Quantum Dissipation in Open Harmonic Systems: Operator Solution

A finite number of harmonic oscillators coupled to infinitely many environment oscillators is fundamental to the problem of understanding quantum dissipation of a small system immersed in a large environment. Exact operator solution as a function of time is given to this problem, by using diagonalized dynamical variable of the entire system, the small system plus the environment. The decay law of prepared initial configuration is worked out in greatest detail. A clear separation of the exponential- and the power-law decay period is made possible by our method. Behavior of physical quantities at asymptotically late times can be understood in terms of the overlap probability of the system variable with the diagonal variable of the entire system.