一维接触排斥相互作用单自旋翻转费米气体的基态和淬火动力学性质

尹相国; 于海如; 郝亚江; 张云波

doi:10.7498/aps.73.20231425

摘要

基于严格解方法, 研究了一维排斥相互作用单自旋翻转费米气体的基态和淬火动力学性质. 借助Bethe波函数, 基态和不同本征态之间的单体关联函数和两体关联函数可以表示为简单函数之和, 这一简洁形式可以极大地降低计算难度. 系统初始状态为无相互作用的基态, 当迅速把相互作用强度调节为有限大时, 动量分布和关联函数出现周期性的振荡行为; 当相互作用调节为比较弱时, 振荡周期性好且振荡幅度小, 系统可以用二能级模型近似; 当相互作用调为非常强时, 振荡周期性变差且振荡幅度大, 但是依然存在主周期. 此时整体偏离初态较远, 但是在时间为$ mL^2/(2\pi\hbar)$时系统非常接近初态.

关键词:

Abstract

Based on the exact solution method, the ground state and quench dynamics properties of one-dimensional single-spin flipped Fermi gas with repulsion interaction are studied. With the Bethe wave function, the single-body correlation function and two-body correlation function of the ground state and those between different eigen-states can be reduced into a summation of simple functions, thereby greatly reducing the computational difficulty. For the system in the ground state, the single-body correlation functions and two-body correlation functions as well as momentum distributions for spin-up particles are investigated in real space with different interaction strengths. As the interaction strength increases, the number of nodes in the single-body correlation function remains unchanged, while the amplitude of oscillation decreases. Meanwhile, the number of peaks in the two-body correlation function increases by one due to interaction, indicating that the spin-down particle behaves as a spin-up particle. The momentum distribution becomes more smooth around Fermi surface with the interaction strength increasing. The interaction quench dynamics is investigated. The system is prepared in the ground state of ideal Fermi gas, and then the interaction strength is quenched to a finite positive value. The system evolves under time-dependent Schrödinger equation. The overlap between the initial state and eigen-state of post-quench interaction strength is expressed in the form of continued multiplication. The square of the modulus of this overlap, which represents the occupation probability, is calculated. We find that the occupation probabilities of the ground state and doubly degenerated excited state always have the first and the second largest value for an arbitrary interaction strength, respectively, which means that the difference in eigenenergy between these two states gives the primary period of oscillation. For relatively large particle number ($ N\geqslant10$), the primary period always does not change under different interaction strengths.It is found that in the case of interaction quenching, the momentum distribution and the correlation function show periodic oscillations. When the interaction strength is adjusted to a relatively small value, the oscillation periodicity is well-defined and the oscillation amplitude is small. The system can be approximated by a two-level model. When the interaction strength increases to a very large value, the oscillation periodicity worsens and the amplitude increases, but a primary period remains unchanged. Although the overall deviation is far from the initial state, it is very close to the initial state at time $ t=mL^2/(2\pi\hbar)$. This is because the difference between most energy eigenvalues is almost an integral multiple of energy unit $ 2\times\left(2\pi/L\right)^{2}$.

Keywords:

作者及机构信息

1.
山西大学理论物理研究所, 量子光学与光量子器件国家重点实验室, 极端光学协同创新中心, 太原　030006

2.
北京科技大学物理系, 理论物理研究所, 北京　100083

3.
浙江理工大学物理系, 浙江省光场调控重点实验室, 杭州　310018

通信作者: 尹相国, yinxiangguo@sxu.edu.cn

基金项目: 山西省基础研究计划(批准号: 202103021224033)、国家自然科学基金(批准号: 11704233, 12074340)和浙江理工大学科研启动基金(批准号: 20062098-Y)资助的课题.

Authors and contacts

1.
Collaborative Innovation Center of Extreme Optics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China

2.
Institute of Theoretical Physics, Department of Physics, University of Science and Technology Beijing, Beijing 100083, China

3.
Key Laboratory of Optical Field Manipulation of Zhejiang Province, Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Corresponding author: Yin Xiang-Guo, yinxiangguo@sxu.edu.cn

Funds: Project supported by the Research Foundation for Basic Research of Shanxi Province, China (Grant No. 202103021224033), the National Natural Science Foundation of China (Grant Nos. 11704233, 12074340), and the Science Foundation of Zhejiang Sci-Tech University, China (Grant No. 20062098-Y).

