Spin Squeezing of One-Axis Twisting Model in The Presence of Phase Dephasing
aa r X i v : . [ qu a n t - ph ] D ec Quantum Information and Computation, Vol. 13, No. 3&4 (2013) 0266–0280c (cid:13)
Rinton Press
SPIN SQUEEZING OF ONE-AXIS TWISTING MODELIN THE PRESENCE OF PHASE DEPHASING
CHEN-GANG JI , YONG-CHUN LIU a , and GUANG-RI JIN b Department of Physics, Beijing Jiaotong University, Beijing 100044, China
Received (July 11, 2012)Revised (October 11, 2012)We present a detailed analysis of spin squeezing of the one-axis twisting model with amany-body phase dephasing, which is induced by external field fluctuation in a two-modeBose-Einstein condensates. Even in the presence of the dephasing, our analytical resultsshow that the optimal initial state corresponds to a coherent spin state | θ , φ i with thepolar angle θ = π/
2. If the dephasing rate γ ≪ S − / , where S is total atomic spin, wefind that the smallest value of squeezing parameter (i.e., the strongest squeezing) obeysthe same scaling with the ideal one-axis twisting case, namely ξ ∝ S − / . While fora moderate dephasing, the achievable squeezing obeys the power rule S − / , which isslightly worse than the ideal case. When the dephasing rate γ > S / , we show that thesqueezing is weak and neglectable. Keywords : Quantum noise, Bose-Einstein condensates, Phase dephasing, Spin squeezedstates
Communicated by : D Wineland & K Moelmer
Spin squeezing of an ensemble of spin-1 / π/ V − below standardquantum limit (SQL)— S/
2, where S is total atomic spin. The degree of spin squeezing ξ (= 2 V − /S ) can reach the power rule S − / , which is the ideal OAT result [4]. Experimentalrealizations of the OAT model has been proposed [10, 11, 12, 13, 14, 15, 16, 17] and demon-strated [18, 19, 20, 21, 22, 23, 24, 25, 26, 27] in a two-mode Bose-Einstein condensate (BEC). a Present address: State Key Lab for Mesoscopic Physics, School of Physics, Peking University, Beijing 100871,China. b Corresponding author: [email protected] 1
SPIN SQUEEZING OF ONE-AXIS TWISTING MODEL... . . .
Due to experimental imperfections in coupling pulses, atom losses, technique and quantumnoises, etc., the achievable squeezing is worse than the theoretical prediction [28]. Recently,we investigated the dependence of spin squeezing on the initial CSS | θ , φ i . Our results showthat the scaling of ξ depends sensitively upon the polar angle θ ; it becomes ξ ∝ S − / [29]even when θ is slightly deviated from its optimal value π/ ξ . For the optimal initial CSS with the polar angle θ = π/
2, we find that the dephasing effect can be negligible as long as the dephasing rate γ ≪ S − / . The ideal one-axis twisting is attainable with the best squeezing ξ ∝ S − / [4].For a moderate dephasing rate (i.e., S − / < γ < S / ), our analytical result indicates thatthe achievable squeezing scales as S − / , which is slightly worse than the ideal OAT case. Asthe dephasing rate increases up to S / , we find that the squeezing becomes very weak as ξ ∼ h ˆ S z i , h ˆ S + i , h ˆ S z i , h ˆ S i ,and h ˆ S + (2 ˆ S z + 1) i . With these expectation values at hands, one can numerically calculatethe squeezing parameter ξ . To get scaling behavior of ξ , in Sec. IV, we consider short-timelimit and large enough particle number. Analytical expression of the squeezing parameter isobtained, with which we analyze power rules of ξ according to the role of the phase dephasing.Finally, we conclude in Sec. V with the main results of our work. We focus on a two-component Bose-Einstein condensates with the internal states | ↑i and | ↓i that is confined in a deep three-dimensional harmonic potential. Quantum dynamics of thetotal system can be described by the Lindblad equation (¯ h = 1): d ˆ ρdt = i [ˆ ρ, ˆ H ] + Γ p (2 ˆ S z ˆ ρ ˆ S z − ˆ S z ˆ ρ − ˆ ρ ˆ S z ) , (1)where ˆ H = χ ˆ S z , known as the one-axis twisting Hamiltonian [4], can be realized in a two-mode BECs [10, 11, 12] with the interaction strength χ tunable via the Feshbach resonancetechnique [25] or the BEC spatial splitting [26]. Atomic spin operators ˆ S z = (ˆ b †↑ ˆ b ↑ − ˆ b †↓ ˆ b ↓ ) / S + = ( ˆ S − ) † = ˆ b †↑ ˆ b ↓ represent atomic population imbalance and phase coherence betweenthe two bosonic modes. The second term in Eq. (1) simulates a phase dephasing of the BECdue to magnetic-field fluctuations [30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Such a kind ofmany-body decoherence has also been studied in cavity-QED system [40, 41].Due to particle-number conservation, the OAT Hamiltonian and the super-operator in theEq. (1) commutes with total angular momentum operator ˆS . Consequently, the state at anytime t can be expanded in terms of common eigenstates of ˆS and ˆ S z , i.e., {| S, m i} with total i CG, . . . angular momentum S = N/ m = − S , − S +1, · · · , S . Using the basis, the density-matrixoperator ˆ ρ reads ˆ ρ = S X m,n = − S ρ m,n | S, m i h
S, n | , (2)with the elements ρ m,n = h S, m | ˆ ρ | S, n i satisfying dρ m,n dt = (cid:2) iχ ( n − m ) − Γ p ( n − m ) (cid:3) ρ m,n . (3)Assume that the BEC system evolves from a coherent spin state (CSS) [42]: | θ , φ i ≡ exp { iθ [ ˆ S x sin( φ ) − ˆ S y cos( φ )] }| S, S i = P m c m | S, m i , with the probability amplitudes c m = (cid:18) SS + m (cid:19) / (cid:18) cos θ (cid:19) S + m (cid:18) sin θ (cid:19) S − m e i ( S − m ) φ . (4)Exact solutions of Eq. (3) can be obtained with the density-matrix elements ρ m,n ( τ ) = ρ m,n (0) e − i ( m − n ) τ e − ( m − n ) γτ , (5)where τ = χt and γ = Γ p /χ . The first term on the right-hand side of Eq. (5) ρ m,n (0) = c m c ∗ n ,with c m given by Eq. (4). The second term arises from time evolution of the density matrixunder the OAT Hamiltonian χ ˆ S z . The last one is the dephasing term due to magnetic-field fluctuation and has been obtained previously [43, 44, 45, 46]. Particularly, Takeuchi etal. [43] considered a light-induced spin squeezing in an atomic gas and obtained almost thesame result with ours. However, their result is based upon an approximated commutationrelation of the Stokes operators of light. Here we present exact solution of the density-matrixelements, which describes quantitatively the OAT-induced spin squeezing in a two-mode BECin the presence of the phase dephasing. Starting from a CSS | θ , φ i , unitary evolution of the spin system under the OAT Hamiltonianleads to spin squeezing and multipartite entanglement. Both of them can be quantified bythe variances of a spin component ˆ S ψ = ˆS · n ψ that is normal to the mean-spin vector h ˆS i ≡ ( h ˆ S x i , h ˆ S y i , h ˆ S z i ), i.e., n ψ · h ˆS i = 0. As usual, the expectation value of an operator ∧O is defined by h ∧Oi = Tr(ˆ ρ ∧O ). For a given state ˆ ρ , the mean spin and its direction n = h ˆS i / |h ˆS i| can be determined uniquely, which in turn gives n ψ = n cos ψ + n sin ψ , withthree orthogonal vectors n , n , and n (for details see Ref. [29]). The increased and thereduced variances of the spin component ˆ S ψ are defined, respectively, as V + = max ψ (∆ ˆ S ψ ) and V − = min ψ (∆ ˆ S ψ ) , with V ± = 12 (cid:16) C ± p A + B (cid:17) , (6)where A , B , and C depend only on five expectation values (see Appendix A): h ˆ S z i , h ˆ S + i , h ˆ S z i , h ˆ S i , and h ˆ S + (2 ˆ S z + 1) i . According to Kitagawa and Ueda [4], a spin state is squeezed onlyif the variance V − is smaller than the SQL, S/
2, namely the squeezing parameter ξ = 2 V − S < . (7) SPIN SQUEEZING OF ONE-AXIS TWISTING MODEL... . . .
