Equation for a Damped Quantum Oscillator and its Equivalence to the Semiclassical Fokker-Planck Equation

R. Bonifacio, Fritz Haake

Marcelo A. Despósito, Silvina M. Gatica, E. Susana Hernández

We analyze the irreversible behavior of a quantum harmonic oscillator immersed in an arbitrary thermal reservoir. For the fully coupled (FC) model, i.e., the coupling between subsystems linear in the coordinate of the oscillator, we construct a generalized master equation for the reduced density operator of the oscillator. We apply the rotating-wave approximation (RWA) and investigate its connection with the FC master equation. By means of the Wigner distribution, we obtain the corresponding Fokker-Planck equations in the semiclassical representation. We find that, when the RWA is not imposed, the correct classical limit is obtained (Kramer's equation). Then, we establish the conditions for the equivalence between the FC Fokker-Planck equation and that obtained in the RWA. Quantum maps for a kicked oscillator immersed in a heat bath are obtained in both cases. These maps are studied in the classical and semiclassical limits, and it is shown that they coincide for low kicking frequency compared to the damping rate.

W. H. Louisell, John H. Marburger

The quantum theory of damping is presented and illustrated by means of a driven damped harmonic oscillator. The theory is formulated in the coherent state representation which illustrates very vividly the nearly classical nature of the problem. In this representation the reduced system density operator equation becomes a Fokker-Planck equation. Green's function solutions are found for the damped oscillator in closed form and as an eigenfunction expansion. In addition, a quantum regression theorem due to Lax is derived in the coherent state representation. The theorem allows two-time averages to be computed from one-time averages.

Vladimir I. Man’ko, Sergey S. Safonov

Ye‐Jun Xu, Chao Li, Yu-hang Ma +1 more

Abstract We exploit the so-called characteristic operator method to derive the Fokker–Planck equation in the Wigner representation. We introduce the characteristic operator of the Wigner representation and establish a set of fundamental differential equations for the action of the creation and annihilation operators on it. As an example, we show the derivation process of the Fokker–Planck equation for the damped harmonic oscillator in the Wigner representation by using the characteristic operator approach. This method provides us with a new convenient approach to deriving the phase-space Fokker–Planck equation in the Wigner representation.

Takeshi Saito, Toshihico Arimitsu