On the master equation and pure state representations for the quantized damped oscillator

H. Dekker

Gang Ren, Jian-guo Ma, Jian-ming Du +1 more

Аурелиан Исар, A. Sãndulescu

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.

I. A. Pedrosa, B. Baseia

Аурелиан Исар, A. Sãndulescu, W. Scheid

For the generalized master equations derived by Karrlein and Grabert for the microscopic model of a damped harmonic oscillator, the conditions for purity of states are written, in particular for different initial conditions and different types of damping, including Ohmic, Drude, and weak coupling cases, and the Agarwal and Weidlich-Haake models. It is shown that the states which remain pure are the squeezed states with variances that are constant in time. For pure states, generalized nonlinear Schrödinger-type equations corresponding to these master equations are also obtained. Then the condition for purity of states of a damped harmonic oscillator is considered in the framework of Lindblad theory for open quantum systems. For a special choice of the environment coefficients, correlated coherent states with constant variances and covariance are shown to be the only states which remain pure all the time during the evolution of the considered system. In Karrlein-Grabert and Lindblad models, as well as in the particular models considered, expressions for the rate of entropy production are written, and it is shown that state which preserve their purity in time are also states which minimize entropy production and, therefore, are the most stable state under evolution in the presence of the environment, and play an important role in the description of decoherence phenomenon.

Mario Cesar Baldiotti, R. Fresneda, Dmitri M Gitman