冷原子物理中的一维少体问题

刘彦霞; 张云波

doi:10.7498/aps.68.20181993

摘要

作为构成量子多体系统的基本单元, 一维少体系统的研究不仅可以在理论上为多体系统的量子关联及动力学等性质提供更为基本的理解, 也可以为实验上制备多体系统提供更加方便和功能更加全面的方法. 本文回顾了冷原子物理中一维少体系统最新的实验和理论进展. 首先介绍了少体实验中实现的谐振子势阱中确定原子数的精确制备, 亚稳态势阱和双阱系统中原子的隧穿, 以及强相互作用下等效自旋链的实验结果. 然后深度解析了理论研究方面, 特别是基于精确可解模型的一些重要结果, 包括亚稳态势阱中相互作用原子的隧穿概率, 以及相应实验上常见势阱的能谱分析、密度分布、隧穿动力学以及强相互作用极限下的有效自旋链模型等.

关键词:

Abstract

We review some recent theoretical and experimental developments of one-dimensional few-body problems in ultracold atomic system. The experiments have so far realized the deterministic loading of few atoms in the ground state of a potential well, the observation of tunneling dynamics out of the metastable trap controlled by a magnetic gradient for a repulsively or attractively interacting system, the preparation of two fermionic atoms in an isolated double-well potential with a full control over the quantum state of the system, the formation of a Fermi sea by studying quasi-one-dimensional systems of ultracold atoms consisting of a single impurity interacting with an increasing number of identical fermions, and the deterministic preparation of antiferromagnetic Heisenberg spin chains consisting of up to four fermionic atoms in a one-dimensional trap. These achievements make the ultracold atoms an ideal platform to study many-body physics in a bottom-up approach, i.e., one starts from the fundamental building block of the system and observes the emergence of many-body effects by adding atoms one by one into the system. Corresponding theoretical models have been developed to explain the experimental data, to tackle the crossover boundary between few and many particles, and even explore the solvability and integrability of the models, especially the energy spectrum of interacting few atoms such as two atoms in a harmonic trap, two heteronuclear atoms of unequal mass in a ring trap, and two atoms in a $\delta$-barrier split double well potential. After a brief review of Bethe-Ansatz method, a theory for the tunneling of one atom out of a trap containing two interacting cold atoms is developed based on the calculation of the quasiparticle wave function, and the tunneling dynamics of two atoms starting from the NOON state is explored from the exactly solved model of $\delta$-barrier split double well based on a Bethe ansatz type hypothesis of the wave functions. It was shown that the spectroscopy and spin dynamics for strongly interacting few atoms of spin-1/2 and spin-1 can be described by effective spin chain Hamiltonians, which serves as a useful and efficient tool to study the quantum magnetism with clod atoms.

Keywords:

作者及机构信息

刘彦霞^1,2,
张云波¹

1.
山西大学, 理论物理研究所, 太原　030006

2.
中国科学院物理研究所, 北京　100190

通信作者: 张云波, ybzhang@sxu.edu.cn

基金项目: 国家自然科学基金(批准号: 11674201, 11474189)资助的课题.

Authors and contacts

Liu Yan-Xia^1,2,
Zhang Yun-Bo¹

1.
Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China

2.
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

Corresponding author: Zhang Yun-Bo, ybzhang@sxu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11674201, 11474189).

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施引文献

图 1 双原子的两种隧穿过程: 单原子次序隧穿及两原子配对隧穿. 本图摘自参考文献[25]

Fig. 1. The loss processes include two tunneling processes of two atoms out of a metastable potential: subsequent single-particle tunneling and direct pair tunneling (Reproduced with permission from Ref. [25]).

下载: 全尺寸图片幻灯片

图 2 三种相互作用的两原子系统的能量E与相互作用强度g的关系: (a)谐振子势阱中的两原子, 黑实线和红虚线分别表示相对运动奇宇称和偶宇称波函数所对应的能量; (b)周期边界条件下两异核原子, 黑实线和红虚线分别表示质量相等两原子和质量比为2.175情况下对应的能量; (c)双势阱中相互作用两原子的奇宇称态, 一维无限深方势阱中心为一个强度为d = 0.5的$ \delta $势垒劈开, 黑实线和红虚线分别表示准动量为实数的原子本征态和准动量为复数的分子态对应的能量. 这里能量的单位分别是$ \hbar \omega, 8\hbar^2 /m_1 L^2, 2\hbar^2 /m L^2 $, 相互作用强度g的单位分别是$ \sqrt{\hbar^3\omega/\mu}, 8\hbar^2 /m_1 L, 2\hbar^2 /m L $

