related to its intrinsic differential geometry and topology, we mention one that should be of importance for further analysis. This regards the issue of what we call a “Skyrmion” structure in a spin texture. That is, the emphasis is always in the role of the location where one finds the origin of the Skyrmion f_z(r⃗_0) = ±ẑ and, therefore, one may believe that the determination of the topological charge needs of a boundary curve that encloses such a point or points. However, this is not completely true or necessary. For this we point out expression (<ref>) where we identified the Skyrmion charge in terms of the angle difference α(T)-α(0) and the contour integral over ϕ. We recall that we derived our results by assuming a contour that enclosed the singularity yielding  α(T)-α(0) = 0 and the topological charge was contained in the ϕ contour integral. However, if we choose a contour that does not enclose the singularity, then the Index of the transverse spin texture vanishes, but now α(T)-α(0)  0. And, for sure, the topological charge is now accounted for by the angle α, namely 2π Q_S = α(T)-α(0). This indicates that if we take a fully arbitrary boundary curve r⃗_c, yet closed and positively oriented, that does not necessarily enclose singularities, one still finds the topological charge contained within the encircled spin texture. In other words, the topological content of the Skyrmion is within the whole spin texture, as Topology may have indicated us in advance.


Acknowledegment. This work was partially funded by grant PAPIIT-UNAM IN117623. VRR acknowledges fruitful discussions with Leonardo Patiño at the early stages of this work.




§ APPENDIX: A GALLERY OF VORTICES AND SKYRMIONS, SOLUTIONS TO GP EQUATIONS (<REF>).


The figures throughout the text and in this Appendix are found from numerical solutions of the two coupled Gross-Pitaevskii equations (<ref>). We use a standard method described in Refs. <cit.>, in a GPU processor, within a 256 × 256 grid. We consider arbitrary dimensionless units such that the solutions are numerically stationary and the desired configuration is clearly observed. We use ħ = m = 1, the interatomic coupling g = 15000 and, in all cases, the computational box is a square of side L_0 = 56.72. The values of the isotropic frequency trap ω, the magnetic fields and the corresponding chemical potential μ are given in the figures captions. 



 Recalling that the full magnetic field is B⃗ = B⃗_T + B_z ẑ, an interesting point to highlight is that both solutions (i) and (ii) in (<ref>) are possible in each position where the transverse magnetic field vanishes. However, if B_z = 0, we find numerically that these solutions appear at random, each 50% of the time, because our numerical methods requires a random guess of the initial wavefunctions <cit.>. However, if we turn on the constant B_z field, we find, as expected, that the sign of this field component forces one of the solutions, namely, if B_z > 0, then solution (i) occurs, while (ii) ensues for B_z < 0. 



The solutions of figures <ref> and <ref>, as well as those below <ref> and <ref>, have been obtained with transverse magnetic fields that become zero at appropriate locations on the XY-plane within the BE condensate. These fields are in turn tailored by superposing magnetic fields of a set of long (infinite) wires at appropriate locations, carrying also an appropriate current I. We recall that the field of a long straight wire in the ẑ-direction, located at  (x_0,y_0), with current I, is

    B⃗_ wire(r⃗) = 2I/c-(y-y_0) x̂ + (x-x_0) ŷ/(x-x_0)^2 + (y-y_0)^2.

In the figures captions we indicate the values of the currents and the locations of the wires. The wires are placed outside of the numerical calculation box.


In the following gallery of figures we show the condensate superfluid density ρ_α and velocity v⃗_α, for component α = +, panels (A) and (B); and for component α = -, panels (C) and (D), respectively. In panel (E), we show the Skyrmion spin texture and in panel (F) the transverse spin texture field lines f⃗_T = f_x x̂ + f_yŷ. In panel (G) we show the transverse magnetic field lines B⃗_T =  B_x x̂ +  B_y x̂ that produces the corresponding condensate. Note that the transverse spin texture streamlines follow those of the transverse magnetic field. Details of the corresponding cases are given in the figures captions. 













§ REFERENCES





 Paper Title: Experimentally probing Landauer's principle in the quantum many-body regime.






These authors contributed equally to this work: 
mailto:stefan.aimet@gmail.comstefan.aimet@gmail.com, mailto:amintajik.physics@gmail.comamintajik.physics@gmail.com
Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

These authors contributed equally to this work: 
mailto:stefan.aimet@gmail.comstefan.aimet@gmail.com, mailto:amintajik.physics@gmail.comamintajik.physics@gmail.com
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
Department of Physics and Astronomy, and Stewart Blusson Quantum Matter Institute,
University of British Columbia, V6T1Z1 Vancouver, Canada

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Institute of Theoretical and Computational Physics, University of Crete, 71003 Heraklion, Greece
Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Dipartimento di Fisica,  Università di Pavia, 27100 Pavia, Italy
Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

§ ABSTRACT 
Landauer’s principle bridges information theory and thermodynamics by linking the entropy change of a system during a process to the average energy dissipated to its environment. Although typically discussed in the context of erasing a single bit of information, Landauer’s principle can be generalised to characterise irreversibility in out-of-equilibrium processes, such as those involving complex quantum many-body systems. 
Specifically, the relationship between the entropy change of the system and the energy dissipated to its environment can be 
decomposed into changes in quantum mutual information and a difference in relative entropies of the environment. 
Here we experimentally probe Landauer’s principle in the quantum many-body regime using a quantum field simulator of ultracold Bose gases. Employing a dynamical tomographic reconstruction scheme, we track the temporal evolution of the quantum field following a global mass quench from a Klein-Gordon to a Tomonaga-Luttinger liquid model and analyse the information-theoretic contributions to Landauer’s principle for various system-environment partitions of the composite system. Our results agree with theoretical predictions, interpreted using a semi-classical quasiparticle picture. Our work demonstrates the potential of ultracold atom-based quantum field simulators to experimentally investigate quantum thermodynamics.



