Phonon-mediated quantum gates in trapped ions coupled to an ultracold atomic gas
Lorenzo Oghittu Arghavan Safavi-Naini Antonio Negretti Rene Gerritsma











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§ ABSTRACT 
We study the dynamics of phonon-mediated qubit-qubit interactions between trapped ions in the presence of an ultracold atomic gas. By deriving and solving a master equation to describe the combined system, we show that the presence of the atoms causes the quantum gate quality to reduce because of motional decoherence. On the other hand, we calculate that the gas may be used to keep the ion crystal cold in the presence of external heating due to electric field noise. We show that tuning the atom-ion scattering length allows to tune the cooling rate of the ions and would make it possible to temporarily reduce the effects of the gas during a quantum gate while keeping the ions cold over long timescales. In this way, the trapped ion quantum computer may be buffer gas cooled. The system may also be used for quantum-enhanced measurements of the atom-ion interactions or properties of the atomic bath. 


Phonon-mediated quantum gates in trapped ions coupled to an ultracold atomic gas
    Lorenzo Oghittu^1,2, Arghavan Safavi-Naini^1,2, Antonio Negretti^3 and Rene Gerritsma^1,4
    September 9, 2024
=============================================================================================





§ INTRODUCTION

Over the last decade, rapid progress has occurred in controlling laboratory systems that explore the quantum dynamics of mixtures of ultracold atoms and trapped ions <cit.>. These systems offer new opportunities in studying quantum impurity physics, ultracold collisions and quantum chemistry <cit.>. On the theoretical side, efforts have been made to study and characterize ionic polarons <cit.> and bath-mediated interactions between charged impurities <cit.>. Moreover, systems involving ionic impurities in ultracold gases have been proposed as quantum simulators and sensors <cit.>. Combining atom-ion systems with precise quantum control of individual trapped ions makes it possible to study the decoherence of non-classical states of ion qubits and motion <cit.> as well as phonon-mediated interactions between ion qubit states <cit.> inside a bath of ultracold atoms. These mixed atom-ion systems may find applications in quantum-enhanced sensing of atom-ion interactions <cit.> and the properties of the atomic bath <cit.>. Furthermore, the ultracold atomic gas may be used to buffer gas cool the trapped ions, with recently reported buffer gas cooling even outperforming laser (Doppler) cooling <cit.>. The discovery of atom-ion Feshbach resonances has greatly enhanced the possibilities of atom-ion combined quantum systems <cit.>. In particular, these resonances allow for tuning the atom-ion interactions by simple tuning of an external magnetic field, in analogy to the case of Feshabch resonances between neutral atoms <cit.>.  

In this article, we consider the case where ionic qubits are coupled via laser-driven phonon-mediated interactions inside a quantum bath of ultracold atoms. We study how the atoms change the dynamics of this system and affect the quality of quantum gates between these qubits. We find that the main source of error occurs due to decoherence in the ion motion in phase space during the gate. We consider the benefits of the atomic bath to keep the ions cool in the presence of an external heating source due to electric field noise. We study the role of the atom-ion s-wave scattering length - which may be tuned using Feshbach resonances. We find that it is possible to switch the interactions in such a way, that either the adverse effects of the bath on the quantum gate, or the negative effects of heating  may be mitigated. These results show that it is possible to buffer gas cool the trapped ion quantum system <cit.>, while maintaining the quality of quantum operations in the system.



The paper is organized as follows: in Sec. <ref> we describe the Hamiltonian of the system, which is used in Sec. <ref> to derive a master equation for the ion chain. In Sec. <ref> we provide some notions about the quantum gate and we define the process fidelity. In Sec. <ref> we comment the results obtained by numerically solving the master equation. Finally, conclusions and outlook are provided in Sec. <ref>.






§ SYSTEM DESCRIPTION


We consider the scheme depicted in Fig. <ref>: it consists of a linear crystal of three ions, where the outer two ions experience a spin-dependent force that couples their (internal) spin states to the (external) vibrational phononic modes of the chain <cit.>. The entire chain is coupled to an external bath by immersing the central ion in a cloud of ultracold atoms. By keeping the two qubits out of the gas, we avoid possible spin changing collisions between atoms and ions such as spin relaxation. The latter, in which the total spin is not conserved, has been observed in atom-ion experiments <cit.>. Additionally, the center of mass mode is subjected to external heating due to electric field noise at room temperature.

