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Physical Review Letters

Quantum signatures of classical chaos: Sensitivity of wave functions to perturbations

Luis Benet, T. H. Seligman, Hans A. Weidenmüller

Published Jul 26, 1993• 31 citations
Publisher

Abstract

We compare the influence of perturbations of classically regular respectively chaotic Hamiltonians on their eigenfunctions. A generic measure of the perturbation strength is given by the average spreading width, which semiclassically is a phase-space integral. Thus, contrary to common assumption, the spreading width cannot be an indicator of regularity or chaos. The distribution of expansion coefficients of perturbed eigenfunctions in terms of unperturbed ones is markedly different for the two cases and may serve as a quantum signature of chaos referring to states rather than to spectra.

DOI
10.1103/physrevlett.71.529

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We compare the influence of perturbations of classically regular respectively chaotic Hamiltonians on their eigenfunctions. A generic measure of the perturbation strength is given by the average spreading width, which semiclassically is a phase-space integral. Thus, contrary to common assumption, the spreading width cannot be an indicator of regularity or chaos. The distribution of expansion coefficients of perturbed eigenfunctions in terms of unperturbed ones is markedly different for the two cases and may serve as a quantum signature of chaos referring to states rather than to spectra.
We demonstrate that a system's rate of fidelity decay under repeated perturbations may be measured efficiently on a quantum information processor, and analyze the conditions under which this indicator is a reliable probe of quantum chaos. The type and rate of the decay are not dependent on the eigenvalue statistics of the unperturbed system, but depend on the system's eigenvector statistics in the eigenbasis of the perturbation. For random eigenvector statistics, the decay is exponential with a rate fixed by the variance of the perturbation's energy spectrum. Hence, even classically regular models can exhibit an exponential fidelity decay under generic quantum perturbations. These results clarify which perturbations can distinguish classically regular and chaotic quantum systems.
Numerical results show that, for quantum autonomous chaotic system, the evolution of initially coherent states are sensitive to perturbation. The overlap of a perturbed state with the unperturbed one decays exponentially, which is followed by fluctuation around N-1, N being the dimension of the Hilbert space. The matrix elements of the evolution operator in interaction picture tend to be a random distribution after sufficiently long time, where the interaction is the perturbation, even when the perturbation is very weak. The difference between a regular system and the chaotic one is shown clearly. In a regular system, the overlap shows strong revival. The distribution of the evolution matrix has only a few dominant terms.
Numerical experiment in Lipkin model shows that, in quantum system with global chaotic classical limit, the temporal mean of the expectation value of an observable is approximately equal to the average over the basic states of Hilbert space, if the wavefunction is initially either a coherent wave packet or the common eigenstates of a complete set of observables, and the observable is independent of the Hamiltonian. The mechanism is the absence of KAM barrier which prevents the spread of wavefunction. This can serve as a quantum signature of classical chaos.
We study signatures of quantum chaos in (1+1)D quantum field theory (QFT) models. Our analysis is based on the method of Hamiltonian truncation, a numerical approach for the construction of low-energy spectra and eigenstates of QFTs that can be considered as perturbations of exactly solvable models. We focus on the double sine-Gordon, also studying the massive sine-Gordon and ϕ^{4} model, all of which are nonintegrable and can be studied by this method with sufficiently high precision from small to intermediate perturbation strength. We analyze the statistics of level spacings and of eigenvector components, which are expected to follow random matrix theory predictions. While level spacing statistics are close to the Gaussian orthogonal ensemble (GOE) as expected, on the contrary, the eigenvector components follow a distribution markedly different from the expected Gaussian. Unlike in the typical quantum chaos scenario, the transition of level spacing statistics to chaotic behavior takes place already in the perturbative regime. Moreover, the distribution of eigenvector components does not appear to change or approach Gaussian behavior, even for relatively large perturbations. Our results suggest that these features are independent of the choice of model and basis.
Two fundamental quantum signatures of classically chaotic behaviour, large second derivatives and the logarithmic time barrier separating classical from quantum dynamics, have been observed near the separatrix of an integrable spin system. A systematic study in the neighbourhood of the separatrix reveals that the error in semiclassical eigenvalues (measured in units of the quantum level spacing) approaches a finite value at the classical limit. The generic factor which limits the validity of the correspondence principle is the presence of classical instabilities; chaotic behaviour is one such case; the separatrix of an integrable system is another.