Experimentally probing Landauer's principle in the quantum many-body regime
    Jens Eisert
    Received ..... .., ....; accepted ......, ....
===========================================================================

Information theory and thermodynamics constitute foundational pillars of modern technology, underpinning our understanding of modern computers and heat engines, respectively. While these disciplines may seem distinct, they are intricately connected. This connection is encapsulated in Landauer's principle, first articulated in 1961 <cit.>. Landauer understood that erasing a bit of information in a computer is not for free but is instead accompanied by a minimal energy cost dissipated to the environment. Subsequently, various experimental studies <cit.> have confirmed this lower bound on energy dissipation for near-reversible bit erasure. Logical operations thus imply irreversibility, posing a fundamental theoretical limitation to the design of any small-scale energy-efficient information-processing technologies <cit.>. However, extending beyond the case of bit erasure, recent influential work <cit.> has generalised the link between information theory and thermodynamics. Using a quantum statistical mechanics framework, their work reinterprets Landauer's principle to relate the entropy change of a system to the energy dissipated to its environment in very general out-of-equilibrium processes. This relationship can be quantified by a measure of process irreversibility <cit.>. Such an extension is thus particularly relevant for quantum many-body systems, where contributions to irreversibility remain an area of active research, notably in phenomena such as equilibration and thermalisation <cit.>. 

In this work, we experimentally probe Landauer's principle for an out-of-equilibrium process in the quantum many-body regime by measuring the time evolution of quantum information-theoretic quantities that characterise irreversibility. We present a crisp information-inspired interpretation of correlations present in states of quantum many-body systems in terms of entropic expressions. 

Concretely, we consider a system-environment composite initially in the state ϱ_ SE(0), evolving under a global unitary time evolution U to the state ϱ_ SE(t)=Uϱ_ SE(0)U^†. The reduced state of the system S (environment E) is ϱ_ S(E)(t)=_E(S)[ϱ_ SE(t)]. 

First, let us assume that there are no initial correlations between the system and environment, and the environment is in a thermal state γ_ E^β_ E=e^-β_ EH_ E/[e^-β_ EH_ E] at inverse temperature β_ E with respect to its Hamiltonian H_ E. In this case, the entropy production Σ(t), which is a measure of irreversibility <cit.>, can be decomposed as

    Σ(t):=I_ SE(t)+D(ϱ_ E(t)||γ_ E^β_ E)  .

Here, the quantum relative entropy

    D(ϱ_ E(t)||γ_ E^β_ E)=[ϱ_ E(t)logϱ_ E(t)]-[ϱ_ E(t)logγ_ E^β_ E]   ,

quantifies the deviation of the environment from its initial thermal state. The quantum mutual information 

    I_ SE(t)=S(ϱ_S(t))+S(ϱ_E(t))-S(ϱ_SE(t))

measures the system-environment correlations, where 

    S(ϱ)=-[ϱlogϱ]

denotes the von Neumann entropy of a state ϱ.



Now we move on to a more general scenario where the system and the environment are initially correlated, and the state of the environment deviates from thermal equilibrium. We thereby introduce the generalised entropy production ΔΣ :=Σ(t)-Σ(0) where the initial correlations and deviations from thermal equilibrium are accounted for by a resource cost Σ(0)=I_ SE(0)+D(ϱ_ E(0)||γ_ E^β_ E) <cit.>, with β_ E being an effective inverse temperature <cit.>. Landauer's principle can now be formulated <cit.> as an equality as 

    ΔΣ=β_ EΔ E_ E+Δ S=Δ I+Δ D   ,

relating the system's entropy 
change Δ S:=S(ϱ_ S(t))-S(ϱ_ S(0)) to the energy dissipated to the environment Δ E_ E:=[(ϱ_ E(t)-ϱ_ E(0))H_ E].
Alternatively, ΔΣ also decomposes into the change of quantum mutual information, Δ I=I_ SE(t)-I_ SE(0), expressing how much the system-environment correlations change along the out-of-equilibrium process, as well as the term Δ D=D(ϱ_E(t)||γ_E^β_E)-D(ϱ_E(0)||γ_E^β_E) that quantifies how much both the initial and final state of the environment are different from the reference equilibrium thermal state. We provide a more thorough motivation for the generalised entropy production ΔΣ in the Methods section. 

Expressed in this form, Landauer's principle serves as a means of tracking changes in information-theoretic quantities, which compose generalised entropy production ΔΣ in two different ways via a single equation. Herein lies the power at the heart of thermodynamics: just as knowing the precise microscopic degrees of freedom, such as the position and momentum of each particle in a classical gas, is not useful for drawing meaningful conclusions about its dynamics, knowing the density matrix of a quantum many-body system is similarly not descriptive. In traditional thermodynamics, macroscopic variables like changes in pressure or volume effectively characterise the dynamics. Similarly, analysing the information-theoretic quantities in Landauer's principle allows for a simple characterisation of the irreversibility in the out-of-equilibrium dynamics of a quantum many-body system.

So far, there has been no systematic experimental pursuit of probing Landauer's principle in the quantum many-body regime. Here, we address this gap by measuring the time evolution of the different terms in Eq. (<ref>) in a (1+1)D quantum field simulator of tunnelling-coupled ultracold Bose gases. This experimental platform has proven to be a suitable testbed for simulating one-dimensional quantum field theories. Previous studies using this setup have examined the area law of quantum mutual information in thermal equilibrium <cit.> 
and the propagation of second- <cit.> or higher-order correlations <cit.>. 

