Heisenberg-limit spin squeezing with spin Bogoliubov Hamiltonian
Jun Zhang Sheng Chang Wenxian Zhang

	
	

	
	
	Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education,
		and School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China
	
	
	Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education,
		and School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China
	
	


	
	
	[Corresponding email:]wxzhang@whu.edu.cn
	Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education,
		and School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China
	Wuhan Institute of Quantum Technology, Wuhan, Hubei 430206, China
	
	
§ ABSTRACT 
It is well established that the optimal spin squeezing under a one-axis-twisting Hamiltonian follows a scaling law of J^-2/3 for J interacting atoms after a quench dynamics. Here we prove analytically and numerically that the spin squeezing of the ground state of the one-axis-twisting Hamiltonian actually reaches the Heisenberg limit J^-1. By constructing a bilinear Bogoliubov Hamiltonian with the raising and lowering spin operators, we exactly diagonalize the spin Bogoliubov Hamiltonian, which includes the one-axis twisting Hamiltonian as a limiting case. The ground state of the spin Bogoliubov Hamiltonian exhibits wonderful spin squeezing, which approaches to the Heisenberg limit in the case of the one-axis twisting Hamiltonian. It is possible to realize experimentally the spin squeezed ground state of the one-axis-twisting Hamiltonian in dipolar spinor condensates, ultracold atoms in optical lattices, spins in a cavity, or alkali atoms in a vapor cell.

	Heisenberg-limit spin squeezing with spin Bogoliubov Hamiltonian
    Wenxian Zhang
    September 9, 2024
================================================================



§ INTRODUCTION

 Development of the methods for realizing squeezed spin states (SSS) or entangled spin states has been a vigorous frontier in quantum enhanced precision measurement for more than three decades <cit.>, because such strongly correlated quantum states may lead to significant improvement in atomic clock <cit.>, optical or atomic interferometer <cit.>, frequency standards <cit.>, and gravitational wave detection <cit.>. The uncertainty principle predicts that relative measurement precision approaches to standard quantum limit (SQL), J^-1/2 for uncorrelated J particles in a quantum interferometer, and Heisenberg limit (HL), J^-1 for some special squeezed or entangled quantum states <cit.>. Generating atomic spin squeezing has been proposed and demonstrated with a variety of methods, including quantum nondemolition measurements <cit.>, squeezing transferring <cit.>, and dynamical evolution <cit.>, in systems such as atom-cavity interaction systems <cit.>, interacting trapped ions <cit.>, and Bose-Einstein condensates (BECs) <cit.>.

Nonlinear atomic interaction induces spin squeezing after a quench dynamics from a coherent spin state (CSS). It is generally believed that the spin squeezing parameter of J interacting atoms may achieve J^-2/3 under a one-axis-twisting (OAT) Hamiltonian and J^-1 under a two-axis-twisting (TAT) Hamiltonian <cit.>. However, the SSSs generated with the dynamical evolution slip away quickly after the optimal time. The long-lived ground SSSs are thus desired and have been explored recently in a spinor BEC by Chapman's group <cit.> and in a trapped-atom clock system by Reichel's group <cit.>.

We investigate in this Letter the ground SSS of a spin Bogoliubov Hamiltonian. By constructing a bilinear Bogoliubov Hamiltonian with spin operators, which includes the OAT Hamiltonian as a limiting case, we prove analytically and numerically that the spin squeezing of the ground SSS of the spin Bogoliubov Hamiltonian (as well as the OAT Hamiltonian) reaches the HL J^-1 with a weak external field. Our theoretical results may be realized potentially in many experiments, such as dipolar spinor BEC <cit.>, ultracold atoms in an optical lattice<cit.>, and spins in a cavity or in a vapor cell <cit.>.