The spin squeezed state is useful to improve frequency resolution in spectroscopy provided that ζ = 2 SV − / |h ˆS i| < ζ = 2 V − / |h ˆS i| =( S/ |h ˆS i| ) ξ [47]. The three definitions are slightly different in magnitude, ξ ≤ ζ ≤ ζ dueto |h ˆS i| ≤ S . For large enough particle number N ( > ), the minimum values of them obeyalmost the same power rule [43], so we only focus on the squeezing parameter ξ .Based upon Eq.(5), we now calculate exact solutions of the mean spin and the variances V ± . After some straightforward calculations, we can obtain (see Append. A, or Ref.[29]) h ˆ S z i = S cos( θ ) , h ˆ S + i = S sin( θ ) e iφ e − γτ [ R ( τ )] S − , (8)where the population imbalance h ˆ S z i is time-independent, and R ( τ ) = cos( τ ) + i cos( θ ) sin( τ ) = q − sin ( θ ) sin ( τ ) · e i tan − [cos( θ ) tan( τ )] . (9)From Eq. (8), we note that the phase dephasing considered in Eq. (1) imposes an exponentialdecay term e − γτ to the phase coherence |h S + i| , but maintains the population imbalance h ˆ S z i intact. Moreover, it is easy to obtain h ˆ S x i = Re h ˆ S + i = |h ˆ S + i| cos( φ ) and h ˆ S y i = Im h ˆ S + i = |h ˆ S + i| sin( φ ), with the argument of h ˆ S + i : φ ≡ arg h ˆ S + i = φ + (2 S −
1) tan − [cos( θ ) tan( τ )] . (10)Here, φ is the azimuth angle of the initial CSS. The variances V ± depend upon the coefficients A , B , and C (see Append. A). In real calculations of them, only cos( φ ) and sin( φ ) are required,which depends on h ˆ S + i . In addition, we need to solve the following expectation values: h ˆ S z i = S S (cid:18) S − (cid:19) cos ( θ ) , (11) h ˆ S i = S (cid:18) S − (cid:19) sin ( θ ) e iφ e − γτ [ R (2 τ )] S − , (12)and h ˆ S + (2 ˆ S z + 1) i = 2 S (cid:18) S − (cid:19) sin( θ ) e iφ e − γτ [ R ( τ )] S − [ i sin ( τ ) + cos( θ ) cos ( τ )] . (13)Substituting the above results into the coefficients A , B , and C , one can obtain the variances V ± and also the squeezing parameter ξ . In Fig. 1, we plot exact numerical results of ξ (solidlines) for different values of the dephasing rate γ . We find that the strongest squeezing occursat a certain time τ min (= χt min ), with its position indicated by the arrows of Fig. 1. In order to analyze scaling behaviors of ξ , we now consider the short-time limit (i.e., τ = χt ≪
1) and large enough particle-number ( S = N/ ≫ R ( τ ) ≈ S exp( − τ sin θ ) e iτ cos θ , which in turn yields h ˆ S + i ≈ S sin( θ ) e iφ e − β , (14) i CG, . . . ξ as a function of scaled time τ (= χt ) forvarious dephasing rates γ = Γ p /χ = 0 (a), 1 (b), and 10 (c). Solid blue lines are given by exactnumerical simulations and red dashed lines are predicated by Eq. (18). Other parameters are S = N/ , θ = π/
