Non-Markovian stochastic Liouville equation and anomalous relaxation kinetics

The kinetics of phase and population relaxation in quantum systems induced by noise with anomalously slowly decaying correlation function P (t) ∼ (wt)−α, where 0 < α < 1 is analyzed within continuous time random walk approach (CTRWA). The relaxation kinetics is shown to be anomalously slow. Moreover for α < 1 in the limit of short characteristic time of fluctuations w the kinetics is independent of w. As α → 1 the relaxation regime changes from the static limit to fluctuation narrowing. Simple analytical expressions are obtained describing the specific features of the kinetics. Introduction. The noise induced relaxation in quantum systems is very important process observed in: magnetic resonance [1], quantum optics and nonlinear spectroscopy [2], etc. Often these processes are analyzed assuming conventional stochastic properties of the noise: fast decay of correlation functions and short correlation time τc [1]. In the absence of memory the relaxation is described by very popular Bloch-type equations. As for memory effects, they are also discussed (within the Zwanzig projection operator approach [3]), however, in the lowest orders in the fluctuating interaction V inducing relaxation [4]. Recently strong attention has been drawn to the processes governed by noises with anomalously slowly decaying correlation functions P (t) ∼ t−α with α < 1. They are discussed in relation to spectroscopic studies of quantum dots ([5, 6] and references therein). Similar problems are analyzed in the theory of stochastic resonances [7]. Such anomalous processes cannot be properly described by methods based on expansion in powers of V . The goal of this work is to analyze the corresponding anomalous relaxation within the continuous time random walk approach (CTRWA) [8] with the use of the recently derived non-Markovian stochastic Liouville equation (SLE) [9] which enables one to describe relaxation kinetics without expansion in V . In some physically reasonable models it allows for describing the phase and population relaxation kinetics in analytical form even for multilevel systems. In particular, the kinetics is shown to be strongly nonexponential . General formulation. We consider noise induced relaxation in the quantum system whose dynamical evolution is governed by the hamiltonian H(t) = Hs + V (t), (1)