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Waiting Times and Noise in Single Particle Transport

Tobias Brandes

TL;DR

Waiting times offer a complementary lens to quantum transport, linking residence times between jumps to both noise spectra and full counting statistics within a generalized Master equation framework. The paper develops a formalism where the waiting-time matrix $\mathbf{W}(z)$ and relaxation currents encode richer information about coherence, internal state structure, and current-channel entropy, beyond what $S(\omega)$ and FCS alone reveal. Through diverse examples (rings, multi-level dots, double dots, and classical transport), it demonstrates how $w(\tau)$ detects tunnel-barrier specifics, reset structure, and dynamical coherences, and provides practical relations to FCS and noise. The approach offers a versatile diagnostic for transport measurements and suggests extensions to non-Markovian dynamics and higher-order waiting-time distributions.

Abstract

The waiting time distribution $w(τ)$, i.e. the probability for a delay $τ$ between two subsequent transition (`jumps') of particles, is a statistical tool in (quantum) transport. Using generalized Master equations for systems coupled to external particle reservoirs, one can establish relations between $w(τ)$ and other statistical transport quantities such as the noise spectrum and the Full Counting Statistics. It turns out that $w(τ)$ usually contains additional information on system parameters and properties such as quantum coherence, the number of internal states, or the entropy of the current channels that participate in transport.

Waiting Times and Noise in Single Particle Transport

TL;DR

Waiting times offer a complementary lens to quantum transport, linking residence times between jumps to both noise spectra and full counting statistics within a generalized Master equation framework. The paper develops a formalism where the waiting-time matrix and relaxation currents encode richer information about coherence, internal state structure, and current-channel entropy, beyond what and FCS alone reveal. Through diverse examples (rings, multi-level dots, double dots, and classical transport), it demonstrates how detects tunnel-barrier specifics, reset structure, and dynamical coherences, and provides practical relations to FCS and noise. The approach offers a versatile diagnostic for transport measurements and suggests extensions to non-Markovian dynamics and higher-order waiting-time distributions.

Abstract

The waiting time distribution , i.e. the probability for a delay between two subsequent transition (`jumps') of particles, is a statistical tool in (quantum) transport. Using generalized Master equations for systems coupled to external particle reservoirs, one can establish relations between and other statistical transport quantities such as the noise spectrum and the Full Counting Statistics. It turns out that usually contains additional information on system parameters and properties such as quantum coherence, the number of internal states, or the entropy of the current channels that participate in transport.

Paper Structure

This paper contains 18 sections, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Method
  3. Relation to the Noise Spectrum
  4. Relaxation Currents
  5. Single and Multiple Reset Systems
  6. High-frequency and short time expansions
  7. Relation to Full Counting Statistics (FCS)
  8. Information Contained in the Waiting Time Distribution, Examples
  9. Waiting Times for Two Different Splittings
  10. Entropy of Current Distribution
  11. Transitions in a Ring
  12. Single Level Quantum Dot
  13. Large Ring $N\to \infty$
  14. Multi-Level Single Dot
  15. Double Quantum Dot (Strong Coulomb Blockade)
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Waiting time distribution $w(\tau)$ for transport through double quantum dot at temperatures $T=4,8,16$ K at $\varepsilon=1$meV, $0.2$meV, $0.1$meV, $-1.0$meV (clockwise). Right tunnel rate $\hbar\Gamma_R=2.5\mu$eV, other parameters $\hbar\Gamma_L=0.1$meV, $T_c=0.1$meV, electron-phonon coupling parameter $g=0.002$, cutoff $\hbar\omega_c=5$meV.
  • Figure 2: Comparison between current entropy $E$ and waiting time parameter $\eta$ as a function of $\gamma_R/\Gamma_L$ [$\Gamma_L/\Gamma_R$] for the three-state system Eq. (\ref{['Anderson']}) [the classical model Eq. (\ref{['classical']}) ].
  • Figure 3: Ratio of the spectrum $\tilde{S}(\omega)$, Eq. (\ref{['F0def']}), and the proper (reduced) noise spectrum $S^{(r)}(\omega)$, Eq. (\ref{['Sreduced']}), for the three-state system Eq. (\ref{['Anderson']}). Frequency $\omega$ and tunnel rates are in units of $\Gamma_L=\gamma_L$.