2, and φ = 0. The arrows indicate the location of maximal-squeezingtimes τ min = 7 . × − (a), 1 . × − (b), and 1 . × − (c), at which the strongestsqueezing occurs with ξ = 6 . × − (a), 9 . × − (b), and 0 . SPIN SQUEEZING OF ONE-AXIS TWISTING MODEL... . . . h ˆ S i ≈ S (cid:18) S − (cid:19) sin ( θ ) e iφ e − β , (15)and h ˆ S + (2 ˆ S z + 1) i ≈ S (cid:18) S − (cid:19) sin( θ ) e iφ e − β (cos θ + iτ ) , (16)where the argument of h ˆ S + i now becomes φ ≈ φ + 2 Sτ cos( θ ), and β = Sτ sin θ + γτ. (17)Note that without the dephasing, the phase coherence reduces to |h S + i| ≈ S sin( θ ) e − ( τ/τ d ) ,with the coherent time τ d = χt d = 1 / ( √ S sin θ ) [29]. Such a kind of exponential decay iswell known as the so-called phase diffusion of a two-mode BEC. For the BEC atoms, thenonlinearity χ ∝ ( a ↑↑ + a ↓↓ − a ↑↓ ) depends upon the intra- and the inter-species atom-atomscattering lengthes. When the three coupling constants are close to each other, the coherencetime t d increases dramatically due to χ → γ = 0), the phase diffusion process will speed up, as demonstrated recently in Ref. [39].In what’s following, we will investigate the role of the phase dephasing on the spin squeezing.Firstly, using Eq. (15) and Eq.(16), as well as the exact result of h ˆ S z i , we obtain theshort-time solutions of the coefficients A , B , and C [see Eq. (B.1)-Eq.(B.2)]. Next, we focuson a time regime: τ < Sτ ≪ γτ ≪
1, which allows us to expand the above resultsin terms of β ( ≪
1) [4]. From Eq. (6), it is easy to find that the product of the variances V + V − = [( C + A )( C − A ) − B ] /
4. To simplify it, we expand
C ± A , B , and hence V + V − up to the third-order of β [see Eq. (B.4)-Eq. (B.5)]. On the other hand, we can reduce theincreased variance V + by keeping the lowest-order of β (see Appendix B). Finally, using therelation V − = ( V + V − ) /V + , we obtain analytical result of the reduced variance and also thesqueezing parameter: ξ ≈ γτβ + 14 Sβ sin ( θ ) + 2 β (cid:2) S sin ( θ ) cos ( θ ) (cid:3) , (18)where β is given by Eq. (17) and θ is polar angle of the initial state. In Fig. 1, we compareour analytical result of ξ (red dash) with its exact solution (solid line) for different values ofthe dephasing rate γ . When γ is not too large, we find that Eq. (18) works well to predictthe minimal value of the squeezing parameter ξ = 2 V − ( τ min ) /S . This is because both theanalytical and the exact results almost merge with each other around the maximal-squeezingtime τ min .Based upon Eq. (17) and Eq. (18), we now analyze in detail the role of the phase dephasingon the spin squeezing. If the first term on right-hand side of Eq. (18) is comparable with thesecond one, we obtain γτ ∼ [4 S sin ( θ )] − . On the other hand, we compare the two termsof Eq. (17) and get γ ∼ Sτ sin ( θ ).Obviously, the dephasing effect can be neglected only if γ ≪ Sτ sin ( θ ) and γτ ≪ [4 S sin ( θ )] − , for which Eq. (17) becomes β ≈ Sτ sin ( θ ) and the first term of Eq. (18)can be omitted. As a result, we obtain the analytical result of the squeezing parameter [29]: ξ ≈ Sβ sin ( θ ) + 2 β (cid:2) S sin ( θ ) cos ( θ ) (cid:3) . (19) i CG, . . . From the relation ( dξ /dτ ) (cid:12)(cid:12) τ min = 0, we obtain the maximal-squeezing time: τ min = χt min ≈ / [2 S sin ( θ )] − / [1 + 9 S sin ( θ ) cos ( θ )] / . (20)Substituting Eq. (20) back to Eq. (19), we further obtain the smallest value of ξ : ξ ≈ ( (cid:2) S sin ( θ ) cos ( θ ) (cid:3) S sin ( θ ) ) / . (21)For the initial CSS with θ = π/
