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arXiv:0911.5587 · Physical Review Letters

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

Vladimir Yurovsky, Maxim Olshanii

Published Jan 13, 2011• 44 citations
arXivHTMLPDF

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 well-developed 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.

DOI
10.1103/physrevlett.106.025303

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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 well-developed 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.
A set of zero-range scatterers along its axis lifts the integrability of a harmonic waveguide. Effective solution of the Schrödinger equation for this model is possible due to the separable nature of the scatterers and millions of eigenstates can be calculated using modest computational resources. Integrability-chaos transition can be explored as the model chaoticity increases with the number of scatterers and their strengths. The regime of complete quantum chaos and eigenstate thermalization can be approached with 32 scatterers. This is confirmed by properties of energy spectra, the inverse participation ratio, and fluctuations of observable expectation values.
In this article, we study the thermalizability of a system consisting of two atoms in a circular, transversely harmonic waveguide in the multimode regime. While showing some signatures of quantum-chaotic behavior, the system fails to reach a thermal equilibrium in a relaxation from an initial state, even when the interaction between the atoms is infinitely strong. We relate this phenomenon to the previously addressed unattainability of a complete quantum chaos in the \ifmmode \check{S}\else \v{S}\fi{}eba billiard [P. \ifmmode \check{S}\else \v{S}\fi{}eba, Phys. Rev. Lett. 64, 1855 (1990)], and we conjecture the absence of a complete thermalization to be a generic property of integrable quantum systems perturbed by a nonintegrable but well-localized perturbation.
The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.
This paper is an attempt to establish a link between the quantum nonequilibrium dynamics of cold gases and 50 years of progress in low-dimensional quantum chaos. We identify two atomic systems lying in the interface: two interacting atoms in a harmonic multimode waveguide and an interacting two-component Bose–Bose mixture in a double-well potential. In particular, we study the level spacing distribution, the wavefunction statistics, the eigenstate thermalization and the ability to thermalize in a relaxation process as such.
If a many-body quantum system approaches thermal equilibrium from a generic initial state, then the expectation value ⟨ψ(t)∣Ai∣ψ(t)⟩\langleψ(t)|A_i|ψ(t)\rangle⟨ψ(t)∣Ai​∣ψ(t)⟩, where ∣ψ(t)⟩|ψ(t)\rangle∣ψ(t)⟩ is the system's state vector and AiA_iAi​ is an experimentally accessible observable, should approach a constant value which is independent of the initial state, and equal to a thermal average of AiA_iAi​ at an appropriate temperature. We show that this is the case for all simple observables whenever the system is classically chaotic.
We consider many-body quantum systems that exhibit quantum chaos, in the sense that the observables of interest act on energy eigenstates like banded random matrices. We study the time-dependent expectation values of these observables, assuming that the system is in a definite (but arbitrary) pure quantum state. We induce a probability distribution for the expectation values by treating the zero of time as a uniformly distributed random variable. We show explicitly that if an observable has a nonequilibrium expectation value at some particular moment, then it is overwhelmingly likely to move towards equilibrium, both forwards and backwards in time. For deviations from equilibrium that are not much larger than a typical quantum or thermal fluctuation, we find that the time dependence of the move towards equilibrium is given by the Kubo correlation function, in agreement with Onsager’s postulate. These results are independent of the details of the system’s quantum state.