In this section we describe the Hamiltonian of the system. In Sec. <ref> we focus on the the ion chain and the spin-dependent force acting on the two outer ions, while in  Sec. <ref> we describe the interaction of the central ion with the ultracold quantum gas.





 §.§ Ion chain and spin-dependent force


We consider three identical ions in a linear Paul trap, and we assume the confinement along y and z to be much stronger than the confinement along the direction of the trap axes x. In such a configuration, the motion along y and z can be neglected, and the interplay between the harmonic confinement and the Coulomb repulsion along x allows to describe the three ions as a one-dimensional crystalline chain. The motion of the ions is therefore approximated as a displacement around their equilibrium position and decomposed into three normal modes of oscillation <cit.>. On top of the motional (phononic) degrees of freedom, the two outer ions are equipped with two internal spin states and will therefore be referred to as qubits.






The phononic modes are coupled to the spin states by a spin-dependent force along the direction perpendicular to the chain. This is done with lasers in a Raman scheme with wave vector k_R <cit.>. When the displacement of the ions is much smaller than the wavelength of the lasers, the corresponding Hamiltonian can be expanded to lowest order in the Lamb-Dicke parameters η_μ=k_R√(ħ/(2Mω_μ)), with ω_μ the frequency of mode μ. In the frame rotating with the Raman beatnote ω_R, we obtain the following spin-phonon Hamiltonian:

    Ĥ_sp=-ħ/2∑_j=1,3∑_μΩ_μ b_j,μ(â_μ+â_μ^†)σ̂_j^z,

where we performed the rotating wave approximation. Here, σ̂_j^z is the z Pauli matrix acting on spin j, while â_μ^(†) annihilates (creates) a phonon of mode μ, whose amplitude vector is defined by 𝐛_μ. The amplitude vectors for the case of three particles in 1D are reported in Tab. <ref>. Note that the sum over the ions in Eq. (<ref>) does not include the central ion j=2, as this is not affected by the spin-dependent force. Finally, the driving strength of mode μ is quantified by Ω_μ=Fη_μ/k_R, where F is the magnitude of the spin-dependent force.

In the same frame, the total Hamiltonian of the ion chain with spin-phonon coupling is given by Ĥ_chain=Ĥ_s+Ĥ_p+Ĥ_sp. The spin Hamiltonian is Ĥ_s=0, while the phonon Hamiltonian reads

    Ĥ_p=-ħ∑_μδ_μn̂_μ

where δ_μ=ω_R-ω_μ is the detuning of the Raman beat note from the mode μ.

In this work, we consider the parameters reported in Tab. <ref>.

The time evolution corresponding to the interaction Hamiltonian in Eq. (<ref>) can be computed analytically. This is done by means of the Magnus expansion <cit.>, which truncates exactly at second order in this specific case. We have

    Û_I(t,0)=   exp{-i/ħ∫_0^tdt' H_sp(t')
       -i/2ħ^2∫_0^tdt'∫_0^t'dt”H_sp(t')H_sp(t”)}
    
            =   Û_sp(t)Û_ss(t)

where H_sp(t) is the spin-phonon Hamiltonian in the interaction picture with respect to Ĥ_p [Note that we are combining the transformation to the frame rotating with ω_R with the one to the interaction picture with respect to Ĥ_p. This is equivalent to transforming to the interaction picture in the laboratory frame, provided that all Hamiltonian operators are expressed in the same frame.]. The last line shows that the total propagator factorizes as the product of the spin-phonon and spin-spin coupling contributions, which are given, respectively, by

    Û_sp(t)=exp{∑_j=1,3∑_μ(ϕ_j,μ(t)â_μ^†-ϕ_j,μ^*(t)â_μ)σ̂_j^z},

and

    Û_ss(t)=exp{-i∑_i,jJ_i,j(t)σ̂_i^zσ̂_j^z}.