2, Eq. (21) shows ξ ≈ ( S ) − / , which is the bestsqueezing that the one-axis twisting scheme can reach [4, 29]. Considering a large enoughparticle-number with S = N/ , we can obtain τ min ≈ . × − and ξ ≈ . × − , fitting very well with numerical simulations [see Fig. 1(a)]. From Eq. (20), wefind that the time scales as τ min ∝ S − / for θ = π/ τ min ∝ S − / for θ = π/ γτ ≪ [4 S sin ( θ )] − , we make a conclusionthat our analytical results, Eq. (20) and Eq. (21), are valid for the dephasing rate γ ≪ S − / ( θ = π/ γ ≪ S − / ( θ = π/ γ ) / ln( S ) < − . -1.0 -0.5 0.0 0.5 1.010 -4 -3 -2 -1 = /2ln( )/ln(S) = /4 Fig. 2. (color online) The minimal value of the squeezing parameter ξ as a function ofln( γ ) / ln( S ) for θ = π/ θ = π/ ξ ≈ γ ) / ln( S ) > . θ = π/ γ ) / ln( S ) > . θ = π/
4. Other parameters: χ = 1, S = N/ and φ = 0. To proceed, let us consider the case γ < Sτ sin ( θ ), but γτ > [4 S sin ( θ )] − , for whichthe first term of Eq. (18) becomes important in a comparison with the second one so we get ξ ≈ γτβ + 2 β (cid:2) S sin ( θ ) cos ( θ ) (cid:3) , (22) SPIN SQUEEZING OF ONE-AXIS TWISTING MODEL... . . . where β ≈ Sτ sin ( θ ). Minimizing ξ with respect to τ , we obtain τ min ≈ (3 γ ) / (cid:2) S sin ( θ ) (cid:3) − / [1 + 9 S sin ( θ ) cos ( θ )] / , (23)and ξ ≈ (cid:26) γ [1 + 9 S sin ( θ ) cos ( θ )]3 S sin ( θ ) (cid:27) / . (24)From Eq. (23), we find that the strongest squeezing appears at τ ∝ γ / S − / for θ = π/
2, and τ ∝ γ / S − / for θ = π/
2. Using the conditions γτ > [4 S sin ( θ )] − and γ < Sτ sin ( θ ), it is easy to find that Eq. (24) works quite well for a relatively weak dephasingrate with S − / < γ < S / ( θ = π/ S − / < γ < S / ( θ = π/ -4 -3 -2 -1 -4 -3 -2 -1 (b) (a) /4 S Fig. 3. (Color online) The minimal value of the squeezing parameter ξ against θ for S = 10 (a), and S for θ = π/ γ = 3 . χ = 1 and φ = 0. As shown in Fig. 1(b), for the case S = 10 and γ = 1, Eq. (23) and Eq. (24) predict τ min ≈ . × − and ξ ≈ . × − , respectively. Both of them fit with the exactnumerical results τ min = 1 . × − and ξ = 9 . × − . As the dephasing rateincreases up to 10, our analytical results give τ min ≈ . × − and ξ ≈ . i CG, . . . the red solid line of Fig. 2, given by Eq. (24) for θ = π/
4, shows clearly that ξ increasesrapidly with γ in the regime − . < ln( γ ) / ln( S ) < . γ becomes large. For instance, let us consider γτ > [4 S sin ( θ )] − and γ > Sτ sin ( θ ).In this case, Eq. (22) reduces to ξ ≈ β (cid:2) S sin ( θ ) cos ( θ ) (cid:3) , (25)with β ≈ γτ . Actually, when γ > S / ( θ = π/ γ > S / ( θ = π/ ξ ≈
