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Variational Principle for Classical-Quantum Systems

M. Grigorescu

Published Oct 2, 2006• 1 citation
OpenAlex

Abstract

The evolution of a quantum particle interacting with a classical system is described by a generalized variational principle. The dynamical variable is a quantum state vector which includes the classical action as a phase factor, and the common time is treated as a collective variable. Combined with the model of bilinear coupling, the variational principle is applied to the problem of a quantum system in a thermal environment. It is shown that the statistical ensemble of Brownian state vectors is described by the solution of a nonlinear quantum Fokker-Planck equation for the density matrix. Exact solutions of this equation are obtained for the case of a two-level system, considering both stationary and nonstationary initial states.

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The evolution of a quantum particle interacting with a classical system is described by a generalized variational principle. The dynamical variable is a quantum state vector which includes the classical action as a phase factor, and the common time is treated as a collective variable. Combined with the model of bilinear coupling, the variational principle is applied to the problem of a quantum system in a thermal environment. It is shown that the statistical ensemble of Brownian state vectors is described by the solution of a nonlinear quantum Fokker-Planck equation for the density matrix. Exact solutions of this equation are obtained for the case of a two-level system, considering both stationary and nonstationary initial states.
An extended variational principle providing the equations of motion for a system consisting of interacting classical, quasiclassical and quantum components is presented, and applied to the model of bilinear coupling. The relevant dynamical variables are expressed in the form of a quantum state vector which includes the action of the classical subsystem in its phase factor. It is shown that the statistical ensemble of Brownian state vectors for a quantum particle in a classical thermal environment can be described by a density matrix evolving according to a nonlinear quantum Fokker-Planck equation. Exact solutions of this equation are obtained for a two-level system in the limit of high temperatures, considering both stationary and nonstationary initial states. A treatment of the common time shared by the quantum system and its classical environment, as a collective variable rather than as a parameter, is presented in the Appendix. PACS 03.65.-w, 03.65.Sq, 05.30.-d, 45.10.Db 1 I.
An extended variational principle providing the equations of motion for a system consisting of interacting classical, quasiclassical, and quantum components is presented, and applied to the model of bilinear coupling. The relevant dynamical variables are expressed in the form of a quantum state vector that includes the action of the classical subsystem in its phase factor. It is shown that the statistical ensemble of Brownian state vectors for a quantum particle in a classical thermal environment can be described by a density matrix evolving according to a nonlinear quantum Fokker–Planck equation. Exact solutions of this equation are obtained for a two-level system in the limit of high temperatures, considering both stationary and nonstationary initial states. A treatment of the common time shared by the quantum system and its classical environment as a collective variable, rather than as a parameter, is presented in the Appendix. PACS Nos.: 03.65.–w, 03.65.Sq, 05.30.–d, 45.10.Db
The Schrodinger equation with Gaussian dynamic fluctuating potentials is considered. This model gives a convenient example of an open quantum system coupled to a given external environment. The evolution equations for averaged correlators are derived and it is shown that a system coupled to a thermoequilibrium environment tends asymptotically with time to a thermodynamic density matrix. At high temperatures the quantum evolution equation for the density matrix can be approximated by the quasiclassical Fokker-Planck equation. The generalisation for multiparticle systems is also given. The results are illustrated by the particular examples of coupling with phonons and of two-level systems. Finally, it is shown how the problem can be reformulated in terms of continuous stochastic evolution in the space of psi functions and a corresponding functional formalism based on the continuous Fokker-Planck equation in the space of psi functions.
: The Brownian dynamics of the density operator for a quantum system interacting with a classical heat bath is described using a stochastic, non-linear Liouville equation obtained from a variational principle. The rate of the noise-induced transitions is expressed as a function of the environmental spectral density, and is discussed for the case of white noise and blackbody radiation. The time-scale of decoherence and dissipation is obtained numerically for a system of two quantum states. These are the ground and first excited state of the center of mass vibrations for a single ion confined in a harmonic trap. PACS numbers: 42.50.Lc,05.40.+j,32.80.Pj 1 Introduction The evolution of a quantum system surrounded by a thermal environment represents a fundamental problem of the quantum theory, increasingly important for applications. In the classical mechanics, a particle interacting with a heat bath has a Brownian evolution towards thermalization, which reduces to a purely dissipative moti...
It has long been recognized that the dynamics of linear quantum systems is classical in the Wigner representation. Yet many conceptually important linear problems are typically analyzed using such generally applicable techniques as influence functionals and Bogoliubov transformations. In this Letter we point out that the classical equations of motion provide a simpler and more intuitive formalism for linear quantum systems. We examine the important problem of Brownian motion in the independent oscillator model, and show that the quantum dynamics is described directly and completely by a c-number Langevin equation. We are also able to apply recent insights into quantum Brownian motion to show that the classical Fokker-Planck equation is always local in time, regardless of the spectral density of the environment.
Quantum and classical stochastic evolution equations are derived for the statistical state of a system subject to continuous observation of a position variable. In the classical case this results in a stochastic Fokker - Planck equation for the conditional state of the system conditioned on observation records. In the quantum case a stochastic master equation results, or a stochastic Schrödinger equation if the system starts in a pure state. In both cases the resulting equations are nonlinear. A simple modification of the model results in a linear stochastic Schrödinger equation which can also be interpreted as a conditional evolution equation for observations which give no information about the measured system but indicate the size of the random perturbations induced by the environment coupling.