In this experiment, two parallel ultracold clouds of  atoms (illustrated in Fig. <ref>) are confined in highly anisotropic magnetic traps produced by an atom chip <cit.>. In the axial direction (z), the clouds are confined in a parabolic trap superposed with an optical dipole potential to produce hard walls, while in the radial directions (x,  y), by a double-well trap with an adjustable barrier in between, created by RF dreessing <cit.>. The single-particle tunnelling rate can be modified by the amplitude of a radio-frequency field created by two parallel wires on the atom chip.

The bosonic quantum field operator for each condensate can be written using the phase-density representation as ψ_n(z) = √(ρ_n(z))e^iθ_n(z), where θ_n and ρ_n denote the phase and density of the respective condensate, indexed by n=1,2. We prepare the Bose-Einstein condensates in a global thermal state of a massive Klein-Gordon Hamiltonian 

    H_ KG   = ∫_0^L dz [ g_ 1Dδρ^2(z) + ħ^2 n_ 1D/4m (∂_zφ(z))^2 
        + ħ J n_ 1Dφ^2(z) ]

which models low-energy phononic excitations. Here, ħ is the reduced Planck constant, m is the atomic mass, g_ 1D is the effective one-dimensional atomic interaction strength, L=49 is the axial length of the condensates, n_ 1D≈ 70^-1 is the average linear density and J≈ 2π× 0.8 Hz is the tunnelling rate introduced before. The operators φ(z) = θ_1(z) - θ_2(z) and δρ(z) = [ρ_1(z) - ρ_2(z)]/2 represent the relative phase and relative density, respectively, and satisfy the commutation relation [φ(z), δρ(z')] = iδ(z-z'). 

To initiate out-of-equilibrium dynamics in the experimental system and measure the evolution of information-theoretic quantities, we rapidly quench J to zero by ramping up the barrier between the condensates within approximately 2. This change corresponds to a global mass quench of the Klein-Gordon Hamiltonian. The condensates then evolve independently under the post-quench Tomonaga-Luttinger liquid Hamiltonian <cit.> (H_KG with J=0) for times t, up to 65.



 

For every time step t, we turn off all the traps and let the atoms fall freely for 15.6. The clouds will expand and interfere, resulting in absorption pictures similar to the one shown in Fig. [fig:setupandtomography]<ref>fig:setupandtomographyb, allowing us to measure the spatially resolved relative phase φ(z) between them. Since the detection process is destructive, the measurements are repeated to gather statistics. In our current experimental setup, relative density fluctuations δρ are not directly measurable, which prompted the development of a dynamical tomographic reconstruction technique <cit.> to access all of the elements of the covariance matrix 

    Γ(t)=[ Γ_φφ(t) Γ_φρ(t); Γ_ρφ(t) Γ_ρρ(t);         ]  .

Here, the elements are defined as [Γ_φφ(t)]_m,n=⟨φ(z_m, t)φ(z_n, t)⟩, [Γ_ρρ(t)]_m,n=⟨δρ(z_m,t)δρ(z_n,t)⟩, and [Γ_φρ(t)]_m,n=[Γ_ρφ(t)]^T_m,n=⟨1/2{φ(z_m, t),δρ(z_n, t)}⟩, all on a discrete grid with N pixels (m,n∈{1,…,N}), which are determined by resolution constraints of our imaging system, thus introducing an ultraviolet (UV) cutoff.

The phase–phase correlations for different evolution times after the quench can be measured directly from the extracted relative phase profiles. Assuming that the short-time dynamics after the quench are governed by the Tomonaga–Luttinger liquid (TLL) Hamiltonian, as time progresses, the initial eigenmodes of the relative density transform into the phase quadrature, and the phase quadrature transforms into the relative density. This transformation allows us to extract information about these eigenmodes by fitting the initial second-order correlation functions of phase–density, and density–density with the observed evolution of the phase–phase correlations in momentum space. 

The schematics in Fig. [fig:setupandtomography]<ref>fig:setupandtomographyc illustrate our dynamical tomographic reconstruction scheme. We follow the technique in Ref. <cit.> for various input intervals with varying starting points. By scanning the starting points of these input intervals throughout the trapping times, we can reconstruct the full covariance matrix at every time t, as depicted in Fig. <ref>. The interval length (32.5 close to L/c ≈ 27) is selected to be sufficiently long for the slowest eigenmode to acquire enough dynamical phase for a stable reconstruction, yet short enough to prevent mode interactions from affecting the reconstruction. A detailed overview of this reconstruction process is provided in the Methods section.

The quadratic form of the pre- and post-quench Hamiltonians allows us to work within the framework of Gaussian quantum information theory <cit.>. In this framework, the covariance matrix Γ captures all the accessible information about the state of the composite system during the dynamics, from which we can extract the information-theoretic quantities of interest in Eq. (<ref>). As has been demonstrated in various experiments <cit.>, the quadratic approximation of the Hamiltonian accurately captures the dynamics for the considered time scales. Even if the true dynamics deviate from the Gaussian regime and violate Tomonaga-Luttinger liquid theory, a Gaussian extremality argument presented in the Supplementary Material justifies the Gaussian tomography scheme and provides bounds on the information-theoretic quantities. Thus, based on our technique of dynamical tomographic reconstruction of covariance matrices, Landauer's principle can also be meaningfully experimentally probed for interacting models.

