Elsevier

Annals of Physics

Volume 421, October 2020, 168289
Annals of Physics

Nonequilibrium steady state and heat transport in nonlinear open quantum systems: Stochastic influence action and functional perturbative analysis

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Abstract

In this paper, we show that a nonequilibrium steady state (NESS) exists at late times in open quantum systems with weak nonlinearity by following its nonequilibrium dynamics with a perturbative analysis. We consider an oscillator chain containing three-types of anharmonicity: cubic α- and quartic β-type Fermi–Pasta–Ulam–Tsingou (FPUT) nearest-oscillator interactions and the on-site (pinned) Klein–Gordon (KG) quartic self-interaction. Assuming weak nonlinearity, we introduce a stochastic influence action approach to the problem and obtain the energy flows in different junctures across the chain. The formal results obtained here can be used for quantum transport problems in weakly nonlinear quantum systems. For α-type anharmonicity, we observe that the first-order corrections do not play any role in the thermal transport in the NESS of the configuration we considered. For KG and β-types anharmonicity, we work out explicitly the case of two weakly nonlinearly coupled oscillators, with results scalable to any number of oscillators. We examine the late-time energy flows from one thermal bath to the other via the coupled oscillators, and show that both the zeroth- and the first-order contributions of the energy flows become constant in time at late times, signaling the existence of a late-time NESS to first order in nonlinearity. Our perturbative calculations provide a measure of the strength of nonlinearity for nonlinear open quantum systems, which may help control the mesoscopic heat transport distinct from or close to linear transport. Furthermore, our results also give a benchmark for the numerical challenge of simulating heat transport. Our setup and predictions can be implemented and verified by investigating heat flow in an array of Josephson junctions in the limit of large Josephson energy with the platform of circuit QED.

Introduction

Inasmuch as the equilibrium state of a system in contact with one heat bath is of fundamental importance in the conceptualization and application of the powerful canonical ensemble in statistical thermodynamics, the existence of a nonequilibrium steady state (NESS) is of similar importance for the understanding of nonequilibrium processes which are widely present in Nature. For example NESS is the arena in the discovery of the fluctuation theorems of various forms from Gallivotti and Cohen [1], [2], [3], [4], [5] to Jarzynsky and Crook [6], [7], [8], [9], [10] and the basis for the investigation of quantum thermodynamics (e.g., [11], [12], [13], [14]). One common feature of the equilibrium state and nonequilibrium steady state is their ubiquitous appearance—many, but not all, systems approach an equilibrium or steady state given sufficient time. While the exceptions are not so easily captured but of special interest, the “norm” also needs to be proven or demonstrated for different types of systems and their environments before one can use the many attractive features of NESS to exert control of nonequilibrium systems in such states.

For classical many-body systems the existence and uniqueness of NESS have been studied by mathematical physicists in statistical mechanics for decades. For Gaussian systems (such as a chain of harmonic oscillators with two heat baths at the two ends of the chain) [15] and anharmonic oscillators under general conditions [16] there are definitive answers in the form of proven theorems. Answering this question for quantum many-body systems is not so straightforward. This is the challenge two of the present authors have undertaken [17], showing the existence of NESS at late times for a quantum harmonic chain with two ends coupled linearly to two separate heat baths at different temperatures. (see also [18]). The goal of this work, is to demonstrate the existence of NESS for a system of two weakly nonlinearly coupled (generalizable to a chain of ) quantum oscillators linearly coupled to two heat baths of different temperatures. The forms of interaction amongst the system oscillators correspond to the Fermi–Pasta–Ulam–Tsingou (FPUT) α (cubic nearest neighbor interaction), β (quartic nearest neighbor interaction) [19], [20] or the Klein–Gordon (KG) (‘pinned’, or quartic self-interaction) [21], [22] models. The current work approaches to this problem using a stochastic influence action which is a part of the functional techniques systematically developed in [23], [24], [25], [26] and applied to transport problems in [17], [27]. A fluctuation–dissipation relation is shown to exist in such open nonlinear quantum systems by nonperturbative methods in [28]. In this paper using the stochastic influence action, we aim to demonstrate the existence of a NESS at late times for the FPUT and KG models with weak nonlinearity.

In terms of physical contexts and applications, we give a brief sketch of the problem of quantum transport in nonlinear systems, motivating the usefulness of analytic results, perturbative notwithstanding. We then give a brief summary of the functional perturbative method developed in [29] and a description of our approach. We illustrate this by calculating the energy transfer between the weakly nonlinear system and its environment, showing that at late times indeed they balance to first order of nonlinearity, signifying the existence of a NESS. Further background descriptions can be found in the Introductions of [17] and [27].

Quantum transport is an important class of problems where the nonequilibrium evolution of an open quantum system has been followed and applicable laws examined. A well-known case is the Fourier law for heat conduction in low-dimensional lattices [30], [31]. Amongst the vast literature on this subject suffice it for our purpose here to mention three reviews [32], [33], [34]. As for methodologies closest to ours, we mention the density matrix approach (e.g., [35]), the nonequilibrium Green function approach (e.g., [36], [37], [38], [39]), the stochastic path integral developed in the context of charge transport in mesoscopic systems [40], [41], the closed time path (CTP, Schwinger–Keldysh, or ‘in-in’) [42], [43], [44], the influence functional [45], [46], [47] and the stochastic influence action formalisms [23], [24], [25], [26] which are the methods of choice for two of the present authors.

