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Physical Review A

Ergodicity and mixing in quantum theory. II

Mario Feingold, Nimrod Moiseyev, Asher Peres

Published Jul 1, 1984• 114 citations
Publisher

Abstract

The approach proposed in the first part of this paper is illustrated by two examples. The first one involves the H\'enon-Heiles model. A small wave packet, initially localized in the classically chaotic region, evolves so that $〈x〉$ and $〈{x}^{2}〉$ rapidly reach their equilibrium values. The second example makes use of a pair of rotators with nonlinear coupling. The matrix elements of ${J}_{z}$, in the energy representation, are pseudorandom in the energy range which is classically chaotic, but obey selection rules in the classically regular energy range.

DOI
10.1103/physreva.30.509

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The approach proposed in the first part of this paper is illustrated by two examples. The first one involves the H'enon-Heiles model. A small wave packet, initially localized in the classically chaotic region, evolves so that 〈x〉〈x〉〈x〉 and 〈x2〉〈{x}^{2}〉〈x2〉 rapidly reach their equilibrium values. The second example makes use of a pair of rotators with nonlinear coupling. The matrix elements of Jz{J}_{z}Jz​, in the energy representation, are pseudorandom in the energy range which is classically chaotic, but obey selection rules in the classically regular energy range.
In this paper we consider some of the consequences of classical stochasticity phenomena in the quantum mechanical Henon–Heiles and Barbanis systems. To explore classical–quantum analogies we have introduced a quantum mechanical phase space based on the coherent state representation. Quantum Poincaré maps (QPMs) were constructed from contour plots of the stationary phase-space densities. We have observed a correlation between the topological features of a QPM at an eigenenergy E and the sensitivity ‖dE/dε‖ of that energy to the strength ε of the nonlinear coupling. States with extreme values of ‖dE/dε‖, corresponding to both high and to low values of ‖dE/dε‖, show regular regions in the quantum phase space, while states with intermediate values of ‖dE/dε‖ span large parts of the quantum phase-space regions. These general features of the QPMs apply for almost each energy multiplet, and no qualitative change in the character of the QPMs was observed above the classical critical energy Ec for the stochastic transition. To investigate the quantum analog of classical trajectories, we have studied the dynamics of initially coherent Gaussian wave packets. We have discovered two limiting types of time evolution of intially coherent wave packets, which exhibit quasiperiodic time evolution or rapid dephasing. Quasiperiodic quantum dynamics is analogous to quasiperiodic classical trajectory dynamics, while rapid dephasing of the wave packet reflects the effects of spreading of the wave packet or/and stochastic classical dynamics. We were able to establish a correspondence between the topology of the quantum phase-space density and the wave packet dynamics. Quasiperiodic time evolution and rapid dephasing are exhibited by wave packets initially located in the regular and in the irregular regions of the quantum phase space, respectively.
The evolution of wave packets under the influence of a Hénon–Heiles potential has been investigated by direct numerical solution of the time-dependent Schrödinger equation. Coherent state Gaussians with a variety of mean positions and momenta were selected as initial wave functions. Three types of diagnostics were used to identify chaotic behavior, namely, phase space trajectories computed from the expected values of coordinates and momenta, the correlation function P(t)=‖〈ψ(0)‖ψ(t)〉‖2, and the uncertainty product or phase space volume V(t)=ΔxΔyΔpxΔpy. The three approaches lead to a consistent interpretation of the system’s behavior, which tends to become more chaotic as the energy expectation value of the wave packet increases. The behavior of the corresponding classical system, however, is not a reliable guide to regular or chaotic behavior in the quantum mechanical system.
Abstract The contributions of different eigenstates of a nonlinearly coupled oscillator system to the expansion of a local wave packet are analyzed from an information theoretical point of view. Such a wave packet can be considered as a nonstationary vibrational state of an electronically excited manifold of a molecule after Franck–Condon type initial preparation. The distributions of these contributions are compared to their individual stochastic ideals using Ruch's concept of “mixing distances.” The stochastic ideals are constructed via a probability diffusion process between neighboring states of the original distributions, representing an initial preparation corresponding to a Hamiltonian with only irregular eigenstates. Gaussian minimum uncertainty wave packets as initial states in a two‐dimensional nonlinear oscillator system with classically regular and chaotic energy ranges are studied numerically. It is found that distance measures, partly reflecting the “mixing distance” of a distribution from its stochastic ideal, show a large fluctuation in the classical regular energy range and a small fluctuation in the range where most of the classical trajectories move chaotically. This indicates that for this type of initial preparation process the actual location of the initial state in space plays the dominant role for the dynamics in the low‐energy range while for wave packets starting near the dissociation energy of the model system this location becomes unimportant.