Here, we used the definitions

    ϕ_j,μ(t)   =Ω_μ/2δ_μ b_j,μ(1-e^-iδ_μ t),
    
            J_i,j(t)   =1/4∑_μΩ_μ^2b_j,μb_i,μ/δ_μ^2(δ_μ t-sin(δ_μ t)).

The term in Eq. (<ref>) corresponds to a spin-dependent displacement and it is responsible for the entanglement between spin and motion. When |δ_μ|≫Ω_μ for all modes, however, we see from Eq. (<ref>) that ϕ_j,μ≈0 and the evolution reduces to the propagator in Eq. (<ref>). In this regime, the interaction Hamiltonian is effectively described by the spin-spin interaction in the exponent of Û_ss at all times. When only the mode ν is driven substantially, i.e. δ_ν≳Ω_ν, the other modes can be neglected and ϕ_j,ν=0 when t is an integer multiple of 2π/δ_ν. At these values of time, the motion is disentangled from the spin, meaning that the internal state does not depend on the external vibrational state <cit.>. Note however, that for larger vibrational state occupation the Lamb-Dicke approximation breaks down. This  causes a reduction in the quantum gate quality. In practice, high fidelity quantum gate operation relies on sub-Doppler cooling <cit.> and minimal background heating <cit.>.





































 §.§ Coupling to the quantum gas


Atom-ion potential – The atom-ion interaction is represented asymptotically at large separation by the polarization potential

    V_ai(r)=-C_4/r^4,      C_4=α e^2/21/4πϵ_0,

where α is the static polarizability of the atom, e is the electronic charge and ϵ_0 the vacum permittivity.
The characteristic length and energy scales of the polarization potential are R^⋆=(2μ C_4/ħ^2)^1/2 and E^⋆=ħ^2/[2μ(R^⋆)^2], respectively.

For analytical purposes, we consider a regularization of the atom-ion polarization potential <cit.>:

    V_ai^r(𝐫)=-C_4r^2-c^2/r^2+c^21/(b^2+r^2)^2

where b and c are parameters that can be associated to the s-wave atom-ion scattering length a_ai and the number of two-body bound states supported by the potential. One useful property of Eq. (<ref>) is that its Fourier transform can be computed analytically, allowing the scattering amplitude in the first-order Born approximation to be written as follows:

    f(q)=c^2π(R^⋆)^2/(b^2-c^2)^2q{e^-bq[1+(b^4-c^4)q/4bc^2]-e^-cq}.

This will be used in the following sections.


Coupling Hamiltonian – The central ion (labeled with j=2) is immersed in an homogeneous ultracold gas of bosons or fermions confined in a box of length L, which is assumed to be much larger than any of the other lengths involved in the description. The bath Hamiltonian is given by

    Ĥ_bath=∫_ℝ^3d𝐫_bΨ̂^†_b(𝐫_b)[𝐩̂_b^2/2m+g/2Ψ̂^†_b(𝐫_b)Ψ̂_b(𝐫_b)]Ψ̂_b(𝐫_b)

where the subscript b indicates that the quantities refer to the bath, Ψ̂^†_b and Ψ̂_b are the bath field operators and g is the atom-atom interaction strength.

In the lab frame, the Hamiltonian accounting for the coupling between the ion chain and the atomic bath reads

    Ĥ_int=∫_ℝ^3d𝐫_bΨ̂^†_b(𝐫_b)V_ai(𝐫_b-𝐫̂_2)Ψ̂_b(𝐫_b),

where 𝐫̂ is the position operator of the ion. Following the procedure and notation adopted in Ref. <cit.>, we express the fluctuations around the condensate in terms of Bogoliubov modes. Let us note that we will refer to phonons in the bath as atomic phonons, in order to avoid confusion with the phonons of the vibrational modes of the ion chain. In the bosonic case, we obtain the following Hamiltonian

    H_int^B=ħ∑_𝐪(Ŝ_𝐪b̂_𝐪+Ŝ_𝐪^†b̂_𝐪^†)+ħ∑_𝐪,𝐪'Ŝ_𝐪,𝐪'b̂_𝐪^†b̂_𝐪'

where b̂_𝐪^† and b̂_𝐪 are the creation and annihilation operators of an atomic phonon excitation with momentum 𝐪 in the bosonic bath and obey the commutation relation [b̂_𝐪,b̂_𝐪'^†]=δ_𝐪,𝐪'. Moreover, we defined 

