Distribution of waiting times between electron cotunneling events

Abstract

In the resonant tunneling regime sequential processes dominate single electron transport through quantum dots or molecules that are weakly coupled to macroscopic electrodes. In the Coulomb blockade regime, however, cotunneling processes dominate. Cotunneling is an inherently quantum phenomenon and thus gives rise to interesting observations, such as an increase in the current shot noise. Since cotunneling processes are inherently fast compared to the sequential processes, it is of interest to examine the short time behaviour of systems where cotunneling plays a role, and whether these systems display nonrenewal statistics. We consider three questions in this paper. Given that an electron has tunneled from the source to the drain via a cotunneling or sequential process, what is the waiting time until another electron cotunnels from the source to the drain? What are the statistical properties of these waiting time intervals? How does cotunneling affect the statistical properties of a system with strong inelastic electron-electron interactions? In answering these questions, we extend the existing formalism for waiting time distributions in single electron transport to include cotunneling processes via an n-resolved Markovian master equation. We demonstrate that for a single resonant level the analytic waiting time distribution including cotunneling processes yields information on individual tunneling amplitudes. For both a SRL and an Anderson impurity deep in the Coulomb blockade there is a nonzero probability for two electrons to cotunnel to the drain with zero waiting time inbetween. Furthermore, we show that at high voltages cotunneling processes slightly modify the nonrenewal behaviour of an Anderson impurity with a strong inelastic electronelectron interaction.

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.…”

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

“…There are tunneling events that do not appear in the standard master equation, yet change the drain number n. To include these events in the WTD, one must use the n-resolved master equation and the definition of the WTD from the idle time probability 36,44,53,77 :…”

“…The WTD in quantum electron transport can be calculated via scattering theory [33][34][35][44][45][46][47] and non-equilibrium Green's functions 48,49 . In order to connect it with other fluctuating time statistics, however, we focus on WTDs calculated from quantum master equations, which rely on quantum jump operators defined from the Liouvillian 4,14,15,36,37,[50][51][52][53][54][55][56][57][58][59] . Unfortunately, WTDs calculated in this manner are so far only able to reproduce I k (k > 1), as according to renewal theory, for unidirectional transport.…”

“…In some cases, furthermore, the processes that contribute to the current do not appear in the Liouvillian; for example, elastic cotunneling transfers electrons from the source to the drain in the same quantum process without altering the state of the intermediate system 17,[60][61][62] . To fully describe the transport in these cases, one must use the n-resolved master equation 63 ; but waiting times calculated from this method 49,53,57 are only defined for unidirectional transport 36 . Bidirectional transitions play an important role in mesoscopic electron transport 64 , so it is desirable to have a fluctuating time statistic applicable to this regime.…”

Counting quantum jumps: A summary and comparison of fixed-time and fluctuating-time statistics in electron transport

“…Alternatively, one could repeatedly measure the time τ it takes for the number of measured quantum jumps to reach n and construct a probability density distribution P (τ (n)), as we demonstrate in Fig.(1). The first quantity, n(t), is an example of a fixed-time statistic [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], while τ (n) is an example of a fluctuating-time statistic [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. Considering the important role time-dependent fluctuations have, analysis of quantum fluctuations has therefore been focused on calculating either fixed-time and fluctuating-time statistics, and exploring the relationship between the two [42][43][44][45].…”