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In this paper, we show that a nonequilibrium steady state (NESS) exists atlate times in open quantum systems with weak nonlinearity by following itsnonequilibrium dynamics with a perturbative analysis. Here we consider anoscillator chain containing three-types of anharmonicity in theFermi-Pasta-Ulam-Tsingou (FPUT) model: cubic $\alpha$- and quartic $\beta$-typenearest-oscillator interactions and the on-site (pinned) Klein-Gordon (KG)quartic self-interaction. Assuming weak nonlinearity, we introduce a stochasticinfluence action approach to the problem and obtain the energy flow indifferent junctures of the chain. The formal results obtained here can be usedfor quantum transport problems in weakly nonlinear quantum systems. For$\alpha$-type anharmonicity, we observe that the first-order corrections do notplay any role in the thermal transport in the NESS of the configuration weconsidered. For KG and $\beta$-types anharmonicity, we work out explicitly thecase of two weakly nonlinearly coupled oscillators, with results scalable toany number of oscillators. We examine the late-time energy flows from onethermal bath to the other via the coupled oscillators, and show that both thezeroth- and the first-order contributions of the energy flow become constant intime at late times, signaling the existence of a late-time NESS to first orderin nonlinearity. Our perturbative calculations provide a measure of thestrength of nonlinearity for nonlinear open quantum systems, which may helpcontrol the mesoscopic heat transport distinct from or close to lineartransport. Furthermore, our results also give a benchmark for the numericalchallenge of simulating heat transport. Our setup and predictions can beimplemented and verified by investigating heat flow in an array of Josephsonjunctions in the limit of large Josephson energy with the platform of circuitQED.

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