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arXiv:2211.03070 · Physical Review Letters

Violation of Detailed Balance in Quantum Open Systems

Robert Alicki, Milan Šindelka, David Gelbwaser-Klimovsky

Published Jul 26, 2023• 13 citations
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Abstract

We consider the dynamics of a quantum system immersed in a dilute gas at thermodynamic equilibrium using a quantum Markovian master equation derived by applying the low-density limit technique. It is shown that the Gibbs state at the bath temperature is always stationary while the detailed balance condition at this state can be violated beyond the Born approximation. This violation is generically related to the absence of time-reversal symmetry for the scattering T matrix, which produces a thermalization mechanism that allows the presence of persistent probability and heat currents at thermal equilibrium. This phenomenon is illustrated by a model of an electron hopping between three quantum dots in an external magnetic field.

DOI
10.1103/physrevlett.131.040401

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We consider the dynamics of a quantum system immersed in a dilute gas at thermodynamic equilibrium using a quantum Markovian master equation derived by applying the low-density limit technique. It is shown that the Gibbs state at the bath temperature is always stationary while the detailed balance condition at this state can be violated beyond the Born approximation. This violation is generically related to the absence of time-reversal symmetry for the scattering T matrix, which produces a thermalization mechanism that allows the presence of persistent probability and heat currents at thermal equilibrium. This phenomenon is illustrated by a model of an electron hopping between three quantum dots in an external magnetic field.
Open quantum systems that comply with the master equation and detailed balance decay in a non-oscillatory manner to thermal equilibrium. Beyond the weak coupling limit, systems that break microreversibility (e.g., in the presence of magnetic fields) violate detailed balance but still thermalize. We study the thermalization of these systems and show that a temperature increase produces novel exceptional points that indicate a sharp transition in the thermalization dynamics. A further temperature increase fuels oscillations of the energy level populations, even without quantum coherences. Moreover, the violation of detailed balance introduces an energy scale that characterizes the oscillatory regime at high temperatures.
Abstract We show that, in general, the unequal time correlation function of system observables, when computed within the Born approximation in open quantum systems, fails to produce the temperature-dependent contributions required for thermalization in the long-time limit. As a consequence, Born approximation fails to respect the detailed balance condition in the long-time limit for arbitrary time difference. This occurs precisely due to the factorization of the total density matrix in the Born approximation, which neglects finite system–bath correlations. We demonstrate how incorporating the correction term recovers the temperature-dependent thermalization term and restores detailed balance. In contrast, we find that so-called spectral correlators, which are unrelated to detailed balance often correctly captured at the Born level to leading order in the system–bath coupling. We also provide a generic condition under which spectral correlators can be correctly predicted under Born approximation. Our approach is illustrated for two paradigmatic models of open quantum systems: (i) the dissipative Caldeira–Leggett model and (ii) the dissipative spin-boson model. We further extend our analysis for multi-bath non-equilibrium quantum transport scenario.
Starting from a microscopic description of weak system-bath interactions, we derive from first principles a quantum master equation that does not rely on the well-known rotating wave approximation. This includes generic many-body systems, with Hamiltonians with vanishingly small energy spacings that forbid that approximation. The equation satisfies a general form of detailed balance, called KMS (Kubo-Martin-Schwinger), which ensures exact convergence to the many-body Gibbs state. Unlike the more common notion of GNS (Gelfand-Naimark-Segal) detailed balance, this notion is compatible with the absence of the rotating wave approximation. We show that the resulting Lindbladian dynamics not only reproduces the thermal equilibrium point up to a small renormalization of the system Hamiltonian, but it also approximates the true system evolution with an error that grows at most linearly in time, giving an exponential improvement upon previous estimates. This master equation has quasilocal jump operators, can be efficiently simulated on a quantum computer, and reduces to the usual Davies dynamics in the limit of a coarse-graining time much larger than the inverse of the smallest frequency difference. With it, we provide a rigorous model of many-body thermalization relevant to both open quantum systems and quantum algorithms.
Kramers' degeneracy theorem underpins many interesting effects in quantum systems with time-reversal symmetry. We show that the generator of dynamics for Markovian open fermionic systems can exhibit an analogous degeneracy, protected by a combination of time-reversal symmetry and the microreversibility (detailed balance) property of systems at thermal equilibrium---the degeneracy is lifted if either condition is not met. We provide simple examples of this phenomenon and show that the degeneracy is reflected in the single-particle Green's functions. Furthermore, we show that certain experimental signatures of topological edge modes in open many-body systems can be protected by microreversibility in the same way. Our results highlight the importance of detailed balance in characterizing open topological matter.
Pseudomode approaches allow for an exact and unapproximated description of a quantum system interacting arbitrarily strongly with a bath. In general, a system coupled to pseudomodes will not thermalize to the system's Gibbs state: This is to be expected when the system-bath coupling is non-perturbative, but conflicts with common thermodynamic intuition when the system-bath coupling is asymptotically weak. We explore under which circumstances pseudomode models satisfy detailed balance (and subsequently thermalize to the system Gibbs state) and how specific choices of parameters can force "weak" detailed balance that is restricted to a limited frequency range. A combination of Hermitian and non-Hermitian pseudomodes that yields a flat effective-temperature profile is also considered. The results and criteria established here are relevant for the construction of pseudomode models in contexts where thermodynamic consistency is required.
We investigate the dynamic evolution and thermodynamic process of a driven quantum system immersed in a finite-temperature heat bath. A Born-Markovian quantum master equation is formally derived for the time-dependent system with discrete energy levels. This quantum master equation can be applied to situations with a broad range of driving speeds and bath temperatures and thus be used to study the finite-time quantum thermodynamics even when nonadiabatic transition and dissipation coexist. The dissipative Landau-Zener model is analyzed as an example. The population evolution and transition probability of the model reveal the importance of the competition between driving and dissipation beyond the adiabatic regime. Moreover, local maxima of irreversible entropy production occur at intermediate sweep velocity and finite temperature, which the low-dissipation model cannot describe.