Nonequilibrium steady state in open quantum systems: Influence action, stochastic equation and power balance

Highlights

• Nonequilibrium steady state (NESS) for interacting quantum many-body systems.

• Derivation of stochastic equations for quantum oscillator chain with two heat baths.

• Explicit calculation of the energy flow from one bath to the chain to the other bath.

• Power balance relation shows the existence of NESS insensitive to initial conditions.

• Functional method as a viable platform for issues in quantum thermodynamics.

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 classical harmonic oscillators connected at each end to a heat bath, and for classical anharmonic oscillators 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 quantum operators of the system and the distributional averages of stochastic variables stemming from the coarse-grained environment. This method though formal in appearance is compact and complete. It can also easily accommodate perturbative techniques and diagrammatic methods from field theory. Taken all together it provides a solid platform for carrying out systematic investigations into the nonequilibrium dynamics of open quantum systems and quantum thermodynamics.

Introduction

Nonequilibrium stationary states (NESS) play a uniquely important role in many-body systems in contact with two or more heat baths at different temperatures, similar in importance to the equilibrium state of a system in contact with one heat bath which is the arena for the conceptualization and utilization of the canonical ensemble in statistical thermodynamics. The statistical mechanics  [1] and thermodynamics  [2] of open systems¹ in NESS have been the focus of investigation into the important features of nonequilibrium processes of theoretical interests, such as providing the context for the celebrated classical and quantum fluctuation theorems, and acting as the fountainhead of a new field known as quantum thermodynamics  [7], [8], where the laws of classical thermodynamics are now scrutinized for smaller quantum systems (e.g.,  [9], [10]), and with a wide range of practical applications, extending from physics and chemistry to biology.

For classical many body systems the existence and uniqueness of NESS is a fundamental subject and a main theme of research 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)  [11] and anharmonic oscillators under general conditions  [12] there are definitive answers in the form of proven theorems. Answering this question for quantum many body systems is not so straightforward and poses a major challenge for the present. For quantum many body systems a new direction of research is asking whether closed quantum systems can come to equilibrium and thermalize  [13]. Equilibration of open quantum systems  [14] with strong coupling to a heat bath also shows interesting new features  [15]. Transport phenomena in open spin systems has also seen a spur of recent activities  [16]. Noteworthy in the mathematical properties is the role played by symmetry in the nonequilibrium dynamics of these systems  [17].

Our current research program on the nonequilibrium dynamics of quantum open systems attempts to address four sets of issues with shared common basis pertaining to NESS:

A. The approach to NESS. Instead of seeking mathematical proofs for these basic issues which are of great importance but not easy to come by it is helpful to see how these systems evolve in time and find out under what conditions one or more NESS may exist. For this we seek to derive the quantum stochastic equations (master, Langevin, Fokker–Planck) for prototypical quantum open systems (e.g., for two oscillators in contact with two heat baths and extension to chains and networks) so one can follow their dynamics explicitly, to examine whether NESS exist at late times, by checking if energy fluxes reach a steady state and whether under these conditions a energy flow (power) balance relation exists. This is probably the most explicit demonstration of the NESS possible. In addition, the stochastic equations can be used to calculate the evolution of key thermodynamic and quantum quantities such as entropy for equilibration/thermalization considerations and quantum entanglement for quantum information inquiries.

B. Quantum transport. Since the seminal paper of  [18], the role of nonlinearity and nonintegrability in the violation of Fourier law  [19] has been explored in a wide variety of representative classical systems with different nonlinear interactions, such as the Fermi–Pasta–Ulam (FPU) models  [20] or the Frenkel–Kontorova (FK) model  [21] and baths of different natures  [19], [22], [23]. For the original papers and current status we refer to two nice reviews  [22], [23]. For applications of heat conduction to phononics, see  [24]. Note also the recent work on anomalous heat diffusion  [25]. Numerical results are usually a great help in regimes not easily accessible by analytical approaches and can bring forth stimulating surprises (e.g.,  [26], [27], [28]). They are nevertheless a lot more difficult to come by for quantum many body systems, thus analytic results, even perturbative, for weak nonlinearity, are valuable. Finding solutions to the quantum stochastic equations have been attempted for simple systems like a quantum anharmonic oscillator chain coupled to two heat baths at the ends or harmonic oscillators coupled nonlinearly, each with its own heat bath (namely, with or without pinning potentials). The related problem of equilibration of open quantum systems with nonlinearity remains an open issue. Even at the classical level this is not a straightforward issue. The existence of breather modes  [29] and ‘strange’ behavior  [30] have been noted earlier. Nonlinearity in quantum system adds a new dimension bearing some similarity or maybe sharing same origins with the issue of how to decipher “scars” of classical chaos in corresponding quantum systems. Quantum transport in higher dimensional systems has seen much development. Several previous works  [22], [31], [32], [9], [33] have extended the established analytical techniques to higher dimensional quantum harmonic network in an effort to identify the thermal natures of the networks such as the scaling law of the steady current, the roles of topological structures of the network in transport.

