Fate of photon blockade in the deep strong-coupling regime

We investigate the photon emission properties of the driven-dissipative Rabi model in the so-called ultrastrong- and deep strong-coupling regimes where the atom-cavity coupling rate $g$ becomes comparable or larger than the cavity frequency ${\omega }_c$. By solving numerically the master equation in the dressed-state basis, we compute the output field correlation functions in the steady-state for a wide range of coupling rates. We find that, as the atom-cavity coupling strength increases, the system undergoes multiple transitions in the photon statistics. In particular, a first sharp antibunching-to-bunching transition, occurring at $g\sim 0.45{\omega }_c$, leading to the breakdown of the photon blockade due to the counter-rotating terms, is shown to be the consequence of a parity shift in the energy spectrum. A subsequent revival of the photon blockade and the emergence of the quasicoherent statistics for even larger coupling rates are attributed to an interplay between the nonlinearity in the energy spectrum and the transition rates between the dressed states.

Figure 1

(a) Intensity $I_{\mathrm{out}}$ as a function of $g/{\omega }_c$ and ${\omega }_d/{\omega }_c$. (b) Second-order correlation function $g^{\left(2\right)}\left(0\right)$. To improve readability, the color scale for the autocorrelation function saturates at $g^{\left(2\right)}\left(0\right)=2$. The dissipation and driving parameters are $\gamma /{\omega }_c={10}^{-2}$ and $F/{\omega }_c={10}^{-3}$.

Figure 2

(a) Intensity $I_{\mathrm{out}}$ and (b) second-order correlation function $g^{\left(2\right)}\left(0\right)$ as a function of $g/{\omega }_c$ for an external driving resonant with the transition $|{\mathrm{\Psi }}_0^{+}\rangle \to |{\mathrm{\Psi }}_1^{-}\rangle$. Results are shown for two values of the dissipation rate $\gamma /{\omega }_c={10}^{-2}$ (solid blue lines) and $\gamma /{\omega }_c={10}^{-3}$ (dashed red lines). In both cases $F/\gamma ={10}^{-1}$.

Figure 3

(a) Energy spectrum of the Rabi Hamiltonian (without driving). The black dotted lines indicate energy levels with an even number of excitations whereas the red solid lines correspond to an odd number of excitations. The parity shift of the second excited state is denoted by a horizontal line. The blue arrows show the available decay channels when driving the second transition $|{\mathrm{\Psi }}_0^{+}\rangle \to |{\mathrm{\Psi }}_1^{-}\rangle$. (b) Transition rates between the different dressed-states ${\chi }_{00}^{+-}$ (yellow squares), ${\chi }_{11}^{+-}$ (blue circles), ${\chi }_{01}^{-+}$ (inverted green triangles), and ${\chi }_{01}^{+-}$ (red triangles) as a function of the coupling strength. (c) Anharmonicity of the Rabi Hamiltonian, measured by the quantity ${\mathrm{\Delta }}_{21}^{+-}-{\mathrm{\Delta }}_{10}^{-+}$. The intersection with the vertical line shows the value of $g$ for which the anharmonicity is equal to $\gamma$ (here, $\gamma /{\omega }_c={10}^{-2}$).

Figure 4

(a) Intensity $I_{\mathrm{out}}$ and (b) $g^{\left(2\right)}\left(0\right)$ as a function of the driving frequency ${\omega }_d$ for $g/{\omega }_c=0.75$. $F$ and $\gamma$ are the same as in Fig. 2.