Nonequilibrium Nonlinear Open Quantum Systems I. Functional Perturbative Analysis of a Weakly Anharmonic Oscillator

Abstract

We introduce a functional perturbative method for treating weakly nonlinear systems such as a chain of coupled quantum oscillators with own baths. We demonstrate using this method to obtain the correlation functions of a quantum anharmonic oscillator interacting with a heat bath. These results are useful for studying the nonequilibrium physical processes of nonlinear quantum systems such as heat transfer or electron transport.

Nonequilibrium Steady State and Heat Transport in Nonlinear Open Quantum Systems: Stochastic Influence Action and Functional Perturbative Analysis

“…The stochastic equations such as the master equation (see, e.g., [47]) and the Langevin equations (see, e.g., [25]) can be obtained from taking the functional variations of these effective actions. We then invoke the functional perturbative approach of [26] 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.…”

“…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.…”

“…Following the procedures outlined in [17,27], we find the classical stochastic action S(0) ξ [ q, r, j, j ] given by…”

“…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] and applied to transport problems in [26,27]. See also [28,29].…”

“…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].…”