§ LMG MODEL



 The Lipkin-Meshkov-Glick (LMG) model originally introduced in nuclear physics in 1965, has been widely studied in many quantum spin systems, such as spinor BEC, Rydberg-dressed atomic gas and cold atoms in optical lattices <cit.>. The entanglement properties as well as spin squeezing of this model in its ground state have been discussed in many literatures with mean field theory <cit.>, by treating the quantum effect as small fluctuations and using the Holstein-Primakoff transformation in the thermodynamic limit <cit.>. Different from previous methods, we explore an exact analytical approach to diagonalize the Hamiltonian for a certain parameters and focus on the spin-squeezed ground state.

The Hamiltonian of the LMG model reads

    H=η(J_x^2+γ J_y^2)+Ω J_z,

where J_x,y,z are collective spin operator, η the interaction strength, γ the anisotropic parameter, and Ω the effective external field. At zero external field Ω = 0, this Hamiltonian becomes an OAT Hamiltonian H_OAT∝ J_x,z^2 if γ=0, 1 and a TAT Hamiltonian H_TAT∝ (J_x^2-J_y,z^2) if γ = -1, 1/2, after dropping the constant term with 𝐉^2 = J_x^2 + J_y^2 + J_z^2.




§ CONSTRUCTION OF BOGOLIUBOV HAMILTONIAN

 To diagonalize the Hamiltonian Eq. (<ref>), we construct a diagonal “Bogoliubov" Hamiltonian H_B through the Bogoliubov transformation of the spin lowering/raising operator J_±=J_x ± iJ_y, inspired by the concept of squeezed light <cit.>. We prove that the constructed Bogoliubov Hamiltonian H_B is a special case of the LMG model Eq. (<ref>).

Spin lowering and raising operators obey the canonical commutation relation,

[J_+, J_-]=2J_z and [J_±,  J_z]= ∓ J_±. A Dicke state |J,M⟩ (M=-J,-J+1,⋯, J) is the eigenstate of the spin operator J_z, satisfying the relation J_z|J,M⟩=M|J,M⟩. Since J_z=1/2(J_+J_--J_-J_+), a Dicke state is

    |J,M⟩   =   1/(J+M)!(2J/J+M)^1/2J_+^J+M|J,-J⟩.

Unlike the annihilation operator â of a photon, the spin lowering operator J_- has only one eigenstate |J,-J⟩, which is the vacuum state and satisfies J_-|J,-J⟩=0.

Similar to generalized creation and annihilation operators, we define the following generalized spin lowering and raising operators,
A_+=μ^* J_+-ν J_-, A_-=A_+^†=μ J_--ν^∗J_+.
To maintain the commutation relation [J_+,  J_-]=[A_+, A_-], we require
|μ|^2-|ν|^2=1. We construct the following diagonal Hamiltonian using A_±, the Bogoliubov Hamiltonian,

    H_B    =    A_+A_-.

By assuming μ=coshθ and ν=e^iφsinhθ, one finds
H_B
	=J_z + cosh2θ(J_x^2+J_y^2)
	-cosφsinh2θ(J_x^2-J_y^2)
	-i/2sinφsinh2θ(J^2_+-J^2_-). When φ=π, we obtain a specific spin Bogoliubov Hamiltonian as

    H_B   =   J_z + cosh2θ(J_x^2+J_y^2) + sinh2θ(J_x^2-J_y^2).

It is obvious that H_B in Eq. (<ref>) is a special case of the LMG Hamiltonian Eq. (<ref>), where γ = exp(-4θ) and η/Ω = exp(2θ) must be satisfied. For a large enough θ, the Bogoliubov Hamiltonian becomes effectively the OAT Hamiltonian, H_OAT = exp(2θ) J_x^2.




 §.§ Spin vacuum state


The ground state of the Bogoliubov Hamiltonian H_B is the eigenstate of the operator A_-,which is a spin vacuum state satisfying

    A_-|χ⟩   =   0|χ⟩.