Having experimentally reconstructed the post-quench time evolution of the covariance matrices, we partition the one-dimensional field of length L into two distinct subregions and split the covariance matrix accordingly (Fig. [fig:setupandtomography]<ref>fig:setupandtomographyc). Since the quantum field is isolated from its surroundings, one subregion serves as the system S with length L_S, while the other subregion functions as the environment E with length L_E = L - L_S. We probe Landauer's principle (Eq. (<ref>)) for various system-environment bipartitions by calculating β_ EΔ E_ E, Δ S, Δ I and Δ D. The results are presented in Fig. <ref> as a function of time for different subregion size ratios and in Fig. <ref> as a function of subregion size for different times. The time scales are shown in units of ct/L, where c=√(g_ 1Dn_ 1D/m)≈ 1.8  / is the speed of sound. Overall, a very good fit of the data (circles) is obtained compared to the theoretical calculations (shaded areas), which use the lowest N=7 modes and consider the imaging resolution of the experiment. The low-lying modes already capture the dynamics of the continuum theory very well. The error bars representing the 68% confidence intervals are obtained via bootstrapping <cit.> with 999 samples, considering the uncertainty in the tunnelling rate and the estimated initial global temperature. The effective inverse temperature of the environment β_ E with respect to the post-quench Tomonaga-Luttinger liquid Hamiltonian is computed using field-theoretic simulations by constructing a global thermal state of the Klein-Gordon Hamiltonian with the estimated initial global temperature from the experimental data. 

First, we examine the decomposition of the generalized entropy production ΔΣ into β_ EΔ E_ E and Δ S. The term associated with the energy dissipated to the environment, β_ EΔ E_ E, reveals that a small amount of energy flows into (or out of) the environment for small (or large) systems around ct/L_ S=1. However, this term contributes only minimally to the irreversible post-quench dynamics of the quantum field. In contrast, the entropy change of the system, Δ S, dominates the dynamics and shows a clear linear increase up to ct/L_ S<1, followed by more pronounced growth.

In comparison, the second decomposition involving Δ I and Δ D sheds a slightly different focus on the out-of-equilibrium dynamics. The largest contribution to ΔΣ, the change in quantum mutual information, Δ I, exhibits similar behaviour to Δ S but with a higher magnitude. This increased magnitude is accounted for by the additional entropy change in the environment. However, Δ I competes with the term proportional to the change of the environment, Δ D, which accounts for both the entropic and energetic changes to the environment. This makes Δ D a particularly interesting quantity to characterise the out-of-equilibrium dynamics of the environment experimentally. In our case, Δ D increases negatively, mirroring otherwise the behaviour of Δ S, due to the small energetic contribution. 

To interpret the results, we need to consider the effects of Neumann and Dirichlet boundary conditions on the quantum field, as shown with field-theoretic simulations in Fig. <ref>. These boundary conditions may be contrasted with the curved
backgrounds giving rise to effective boundary conditions discussed in Refs. <cit.>. The experimental system exhibits Neumann boundary conditions (∂_zφ(z)|_z=0, L=0) due to the vanishing particle current at the edges. Plots contrasting the scaling with subregion size are shown in the Supplementary Material. 

However, we first consider simulations for Dirichlet boundary conditions (φ(z)|_z=0, L=0), which are simpler to understand. Here, the well-established semi-classical quasiparticle picture <cit.> offers a clear and intuitive model for the post-quench dynamics of global mass quenches <cit.>, depicting the propagation of short-range initial correlations via ballistically moving quasiparticles. Following the homogeneous global quench, the initial Klein-Gordon thermal state becomes a non-equilibrium state with excess energy relative to the post-quench Tomonaga-Luttinger liquid Hamiltonian, which governs the time evolution. This initial state acts as a source of quasiparticle pairs emitted globally from every point across the length of the composite system. Each pair in the bulk consists of two quasiparticles correlated with each other and moving in opposite directions at the same speed. For our set of parameters, due to the finite initial correlation length, there exists a small localised region where quasiparticles are correlated.

In this quasiparticle picture, the change of quantum mutual information Δ I between the system and environment is proportional to the number of pairs of quasiparticles that are shared between the two different subregions. The spread of correlations of short-range interacting models is described by a linear effective light cone originating at the system-environment boundary. The increase in correlations is thus proportional to the section of the light cone, or in other words, it is proportional to the distance between the quasiparticles of the pair emitted at the system-environment boundary (Fig. [fig:DBC_main]<ref>fig:DBC_mainc). If the system is smaller than the environment, once the first quasiparticle of this boundary pair reaches the edge of the system at ct/L_ S=1 and undergoes reflection, the section of the system-environment boundary pair light cone remains constant. The behaviour of Δ I transitions to a plateau value. From a system perspective, the composite system appears to have locally equilibrated to a generalised Gibbs ensemble steady state <cit.>. For finite-size composite systems, the plateau eventually ends again at ct/L_ E=1 when the other quasiparticle of the boundary pair reaches the edge. The distance between the two quasiparticles decreases until they meet again at ct/L=1, explaining the phenomenon of recurrences <cit.> (with respect to the correlation functions). 



On the other hand, Δ I behaves differently for Neumann boundary conditions. Neumann boundary conditions introduce a zero mode, whose variance does not evolve harmonically like the other momentum modes, but quadratically (phase diffusion <cit.>). The global fluctuations of the zero mode contribute to entropic quantities <cit.> and as they increase with time after a quench they eventually dominate over the contribution of other modes. Since the quasiparticle picture cannot capture this zero-mode feature, its predictions for ct/L_ S>1, when a plateau value should be reached, need to be modified under the present Neumann boundary conditions. The above features of the dynamics also apply to the change of system entropy Δ S. In addition, the zero mode restricts the validity of the tomography scheme to ct/L<1. 

