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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Is semiquantum chaos real?

L. E. Ballentine

Published Apr 16, 2001• 18 citations
Publisher

Abstract

A semiquantum system, composed of a quantum part coupled to a classical part, can exhibit dynamical chaos in the motion of the quantum state vector. However, there has been disagreement as to whether this mathematical chaos is physically real or merely an artifact of the semiquantum approximation. It is shown, for a model of a quantum spin coupled to an approximately classical nonlinear oscillator, that the semiquantum chaos disappears rapidly as the mass of the oscillator is increased to make it more classical. The time interval during which the semiquantum approximation remains accurate increases with the mass, but is not closely related to the Lyapunov time.

DOI
10.1103/physreve.63.056204

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A semiquantum system, composed of a quantum part coupled to a classical part, can exhibit dynamical chaos in the motion of the quantum state vector. However, there has been disagreement as to whether this mathematical chaos is physically real or merely an artifact of the semiquantum approximation. It is shown, for a model of a quantum spin coupled to an approximately classical nonlinear oscillator, that the semiquantum chaos disappears rapidly as the mass of the oscillator is increased to make it more classical. The time interval during which the semiquantum approximation remains accurate increases with the mass, but is not closely related to the Lyapunov time.
A direct comparison of quantum and classical dynamical systems can be accomplished through the use of distribution functions. This is useful for both fundamental investigations such as the nature of the quantum-classical transition as well as for applications such as quantum feedback control. By affording a clear separation between kinematical and dynamical quantum effects, the Wigner distribution is particularly valuable in this regard. Here we discuss some consequences of the fact that when closed-system classical and quantum dynamics are treated in Gaussian approximation, they are in fact identical. Thus, it follows that several results in the so-called `semiquantum' chaos actually arise from approximating the classical, and not the quantum dynamics. (Similarly, opposing claims of quantum suppression of chaos are also suspect.) As a simple byproduct of the analysis we are able to show how the Lyapunov exponent appears in the language of phase space distributions in a way that clearly underlines the difference between quantum and classical dynamical situations. We also informally discuss some aspects of approximations that go beyond the Gaussian approximation, such as the issue of when quantum nonlinear dynamical corrections become important compared to nonlinear classical corrections.
We consider a system in which a classical oscillator is interacting with a purely quantum mechanical oscillator, described by the Lagrangian L=1/2x${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}^{2}$+1/2A${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}^{2}$-1/2(m2{\mathit{m}}^{2}m2+e2{\mathit{e}}^{2}e2A2{\mathit{A}}^{2}A2)x2{\mathit{x}}^{2}x2, where A is a classical variable and x is a quantum operator. With 〈x(t)〉=0, the relevant variable for the quantum oscillator is 〈x(t)x(t)〉=G(t). The classical Hamiltonian dynamics governing the variables A(t),${\mathrm{\ensuremath{\Pi}}}_{\mathit{A}}$(t),G(t), and ${\mathrm{\ensuremath{\Pi}}}_{\mathit{G}}$(t) is chaotic so that the results of making measurements on the quantum system at later times are sensitive to initial conditions. This system arises as the zero momentum part of the problem of pair production of charged scalar particles by a strong external electric field.
Dynamical chaos has recently been shown to exist in the Gaussian approximation in quantum mechanics and in the self-consistent mean field approach to studying the dynamics of quantum fields. In this study, we first note that any variational approximation to the dynamics of a quantum system based on the Dirac action principle leads to a classical Hamiltonian dynamics for the variational parameters. Since this Hamiltonian is generically nonlinear and nonintegrable, the dynamics thus generated can be chaotic, in distinction to the exact quantum evolution. We then restrict our attention to a system of two biquadratically coupled quantum oscillators and study two variational schemes, the leading order large-NNN (four canonical variables) and Hartree (six canonical variables) approximations. The chaos seen in the approximate dynamics is an artifact of the approximations: this is demonstrated by the fact that its onset occurs on the same characteristic time scale as the breakdown of the approximations when compared to numerical solutions of the time-dependent Schr\"odinger equation.
The dynamics of systems consisting of coupled quantum–classical oscillators is numerically investigated. It is shown that, under certain conditions, the quantum oscillator exhibits chaos. When the mass of the classical oscillator increases, the chaos will be suppressed; if the energy of the system and/or the coupling strength between the two oscillators increases, chaotic behaviour of the system appears. This result will be helpful to understand the probability of the emergence of quantum chaos and may be applied to explain the spectra of complex atoms qualitatively.
The relation between the dynamical properties of a coupled quasiparticle-oscillator system in the mixed quantum-classical and fully quantized descriptions is investigated. The system is considered as a model for applying a stepwise quantization. Features of the nonlinear dynamics in the mixed description such as the presence of a separatrix structure or regular and chaotic motion are shown to be reflected in the evolu- tion of the quantum state vector of the fully quantized system. In particular, it is demonstrated how wave packets propagate along the separatrix structure of the mixed description, and that chaotic dynamics leads to a strongly entangled quantum state vector. Special emphasis is given to viewing the system from a dyn- amical Born-Oppenheimer approximation defining integrable reference oscillators, and elucidating the role of the nonadiabatic couplings which complement this approximation into a rigorous quantization scheme.