    Ŝ_𝐪=√(n_0L^3)/ħe^i𝐪·𝐫̂c_𝐪,    Ŝ_𝐪,𝐪'=1/ħe^i(𝐪'-𝐪)·𝐫̂c_𝐪'-𝐪,

where n_0 is the density of the condensate, and the coefficient c_𝐪 is related to the scattering amplitude f(q) of the atom-ion potential by

    c_𝐪=-2πħ^2/μ L^3f(q).

In the case of the regularized polarization potential that we consider in this work, f(q) is given in Eq. (<ref>). 

We note that the definitions in Eq. (<ref>) and the second sum in Eq. (<ref>) are obtained by considering that only atomic phonons with energy comparable to ħω_μ will couple to the ion motion. For instance, let us consider the center-of-mass frequency ω_cm=2π·500 kHz, corresponding to an atom velocity √(2ħω_cm/m)≃0.24 m/s. This is much faster than the speed of sound in the ^7Li condensate, which for a condensate density n_0=10^14 cm^-3 is on the order of 0.005 m/s. Therefore, we can safely assume a particle-like dispersion relation ħω_𝐪=ħ^2q^2/(2m) for the atomic phonons, and treat the bath as a non-interacting Bose gas. We refer to Ref. <cit.> for the general expressions before the particle-like approximation.

In the case of a Fermi gas, the coupling Hamiltonian simply becomes

    H_int^F=ħ∑_𝐪,𝐪'Ŝ_𝐪,𝐪'f̂_𝐪^†f̂_𝐪'

where the operators f̂_𝐪^† and f̂_𝐪 create and annihilate a fermion with momentum 𝐪 and obey the anticommutation relation {f̂_𝐪,f̂_𝐪'^†}=δ_𝐪,𝐪'.

The derivation of the master equation (see Sec. <ref>), requires the system-bath Hamiltonian to be expressed in the same rotating frame of the system Hamiltonian. This is obtained by transforming Eq. (<ref>) and Eq. (<ref>) with the operator 𝒰̂_R(t)=exp[-iω_R∑_μn̂_μ t]. Let us consider the coupling to the condensed fraction of a Bose gas, which is given by the first sum in Eq. (<ref>). The transformation only acts on the phonon operators, and the Hamiltonian has the same form, but with the time-dependent operator

    Ŝ_𝐪(t)=√(n_0 L^3)/ħe^iq_xx̂_R(t),

where we made use of the fact that the chain oscillates along the x direction and we defined the position operator of the central ion in the rotating frame as

    x̂_R(t)=∑_μb_2,μλ_μ(â_μ e^-iω_Rt+â_μ^† e^iω_Rt),

with λ_μ=√(ħ/(2Mω_μ)) the length associated to mode μ. The coupling to the non condensed Bose gas, represented by the double sum in Eq. (<ref>), and to the Fermi gas, reported in Eq. (<ref>), are transformed to the rotating frame in an analogous manner.

We finally remark that working in the frame rotating with ω_R allows the ion chain Hamiltonian to be time-independent, as long as the rotating-wave approximation holds. This simplifies the derivation of the master equation, although a time dependence is gained by the coupling to the bath. We refer to Sec. <ref> for more details.



§ MASTER EQUATION


We describe the evolution of the system with a master equation  <cit.>, in analogy to what has been done in Ref. <cit.>. Here, we only report the most important steps of the derivation, focusing on the differences with respect to the previous works. In Sec. <ref> we derive the contribution from the coupling to the quantum gas, while in Sec. <ref> the external heating term is considered.



