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SEMIQUANTUM CHAOS AND THE LARGE N EXPANSION

Fred Cooper, S. Habib, Y. Kluger, J. Dawson, Dawn C. Meredith, Harvey K. Shepard

Published Jan 1, 1994• 3 citations
OpenAlex

Abstract

We consider the dynamical system consisting of a quantum degree of freedom A interacting with N quantum oscillators described by the Lagrangian L = 1 2 ˙ A 2 +

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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.
We study the nonequilibrium dissipative dynamics of the center of mass of a particle coupled to a field via its internal degrees of freedom. We model the internal and external degrees of freedom of the particle as quantum harmonic oscillators in 1+1 D1+1\text{ }\mathrm{D}1+1 D, with the internal oscillator coupled to a scalar quantum field at the center of mass position. Such a coupling results in a nonlinear interaction between the three pertinent degrees of freedom---the center of mass, internal degree of freedom, and the field. It is typically assumed that the internal dynamics is decoupled from that of the center of mass owing to their disparate characteristic timescales. Here we use an influence functional approach that allows one to account for the self-consistent backaction of the different degrees of freedom on each other, including the coupled nonequilibrium dynamics of the internal degree of freedom and the field, and their influence on the dissipation and noise of the center of mass. Considering a weak nonlinear interaction term, we employ a perturbative generating functional approach to derive a second order effective action and a corresponding quantum Langevin equation describing the non-equilibrium dynamics of the center of mass. We analyze the resulting dissipation and noise arising from the field and the internal degree of freedom as a composite environment. Furthermore, we establish a generalized fluctuation-dissipation relation for the late-time dissipation and noise kernels. Our results are pertinent to open quantum systems that possess intermediary degrees of freedom between system and environment, such as in the case of optomechanical interactions.
Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behavior of expectation values of simple observables and of eigenvalues of the Groenewold operator are calculated numerically and compared for the various semiclassical and semiquantum approximations.