Damped quantum harmonic oscillator density operator and related quantities.pdf

a r X i v : h e p - t h / 9 4 0 6 1 4 2 v 1 2 2 J u n 1 9 9 4 IFA-FT-396-1994 Damped quantum harmonic oscillator: density operator and related quantities A. Isar Department of Theoretical Physics, Institute of Atomic Physics POB MG-6, Bucharest-Magurele, Romania Internet: isar@ifa.ro Abstract A closed expression for the density operator of the damped harmonic oscillator is extracted from the master equation based on the Lindblad theory for open quantum systems. The entropy and effective temperature of the system are subsequently calcu- lated and their temporal behaviour is surveyed by showing how these quantities relax to their equilibrium values. The entropy for a state characterized by a Wigner distri- bution function which is Gaussian in form is found to depend only on the variance of the distribution function. Shortened title: Damped quantum harmonic oscillator PACS numbers: 02.50.Ga, 05.20, 05.40.+j 1 Introduction In the last two decades, the problem of dissipation in quantum mechanics, i.e. the consis- tent description of open quantum systems, was investigated by various authors [1-5] (for a recent review on open quantum systems see ref. [6]). It is commonly understood [4, 7] that dissipation in an open system results from microscopic reversible interactions between the observable system and the environment. Because dissipative processes imply irreversibility and, therefore, a preferred direction in time, it is generally thought that quantum dynamical semigroups are the basic tools to introduce dissipation in quantum mechanics. In the Markov approximation the most general form of the generators of such semigroups was given by Lind- blad [8]. This formalism has been studied for the case of damped harmonic oscillators [9, 10] and applied to various physical phenomena, for instance, the damping of collective modes in deep inelastic collisions in nuclear physics [11]. In [12] the Lindblad master equation for the harmonic oscillator was transformed into Fokker