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Journal of Contemporary Physics (Armenian Academy of Sciences)

Quantum distributions for localized quantum states

T. V. Gevorgyan

Published Jan 4, 2010• 0 citations
Publisher

Abstract

The principle of ergodicity of the quantum theory has been used for elaboration of a new technique for numerical simulation of the Wigner function of open dissipative quantum systems. With this purpose the density matrix of a quantum system is represented via averaging over the ensemble of quantum states in time intervals instead of averaging over the ensemble of stochastic variables. It is shown that this approach leads to new approximate expressions for quantum distributions in the phase space, in particular, Wigner functions for systems localized in the region of classical phase trajectories. As an application, the Wigner functions are calculated for the process of intracavity second harmonic generation in the region of Hopf bifurcations.

DOI
10.3103/s1068337210010020

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The principle of ergodicity of the quantum theory has been used for elaboration of a new technique for numerical simulation of the Wigner function of open dissipative quantum systems. With this purpose the density matrix of a quantum system is represented via averaging over the ensemble of quantum states in time intervals instead of averaging over the ensemble of stochastic variables. It is shown that this approach leads to new approximate expressions for quantum distributions in the phase space, in particular, Wigner functions for systems localized in the region of classical phase trajectories. As an application, the Wigner functions are calculated for the process of intracavity second harmonic generation in the region of Hopf bifurcations.
The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of the strongly interacting systems of particles has been developed for extreme conditions, when there are no small physical parameters and analytical approximations used in different kind of perturbation theories can not be applied. The new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in linear and harmonic approximations. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Preliminary calculations of the momentum distribution function of the Coulomb systems of particles have been carried out. Comparison with classical Maxwell-Boltzmann distribution shows the significant influence of quantum effects on the high energy asymptotics ("tails') of the calculated momentum distribution functions, which resulted in appearance of sharp oscillations.
This paper is concerned with certain extensions of a formal technique devised by Wigner for handling problems in quantum-statistical mechanics, especially to problems in quantum-mechanical transport processes. The approach is to find the closest possible analogy between classical and quantum-statistical mechanics, so that the extensive work in classical statistical mechanics can be utilized. This analogy is attained with the Wigner distribution function, with which averages of dynamical variables in quantum mechanics may be calculated by integrations in phase space. We will first state some basic properties of distribution functions in classical statistical mechanics, and then state the corresponding properties of the density matrix in quantum mechanics. We will define and discuss the Wigner distribution function, show that it has the desired averaging properties, and obtain the analog of the Liouville equation satisfied by this function. We will derive the analog of the Liouville equation in reduced phase space, and then obtain the equations of hydro-dynamics from quantum-statistical mechanics.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad's theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function is also solved by using the Wang-Uhlenbeck method of transforming it into a linearized partial differential equation for the Wigner function, subject to either the Gaussian type or the δδδ-function type of initial conditions. The Wigner functions which we obtain are two-dimensional Gaussians with different widths.
In this paper we formulate a new stochastic description of quantum mechanics in phase space. The theory of phase-space representations of quantum mechanics, initiated by Wigner, Groenewold, and Moyal and systematized recently by Agarwal and Wolf is essentially a single-time theory, in that it deals only with the quantum-mechanical joint distribution functions for position and momentum at a single instant of time. We develop a natural multitime extension of such a single-time theory. We consider a class of multitime phase-space distribution functions such that an arbitrary quantum multitime correlation function can be expressed as a phase-space average of the form encountered in classical stochastic theories. We study the nonclassical features of these multitime distribution functions and show that they may be considered as characterizing a generalized stochastic process in phase space. We demonstrate that the multitime distribution functions that correspond to Hamiltonian evolution of isolated quantum systems satisfy a certain condition that may be regarded as characterizing a generalized Markov process. We also investigate certain special features of the generalized stochastic processes that characterize the evolution of open systems.
In this paper we formulate a new stochastic description of quantum mechanics in phase space. The theory of phase-space representations of quantum mechanics, initiated by Wigner, Groenewold, and Moyal and systematized recently by Agarwal and Wolf is essentially a single-time theory, in that it deals only with the quantum-mechanical joint distribution functions for position and momentum at a single instant of time. We develop a natural multitime extension of such a single-time theory. We consider a class of multitime phase-space distribution functions such that an arbitrary quantum multitime correlation function can be expressed as a phase-space average of the form encountered in classical stochastic theories. We study the nonclassical features of these multitime distribution functions and show that they may be considered as characterizing a generalized stochastic process in phase space. We demonstrate that the multitime distribution functions that correspond to Hamiltonian evolution of isolated quantum systems satisfy a certain condition that may be regarded as characterizing a generalized Markov process. We also investigate certain special features of the generalized stochastic processes that characterize the evolution of open systems.