We analyze the spectral properties of a resonator coupled to asuperconducting single electron transistor (SSET) close to the Josephsonquasiparticle resonance. Focussing on the regime where the resonator is driveninto a limit-cycle state by the SSET, we investigate the behavior of theresonator linewidth and the energy relaxation rate which control the widths ofthe main features in the resonator spectra. We find that the linewidth becomesvery narrow in the limit-cycle regime, where it is dominated by a slow phasediffusion process, as in a laser. The overall phase diffusion rate isdetermined by a combination of direct phase diffusion and the effect ofamplitude fluctuations which affect the phase because the resonator frequencyis amplitude dependent. For sufficiently strong couplings we find that a regimeemerges where the phase diffusion is no longer minimized when the averageresonator energy is maximized. Finally we show that the current noise of theSSET provides a way of measuring both the linewidth and energy relaxation rate.
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