Lasing without Inversion in Circuit Quantum Electrodynamics

We study the photon generation in a transmission line oscillator coupled to a driven qubit in the presence of a dissipative electromagnetic environment. It has been demonstrated previously that a population inversion in the qubit can lead to a lasing state of the oscillator. Here we show that the circuit can also exhibit the effect of “lasing without inversion.” It arises since the coupling to the dissipative environment enhances photon emission as compared to absorption, similar to the recoil effect predicted for atomic systems. While the recoil effect is very weak, and so far elusive, the effect described here should be observable with realistic circuits. We analyze the requirements for system parameters and environment.

Figure 1

(a) Photon emission and absorption rates ${\Gamma }_{\mathrm{ph}}^{+}$ and ${\Gamma }_{\mathrm{ph}}^{-}$, respectively, as a function of the detuning $\delta \omega$. Without noise, which can be effectively modeled by ${\omega }_R\to \infty$, the two rates are equal (black line). Dashed red and blue lines show the emission and absorption rate, respectively, for ${\omega }_R/{ϵ}_C=0.5$. Full red and blue lines show the rates for ${\omega }_R/{ϵ}_C=0.05$. (b) The average photon number $⟨n⟩$ as a function of the inversion coefficient $D_0$. The black line is the result for a qubit without noise, ${\omega }_R\to \infty$, at $\delta \omega =0$. The red lines are the results under the influence of noise, for ${\omega }_R=0.5$ at $\delta \omega /{ϵ}_C\approx 0.35$ (dashed line) and for ${\omega }_R=0.05$ at $\delta \omega /{ϵ}_C\approx 1$ (full line). The frequency detuning has been chosen to maximize the photon number. We used the parameters ${\Gamma }_{\mathrm{pump}}/{ϵ}_C=0.0325$, $k_{B}T/{ϵ}_C=0.05$, $g/{ϵ}_C=0.01$, and $\kappa /{ϵ}_C=8.125\times {10}^{-5}$.

Figure 2

(a) The phase correlation decay rate ${\kappa }_L$ as a function of the detuning $\delta \omega$. The values of the cutoff frequency are ${\omega }_R/{ϵ}_C=0.05$ (red line), ${\omega }_R/{ϵ}_C=0.2$ (black line), ${\omega }_R/{ϵ}_C=0.5$ (green line), and ${\omega }_R/{ϵ}_C=2$ (blue line). (b) The average photon number $⟨n⟩$ as a function of the frequency detuning $\delta \omega$ with color coding as in (a). Full lines display the results of a numerical solution of the master equation; dashed lines are the approximate analytical solution (11). The system parameters are ${\Gamma }_{\mathrm{pump}}/{ϵ}_C=0.0325$, $k_{B}T/{ϵ}_C=0.05$, $g/{ϵ}_C=0.01$, $\kappa /{ϵ}_C=8.125\times {10}^{-5}$, and $D_0=-0.25$.