Fig. 2. Energy spectrum of three types of interacting two-atom system: (a)Two atoms in a harmonic oscillator potential. Black solid lines and red dashed lines are odd parity and even parity energy level respectively; (b)two heteronuclear atoms in a ring trap. Black solid lines and red dashed lines are energy levels for equal mass and mass ratio $\alpha $ = 2.175 respectively; (c) two atoms in a $\delta $-split hard-wall double well. Five lowest odd parity levels for barrier height d = 0.5. Black solid lines and red dashed lines are the bound states for atoms with real-valued quasimomentum and the molecule states with complex-valued quasimomentum respectively.

下载: 全尺寸图片幻灯片

图 3 (a)衰减时间随磁场的变化. 黑色虚线和实线分别是WKB和考虑准粒子波动方程的修正的结果. 绿色线表示通过微扰论考虑非谐项贡献的结果; (b)相互作用强度和隧穿能量随磁场的变化.本图摘自参考文献[33]

Fig. 3. (a) Decay time $ \tau $ vs magnetic field B. The points with error bars are the experimental data^[24], the dashed and solid lines are, respectively, the WKB $ (\tau_0) $ and QPWF predictions. The green light gray lines include the perturbation theory correction to the tunneling energy $ \varepsilon $; (b) interaction strength (red gray curve) and tunneling energy (black and green light gray curve) vs B (Reproduced with permission from Ref. [33]).

下载: 全尺寸图片幻灯片

图 4 (a)两个原子在同一个阱中的占据概率随时间的演化图. 黑实线, 蓝虚线, 红点线分别表示不同的相互作用强度g = 0.5, 1.5, 20. 在t = 0时刻势垒高度突然从d = 300降到d =0.5. (b)g = 1.5和20时一个周期里不同时刻的密度分布图, 这里坐标 $ x_1 $ 和 $ x_2 $ 的单位是 $ L/2 $. 本图摘自参考文献[98]

Fig. 4. (a) Tunneling dynamics of the occupation probability $ P_2(t) $ of finding both atoms in the same well for g = 0.5 (black solid line), g = 1.5 (blue dashed line), and g = 20 (red dotted line). The $ \delta $ barrier is abruptly lowered from a height d = 300 to 0.5 at time t = 0. (b) the two-body density functions $ \rho (x_1, x_2, t) $ at different times t, for g = 1.5 and 20, respectively. Here the coordinates $ x_1 $ and $ x_2 $ are in units of $ L/2 $ (Reproduced with permission from Ref. [98]).

