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 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.

Figure 1

(Color online) Scaled photon number, $⟨n⟩/N_a$, in the threshold region versus scaled coupling, $g\sqrt{N_a}$, for different values of $N_a$. The other parameters are ${\omega }_p={\omega }_0$, ${\Gamma }_1/{\omega }_0=0.016$, ${\Gamma }_{\phi }^{\ast }/{\omega }_0=0.004$, $D_0=0.975$, $\kappa /{\omega }_0=3\times {10}^{-4}$, and $N_{th}=0$.

Figure 2

(Color online) Diffusion constant versus qubit-oscillator coupling and pure dephasing; parameters as in Fig. 1. Inset: phase diffusion constants, ${\kappa }_d$ and ${\kappa }_d^{\text{fac}}$, calculated with and without taking into account correlation and the average photon number obtained from the master equation for ${\Gamma }_{\phi }^{\ast }/{\omega }_0=0.004$.

Figure 3

(Color online) Left panel: maximum spectrum’s amplitude $\widehat{O}\left({\omega }_0+\delta {\omega }_0\right)$ for different values of $g$. ${\omega }_0=10\text{GHz}$. Right panel: logarithm of the normalized resonator spectrum, $\text{ln}\left[{\omega }_0\widehat{O}\left(\omega \right)\right]$ , for $g/{\omega }_0=0.0045$ in the presence of an external driving at resonance as a function of the driving power. Other parameters as in Fig. 1.