Dynamical Spin Squeezing via Higher Order Trotter-Suzuki ...

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Jul 16, 2014 - twisting Hamiltonian based on high-order Trotter-Suzuki approximation. Compared with the paper by Liu et al. [Phys. Rev. Lett. 107, 013601 ...
Dynamical Spin Squeezing via Higher Order Trotter-Suzuki Approximation Ji-Ying Zhang, Xiang-Fa Zhou,∗ Guang-Can Guo, and Zheng-Wei Zhou† Key Laboratory of Quantum Information, University of Science and Technology of China, Chinese Academy of Sciences, Hefei, Anhui 230026, China and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Here we provide a scheme of transforming the one-axis twisting Hamiltonian into a two-axis twisting Hamiltonian based on high-order Trotter-Suzuki approximation. Compared with the paper by Liu et al. [Phys. Rev. Lett. 107, 013601 (2011)], our method can reduce the number of controlling cycles from 1000 to 50. Moreover, it is also spin number independent and takes a shorter optimal evolution time as compared with the method of Shen et al.[Phys. Rev. A 87, 051801 (2013)]. The corresponding error analysis is also provided.

arXiv:1406.3887v2 [quant-ph] 16 Jul 2014

PACS numbers: 03.75.Gg, 42.50.Dv

I.

INTRODUCTION

Squeezed spin states [1–4] are entangled quantum states of an ensemble of two-level (or spin-half) systems, and they play significant roles in quantum information science [5–13] and quantum metrology [2, 3, 14–20]. People have made much progress in both theory and experiment over the past decades [4, 19, 20, 23–28]. Specifically, the recent experimental success of achieving the one-axis twisting (OAT) scheme in spinor Bose-Einstein condensates (BEC) using two chosen hyperfine states provides an ideal platform to implement such novel states in a highly controllable manner [19, 20]. As is well known, two-axis twisting (TAT) is capable of causing Heisenberg limited noise reduction to scale as 1/N , better than the OAT, whose noise reduction limit scales as 1/N 2/3 [1]. To realize better spin squeezing, several theoretical proposals have been presented to enhance the OAT spin squeezed states [24, 25, 29]. In one scheme [24], one applies a series of subtle Rabi pulses to the system with the purpose of transforming OAT into TAT. Due to a large number of pulses acting on the atoms, it’s unavoidable to bring in accumulated noise and imperfection in control pulses. In another approach [25], only several pulses are needed to obtain much better squeezed spin states. However, to achieve the optimal squeezing it takes a long evolution time, which would be an obstacle in systems with short coherence time. Additionally, this scheme is also spin number dependent, so it naturally brings in certain difficulties when applied to some systems, such as ultracold atomic gases, in which we do not know the spin number N exactly. Here we propose a scheme following the idea of transforming OAT into TAT to enhance the performance of OAT. Compared with the method discussed in the paper of Ref. [24], pulse sequences based on Trotter-Suzuki (TS) expansion [30] are proposed. To achieve this, we

∗ Electronic † Electronic

address: [email protected] address: [email protected]

also introduce another kind of radio frequency (rf) pulses to realize the rotation around the x axis apart from that around the y axis [24]. We note that the scheme can be generalized to implement pulse sequences based on any high order TS expansion within these experimentally available conditions. A numerical investigation of the scheme based on the 2nd-order expansion indicates that only 50 cycles are enough to obtain the ideal spin squeezed states, while more than 1000 cycles are needed in [24] to get the same results. So compared with the previous proposals [24, 25], our idea can overcome their disadvantage to some extent. Moreover, we also provide the corresponding error analysis for a scheme using higherorder TS expansions.

II.

THE SCHEMES AND PULSE SEQUENCES

To clarify our key point in this paper, we first briefly review the TS expansion theory [30]. The standard 1stand 2nd-order TS real decomposition of eα(P +Q) (with the commutation relation [P, Q] 6= 0 in terms of operators P and Q) for small |α|(|α| ≪ 1) are eα(P +Q) = eαP eαQ + O(α2 ) eα(P +Q) = S(α) + O(α3 ) =e

(α/2)P αQ (α/2)P

e

e

(1) 3

+ O(α ).

For the 3rd-order expansion, we begin with eα(P +Q) = esα(P +Q) e(1−2s)α(P +Q) esα(P +Q) .

