-
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:
- ultracold atom physics /
- one dimensioanl system /
- few-body problem /
- exactly solvable model
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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 $ Figure 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.
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]
Figure 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]).
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]Figure 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]). -
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