This state can be expanded in the Dicke basis as |χ⟩=∑_M=-J^JC_M |J,M⟩. By substituting the definition of A_-, we find the following recursion relation








	
	
	








    C_M+2=ν^∗/μC_M√((J-M)(J+M+1)/(J-M-1)(J+M+2)).

It is straightforward to calculate the general formula in terms of the coefficient C_-J,

    C_-J+2K   =   (ν^∗/μ)^KJ!/(J-K)!K!√((2J-2K)!(2K)!/(2J)!)C_-J,

K=0,1,⋯,J. Since ⟨J,J|A_-|χ⟩=0 and C_J-1=0, it is easy to find that C_-J+2K+1=0 for K=0,1,⋯,J-1. By further utilizing the normalization condition ∑_K=0^J|C_-J+2K|^2=1, we obtain

    C_-J   =   1/√(ℱ(
    	1/2,-J,1/2-J, |ν/μ|^2))

where ℱ[a,b,c,x] represents the hypergeometric function.



 §.§ Spin squeezing of the spin vacuum state


Similar to a photon squeezed state, the spin vacuum state is also spin squeezed. To characterize the spin squeezing, two kinds of spin squeezing parameters were introduced by Kitagawa et al. and Wineland et al. respectively <cit.>,

    ξ_S^2=(Δ J_n_⊥)^2/(J/2),   ξ_R^2=2J(Δ J_n_⊥)^2/|⟨J⃗⟩ |^2

where subscript n_⊥ refers to an arbitrary axis perpendicular to the mean spin ⟨J⃗⟩, where the minimum of (Δ J_n_⊥)^2 is obtained. The inequality ξ_i^2<1 (i=S,R) indicates that the state is spin squeezed, compared to a CSS with ξ_i^2 = 1. The relation between the two squeezing parameters is ξ_R^2=(J/|⟨J⃗⟩|)^2ξ_S^2. Since J≥|⟨J⃗⟩ | always holds, one finds ξ_S^2 ≤ξ_R^2 <cit.>. The equality is taken at J=|⟨J⃗⟩ |, when a CSS is considered. The spin squeezing parameter ξ_R^2 relates directly to the phase sensitivity of a quantum interferometer, which takes the general form as

    Δϕ   =   (Δ J_n_⊥)/|⟨J⃗⟩ |
    	=ξ_R/√(J).

Obviously, the phase sensitivity of a CSS is, Δϕ=1/√(J), which is denoted as the SQL <cit.>. By contrast, an SSS is expected to achieve a phase sensitivity below the SQL but above the HL, i.e., 1/J ≤Δϕ < 1/√(J) <cit.>. It follows immediately that 1/J ≤ξ_R^2 <1. On the other hand, ξ_S^2 may approach to zero, not constrained by the HL. For instance, a Dicke state |J,M ≠ 0⟩ has a constant spin size J and zero spin variance along z-direction, resulting in ξ_S^2=0.

To calculate the parameters ξ_S^2 and ξ_R^2 of the spin vacuum state, we need to find spin average ⟨ J_n⟩ and the minimal variance perpendicular to the mean-spin direction (MSD), i.e. Δ J_n_⊥. Since the operators J_x,y are a linear combination of A_±, it follows immediately that ⟨χ|J_x,y|χ⟩ = 0. By further employing the commutator [A_+,A_-]=2J_z, we find that ⟨χ|J_z|χ⟩≠ 0 thus the MSD of the state |χ⟩ is along z-direction. The minimal variance must be in the x-y plane and the covariance matrix is defined as

    Γ_xy=([     ⟨ J^2_x⟩ Cov(J_x,J_y); Cov(J_x,J_y)     ⟨ J^2_y⟩;              ]),

with Cov(J_x,J_y)=(⟨ [J_x, J_y]_+⟩-⟨ J_x⟩⟨ J_y⟩)/2 being the covariance between J_x and J_y, and
[X, Y]_+=XY+YX denoting the anti-commutator. The eigenvalues of the covariance matrix are

    λ_±   =   1/2[⟨ J_x^2+J_y^2⟩±√(⟨ J_x^2-J_y^2⟩^2+4Cov(J_x,J_y)^2) ].

