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arXiv:quant-ph/9605016

Generalized Moyal structures in phase space, kinetic equations and their classical limit: II. Applications to harmonic oscillator models

Constantinos Tzanakis, A.P. Grécos, Polyxeni Hatjimanolaki

Published May 14, 1996• 0 citations
arXivHTMLPDF

Abstract

The formalism of generalized Wigner transformations developped in a previous paper, is applied to kinetic equations of the Lindblad type for quantum harmonic oscillator models. It is first applied to an oscillator coupled to an equilibrium chain of other oscillators having nearest-neighbour interactions. The kinetic equation is derived without using the so called rotating-wave approximation. Then it is shown that the classical limit of the corresponding phase-space equation is independent of the ordering of operators corresponding to the inverse of the generalized Wigner transformation, provided the latter is involutive. Moreover, this limit equation, which conserves the probabilistic nature of the distribution function and obeys an H-theorem, coincides with the kinetic equation for the corresponding classical system, which is derived independently and is distinct from that usually obtained in the litterature and not sharing the above properties. Finally the same formalism is applied to more general model equations used in quantum optics and it is shown that the above results remain unaltered.

DOI
10.48550/arxiv.quant-ph/9605016

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The formalism of generalized Wigner transformations developped in a previous paper, is applied to kinetic equations of the Lindblad type for quantum harmonic oscillator models. It is first applied to an oscillator coupled to an equilibrium chain of other oscillators having nearest-neighbour interactions. The kinetic equation is derived without using the so called rotating-wave approximation. Then it is shown that the classical limit of the corresponding phase-space equation is independent of the ordering of operators corresponding to the inverse of the generalized Wigner transformation, provided the latter is involutive. Moreover, this limit equation, which conserves the probabilistic nature of the distribution function and obeys an H-theorem, coincides with the kinetic equation for the corresponding classical system, which is derived independently and is distinct from that usually obtained in the litterature and not sharing the above properties. Finally the same formalism is applied to more general model equations used in quantum optics and it is shown that the above results remain unaltered.
Generalised Wigner and Weyl transformations of quantum operators are defined and their properties, as well as those of the algebraic structure induced on the phase-space are studied. Using such transformations, quantum linear evolution equations are given a phase-space representation. In particular this is done for the general kinetic equation of the Lindblad type. The resulting expressions are better suited for the passage to the classical limit and for a general comparison of classical and quantum systems. In this context a preliminary discussion of a number of problems of kinetic theory of open systems is given, whereas explicit applications are made in the next paper of the series.
A new approach to perturbation theory in the quantum phase-space formalism is proposed, in order to devise some approximated description of the quantum phase-space dynamics, which is not directly related to the usual semi-classical approximation. A general class of equivalent quasi-distribution functions based on the Wigner-Moyal formalism is considered and a first-order invariant formulation of the dynamics is obtained. The relationship between the various phase-space representation is expressed in term of a pseudo-differential operator defined by the Moyal product. In particular, our theory is applied to the sub-class of representations obtained by a first order perturbation of the Wigner representation. Finally the connection of our approach with some well established gauge-invariant formulation of the Wigner dynamics in the presence of an external magnetic field is investigated.
The relation between the Hilbert space structure and the generalized spaces structure represented by dual states for dissipative kinetic equation is discussed for quantum systems. As working examples, we consider the systems of a harmonic oscillator or a particle interacting with a thermal reservoir and construct analytic solutions to the eigenvalue problem of the quantum collision operators of these systems. The generalized spaces structure of the eigenfunctions indicates that dissipation destroys the Hilbert space structure of the undamped system. In the Wigner representation where the quantum collision operators closely resemble the classical kinetic operators in phase space, the Hilbert space structure can be restored to certain extent by introducing a weighted norm or a similarity transformation on the operators. However, in the position space where the collision operators have no classical counterpart, generalized spaces description cannot be avoided.
The inversion method to derive the quantum kinetic equation is extended to the case of an inhomogeneous system. Within the closed-time-path formalism, the source is introduced to probe the (Fourier transform of the) Wigner distribution function and the equation of motion for the Wigner function is derived perturbatively by the inversion formulae. The equation of motion has the form of the generalized Boltzmann equation, which is non-local in space and time. For the obtained kinetic equation, the local conservation laws of number, energy and momentum are confirmed.
The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger, Heisenberg and Weyl-Wigner-Moyal representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that not all of these equations are satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions and a comparative study is made for the Glauber P representation, the antinormal ordering Q representation, and the Wigner W representation. The density matrix is represented via a generating function, which is obtained by solving a timedependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided. The solution of the master equation in the Weyl-Wigner-Moyal representation is of Gaussian type if the initial form of the Wigner function is taken to be a Gaussian corresponding (for example) to a coherent wavefunction. The damped harmonic oscillator is applied for the description of the charge equilibration mode observed in deep inelastic reactions. For a system consisting of two harmonic oscillators the time dependence of expectation values, Wigner function and Weyl operator, are obtained and discussed. In addition models for the damping of the angular momentum are studied. Using this theory to the quantum tunneling through the nuclear barrier, besides Gamow’s transitions with energy conservation, additional transitions with energy loss are found. The tunneling spectrum is obtained as a function of the barrier characteristics. When this theory is used to the resonant atom-field interaction, new optical equations describing the coupling through the environment of the atomic observables are obtained. With these equations, some characteristics of the laser radiation absorption spectrum and optical bistability are described.
Abstract We derive the semi-classical Lindblad master equation in phase space for both canonical and non-canonical Poisson brackets using the Wigner–Moyal formalism and the Moyal star-product. The semi-classical limit for canonical dynamical variables, i.e. canonical Poisson brackets, is the Fokker–Planck equation, as derived before. We generalize this limit and show that it holds also for non-canonical Poisson brackets. Examples are gyro-Poisson brackets, which occur in spin ensembles, systems of recent interest in atomic physics and quantum optics. We show that the equations of motion for the collective spin variables are given by the Bloch equations of nuclear magnetization with relaxation. The Bloch and relaxation vectors are expressed in terms of the microscopic operators: the Hamiltonian and the Lindblad functions in the Wigner–Moyal formalism.