(2)

The 3rd-order symmetric approximation S3 (α) is given by S3 (α) = S(sα)S((1 − 2s)α)S(sα)

(3)

with the parameter s = 1/(2 − 2(1/3) ) ≃ 1.3512. The 4-th order expansion is the same as the 3rd-order one, S4 (α) = S3 (α) [30]. In general, the (2m − 1)th and 2mth

2 approximants, S2m−1 (α) and S2m (α), are determined recursively as S2m−1 (α) = S2m (α) = S2m−3 (km α)S2m−3 ((1 − 2km )α)S2m−3 (km α) (4) with the parameter km = (2 − 21/(2m−1) )−1 . According to Refs [19, 20], the OAT Hamiltonian existing in two-component BEC controlled by coupling pulses can be written as H=

χJz2

+ G(t)Jx + Ω(t)Jy .

(5)

PN Here Jµ = k=1 σµk /2 is the collective angular momentum operator for the system with N spins, µ = x, y, z. χ indicates the nonlinear interaction strength between the atoms. Ω(t) and G(t) are defined as the coupling pulse amplitudes. The model Hamiltonian in Eq.(5) is the socalled Lipkin-Meshkov-Glick model [21]. Some aspects of this model have been discussed in Ref. [22]. Here and in the following, we assume Ω(t) = Ω0 and G(t) = G0 when the coupling pulses are switched on and Ω(t) = G(t) = 0 when they are turned off. Note that we will ignore the nonlinear interaction χJz2 during the time applying the coupling pulses, because the conditions χN ≪| Ω0 | and χN ≪| G0 | are satisfied when the strong coupling Rabi pulses are switched on. The terms G(t)Jx and Ω(t)Jy can realize the rotation 2 of e−iχJz t around the x and y axis by angle θ ∈ (0, 2π) R∞ x(y) with θ = −∞ dtΩ(t)(G(t)). The rotation operator Rθ is defined as x(y)



= e−iθJx (Jy )

2

x(y)

2

= e−iχ(Jz cos(θ)+Jy(x) sin(θ)) t .

(7)

Using this definition, the combination of θ = π/2 and θ = −π/2 is able to accomplish the following operations 2

2

2

2

x x = e−iχJy t , e−iχJz t Rπ/2 R−π/2

(8)

y y R−π/2 e−iχJz t Rπ/2 = e−iχJx t . 2

2

From Eq. (7) we find out that the terms eiχJx t and eiχJy t can-not be realized directly with the Hamiltonian shown by Eq. (5). 2 2 To generate the TAT evolution e−iχ(Jx −Jy )t , we notice that J 2 is conserved during the dynamics. So up 2 2 to a constant phase factor, we can write e−iτ (Jx −Jy ) as 2 2 2 2 2 e−iτ (J +Jx −Jy ) = e−iτ (2Jx +Jz ) with τ = χδt and δt is a small time interval. Therefore the 1st- and 2nd-order expansion can be obtained as 2

2

2

e

≃e

−i τ2

Jz2

e

−i2τ Jx2

e

−i τ2

τ

2

2

τ

2

y y e−i 2 Jz e−i2τ Jz Rπ/2 U1 = e−i 2 Jz R−π/2 2

(10)

2

= e−iτ (2Jx +Jz ) + O((−iτ )3 ).

If we bring in Nc periods the same as the one described above during a fixed interested time, the complete time (A) evolution operator at time instant t = Nc tc is written as 2

2

U1Nc ≃ e−iNc τ (2Jx +Jz ) = e−i ≃e

−i

2 −J 2 Jx y 3

χt

2 +J 2 2Jx z 3

χt

(11)

.

Jz2

+ O((−iτ )3 ).

2

(9)

2

the TAT evolution UT AT = e−iχ(Jx −Jy )topt , with topt the time when the optimal squeezing state is achieved, our scheme A takes the total time 3topt . To proceed with our scheme B, let us refer to the TS 3rd-order expansion formula Eq. (3). Unfortunately, there exists a term S((1 − 2s)α) on its right side, which can not be realized directly since 2s − 1 > 0. To solve this, we go back to Eq. (2) and transform e(1−2s)α(P +Q) to e(2s−1)α(−P −Q) . Taking into account the property of J 2 , we obtain 2

2

e−iτ (Jx −Jy ) 2

2

2

2

2

2

≃e−isτ (2Jx +Jz ) e−i(2s−1)τ (2Jy +Jz ) e−isτ (2Jx +Jz ) .