After a straightforward calculation the eigenvalues are simplified as
λ_±=1/2[(F-K)±√((F-K)^2-G^2)]
where F=J(J+1), G=⟨ J_z⟩ and K=⟨ J^2_z⟩. Obviously we find that min (Δ J_n_⊥)^2 =λ_- <cit.>. The squeezing parameters become

    ξ_S^2   =   2λ_-/J, 
    ξ_R^2   =   2Jλ_-/|⟨ J_z⟩ |^2 = J/2λ_+.


In the limit θ→∞, we approximate (ν^*/μ)^K ≈ 1-2Kexp(-2θ) and the spin average becomes <cit.>

    ⟨ J_z⟩≈ -J^2 exp(-2θ).

Accordingly, the eigenvalues of covariance matrix are
λ_+ ≈   J^2/2 and
λ_- ≈ (J^2/2) exp(-4θ).
Finally, we find that the squeezing parameters are

    ξ_R^2   ≈   1/J,   ξ_S^2≈ J exp(-4θ).

In addition, the Bogoliubov Hamiltonian in Eq. (<ref>) reduces effectively to the OAT Hamiltonian in the limit of θ→∞. Therefore, we prove that the spin squeezing of the ground state for an OAT Hamiltonian is approaching HL, surpassing the constraint of ∝ J^-2/3 for dynamic evolution governed by the same Hamiltonian. Moreover, the resulting ground state of H_B is categorized as generalized intelligent state (GIS) which minimize the Robertson-Schrödinger uncertainty relation <cit.>. It is  straightforward to find that λ_+λ_-=G^2/4=⟨ J_z⟩^2/4, indicating that |χ⟩ is an intelligent state <cit.>.

We present in Fig. <ref>(a) the squeezing parameters ξ_R^2 and ξ_S^2, as well as the spin average ⟨ J_z⟩/J, of the state |χ⟩ for a spin size J=1000. These quantities vary with the parameter θ. The coefficients C_M of the ground state |χ⟩ are analytically calculated using the recursion relation Eq. (<ref>) and the coefficient C_-J. It is then straightforward to calculate the spin average ⟨χ| J_z|χ⟩ and the spin squeezing parameters ξ_R^2 and ξ_S^2. We also present the spin squeezing parameters ξ_R^2 and ξ_S^2 of the optimal SSS, which is generated numerically by evolving the system from an initial CSS |J,J⟩ under the TAT Hamiltonian H_TAT = J_x^2-J_y^2 <cit.>. As a comparison, we plot in Fig. <ref>(b) the same quantities of the ground state of a special anisotropic LMG Hamiltonian with γ =0, H_A = η J_x^2 + Ω J_z. As η≫Ω, the Hamiltonian H_A effectively reduces to the OAT Hamiltonian. Because it is challenging with analytical method, the ground state of H_A is calculated numerically for J=1000.


		




As shown in Fig. <ref>(a), the spin squeezing parameter ξ_R^2 approaches to the HL quickly in the large θ region (θ > 3). The parameter ξ_S^2 and the spin average ⟨ J_z⟩ decrease exponentially to zero approximately described by Eqs. (<ref>) and (<ref>), respectively. In the small θ region (θ < 3), the spin average is almost a constant while the spin squeezing parameters ξ_S,R^2 decrease exponentially. This feature is especially useful for quantum enhanced metrology <cit.>. At θ≈ 3, the squeezing parameters ξ_S,R^2 of the ground state |χ⟩ and of the optimal SSS of H_TAT are close to the HL. On the other hand, the spin averages are quite different, ⟨χ| J_z |χ⟩ = -0.89J and |⟨ SSS| J_z |SSS⟩ = 0.59J. Similarly in Fig. <ref>(b), the spin squeezing parameters ξ_R^2 also approaches to the HL as η/Ω becomes large. Both panels (a) and (b) indicate that the spin squeezing of the ground state of the OAT Hamiltonian is at the HL, in stark contrast to earlier understanding <cit.>.



