Preserving coherent spin and squeezed spin states of a spin-1 Bose-Einstein condensate with rotary echoes

Jun Zhang, Yingying Han, Peng Xu, and Wenxian Zhang

Phys. Rev. A 94, 053608 – Published 9 November 2016

Abstract

A challenge in precision measurement with squeezed spin state arises from the spin dephasing due to stray magnetic fields. To suppress such environmental noises, we employ a continuous driving protocol, rotary echo, to enhance the spin coherence of a spin-1 Bose-Einstein condensate in stray magnetic fields. Our analytical and numerical results show that the coherent and the squeezed spin states are preserved for a significantly long time, compared to the free induction decay time, if the condition $h τ = m π$ is met with $h$ the pulse amplitude and $τ$ pulse width. In particular, both the spin average and the spin squeezing, including the direction and the amplitude, are simultaneously fixed for a squeezed spin state. Our results point out a practical way to implement quantum measurements based on a spin-1 condensate beyond the standard quantum limit.

Received 13 May 2016

DOI:https://doi.org/10.1103/PhysRevA.94.053608

Physics Subject Headings (PhySH)

Squeezing of quantum noise

Bose-Einstein condensates

Atomic, Molecular & Optical

Authors & Affiliations

Jun Zhang^1,2, Yingying Han¹, Peng Xu¹, and Wenxian Zhang^1,*

¹School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China
²Beijing Computational Science Research Center, Beijing 100084, China

^*Corresponding author: wxzhang@whu.edu.cn

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Issue

Vol. 94, Iss. 5 — November 2016

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Images

Figure 1
Typical spin trajectories in (a) rotary echo and (b) Hahn echo. The initial spin state is marked by a yellow cross. (a) In a rotary echo, the spin rotates about the total field $h \hat{x} + ω \hat{z}$ during the first and the fourth quarter-cycle $τ$ , in purple and blue solid lines, respectively; the spin rotates about the total field $- h \hat{x} + ω \hat{z}$ during the second and the third quarter-cycle, in red dashed lines. (b) In a Hahn echo, the spin evolves in a noise magnetic field $ω \hat{z}$ for a cycle $4 τ$ . At the moments $τ$ and $3 τ$ , the spin is rotated for a $π$ angle about $+ x$ axis, denoted by red solid lines.
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Figure 2
Pulse sequence of a RE. The RE is a symmetric sequence consisting of $N$ cycles. Within each cycle, the driving field direction is reversed between $+ x$ and $- x$ , denoted by $+ h$ and $- h$ , respectively. The two outer segments are along $+ x$ for a duration time $τ$ and the inner segment along $- x$ for a duration time $2 τ$ . The period of a single cycle is $T = 4 τ$ . After $N$ cycles, the total time is $t = 4 N τ$ .
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Figure 3
Spin average evolution under REs for $τ = 0.05$ (red diamonds), 0.1 (blue crosses), 0.12 (black squares), 0.14 (cyan hexagons), and 0.17 (green asterisks) with $h = 20$ . The blue circles are for $τ = π / h$ with $h = 20$ . The FID is denoted with purple triangles. The solid lines, coinciding with the marks, are the corresponding analytical predictions by Eq. (11). The black dots (coinciding with the blue circles) are spin average under Hahn echoes as shown in Fig. 1, where the cycle period is $4 τ = 0.2 π$ , the pulse amplitude is $10 h = 200$ , and the pulse width is $0.005 π$ . Other parameters are $A = 0.02$ and $J = 100$ . The average is performed over $N_{r} = 10^{3}$ random $w$ 's distributed uniformly between $[0, 1]$ . When $τ = π / h$ , the spin average barely decay under REs and Hahn echoes, indicating strong suppression of stray magnetic fields.
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Figure 4
Dependence of the spin average on $t = 4 N τ$ and $h τ$ . The parameters are $J = 1, h = 20$ , and $A = 0.02$ . The spin average is conserved if $h τ = m π (m = 1, 2, 3, ...)$ , as shown by horizontal dashed lines.
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Figure 5
Time evolution of (a) the spin average and (b) the $z$ -axis spin squeezing parameter for $J = 10^{3}$ . The initial direction of the spin is along $y$ axis and the initial optimal squeezing direction is along $z$ axis. The parameters are $h = 2, τ = 0.1$ (blue crosses), $h = 2, τ = 0.5$ (black asterisks), $h = 2, τ = π / h$ (cyan diamonds), $h = 20, τ = 0.1$ (red squares), and $h = 20, τ = π / h$ (blue circles). The FID is denoted with purple triangles. The black dots in (a) and (b) are the evolutions under the Hahn echoes with the cycle period of $4 τ = 0.2 π$ , the pulse amplitude of $10 h = 200$ , and the pulse width of $0.005 π$ . Other parameter is $A = 0.02$ . In (b), the black solid line (coinciding with the purple triangles) denotes the optimal squeezing parameter $ξ_{s}^{2}$ sought with numerical method. The spin average and the squeezing parameter are simultaneously fixed by the RE with $h ≫ 1$ and $τ = m π / h$ . However, the Hahn echoes fail to fix the squeezing parameter.
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Figure 6
Time evolution of (a) the spin average and (b) the $z$ -axis spin squeezing parameter in the presence of $1 / f$ noise (purple crosses), Gaussian noise (black diamonds), and white noise (blue circles). Other parameters are the same as in Fig. 5. The suppression of the $1 / f$ and the Gaussian noise is more efficient than that of the white noise.
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