Quantum dissipation in open harmonic systems - Operator solution and application to decay process

Study of a system of a finite number of harmonic oscillators coupled to an environment consisting of infinitely many oscillators is fundamental to the problem of understanding quantum dissipation of a small system immersed in a large environment. The exact operator solution as a function of time is found, by using a diagonalized dynamical variable representing the entire system, the small system plus the environment. The decay law of the prepared initial configuration in the medium is worked out in great detail. A clear separation of the exponential and 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. The final abundance of decaying particles has the power dependence of the environment temperature at low temperatures, without the Boltzmann suppression factor.