 §.§ Master equation for the ion chain coupled to the ultracold gas


Our open system is represented by the ion chain with spin-phonon coupling, whose Hamiltonian Ĥ_chain is defined in Sec. <ref> [see Eq. (<ref>) and (<ref>)]. As a starting point for the derivation, we consider the Redfield equation:

    d/dtρ_R(t)=-∫_0^tdt'/ħ^2 Tr_b{[H_int(t),[H_int(t'),ρ_R(t)⊗B̂_0]]}

where ρ_R and H_int are, respectively, the reduced density matrix of the system and the system-bath interaction Hamiltonian in the rotating frame and in the interaction picture with respect to Ĥ_chain+Ĥ_bath. Moreover, B̂_0 is the density matrix of the bath and Tr_b{…} represents the trace over the bath degrees of freedom. We note that Eq. (<ref>) relies on the Markov and Born approximations.
Let us now focus on the case where the bath is a Bose-Einstein condensate. By explicitly writing H_int and performing the trace, we get

    
    d/dtρ_R(t)=-∫_0^tdt' ∑_𝐪{(n_𝐪+1)(e^-iω_𝐪(t-t')   S_𝐪(t)S_𝐪^†(t')ρ_R(t)+H. c.)
    
            +   n_𝐪(e^-iω_𝐪(t-t')ρ_R(t)S_𝐪^†(t')S_𝐪(t)+H. c.)}


where ħω_𝐪 is the energy of the atomic phonons in the condensate, H. c. indicates the Hermitian conjugate, S_𝐪 represents the operator in Eq. (<ref>) when transformed to the interaction picture and n_𝐪=[e^β(ħω_𝐪-μ_𝐁)-1]^-1 is the Bose-Einstein occupation number obtained from the thermal averages over the state of the bath B̂_0 

    Tr_b{b̂_𝐪^†b̂_𝐪'B̂_0 }=n_𝐪δ_𝐪,𝐪',

with μ_B the chemical potential of the gas at temperature T and β=1/(k_BT) (k_B the Boltzmann constant).

The next step consists of transforming the master equation back to the Schrödinger picture and perform the Lamb-Dicke approximation for small values of q_xx̂. As an example, we consider the first commutator in Eq. (<ref>), from which we get

    e^iq_xx̂_R(t)e^iq_xx̂_R(t',-τ)ρ̂_R(t)   
    ≃x̂_R(t)x̂_R(t',-τ)ρ̂_R(t)   -1/2(x̂_R(t))^2ρ̂_R(t)

where we neglected the term proportional to q_x as this vanishes after the sum over 𝐪 due to spherical symmetry of the bath. Here, we defined τ=t-t' and

    x̂_R(t,-τ)=e^-iĤ_chain-τ x̂_R(t) e^iĤ_chainτ.

Because of the time-independence of Ĥ_chain (see Sec. <ref>), it is straightforward to calculate the product in Eq. (<ref>) by means of the Baker-Campbell-Hausdorff identity [Baker-Campbell-Hausdorff identity: e^iĜλÂe^-iĜλ=Â+iλ[Ĝ,Â]+(iλ)^2[Ĝ,[Ĝ,Â]]/2!+…]. The explicit form of x̂_R(t,-τ) is reported in Eq. (<ref>).

We refer the interested reader to Ref. <cit.> for more details and for the case of a Fermi gas.
We now report the master equation for the ion chain coupled to a Bose-Einstein condensate in the frame rotating with ω_R: 


    d/dtρ̂_R=-   i/ħĤ_chainρ̂_R
    
            -Γ∑_μ{   (n_q_μx̂_R(α̂_μ+α̂^†_μ)ρ̂_R-ρ̂_R(α̂_μ+α̂^†_μ)+x̂_Rα̂_μρ̂_R-ρ̂_Rα̂_μ^†)
    
            +Ω_μ/2δ_μ   ∑_j=1,3b_j,μ(cos(ω_R t)x̂_Rσ̂_j^zρ̂_R-ρ̂_Rσ̂_j^z2h_μ^(n_q)+e^-iω_Rtx̂_Rσ̂_j^zρ̂_Rh_μ-e^iω_Rtx̂_Rρ̂_Rσ̂_j^zh_μ)},


where we omitted time dependencies for clarity and we defined Γ=2π m ħ n_0/(3μ^2),

    α̂_μ(t)=b_2,μλ_μ|f(q_μ)|^2q_μ^3â_μ e^-iω_Rt
       h_μ^(n_q)=b_2,μλ_μ((n_q_μ)|f(q_μ)|^2q_μ^3-(n_q_R)|f(q_R)|^2q_R^3)

and q_μ(R)=√(2mω_μ(R)/ħ). 

