Nonrenewal statistics in quantum transport from the perspective of first-passage and waiting time distributions

Abstract

The waiting time distribution has, in recent years, proven to be a useful statistical tool for characterising transport in nanoscale quantum transport. In particular, as opposed to moments of the distribution of transferred charge, which have historically been calculated in the long-time limit, waiting times are able to detect non-renewal behaviour in mesoscopic systems. They have failed, however, to correctly incorporate backtunneling events. Recently, a method has been developed that can describe unidirectional and bidirectional transport on an equal footing: the distribution of firstpassage times. Rather than the time between successive electron tunnelings, the first-passage refers to the first time the number of extra electrons in the drain reaches +1. Here, we demonstrate the differences between first-passage time statistics and waiting time statistics in transport scenarios where the waiting time either cannot correctly reproduce the higher order current cumulants or cannot be calculated at all. To this end, we examine electron transport through a molecule coupled to two macroscopic metal electrodes. We model the molecule with strong electron-electron and electron-phonon interactions in three regimes: (i) sequential tunneling and cotunneling for a finite bias voltage through the Anderson model, (ii) sequential tunneling with no temperature gradient and a bias voltage through the Holstein model, and (iii) sequential tunneling at zero bias voltage and a temperature gradient through the Holstein model. We show that, for each transport scenario, backtunneling events play a significant role; consequently, the waiting time statistics do not correctly predict the renewal and non-renewal behaviour, whereas the first-passage time distribution does.

Waiting time distributions in a two-level fluctuator coupled to a superconducting charge detector

“…Electron waiting times have been investigated for a wide range of physical systems including quantum dots, [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] coherent conductors, 36,37 molecular junctions, 38,39 and superconducting systems. [40][41][42][43][44][45][46] Distributions of waiting times contain complementary information on charge transport properties which is not necessarily encoded in the full counting statistics (FCS) and vice versa.…”

Waiting Time Distributions of Transport through a Two-Channel Quantum System

“…The standard expression for the WTD of the two-channel QD system in the time domain is [19,25,26,34]:…”

“…The difference between single and multiple reset QD systems is that in single reset systems, the dot is empty after an electron tunnels out, whereas in a multiple reset system, the dot is empty only after a certain number of electrons has tunneled out. In more complicated systems, like the Anderson-Holstein model, an electron tunneling out can leave the system in various vibrational states, and there are thus multiple possibilities for the state after the electron jumps, even if there are only one or zero electrons involved in the transport [24,25]. Recently, Kosov et al studied the effects of cotunneling on the WTDs of an Anderson single-impurity model in a Coulomb blockade regime and tunneling regime and discovered nonrenewal statistics, which are the short-time correlations between subsequent waiting times and are invisible in the current cumulants [26].…”

Fluctuating-time and full counting statistics for quantum transport in a system with internal telegraphic noise

“…The individual J ϕ F and J ϕ B are also easily defined, as in Ref. [50]. At this point the full Liouvillian remains too large to be written in matrix form, since we have made no assumptions about the underlying phonon distribution.…”

“…The FPTD F (n|τ ) is the conditional probability density that, given an electron has tunneled to the drain, the jump number first reaches n after a time-delay τ . Since the jump number is the total number of forward and backward transitions, it directly relates to bidirectional current, whereas the WTD only works for unidirectional transport 23,[50][51][52] .…”