Phase diffusion and locking in single-qubit lasers

Motivated by recent experiments, which demonstrated lasing and cooling of the electromagnetic field in an electrical resonator coupled to a superconducting qubit, we study the phase coherence and diffusion of the system in the lasing state. We also discuss phase locking and synchronization induced by an additional {\sl ac} driving of the resonator. We extend earlier work to account for the strong qubit-resonator coupling and to include the effects of low-frequency qubit's noise. We show that the strong coupling may lead to a double peak structure of the spectrum, while the shape and width are determined to the low-frequency noise.