Transport problems depend heavily on numerical computations (e.g., [48]), which enable one to identify qualitatively different behaviors in different regimes. Many analytical approaches, e.g., Boltzmann molecular dynamics [49], nonlinear kinetic theory and fluctuating hydrodynamics [50] when combined with numerical methods give reasonably good predictions for classical systems in different parameter regimes. However, for quantum many-body systems, numerical computations are a lot more difficult to carry out. For example, even casting the Langevin equations for non-linear quantum systems suitable for numerical analysis poses a challenge [51].

For this reason analytical results for quantum nonlinear systems, even perturbative in nature, may serve some function, in bridging the gaps in the parameter space regions these theories left behind [52].

The motivation for our investigation using perturbative methods is to explore how analytical methods and results, no matter how unimpressive they are in the face of full scale numerical computations, can still serve some function in clarifying the issues behind energy or charge transport in nonequilibrium weakly-nonlinear systems.

In recent years, several discoveries highlighted the importance of even weak nonlinearity in the thermodynamic and transport properties of classical many-body systems. E.g., it is shown that in the thermodynamic limit a one-dimensional (1D) nonlinear lattice can always be thermalized for arbitrarily small nonlinearity [53]. On the effects of nonlinear interaction on disordered chains there is a tentative suggestion that weak nonlinearity can destroy Anderson localization [54]. Thermalization is an important current subject in quantum many-body physics. It is found that even weak nonlinearity can play a pivotal role for thermalization. Thus, a different perspective from nonlinear quantum open systems is desirable. Thermalization via wave turbulence theory has also generated renewed interest in the FPUT and KG models [55], [56]. These recent developments provided good impetus for us to implement our program on nonequilibrium dynamics of open quantum nonlinear systems. The first step is to explore the conditions for systems in popular models to reach a NESS. Because knowing that a system enters such a state from following its nonequilibrium dynamics can offer great simplification in the analysis of the nonlinear system’s long time behavior.

We use the same mathematical framework as in [17], namely, the path-integral [42], [43], influence functional [45], [46] formalism, under which the influence action, the coarse-grained effective action [23] and the stochastic influence action [24], [25] are defined. The stochastic equations such as the master equation (see, e.g., [47]) and the Langevin equations (see, e.g., [26]) can be obtained from taking the functional variations of these effective actions. We then invoke the functional perturbative approach of [29] developed further in [27] to treat systems with weak nonlinearity. In this approach we first introduce external sources to drive a linear (harmonic oscillator) system and calculate the in-in generating functional. Taking the functional derivatives with respect to the sources gives the expectation values of the covariance matrix elements or the two-point functions of the canonical variables. The perturbative correction to the two-point functions, or the expectation values of quantum observables due to the nonlinear potential can also be found order by order by taking the appropriate functional derivatives of the generating functional of the linear system linked to external sources.

Going beyond the stochastic path integrals [57], [58] which unravels the effect of a Markovian bath with continuous weak measurement, in this work we unravel the full effect of the bath without invoking the Markov approximation but at the price that the unraveled trajectory is fictitious which may not correspond to any real measurement. It has been shown that the stochastic unraveling admits a time-local master equation [59] even in the presence of the nonlinearity of the quantum open systems. In our stochastic influence action approach, we can define the heat flow rigorously among the oscillators for our cases where the system Hamiltonian contains α,β-FPUT and KG nonlinearities. Furthermore, we introduce an additional external source h, which allows one to compute correlation functions involving momentum operators using the stochastic influence action. These new ingredients allow us to compute steady-state heat currents in a more efficient and economic way than [27], which provides fully nonequilibrium evolution from the transient to the relaxation to a steady state of an open nonlinear system.

Instead of seeking mathematical proofs for the existence and uniqueness of NESS in quantum nonlinear systems, which to us is a daunting challenge, we aim at solving for the nonequilibrium dynamics of the model systems mentioned above, examine how they evolve in time and see if one or more NESS exist(s) at late times. This is the same approach used earlier by two of the present authors in [17]. Namely, taking the functional variation of the stochastic influence action yields the quantum stochastic equations. The Langevin equation is then used to obtain expressions for the energy flow from one bath to another through the nonlinear system. We can check if these two energy fluxes reach a steady state, and an energy flow (power) balance relation exists. The main result of this work is that at late times at least to first order in the nonlinear interaction between the oscillators, perturbations do not grow any faster or stronger so as to overtake and disrupt the contributions of the linear (harmonic) order. Thus the NESS is explicitly demonstrated to exist for the weakly nonlinear systems studied in our models.