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参考文献

[1]	Gring M, Kuhnert M, Langen T, Kitagawa T, Rauer B, Schreitl M, Mazets I, Smith D A, Demler E, Schmiedmayer G 2012 Science 337 1318 Google Scholar
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[38]	Chang M, Yin X, Chen L, Zhang Y 2023 Phys. Rev. A 107 053312 Google Scholar
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[44]	Gamayun O, Pronko A G, Zvonarev M B 2016 New J. Phys. 18 045005 Google Scholar

施引文献

图 1 系统基态的自旋向上粒子的(a)单体关联函数、(b)两体关联函数和(c)动量分布. 粒子数和相互作用强度分别取值为$N=6$, $c=0, 10, 100, 1000$. 图(c)动量取离散的值, 短虚线只是为了视觉效果

Fig. 1. (a) Single-body correlation function, (b) two-body correlation function, and (c) momentum distribution of spin-up particle for the ground state with interaction strength $c=0, 10, 100, 1000$ and particle number $N=6$. The momenta in panel (c) are discrete and the dashed-line is for visual effect

下载: 全尺寸图片幻灯片

图 2 (a)系统基态和最主要的两重简并激发态在初态上的各自占据概率和它们的和. GS (实线)代表基态, ES (点虚线)代表激发态中占据概率最大的两重简并态, GS+ES (短虚线)代表基态和上述两重简并态的占据概率之和. (b)物理量振荡的主周期随无量纲相互作用强度的变化. $T_{0}$是无相互作用时的主周期. (c)不同相互作用强度下随时间演化的保真度. 粒子数为$N=6$

Fig. 2. (a) The respective occupation probabilities of the ground state and the most dominant degenerate excited state of the sys tem on the initial state and their sum. GS (solid lines) represents the ground state, ES (dotted lines) represents the double degenerate state that occupies the highest probability of the excited state, GS+ES (short dashed lines) represents the sum of the occupation probabilities of the ground state and the above double degenerate states. (b) The primary period of the oscillation of the physical quantity v.s. the strength of the dimensionless interaction. $T_{0}$ is the primary period in the absence of interaction. (c) Fidelity v.s. time for different interaction strength. The particle number is taken as $N=6$.

下载: 全尺寸图片幻灯片

图 3 相互作用强度$c=100$的动量分布含时演化　(a)横轴和纵轴取对数坐标; (b)取正常坐标. 虚线是动量分布对时间的平均值. 图(b)纵轴上的圆圈是$c=100$时基态的动量分布值

Fig. 3. The evolution of momentum distribution with interaction strength $c=100$. Both axis of (a) is taken logarithm. The axis of (b) is normal. The dotted lines is the average of momentum distribution with respective to time. The circles on Y-axis of panel (b) is momentum distribution of the ground state with $c=100$.

下载: 全尺寸图片幻灯片

图 4 (a)$c=10$和(b)$c=1000$ 动量分布的含时演化. 图(a)中的虚线是二能级模型近似的结果

Fig. 4. Evolution of momentum distribution with (a) $c=10$ and (b) $c=1000$. The dotted lines in panel (a) is the result from two-level model.

下载: 全尺寸图片幻灯片

图 5 自旋向上粒子的(a)单体关联函数和(b)两体关联函数分别与初态对应关联函数的差随时空的变化. (a)中黑色点线表示粒子以一个比较明显的固定速度的运动轨迹, 红色虚线表示时间主周期3倍的位置. 相互作用强度为$c=1000$

Fig. 5. Temporal and spatial evolution of (a) single-body correlation function and (b) two-body correlation function minus the corresponding function of initial state between spin-up particles. The black dotted-line in panel (a) is trajectory of the particle moving at relative obvious velocity and red dashed-line is triple position of primary temporal period. The interaction strength is taken as $c=1000$.

下载: 全尺寸图片幻灯片

图 6 三个主周期内不同自旋粒子间的两体关联函数. 相互作用强度为c = 1000

Fig. 6. Two-body correlation function between spin-up and spin-down particles in three times primary periods. The interaction strength is taken as c = 1000.

