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 tunneling 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 the master equation approach, we compute the joint probability distribution for waiting times of two successive tunneling events. We show that the probability distribution is completely reset after each tunneling event if molecular vibrations are thermally equilibrated. If we treat vibrational dynamics exactly without imposing the equilibration constraint, the statistics of electron tunneling events become non-renewal. Non-renewal statistics between two waiting times τ₁ and τ₂ means that the density matrix of the molecule is not fully renewed after time τ₁ and the probability of observing waiting time τ₂ for the second electron transfer depends on the previous electron waiting time τ₁. The strong electron-vibration coupling is required for the emergence of the non-renewal statistics. We show that in the Franck-Condon blockade regime, extremely rare tunneling events become positively correlated.