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arXiv:0802.2233 · Annalen der Physik

Waiting times and noise in single particle transport

Tobias Brandes

Published Jun 11, 2008• 104 citations
arXivHTMLPDF

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.

DOI
10.1002/andp.200810306

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The waiting time distribution w(τ)w(τ)w(τ), i.e. the probability for a delay τττ 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)w(τ)w(τ) and other statistical transport quantities such as the noise spectrum and the Full Counting Statistics. It turns out that w(τ)w(τ)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.
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In this work, we consider a general feedback protocol based on quantum-jump detections, where the last detected jump channel is stored in a memory and subsequently used to implement a feedback action, such as modifying the system Hamiltonian conditioned on the last jump. We show that the time evolution of this general protocol can be described by a Lindblad master equation defined in a hybrid classical-quantum space, where the classical part encodes the stored measurement record (memory) and the quantum part represents the monitored system. Moreover, we show that this new representation can be used to fully characterize the counting statistics of a system subject to a general jump-based feedback protocol. We apply the formalism to a three-level system coupled to two thermal baths operating as a thermal machine, and we show that jump-based feedback can be used to convert the information obtained from the jump detections into work. Our framework provides analytical tools that enable the characterization of key statistical properties of any counting observable under jump-based feedback, such as the average current, noise, correlation functions, and power spectrum.
Markovian transport is often described by a master equation for the system state. The thermodynamic information measured in transport experiments, however, is carried by reservoir-resolved transfer records, such as particle currents, heat currents, entropy production, and current noise. We identify a thermodynamic incompleteness of state dynamics: a Markovian state generator can fix the occupation probabilities, stationary response, and relaxation without specifying how the underlying transitions are assigned to reservoirs and energy filters. We study a multi-terminal Coulomb-blockaded quantum dot coupled to energy-filtered reservoirs, for which different assignments of reservoir channels can generate the same state master equation. These assignments give identical occupation dynamics, stationary state, and linear response of the dot, but different heat currents, entropy production, and current noise. We formulate a thermodynamic completeness criterion: a transport observable can be reconstructed from state dynamics only when it is invariant under all changes of reservoir-channel assignments that leave the state generator unchanged. The criterion gives a practical diagnostic for Markovian transport models and a measurable prediction: state tomography can be insufficient to predict heat-noise and cross-correlation measurements, even when the full Markovian state dynamics is known. The analysis identifies a concrete limitation of state-only Markovian thermodynamics and shows which additional transport records must be specified to make thermodynamic predictions experimentally complete.