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

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. The concept of an effective temperature is introduced to analyze to what extent a detailed balance relation exists. Explicit expressions are also found for the Lamb-shift. Results for ohmic baths are in agreement with experimental findings, while for structured environments population inversion is predicted that may qualitatively explain recent observations.

Figure 1

${\beta }_{\mathit{QQ}}$ (solid) and ${\beta }_{\mathit{QP}}$ (dashed) versus frequency $\omega$ at $\beta \hslash {\omega }_F=4$ (left) and versus laboratory temperature at $\omega =0.2{\omega }_F$ (right) for an ohmic bath.

Figure 2

${\beta }_{\mathit{xy}}$ versus frequency $\omega$ at $\beta \hslash {\omega }_F=4$ (left) and versus laboratory temperature at $\omega =0.2{\omega }_F$ (right) for a damped harmonic oscillator bath with $\gamma /{\omega }_F=0.5$ at ${\omega }_p=.95{\omega }_F$ (${\beta }_{\mathit{QQ}}$ dashed, ${\beta }_{\mathit{QP}}$ solid) and ${\omega }_p=1.35{\omega }_F$ (${\beta }_{\mathit{QQ}}$ dotted, ${\beta }_{\mathit{QP}}$ dash-dotted).

Figure 3

Bath factor for the Lamb shift in the laboratory $D^L$ (dashed) and the rotating frame ${\mathrm{D̃}}_{\mathit{QQ}}^L$ (solid) and ${\mathrm{D̃}}_{\mathit{QP}}^L$ (dotted) for $\hslash \beta {\omega }_F=4$.