Spin squeezing in a generalized one-axis twisting model

We investigate the dependence of spin squeezing on the polar angle of the initial coherent spin state $|\theta_0, \phi_0>$ in a generalized one-axis twisting model, where the detuning $\delta$ is taken into account. We show explicitly that regardless of $\delta$ and $\phi_0$, previous results of the ideal one-axis twisting is recovered as long as $\theta_0=\pi/2$. For a small departure of $\theta_0$ from $\pi/2$, however, the achievable variance $(V_{-})_{\min}\sim N^{2/3}$, larger than the ideal case $N^{1/3}$. We also find that the maximal-squeezing time $t_{\min}$ scales as $N^{-5/6}$. Analytic expressions of $(V_{-})_{\min}$ and $t_{\min}$ are presented, which agree with numerical simulations.

Possible realization of the OAT-induced squeezing in a two-mode Bose-Einstein Condensates (BECs) has been proposed [4], where the self-interaction parameter χ ∼ (a aa + a bb − 2a ab )/2 is inherently aroused from atomic intra-and inter-species collisions. Atomic collisions lead to both the squeezing and phase diffusion [11,12]. The dephasing process destroys phase coherence of the two-component BECs, and thus sets a limit to the applications of the condensates in high-precision measurement and quantum information processing. A straightforward way to suppress the diffusion is the preparation of number-squeezed state, a special case of the SSS with the reduced variance along the J z component. Such a kind of squeezed states have been investigated both experimentally [13,14,15,16,17,18,19] and theoretically [20,21,22,23,24].
Besides the above schemes that rely on nonlinear interactions of the ultracold atoms, spin squeezing can be generated via light-matter interactions [2,3,25,26,27] and quantum nondemolition measurement [28,29,30,31,32,33,34]. Recently, the OATinduced squeezing has been demonstrated in an ensemble of cesium atoms [31,32] and ytterbium atoms [33,34]. In their experiments, the CSS with θ 0 = π/2 was adopted as the input state, which is the optimal initial state to obtain the strongest squeezing. Via optical pumping, it was shown that 98% atoms are in the CSS [32].
In this paper, we investigate the degree of the OAT-induced squeezing for θ 0 slightly departure from π/2. A generalized one-axis twisting model: H = δJ z +χJ 2 z is considered, which is the most important prototype in studying spin squeezing [1,4] and quantum metrology [35,36]. We prove explicitly that without particle losses, the detuning δ and the azimuth angle φ 0 give vanishing contribution to the squeezing parameter, and the ideal OAT-induced spin squeezing can be reproduced as long as θ 0 = π/2. As the main result of our paper, we investigate the dependence of the variance (V − ) min and the time t min on the particle number N and the polar angle θ 0 . Our results show that even for a small departure of θ 0 from π/2, power rule of the smallest variance (V − ) min changes from N 1/3 to N 2/3 with the increase of particle number N. The maximal squeezing is achievable at the time that scaled as χt min ∼ N −5/6 . Our paper is organized as follows. In Sec. II, we present general formulas of spin squeezing for arbitrary spin-1/2 system. In Sec. III, we study quantum dynamics of the OAT model, which is exactly solvable for any initial CSS. In Sec. IV, we present short-time solutions of the first-and second-order moments of the spin operators.
Approximated expression of the reduced variance V − is presented to obtain power rules of the maximal squeezing and its time scale t min . Finally, a summary of our paper is presented.

