Single-Qubit Lasing and Cooling at the Rabi Frequency

For a superconducting qubit driven to perform Rabi oscillations and coupled to a slow electromagnetic or nanomechanical oscillator we describe previously unexplored quantum optics effects. When the Rabi frequency is tuned to resonance with the oscillator, the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the qubit and a bistability with lasing behavior of the oscillator; for red detuning the qubit cools the oscillator. This behavior persists at the symmetry point where the qubit-oscillator coupling is quadratic and decoherence effects are minimized. There the system realizes a “single-atom-two-photon laser.”

Figure 1

The systems. (a) In the circuit QED setup of Ref. 1 an externally driven three-junction flux qubit is coupled inductively to an $LC$ oscillator. (b) In an equivalent setup a charge qubit is coupled to a mechanical resonator.

Figure 2

Average number of photons in the resonator as a function of the driving detuning $\delta \omega$ and amplitude ${\Omega }_{R0}$. Peaks at $\delta \omega >0$ correspond to lasing, dips at $\delta \omega <0$ to cooling. The inner curve is the one-photon resonance, which exists only away from the symmetry point. (Here we assume $ϵ=0.01\Delta$.) The outer curve is the two-photon resonance, which persists at $ϵ=0$. In domains of bistability the lowest value of $\overline{n}$ is plotted. Sharp drops in the curves are related to the bistability bifurcations. The parameters for the qubit are $\Delta /2\pi =1\mathrm{GHz}$, $ϵ=0.01\Delta$, and ${\Gamma }_0/2\pi =125\mathrm{kHz}$, the frequency and linewidth of the resonator are ${\omega }_T/2\pi =6\mathrm{MHz}$ and $\kappa /2\pi =1.7\mathrm{kHz}$, the coupling constant is $g/2\pi =3.3\mathrm{MHz}$, and the temperature of the resonator $T=10\mathrm{mK}$. The inset shows the bistability of the photon number for ${\Omega }_{R0}/2\pi =7\mathrm{MHz}$. The dashed line represents the unstable solution.