(12)

Therefore, following the same routine, we have the final result as 2

2

e−iτ (Jx −Jy ) s

2

e−iτ (Jx −Jy ) ≃ e−iτ Jz e−i2τ Jx + O((−iτ )2 ), −iτ (Jx2 −Jy2 )

(A)

(A)

are employed at time δt/2 + M tc and 5δt/2 + M tc respectively to realize the rotation shown in Eq. (8). (A) Here tc = 3δt is the time interval of a single period and M = 0, 1, · · · , Nc − 1 with Nc the number of total (A) (A) period. In this case we have Np = 2, where Np is defined as the pulse number added in each period. Without the controlling pulses, the dynamics of the system is determined by the Hamiltonian H = χJz2 , so the evolution operator for one single period U1 is

From Eq. (11) we find that the effective Hamiltonian (A) of the system is Hef f = χ3 (Jx2 − Jy2 ). Hence to realize

2

x(y)

2

realize the evolution e−i2τ Jx , namely, after introducing the Ω(t)Jy pulse we can simulate the TAT based on Eq. (5). The work based on TS 1st-order expansion has been finished by Liu et al. [24], and the pulse sequence is shown in Fig. 1(b). Next we will provide the expansion scheme according to the TS 2nd-order expansion theory shown in Eq. (9). Figure 1(a) shows the pulse sequences of our scheme A. y y Within each period, two strong pulses Rπ/2 and R−π/2

(6)

and it rotates e−iχJz t as follows R−θ e−iχJz t Rθ

y Equation (9) tells us the rotation R±π/2 is required to

2

2

≃e−i 2 τ Jz e−i2sτ Jx e−i ×e

−i (3s−1) τ Jz2 2

e

(3s−1) τ Jz2 2

−i2sτ Jx2

e

−i 2s τ Jz2

2

e−i2(2s−1)τ Jy

+ O((−iτ )4 ).

(13)

3

a

a %$ 2 0

%$ 2 •• (A)

2δ$

••

••

… •!"•

•• (A)



b

••

••

••

••

(B)

n•

•• (n+•)•



(B)

b

•• (A)

•!"•

•• (A)



FIG. 1: (color online). Schematic plot of the repeated pulses Ω0 vs time t in arbitrary unit. (a) The whole pulse sequence (A) of scheme A. One period from 0 to tc (shaded) consists of y y rotations Rπ/2 (red pulse) and R−π/2 (pink pulse). (b) The proposal in paper [24]. Apart from the time at which applying the laser pulses, others are all the same with (a).

From Eq. (13), we find that both the coupling pulses G(t)Jx and Ω(t)Jy are needed to implement the evolution 2 2 e−i2sτ Jx and e−i2(2s−1)τ Jy . Figure 2(a) shows the corresponding pulse sequences within one single period: two (B) y y pulses Rπ/2 and R−π/2 are employed at time T1 + M tc



0

(B)

••

(B)

FIG. 2: (color online) An illustration of the pulse sequences for our scheme B. The red, pink, black and blue rectangles correspond to the π/2, −π/2 pulse around the y axis, and π/2, −π/2 pulse around the x axis respectively. (a) The pulse sequence for one single period, including a total of six pulses. (B) (b) A series of pulse sequence periods. One period, from tc (B) to 2tc , is shaded. 1

10

10

10 −1

10

(B)

U2Nc ≃ e

2 −J 2 Jx y −i 12s−3

χt

(B)

χ with an effective Hamiltonian Hef f = 12s−3 (Jx2 − Jy2 ). Furthermore, in this case the evolution time arriving at the optimal squeezing is (12s − 3)topt . We note that the above method can be generalized to implement the TAT Hamiltonian based on any higher-order TS expansion.

III.

10

−2

−2

Nc = 10

(B)

(14)

−1

10

Nc = 50

−3

−3

10 0 10

100 10

HM t(A) ∼(M +1)t(A) c

c

(A)

−1

10

−1

Heff HM t(A) c

ξ2

10

ξ2

(B)

0

0

10

and T2 + M tc , respectively, to implement a rotation x around the y axis; then another two pulses Rπ/2 and x R−π/2 are added at time T3 + M tc and T4 + M tc ; and finally, a y rotation is applied again with the pulses (B) (B) added at time T5 P + M tc and T6 + M tc , respecν tively. Here Tν = i=1 ti , ti = t8−i , with t1 = sδt/2, t2 = 2sδt, t3 = (3s − 1)δt/2, and t4 = 2(2s − 1)δt. We note that the duration time of one single period is (B) tc = (12s − 3)δt ≃ 13.2δt. In scheme B, the pulse (B) number needed in one single period is Np = 6. Following the similar way of getting the result of our scheme A, we conclude that the effective evolution of our scheme B is



•!"•

ξ2

0





δ$

ξ2

2δ$

−2

10

10

−2

Nc = 1000

Nc = 100 −3

10

0

0.002 0.004 0.006 0.008

χt

10

−3

0

0.002 0.004 0.006 0.008

χt

FIG. 3: (Color online). The resulting spin squeezing vs the evolution time based on the scheme in Ref. [24] for the system with 1250 atoms. The magenta diamonds, the dashed light blue lines, and the solid black lines display the result for (A) the case where M tc is considered, every time instant from (A) (A) M tc to (M + 1)tc (M = 0, 1, 2, · · · , Nc − 1) is taken into (A) account, and where Hef f is concerned. Nc is the number of total periods.

