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Journal of Physics A Mathematical and General

Equal partial revivals of wavepackets in long time scales

Quanlin Jie, Shun-Jin Wang

Published Mar 23, 2000• 2 citations
Publisher

Abstract

In terms of the action-angle Wigner function, we analyse the effects of energy spectrum anharmonicity on the evolutions of wavepackets in long time scales. The formulation shows a close connection between classical periodic motions of phase space points and the quantum (full and partial) revivals of wavepackets. This enables us to find the conditions of (full and partial) revivals in an intuitive way. For one-dimensional cases, we obtain analytic solutions for times of equal partial revivals, i.e. an initial wavepacket splits equally into several small parts in long time scales. Numerical results for the JC model confirm the theoretical predictions.

DOI
10.1088/0305-4470/33/13/305

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In terms of the action-angle Wigner function, we analyse the effects of energy spectrum anharmonicity on the evolutions of wavepackets in long time scales. The formulation shows a close connection between classical periodic motions of phase space points and the quantum (full and partial) revivals of wavepackets. This enables us to find the conditions of (full and partial) revivals in an intuitive way. For one-dimensional cases, we obtain analytic solutions for times of equal partial revivals, i.e. an initial wavepacket splits equally into several small parts in long time scales. Numerical results for the JC model confirm the theoretical predictions.
Partial revivals of wave packets are investigated by means of the action-angle Wigner function, which is formulated in terms of action quantum numbers and angle variables. In the eigenbasis of the Hamiltonian, the phase shifts of the expansion coefficients of the wave packet are approximated by a quadratic function of action quantum numbers. Under this approximation, the evolution of the action-angle Wigner function of the wave packet, similarly to the classical Liouville density, can be viewed as a result of each phase-space point moving along a classical orbit. The partial and full revivals of wave packets are direct consequences of two facts: (i) The action-angle Wigner function is distributed only on discrete tori of the phase space and (ii) phase-space points of nearby tori move against each other with a constant speed.
A unified treatment of quantum mechanical oscillatory motion in one dimension is presented in a phase space formalism which is especially adapted to the semiclassical energy domain and closely related to a `naive' quantization of action and angle variables. Characteristics of this scheme are non-classical symmetry properties of the phase space functions representing density operators and observables and the inclusion of half-Bohr orbits besides the familiar Bohr orbits. Over long time intervals the quantum evolution can be well approximated by a Hamiltonian flow along these distinguished classical orbits. The interplay of this reduced classical evolution and the symmetry properties of the phase space functions results in a consistent quantitative description of quantum interference effects which are most clearly seen in the revivals of wavepackets.
The dynamics of a quantum mechanical particle in a time-independent potential are found to contain many interesting phenomena. These are direct consequences of the (typical) existence of more than one time scale governing the problem. This gives rise to full revivals of initial wavepackets, fractional revivals (multiple wavepackets appearing at fractions of the revival time) and the striking quantum carpets. A variety of analytic techniques are used to consider the interference that gives rise to these phenomena while skirting calculations involving cross-terms. Novel results include a new theorem on the weighting coefficients ama_mam​ that govern fractional revivals, a demonstration that ΨclΨ_{cl}Ψcl​, the function that governs the distribution and features of these fractional revivals, really does behave classically, a treatment of the wavepacket dephasing in the infinite square well by means of the Poisson summation formula, and a correct analysis of the spatial distribution of intermode traces. Also, this work presents a coherent treatment of these phenomena, which before now did not exist.
We study numerically the one-dimensional quantum evolution of a coherent state when the particle is bound by a nonlinear force. Snapshots of the Husimi function show that there exist times where the initial Gaussian distribution in phase space splits into chains of 2,3,…2,3,\dots{}2,3,… distributions of similar form. During their finite lifetime the members of such a string are equally distributed along the classical orbit which passes through the center of the initial distribution, and they move along this curve with the corresponding velocity. This effect, which is generic for integrable bounded motion, is explained in terms of the quantum analogs of action and angle variables.