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

TL;DR

This paper develops a perturbative approach to analyze nonequilibrium steady states and heat transport in weakly nonlinear open quantum systems, specifically in oscillator chains with anharmonic interactions, providing insights into quantum thermal transport.

Contribution

It introduces a stochastic influence action method for weakly nonlinear quantum systems and explicitly characterizes energy flow and NESS formation, extending analysis to various anharmonicities.

Findings

Existence of a late-time NESS in weakly nonlinear quantum systems.

First-order corrections do not affect thermal transport for α-type anharmonicity.

Energy flow becomes constant at late times, indicating stable NESS.

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. Here we consider an oscillator chain containing three-types of anharmonicity in the Fermi-Pasta-Ulam-Tsingou (FPUT) model: cubic $\alpha$- and quartic $\beta$-type 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 flow in different junctures of the chain. The formal results obtained here can be used for quantum transport problems in weakly nonlinear quantum systems. For $\alpha$-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 $\beta$-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 flow 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.