The first line of Eq. (<ref>) is the unitary evolution of the system in absence of the bath, while the second and third lines describe the dissipative contribution due to the coupling to the Bose-Einstein condensate. 



 §.§ Contribution from external heating


The external heating is described as a coupling of the center-of-mass mode to an effective bath with mean occupation number N̅ <cit.>. The corresponding contribution to the master equation reads

    γ/2(N̅+1)(2â_cmρ̂_Râ_cm^†-â_cm^†â_cmρ̂_R-ρ̂_Râ_cm^†â_cm)
       +γ/2N̅(2â_cm^†ρ̂_Râ_cm-â_cmâ_cm^†ρ̂_R-ρ̂_Râ_cmâ_cm^†),

where we omitted the time dependencies. For a bath at room temperature, we have N̅≫1, and the prefactor γ(N̅+1)≈γN̅ can be obtained empirically from experiments.




§ QUANTUM GATE


The spin-dependent Hamiltonian described in Sec. <ref> can be used to perform a geometric phase gate. In this section, we briefly summarize some of the basic features of this operation and we define the process fidelity.



 §.§ Gate behavior


Given an initial 4×4 matrix ρ̂_0 describing the state of the two qubits, the final state after the gate operation in the case of an ideal process represented as Λ is given in the {|↑⟩, |↓⟩} basis by

    Λ(ρ̂_0)=𝒰̂^†(t_gate) ρ̂_0 𝒰̂(t_gate),

with 𝒰̂(t_gate)=Diag{1,-i,-i,1}. For instance, let us consider the initial state to be |++⟩≡|+⟩_1⊗|+⟩_3. The corresponding density matrix ρ̂_0=|++⟩⟨++| in the basis {|↑⟩, |↓⟩} is the 4×4 matrix where all the elements are equal to 1/4. According to Eq. (<ref>), the final density matrix is

    Λ(ρ̂_0)=[  1/4  i/4  i/4  1/4; -i/4  1/4  1/4 -i/4; -i/4  1/4  1/4 -i/4;  1/4  i/4  i/4  1/4 ],

which corresponds to the state (|++⟩-i|–⟩)/√(2), i.e., the maximally entangled state.



 §.§ Process fidelity


The scenario explained in Sec. <ref> refers to an ideal gate. However, the coupling to the external heating and to the quantum gas introduces an error. To quantify how close the non-ideal process, indicated as Γ, reproduces the ideal one, we define the process fidelity as <cit.>

    F(Λ,Γ)=1/d^2∑_i=1^d^2Tr{Λ(ρ̂_i^†)Γ(ρ̂_j)},

where {ρ̂_i} is a complete set of mutually orthonormal operators in the Hilbert space of the two qubits with dimension d=4. In the case of a two-qubit gate, a natural choice is to define each operator ρ̂_i as the one of the possible tensor products σ̂_1^ξ⊗σ̂_3^ζ, where σ̂_j^ξ (ξ=0,x,y,z) are the Pauli matrices in the subspace of ion j. Note that these operators do not represent, in general, a quantum state. Therefore, in the case of an experiment, a different definition of the fidelity has to be considered. We refer the interest reader to Ref. <cit.> for a complete treatment of the topic.




§ RESULTS


In this section, we report the results obtained by solving the master equation described in Eq. (<ref>). Specifically, we study the dynamics of the system by deriving a set of equations for the expectation value of the position and momentum associated to the center of mass mode, as well as for the expectation value of the kinetic energy. Moreover we study how the process fidelity of the gate described in Sec. <ref> is affected by the heating and the ultracold gas. Unless stated differently, we consider the frequencies in Tab. <ref> and we fix the atom-ion s-wave scattering length a_ai≃ R^⋆. The ions are ^174Yb^+, while the bath is an ultracold gas of either ^6Li (Fermi) or ^7Li (Bose) with density n=10^13 cm^-3 and temperature T=200 nK.