下载: 全尺寸图片幻灯片

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[2]	Pethick C J, Smith H 2008 Bose-Einstein Condensation in Dilute Gases (2nd Ed.) (Cambridge: Cambridge University Press) p.159
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[4]	Gross E P 1963 J. Math. Phys. 4 195 Google Scholar
[5]	Pitaevskii L P 1961 J. Exp. Theor. Phys. 40 646
[6]	Dalfovo F, Giorgini S, Pitaevskii L P, Stringari S 1999 Rev. Mod. Phys. 71 463 Google Scholar
[7]	Baranov M A 2008 Phys. Rep. 464 71 Google Scholar
[8]	Lahaye T, Menotti C, Santos L, Lewenstein M, Pfau T 2009 Rep. Prog. Phys. 72 126401 Google Scholar
[9]	Serwane F, Zürn G, Lompe T, Ottenstein T B, Wenz A N, Jochim S 2011 Science 332 236
[10]	Greiner M, Mandel O, Esslinger T, Hänsch T W, Bloch I 2002 Nature 415 39 Google Scholar
[11]	Kohl M, Moritz H, Stöferle T, Günter K, Esslinger T 2005 Phys. Rev. Lett. 94 080403 Google Scholar
[12]	Chin C, Grimm R, Julienne P, Tiesinga E 2010 Rev. Mod. Phys. 82 1225 Google Scholar
[13]	Inouye S, Andrews M R, Stenger J, Miesner H J, Stamper-Kurn D M, Ketterle W 1998 Nature 392 151 Google Scholar
[14]	Courteille P, Freeland R S, Heinzen D J, van Abeelen F A, Verhaar B J 1998 Phys. Rev. Lett. 81 69 Google Scholar
[15]	Roberts J L, Claussen N R, Burke J P, Greene C H, Cornell E A, Wieman C E 1998 Phys. Rev. Lett. 81 5109 Google Scholar
[16]	Efimov V 1970 Phys. Lett. B 33 563 Google Scholar
[17]	Efimov V 1970 Sov. J. Nucl. Phys. 12 598
[18]	Efimov V 1973 Nucl. Phys. A 210 157 Google Scholar
[19]	Blume D, Daily K M 2010 Phys. Rev. Lett. 105 170403 Google Scholar
[20]	Naidon P, Endo S 2017 Rep. Prog. Phys. 80 056001 Google Scholar
[21]	Kraemer T, Mark M, Waldburger P, Danzl J G, Chin C, Engeser B, Lange A D, Pilch K, Jaakkola A, Nägerl H C, Grimm R 2006 Nature 440 315 Google Scholar
[22]	刘彦霞 2017 博士学位论文(太原: 山西大学) Liu Y X 2017 Ph. D. Dissertation (Taiyuan: Shanxi University) (in Chinese)
[23]	Blume D 2012 Rep. Prog. Phys. 75 046401 Google Scholar
[24]	Zürn G, Serwane F, Serwane F, Lompe T, Wenz A N, Ries M G, Bohn J E, Jochim S 2012 Phys. Rev. Lett. 108 075303 Google Scholar
[25]	Zürn G, Wenz A N, MurmannS, Bergschneider A, Lompe T, Jochim S 2013 Phys. Rev. Lett. 111 175302 Google Scholar
[26]	Wenz A N, Zürn G, Murmann S, Brouzos I, Lompe T, Jochim S 2013 Science 342 457 Google Scholar
[27]	Murmann S, Bergschneider A, Klinkhamer V M, Zürn G, Lompe T, Jochim S 2015 Phys. Rev. Lett. 114 080402 Google Scholar
[28]	Murmann S, Deuretzbacher F, Zürn G, Bjerlin J, Reimann S M, Santos L, Lompe T, Jochim S 2015 Phys. Rev. Lett. 115 215301 Google Scholar
[29]	Kaufman A M, Lester B J, Regal C A 2012 Phys. Rev. X 2 041014
[30]	Kaufman A M, Lester B J, Reynolds C M, Wall M L, Foss-Feig M, Hazzard K R A, Rey A M, Regal C A 2014 Science 345 306 Google Scholar
[31]	Busch T, Englert B G, Rzazewski K, Wilkens M 1998 Found. Phys. 28 549 Google Scholar
[32]	Olshanii M 1998 Phys. Rev. Lett. 81 938 Google Scholar
[33]	Rontani M 2012 Phys. Rev. Lett. 108 115302 Google Scholar
[34]	Rontani M 2013 Phys. Rev. A 88 043633 Google Scholar
[35]	Bethe H A 1931 Z. Phys. 71 205 Google Scholar
[36]	Lieb E H, Liniger W 1963 Phys. Rev. 1963 130
[37]	McGuire J B 1965 J. Math. Phys. 6 432 Google Scholar
[38]	Gaudin M 1967 Phys. Lett. A 24 55 Google Scholar
[39]	Yang C N 1967 Phys. Rev. Lett. 19 1312 Google Scholar
[40]	Baxter R J 1972 Ann. Phys. 70 193 Google Scholar
[41]	Lieb E H, Wu F Y 1968 Phys. Rev. Lett. 20 1445 Google Scholar
[42]	Korepin V E, Essler F H L 1994 Exactly Solvable Models of Strongly Correlated Electrons (Singapore: World Scientific)
[43]	Mattis D C 1993 The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension (Singapore: World Scientific)
[44]	Takahashi M 1999 Thermodynamics of One-Dimensional Solvable models (Cambridge: Cambridge University Press)
[45]	Baxter R J 1989 Exactly solved models in statistical mechanics (London: Academic Press)
[46]	Sutherland B 2004 Beautiful Models-70 years of exactly solved quantum many-body problems (Singapore: World Scientific)
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冷原子物理中的一维少体问题