Figure 1

Transmitted heterodyne amplitude $|⟨a⟩|$ as a function of drive detuning (normalized by the dispersive shift $\chi =g^2/\delta$) and drive amplitude (normalized by the amplitude to put $n=1$ photon in the cavity in linear response, ${\xi }_1=\kappa /\sqrt{2}$). Dark colors indicate larger amplitudes. (a) Experimental data [18], for a device with cavity at 9.07 GHz and 4 transmon qubits at 7.0, 7.5, 8.0, 12.3 GHz. All qubits are initialized in their ground state, and the signal is integrated for the first $400\mathrm{ns}\simeq 4/\kappa$ after switching on the drive. (b) Numerical results for the JC model of Eq. (3), with qubit fixed to the ground state and effective parameters [20] $\delta /2\pi =-1.0\mathrm{GHz}$, $g/2\pi =0.2\mathrm{GHz}$, $\kappa /2\pi =0.001\mathrm{GHz}$. Hilbert space is truncated at 10 000 excitations (truncation artifacts are visible for the strongest drive), and results are shown for time $t=2.5/\kappa$.Reuse & Permissions

Figure 2

Solution to semiclassical Eq. (5), using the same parameters as Fig. 1b. (a) Amplitude response as a function of drive frequency and amplitude. The region of bifurcation is indicated by the shaded area, and has corners at the critical points $C_1$, $C_2$. The dashed lines indicate the boundaries of the bistable region for a Kerr oscillator (Duffing oscillator), constructed by making the power-series expansion of the Hamiltonian to second order in $N/N_{\mathrm{crit}}$. The Kerr bistability region matches the JC region in the vicinity of $C_1$ but does not exhibit a second critical point. (b) Cut through (a) for a drive of $6.3{\xi }_1$, showing the frequency dependence of the classical solutions (solid blue line). For comparison, the response from the full quantum simulation of Fig. 1b is also plotted (dashed red line) for the same parameters. (c) Cut through (a) for driving at the bare cavity frequency, showing the large gain available close to $C_2$ (the “step”). Faint lines indicate linear response. (d) Same cut as in (c), showing intracavity amplitude on a linear scale.Reuse & Permissions

Figure 3

Symmetry breaking. State-dependent transition frequency versus photon number (a) for the JC model, parameters as in Figs. 1, 2, (b) for the model extended to 2 qubits, ${\delta }_1/2\pi =-1.0\mathrm{GHz}$, ${\delta }_2/2\pi =-2.0\mathrm{GHz}$, $g_1/2\pi =g_2/2\pi =0.25\mathrm{GHz}$, and (c) for the model extended to one transmon qubit [7], tuned below the cavity, ${\omega }_c/2\pi =7\mathrm{GHz}$ $E_C/2\pi =0.2\mathrm{GHz}$, $E_J/2\pi =30\mathrm{GHz}$, $g/2\pi =0.29\mathrm{GHz}$. (For the given parameters, ${\delta }_{01}/2\pi =-0.5\mathrm{GHz}$, ${\delta }_{12}/2\pi =-0.7\mathrm{GHz}$, defining ${\delta }_{ij}=E_j-E_i-{\omega }_c$, with $E_i$ the energy of the $i$th transmon level.) In all panels, the transition frequency asymptotically returns to the bare cavity frequency. In (a) the frequencies within the ${\sigma }_z=\pm 1$ manifolds are (nearly) symmetric with respect to the bare cavity frequency. For (b), if the state of one (“spectator”) qubit is held constant, then the frequencies are asymmetric with respect to flipping the other (“active”) qubit. In (c), the symmetry is also broken due the existence of higher levels in the weakly anharmonic transmon.Reuse & Permissions