A typical ground state of the Bogoliubov Hamiltonian |χ⟩ is presented in Fig. <ref>(a) for θ=3. As a comparison, the optimal spin squeezed state |SSS⟩ under the TAT Hamiltonian is also shown. Both |χ⟩ and |SSS⟩ are real function. Besides the odd M coefficients C_M being zero, the even M coefficients of |SSS⟩ are positive while that of |χ⟩ are alternatively positive and negative. Clearly, these two states are very different, though the spin squeezing parameter ξ_R^2 of them is close to the HL. As shown in the inset, the state |χ⟩ at θ = 9 is similar to a Dicke state |ψ_D⟩ with J_x|ψ_D⟩ = 0. We then plot the infidelity of the state |χ⟩ and |ψ_D⟩ in Fig. <ref>(b) with 1-F = 1-|⟨χ|ψ_D⟩|^2. As θ increases to a large value, the infidelity approaches to zero exponentially, indicating the spin squeezed ground state |χ⟩ of the OAT Hamiltonian is in fact a Dicke state.

Here we note that the spin squeezing parameter ξ_R^2 of the Dicke state was previously considered illy defined because its spin average is zero and other parameters, like Dicke squeezing parameter and Mach-Zehnder phase sensitivity <cit.>, were introduced to characterize the strong entanglement of the state. These parameters exhibit Heisenberg scaling. However, the spin squeezing parameter ξ_R^2 is in fact well-defined and converges to the Heisenberg limit 1/J, even though the spin average is zero <cit.>. Such a finite (nonzero) spin squeezing parameter ξ_R^2 is due to the exactly same asymptotic behavior of the nominator, 2J(Δ J_n_⊥)^2, and the denominator, |⟨ J_z⟩ |^2, as exp(-2θ) approaches zero (see Eqs. (<ref>-<ref>)). In addition to investigating the squeezing and entanglement properties of squeezed states, one may also employ many-body correlators to explore the many-body non-locality of such states <cit.>.






§ POSSIBLE EXPERIMENTAL PLATFORMS


There are a variety of experimental platforms to potentially realize the HL spin-squeezed ground state |χ⟩. First, a dipolar spinor BEC is an ideal test bed of the LMG model, where the atoms macroscopically occupy internal hyperfine states that can be treated as collective spin states. Under single mode approximation, the spin-dependent effective Hamiltonian of the condensate with magnetic dipole-dipole interaction in an external magnetic field along z-direction is <cit.>

    H_e= -D J_z^2+E(J_x^2-J_y^2)+Ω J_z,

where 𝐉=∑_αβâ_α^†𝐅_αβâ_β is the collective condensate spin operator and J_η  (η=x, y, z) its η-component. The constants D and E depends on the density and geometry of the condensate and Ω on the external magnetic field. By tuning the isotropic parameter γ=(D-E)/(D+E) and the field Ω = 0, this Hamiltonian takes the form of OAT model, i.e. H_OAT≈ -D J_z^2 for γ≈ 1 and H_OAT≈ 2E J_x^2 for γ≈ 0, and TAT model H_TAT≈ E(J_x^2-J_y^2) for γ = -1, or H_TAT≈ 2E(J_x^2-J_z^2) for γ = 1/2. We have omitted the constant term proportional to 𝐉^2.

Second, nonlinear atom interaction in a two-component BEC is described by an OAT Hamiltonian H_OAT=ηJ̃_̃z̃^2, where J̃_x,y,z is the pseudo-spin operator constituted by two different modes of the condensate and η the inter-mode interaction strength. The system can be regarded as a BEC with atoms in two internal hyperfine (or Zeeman) states or a BEC in a double-well potential. The interaction between the pseudo-spin modes for both cases is fine tunable in experiment <cit.>.

