Photon Blockade in Weakly Driven Cavity Quantum Electrodynamics Systems with Many Emitters

We use the scattering matrix formalism to analyze photon blockade in coherently driven cavity quantum electrodynamics systems with a weak drive. By approximating the weak coherent drive by an input single- and two-photon Fock state, we reduce the computational complexity of the transmission and the two-photon correlation function from exponential to polynomial in the number of emitters. This enables us to easily analyze cavity-based systems containing $\sim 50$ quantum emitters with modest computational resources. Using this approach we study the coherence statistics of photon blockade while increasing the number of emitters for resonant and detuned multiemitter cavity quantum electrodynamics systems—we find that increasing the number of emitters worsens photon blockade in resonant systems, and improves it in detuned systems. We also analyze the impact of inhomogeneous broadening in the emitter frequencies on the photon blockade through this system.

Figure 1

Schematic of the multiemitter CQED system. An optical cavity mode couples to $N$ emitters with coupling constants $g_i$, $1\le i\le N$. The input and output coupling constants are ${\kappa }_b$ and ${\kappa }_c$. In addition to the cavity, the emitters also couple to loss channels with coupling constants ${\gamma }_i$. The total decay rate of the cavity is given by $\kappa ={\kappa }_b+{\kappa }_c$ (we assume ${\kappa }_b={\kappa }_c=\kappa /2$ throughout this Letter).

Figure 2

Equal-time correlation $g^{(2)}(0;{\omega }_L)$ and transmissivity $T({\omega }_L)$ for (a) strongly coupled resonant emitters ($g=2\kappa ,{\omega }_e={\omega }_c$), (b) weakly coupled resonant emitters ($g=0.2\kappa ,{\omega }_e={\omega }_c$), and (c) weakly coupled detuned emitters ($g=0.2\kappa ,{\omega }_e-{\omega }_c=0.8\kappa$). The insets show the dependence of $\underset{{\omega }_L}{\min }[g^{(2)}(0;{\omega }_L)]$ as a function of $N$ and the dashed lines show $\underset{N\to \infty }{\lim }{g}^{(2)}(0;{\omega }_L)$ computed using Eq. (8). Increasing the number of emitters clearly deteriorates the polaritonic photon blockade observed in the system. For detuned systems, increasing the number of emitters enhances the interference-based blockade. $\gamma =0.01\kappa$ is assumed in all simulations.

Figure 3

(a) Anharmonicity $\mathrm{\Delta }{\omega }_{1,2}$ and the linewidth $\delta {\omega }_2$ of the most harmonic eigenstate in the second excitation subspace as a function of number of emitters. We have used $g=2\kappa$ and only considered those two-excitation eigenstates which have a nonzero overlap with two photons in the cavity while computing $\mathrm{\Delta }{\omega }_{1,2}$ and $\delta {\omega }_2$. (b) Amplitude and phase of the equal-time two-photon emission (${\mathrm{\Gamma }}_{\pm },{\mathrm{\Gamma }}_0$) from the three eigenstates ($|{\varphi }_{\pm }^{(2)}⟩$, $|{\varphi }_0^{(2)}⟩$) that contribute to $g^{(2)}(0;{\omega }_L)$ in a detuned multiemitter CQED system with weakly coupled emitters ($g=0.2\kappa$, ${\omega }_e-{\omega }_c=0.8\kappa$) at the frequency corresponding to the interference-based blockade. $\gamma =0.01\kappa$ is assumed in all simulations.

Figure 4

Impact of inhomogeneous broadening on the photon blockade in multiemitter CQED systems for (a) the emitters, on an average, being resonant with the cavity. (b) emitters that are, on an average, detuned from the cavity resonance by $⟨{\omega }_e⟩-{\omega }_c=0.8\kappa =2\pi \times 20\mathrm{GHz}$. For both cases, we show a typical line shape $g^{(2)}(0;{\omega }_L)$, and the statistics of the frequencies ${\omega }_B$ and the $g^{(2)}(0;{\omega }_B)$ values for the polaritonic and subradiant photon blockade (note that the $y$ axis in the histogram is the unnormalized frequency of occurrence of the sample statistic). Parameter values $\mathrm{\Delta }=25\mathrm{GHz}$, $\kappa =2\pi \times 25\mathrm{GHz}$, $g=0.2\kappa =2\pi \times 5\mathrm{GHz}$, and $\gamma =2\pi \times 0.3\mathrm{GHz}$ are assumed in all simulations.