Quantum breaking of ergodicity in semi-classical charge transfer dynamics

TL;DR

This paper investigates the ergodic behavior of electron transfer kinetics, revealing that quantum effects cause ergodicity breaking in the adiabatic regime, with implications for understanding charge transfer dynamics.

Contribution

It demonstrates the fundamental breaking of ergodicity in semi-classical charge transfer dynamics under adiabatic conditions, contrasting with non-adiabatic regimes.

Findings

Ergodicity holds in non-adiabatic electron transfer regimes.

In adiabatic regimes, single-electron survival probabilities are non-exponential.

Mean transfer times align with nonadiabatic quantum tunneling rates.

Abstract

Does electron transfer (ET) kinetics within a single-electron trajectory description always coincide with the ensemble description? This fundamental question of ergodic behavior is scrutinized within a very basic semi-classical curve-crossing problem of quantum Landau-Zener tunneling between two electronic states with overdamped classical reaction coordinate. 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. The ensemble survival probability remains nearly exponential with the inverse rate given by the sum of the adiabatic curve crossing (Kramers) time and inverse MLD rate. However, near to adiabatic regime, the single-electron survival probability is clearly non-exponential but possesses an exponential tail which agrees well with the ensemble description. Paradoxically, the mean transfer time in this classical on the ensemble level regime is well described by the inverse of nonadiabatic quantum tunneling rate on a single particle level.