Some formulas of the spin squeezing
Assume that an ensemble of N two-level atoms (i.e., spin 1/2 particles) with ground state |a and excited state |b can be described by collective spin operator J = N k=1 1 2 σ (k) , where σ (k) is the Pauli operator of the kth atom. Spin components of J obey SU(2) algebra, [J n 1 , J n 2 ] = iJ n 3 for any three orthogonal vectors n 1 , n 2 , n 3 . The associated uncertainty relation reads (∆J where the variance is defined as usual, (∆Â) 2 = Ψ|Â 2 |Ψ − Ψ|Â|Ψ 2 for any spin state |Ψ and operatorÂ. Considering the mean spin J = ( J x , J y , J z ), we choose the orthogonal vectors as where the azimuth angles φ = tan −1 [ J y / J x ], and the polar angle Fig. 1(a)]. For arbitrary spin state |Ψ , it is easy to prove that the mean spin J is along the n 3 direction, with the length of the mean spin R = | J | = J n 3 [see Append.]. Now, let us consider the CSS [10]: which is eigenstate of J n 3 with eigenvalue j = N/2 (where N is total particle number), and thus J n 3 = | J | = j. In single-particle picture, the CSS can be rewritten as a direct product, |θ, φ = N k=1 [cos(θ/2)|b k + e iφ sin(θ/2)|a k ], where |a k and |b k are ground and excited states of the kth atom. Such a quantum uncorrelated state obeys the minimal uncertainty relationship: (∆J n 1 ) 2 = (∆J n 2 ) 2 = 1 2 | J n 3 | = j/2, where the value j/2 is termed as the SQL.
Since the mean spin is parallel with n 3 , one can introduce any spin component normal to the mean spin as where the unit vector n ψ = n 1 cos ψ + n 2 sin ψ, with ψ, being arbitrary angle between n 1 and n ψ . For any spin state |Ψ , we have J ψ = 0 and therefore, the variance of J ψ reads where the coefficients A = J 2 n 1 − J 2 n 2 , B = J n 1 J n 2 + J n 2 J n 1 , and C = J 2 n 1 + J 2 n 2 = j(j + 1)− J 2 n 3 . Another orthogonal spin component with respect to J ψ and its variance can be obtained by replacing ψ with ψ + π/2. For the CSS |θ, φ , it is easy to verify that the coefficients A = B = 0 and C = j, which gives the variance (∆J ψ ) 2 = j/2, indicating isotropically distributed variances of the CSS [1], as shown in Fig. 1(b). A spin-squeezed state (SSS) is defined if the variance of one spin component normal to the mean spin is smaller than the SQL [1], i.e., (∆J ψ ) 2 < j/2. The SSS has anisotropic variances distribution in a plane normal the mean spin [see Fig. 1 Substituting these results into Eq. (4), we obtain the reduced and the increased variances [7,22,23] where the reduced variance V − = (∆J ψ ) 2 corresponds to the squeezing along n ψ with ψ = ψ op = [π + tan −1 (B/A)]/2; while the increased variance V + gives the so-called antisqueezing for the angle ψ = ψ op + π/2. The degree of spin squeezing can be quantified by the normalized variance For the CSS, the variances V − = V + = j/2 and ξ 2 = 1; while for the SSS, ξ 2 < 1. It should be mentioned that the coefficients A, B, and C depend only on five quantities [see Append. A]: J z , J + , J 2 z , J 2 + , and J + (2J z + 1) , from which one can solve the mean spin J and the squeezing parameter ξ 2 . In addition, there are several definitions of the squeezing parameter. According to Wineland et al. [2], the squeezing parameter is defined as which closely relates to both frequency resolution in spectroscopy [2] and many-body quantum entanglement [4].

Generalized one-axis twisting model and its exact solutions
The above formulas are valid for any spin-1/2 system with SU(2) symmetry. As an example, we consider a two-component BECs [37,38] confined in a deep 3D harmonic potential. The total system can be described by the two-mode Hamiltonian ( = 1) [39]: whereâ,b, andN i (i = a, b) are the annihilation and number operators for the two internal states |a and |b , ω i are single-particle kinetic energies, and U ij = (4πa ij /M) d 3 r|Φ 0 (r)| 4 are atom-atom interaction strengthes. For a conserved total particle number N =N a +N b , the two-mode model can be rewritten as Assumed that the two-mode system evolves from the CSS, where the polar angles θ 0 and φ 0 determine population imbalance and the relative phase between the two internal states [40,41]. The state vector at any time t reads where the self-interaction χ scrambles phase of each number state |j, m , and leads to spin squeezing [1,4] and phase diffusion [11] of the two-mode BEC. In theory, the diffusion is quantified by correlation function b †â (i.e., J + ), which decays exponentially with the time scale χt d = j −1/2 for θ 0 = π/2. Such a kind of the dephasing process has been observed in experiment by extracting the visibility of the Ramsey fringe [12].
As an ideal case, spin squeezing induced by the OAT Hamiltonian χJ 2 z has been investigated for the initial CSS |θ 0 = π/2, φ 0 = 0 [1]. For this special CSS, it was shown the smallest variance (V − ) min ∼ (2j) 1/3 is obtainable at the time t min ∼ (2j) −2/3 . Based upon this, Sørensen et al. studied possible realization of the squeezing in 23 Na atom BECs [4]. More important, they proposed that the squeezing parameter can be used as a probe of many-body entanglement. In this paper, we investigate dynamical generation of the SSS in the generalized OAT model from arbitrary CSS. We find that the power rules change significantly even for θ 0 ∼ π/2.
At first, we determine the mean spin J = ( J x , J y , J z ), where J z = j cos(θ 0 ), J x = Re J + , and J y = Im J + , with It is convenient to rewrite Eq. (11) as J + = r exp(iφ), which yields J x = r cos φ and J y = r sin φ, as defined in Eq. (1). Therefore, we obtain where ϕ(t) = tan −1 [cos(θ 0 ) tan(χt)] is dynamical phase. Note that in real calculations of the squeezing parameters, only cos(φ) and sin(φ) are needed and given by Eq. (A.2) and Eq. (A.3). The explicit form of the phase φ or ϕ is introduced to find out the roles of δ and φ 0 in the squeezing. Obviously, r, ϕ, and also R = (r 2 + J z 2 ) 1/2 do not depend on them.

