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Cambridge University Press eBooks

Transport in Open Quantum Systems

Pierre Gaspard

Published Jul 14, 2022• 0 citations
Publisher

Abstract

The multivariate fluctuation relation is established for the full counting statistics of the energy and particle fluxes across an open quantum system in contact with several reservoirs on the basis of microreversibility The quantum version of the nonequilibrium work fluctuation relation is recovered in the presence of a single reservoir. In the long-time limit, the time-reversal symmetry relation is expressed in terms of the cumulative generating function for the full counting statistics. In systems with independent particles, the symmetry relation can be obtained in the scattering approach for the transport of bosons and fermions. The temporal disorder and its time asymmetry can be characterized by the quantum version of the entropy and coentropy per unit time. Their difference gives the thermodynamic entropy production rate. Furthermore, the stochastic approach is also considered for electron transport in quantum dots, quantum point contacts, and single-electron transistors.

DOI
10.1017/9781108563055.016

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The multivariate fluctuation relation is established for the full counting statistics of the energy and particle fluxes across an open quantum system in contact with several reservoirs on the basis of microreversibility The quantum version of the nonequilibrium work fluctuation relation is recovered in the presence of a single reservoir. In the long-time limit, the time-reversal symmetry relation is expressed in terms of the cumulative generating function for the full counting statistics. In systems with independent particles, the symmetry relation can be obtained in the scattering approach for the transport of bosons and fermions. The temporal disorder and its time asymmetry can be characterized by the quantum version of the entropy and coentropy per unit time. Their difference gives the thermodynamic entropy production rate. Furthermore, the stochastic approach is also considered for electron transport in quantum dots, quantum point contacts, and single-electron transistors.
Scattering theory is complemented by recent results on full counting statistics, the multivariate fluctuation relation for currents, and time asymmetry in temporal disorder characterized by the Connes‐Narnhofer‐Thirring entropy per unit time, in order to establish relationships with the thermodynamics of quantum transport. Fluctuations in the bosonic or fermionic currents flowing across an open system in contact with particle reservoirs are described by their cumulant generating function, which obeys the multivariate fluctuation relation as the consequence of microreversibility. Time asymmetry in temporal disorder is shown to manifest itself out of equilibrium in the difference between a time‐reversed coentropy and the Connes‐Narnhofer‐Thirring entropy per unit time. This difference is shown to be equal to the thermodynamic entropy production for ideal quantum gases of bosons and fermions. The results are illustrated for a two‐terminal circuit. image
A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained, which is the consequence of microreversibility for the probability of the nonequilibrium work and the transfer of particles and energy between the reservoirs. In some appropriate long-time limit, the symmetry relation leads to a steady-state quantum fluctuation theorem for the currents between the reservoirs. On this basis, relationships are deduced which extend the Onsager–Casimir reciprocity relations to the nonlinear response coefficients.
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Abstract The microscopic features of bosonic quantum transport in a nonequilibrium steady state, which breaks time reversal invariance spontaneously, are investigated. The analysis is based on the probability distributions, generated by the correlation functions of the particle current and the entropy production operator. The general approach is applied to an exactly solvable model with a point‐like interaction driving the system away from equilibrium. The quantum fluctuations of the particle current and the entropy production are explicitly evaluated in the zero frequency limit. It is shown that all moments of the entropy production distribution are non‐negative, which provides a microscopic version of the second law of thermodynamics. On this basis a concept of efficiency, taking into account all quantum fluctuations, is proposed and analyzed. The role of the quantum statistics in this context is also discussed.
The entanglement entropy generated by quantum transport, similar to any physical observable quantity, is a stochastic variable that has its distribution and can fluctuate. The fundamental question is how to define the entanglement entropy operator which allows one to discuss entanglement entropy fluctuation. By introducing the entanglement entropy operator, we develop a theoretical framework to calculate the entanglement entropy fluctuation as well as its higher order cumulants generated by electronic transport in open systems. The distribution of entanglement entropy generated by opening or closing a quantum point contact (QPC) is solved exactly. When the transmission coefficient of QPC is one-half, the entanglement entropy is maximized and fluctuationless. We also establish a general relation between the generated entanglement entropy fluctuation and charge fluctuation. We apply our theory to electronic transport through a quantum dot and study the generated entanglement entropy in the transient regime. Universal behavior is found for the cumulants of entanglement entropy at short times.