Dissipation in a rotating frame: master equation, effective temperature and Lamb-shift

TL;DR

This paper derives a master equation for a driven nonlinear resonator in a rotating frame, revealing environmental effects like effective temperature and Lamb-shift, with implications for superconducting circuits and experimental observations.

Contribution

It introduces a comprehensive derivation of the master equation including momentum interactions, providing explicit formulas for effective temperature and Lamb-shift in a rotating frame.

Findings

Agreement with experimental data for ohmic baths

Prediction of population inversion in structured environments

Identification of additional momentum interactions

Abstract

Motivated by recent realizations of microwave-driven nonlinear resonators in superconducting circuits, the impact of environmental degrees of freedom is analyzed as seen from a rotating frame. A system plus reservoir model is applied to consistently derive in the weak coupling limit the master equation for the reduced density in the moving frame and near the first bifurcation threshold. It turns out that additional interactions between momenta of system and bath appear which have been omitted in previous studies. Explicit expressions for the effective temperature and the Lamb-shift are given which for ohmic baths are in agreement with experimental findings, while for structured environments population inversion is predicted that may qualitatively explain recent observations.