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Nonequilibrium Steady State in Open Quantum Systems: Influence Action, Stochastic Equation and Power Balance

J. -t. Hsiang, B. L. Hu

Published Aug 15, 2016• 34 citations
OpenAlex

Abstract

The existence and uniqueness of a steady state for nonequilibrium systems (NESS) is a fundamental subject and a main theme of research in statistical mechanics for decades. For Gaussian systems, such as a chain of harmonic oscillators connected at each end to a heat bath, and for anharmonic oscilla-tors under specified conditions, definitive answers exist in the form of proven theorems. Answering this question for quantum many-body systems poses a challenge for the present. In this work we address this issue by deriving the stochastic equations for the reduced system with self-consistent backaction from the two baths, calculating the energy flow from one bath to the chain to the other bath, and exhibiting a power balance relation in the total (chain + baths) system which testifies to the existence of a NESS in this system at late times. Its insensitivity to the initial conditions of the chain corroborates to its uniqueness. The functional method we adopt here entails the use of the influence functional, the coarse-grained and stochastic effective actions, from which one can derive the stochastic equations and calculate the average values of physical variables in open quantum systems. This involves both taking the expectation values of

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Continuing our inquiry into the conditions when fluctuation-dissipation relations (FDR) may appear in the context of nonequilibrium dynamics of open quantum systems (over and beyond the conventional FDR from linear response theory) we turn to non-Gaussian systems and consider this issue for an anharmonic quantum oscillator interacting with a scalar quantum field bath. We present the general nonperturbative expressions for the rate of energy (power) exchange between the anharmonic oscillator and its thermal bath. For the cases that a stable final equilibrium state exists, and the nonstationary components of the two-point functions of the anharmonic oscillator have negligible contributions to the power balance, we can show nonperturbatively that equilibration implies an FDR for the anharmonic oscillator. We then use a weakly anharmonic oscillator as an example to illustrate the validity of those two assumptions and show that in the weak anhamonicity limit, they are satisfied according to our first-order perturbative results..