Third, both OAT and TAT Hamiltonians may be experimentally realized in a system of spins in a cavity, where the interaction between the spins and the cavity is described by the Hamiltonian H_TC=ω_mâ^†â+ω_B J_z+g(â^†J_-+âJ_+), where â (â^†) is the annihilation (creation) operator of the cavity field with a mode frequency ω_m, J_α (α=z,±) the collective spin operator, ω_B the Zeeman splitting of the spins, and g the interaction strength. After Schrieffer-Wolf transformation e^RH e^-R with R=(g/Δ_m)(â^†J_--âJ_+), Δ_m=|ω_B-ω_m|, the Hamiltonian of the spin part is approximated to H_OAT≈ (2g^2/Δ_m)J_z^2, indicating that the effective spin interaction is mediated by the cavity mode <cit.>. When the system subjects to a parametric two-photon driving, the spin interaction term becomes a TAT Hamiltonian H_TAT≈[2λ g^2/(Δ_mΔ_c)] (J_x^2-J_y^2) with λ being the driving amplitude and Δ_c the detuning between the cavity and driving frequency <cit.>.

Fourth, ultracold atoms loaded in an optical lattice is described by Bose(Fermi)-Hubbard model in which the site-dependent spin operator can be reduced to a collective spin operator in Mott phase. Both OAT and TAT Hamiltonian may be induced by applying an additional weak light to the system <cit.>.
The effective Hamiltonian depends on the phase ϕ of the light, and becomes an OAT Hamiltonian H_OAT≈∓ħχ_ϕJ_x,z^2, for ϕ=π, 2nπ/N with n=±1,±2,⋯±(N/2-1), where N is the atom number and χ_ϕ the effective coupling strength. When the system is driven by two lights, the effective Hamiltonian becomes a TAT model H_TAT≈ħχ_ϕJ_z^2-ħχ_πJ_x^2. This approach is also suitable for preparation of squeezed state in Rydberg atom arrays
  <cit.>.

Fifth, for two species of atoms coupled through dipole-dipole interaction in a vapor cell (represented by their collective spin operator S and J), periodically driving the system can transform the inter-species dipolar interaction into a nonlinear intra-species interaction, resulting in both effective OAT and TAT Hamiltonians <cit.>. As for a continuous driving, the OAT Hamiltonian is realized as H_OAT=χ_effS_z^2, where χ_eff=-g^2/(2Δ_f) with g being the dipolar coupling strength between two species and Δ_f the difference between the magnitudes of two external DC fields applied to the two species of spins respectively. Realization of a TAT Hamiltonian needs an extra AC field applied to the S species, which yields H_TAT=χ_eff(S_x^2-S_y^2).



§ CONCLUSION

 We construct a diagonal bilinear spin Bogoliubov Hamiltonian, which includes the one-axis twisting Hamiltonian as a limiting case, by employing the spin lowering and raising operators. We prove analytically and numerically that the ground state of the spin Bogoliubov Hamiltonian (and the one-axis-twisting Hamiltonian) exhibits Heisenberg-limit spin squeezing, ξ_R^2 ∝ J^-1 for an arbitrary spin J in a certain parameter regime, in contrast to previous scaling ξ_R^2 ∝ J^-2/3 under the one-axis-twisting Hamiltonian by quench dynamics. Such a spin-squeezed ground state may be realized experimentally in dipolar spinor BECs, ultracold atoms in an optical lattice, and spins in a cavity or in a vapor cell.





This work is supported by the National Natural Science Foundation of China (Grant No. 12274331), Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302100), and the NSAF (Grant No. U1930201). The numerical calculations in this paper have been partially done on the supercomputing system in the Supercomputing Center of Wuhan University.