The energetic contribution, β_ EΔ E_ E, can also be explained using the quasiparticle picture. Initially, β_ EΔ E_ E remains constant because quasiparticles traveling between the system and the environment carry equal energy, resulting in zero net energy flux. However, a finite-size energy flow becomes apparent when quasiparticles from the edge of the system cross the system-environment boundary. The translational invariance of the homogeneous quench is broken due to the higher energy density at the edges of the composite system for Neumann boundary conditions (the reverse is true for Dirichlet boundary conditions, where the energy density is lower at the edges). This edge region has a size of the order of the initial correlation length. The resulting behaviour of the change of the environment Δ D reflects both the entropic and energetic fluxes. 

In this study, we have experimentally probed Landauer's principle in the quantum many-body regime following a global mass quench in an ultracold atom-based quantum field simulator. By reconstructing the dynamics of the state of the composite system, we examined the information-theoretic quantities related by Landauer's principle, which we interpret using a semi-classical quasiparticle picture. Our approach underscores the general utility of Landauer's principle for characterising the irreversibility of out-of-equilibrium dynamics in quantum many-body systems. The way we express correlations in terms of entropic quantities with an 
information-theoretic meaning can be viewed as a vehicle to capture many-body correlations in 
non-equilibrium. The Gaussian extremality argument may offer a pathway to extend our methodology to capture features of out-of-equilibrium processes in interacting models, despite the challenges posed by non-Gaussian effects. Looking forward, progress has already been made towards investigating a local quench involving two Bose-Einstein condensates of different temperatures being joined together <cit.>. This protocol represents a crucial primitive towards developing a quantum field thermal machine <cit.>, potentially allowing for Landauer erasure as a mechanism to reduce the entropy of a subregion in the quantum many-body regime, hence functioning as an effective cooling mechanism. Our current work demonstrates the potential of ultracold one-dimensional gases as testbeds for quantum thermodynamics in the many-body regime, where complexity, quantum effects, and finite-size effects play a crucial role.


We thank Bernhard Rauer for conducting the experimental measurements. In addition, we are grateful to Marcus Huber, Patrick Emonts, Ivan Kukuljan, Mauro Paternostro, Philippe Faist, Florent Goulette, Marek Gluza, and Igor Mazets for helpful discussions and comments. This work has been supported by the DFG Research Unit FOR 2724 on `Thermal machines in the quantum world', the FQXi, the Quantum Flagship (`Millenion' and `PasQuans2'), the Einstein Research Unit, the BMBF, Berlin Quantum, and the ERC-AdGs `Emergence in Quantum Physics' and 
`Delineating the boundary
between the computational
power of quantum and
classical devices'.

S.S. acknowledges support by
the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No.  101030988. 



§ METHODS




 §.§ Landauer's principle

Consider a system S and an environment E with a bipartite Hilbert space ℋ=ℋ_S⊗ℋ_E. For an initial product state ϱ_SE=ϱ_S⊗γ_E^β_E, where the environment is in a thermal state γ_E^β_ E=e^-β_ E H_ E/[e^-β_E H_E], the entropy production under global unitary dynamics is given by <cit.>

    Σ(t)=I_ SE(t)+D(ϱ_E(t)||γ_E)=D(ϱ_ SE(t)||ϱ_S(t)⊗γ_E^β_ E)  .

Since it is expressed as a quantum relative entropy, Σ is non-negative due to Klein's inequality <cit.>, which is a manifestation of the second law of thermodynamics. More generally, as in our work, we start with a resourceful initial state that includes both initial correlations I_ SE(0) and athermality of the environment D(ϱ_E(0)||γ_E^β_ E). While defining temperature for a non-equilibrium state is ambiguous, we determine the effective inverse temperature of the environment β_ E by demanding that the environment's energy is equal to that 
of a Gibbs state, such that [ϱ_ EH_ E]≡[γ_E^β_ EH_ E] <cit.>. This choice also minimises the athermality term. Thus, we define the resource cost of the initial state <cit.>

    Σ(0)=I_ SE(0)+D(ϱ_E(0)||γ_E^β_ E)=D(ϱ_ SE(0)||ϱ_S(0)⊗γ_ E^β_ E).


Although there is no unique consensus on entropy production for initial non-equilibrium environment states <cit.> and initial correlations <cit.>, we define Landauer's principle Eq. (<ref>) in terms of generalised entropy production ΔΣ:=Σ(t)-Σ(0) as

    ΔΣ :=β_EΔ E_ E+Δ S=Δ I+Δ D  .

This can be derived similarly to Ref. <cit.>. One caveat is that ΔΣ may not necessarily be non-negative, which is a necessary criterion for a strictly valid definition of entropy production. However, this makes sense as ΔΣ may be considered a finite-difference version of the entropy production rate ΔΣ=Σ(t)-Σ(0)=∫_0^t dt' dΣ(t')/dt', which can also be negative <cit.>. Thus, ΔΣ intuitively measures the "extra" irreversibility associated with the protocol, excluding the initial resources.



 §.§ Experimental implementation

The experimental setup consists of a pair of tunnelling-coupled one-dimensional  quasicondensates, following a protocol implemented previously <cit.>. The magnetic fields required to trap the gas in a double-well potential are generated by an atom chip <cit.>, achieving transverse trapping frequencies of ω_⊥/2π = 1.4 kHz. After preparing a thermal equilibrium state at a temperature of 49 nK, all traps are turned off, and absorption imaging of the atoms is performed during a free-fall time of 15.6 ms to measure the two-dimensional atomic density distributions. This procedure enables us to detect the relative phase between the condensates across the length of the quasicondensate using matter interferometry (Fig.  [fig:setupandtomography]<ref>fig:setupandtomographyb). Due to the destructive nature of the detection process, many repetitions of the experiment are necessary to accumulate numerous phase correlation images throughout the time evolution, thereby also ensuring statistically accurate expectation values.



