We consider a quantum quench of the trap frequency in a system of bosons interacting through an inverse-square potential and confined in a harmonic trap (the harmonic Calogero model). We determine exactly the initial state in terms of the post-quench eigenstates and derive the time evolution of simple physical observables. Since this model possesses an infinite set of integrals of motion (IoM) that allow its exact solution, a generalised Gibbs ensemble (GGE), i.e. a statistical ensemble that takes into account the conservation of all IoM, can be proposed in order to describe the values of local physical observables long after the quench. Even though, due to the presence of the trap, physical observables do not exhibit equilibration but periodic evolution, such a GGE may still describe correctly their time averaged values. We check this analytically for the local boson density and find that the GGE conjecture is indeed valid, in the thermodynamic limit.
We show that the quasiparticle kinetic theory for quantum and classical Calogero models reduces to the free-streaming Boltzmann equation. We reconcile this simple emergent behaviour with the strongly interacting character of the model by developing a Bethe-Lax correspondence in the classical case. This demonstrates explicitly that the freely propagating degrees of freedom are not bare particles, but rather quasiparticles corresponding to eigenvectors of the Lax matrix. We apply the resulting kinetic theory to classical Calogero particles in external trapping potentials and find excellent agree…
Analytic results for thermal correlation enhancement after a trap quench
King’s College London（伦敦国王学院）；Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & Université PSL, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎高等师范学院，巴黎文理研究大学，索邦大学，巴黎西岱大学）
Dynamically generated correlations in a trapped bosonic gas via frequency quenches
Raman Research Institute（拉曼研究所）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究所国际理论科学中心）；LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay（巴黎南大学萨克雷大学法国国家科学研究中心理论与统计物理实验室）
In this work the diffusion in the quenched trap model with diverging mean waiting times is examined. The approach of randomly stopped time is extensively applied in order to obtain asymptotically exact representation of the disorder averaged positional probability density function. We establish that the dimensionality and the geometric properties of the lattice, on top of which the disorder is imposed, dictate the plausibility of a mean-filed approximation that will only include annealed disorder. Specifically, for any case when the probability to return to the origin ($Q_0$) is less than $1$,…
We provide exact results for the mean and variance of first-passage times (FPTs) of making a directed revolution in the presence of a bias in heterogeneous quenched environments where the disorder is expressed by random traps on a ring with period $L$. FPT statistics are crucially affected by the disorder realization. In the large-$L$ limit, we obtain exact formulae for the FPT statistics, which are described by the sample mean and variance for waiting times of periodically arranged traps. Furthermore, we find that these formulae are still useful for nonperiodic heterogeneous environments; i.e…
We investigate the biased quenched trap model on top of a two-dimensional lattice in the case of diverging expected dwell times. By utilizing the double-subordination approach and calculating the return probability in $2$d, we explicitly obtain the disorder averaged probability density function of the particle's position as a function of time (for any given bias) in the limit of large times ($t \rightarrow \infty$). The first and second moments are calculated, and a formula for a general $μ$-th moment is found. The behavior of the first moment, i.e. $\langle x(t)\rangle$, presents non-linear r…
We investigate a class of exactly solvable quantum quench protocols with a finite quench rate in systems of one dimensional non-relativistic fermions in external harmonic oscillator or inverted harmonic oscillator potentials, with time dependent masses and frequencies. These hamiltonians arise, respectively, in harmonic traps, and the $c=1$ Matrix Model description of two dimensional string theory with time dependent string coupling. We show how the dynamics is determined by a single function of time which satisfies a generalized Ermakov-Pinney equation. The quench protocols we consider asympt…
We consider the nonadiabatic energy fluctuations of a many-body system in a time-dependent harmonic trap. In the presence of scale-invariance, the dynamics becomes self-similar and the nondiabatic energy fluctuations can be found in terms of the initial expectation values of the second moments of the Hamiltonian, square position, and squeezing operators. Nonadiabatic features are expressed in terms of the scaling factor governing the size of the atomic cloud, which can be extracted from time-of-flight images. We apply this exact relation to a number of examples: the single-particle harmonic os…
Universality of monopole mode and time evolution of a d-dimensional trapped interacting Bose gas
本文研究了一个广义Gross-Pitaevskii方程，该方程描述的是一个d维谐振子（陷阱频率为$ω_{0}$）与弱相互作用玻色气体的相互作用势具有（2 k + 1）阶和标度指数（n）的非线性.使用含时变分分析，我们明确地表明，对于一个特定的组合的n，k和d，广义GP方程具有普遍的振荡频率$2 ω_{0}$。我们还发现，如果系统满足特定的组合，宽度的时间演化可以普遍描述相同的希尔方程。我们还得到了Gross-Pitaevskii方程的精确自相似解的条件。作为应用，我们讨论了低维玻色凝聚态和Calogero模型。
By exploring a phase space hydrodynamics description of one-dimensional free Fermi gas, we discuss how systems settle down to steady states described by the generalized Gibbs ensembles through quantum quenches. We investigate time evolutions of the Fermions which are trapped in external potentials or a circle for a variety of initial conditions and quench protocols. We analytically compute local observables such as particle density and show that they always exhibit power law relaxation at late times. We find a simple rule which determines the power law exponent. Our findings are, in principle,…
In recent times it has been observed that signatures of equilibrium quantum criticality surprisingly show up in many-body systems which are manifestly far from equilibrium. We explore such scenarios in interacting spin systems subject to a quench and develop a robust method to systematically probe ground state critical physics through nonequilibrium post-quench dynamics. Analyzing late-time behavior of finite string-like observables, we find emerging sharp signatures of equilibrium criticality. Specifically, these observables accurately detect equilibrium critical points and universal scaling …
Probing quantum statistical mechanics with Bose gases: Non-trivial order parameter topology from a Bose-Einstein quench
这是quant-ph/9804035 [Phys Rev Lett 83（1999），1707-1710]中思想的技术性较低的表述。由快速淬火引起的二级相变可以锁定密度远远超过其平衡期望值的拓扑缺陷。这种现象是非平衡统计力学的一般预测，并且可以出现在广泛的物理系统中。我们定性地讨论它的背景下，被困的稀玻色-爱因斯坦凝聚体，概述了一个简单的定量理论的基础上的时间依赖的金兹伯格-朗道方程，并简要地比较量子动力学理论的结果。
