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Quantum Mechanical Cumulant Dynamics near Stable Periodic Orbits in Phase Space

Radka Bach, D'Arcy, M B, S. A. Gardiner, Keith R. Burnett

Published Oct 29, 2003• 0 citations
OpenAlex

Abstract

We formulate a general method for the study of semiclassical-like dynamics in stable regions of a mixed phase-space. This involves determining stable Gaussian wavepacket solutions, and then propagating them using a cumulant-based formalism. We apply our method to the problem of quantum accelerator modes, determining their relative longevity under different parameter regimes, and obtaining good qualitative agreement with exact wavefunction dynamics.

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Publisher's Note: Quantum-mechanical cumulant dynamics near stable periodic orbits in phase space: Application to the classical-like dynamics of quantum accelerator modes [Phys. Rev. A71, 033417 (2005)]

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We formulate a general method for the study of semiclassical-like dynamics in stable regions of a mixed phase-space. This involves determining stable Gaussian wavepacket solutions, and then propagating them using a cumulant-based formalism. We apply our method to the problem of quantum accelerator modes, determining their relative longevity under different parameter regimes, and obtaining good qualitative agreement with exact wavefunction dynamics.
We formulate a general method for the study of semiclassical-like dynamics in stable regions of a mixed phase space, in order to theoretically study the dynamics of quantum accelerator modes. In the simplest case, this involves determining solutions, which are stable when constrained to remain pure-state Gaussian wave packets, and then propagating them using a cumulant-based formalism. Using this methodology, we study the relative longevity, under different parameter regimes, of quantum accelerator modes. Within this attractively simple formalism, we are able to obtain good qualitative agreement with exact wave-function dynamics.
A simple and elegant semiclassical theory for propagating wavepackets is presented. The development of the theory avoids the explicit use of the Green-function formalism. It involves expanding the potential to a quadratic time-dependent form and the use of multiple reference trajectories. The classical analogue in phase space predicts the accuracy of the results. The theory is applied to the calculation of the autocorrelation function in a Coulomb field in one dimension. Remarkable agreement with an exact quantum calculation is obtained in many circumstances in spite of the Coulomb singularity. All the detailed quantum behaviour, including the long-time recurrences and the spectrum, is reproduced.
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.
In this thesis, we study semiclassical phase-space methods for quantum evolution in Hermitian and non-Hermitian systems. We first present the dynamics of Gaussian wave packets under non-Hermitian Hamiltonians and interpret them as a classical dynamics for complex Hamiltonians. We use these to derive exact dynamics for wave packets in the quadratic Swanson oscillator. We show that in the case of unbroken PT-symmetry there can be periodic divergences in this system and relate this to the fact that any operator mapping the system to a Hermitian counterpart is unbounded. We apply the semiclassical wave-packet dynamics to two further anharmonic example systems: a PT-symmetric wave guide, a version of which we propose as a filtering device for optical beams, and a non-Hermitian single-band tight-binding model, for which we use classical equations of motion to model both narrowly and widely distributed initial states. We further develop an exact quantum propagator for non-Hermitian dynamics using lattices of wave packets, whose evolution is governed by the semiclassical equations of motion. We demonstrate that this accurately reproduces quantum dynamics compared to the split operator method. Finally we study a Hermitian two-mode many-particle model for bosonic atom-molecule conversion, for which the classical phase-space structure is an orbifold. We show that standard semiclassical tools may be applied to recover features such as the dynamics, spectrum and density of states of the many-particle system.
Quantum accelerator modes were discovered in experiments with kicked cold atoms in the presence of gravity. They were shown to be tightly related to resonances of the quantum kicked rotor. In this paper a spinor formalism is developed for the analysis of modes associated with resonances of arbitrary order $q\ensuremath{\ge}1$. Decoupling of spin variables from the orbital ones is achieved by means of an ansatz of the Born-Oppenheimer type that generates qqq independent band dynamics. Each of these is described, in classical terms, by a map, and the stable periodic orbits of this map give rise to quantum accelerator modes, which are potentially observable in experiments. The arithmetic organization of such periodic orbits is briefly discussed.