Based on the perturbative formalism in the framework of quantum open systems, we give formal expressions for the energy currents up to the first-order of the nonlinear couplings for an FPUT and KG chain. It consists of nonlinearly coupled anharmonic oscillators via the α-, β-FPUT and KG nonlinearities. Each oscillator also linearly couples to its own private bath. We find for α-FPUT nonlinearity, the first-order corrections to the energy currents vanish. We demonstrate that for the two-oscillator chain, the NESS is established in the late time up to the first-order of the nonlinear coupling constant. This procedure can be straightforwardly extended to a chain of arbitrary length. In addition, we devise diagrammatic representations for the energy currents in the NESS, which may provide an intuitive understanding of the heat transport at NESS. As a physical application, the ratio between the first-order and zeroth correction of the energy currents in the steady-state provides a measure of the strength of the nonlinearity in the context of thermal transport, which may be applied to controlling thermal transport either distinct from or close to the linear transport. Our analytical results suggest that when the coupling between the chain and the baths is strong or when the temperature bias across the chain is large, the heat transport behaves like a linear one in the presence of the nonlinearities, offering a wider latitude to manipulate nonlinear couplings. Our results can provide a useful benchmark for the numerical simulations of anomalous heat transport in low dimensions in the weakly nonlinear regime [32]. Our setup can be implemented by e.g. engineering an array of Josephson junctions in circuit quantum electrodynamics in the large Josephson energy limit [60], [61].

This paper is organized as follows: in Section 2 we first present the general formalism of the influence functional and functional perturbation, aiming at the problem of heat transport in a network of oscillators and baths. As a first application of the functional perturbation formalism, in Section 3, we apply the functional perturbation formalism to compute the zeroth-order steady-state energy currents in a chain configuration with two oscillators and two baths. In Section 4, we compute the first-order correction for the KG, β-FPUT and α-FPUT models respectively. Upon replacing causal propagator for two-oscillator and two-bath configuration with the one for the general N-oscillator and N-bath configuration, the results in Sections 3 The zeroth-order of the heat exchange between an oscillator and their private baths, 4 The first-order corrections can be generalized straightforwardly to the case of a chain of anharmonic oscillators. In Section 5, we give the diagrammatic representation of all the energy currents, which provides an intuitive understanding of the heat transport up to the first-order. In Section 6, we prove for the two-oscillator two-bath configuration that, the NESS exists at the late times for both KG and β-FPUT nonlinearities. In Section 7, we discuss the possible physical applications of our perturbative results and summarize the findings.

Section snippets

The Feynman–Vernon formalism of a network of oscillators and baths

We consider N coupled oscillators interacting with their private thermal baths, as shown in Fig. 1. Each oscillator couples to only its nearest neighbors via the linear and the nonlinear mutual couplings, so that these oscillators form a general FPUT chain threading through thermal baths of various temperatures. The dynamics of this entire system can be described by the action Stot=S+Sϕ+SI.The action Sχ for the chained oscillators contains linear and nonlinear mutual couplings between the

The zeroth-order of the heat exchange between an oscillator and their private baths

With the functional perturbative approach sketched in Section 2, we are in a position to compute the heat flow across the nonlinear chain in the weak nonlinearity regime. For the sake of simplicity, we will demonstrate in detail only for the case of two anharmonic oscillators. The argument generalizes to N>2 anharmonic oscillators. We outline the steps as follows:

  • 1.

    We show in Appendix C that thanks to the natures of the linear dynamics at late time, tf, the stochastic generating functional Z0ξ[j

The first-order corrections

From (41), we observe that for the first-order corrections, the three types of nonlinearities contribute independently to the first-order corrections, so in what follows, we will first discuss the contributions from the β-FPUT and KG nonlinearities, and then the α-FPUT nonlinearity. In the end, we put them together to form the overall first-order corrections, and examine the effects of these nonlinearities on the heat transport in NESS.

Diagrammatic representations

From now on, we will discuss the steady-state energy and will implicitly assume tf=. Therefore we suppress all the time dependence in all the relevant energy currents, which are given by Eqs. (51), (53), (56), (74), (77), (95), (96), (112) respectively. Although they may look formidable, all these energy currents admit time-domain diagrammatic representations, which can be seen directly from their respective analytic expressions through definite rules. Furthermore, one can easily convert the

The non-equilibrium steady state (NESS)

For the configuration we are interested, if the NESS exists, then we expect when the NESS is reached we will have a steady, time-independent energy current along the chain. Thus, in principle, in order to demonstrate the existence of the NESS, we would like to show for the configuration in consideration that in the late time limit tf, the time rate of the internal energy of each oscillator vanishes. In other words, we will show that the ensemble average of Eq. (22) vanishes in the late time

Discussion and conclusion

Having established the NESS in the late time limit in the last section, we now show how our formal perturbative results presented in Sections 3 The zeroth-order of the heat exchange between an oscillator and their private baths, 4 The first-order corrections naturally give a measure of the strength of nonlinearity for nonlinear open quantum systems with KG and β-FPUT type nonlinearity. This should provide insights for understanding and controlling the mesoscopic quantum heat transport. We

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Work by JY and ANJ was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award No. DE-SC0017890. JY would like to thank Professors A. Das and S. G. Rajeev for helpful discussions. JTH and BLH are thankful to Prof. Hong Zhao for his insight in nonlinear transport problems based on his extensive numerical work.

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