A new method to treat open systems

Abstract

Let Q be a quantum system composed of two mutual ly interacting parts S and R. If only the dynamics S is of interest, we are natural ly led to the problem of deducing the evolution of the reduced statistical operator ~(t) of the open system S from the Von Neumann equation (or the master equation, if Q is not isolated) which gives the evolution of the full statistical operator W(t) of Q. Usually this goal is achieved by means of the projection technique (1) with the projection of ARGYRES and KELLEr (3). Such projection is characterized by the introduction of a (~ reference state ~) for the reservoir R, which a pr ior i can be chosen arbitrarily. Introducing such projection into the usual machinery one gets a generalized master equation (GME) for Q(t), i.e. an integro-differential equation with a kernel formally expressed as a power expansion in the coupling between S and R. However, the GME is meaningful only if one can truncate this expansion at a low order. This is possible when the state of R does never deviate appreciably from the reference state. Hence such state is chosen as the equil ibrium or stat ionary state of R (typically the initial state of R) which natural ly arises from each particular problem. However there are eases in which the state of the reservoir is changed by the interaction with S in a way relevant for the dynamics of S itself. Typical examples are laser ~)scillations and, under suitable conditions, co-operative spontaneous emission (superradiance and superfluorescence). In such cases the projection technique becomes unsuitable, because one must take into account several or rather infinite terms of the power expansion of the kernel of the GME. In this paper we very briefly sketch a different method which avoid the introduction of a projection operator and allows one to go far beyond the Born approximation of the GME, bypassing resummation difficulties. The details are given elsewhere (3). Let the statistical operator W(t) of the system Q = s % R obey the equation