THE RESULT AND ANALYSIS

To get the numerical result, we follow Kitagawa and Ueda’s criteria that choose the squeezing parameter ξ 2 = 2(△J⊥ )2min as the measurement of the squeezing, where J (△J⊥ )2min is the smallest variance normal to the mean spin vector and J = N/2 is the total spin of the system. The initial state we choose is |J, Ji, where all the spins

are polarized along the z axis. This state is favorable when the twisting Hamiltonian H ∝ Jx2 − Jy2 . It is shown in [24] that for the scheme based on the 1st-order expansion, 1000 pulse pairs are enough to get the optimal spin squeezing with required precision. This result is obtained by taking into account every time in(A) (A) stant in the time period from M tc to (M + 1)tc

4 1

10

10

0

0

10

a

0

10 10

10

−2

10

10 0 10

−3

b

0

10

HM t(A) ∼(M +1)t(A) c

−1

c

10

Heff HM t(A)

−1

−1

−2

10

ξ2

ξ2

(A)

c

10

10

ξ2

(A)

0

0.002

0.004

χt

Nc = 570

Nc = 100 0

Nc = 1000

−3

10

10

−3

Hef f HM t(A) c

−2

−2

10

10

Nc = 50

−3

Nc = 50

−3

Hef f HM t(A) c

Nc = 10

10 0 10

(A)

−2

10

−2

10

ξ2

ξ2

ξ2

10 −1

−1

10

−1

0.006

0.008

−3

0.002 0.004 0.006 0.008

χt

10

0

0.002 0.004 0.006 0.008

χt

FIG. 4: (Color online). Same as in Fig. 3, except this time scheme A is considered. The red squares are used for the (A) result at time M tc . The dashed dark green lines show the (A) (A) results when every time instant from M tc to (M + 1)tc (M = 0, 1, 2, · · · , Nc − 1) is considered.

FIG. 5: (Color online) The numerical result of spin-squeezing parameter of (a) our scheme A and (b) the proposal in [24]. The red squares and the magenta diamonds represent the re(A) sults at time M tc using the dynamics controlling pulses in the schemes. The black lines denote the results derived (A) from the ideal effective TAT Hamiltonian Hef f directly. Here N = 1250 is used.

0

10

−1

10

(B)

Hef f HM t(B)

ξ2

(M = 0, 1, 2, · · · , Nc − 1). That is, at every time in(A) (A) stant t satisfying M tc ≤ t ≤ (M + 1)tc , the approximated time evolution almost overlaps with the dynamics driven by the ideal TAT Hamiltonian. However, according to the theoretical analysis presented in Eqs. (10) and (11), we notice that only the result at time instant (A) M tc is necessary to be calculated. This reminds us of searching for a more efficient way to obtain the optimal spin squeezing states based on these dynamics controlling procedures. This is also the way discussed in paper [25]. Figures 3 and 4 show the numerical time evolution of the corresponding two schemes depicted in Fig. 1(a) and Fig. 1(b). One can see that in both cases, the numerical results exhibit oscillation behaviors away from the ideal dynamics when χt is large. For the scheme based on the 1st-order TS expansion, the spin squeezing pa(A) rameter ξ 2 at time instant M tc is always on the top of the evolution curve, even for large Nc . Therefore, to achieve the ideal spin squeezing at time topt , δt should be sufficiently small, which indicates a relatively large Nc . However, for scheme A based on the 2nd-order TS expansion, the corresponding values ξ 2 at time instant (A) M tc moves to the bottom of the evolution curve as Nc increases, as shown in Fig. 4. So with much smaller Nc , we can obtain a good approximation of the optimal spin squeezing ξ 2 by controlling the total evolution time. Fig(A) ure 5 shows the tracks of ξ 2 at time instant M tc for different schemes. As a result, we conclude while it re-

c

−2

10

Nc = 17 −3

10

0

0.01

0.02

0.03

0.04

0.05

χt

FIG. 6: (Color online) Squeezing parameter ξ 2 as a function of evolution time χt calculated with N = 1250 and Nc = 17 (Np ≈ 100). The magenta circles represent the values derived from the actual process of the pulse sequence shown in Fig. (B) 2(b) and the dashed blue line indicates the result for Hef f .