We remark that the choice of the frequencies would allow us to neglect the stretching and wobbling mode in the description of the gate operation, strongly reducing the computational effort. In fact, δ_μ≫Ω_μ for said modes (see Sec. <ref>). The gate, therefore, relies on the entanglement of the spin with the center-of-mass mode and the gate time is given by 2π/δ_cm=0.25 ms. However, the wobbling mode has to be taken into account in order to properly describe the motion of the central ion inside the bath. The sums over the modes μ in all of the derived equations will hence be restricted to μ=cm,wb.



 §.§ Cooling dynamics



We consider the temperature T_chain of the ion chain and we study how this is affected by the coupling to the ultracold quantum gas and to the external heating. The temperature is defined as

    T_chain=1/k_B1/2M∑_μ⟨p̂_μ^2⟩,

where p̂_μ=i√(ħ Mω_μ/2)(â_μ^†-â_μ) is the momentum operator corresponding to mode μ. Note that, in neglecting the stretching mode, we are not considering its contribution to the chain temperature. However, this is uncoupled from the other modes and is not affected by either heating or the quantum gas, meaning that it would add to the total temperature as a mere shift.

The spin of the two outer ions is set to the state |↑↑⟩, while the central ion is initialized in the approximate thermal motional state c_0|0⟩⟨0|_μ+c_1|1⟩⟨1|_μ for both the center-of-mass and wobbling mode, with c_0=0.9 and c_1=0.1. Here, |n⟩_μ represents the n-th oscillator state of mode μ, which contributes to the initial temperature with c_nħω_μ(n+1/2)/(2k_B), where the factor 1/2 is due to the equipartition theorem. 

In Fig. <ref>(a), we compare the behavior of the temperature in different scenarios. We first note that, in absence of spin-dependent force acting on the outer ions, the time evolution of T_chain is monotonous (dashed lines). 
On the other hand, when the spin-dependent force explained in Sec. <ref> is considered, each mode is driven with frequency ω_R and the ion temperature oscillates accordingly. In particular, we observe that this oscillation has a slow and a fast component [see Fig. <ref>(b)], corresponding to the momentum of the center-of-mass mode oscillating with frequencies δ_cm/2 and ω_R+δ_cm/2, respectively <cit.>. We remark that the wobbling mode has an analogous behavior, but its contribution to the oscillation of T_chain is not perceptible due to the much weaker driving strength compared to the center-of-mass mode.
In the case of unitary evolution (gray), the system occupies the same state after every gate cycles. Hence, the temperature retrieves its initial value at every multiple of t_gate. In the presence of heating (green) a similar behavior is observed. In this case, however, the temperature increases linearly with a rate proportional to γN. 
Interestingly, when the chain is coupled to the ultracold quantum gas, the latter guarantees the cooling of the chain. The dashed blue line shows that the ion temperature without spin-dependent force and without heating converges to a final value which is given by the ground state energy of the ion in the trap. In our case, it corresponds to ħ(ω_cm+ω_wb)/(4k_B)≃20.45 μK. 
Note that, if micromotion was considered, the ion would converge to a higher temperature <cit.>. 
When both the heating and ultracold gas are considered (red), a dynamical equilibrium is reached where the ion temperature stabilizes at a value which is higher than the ground state temperature. We observe that even in the case of high heating rate compared to typical values in experiments, the cooling action of the gas is remarkably fast. For the parameters considered in Fig. <ref> and without spin-dependent force, the final temperature is reached in around 0.5 ms, i.e. the time of two gate cycles. The blue and red shaded areas in Fig. <ref>(a) show that the coupling to the gas damps the oscillation due to the spin-dependent driving force. In other words, the system behaves effectively as a driven damped harmonic oscillator and converges to a steady state where the temperature oscillates with frequency ω_R and constant amplitude.
 



 §.§ Effect on process fidelity



The error induced on the gate operation by the coupling to the ultracold gas can be understood by observing the phase space behavior of the system. As shown in Fig. <ref>, the unitary evolution (gray) in the frame rotating with ω_R corresponds to a circle in the phase space of the center-of-mass mode. On the other hand, the presence of the gas (blue) deforms the trajectory and does not allow the circle to close after the gate time t_gate=0.25 ms, indicating decoherence of motional states.