1, as shown by the blue dotted line of Fig. 2.For a fixed S (= N/ θ = π/ ξ even in the presence of phase dephasing. In Fig. 3(b), we focus on the optimalcase θ = π/ ξ on S for the dephasing rates γ = 0 (opencircles), 1 (red crosses), and 3 . S = N/ > ). More interestingly, it is also found that the slop of the solidcurve is different with that of other two lines. This is because our analytical result for thecase γ = 0, Eq. (21), predicts ξ ∝ S − / [4]; while for a small but nonzero γ , Eq. (24) gives ξ ∝ S − / . Such a power rule has been also obtained by Takeuchi et al. [43]. However, theirscheme is based upon a double-pass Faraday interaction between atoms and far-off-resonantlight. In addition, the starting point of their work, though quite similar with Eq. (5), isderived by an approximated commutation relation of the light-field Stokes operators [43]. In summary, we have investigated the role of phase diffusion on spin squeezing of the one-axistwisting model. Our results show that the spin squeezing depends upon the initial state | θ , i (= e − iθ ˆ S y | S, S i ). The optimal initial state corresponds to the polar angle θ = π/
2, even inthe presence of phase dephasing. If the dephasing rate γ = Γ p /χ ≪ S − / , the dephasingeffect is negligible and the ideal one-axis twisting is restored. The strongest squeezing scalesas ξ ∝ S − / . For a moderate dephasing rate (i.e., S − / < γ < S / ), the achievablesqueezing obeys the power rule ξ ∝ S − / , which is slightly worse than the ideal case.When the dephasing rate γ > S / , we show that the squeezing becomes very weak due to ξ ∼ Acknowledgements
We thank Professor L. You for helpful discussions. This work is supported by Natural ScienceFoundation of China (NSFC, Contract No. 10804007 and No. 11174028), the FundamentalResearch Funds for the Central Universities (Contract No. 2011JBZ013), and the Program forNew Century Excellent Talents in University (Contract No. NCET-11-0564). C.G.J is par-tially supported by National Innovation Experiment Program for University Students (BJTUNo. 1270021 and No. 1270037).
Appendix A* SPIN SQUEEZING OF ONE-AXIS TWISTING MODEL... . . .
Exact solutions of the squeezing parameter
As Eq. (6), the reduced and the increased variances depend upon the coefficients A , B , and C (for details, see Ref. [29]): A = 12 n sin ( θ ) h S ( S + 1) − h ˆ S z i i − [1 + cos ( θ )] Re h h ˆ S i e − iφ i + sin(2 θ ) Re h h ˆ S + (2 ˆ S z + 1) i e − iφ io , (A.1) B = − cos( θ ) Im h h ˆ S i e − iφ i + sin( θ ) Im h h ˆ S + (2 ˆ S z + 1) i e − iφ i , (A.2) C = S ( S + 1) − h ˆ S z i − Re h h ˆ S i e − iφ i − A , (A.3)where the angles θ and φ are determined by the mean spin h S i = ( h S x i , h S y i , h S z i ), withsin( θ ) = |h ˆ S + i||h ˆS i| , cos( θ ) = h ˆ S z i|h ˆS i| , (A.4)cos( φ ) = h ˆ S x i|h ˆ S + i| , sin( φ ) = h ˆ S y i|h ˆ S + i| . (A.5)Here, h ˆ S x i = Re h ˆ S + i and h ˆ S y i = Im h ˆ S + i , as mentioned above. Note that the above formulaeare valid for any SU(2) system. Moreover, one can find that A , B , and C depend only onfive expectation values: h ˆ S z i , h ˆ S + i , h ˆ S z i , h ˆ S i , and h ˆ S + (2 ˆ S z + 1) i . Although the argument φ appears in the coefficients, we do not need to know its explicit expression. Instead, onlycos( φ ) and sin( φ ) are needed in real calculations of the coefficients and hence the variances V ± .Starting with an initial CSS | θ , φ i , the OAT model can be solved exactly. For instance,we calculate the expectation value h ˆ S l + i ≡ Tr(ˆ ρ ˆ S l + ), with an integer l = 1 , , · · · , [ S ]. Here,[ S ] denotes the greatest integer of any real number S . Using Eq. (5), we obtain h ˆ S l + i = S − l X m = − S ρ m,m + l (0) X − m X − m − · · · X − m − l +1 e i (2 ml + l ) τ e − γl τ = e − γl τ h ˆ S l + i , (A.6)where X m = p ( S + m )( S − m + 1), and ρ m,n (0) = c m c ∗ n represent the density-matrix ele-ments of the initial CSS, with c m given by Eq. (4). In addition, we have introduced h ˆ S l + i ≡ S − l X m = − S ρ m,m + l (0) e i (2 ml + l ) τ X − m X − m − · · · X − m − l +1 = S − l X m = − S (2 S )!( S + m )!( S − m − l )! e ilφ e i (2 ml + l ) τ (cid:18) cos θ (cid:19) S +2 m + l (cid:18) sin θ (cid:19) S − m − l = (2 S )!2 l (2 S − l )! sin l ( θ ) e ilφ X m (cid:18) S − lS + m (cid:19) (cid:18) e ilτ cos θ (cid:19) S + m (cid:18) e − ilτ sin θ (cid:19) S − m − l = (2 S )!2 l (2 S − l )! sin l ( θ ) e ilφ [ R ( lτ )] S − l , (A.7) i CG, . . . where R ( lτ ) = cos( lτ ) + i cos( θ ) sin( lτ ), and we have used the binomial formula: S − l X m = − S (cid:18) S − lS + m (cid:19) a S + m b S − m − l = S − l X n =0 (cid:18) S − ln (cid:19) a n b S − n − l = ( a + b ) S − l . (A.8)With the help of Eq. (A.6) and Eq. (A.7), we can obtain the exact solutions of h ˆ S + i and h ˆ S i ,given by Eq. (8) and Eq. (12), respectively.We note that without the dephasing, the spin system evolves under governed by the OATHamiltonian ˆ H = χ ˆ S z , so we have d h ˆ S + i /dτ = − i h [ ˆ S + , ˆ H ] i /χ = i h ˆ S + (2 ˆ S z + 1) i , wherethe subscript 0 denotes the expectation values in the absence of phase dephasing (i.e., γ = 0),and h ˆ S + i has been given in Eq. (A.7) with l = 1. Therefore, we obtain h ˆ S + (2 ˆ S z + 1) i = e − γτ h ˆ S + (2 ˆ S z + 1) i = − ie − γτ d h ˆ S + i dτ , (A.9)which gives Eq. (13). Finally, we calculate the population imbalance and its variance: h ˆ S z i = S X m = − S +1 ( S + m ) ρ m,m (0) − S X m = − S Sρ m,m (0)= 2 S cos (cid:18) θ (cid:19) X m (cid:18) S − S + m − (cid:19) (cid:18) cos θ (cid:19) S + m − (cid:18) sin θ (cid:19) S − m − S = 2 S cos (cid:18) θ (cid:19) − S = S cos( θ ) , (A.10)and h ˆ S z i = S X m = − S S ρ m,m (0) − S − X m = − S +1 ( S + m )( S − m ) ρ m,m (0)= S − S (cid:18) S − (cid:19) sin ( θ ) X m (cid:18) S − S + m − (cid:19) (cid:18) cos θ (cid:19) S + m − (cid:18) sin θ (cid:19) S − m − = S − S (cid:18) S − (cid:19) sin ( θ ) = S S (cid:18) S − (cid:19) cos ( θ ) , (A.11)where we have used the normalization condition P m ρ m,m (0) = 1. So far we have solved allthe quantities that relevant to get the coefficients A , B , and C , with which we can calculateexactly the variances V ± , and hence the squeezing parameter ξ . Appendix B*Short-time solutions of the squeezing parameter
To obtain analytical results of the coefficients A , B , and C , we have to make further approx-imations [29], sin( θ ) = |h ˆ S + i| / |h ˆS i| ≈ sin( θ ) and cos( θ ) = h ˆ S z i / |h ˆS i| ≈ cos( θ ), where θ ispolar angle of the initial CSS. Substituting Eq. (15) and Eq.(16), as well as Eq.(A.11) into SPIN SQUEEZING OF ONE-AXIS TWISTING MODEL... . . .
