Non-renewal statistics for electron transport in a molecular junction with electron-vibration interaction

Quantum transport of electrons through a molecule is a series of individual electron tunnelling events separated by stochastic waiting time intervals. We study the emergence of temporal correlations between successive waiting times for the electron transport in a vibrating molecular junction. Using master equation approach, we compute joint probability distribution for waiting times of two successive tunnelling events. We show that the probability distribution is completely reset after each tunnelling event if molecular vibrations are thermally equilibrated. If we treat vibrational dynamics exactly without imposing the equilibration constraint, the statistics of electron tunnelling events become non-renewal. Non-renewal statistics between two waiting times $\tau_1$ and $\tau_2$ means that the density matrix of the molecule is not fully renewed after time $\tau_1$ and the probability of observing waiting time $\tau_2$ for the second electron transfer depends on the previous electron waiting time $\tau_1$. The strong electron-vibration coupling is required for the emergence of the non-renewal statistics. We show that in Franck-Condon blockade regime the extremely rare tunnelling events become positively correlated.