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Physical Review Letters

Semiclassical quantization of classically chaotic systems

John R. Klauder

Published Aug 17, 1987• 17 citations
Publisher

Abstract

It is argued that the global, uniform semiclassical approximation to quantization afforded by a coherent-state approach is well suited for an analysis of classically chaotic systems. Special attention is given to the interpolating action which incorporates in a single expression the classical action for each and every contributing classical trajectory. Approximate expressions for the configuration-space propagator are presented for small, intermediate, and large propagation times.

DOI
10.1103/physrevlett.59.748

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Physical Review E•Jun 4, 2004•48 citations
It is argued that the global, uniform semiclassical approximation to quantization afforded by a coherent-state approach is well suited for an analysis of classically chaotic systems. Special attention is given to the interpolating action which incorporates in a single expression the classical action for each and every contributing classical trajectory. Approximate expressions for the configuration-space propagator are presented for small, intermediate, and large propagation times.
In this work, we derived a semiclassical approximation for the matrix elements of a quantum propagator in coherent states (CS) basis that avoids complex trajectories; it only involves real ones. For that purpose, we used the symplectically invariant semiclassical Weyl propagator obtained by performing a stationary phase approximation (SPA) for the path integral in the Weyl representation. After that, for the transformation to CS representation SPA is avoided; instead a quadratic expansion of the complex exponent is used. This procedure also allows us to express the semiclassical CS propagator uniquely in terms of the classical evolution of the initial point without the need of any root search typical of van Vleck--Gutzwiller-based propagators. For the case of chaotic Hamiltonian systems, the explicit time dependence of the CS propagator has been obtained. The comparison with a realistic chaotic system that derives from a quadratic Hamiltonian, the cat map, reveals that the expression here derived is exact up to quadratic Hamiltonian systems.
The semiclassical Herman–Kluk initial value approximation of the quantum-mechanical propagator, which is heavily used in dynamical studies of atomic and molecular systems, takes into account nonlinearity of the underlying classical dynamics by using multiple classical initial value solutions. In applications to the propagation of Gaussian wavepackets non-Gaussian distortions in the course of time and therefore a realistic description of the quantum dynamics are possible. This is in contrast to the single-trajectory approximations investigated in a recent paper by Baranger et al , one of which was erroneously termed Herman–Kluk approximation.
We examine semiclassical propagation of nonstationary states. The fundamental object is the 1928 time dependent Van Vleck semiclassical Green`s function, as modified by Gutzwiller in 1967. This direct approach to semiclassical quantum mechanics has received very little attention over the years. The problems of evaluating the Green`s function (finding all trajectories which go from x{prime} to x in time t, for every x, x{prime}, and t sounds foreboding) and the development of caustics at an alarming rate in chaotic systems, may have suppressed its use. We have developed several strategies for evaluating it efficiently, including for chaotic dynamics. We found it can be used to accurately propagate smooth wave functions in spite of the caustics. The smooth wave function has the effect of strongly damping the caustics and leaving an accurate propagated state. Moreover many of the contributions to the semiclassical amplitude are legal, and allow a much longer time scale for breakdown of the semiclassical propagation than had previously been thought. This has allowed us to extract accurate spectra and eigenstates for chaotic dynamics in the stadium billiard and to calculate a complete molecular electronic absorption spectrum for two nonlinearly interacting vibrational modes, including line intensities and positions, usingmore » only classical trajectory input.« less
The relationship between the diverging of classical trajectories in chaotic many-body systems, the spreading of quantum wave packets, and the validity and use of classical molecular dynamics is explored. This analysis, which is based on the semiclassical description of wave function propagation in terms of a weighted integration over a traveling fixed width coherent state basis, suggests that the exponential divergence of nearby classical trajectories in chaotic many-body systems should result in the rapid delocalization of an initially localized quantum wave packet describing the state of the system. Thus the justification for the use of classical molecular dynamics procedures for these supposedly classical systems cannot be based on the picture of the system wave function remaining localized as its center follows a nearly classical trajectory. The quantum evolution of the system density, on the other hand, requires two propagators, and each of these propagators is represented as an integration over trajectories in the semiclassical picture. The interference between the contributions from these two integrations over classical trajectories focuses the analysis on the most important points in this trajectory pair space, which are shown to occur when both trajectories in the pair are the same. Given reasonable assumptions for the initial density for a system that is expected to be well described by classical molecular dynamics, and given an appropriate choice for the width of the coherent state basis which is employed in the semiclassical description, it is shown that the semiclassical expressions for time dependent observables and correlation functions reduce the purely classical expressions, despite the fact that an initially localized wave packet would rapidly delocalize for the same system.
We present a new method for the semiclassical quantization of classically integrable as well as nonintegrable systems. The method is based on the semiclassical approximation of the equilibrium density matrix, using classical trajectories on the upside down potential surface. Periodic orbits do not play any special role. Explicit results are given for the case of the classically chaotic potential kx2y2/2{\mathrm{kx}}^{2}{y}^{2}/2kx2y2/2.
The semiclassical limit of the coherent state propagator $⟨{\mathbf{z}}^{\ensuremath{''}}\ensuremath{\mid}{e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}T∕\ensuremath{\hbar}}\ensuremath{\mid}{\mathbf{z}}^{\ensuremath{'}}⟩$ involves complex classical trajectories of the Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}(\mathbf{u},\mathbf{v})=⟨\mathbf{v}\ensuremath{\mid}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\mid}\mathbf{u}⟩$ satisfying $\mathbf{u}(0)={\mathbf{z}}^{\ensuremath{'}}$ and $\mathbf{v}(T)={\mathbf{z}}^{\ensuremath{''}}*$. In this work we study mostly the case ${\mathbf{z}}^{\ensuremath{'}}={\mathbf{z}}^{\ensuremath{''}}$. The propagator is then the return probability amplitude of a wave packet. We show that a plot of the exact return probability brings out the quantal images of the classical periodic orbits. Then we compare the exact return probability with its semiclassical approximation for a soft chaotic system with two degrees of freedom. We find two situations where classical trajectories satisfying the correct boundary conditions must be excluded from the semiclassical formula. The first occurs when the contribution of the trajectory to the propagator becomes exponentially large as $\ensuremath{\hbar}$ goes to zero. The second occurs when the contributing trajectories undergo bifurcations. Close to the bifurcation the semiclassical formula diverges. More interestingly, in the example studied, after the bifurcation, where more than one trajectory satisfying the boundary conditions exist, only one of them in fact contributes to the semiclassical formula, a phenomenon closely related to Stokes lines. When the contributions of these trajectories are filtered out, the semiclassical results show excellent agreement with the exact calculations.