Eq. (A.1), we obtain the short-time solution of the coefficient
A ≈ sin θ (cid:26)(cid:20) S ( S + 1) − (cid:18) S S (cid:18) S − (cid:19) cos θ (cid:19)(cid:21) − S (cid:18) S − (cid:19) (cid:0) θ (cid:1) e − β + 4 S (cid:18) S − (cid:19) e − β cos θ (cid:27) = S (cid:18) S − (cid:19) sin θ (cid:8) − e − β − (cid:0) e − β − e − β (cid:1) cos θ (cid:9) . (B.1)Hereafter, we assume S ( S − / ≈ S for large enough particle number N (= 2 S >
B ≈ S τ sin ( θ ) e − β , and C ≈ S sin θ (cid:8) − e − β + (3 + e − β − e − β ) cos θ (cid:9) + S. (B.2)Following Kitagawa and Ueda [4], we focus on a time regime τ < Sτ ≪ γτ ≪ A , B , C in terms of β because of β ≪
1. Firstly,we calculate the product of the variances V + V − , which is given by Eq. (6), V + V − = 14 (cid:2) ( C + A )( C − A ) − B (cid:3) . (B.3)To calculate it, we expand C ± A and B up to the third-order of β (also γτ ) and obtain B ≈ S sin ( θ ) ( β − γτ ) (cid:0) − β + 2 β (cid:1) , C + A ≈ S sin ( θ ) β (cid:18) − β + 83 β (cid:19) + S, (B.4) C − A ≈ S sin ( θ ) cos ( θ ) β (cid:18) − β (cid:19) + S. Keeping the terms up to O [( Sβ ) ], we obtain V + V − ≈ S (cid:26) S sin ( θ ) (cid:20) β + 6 S sin ( θ ) cos ( θ ) β + γτ + 32 cos ( θ ) β (cid:18) − β (cid:19)(cid:21)(cid:27) . (B.5)For brevity, we will omit the last term 3 cos ( θ ) β ( · · · ) /
2, though its contribution may belarger than that of the term 2 β / V + . From Eq. (B.1)-Eq. (B.2), we note that theleading terms of the coefficients A ∝ S β ( ∝ S τ ) and B ∝ S τ . In the time scale with S τ >
1, it is easy to find that A > B , so the increased variance can be simplified as V + ≈
12 ( C + A ) ≈ S sin ( θ ) β. (B.6)where we only keep the lowest-order of β in the last step [see also Eq. (B.4)]. Finally, using V − = ( V + V − ) /V + , we obtain analytical result of the reduced variance V − ≈ S (cid:26) S sin ( θ ) β + γτβ + 23 β (cid:20) S (2 θ ) (cid:21)(cid:27) , (B.7) i CG, . . . which gives the analytical result of the squeezing parameter, i.e. , Eq. (18). References
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