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Fluctuation-Dissipation Relation from the Nonequilibrium Dynamics of a Nonlinear Open Quantum System

Continuing our inquiry into the conditions when fluctuation-dissipation relations (FDR) may appear in the context of nonequilibrium dynamics of open quantum systems (over and beyond the conventional FDR from linear response theory) we turn to nonGaussian systems and consider this issue for an anharmonic oscillator interacting with a scalar quantum field bath. We present the general {nonperturbative} expressions for the rate of energy (power) exchange between the anharmonic oscillator and the thermal bath. For the cases that a stable final equilibrium state exists, and the nonstationary components of the two-point functions of the anharmonic oscillator have negligible contributions to the evaluation of the power balance, we can show nonperturbatively that equilibration implies an FDR for the anharmonic oscillator. We then use a weakly anharmonic oscillator as an example to illustrate that those two assumptions indeed are satisfied according to our first-order perturbative results: that the net energy exchange vanishes after relaxation in the open system dynamics and an equilibrium state exists at late times.