C. Fluctuation theorems, work relations and entropy production. We refer to the fluctuation theorems both in the Gallavotti–Cohen vein  [34], [35], [36], [37], [38] and the Jarzynski–Crook work relations  [39], [40], [41], [42] applied to quantum systems. A related issue is entropy production and the role of large deviations in currents in nonequilibrium systems. Probing deep into the quantum regime where traditional methods in nonequilibrium classical thermodynamics could not reach (low temperature, non-Ohmic bath, strong coupling) can be facilitated by the use of microphysics, such as quantum Brownian motion, models and open quantum systems techniques, including the decoherence history concepts (e.g., to locate quasi-classical domains where trajectories and work can be well-defined, see  [43] and references therein). Constructing a viable theory for the nonequilibrium thermodynamics of quantum open systems poses interesting new challenges. Noteworthy is the investigation of foundational issues in quantum thermodynamics such as the validity of the third law  [10]. An interesting result of relevance to our present study is Martinez and Paz’s proof  [9], using quantum harmonic network models, that nonlinearity is an essential resource for quantum absorption refrigerators  [10], [44]. See  [45] for a nice review of this subject.

D. Quantum entanglement at finite temperature   [46], [47]. It is generally believed that at high temperatures thermal fluctuations will overshadow quantum entanglement. This problem was explored by Audenaert et al.  [48] who work out exact solutions for a bisected closed harmonic chain at ground and thermal states, and by Anders  [49] who derived a critical temperature for a harmonic lattice in 1–3 dimensions, above which the quantum system becomes separable. Anders and Winters  [50] further provided proof of theorems and a phase diagram on this issue. Entanglement of a two-particle Gaussian state interacting with a single heat bath is investigated recently in  [51]. What stirs up this issue is the suggestion  [52] that quantum entanglement can persist at high temperature in a NESS or even merely a non-equilibrium setting. Recently  [53] showed by a coupled oscillator model in NESS that no thermal entanglement is found in the high temperature limit. In a parallel series of papers we have investigated the entanglement dynamics of a system of two coupled quantum oscillators under two conditions: (A) system interacting with a common bath  [54], [55], (b) each oscillator interacting with its own bath  [56]. The first case explores the nonequilibrium dynamics of the system, the second explores the entanglement of quantum systems under NESS, with the same setup as in this paper. However, the existence of quantum entanglement in NESS for driven systems as claimed by the experiment of Galve et al.  [57], and the calculations for spin systems  [58], remains an interesting open issue which merits further investigation.

Placed in perspective, this issue is only one exciting topic out of a much broader and fast-growing field of quantum information and thermodynamics (where workshops such as kT log2 are dedicated to  [59]). We can only mention here some noticeable recent work  [60], [61] as samples.

The generic quantum open system we study is a simple 1-dimensional quantum oscillator chain, with the two end-oscillators interacting with its own heat bath, each described by a scalar field. The two baths combined make up the environment. We begin our analysis with two oscillators linearly coupled and explore whether a NESS exists for this open system at late times. We do this by solving for the stochastic effective action and the Langevin equations, which is possible for a Gaussian system. From this we can derive the expressions for the energy flow from one bath to another through the system. This is the reason why we begin our study with this model, since in addition to its generic character and versatility, it provides a nice platform for explaining the methodology we adopt. For the sake of clarity we will work out everything explicitly, so as to facilitate easier comparison with other approaches. We name two papers which are closest to ours, either in the model used or in the concerns expressed: the paper by Dhar, Saito and Hanggi  [62] uses the reduced density matrix approach to treat quantum transport, while that of Ghsquiere, Sinayskiy and Petruccione  [53] uses master equations to treat entropy and entanglement dynamics.

