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arXiv:0811.3717 · Reviews of Modern Physics

Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems

Massimiliano Esposito, Upendra Harbola, Shaul Mukamel

Published Dec 2, 2009• 1,414 citations
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Abstract

Fluctuation theorems (FTs), which describe some universal properties of nonequilibrium fluctuations, are examined from a quantum perspective and derived by introducing a two-point measurement on the system. FTs for closed and open systems driven out of equilibrium by an external time-dependent force, and for open systems maintained in a nonequilibrium steady state by nonequilibrium boundary conditions, are derived from a unified approach. Applications to fermion and boson transport in quantum junctions are discussed. Quantum master equations and Green's functions techniques for computing the energy and particle statistics are presented.

DOI
10.1103/revmodphys.81.1665

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Fluctuation theorems (FTs), which describe some universal properties of nonequilibrium fluctuations, are examined from a quantum perspective and derived by introducing a two-point measurement on the system. FTs for closed and open systems driven out of equilibrium by an external time-dependent force, and for open systems maintained in a nonequilibrium steady state by nonequilibrium boundary conditions, are derived from a unified approach. Applications to fermion and boson transport in quantum junctions are discussed. Quantum master equations and Green's functions techniques for computing the energy and particle statistics are presented.
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We develop a comprehensive framework for characterizing fluctuations in quantum transport and nonequilibrium thermodynamics using two complementary approaches: full counting statistics and first-passage times. Focusing on open quantum systems governed by Markovian Lindblad dynamics, we derive general ensemble relations that connect the two approaches at all times, and we clarify how the steady states reached at long times relate to those reached at large jump counts. In regimes of metastability, long-lived intermediate states cause violations of experimentally testable cumulant relations, as we discuss. We also formulate a fluctuation theorem governing the probability of rare fluctuations in the first-passage time distributions based on results from full counting statistics. Our results apply to general integer-valued trajectory observables that do not necessarily increase monotonically in time. Three illustrative applications, a two-state emitter, a driven qubit, and a variant of the Su-Schrieffer-Heeger model, highlight the physical implications of our results and provide guidelines for practical calculations. Our framework provides a complete picture of first-passage time statistics in Markovian quantum systems, encompassing multiple earlier results, and it has direct implications for current experiments in quantum optics, superconducting circuits, and nanoscale heat engines.
We demonstrate that the probability distribution of the net number of electrons passing through a quantum system in a junction obeys a steady-state fluctuation theorem (FT) which can be tested experimentally by the full counting statistics (FCS) of electrons crossing the lead-system interface. The FCS is calculated using a many-body quantum master equation combined with a Liouville space generating function formalism. For a model of two coupled quantum dots, we show that the FT becomes valid for long binning times, and provide an estimate for the finite-time deviations. The Mandel (or Fano) parameter associated with the incoming or outgoing electron transfers shows sub-Poissonian (antibunching) statistics.
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.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.