Collective Enhancement of Photon Blockade via Two-Photon Interactions

TL;DR

The work addresses achieving photon blockade when single-emitter coupling is not in the strong-coupling regime by introducing a collective two-photon light–matter interaction in a cavity–emitter ensemble. A hierarchical modeling framework (Full Quantum, Holstein–Primakoff, and Non-Hermitian) is developed to analyze both cavity-drive and emitter-drive scenarios for linear and two-photon couplings. In the nonlinear two-photon case, the minimum $g^{(2)}(0)$ scales as $g^{(2)}(0) \approx \gamma^4/(4 g^4 N^2)$ at resonance in the large-$N$ limit, demonstrating a robust collective PB in high-transmission regimes. The results show PB persists under moderate dephasing and point to practical applications in quantum light sources for sensing, communication, and computing, even when individual strong coupling is unattainable.

Abstract

Analogous to Coulomb blockade for electrons, photon blockade is a key quantum optical effect in which the presence of one photon prevents the transmission of subsequent ones through a nonlinear medium. Beyond its fundamental interest, photon and multi-photon blockade are actively studied as mechanisms for generating technologically-relevant quantum states of light. Although photon blockade typically requires achieving strong light-matter coupling, increasing the number of atoms fails to enhance antibunching. Here, we analyze the optical transmission properties of a quantum resonator that embeds a two-photon-coupled ensemble of emitters, combining an approximate analytical approach with full quantum numerical simulations. We show that when light and matter are coupled via a two-photon interaction, both single- and multi-photon blockade can benefit from a collective enhancement. We propose different driving schemes in which the second or third-order correlation functions are strongly suppressed with increasing atom number. Differently from established methods, this collective enhancement of non-classical properties occurs with unitary transmission and is ultimately constrained only by decoherence. This demonstrates that collective two-photon couplings are a powerful mechanism for realizing photon blockade even in platforms where individual strong coupling is not achievable.

Figures (7)

• Figure 1: Modeling approach and energy-level structure of the coupled cavity--emitter system.(a) Physical scheme: A cavity mode $a$ (green) couples to $N$ identical two-level emitters $\sigma_i^z$ (pink) with individual coupling strength $g$. The cavity interacts with input/output ports at rates $\gamma_{\mathrm{in}}$ and $\gamma_{\mathrm{out}}$, and each emitter undergoes individual decay at rate $\gamma_{e}$. (b) An effective bosonic model used in the HP approximation. The emitter ensemble is mapped to a collective bosonic mode $b$(pink), which interacts with the cavity mode $a$ via collective coupling $g_N = g\sqrt{N}$. The effective decay rate of the bosonic mode is $\gamma_e$. (c),(d) Level diagrams for the one-photon ($\alpha = 1$)and two-photon ($\alpha = 2$) coupling cases, respectively. In (c), the first excited doublet $|\pm\rangle_1$ enables resonant excitation via either cavity or emitter drive when $\omega_c = \omega_e$, leading to 1PB. In (d), cavity drive still enables 1PB via the $|\pm\rangle_2$ polariton, while a resonant emitter drive at $2\omega_c - \sqrt{2}g_N$ induces 2PB with photon-pair emission at $\omega_c - \frac{g_N}{\sqrt{2}}$. Photons are emitted through the cavity decay channel $\gamma_c = \gamma_{\mathrm{in}} + \gamma_{\mathrm{out}}$.
• Figure 2: One- and two-photon TC ($\alpha=1, 2$). Comparison of NH analytical and HP numerical spectra under coherent cavity drive. One-photon coupling ($\alpha = 1$) and two-photon coupling ($\alpha = 2$) are shown in the first and second rows, respectively. For each case, the normalized transmitted intensity $T$ and second-order correlation function $g^{(2)}(0)$ are plotted as functions of detuning $\Delta_c$ for varying drive amplitudes $D_c$. Solid lines (HP Num.) show numerical results from the HP model, while dashed lines (NH Ana.) correspond to analytical results from the NH model. Color shading indicates emitter number: $N = 10$ (light), $N = 100$ (medium), and $N = 1000$ (dark). Frequencies are rescaled by $\sqrt{N}$ to account for collective enhancement. Insets show zoomed-in plots near the polariton resonance. Analytical curves are overlaid but become distinguishable only where deviations from HP numerical results are significant. Parameters: $D_c = 0.01\gamma$ for (a), (b), (d), (e), and $D_c = 0.1\gamma$ for (c), (f); $g= 0.01\,\omega_c$, $\gamma_c = 2\gamma$, $\gamma_e = \gamma$, with $\gamma = 0.001\,\omega_c$.
• Figure 3: Two-photon TC ($\alpha=2$). Comparison of FQ numerical simulations with NH analytical results. Panels (a), (b) show the transmitted intensity $T$, while (c), (d) present the second-order correlation function $g^{(2)}(0)$, under weak and strong cavity driving. NH analytical predictions are shown in orange (light for $N = 10$, dark for $N = 20$); FQ numerical results are shown in green (light for $N = 10$, dark for $N = 20$). Parameters: Panels (a), (c): weak drive $D_c = 0.1\gamma$; panels (b), (d): strong drive $D_c = \gamma$; $g= 0.01\,\omega_c$, $\gamma_c = \gamma_e = \gamma = 0.01\,\omega_c$,$\gamma = 0.01\,\omega_c$, same values for Fig. \ref{['fig4: g2-min-vs-N']} and Fig. \ref{['fig:dep']}.
• Figure 4: Two-photon TC ($\alpha=2$). Minimum of $g^{(2)}(0)$ for varying emitter number $N$. Minimum values of the second-order correlation function $g^{(2)}(0)$ as a function of emitter number $N$, for (a) weak drive $D_c = 0.1\gamma$ and (b) strong drive $D_c = \gamma$.
• Figure 5: Two-photon TC ($\alpha=2$). Effect of emitter dephasing on NH analytical results. Second-order correlation function $g^{(2)}(0)$ obtained from NH analytical results (orange solid lines) and FQ simulations (dots), for $N = 20$ emitters. Parameters: $D_c = 0.1\gamma$; emitter dephasing rates $\gamma_d = 0$, $0.1\gamma$, and $\gamma$.