Memory of the Initial Conditions in an Incompletely Chaotic Quantum System: Universal Predictions with Application to Cold Atoms

Abstract

Two zero-range-interacting atoms in a circular, transversely harmonic waveguide are used as a test-bench for a quantitative description of the crossover between integrability and chaos in a quantum system with no selection rules. For such systems we show that the expectation value after relaxation of a generic observable is given by a linear interpolation between its initial and thermal expectation values. The variable of this interpolation is universal; it governs this simple law to cover the whole spectrum of the chaotic behavior from integrable regime through the welldeveloped quantum chaos. The predictions are confirmed for the waveguide system, where the mode occupations and the trapping energy were used as the observables of interest; a variety of the initial states and a full range of the interaction strengths have been tested. Two distinct types of evolution of an isolated dynamical system are usually identified: a predictable evolution, strongly correlated with the initial state on one hand and a relaxation to the thermal equilibrium with no memory of the initial state on the other. An ideal gas of noninteracting atoms is a trivial example of a predictable evolution. In cold quantum gases, this type of behavior was observed for the single-mode excitations in BoseEinstein condensates [1, 2] and a one-dimensional gas of interacting atoms [3]. Such systems can be described by integrable models, i.e. models that have as many integrals of motion as there are degrees of freedom. Lifting of integrability leads to a chaotic motion. For ultracold atoms, notable examples of the quantum-chaotic motion include both a non-stationary δ-kicked rotor [4,5] and a stationary billiard [6].Interactions between the trapped atoms are the most common cause of integrability lifting (with the exception of the one-dimensional gas, see [7]). Even for the case of just two trapped atoms, interactions can already destroy integrability [8]. For ultracold atoms, the atomic de Brogle wavelength typically substantially exceeds the interaction range. In this case, the interaction couples every two eigenstates of non-interacting atoms with approximately the same strength, yielding no selection rules interaction can obey.In this Letter, we analyze the case of two interacting atoms in a circular, transversely harmonic waveguide in the multimode regime [8]. Similar models, thě Seba billiard [9] and a cylindrically-symmetric harmonic trap with a δ-scatterer [10], were considered by other authors as well. These models, as well as the waveguide model, are non-integrable and show some signatures of a quantum-chaotic behavior as a result. However, their behavior is only incompletely-chaotic, since the systems demonstrate some substantial deviations from quantumchaotic predictions as well (see [8,11] and the references therein). In these systems, the interaction strength can be used to tune the "chaoticity".Trajectories of a completely-chaotic classical system fill all the available phase space in a "mixing" motion. This leads to a relaxation to t...

Glimmers of a Quantum KAM Theorem: Insights from Quantum Quenches in One-Dimensional Bose Gases

“…This process is then considerably different than the integrability breaking considered in Refs. [66,67], where they considered integrability that respected no selection rules and correspondingly saw an extremely rapid crossover from quantum integrable to quantum chaoticity. However, this does not mean our construction of Q does not work if the potential induces nontrivial mixing between wave vectors.…”

Finite-size scaling of eigenstate thermalization

“…Despite intense recent research [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], understanding of several aspects of the ETH remains incomplete. For example, it is not fully known exactly which observables will or will not serve as "typical" observables.…”

Chaos-driven dynamics in spin-orbit-coupled atomic gases

“…While true in most cases studied so far, exceptions to this hypothesis has been found [28]. Moreover, the behavior near the transition from regular to chaotic dynamics, classically explained by KolmogorovArnold-Moser theory [29], is not well understood for a quantum system [30]. It is therefore desirable to study a system where these two regimes can be explored by tuning an external parameter, and for which the experimental methods in terms of preparation and detection arXiv:1208.2923v2 [quant-ph] 28 Jan 2013 are already well developed.…”