  Motivated by recent ``circuit QED'' experiments we study the lasing
transition and spectral properties of single-qubit lasers. In the strong
coupling, low-temperature regime quantum fluctuations dominate over thermal
noise and strongly influence the linewidth of the laser. When the qubit and the
resonator are detuned, amplitude and phase fluctuations of the radiation field
are coupled, and the phase diffusion model, commonly used to describe
conventional lasers, fails. We predict pronounced effects near the lasing
transition, with an enhanced linewidth and non-exponential decay of the
correlation functions. We cover a wide range of parameters by using two
complementary approaches, one based on the Liouville equation in a Fock state
basis, covering arbitrarily strong coupling but limited to low photon numbers,
the other based on the coherent-state representation, covering large photon
numbers but restricted to weak or intermediate coupling.