下载: 全尺寸图片幻灯片

[1]	Gring M, Kuhnert M, Langen T, Kitagawa T, Rauer B, Schreitl M, Mazets I, Smith D A, Demler E, Schmiedmayer G 2012 Science 337 1318 Google Scholar
[2]	Kaufman A M, Tai M E, Lukin A, Rispoli M, Schittko R, Preiss P M, Greiner M 2016 Science 353 794 Google Scholar
[3]	Mitra A 2018 Annu. Rev. Condens. Matter Phys. 9 245 Google Scholar
[4]	Gouraud G, Doussal P L, Schehr G 2022 J. Phys. A: Math. Theor. 55 395001 Google Scholar
[5]	王欢, 贺夏瑶, 李帅, 刘博 2023 物理学报 72 100309 Google Scholar Wang H, He X Y, Li S, Liu B 2023 Acta Phys. Sin. 72 100309 Google Scholar
[6]	Haller E, Gustavsson M, Mark M J, Danzl J G, Hart R, Pupillo G, Nägerl H-C 2009 Science 325 1224 Google Scholar
[7]	Du R, Xing J C, Xiong B, Zheng J H, Yang Tao 2022 Chin. Phys. Lett. 39 070304 Google Scholar
[8]	Erne S, Bücker R, Gasenzer T, Berges J, Schmiedmayer J 2018 Nature 563 225 Google Scholar
[9]	Chen J, Mistakidis S I, Schmelcher P 2023 J. Phys. B: At. Mol. Opt. Phys. 56 075003 Google Scholar
[10]	Zhang Z, Jiang Y, Lin H, Guan X 2023 arXiv: 2307.05955 v1 [cond-mat.quant-gas
[11]	Le Y, Zhang Y, Gopalakrishnan S, Rigol M, Weiss D S 2023 Nature 618 494 Google Scholar
[12]	Amico L, Boshier M, Birkl G, et al. 2021 AVS Quantum Sci. 3 039201 Google Scholar
[13]	Pereira D, Mueller E J 2022 Phys. Rev. A 106 043306 Google Scholar
[14]	Tylutki M, Astrakharchik G E, Recati A 2017 Phys. Rev. A 96 063603 Google Scholar
[15]	Dolgirev P E, Qu Y F, Zvonarev M B, Shi T, Demler E 2021 Phys. Rev. X 11 041015 Google Scholar
[16]	Carleo G, Cevolani L, Sanchez-Palencia L, Holzmann M 2017 Phys. Rev. X 7 031026 Google Scholar
[17]	Cao L, Krönke S, Vendrell O, Schmelcher 2013 J. Chem. Phys. 139 134103 Google Scholar
[18]	Nardis J D, Wouters B, Brockmann M, Caux J S 2014 Phys. Rev. A 89 033601 Google Scholar
[19]	Zill J C, Wright T M, Kheruntsyan K V, Gasenzer T, Davis M J 2015 Phys. Rev. A 91 023611 Google Scholar
[20]	Liu W, Andrei N 2014 Phys. Rev. Lett. 112 257204 Google Scholar
[21]	Rylands C, Bertini B, Calabrese P 2022 J. Stat. Mech. 2022 103103 Google Scholar
[22]	Lieb E H, Liniger W 1963 Phys. Rev. 130 1605 Google Scholar
[23]	Chen S, Guan L, Yin X, Hao Y, Guan X W 2010 Phys. Rev. A 81 031609(R Google Scholar
[24]	Chen H H 2020 Phys. Lett. B 808 135631 Google Scholar
[25]	Zill J C, Wright T M, Kheruntsyan K V, Gasenzer T, Davis M J 2016 New J. Phys. 18 045010 Google Scholar
[26]	Piroli1 L, Calabrese P, Essler F H L 2016 SciPost Phys. 1 001 Google Scholar
[27]	Zill J C, Wright T M, Kheruntsyan K V, Gasenzer T, Davis M J 2018 SciPost Phys. 4 011 Google Scholar
[28]	Piroli L, Calabrese P 2017 Phys. Rev. A 96 023611 Google Scholar
[29]	Collura M, Kormos M, Calabrese P, 2018 Phys. Rev. A 97 033609 Google Scholar
[30]	Alba V, Calabrese P 2018 SciPost Phys. 4 017 Google Scholar
[31]	Robinson N J, Klerk A J J M, Caux J S 2021 SciPost Phys. 11 104 Google Scholar
[32]	Guan X W, Batchelor M T, Lee C 2013 Rev. Mod. Phys. 85 1633 Google Scholar
[33]	McGuire J B 1965 J. Math. Phys. 6 432 Google Scholar
[34]	McGuire J B 1965 J. Math. Phys. 7 123 Google Scholar
[35]	Guan X W 2012 Front. Phys. 7 8 Google Scholar
[36]	Mao R, Guan X, Wu B 2016 Phys. Rev. A 94 043645 Google Scholar
[37]	张瑞江, 尹相国, 陈立, 张云波 2019 山西大学学报(自然科学版) 42 2347 Google Scholar Zhang R, Yin X, Chen L, Zhang Y 2019 J. Shanxi Univ. (Nat. Sci. Ed.) 42 2347 Google Scholar
[38]	Chang M, Yin X, Chen L, Zhang Y 2023 Phys. Rev. A 107 053312 Google Scholar
[39]	Song Y, Cai X 2018 Chin. Phys. Lett. 35 110301 Google Scholar
[40]	Song Y, Zhang H 2019 Eur. Phys. J. D 73 106 Google Scholar
[41]	Song Y, Barthwal S 2019 Commun. Theor. Phys. 71 617 Google Scholar
[42]	Li W, Cui X 2017 Phys. Rev. A 96 053609 Google Scholar
[43]	Gamayun O, Lychkovskiy O, Zvonarev M B 2020 SciPost Phys. 8 053 Google Scholar
[44]	Gamayun O, Pronko A G, Zvonarev M B 2016 New J. Phys. 18 045005 Google Scholar

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一维接触排斥相互作用单自旋翻转费米气体的基态和淬火动力学性质