Similar in spirit is an earlier paper by Chen, Lebowitz and Liverani  [63] which use the Keldysh techniques in a path integral formalism to consider the dissipative dynamics of an anharmonic oscillator in a bosonic heat bath, and recent papers of Zoli  [64], Aron et al.  [65] for instance. The main tools in nonequilibrium quantum many-body dynamics such as the closed time path (CTP, in–in, or Schwinger–Keldysh)  [66] effective action, the two-particle irreducible (2PI) representation, the large $N$ expansion were introduced for the establishment of quantum kinetic field theory a quarter of centuries ago  [67] and perfected along the way  [68], [69], [70]. Applications to problems in atomic–optical  [71], condensed matter  [72], nuclear-particle  [73] and gravitation–cosmology  [74] have been on the rise in the last decade. A description of quantum field theoretic methods applied to nonequilibrium processes in a relativistic setting can be found in  [75]. By contrast, there is far less applications of these well-developed (powerful albeit admittedly heavy-duty) methodology for the study of nonequilibrium steady state in open quantum systems in contact with two or more baths. We make such an attempt here for the exploration of fundamental issues of nonequilibrium statistical mechanics for quantum many-body systems and to provide a solid micro-physics foundation for the treatment of problems in quantum thermodynamics which we see will span an increasingly broader range of applications in physics, chemistry and biology. Below we explain our methodology and indicate its advantage when appropriate, while leaving the details of how it is related to other approaches in the sections proper.

The mathematical framework of our methodology is the path-integral influence functional formalism  [76], [77], [78], under which the influence action  [79], the coarse-grained effective action  [80] and the stochastic effective action  [81], [82] are defined. The stochastic equations such as the master equation (see, e.g.,  [79]) and the Langevin equations (see, e.g.,  [83]) can be obtained from taking the functional variations of these effective actions.

There are two main steps in this approach we devised:

1. The derivation of the influence action $S_{IF}$ and coarse-grained effective actions $S_{CG}$ for the reduced system (composed of two linearly interacting oscillators, then extended to a harmonic chain) obtained by coarse-graining or integrating over the environmental variables (composed of two baths, coupled to the two end oscillators of a harmonic chain). The baths are here represented by two scalar fields [84], [85]. Noise does not appear until the second stage. This material is contained in Section  2.

2. For Gaussian systems the imaginary part of the influence action can be identified via the Feynman–Vernon integral identity with a classical stochastic force (see, e.g.,  [86], [87]). Expressing the exponential of the coarse-grained effective action $S_{CG}$ in the form of a functional integral over the noise distribution, the stochastic effective action $S_{SE}$ is identified as the exponent of the integrand. Taking the functional variation of $S_{SE}$ yields a set of Langevin equations for the reduced system. Alternatively one can construct the stochastic reduced density matrix. The averages of dynamical variables in a quantum open system includes taking the expectation values of the canonical variables as quantum operators and the distributional averages of stochastic variables as classical noises. We illustrate how to calculate these quantities with both methods in Sections  3 Stochastic effective action and Langevin equations, 4 Energy transport, power balance and stationarity condition.

Our methodology includes as subcomponents the so-called reduced density matrix approach (e.g.,  [62], [88]), and the quantum master equation and quantum Langevin equation approaches. It is intimately related to the closed-time-path, Schwinger–Keldysh or in–in effective action method, where one can tap into the many useful field theoretical and diagrammatic methods developed, as useful in the nonequilibrium Green’s function approach (NEGF)  [89], [90], [91]. The stochastic equations of motion² obtained from taking the functional variation of the stochastic effective action enjoy the desirable features that (a) they are real and causal, which guarantee the positivity of the reduced density matrix, and (b) the backaction of the environment on the system is incorporated in a self-consistent way. These conditions are crucial for the study of nonequilibrium quantum processes including the properties of NESS.

The physical question we ask is whether a NESS exists at late times. Since we have the evolutionary equations and their solutions for this system we can follow the quantum dynamics (with dissipation and decoherence) of physical quantities under the influence of the environment (in the form of two noise sources). We describe the behavior of the energy flux and derive the balance relations in Section  4.

Paper II  [93] will treat the same system but allow for nonlinear interaction between the two oscillators. For this we shall develop a functional perturbation theory for treating weak nonlinearity. Entanglement at high temperatures in quantum systems in NESS and equilibration in a quantum system with weak nonlinearity are the themes of planned Papers III, IV respectively  [56], [94].

For the description of the dynamics of an open quantum system obtaining the time development of the reduced density operator pretty much captures its essence and evolution. We derive the reduced density operator with the influence functional and closed-time-path formalisms (for a ‘no-thrill’ introduction, see, e.g., Chapters 5, 6 of  [75]).

