A novel approach to quantum transition rate theory using open quantum dynamics

Abstract

This work is about a quantum-mechanical generalisation of transition rate theory that accounts for temperature-dependent quantum-dynamical effects. The starting point of this investigation is a formulation of the classical transmission coefficient where the activated dynamics is needed to obtain the time evolution of the classical state function and the rate is expressed as an average flux in momentum space. Traditional approaches to quantum transition rate theory are briefly discussed, with emphasis on the fact that these do not generally require knowledge of the open quantum dynamics. The latter can be studied by means of the Lindblad equation for mixed states, which is shown to be analogous to an average over realisations of a certain type of stochastic Schrödinger dynamics. The stochastic evolution in Hilbert space is used in combination with an ansatz for the wave function as a Gaussian wave packet to show that under appropriate assumptions the stochastic Schrödinger equation converges to the classical Langevin equation. A novel formulation of the quantum transmission coefficient is presented which features a quantum-mechanical version of the state function obtained via the density operator, so that the activated quantum dynamics can be taken into account by solving the Lindblad equation. A numerical investigation of the quantum transmission coefficient thus obtained for different temperatures is carried out, along with a comparison to the classical results. The quantum-mechanical transmission coefficient is found to display some expected features, such as vanishing in the limit as t → 0, displaying a “plateau” value for certain values of the temperature, and decaying to zero exponentially fast when the fundamental assumptions of TRT do not hold. Additionally the plateau value of the quantum transmission coefficient is found to be smaller than the classical counterpart in all regimes, and to approach the corresponding classical value in the limit of large temperatures.