有关量子相变、冷原子、玻色-爱因斯坦凝聚体（BEC）的若干问题与进展 王文阁 中国科技大学，近代物理系 合作者: 秦品权，王骞，何乐为，王评，杨寅彪

（I）量子相变 Quantum phase transition (QPT) Coleman and Schofield, Nature 433, 226 (2005).

量子相变定义 热相变：在有限温度下发生的相变，巨大的热涨落伴随其发生、为其发生的重要动因。 例子：液固相变，铁磁-顺磁相变。 量子相变：量子涨落起主要作用的相变。 为尽可能将量子涨落因素与热涨落因素区分开来，人们首先考虑零温、即基态情况。 研究量子相变的意义：为理解许多重要实验现象，比如有关高温超导、反铁磁膜、量子伊辛链等，提供重要、甚至是关键性的思路

Non-analyticity of ground level at critical point（临界点） λc QPT 狭义的量子相变定义 Non-analyticity of ground level at critical point（临界点） λc QPT Drastic change of fundamental properties of the ground state (GS), when the parameter λ passes λc Some low lying excited states also change drastically

量子相变的研究策略与方法 就其一般方法策略而言， （a）利用传统统计力学方法，计算系统在非零温下的配分函数，然后，令温度趋于零。 （b）非传统方法（传统统计力学方法并不能揭示出量子相变点附近的所有性质）。例如，量子信息领域中的一些概念：纠缠度与保真度在许多系统中可以刻画量子相变的发生。 就与时间的关系而言， （1）有关平衡态的性质，包括对配分函数的计算，以及对定态的纠缠度和保真度的研究等。 （2）有关非平衡态的性质，例如，对系统之动力学性质随时间变化的研究。

Equilibrium and non-equilibrium regions τ: time of interest τr: relaxation time. Unitary dynamics should be considered

Survival probability (SP): in the non-equilibrium region λ’ λ t : the GS of H(λ) Survival probability

H.T.Quan et al., Phys.Rev.Lett. 96, 140604 (2006) Our problem: What is the decaying law of SP at QPT and whether it may reveal some characteristic properties of QPT?

A type of QPT that allows a semiclassical approach En The ground level has infinite degeneracy at the critical point For example, due to avoided level crossings of infinite levels. λ λc (1) Density of states at the ground level, ρ(λ)→ ∞ when λ→λc.

For any fixed small value of |λ-λ’|, the overlap |<0λ|0λ’>|2 usually decreases significantly when λ’ approaches λc. may be in a relatively “high” energy region in the system H(λ’). The semiclassical theory may be applicable in the study of the survival probability, when λ’ is sufficiently close to λc.

SP as a special case of quantum Loschmidt echo (Peres fidelity) Quantum Loschmidt echo or (Peres) fidelity is a measure of the stability of quantum motion under small perturbation in the Hamiltonian, For sufficiently small ε,

m(t) is the overlap of the evolution of the same initial state under two slightly different Hamiltonians. H initial state M(t) H0

All classically permissible trajectories 半经典理论的基本思路 An initial wave function in a d-dimensional configuration space, ψ0(r0),propagated by the semiclassical Van Vleck-Gutzwiller propagator, Final state 稳相近似 量子力学的路径积分表示： Initial state All classically permissible trajectories

波函数的半经典表示式 r s r0

is the action difference along For a ξsmall enough, R.A. Jalabert and H.M. Pastawski, Phys. Rev. Lett. 86, 2490 (2001). is the action difference along nearby trajectories in the two systems.

M(t) expressed as an integral of p0 Changing variables r→p0, = ΔS=

Loschmidt echo decay in regular systems with In a very low energy region, the classical counterpart of the system (if exists) is usually a regular system. The ground state has approximately a Gaussian shape. (The main results to be derived do not depend on the Gaussian shape.) U=V|initial point for t<<T d=1. ΔS ≈ εUt. for t>T T is the period of the classical motion.

Expanding U to the second order term in p0, WGW, G. Casati, and B. Li, Phys. Rev. E 75, 016201 (2007). Expanding U to the second order term in p0, For short times, it has an initial Gaussian decay; for long times, it has a 1/t decay.

WGW, P.Qin, L.He, and P.Wang, Phys. Rev. E, 81, 016214 (2010) Very large d. (2) For a very large d, there are many different frequencies, hence, T is very large and τ<<T. For times of interest, the classical motion looks random due to the many different frequencies. where P(ΔS) is the distribution of ΔS and is close to a Gaussian distribution for times much shorter than T. Similar to the chaotic case discussed in N.R.Cerruti and S.Tomsovic, Phys. Rev.Lett. 88, 054103 (2002).

semiclassical predictions for the decay law of SP at QPT. WGW, P.Qin, L.He, and P.Wang, Phys. Rev. E, 81, 016214 (2010) Summary of semiclassical predictions for the decay law of SP at QPT. For systems with classical counterparts in the very low energy region, the SP may have two qualitatively different decays for relatively long times: Power law decay may appear for d=1. Exponential decay for sufficiently large d.

Numerical study (1): single-mode Dicke model Model: Interaction of a single bosonic mode and a collection of N two-level systems (collective motion). C. Emary and T. Brandes, Phys. Rev. E 67, 066203 (2003).

Properties of Dicke model at QPT In the normal phase, At the critical point, Therefore, at the QPT, effectively the system has a classical counterpart with d=1.

Properties of Dicke model at QPT

Numerical results in Dicke model

Numerical study (2): 1-dimensional Ising chain Model: A 1-dimensional Ising chain in a transverse field. Using Jordan-Wigner and Bogoliubov transformations, the Hamiltonian can be diagonalized, with m=-M, 1-M, …,M, N=2M+1

Properties of Ising chain at QPT In the large N limit, for fixed m (low energy region), having no singularity.

