Quantum ergodicity breaking in semi-classical electron transfer dynamics

Can the statistical properties of single-electron transfer events be correctly predicted within a common equilibrium ensemble description? This fundamental in nanoworld question of ergodic behavior is scrutinized within a very basic semi-classical curve-crossing problem. It is shown that in the limit of non-adiabatic electron transfer (weak tunneling) well-described by the Marcus-Levich-Dogonadze (MLD) rate the answer is yes. However, in the limit of the so-called solvent-controlled adiabatic electron transfer, a profound breaking of ergodicity occurs. Namely, a common description based on the ensemble reduced density matrix with an initial equilibrium distribution of the reaction coordinate is not able to reproduce the statistics of single-trajectory events in this seemingly classical regime. For sufficiently large activation barriers, the ensemble survival probability in a state remains nearly exponential with the inverse rate given by the sum of the adiabatic curve crossing (Kramers) time and the inverse MLD rate. In contrast, near to the adiabatic regime, the single-electron survival probability is clearly non-exponential, even though it possesses an exponential tail which agrees well with the ensemble description. Initially, it is well described by a Mittag-Leffler distribution with a fractional rate. Paradoxically, the mean transfer time in this classical on the ensemble level regime is well described by the inverse of the nonadiabatic quantum tunneling rate on a single particle level. An analytical theory is developed which perfectly agrees with stochastic simulations and explains our findings.