Nevertheless, this behavior can be mitigated by properly tuning the atom-ion scattering length. In Fig. <ref>, we observe that for a_ai≃-1.3 R^⋆, the effect of the gas on the process fidelity is strongly suppressed. The same behavior is observed for both a ^7Li Bose-Einstein condensate (blue circles) as well as a spin-Polarized ultracold Fermi gas of ^6Li atoms (orange triangles), although a better fidelity can be achieved in the bosonic environment. This difference may be due to the stronger coupling of the ion to a fermionic bath, which is also responsible for the lower temperatures predicted by our model for a Paul-trapped ion immersed in ^6Li compared to ^7Li <cit.>. Moreover, we note that the interaction tune-out value of the scattering length does not correspond to a_ai=0 due to the interplay between molecular and confinement-induced resonances <cit.>.
This result, combined with the cooling dynamics presented in Sec. <ref>, may have interesting applications in experiments. There, the atom-ion scattering rate can be controlled by tuning a magnetic field close to a Feshbach resonance <cit.>, making it possible to alternate from the regime were the coupling to the bath is suppressed and gate operations are performed, to the one where the effect of the ultracold gas is maximized and the ions are cooled.



§ CONCLUSIONS AND OUTLOOK


We studied phonon-mediated interactions between ionic qubits coupled to an ultracold quantum gas and external heating. 
We derived a master equation for the reduced density matrix of the chain. 
We find that the presence of the atomic bath has a negative effect on the quality of quantum gates between the ion qubits. On the other hand, we calculate that the atomic bath can be used to keep the ion crystal cool in the presence of external heating due to electric field noise. Tuning of the atom-ion scattering length using a magnetic field allows to set the cooling rate of the atoms. In particular, the adverse effects during quantum gates may be limited while the buffer gas cooling may be maximized in between gates. This shows that buffer gas cooling is a viable technique for trapped ion quantum systems. 

The system offers further opportunities in studying the decoherence of non-classical states of ion motion and spin-motion entangled states in the presence of a fermionic or bosonic quantum bath. The system may be used to explore the crossover between classical and quantum dynamics <cit.> and the occurrence of classical and quantum chaos in mixtures of ultracold atoms and trapped ions <cit.>. Moreover, the coupling of the trapped ion atom mixture to qubits may allow for quantum-enhanced measurements of the properties of this system in analogy to e.g. Ref. <cit.>. There, a fragile spin-motion entangled state was used to detect the scattering of a single photon. The momentum transfer between an ultracold Li atom and an Yb^+ is comparable to that of scattering infrared photons such that the scheme could be extended to quantum enhanced detection of atom-ion scattering.






§ ACKNOWLEDGEMENTS

This work was supported by the Dutch Research Council (Grant Nos. 680.91.120 and VI.C.202.051), (R.G.). A.S.N is supported by the Dutch Research Council (NWO/OCW), as part of the Quantum Software Consortium programme (project number 024.003.037), Quantum Delta NL (project number NGF.1582.22.030) and ENW-XL grant (project number OCENW.XL21.XL21.122). L.O. acknowledges the European Cooperation in Science & Technology (COST Action CA17113).









§ TIME DEPENDENCE OF ION POSITION


Here we provide the explicit form of the operator defined in Eq. (<ref>), describing the motion of the central ion of the chain in the frame rotating with ω_R and in absence of the bath:


    x̂_R(t,-τ)=∑_μb_2,μλ_μ   {â_μ e^-iδ_μτe^-iω_Rt+â_μ^† e^iδ_μτe^iω_Rt
    
            +   Ω_μ/2δ_μ∑_j=1,3b_j,μσ̂_j[e^-iδ_μτe^-iω_R t+e^iδ_μτe^iω_R t-2cos(ω_R t)]}.


We remark that this result relies on the Hamiltonian Ĥ_chain being independent of time, hence on the rotating wave approximation (Sec. <ref>). In case of a time-dependent chain Hamiltonian, the calculation would require a different procedure, as in the case of the Paul-trapped ion <cit.>.