Power rules of the strongest squeezing and its time scale
As shown by the red lines of Fig. 2(b), both ξ 2 min and t min change significantly in comparison with the idea case (i.e., θ 0 = π/2). As a result, it is necessary to determine power rules of the variance (V − ) min and the time t min for θ 0 = π/2. In this section, we calculate analytically the power rules by using standard treatments of Ref. [1]. We will focus on a small departure of θ 0 from π/2 due to the fact that a relatively small population imbalance between two internal states favors the one-axis twisting effect.

Ideal OAT case with
In the short-time limit (χt << 1) and large particle number (j >> 1), the increased and reduced variances Eq. (19) can be approximated as [1]: where α 0 = jχt > 1 and β 0 = j(χt) 2 << 1. Eq. (20) is the key point to obtain the strongest squeezing ξ min and its time scale t min . Previously, the time t min was obtained by comparing the second term of V − with that of the first one [1]. Here, we solve t min via minimizing V − with respect to t, i.e., which yields power rule of the maximal-squeezing time: Inserting χt min into Eq. (20), we further obtain the reduced variance as and also, the smallest squeezing parameter ξ 2 min = 2j −1 (V − ) min ≃ 1 2 ( 2j 3 ) −2/3 . Power exponents of Eq. (22) and Eq. (23) are consistent with Ref. [1], but different in the coefficients. As shown by the black solid lines of Fig. 3, the revised results fit very well with their numerical results (empty circles).