 §.§ Quantum field simulation

We have the Klein-Gordon Hamiltonian as a model for our quantum field simulator

    H_ KG=∫_0^Ldz [g_ 1Dδρ(z)^2 +ħ Jn_ 1Dφ(z)^2+ħ^2n_ 1D/4m(∂_zφ(z))^2] ,

where the length of the composite system is set to L =49 with a mean density of n_ 1D = 70^-1. The coupling strength between the particles is chosen as g_ 1D = 8.594× 10^-39kg·m^3·s^-2, and we have an atomic mass of m = 1.433× 10^-25kg.
The initial single-particle 
tunnelling rate is J= 2π× 0.76Hz. The post-quench evolution is described by the Tomonaga-Luttinger liquid Hamiltonian

    H_ TLL=∫_0^L dz[g_ 1D(δρ(z))^2+ħ^2 n_ 1D/4m(∂_zφ(z))^2
     ]  .




 §.§ Dynamical tomographic reconstruction of covariance matrices

The tomography scheme for reconstructing the initial thermal covariance matrix is detailed in Refs. <cit.>. Here, we extend this method to dynamically reconstruct the state over time. We provide a brief overview below.
Given Neumann boundary conditions ∂_zφ(z)|_z=0,L=0, the relative phase φ(z) and relative density fluctuations δρ(z) can be expanded in Fourier space in terms of cosine eigenfunctions

    f_k^φ(z) = 
    2(1/kħπ√(mg_ 1D/n_ 1D))^1/2cos(kπ/Lz)   if  k > 0 ,
    
    1    if  k = 0,
    
    
    f_k^ρ(z) = 1/L(kħπ√(n_ 1D/mg_ 1D))^1/2cos(kπ/Lz)   if  k > 0 ,
    1/L   if  k = 0.


The Hamiltonian of the post-quench Tomonaga-Luttinger liquid Eq. (<ref>) is then diagonalised as 
(indices omitted for brevity from now on)

    H=ħ u/2δρ_0^2+∑_k=1^∞ħω_k/2[δρ_k^2+φ_k^2]

with u:=2g_ 1D/ħ L and 

    ω_k:=√(g_ 1Dn_ 1D/m)kπ/L  .

The interaction strength g_ 1D is given by 

    g_ 1D:=ħω_⊥a_s2+3a_s n_ 1D/(1+2a_s n_ 1D)  ,

where a_s=5.2 nm is the three-dimensional scattering length.

Consequently, the time evolution of modes (k>0) in Fourier space is

    φ_k(t)=φ_k(0)cos(ω_k t)+δρ_k(0)sin(ω_k t),
    δρ_k(t)=δρ_k(0)cos(ω_k t)-φ_k(0)sin(ω_k t),

with non-harmonic time evolution of the zero mode

    φ_0(t)= utδρ_0(t)+φ_0(0),
    δρ_0(t)=δρ_0(0)= const   .

As mentioned in the main text and shown in Fig. <ref>, the covariance matrices can be dynamically tomographically reconstructed by measuring the relative phase profiles using matter interferometry at various times. Subsequently, the relative phase correlations are computed as
Φ^2_m,n(t)=⟨(φ(z_m,t)-φ(z_0, t))(φ(z_n,t)-φ(z_0, t))⟩ with z_0 as a reference position. 
We then expand in Fourier space, see Eq. (<ref>), and expand the time evolved operators to express the measured phase correlation function in terms of the covariance matrix elements prior to the time evolution

    Φ^2_m,n(t)=⟨(φ(z_m,t)-φ(z_0, t))(φ(z_n,t)-φ(z_0, t))⟩
    
    =∑_k,l=1^N-1f_k,l^m,n⟨φ_k(t)φ_l(t)⟩

where

    f_k,l^m,n=(f_k^φ(z_m)-f_k^φ(z_0))(f_l^φ(z_n)-f_l^φ(z_0))

and so we see that the covariance matrix Γ(t) can be obtained by fitting the coefficients f_k,l^m,n using the phase-space correlations

    Φ^2_m,n(t+Δ t)
    		=∑_k,l=1^N-1f_k,l^m,ncos(ω_k Δ t)cos(ω_lΔ t)Γ_φφ; k,l(t)
    +∑_k,l=1^N-1f_k,l^m,nsin(ω_k Δ t)sin(ω_l Δ t)Γ_ρρ; k,l(t)
    +∑_k,l=1^N-1(f_k,l^m,n+f_l,k^m,n)cos(ω_k Δ t)sin(ω_l Δ t)Γ_φρ; k,l(t)  .

By gathering statistics, we employ the same method as in Ref. <cit.>, but simply shift the window of observation.

We obtain the covariance matrix in momentum space for the first N modes that give physical (positive) occupation numbers. The imaging system is additionally accounted for by a Gaussian point spread function with standard deviation of 3μ m to estimate the initial state. In this way, we can also confirm the global initial thermal state and extract a pre-quench temperature. We use discrete cosine eigenfunctions to convert the momentum space correlators to real-space correlators, from which we compute the quantities of interest.






 §.§ Probing Landauer's principle from covariance matrices

In this section, we delineate how the quantities in Landauer's principle, as expressed in Eq. (<ref>), are computed using the full covariance matrix Γ in real space, defined on a grid of N pixels.
First, the von Neumann entropy S(Γ) is computed using the symplectic eigenvalues λ_n of the covariance matrix Γ <cit.>. These eigenvalues correspond to the eigenvalues of iΩΓ, where 

    Ω:=[    0  1_N; -1_N    0 ]

is the 
standard kernel of the symplectic form capturing the canonical commutation relations. The expression for 
the von Neumann entropy 
S(Γ) of a Gaussian state is given by

    S(Γ)=∑_n=1^N[(λ_n+1/2)ln(λ_n+1/2)-(λ_n-1/2)ln(λ_n-1/2)].