With this reduced density operator one can compute the time evolution of the expectation values of the operators corresponding to physical variables in the reduced system.³ Here we are interested in the energy flux (heat current) flowing between a chain of $n$ identical coupled harmonic oscillators which together represent the system ($\mathfrak{S}={\sum }_{k=1}^n{O}_k$). Let us call $B_1$ the bath which $O_1$ interacts with, and $B_2$ the bath oscillator $O_2$ interacts with. Thus $B_1$, $B_2$ are affectionately named our oscillators’ ‘private’ baths.

Writing the reduced density operator in terms of the stochastic effective action + the probability functional, one can compute the energy current between each oscillator and its private bath in the framework of the reduced density operator. This functional method provides a useful platform for the construction of a perturbation theory, which we shall show in the next paper, in treating weakly nonlinear cases.

Alternatively, from the influence action one can derive the Langevin equation describing the dynamics of the reduced system under the influence of a noise obtained from the influence functional. This is probably a more intuitive and transparent pathway in visualizing the energy flow between the system and the two baths.

The fundamental solutions which together determine the evolutionary operator of the reduced density operator all have an exponentially decaying factor. This has the consequences that

(1) the dependence on the system’s initial conditions will quickly become insignificant as the system evolves in time. Because of the exponential decay, only during a short transient period are the effects of initial conditions observable. At late times, the behavior of the system is governed by the baths. In other words, for Gaussian initial states, the time evolution of the system is always attracted to the behavior controlled by the bath, independent of the initial conditions of the system;

(2) the physical variables of interest here tend to relax to–becoming exponentially close to–a fixed value in time. For example, the velocity variance will asymptotically go to a constant on a time scale longer than the inverse of the decay constant. In addition all oscillators $O_k$ along the chain have the same relaxation time scale.

The energy currents between $B_1$–$O_1$, or $O_k$–$O_{k+1}$, or $O_n$–$B_n$ in general all evolve with time, and will depend on the initial conditions. However, after the motion of the oscillators along the chain is fully relaxed, the energy currents between components approach time-independent values, with the same magnitude.

This time-independence establishes the existence of an equilibrium steady state. Its insensitivity to the initial conditions of the chain testifies to its uniqueness, the same magnitude ensures there is no energy buildup in any component of the open system: Heat flows from one bath to another via the intermediary of the subsystems. To our knowledge, unlike for classical harmonic oscillators where mathematical proofs of the existence and uniqueness of the NESS have been provided, there is no such proof for quantum harmonic systems.⁴ We have not provided a mathematical proof of the existence and uniqueness of a NESS for this generic system under study. What we have is an explicit demonstration, drawing our conclusions from solving the dynamics of this system under very general conditions, with the belief that the full time evolution of the nonequilibrium open system is perhaps more useful for solving physical problems.

• We have obtained the full nonequilibrium time evolution of the reduced system (in particular, energy flow along a harmonic chain between $B_1$–$O_1$, or $O_k$–$O_{k+1}$, or $O_n$–$B_n$ in a harmonic chain) at all temperatures and couplings with permissible but arbitrary strength⁵.

  The formal mathematical expressions of the energy current are given in

  • Eqs. (4.13), (4.17), (4.20) for a two-oscillator chain, and

  • Eqs. (5.4), (5.5), (5.9) for an $n$-oscillator chain,

  from which we can obtain a profile of energy currents between the components.

• We have established the steady state value of the energy flux in (5.4), (5.5), (5.9). Manifest equality and time-independence of these expressions implies stationarity. There is no buildup or deficit of energy in any of the components.

• We have demonstrated that the NESS current is independent of the initial (Gaussian) configurations of the chain after the transient period. It thus implies uniqueness.

• We have obtained a Landauer-like formula in

  • Eq. (4.27) for a two-oscillator chain, and

  • Eq. (5.10) for an $n$-oscillator chain.

• In particular for the case of two oscillators ($n=2$), we define heat conductance (4.32), and have shown that

  • in the high temperature limit,

    • the steady energy current is proportional to the temperature difference between the baths, in (4.31),

    • the heat conductance is independent of the temperature of either bath, as is seen in (4.33),

    • the dependence of the conductance on two types of coupling constants is shown in (4.34) and in Fig. 4.1.

  • in the low temperature limit

    • the temperature dependence of the steady energy current, (4.35), (4.36) and in Fig. 4.2, and

    • the temperature dependence of the conductivity in (4.38).

• We also plot the general dependence of the NESS energy current on the length of the chain $n$ in Fig. 5.2, based on our analytical expressions (5.11), (5.12), (5.13). It shows that

  • for small $n$, the NESS current does depend on the length in a nontrivial way; however

  • for sufficiently large $n$, the NESS current oscillates but converges to a constant independent of $n$.