Classical counterpart of Ising chain in low energy region For λ=λc, large N, and relatively small m, Due to the linear dependence of ek on m, the method of bosonization can be used to express the fermionic states in terms of bosonic modes. The system has a classical counterpart with a large d in the low energy region.

Numerical results in Ising chain

Two other models (3) LMG model: d=1 and power law decay 1/t of the SP has been found. (4) XY model Its classical counterpart has a large d and an exponential decay of SP has been found.

(II) 冷原子系统——拾零 Laser cooling (Stevens Chu, 1997 Nobel Prize in Physics) Evaporative cooling

First experimental realizations of Bose-Einstein condensates (BEC) Anderson et al., Science, 269 (1995), 198: JILA Group; Rb Davis et al., Phys. Rev. Lett., 75 (1995), 3969: MIT Group; Rb Bradly et al., Phys. Rev. Lett., 75 (1995), 1687, Rice Group; Li 2001 Nobel prize in physics: C. Wiemann: U. Colorado E. Cornell: NIST W. Ketterle: MIT

Ultracold Fermi Gas (1999) Ultracold molecules formed from an ultracold Fermi gas (2003) Molecular Bose-Einstein Condensate (2003) Cold atoms with Fermi-Dirac statistics, 1999 Fermionic atom  diatomic molecular BEC, 2003

Munich: I. Bloch, T. Haensch et al.

Formation of 1D, 2D, 3D optical lattices laser 1D 2D square lattice 3D laser triangular lattice

What can cold-atoms do? Quantum simulation of condensed-matter physics (electrons in periodic potentials, electrons in strong magnetic fields, superconductivity physics, spintronics) Exotic quantum materials (such as those with long-range dipolar interactions made from dipolar BECs, Bose-Fermi mixture, etc) Fundamental studies of ultracold physics, ultracold chemistry, and quantum physics (macroscopic quantum coherence, chemical reactions in the BEC regime) Quantum simulation of other physics areas, such as relativistic quantum mechanics, quantum field theory, nonlinear physics. Precision measurement, novel interferometry, better atomic clocks, novel atom devices…

Example: Cold atoms in periodic optical potentials (optical lattices) Strongly correlated gas (e.g., Bose-Hubbard model, Fermi-Hubbard model) Bloch oscillations (extremely hard to observe in solid-state systems) Quasi-crystal Anderson localization …

(III) 稀薄气体的玻色-爱因斯坦凝聚体(BEC) 1924

Bose-Einstein condensates become an ultralow-temperature laboratory

First experimental realizations of Bose-Einstein condensates (BEC) Anderson et al., Science, 269 (1995), 198: JILA Group; Rb Davis et al., Phys. Rev. Lett., 75 (1995), 3969: MIT Group; Rb Bradly et al., Phys. Rev. Lett., 75 (1995), 1687, Rice Group; Li 2001 Nobel prize in physics: C. Wiemann: U. Colorado E. Cornell: NIST W. Ketterle: MIT

Typical parameters of BEC, Density 1011 ～ 1015 cm-3 macroscopic quantum fluid phenomena interference tunnelling Typical parameters of BEC, Density 1011 ～ 1015 cm-3 Temperature nK ～ μK

Example: Macroscopic quantum coherence: Matter transport without transit (Rab et al, 2008) Tunneling rate from well 1 to well 2 is Tunneling rate from well 2 to well 3 is Assuming the same on-site energy for the three wells, the Hamiltonian is

More example: Cold-atom and Superconductivity: Using rotating BEC to simulate superconductors in strong magnetic fields: formation of vortex lattices From Abrikosov vortex lattice to vortex lattice in a rotating BEC A.A. Abrikosov, Nobel Prize 2003

BEC的基本特性 T>Tc, 热原子 T<Tc，达到平衡态之后，为BEC。 BEC的特点：众多（宏观数量）粒子处于同一个宏观波函数所描述的状态，可有超流（超导）性。 BEC的多体波函数描述：

BEC的数学描述：Gross-Pitaevskii (GP)方程

BEC的形成过程如何描述？ —— 对理论物理学家的挑战 实验 Nature 455, 948 (2008)

Ketterle小组的实验及分析 利用Bragg脉冲，制造空间位置不同的两个态的叠加态，形成类似于双缝干涉的条纹。 Phys. Rev A 71, 043615 (2005) Ketterle小组的实验及分析 利用Bragg脉冲，制造空间位置不同的两个态的叠加态，形成类似于双缝干涉的条纹。           通过光照观察干涉条纹

实验结果     这里存在额外的量子相干性

速度选择性：脉冲持续时间越长，单色性越好，选择特定速度附近的原子。 作者认为，这个现象可能跟速度选择性有关   速度选择性：脉冲持续时间越长，单色性越好，选择特定速度附近的原子。

在有速度选择性的情况下，条纹对比度的确在极低温就偏离了热原子模型的预计。   在有速度选择性的情况下，条纹对比度的确在极低温就偏离了热原子模型的预计。

我们假设： 在BEC临界温度以上，仍然可能存在部分原子，共享同一个波函数，形成类似于凝聚的现象，我们称之为相干凝聚态（coherently condensed state）

BEC的启示 我们的假设: 在BEC临界温度以上，仍然可能存在部分原子，共享同一个波函数，形成类似于凝聚的现象，我们称之为相干凝聚态（coherently condensed state） 混杂模型: 相干长度超过粒子间平均间距的那部分粒子将会形成相干凝聚态；而对于相干长度不超过粒子间平均间距的那部分粒子，假设可分辨粒子近似仍然能够成立。

对应上图的对比度拟合结果