Given two full covariance matrices corresponding to different times and partitioning it into system and environment subspace covariance matrices (Fig. [fig:setupandtomography]<ref>fig:setupandtomographyc), this allows us to compute the entropy change of the system Δ S and the mutual information Δ I.

Next, we illustrate how to calculate the energy of the environment with respect to the post-quench Tomonaga-Luttinger liquid Hamiltonian. To obtain the correct environment Hamiltonian, we need to proceed with care. The field corresponding to the environment is discretely defined with N_ E=N (L_ E/L) pixels. A naive approach would discretise the continuous Hamiltonian defined over the length of the environment, but this nearest-neighbour phase coupling Hamiltonian would yield a non-linear dispersion relation, inadequately describing the physics of the environment for the continuum model used in the experiment. Instead, we start from the environment Hamiltonian of the continuous theory, diagonalise it to obtain a momentum decomposition of infinitely many modes, and then truncate the Hamiltonian <cit.> of the environment to a form with a linear dispersion relation, retaining only N_ E modes

    H_ E=ħ u_ E/2(δρ^ E_0)^2+∑_k=1^N_ E-1[(δρ_k^ E)^2+(φ_k^ E)^2]

with u_ E:=2g_ 1D/ħ L_ E and ω_k^ E:=√(g_ 1Dn_ 1D/m)kπ/L_ E. Converting back to real-space operators, we derive an effective resolution scale-dependent Hamiltonian

    H_ E=g_ 1DΔ z∑_n=N_ S^N-1δρ(z_n)^2
    +ħ^2 n_ 1D/4mΔ z∑_n,m=N_ S^N-1φ(z_n)φ(z_m)2π^2/N_ E^3κ_n,m^ E  ,

where 

    κ_n,m^ E   :=   ∑_k=1^N_ E-1{ k^2cos(kπ/N_ E(n-N_ S+1/2))
       ×   cos(kπ/N_ E(m-N_ S+1/2))}

encodes the long-range coupling of the effective Hamiltonian. Here, Δ z:=L/N is the pixel size, N_ S(E) is the number of pixels of the system (environment).

The energy of the environment can then be calculated <cit.> as [ϱ_ E H_ E]=[Γ^ EH̃_ E], where H̃_ E is the matrix expression related to H_ E=X^TH̃_ EX, with X=(φ(z_N_S), …,  φ(z_N-1), δρ(z_N_S), …, δρ(z_N-1))^T. Thus, to calculate the energetic contribution β_ EΔ E_ E, we need to obtain the inverse temperature of the environment. As indicated in the previous section and detailed in Ref. <cit.>, we have obtained the inverse temperature of the composite system with respect to the pre-quench Klein-Gordon Hamiltonian experimentally. Subsequently, we construct the thermal state corresponding to this temperature in the continuum limit simulation. We obtain the inverse temperature of the environment β_ E for different subregion sizes, by demanding the environment's energy to match that of a Gibbs state. We then interpolate the profile for each pixel. We note that the post-quench temperature is slightly higher than the temperature measured with respect to the pre-quench Hamiltonian and β_ E>0 
is nearly homogeneous across different subregion sizes.

To calculate the energy of a Gibbs state for the environment in the Tomonaga-Luttinger liquid model, we begin with the partition function Z_ E(β_ E)=Z^ zm_ E(β_ E)Z^ ho_ E(β_ E) <cit.>.
Due to the compactified nature of the field, the zero mode partition function can be expressed in terms of the Jacobi-theta function. Given our experimental parameters, it approximates to

    Z^ zm_ E(β_ E)=∑_n=-∞^∞e^-β_ Eħ u_ E/2n^2≈√(2π/β_ Eħ u_ E)  .

The harmonic oscillator partition function is

    Z_ E^ ho(β_ E)=∏_k=1^N_ E-11/2sinh(βħω^ E_k/2)  .

The energy is then E_ E(β_ E)=-∂_β_ E[logZ_ E(β_ E)]=E_ E^ zm(β_ E)+E_ E^ ho(β_ E). The zero-mode contribution to the energy is

    E^ zm_ E = 1/Z_ 0^ E∑_n=-∞^∞e^-β_ Eħ u_ E/2 n^2(ħ u_ E/2n^2 ) ,

which approximates to 1/(2β_ E) for our experimental parameters. The harmonic oscillator contribution to the energy is

    E_ E^ ho=∑_k=1^N_ E-1ħω_k^ E/2(β_ Eħω_k^ E/2)  .

Consequently, we can express then quantum relative entropy of the environment state ϱ_ E with respect to a Gibbs state at inverse temperature β_ E as

    D(ϱ_ E||γ_ E^β_ E)
    		=-S(ϱ_ E)+β_ E[ϱ_ E H_ E]+logZ_ E(β_ E)

and we obtain Δ D accordingly.






Supplementary Material













 §.§ Continuum limit and role of boundary conditions

In this section we compare the effect of Neumann boundary conditions (∂_zφ(z)|_z=0, L=0) and Dirichlet boundary conditions (φ(z)|_z=0, L=0) on the information-theoretic quantities within Landauer's principle Eq. (<ref>) in the continuum limit. The quantum field is discretised on a  sufficiently fine grid to ensure convergence, achieved when higher momentum modes no longer affect the results significantly. The simulation parameters closely match experimental values, assuming a homogeneous density profile n_ 1D.