Small departure case with
The power rules, Eq. (22) and Eq. (23), are valid only for θ 0 = π/2. Now, we generalize them for θ 0 = π/2 case. To obtain the approximated expressions of the variances as Eq. (20), we calculate short-time solutions of J + , J 2 + , and J + (2J z + 1) . In the short-time limit (χt << 1), the dynamical phase ϕ(t) = tan −1 [cos(θ 0 ) tan(χt)] ≃ χt cos(θ 0 ), and Eq. (11) can be approximated as where β = β 0 sin 2 (θ 0 ) = j(χt) 2 sin 2 (θ 0 ), and φ ≃ φ 0 +δt+2jχt cos(θ 0 ). We have assumed that particle number is large enough so 2j − 1 ≃ 2j. The length of the correlation reads r = | J + | ≃ j sin(θ 0 )e −β , which indicates that phase coherence of the two-mode BEC decays exponentially (i.e., phase diffusion [11]) with the coherence time scaled as χt d = sin −1 (θ 0 )j −1/2 [42,43]. Similarly, short-time solutions of Eq. (15) and Eq. (16) can be written approximately as and where the factor cos θ 0 can not be neglected since it is comparable with χt. In fact, it is the presence of cos θ 0 that leads to significant change of t min and (V − ) min even for θ 0 ∼ π/2. To simplify the calculations, we make further approximations to the angles of Eq. (A.4)-Eq. (A.6): sin θ = r/R ≃ sin θ 0 and cos θ = J z /R ≃ cos θ 0 , where θ 0 is polar angle of the initial CSS. This approximation is equivalent with r ≃ j sin(θ 0 ), i.e., neglecting the the diffusion within the squeezing time due to t d > t min . Now, we expand the coefficients A, B, and C in terms of β. In calculating the increased variance, we only keep the lowest order of β, and get V + ≃ j 2 (4α 2 ), where α = α 0 sin 2 θ 0 = jχt sin 2 θ 0 . Next, we solve power series of 4V + V − up to the third order of β, from which we obtain the reduced variance as where the j-dependent additional term gives significant contribution to the squeezing for θ 0 = π/2. By minimizing V − with respect to t, we obtain power rule of the time as and that of the decreased variance: For θ 0 = π/2, our results reduce to the ideal OAT case, i.e., Eq. (22) and Eq. (23); while for θ 0 = π/2 and large j, Eq. (28) and Eq. (29) predict that the power rules change to which are confirmed by numerical simulations. To see this more clearly, let us focus on red lines of Fig. 3. For θ 0 ∼ π/2 and small j, both the time χt min and the variance (V − ) min follow the same rule with the θ 0 = π/2 case. With the increase of j, however, the red line (the crosses) of Fig. 3(a) decreases faster than the ideal OAT case [see also Fig. 2(b)]. The change of the power rule is shown more clearly in Fig. 3(b). In Fig. 4, we show the dependence of t min and (V − ) min on θ 0 for a fixed value j. It was show that both t min and (V − ) min are symmetrical with respect to θ 0 = π/2. The most strongest squeezing [i.e., the smallest value of ξ 2 min ] occurs for the optimal initial state θ 0 = π/2. Our analytic results, Eq. (28) and Eq. (29), agree quite well with numerical simulations except θ 0 = 0 or π. In this case, the state vector |Ψ(t) = exp[−i(χj 2 ± δj)t]|j, ±j , which is the CSS with the variances (V + ) = (V − ) = j/2 and ξ 2 = 1. However, Eq. (29) diverges as θ 0 → 0 or π, inconsistent with the real situation. Eq. (28) gives relatively good estimate of the maximal-squeezing time. As shown in Fig. 4(a), t min decreases monotonically in the small departure regime |θ 0 − π/2| < 0.27π/2, which implies that the maximal squeezing occurs more and more earlier [see also Fig. 2(b)]. Out of the regime, t min increases with the departure of θ 0 , and goes infinity as θ 0 → 0 or π.

Dissipation effect due to atomic decay
So far, we have neglected the effects of dissipation on the spin squeezing, such as particle losses and center-of-motion of the atoms in the BECs [4,40,41]. For the squeezing generated in atomic ensemble, the dominant dissipation source is atomic decay due to spontaneous emission [32], which can be described by the master equation [26]: where ρ is the density operator, and γ is the decay rate of the atoms. In the basis of |j, m , the elements ρ m,n = j, m|ρ|j, n could be solved numerically by using the Runge-Kutta routine [26]. In real calculations of the squeezing parameters, only 6j elements like ρ m,m , ρ m,m+1 , and ρ m,m+2 are needed. In Fig. 5, we plot time evolution of ξ 2 for small decay rate, e.g., γ/χ = 0.01 and 0.1. Such a small dissipation can be realized by increasing χ, which in turn leads to the preparation of the SSS within the lifetime of the atoms γ −1 [32]. For relatively small decay rate γ/χ = 0.01 (red curves), both the maximal squeezing and its time scale change slightly in comparison with γ = 0 case. The initial state with θ 0 = π/2 looks more sensitive to atomic decay than θ 0 = 0.8 × π/2 case. From the blue dotted lines of Fig. 5(b), we find that even for γ/χ = 0.1, a considerable squeezing with ξ 2 (and also ζ 2 )∼ 0.22 could be reached in an ensemble of 200 atoms, which occurs at a time scale given by Eq. (28).

Conclusion
In summary, we have presented general formulas to study spin squeezing in spin-1/2 system. Instead of six fluctuation parameters as Refs. [40,41], only five parameters, i.e., J z , J + , J 2 z , J 2 + , and J + (2J z + 1) are needed to determine the mean spin and the squeezing parameters.