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arXiv:1202.1700 · Physical Review B

Work fluctuation theorem for a classical circuit coupled to a quantum conductor

Yasuhiro Utsumi, Dmitry S. Golubev, Michael Marthaler, Gerd Schön, Kensuke Kobayashi

Published Aug 8, 2012• 9 citations
arXivHTMLPDF

Abstract

We propose a setup for a quantitative test of the quantum fluctuation theorem. It consists of a quantum conductor, driven by an external voltage source, and a classical inductor-capacitor circuit. The work done on the system by the voltage source can be expressed by the classical degrees of freedom of the $LC$ circuit, which are measurable by conventional techniques. In this way, the circuit acts as a classical detector to perform measurements of the quantum conductor. We prove that this definition is consistent with the work fluctuation theorem. The system under consideration is effectively described by a Langevin equation with non-Gaussian white noise. Our analysis extends the proof of the fluctuation theorem to this situation.

DOI
10.1103/physrevb.86.075420

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We propose a setup for a quantitative test of the quantum fluctuation theorem. It consists of a quantum conductor, driven by an external voltage source, and a classical inductor-capacitor circuit. The work done on the system by the voltage source can be expressed by the classical degrees of freedom of the LCLCLC circuit, which are measurable by conventional techniques. In this way, the circuit acts as a classical detector to perform measurements of the quantum conductor. We prove that this definition is consistent with the work fluctuation theorem. The system under consideration is effectively described by a Langevin equation with non-Gaussian white noise. Our analysis extends the proof of the fluctuation theorem to this situation.
When measuring quantum mechanical properties of charge transport in mesoscopic conductors, backaction effects occur. We consider a measurement setup with an elementary quantum circuit, composed of an inductance and a capacitor, as detector of the current flowing in a nearby quantum point contact. A quantum Langevin equation for the detector variable including backaction effects is derived. Differences with the quantum Langevin equation obtained in linear response are pointed out. In this last case, a relation between fluctuations and dissipation is obtained, provided that an effective temperature of the quantum point contact is defined.
Fluctuation theorems for work provide universal constraints on nonequilibrium fluctuations, yet their quantum generalizations often rely on externally prescribed classical driving protocols. While for classical systems, fluctuation theorems have been extended to autonomous work, where the dynamics of the work source is subject to the backaction of the system, their generalization to the quantum regime is constrained by the uncertainty principle. Here, we extend fluctuation theorems for autonomous work from the classical regime to the quantum regime. By performing successive projective measurements over the work source and the system, we derive Jarzynski-type and Crooks-type fluctuation theorems for autonomous inclusive work from initial mixed thermal states. These relations are analogous to fluctuation theorems for autonomous work in the classical regime and explicitly incorporate the fluctuations of the work source. However, quantum noncommutativity prevents a consistent reduction to the nonautonomous counterparts, even in the limit of a large work source and correspondingly negligible backaction. By contrast, under the exclusive work definition, the nonautonomous limit is recovered when the measured observable of the work source commutes with its bare Hamiltonian and the backaction of the system on the work source is negligible. Our results are illustrated with the Dicke model, where a single-mode radiation field and an ensemble of two-level atoms act as the system of interest and the work source, respectively.
We study how Thomson's formulation of the second law: no work is extracted from an equilibrium ensemble by a cyclic process, emerges in the quantum situation through the averaging over fluctuations of work. The latter concept is carefully defined for an ensemble of quantum systems interacting with macroscopic sources of work. The approach is based on first splitting a mixed quantum ensemble into pure subensembles, which according to quantum mechanics are maximally complete and irreducible. The splitting is done by filtering the outcomes of a measurement process. A critical review is given of two other approaches to fluctuations of work proposed in the literature. It is shown that in contrast to those ones, the present definition {\it i)} is consistent with the physical meaning of the concept of work as mechanical energy lost by the macroscopic sources, or, equivalently, as the average energy acquired by the ensemble; {\it ii)} applies to an arbitrary non-equilibrium state. There is no direct generalization of the classical work-fluctuation theorem to the proper quantum domain. This implies some non-classical scenarios for the emergence of the second law.
We propose a definition of externally measurable quantum work in driven systems.Work is given as a quantum observable on a control device which is forcing the system and can be determined without knowledge of the system Hamiltonian ĤS .We argue that quantum work fluctuation theorems which rely on the knowledge of ĤS are of little practical relevance, contrary to their classical counterparts.Using our framework, we derive a fluctuation theorem which is operationally accessible and could in principle be implemented in experiments to determine bounds on free energy differences of unknown systems.
We study how Thomson’s formulation of the second law: no work is extracted from an equilibrium ensemble by a cyclic process, emerges in the quantum situation through the averaging over fluctuations of work. The latter concept is carefully defined for an ensemble of quantum systems interacting with macroscopic sources of work. The approach is based on first splitting a mixed quantum ensemble into pure subensembles, which according to quantum mechanics are maximally complete and irreducible. The splitting is done by filtering the outcomes of a measurement process. A critical review is given of two other approaches to fluctuations of work proposed in the literature. It is shown that in contrast to those ones, the present definition i) is consistent with the physical meaning of the concept of work as mechanical energy lost by the macroscopic sources, or, equivalently, as the average energy acquired by the ensemble; ii) applies to an arbitrary nonequilibrium state. There is no direct generalization of the classical work-fluctuation theorem to the proper quantum domain. This implies some non-classical scenarios for the emergence of the second law.
While the fluctuation theorem in classical systems has been thoroughly generalized under various feedback control setups, an intriguing situation in quantum systems, namely under continuous feedback, remains to be investigated. In this work, we derive the generalized fluctuation theorem under quantum jumps with continuous measurement and feedback. The essence for the derivation is to newly introduce the operationally meaningful information, which we call quantum-classical-transfer (QC-transfer) entropy. QC-transfer entropy can be naturally interpreted as the quantum counterpart of transfer entropy that is commonly used in classical time series analysis. We also verify our theoretical results by numerical simulation and propose an experiment-numerics hybrid verification method. Our work reveals a fundamental connection between quantum thermodynamics and quantum information, which can be experimentally tested with artificial quantum systems such as circuit quantum electrodynamics.