The plots showing the scaling with subregion size at various times are presented in Fig. <ref>. The scaling of β_ EΔ E_ E can be explained by the initial symmetric energy density profile, which is higher at the edges than in the bulk for Neumann boundary conditions and lower for Dirichlet boundary conditions (Fig. [fig:DBC_main]<ref>fig:DBC_maina-b). The dominant contribution for the considered quench, Δ I, shows scaling symmetrically about L_ S/L=0.5 by construction. For Dirichlet boundary conditions, small systems (or environments) do not increase in correlations with its environment for ct/L_ S>1 as correlations reach a plateau phase in time (Fig. [fig:DBC_main]<ref>fig:DBC_maine). Neumann boundary conditions, however, actually yield Δ I to be higher for the case of a small system than for the case of system and environment being of equal size. As discussed in the main text, the zero mode causes the results for Dirichlet boundary conditions, which are well described by the quasiparticle picture, to diverge from those for Neumann boundary conditions after ct/L_ S > 1.  



 §.§ Exploiting the extremality of Gaussian states

The validity of the Gaussian tomography scheme has been confirmed by measuring the normalised averaged connected fourth-order correlation functions, as demonstrated in previous works <cit.>. However, leveraging properties of Gaussian quantum information, we argue that the information-theoretic quantities in Landauer's principle can also be estimated using our second-moment tomography scheme, even for time evolution under a non-Gaussian model with significant higher-order correlation functions. This justifies the potential use of our experimental protocol under the lens of Landauer's principle for more complex protocols and interacting models.

To begin, we define G as the Gaussian state that shares the same second moments as the true state ϱ that may be non-Gaussian, reflecting a preparation involving interactions. Central to the argument are several extremality properties of Gaussian states. Firstly, Gaussian states G have the largest von Neumann entropy S(G)≥ S(ϱ) among all states ϱ with a given covariance matrix.
Secondly, we can make use of the property

    [ϱ(t)logG(0)]=[G(t)logG(0)]  ,

as any Gaussian state can be written as a matrix exponent of a quadratic polynomial, thereby causing the matrix logarithm of the relative entropy to cancel out. Thirdly, Ref. <cit.> has demonstrated that the quantum conditional entropy is also maximised for Gaussian states, and the  inequality 

    S(ϱ_ S(t))-S(ϱ_ SE(t))≥ S(G_ S(t))-S(G_ SE(t))

is true. This is less obvious, as the expression under consideration is 
a difference of von Neumann entropies. 
The generalised entropy production of the true dynamics ΔΣ:=β_ EΔ E_ E+Δ S= Δ I+Δ D can be written as

    ΔΣ:=Δ I+Δ D
    
    =S(ϱ_ S(t))+S(ϱ_ E(t))-S(ϱ_ SE(t))-S(ϱ_ S(0))-S(ϱ_ E(0))+S(ϱ_ SE(0))
    -S(ϱ_ E(t))-[ϱ_ E(t)logγ_ E^β_ E]+S(ϱ_ E(0))+[ϱ_ E(0)logγ_ E^β_ E]
    
    =S(ϱ_ S(t))-S(ϱ_ S(0))-[ϱ_ E(t)logγ_ E^β_ E]+[ϱ_ E(0)logγ_ E^β_ E]
    
    =Δ S-[ϱ_ E(t)logγ_ E^β_ E]+[ϱ_ E(0)logγ_ E^β_ E]  ,

where we have used the unitary invariance of the von Neumann entropy. Clearly, the generalised entropy production ΔΣ^G:=β_ EΔ E_ E^G+Δ S^G=Δ I^G+Δ D^G assuming an initially Gaussian state and assuming Gaussian dynamics is given by

    ΔΣ^G:=Δ I^G+Δ D^G
    =Δ S^G-[G_ E(t)logγ_ E^β_ E]+[G_ E(0)logγ_ E^β_ E]

where Δ E_ E^G=[(G_ E(t)-G_ E(0))H_ E], Δ S^G=S(G_ S(t))-S(G_ S(0)), Δ I^G=I^G_ SE(t)-I^G_ SE(0) and Δ D^G=D(G_ E(t)||γ_ E^β_ E)-D(G_ E(0)||γ_ E^β_ E). By utilising the property in Eq. (<ref>) and considering that γ_ E^β_ E is Gaussian we can write

    ΔΣ = Δ S-[G_ E(t)logγ_ E^β_ E]+[G_ E(0)logγ_ E^β_ E]=Δ S-Δ S^G+Σ^G  .

This also implies that

    Δ E_ E = Δ E_ E^G

demonstrating that covariance matrices are entirely sufficient for capturing the energetic dynamics, even when the true dynamics are not Gaussian. This is important, as in this way, one can meausure second moments experimentally and still get bounds for the true state that may show signatures of interactions.

To proceed further, we must assume that the initial state post-quench is also Gaussian, i.e., ϱ_SE,S,E=G_SE,S,E. At this step, we have not made the additional assumption of Gaussian dynamics to bound the quantities in Landauer's principle. By the maximality of entropy for Gaussian states, we have Δ S≤Δ S^G, we quickly deduce that the generalised entropy production is upper-bounded by the Gaussian value, i.e., ΔΣ≤ΔΣ^G. Similarly, we infer that the change in mutual information is upper-bounded by the Gaussian value <cit.>, i.e., Δ I≤Δ I^G. Utilising Eq. (<ref>), we find that the change in relative entropy is lower-bounded by the Gaussian value, i.e., 

    Δ D≥Δ D^G  .

However, in both cases, these inequalities alone do not provide quantitative information on the extent to which these quantities dominate each other in the true state. To gain a more precise understanding of the relative contributions of mutual information and relative entropy in this context, a more comprehensive theoretical model and advanced experimental methods are required for an experiment modelled by an arbitrary interacting theory.









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