Photon Blockade in the Ultrastrong Coupling Regime

We explore photon coincidence counting statistics in the ultrastrong coupling regime, where the atom-cavity coupling rate becomes comparable to the cavity resonance frequency. In this regime, usual normal order correlation functions fail to describe the output photon statistics. By expressing the electric-field operator in the cavity-emitter dressed basis, we are able to propose correlation functions that are valid for arbitrary degrees of light-matter interaction. Our results show that the standard photon blockade scenario is significantly modified for ultrastrong coupling. We observe parametric processes even for two-level emitters and temporal oscillations of intensity correlation functions at a frequency given by the ultrastrong photon emitter coupling. These effects can be traced back to the presence of two-photon cascade decays induced by counterrotating interaction terms.

Figure 1

Energy spectrum of $H_0$ as a function of the coupling strength for $\theta =\pi /2$ (left panel) and $\theta =0.93$ (right panel). In both panels ${\omega }_x={\omega }_0$. The vertical line marks the coupling strength ($g=0.2{\omega }_0$) that is used in what follows. For $\theta =\pi /2$ the level structure is analogous to that of the JC model. The arrows indicate possible transitions of radiative decay.

Figure 2

Emission characteristics of our system. (a) $g^{(2)}(0)$ as a function of the frequency ${\omega }_d$ of the coherent drive for $\theta =0.93$ (red solid line) and $\theta =\pi /2$ (blue dashed line). (b) Resonance fluorescence spectra calculated for a driving field tuned with the transition $|0⟩\to |1⟩$, i.e., the lower polariton (green dashed line), and with the transition $|0⟩\to |2⟩$, i.e., the upper polariton (orange dash-dotted line), for $\theta =0.93$. The remaining parameters are ${\omega }_x={\omega }_0$, ${\gamma }_a={\gamma }_x={10}^{-2}{\omega }_0$, $\Omega ={10}^{-4}{\omega }_0$, and $g=0.2{\omega }_0$ for all plots.

Figure 3

$g^{(2)}(\tau )$ for a driving frequency resonant to the energy of the first (blue dashed line) and on the second polariton (red solid line). ${\omega }_x={\omega }_0$, ${\gamma }_a={\gamma }_x={10}^{-2}{\omega }_0$, $\Omega ={10}^{-4}{\omega }_0$, $g=0.2{\omega }_0$, and $\theta =0.93$ for both cases.