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Physical Review Letters

Long-Time Evolution of Semiclassical States in Anharmonic Potentials

Peter Kasperkovitz, Momtchil Peev

Published Aug 7, 1995• 24 citations
Publisher

Abstract

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 $2,3,\dots{}$ 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.

DOI
10.1103/physrevlett.75.990

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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.
The authors derive a semiclassical formula for the Wigner function W(q,p,t) describing the evolution in the two-dimensional phase space qp of a nonstationary quantum state psi (q,t) for a system with one degree of freedom. The initial state psi (q,0) corresponds to a family of classical orbits represented by a curve C 0 in qp. Under the classical motion C 0 evolves into a curve C t ; it is shown that the region where W is large hugs C t in an adiabatic fashion, and that W has semiclassical oscillations depending only on the geometry of C t and neighbouring curves.
The dynamics generated by non-Hermitian Hamiltonians are often less intuitive than those of conventional Hermitian systems. Even for models as simple as a complexified harmonic oscillator, the dynamics for generic initial states shows surprising features. Here we analyze the dynamics of the Husimi distribution in a semiclassical limit, illuminating the foundations of the full quantum evolution. The classical Husimi evolution is composed of two factors: (i) the initial Husimi distribution evaluated along phase-space trajectories and (ii) the final value of the norm corresponding to each phase-space point. Both factors conspire to lead to intriguing dynamical behaviors. We demonstrate how the full quantum dynamics unfolds on top of the classical Husimi dynamics for two instructive examples.
In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.
We derive a continuity equation for the Husimi function evolving under a general non-Hermitian Hamiltonian and identify the phase space flow associated with it. For the case of unitary evolution we obtain explicit formulas for the quantum flow, which can be written as a classical term plus quantum corrections. The quantum terms can be expanded in powers of Planck's constant providing a series of semiclassical corrections to the classical flow. We test the exact and semiclassical formulas for a particle in a double well potential and find numerical evidence that the zeros of the Husimi function are always saddle points of the flow. Merging or splitting of stagnation points, reported for the Wigner flow, was not observed.
Coherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schr\"odinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: The center follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satisfies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape or of the confining potential. A rich variety of effects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the wavepacket is transferred toward incoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped toward the bottom of the well.
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.