quires as many as Nc = 1000 periods to get a good result using the proposal in [24], a much smaller Nc (Nc = 50) is sufficient when scheme A is employed without introducing new controlling pulses. To investigate the efficiency of the scheme based on higher-order TS expansions, in Fig. 6 we plot the squeez-

5 0

10

6

N p = 300

N p = 300

ξ2

RE

4

−2

2

10

d

0

a

−8

0

10

10

N p = 30000

x 10

ξ2

RE

N p = 30000 5

−2

10

b

e

0

0

−9

N p = 60000

(B)

Heff

20

(A) Heff

15

ξ2

HM t(B) c

HM t(A)

−2

c

10

RE

10

x 10

N p = 60000

RE (A) RE (B)

10 5

c 0

f

0 0.01

0.02

χt

0.03

0.04

0

0.01

0.02

χt

0.03

0.04

FIG. 7: (Color online) The spin squeezing (a, b, c) and associated relative errors (d, e, f). In (a), (b), and (c) the red squares, the solid black line, the magenta circles, and the dashed blue line correspond to the evolution of (A) (B) scheme A, Hef f , and scheme B, Hef f , respectively. The solid green line and the dashed magenta line in (d), (e), and (f) are the relative errors of scheme A and B getting from 2(A) 2(A) 2(A) 2(B) 2(B) 2(B) |ξ (A) − ξef f |/ξef f and |ξ (B) − ξef f |/ξef f . M tc

than scheme A because the error in the 3-rd order expansion decreases faster than that of the 2nd-order expansion. Figure 7 shows the error analysis for both schemes with a different total number of pulses Np . One can see that scheme A always has a relatively lower error rate and shorter evolution time until Np reaches 60 000. Such a large Np requires too many resources. We conclude that scheme B may have higher precision when Np is large enough, but it requires too many controlling pulses and is experimentally impractical. A simplified scheme A based on 2nd-order TS expansion is enough for our purposes. After the paper of Ref.[24], Shen et al. also presented an idea [25] to enhance the performance of OAT to get spin squeezed states close to the Heisenberg limit. Compared with their proposal, our result takes a shorter evolution time. Taking N = 2000 as an example, the time cost of our scheme A(B) is around 0.006/χ(0.027/χ), shorter than their 0.1/χ. Besides, our scheme is also spin number independent.

M tc

ing parameter following scheme B for N = 1250 with pulse number Np = 100. One can see that a large de(B) viation from the ideal evolution derived by Hef f appears as the duration time grows. In principle, when the duration time of the single period of TS expansion is fixed, for the same evolution time, the higherorder TS expansion will lead to the higher precision compared with the lower-order one. However, in this problem our scheme B takes a longer evolution time to achieve the optimal squeezing compared with scheme A. With the effective Hamiltonian χ(Jx2 − Jy2 )/(12s − 3), scheme B needs an evolution time (12s − 3)topt to realize the TAT optimal squeezing, while scheme A only takes time 3topt . So for the fixed pulse number Np = 100 in Fig. 5 and 6, the duration times of one single period (B) for the two schemes satisfy (12s − 3)topt /(Np /Np ) ≈ (A) 0.79topt ≫ 3topt /(Np /Np ) = 0.06topt. This is the reason scheme A has a better result. With the increase of the pulse number Np , we can make the duration times (B) (A) (12s − 3)topt /(Np /Np ) and 3topt /(Np /Np ) smaller and smaller, and finally scheme B will have a better result

[1] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).

IV.

CONCLUSION

In conclusion, we have developed a scheme using a series of rf pulses to transform an OAT to a TAT Hamiltonian. In contrast to the proposal in Ref.[24], our scheme A reduces the pulse number from Nc = 1000 to 50 for N = 1250 atoms, which is very experimentally friendly. With the help of the terms Ω(t)Jy and G(t)Jx , pulse sequences designed according to higher order TrotterSuzuki expansion can be realized. We find that while scheme B can reach optimal spin squeezing with high precision during the whole evolution, it needs too many controlling pulses and is experimentally impractical. We note that our scheme is spin-number independent, and it can be generalized in other systems where only an OAT Hamiltonian [31–33] is realized. Moreover, compared to the known work [25], our schemes also have a relatively shorter evolution time. Therefore they should be realizable with current techniques, such as those reported in Refs. [19] and [20].

V.

ACKNOWLEDGEMENT

This work was funded by the National Natural Science Foundation of China (Grant No. 11174270), National Basic Research Program of China (Grants No. 2011CB921204 and No. 2011CBA00200), the Fund of CAS, and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20103402110024). Z. W. Z. gratefully acknowledges the support of the K. C. Wong Education Foundation, Hong Kong.

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