The averaging method for asymptotic evolutions. II. Quantum open systems (Q1065236)

The paper is the second part of a work whose first part, by the same authors, appeared in Adv. Appl. Math. 2, 456-481 (1981; Zbl 0484.34041). A variant of the averaging method of Bogoljubov and Mitropolskij is applied to derived successive approximations to the reduced evolution of a quantum system coupled to a quasi-free reservoir whose correlation functions are exponentially decreasing in time. These approximate evolutions are of the form \[ y_ n(t)=[1+\sum^{n- 2}_{k=1}\lambda^{2k}M^{(2k)}(t)]\exp [\sum^{n}_{k=\quad 1}\lambda^{2k}G^{(2k)}t]y(0), \] where y(0) is the initial condition, \(\lambda\) measures the strength of the coupling between system and reservoir, the \(G^{(2k)}\) are time-independent operators, and the operators \(M^{(2k)}(t)\) vanish at \(t=0\) and satisfy \(\lim_{t\to \infty}t^{-1}M^{(2k)}(t)=0\) for all k. Explicit expressions and error bounds are given for \(n=2\) and \(n=4\). For \(n=2\), the result coincides with the familiar semigroup approximation obtained in the weak coupling limit \(\lambda\) \(\to 0\), \(t\to \infty\), \(\lambda^ 2t\) constant. Beyond second order in \(\lambda\), a semigroup approximation is no longer adequate, the operators \(M^{(2k)}(t)\) providing ''non-Markovian'' corrections. However, the time averages of the \(M^{(2k)}\) may be used to define a modified initial condition from which to start an approximate semigroup evolution (''initial slip'').