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	[app()]approx
	@noop  note At a large enough θ, tanhθ
		can firstly be approximated by (1-2exp(-2θ)). Furthermore, the
		approximation (ν^*/μ)^K ≈ 1-2Kexp(-2θ) is valid when
		exp(-2θ)≪ 1/(2K-2). Since maxK=J, the minimum requirement for
		this approximation to be valid for all Ks is exp(-2θ)≪1/(2J-2).
		Specifically, for J=1000, the valid regime for the the exponential decay
		formula of spin average(Eq.(13)) is θ>3.8, which coincides with the
		results shown in Fig.1(a).Stop
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			Phys. Rev volume 35, pages 667
		(year 1930)NoStop
	[Schrödinger(1930)]schrodinger1930
	author author E. Schrödinger, @noop  journal journal Phys.-Math. Klasse volume 14, pages 296 (year 1930)NoStop
	[Kinani and Daoud(2001)]Kinani_2001GIS
	author author A. H. E.			Kinani and author M. Daoud, @noop  journal journal J. Phys. A: Math. Gen. volume 34,		pages 5373 (year 2001)NoStop
	[not()]note1
	@noop  note We note that the phase sensitivity is in
		principle determined by the quantum Cramer-Rao bound
		Δϕ=ξ_R/√(J)≥1/√(F_Q), where F_Q denotes the quantum
		Fisher information. From this inequality, one finds that the spin squeezing
		is limited by F_Q. It is evident that the GIS |χ⟩ saturates the
		quantum Cramer-Rao bound, resulting in
		ξ_R^2=2J/F_Q <cit.>.Stop
	[Giovannetti et al.(2004)Giovannetti, Lloyd, and Maccone]Giovannetti2004Science
	author author V. Giovannetti, author S. Lloyd,		and author L. Maccone, https://doi.org/10.1126/science.1104149 journal journal Science volume 306, pages
		1330 (year 2004)NoStop
	[Giovannetti et al.(2006)Giovannetti, Lloyd, and Maccone]Giovannetti2006PRL
	author author V. Giovannetti, author S. Lloyd,		and author L. Maccone, https://doi.org/10.1103/PhysRevLett.96.010401 journal
		journal Phys. Rev. Lett. volume 96,		pages 010401 (year 2006)NoStop
	[Giovannetti et al.(2011)Giovannetti, Lloyd, and Maccone]Giovannetti2011NP
	author author V. Giovannetti, author S. Lloyd,		and author L. Maccone, https://doi.org/10.1038/nphoton.2011.35 journal journal Nat. Photonics volume 5, pages 222 (year 2011)NoStop
	[Tóth and Apellaniz(2014)]Toth2014
	author author G. Tóth and author I. Apellaniz, https://doi.org/10.1088/1751-8113/47/42/424006
	journal journal J. Phys. A: Math. Theor.		volume 47, pages 424006 (year 2014)NoStop
	[Zhang and Duan(2014)]Zhang_2014NJP
	author author Z. Zhang and author L. M. Duan, @noop  journal journal New J.
					Phys., volume 16, pages 103037
		(year 2014)NoStop
	[two()]two_mode
	@noop  note The same effective Hamiltonian H_A is
		derived approximately in a Bose-Einstein condensate trapped in a double-well
		potential and the Heisenberg-limit spin squeezing may be achieved under the
		two-mode, large J, and mean-field approximations. However, our results on
		the spin Bogoliubov Hamiltonian H_B are free of these approximations thus
		are applicable to many other systems.Stop
	[Płodzie ńń et al.(2022)Płodzie ńń,
		Lewenstein, Witkowska, and Chwede ńńczuk]Plodzie2022PRL
	author author M. Płodzie ńń, author
		M. Lewenstein, author
		E. Witkowska, and author
		J. Chwede ńńczuk, https://doi.org/10.1103/PhysRevLett.129.250402
	journal journal Phys. Rev. Lett. volume 129, pages 250402 (year
		2022)NoStop




