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Physical Review A

Time-driven quantum master equations and their compatibility with the fluctuation dissipation theorem

Jérôme Flakowski, Maksym Osmanov, David Taj, Hans Christian Öttinger

Published Oct 17, 2014• 5 citations
Publisher

Abstract

We contribute to a long-standing debate on the supposed failure of the fluctuation dissipation theorem (FDT) for the Davies master equation (DME), an important class of Lindblad quantum master equations, describing time-driven quantum systems weakly coupled to a heat bath. First we propose two simple and natural criteria on the driving which guarantee compatibility with the FDT. We show through our setting that, contrary to what is often stated in the literature, the DME is fully compatible with the FDT. We thus argue that the cause of the dispute lies in the adopted perturbation scheme, rather than in the Lindblad character of the master equation itself. We confirm our statement by proving that the Grabert master equation, first proposed by Grabert [Projection Operator Techniques in Nonequilibrium Statistical Mechanics (Springer, Berlin, 1982)] as an alternative linear dynamics fulfilling the FDT, is nothing else than the incriminated DME. Our criteria for the FDT can also be used in the analysis of the nonlinear thermodynamical master equation, first obtained in the Brownian motion limit [H. Grabert, Z. Phys. B 49, 161 (1982)] and later independently rediscovered and generalized on purely thermodynamic grounds [H. C. \"Ottinger, Europhys. Lett. 94, 10006 (2011)].

DOI
10.1103/physreva.90.042110

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We contribute to a long-standing debate on the supposed failure of the fluctuation dissipation theorem (FDT) for the Davies master equation (DME), an important class of Lindblad quantum master equations, describing time-driven quantum systems weakly coupled to a heat bath. First we propose two simple and natural criteria on the driving which guarantee compatibility with the FDT. We show through our setting that, contrary to what is often stated in the literature, the DME is fully compatible with the FDT. We thus argue that the cause of the dispute lies in the adopted perturbation scheme, rather than in the Lindblad character of the master equation itself. We confirm our statement by proving that the Grabert master equation, first proposed by Grabert [Projection Operator Techniques in Nonequilibrium Statistical Mechanics (Springer, Berlin, 1982)] as an alternative linear dynamics fulfilling the FDT, is nothing else than the incriminated DME. Our criteria for the FDT can also be used in the analysis of the nonlinear thermodynamical master equation, first obtained in the Brownian motion limit [H. Grabert, Z. Phys. B 49, 161 (1982)] and later independently rediscovered and generalized on purely thermodynamic grounds [H. C. "Ottinger, Europhys. Lett. 94, 10006 (2011)].
Master equations of Lindblad type have attained prominent status in the fields of quantum optics and quantum information since they are guaranteed to satisfy fundamental notions of quantum dynamics such as complete positivity. When Lindblad operators are used to describe thermal reservoirs in contact with an open quantum system, the fundamental laws of thermodynamics and the fluctuation‐dissipation theorem provide additional mandatory criteria. We show several examples of innocent‐looking Lindblad operators which have questionable properties in this regard. Compatibility criteria between Hamiltonian and Lindblad terms as well as consequences of their violation are discussed. An alternative stochastic approach to dissipative quantum dynamics is outlined and illustrated through a harmonic‐chain model for which the approach of local Lindblad operators fails.
We derive a general set of fluctuation relations for a nonequilibrium open quantum system described by a Lindblad master equation. In the special case of conservative Hamiltonian dynamics, these identities allow us to retrieve quantum versions of Jarzynski and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a stationary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo formula.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolutions, as given by quantum maps. Third, it brings out the full connection between the FDT and the Quantum Fisher information, the figure of merit in quantum metrology.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolution, as given by quantum maps. Third, it clarifies the connection between the FDT and quantum metrology in systems with a non-equilibrium steady state.
We report an alternate formulation of the quantum master equation (QME) to describe the dynamics of a quantum system weakly coupled to a heat bath, in the presence of weak external driving. A key feature of this approach is the introduction of an explicit Hamiltonian to model the thermal fluctuations in the heat bath. We show that the resulting time coarse-grained dynamical equation for the quantum system has dissipators with a natural regulator, which emerges from an ensemble average over the fluctuations. Importantly, such regularized dissipators arise from the second-order contributions of both the external drive as well as the system-environment coupling. We show that the second-order drive terms, regularized to time-scales set by the fluctuations, result in dynamic drive-induced frequency shifts as well as drive-dependent relaxation phenomena. Considering the specific case of an ensemble of two-level systems, subjected to a linearly polarized external drive, we derive the modified Bloch Equations with such drive-dependent shift and damping terms. The resulting drive-induced frequency shifts converge to the known forms of dynamic frequency shifts, such as the Bloch-Siegert shift or the dynamic Stark shift, in appropriate limits. The Kramers-Kronig pairs of these frequency shifts - manifest as drive-dependent damping terms in the modified Bloch Equations and help explain the Redfield limit of free-induction-decay (FID) rates as well as the non-Bloch decay of Rabi oscillations in isotropic medium. Our method predicts correct orders of magnitudes of non-Bloch decay rates in isotropic medium as well as their observed temperature dependence. The QME reported here, correctly describes all known aspects of the driven-dissipative dynamics up to second-order of an open quantum system with appropriate thermal signatures and as such is expected to provide deeper insights into the study of quantum information processing on real systems.
We present a non-equilibrium quantum master equation for a driven open quantum system in the presence of a continuously applied electromagnetic field. Starting from a driven Caldeira-Leggett (CL) model in which the external electromagnetic field couples simultaneously to the subsystem and reservoir degrees of freedom, the canonical fluctuation-dissipation theorem (FDT) relations that encode the coefficients of the master equation can no longer be expected to hold. The bath statistics acquire an explicit dependence on the two-time autocorrelation function of the applied field, leading to drive-biased noise correlations and the potential for non-Markovian dynamics. By eliminating the reservoir degrees of freedom at the operator level, we obtain a modified Hu-Paz-Zhang (HPZ) master equation, in which the diffusion coefficients and coherent forces inherit an explicit memory dependence on the external field. We demonstrate that the physically observable resonant frequency remains encoded in the homogeneous Green's function of the Generalized Langevin equation (GLE), while the drive-induced corrections manifest exclusively through modified diffusion and drift terms, should the drive be treated classically. The resultant field-modified HPZ master provides a unified microscopic framework for understanding field-biased open quantum systems with direct pertinence to a wide variety of experiments in quantum optics and microscopic quantum circuits.