Damped Quantum Harmonic Oscillator

In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. 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 and Heisenberg 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 the majority of these equations are not satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are also obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions. A comparative study is made for the Glauber P representation, the antinormal ordering Q representation and the Wigner W representation. It is proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker-Planck equations in the steady state, it is shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided. 1. Introduction In the last two decades, more and more interest arose about the problem of dissipation in quantum mechanics, i.e. the consistent description of open quantum systems [1-4].The quantum description of dissipation is important in many different areas of physics. In quantum optics we mention the quantum theory of lasers and photon detection. There are some directions in the theory of atomic nucleus in which dissipative processes play a basic role. In this sense we mention the nuclear fission, giant resonances and deep inelastic collisions of heavy ions. Dissipative processes often occur also in many body or field-theoretical systems. The irreversible, dissipative behaviour of the vast majority of physical phenomena comes into an evident contradiction with the reversible nature of our basic models. The very restrictive principles of conservative and isolated systems are unable to deal with more complicated situations which are determined by the features of open systems. The fundamental quantum dynamical laws are of the reversible type. The dynamics of