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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:
- one-dimensional Fermi gas /
- momentum distribution /
- correlation function /
- quench dynamics
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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
Figure 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
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$N=6$
Figure 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 $N=6$.
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
Figure 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 $c=100$.
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
Figure 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.
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$c=1000$
Figure 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 $c=1000$.
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
Figure 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.
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[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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