Inducing Nonclassical Lasing via Periodic Drivings in Circuit Quantum Electrodynamics

We show how a pair of superconducting qubits coupled to a microwave cavity mode can be used to engineer a single-atom laser that emits light into a nonclassical state. Our scheme relies on the dressing of the qubit-field coupling by periodic modulations of the qubit energy. In the dressed basis, the radiative decay of the first qubit becomes an effective incoherent pumping mechanism that injects energy into the system, hence turning dissipation to our advantage. A second, auxiliary qubit is used to shape the decay within the cavity, in such a way that lasing occurs in a squeezed basis of the cavity mode. We characterize the system both by mean-field theory and exact calculations. Our work may find applications in the generation of squeezing and entanglement in circuit QED, as well as in the study of dissipative few- and many-body phase transitions.

Figure 1

(Left panel) Circuit QED architecture of the system: A superconducting qubit (transmon [25, 26]) is capacitively coupled to an LC resonator of frequency $\omega$, while its transition frequency between the ground $|g⟩$ and excited $|e⟩$ states is modulated via the flux generated by an external circuit; the qubit is coupled to an open transmission line to which it can radiate excitations, while the resonator experiences leakage to a readout circuit. (Right panel) A biperiodic modulation with frequencies matching the lower and upper sidebands of the qubit-resonator system allows us to independently tune the relative amplitudes of the rotating and counterrotating terms of the qubit-field interaction.

Figure 2

(a) Fidelity between the mean-field ansatz (4) and the exact steady state of the system, as a function of the cooperativity $\tilde{C}$, for ${\kappa }_{\mathrm{eff}}/\gamma =0.02$, $r\approx 1.15$ (90% of quadrature squeezing), and ${\tilde{C}}^{\prime }=10$. The solid blue curve corresponds to the exact steady state of Eq. (2), while in the other curves the effect of the second qubit is considered for ${\tilde{g}}^{\prime }/{\gamma }^{\prime }=0.02$ (dashed red line), 0.05 (dotted yellow line), and 0.07 (dash-dotted green line); the inset shows the fidelity as a function of the ratio ${\tilde{g}}^{\prime }/{\gamma }^{\prime }$, fixing the parameters as in the main plot, plus $\tilde{C}=5$. We also show density plots of the Wigner functions corresponding to the steady state of (2) for $\tilde{C}=5$ and ${\kappa }_{\mathrm{eff}}/\gamma =0.02$, and two values of ${\tilde{C}}^{\prime }$, 10 (b),(c) and 0.01 (d),(e), in which the states are well approximated by (5) and (6), respectively. Note that since (b) approximately corresponds to a mixture of coherent states of equal amplitude, its width can be used as the unit for shot noise, thus making apparent the squeezing in (c).