Theory of nonstationary activated rate processes: Nonexponential kinetics

Abstract

We have explored a simple microscopic model to simulate a thermally activated rate process where the associated bath which comprises a set of relaxing modes is not in an equilibrium state. The model captures some of the essential features of non-Markovian Langevin dynamics with a fluctuating barrier. Making use of the Fokker-Planck description we calculate the barrier dynamics in the steady state and non-stationary regimes. The Kramers-Grote-Hynes reactive frequency has been computed in closed form in the steady state to illustrate the strong dependence of the dynamic coupling of the system with the relaxing modes. The influence of nonequilibrium excitation of the bath modes and its relaxation on the kinetics of activation of the system mode is demonstrated.We derive the dressed time-dependent Kramers rate in the nonstationary regime in closed analytical form which exhibits strong non-exponential relaxation kinetics of the reaction co-ordinate. The feature can be identified as a typical non-Markovian dynamical effect.

Modified Bloch equations in the presence of a nonstationary bath

“…where the amplitude c s µ (operator) and the phases φ s µ (c-number) are assumed to be randomly distributed [6]. The random distribution of phases and amplitudes in the stationary regime makes Eq.…”

“…The Hamiltonian is essentially a simpler quantum version of the model used in [6] with two-level atom as the system. The first term on the right-hand side describes the system mode with characteristic frequency ω 0 .…”

“…In the present paper we extend this treatment to a quantum optical context. Since the initial excitation creates a nonequilibrium energy density fluctuation distribution which imparts nonstationarity of the bath, it is expected that optical Bloch equations which take into account of both the coherent interaction and the relaxation processes within a simplified description of a twolevel scheme, are likely to be modified by the nonstationarity of the bath [6,7]. Based on a quantum version of the model we study this essential modification of the optical Bloch equations and explore some of the relevant consequences.…”

“…This nonequilibrium bath is effectively realized in terms of a semi-infinite dimensional broad-band reservoir which is subsequently kept in contact with a standard thermal bath which allows the nonthermal bath to relax with a characteristic time scale. The important separation of the time scales of fluctuations of the nonequilibrium and the thermal bath is that [6] the former remains effectively stationary on the fast correlation of the thermal noise. The model captures the essential features of a nonstationary quantum Markov process.…”

“…The essential underlying assumption about the bath, be it thermal or nonthermal, is that it is considered to be in a state of equilibrium throughout the process. Very recently a solvable microscopic model for a nonequilibrium bath has been proposed [6] to explore classically, the influence of an initial nonequilibrium excitation in a complex system on the relaxation of a specific quantity of interest. In the present paper we extend this treatment to a quantum optical context.…”