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The existence and uniqueness of a steady state for nonequilibrium systems (NESS) is a fundamental subject and a main theme of research in statistical mechanics for decades. For Gaussian systems, such as a chain of harmonic oscillators connected at each end to a heat bath, and for anharmonic oscilla-tors under specified conditions, definitive answers exist in the form of proven theorems. Answering this question for quantum many-body systems poses a challenge for the present. In this work we address this issue by deriving the stochastic equations for the reduced system with self-consistent backaction from the two baths, calculating the energy flow from one bath to the chain to the other bath, and exhibiting a power balance relation in the total (chain + baths) system which testifies to the existence of a NESS in this system at late times. Its insensitivity to the initial conditions of the chain corroborates to its uniqueness. The functional method we adopt here entails the use of the influence functional, the coarse-grained and stochastic effective actions, from which one can derive the stochastic equations and calculate the average values of physical variables in open quantum systems. This involves both taking the expectation values of
Continuing our work on the nature and existence of fluctuation-dissipation relations (FDR) in linear and nonlinear open quantum systems [J.-T. Hsiang, B. L. Hu, and S.-Y. Lin, Phys. Rev. D 100, 025019 (2019); J.-T. Hsiang, B. L. Hu, S.-Y. Lin, and K. Yamamoto, Phys. Lett. B 795, 694 (2019); J.-T. Hsiang and B. L. Hu, Physics (Utrecht) 1, 430 (2019); J.-T. Hsiang and B. L. Hu, Phys. Rev. D 101, 125003 (2020)], here we consider such relations when a linear system is in a nonequilibrium steady state (NESS). With the model of two-oscillators (considered as a short harmonic chain with the two ends) each connected to a thermal bath of different temperatures we find that when the chain is fully relaxed due to interaction with the baths, the relation that connects the noise kernel and the imaginary part of the dissipation kernel of the chain in one bath does not assume the conventional form for the FDR in equilibrium cases. There exists an additional term we call the ``bias current'' that depends on the difference of the bath's initial temperatures and the interoscillator coupling strength. We further show that this term is related to the steady heat flow between the two baths when the system is in an NESS. The ability to know the real-time development of the interheat exchange (between the baths and the end-oscillators) and the intraheat transfer (within the chain) and their dependence on the parameters in the system offers possibilities for quantifiable control, and in the design of quantum heat engines, or thermal devices.
A definition of nonequilibrium free energy Fs{\mathcal{F}}_{\mathrm{s}}Fs​ is proposed for dynamical Gaussian quantum open systems strongly coupled to a heat bath and the formal relation with the generating functional, the coarse-grained effective action and the influence action is indicated. For Gaussian open quantum systems exemplified by the quantum Brownian motion model studied here, a time-varying effective temperature can be introduced in a natural way, and, with it, the nonequilibrium free energy Fs{\mathcal{F}}_{\mathrm{s}}Fs​, von Neumann entropy SvN{\mathcal{S}}_{vN}SvN​ and internal energy Us{\mathcal{U}}_{\mathrm{s}}Us​ of the reduced system (SSS) can be defined accordingly. In contrast to the nonequilibrium free energy found in the literature which references the bath temperature, the nonequilibrium thermodynamic functions we find here obey the familiar relation ${\mathcal{F}}_{\mathrm{s}}(t)={\mathcal{U}}_{\mathrm{s}}(t)\ensuremath{-}{T}_{\mathrm{EFF}}(t){\mathcal{S}}_{vN}(t)$ at any and all moments of time in the system's fully nonequilibrium evolution history. After the system equilibrates they coincide, in the weak coupling limit, with their counterparts in conventional equilibrium thermodynamics. Since the effective temperature captures both the state of the system and its interaction with the bath, upon the system's equilibration, it approaches a value slightly higher than the initial bath temperature. Notably, it remains nonzero for a zero-temperature bath, signaling the existence of system-bath entanglement. Reasonably, at high bath temperatures and under ultraweak couplings, it becomes indistinguishable from the bath temperature. The nonequilibrium thermodynamic functions and relations discovered here for dynamical Gaussian quantum systems should open up useful pathways toward establishing meaningful theories of nonequilibrium quantum thermodynamics.
We study the nonequilibrium steady state (NESS) in a quantum system in contact with two heat baths at different temperatures. We use a time-independent perturbative expansion with respect to the coupling with the two heat baths to obtain the density matrix for the NESS. In particular, we show an explicit representation of the density matrix for the reflection symmetric and weakly nonequilibrium case. We also calculate the expectation value of the energy current and show that the Kubo formula holds in this case.
We introduce a response theory for open quantum systems within nonequilibrium steady states subject to a Hamiltonian perturbation.Working in the weak system-bath coupling regime, our results are derived within the Lindblad-Gorini-Kossakowski-Sudarshan formalism.We find that the response of the system to a small perturbation is not simply related to a correlation function within the system, unlike traditional linear response theory in closed systems or expectations from the fluctuation-dissipation theorem.In limiting cases, when the perturbation is small relative to the coupling to the surroundings or when it does not lead to a change of the eigenstructure of the system, a perturbative expansion exists where the response function is related to a sum of a system correlation functions and additional forces induced by the surroundings.Away from these limiting regimes, however, the secular approximation results in a singular response that cannot be captured within the traditional approach but can be described by reverting to a microscopic Hamiltonian description.These findings are illustrated by explicit calculations in coupled qubits and anharmonic oscillators in contact with bosonic baths at different temperatures.
We introduce a functional perturbative method for treating weakly nonlinear systems coupled with a quantum field bath. We demonstrate using this method to obtain the covariance matrix elements and the correlation functions of a quantum anharmonic oscillator interacting with a heat bath. We identify a fluctuation-dissipation relation based on the nonequilibrium dynamics of this nonlinear open quantum system. To establish its connection with dynamical equilibration, we further examine the energy flows between the anharmonic oscillator and the bath field. The vanishing of the net flow is an indication of the existence of an equilibrium state for such an open-system configuration. The results presented here are useful for studying the nonequilibrium physical processes of nonlinear quantum systems such as heat transfer or electron transport.
Continuing our inquiry into the conditions when fluctuation-dissipation relations (FDR) may appear in the context of nonequilibrium dynamics of open quantum systems (over and beyond the conventional FDR from linear response theory) we turn to non-Gaussian systems and consider this issue for an anharmonic quantum oscillator interacting with a scalar quantum field bath. We present the general nonperturbative expressions for the rate of energy (power) exchange between the anharmonic oscillator and its thermal bath. For the cases that a stable final equilibrium state exists, and the nonstationary components of the two-point functions of the anharmonic oscillator have negligible contributions to the power balance, we can show nonperturbatively that equilibration implies an FDR for the anharmonic oscillator. We then use a weakly anharmonic oscillator as an example to illustrate the validity of those two assumptions and show that in the weak anhamonicity limit, they are satisfied according to our first-order perturbative results..