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Physical Review A

Random perturbation of systematic degeneracies and quantum chaos

W. D. Heiss, John C. H. Chiang

Published Apr 1, 1993• 7 citations
Publisher

Abstract

The effect of a random perturbation is investigated for a Hamiltonian with systematic degeneracies. It turns out that the influence of the degeneracies remains significant under a perturbation. In particular, the eigenvectors manifest specific patterns when the perturbation produces all characteristics of quantum chaos for the spectrum.
DOI
10.1103/physreva.47.2533

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The effect of random perturbation is investigated for a Hamiltonian with systematic degeneracies. It turns out that the influence of the degeneracies remains significant under perturbation. The model simulates quantized versions of classically chaotic systems in that the Fourier transform of the level density shows the characteristic lines which would be associated with the periods of periodic orbits if the system had a classical analogy. The decaying magnitude of the recurring lines is related to the spread of the probability distribution under the influence of perturbation of the degenerate levels.
The influence of quantum chaos on small atomic systems is reviewed. It now seems clear that chaos in the usually understood sense (e.g. exponential sensitivity to perturbations) is not found in isolated quantum systems. However, there are phenomena which appear only when the corresponding classical system is chaotic. The stability of the classical dynamics, in other words whether it is regular or chaotic, has a profound effect on the character of the corresponding quantum spectrum and wavefunctions.
We compare the influence of perturbations of classically regular respectively chaotic Hamiltonians on their eigenfunctions. A generic measure of the perturbation strength is given by the average spreading width, which semiclassically is a phase-space integral. Thus, contrary to common assumption, the spreading width cannot be an indicator of regularity or chaos. The distribution of expansion coefficients of perturbed eigenfunctions in terms of unperturbed ones is markedly different for the two cases and may serve as a quantum signature of chaos referring to states rather than to spectra.
We provide compelling evidence for the presence of quantum chaos in the unitary part of the operator usually employed in Shor's factoring algorithm. In particular we analyze the spectrum of this part after proper desymmetrization and show that the fluctuations of the eigenangles as well as the distribution of the eigenvector components follow the circular unitary ensemble of random matrices, of relevance to quantized chaotic systems that violate time-reversal symmetry. However, as the algorithm tracks the evolution of a single state, it is possible to employ other operators; in particular, it is possible that the generic quantum chaos found above becomes of a nongeneric kind such as is found in the quantum cat maps and in toy models of the quantum baker's map.
Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector σ in the Bloch sphere.
We consider a system of linear oscillators, or quantum states, described by random matrix theory and analyze how its time evolution is affected by a nonlinear perturbation. Our numerical results show that above a certain chaos border a weak or moderate nonlinearity leads to a dynamical thermalization of a finite number of degrees of freedom with energy equipartition over linear eigenmodes as expected from the laws of classical statistical mechanics. The system temperature is shown to change in a broad range from positive to negative values, and the dependence of system characteristics on the initial injected energy is determined. Below the chaos border the dynamics is described by the Kolmogorov-Arnold-Moser integrability. Owing to universal features of random matrix theory we argue that the obtained results describe the generic properties of its nonlinear perturbation.