Origin of strong photon antibunching in weakly nonlinear photonic molecules

In a recent work [Liew and Savona, Phys. Rev. Lett. 104, 183601 (2010)] it was numerically shown that a resonantly driven photonic “molecule” consisting of two coupled cavities can exhibit strong photon antibunching with a surprisingly weak Kerr nonlinearity. Here, we analytically identify the subtle quantum interference effect that is responsible for the predicted efficient photon blockade effect. We then extend the theory to the experimentally relevant Jaynes-Cummings system consisting of a single quantum emitter in a coupled-cavity structure and predict the strong antibunching even for single-atom cooperativity on the order of or smaller than unity. The potential of this quantum interference effect in the realization of strongly correlated photonic systems with only weak material nonlinearities is assessed by comparing on-site and inter-site correlations in a ring of three coupled photonic molecules.

Figure 1

(a) Sketch of the two coupled photonic modes. The coupling strength is $J$, and the antibunching is obtained with a small nonlinear energy $U$ compared to mode broadening $\gamma$. (b) Equal-time second-order correlation functions $g_{\mathit{ij}}^{(2)}(\tau =0)$ plotted as functions of nonlinearity $U=U_1=U_2$ normalized to $\gamma$. The nearly perfect antibunching is obtained at the pumped mode [$g_{11}^{(2)}(\tau =0)\simeq 0$] for $U=0.0428\gamma$. The parameters are ${\gamma }_1={\gamma }_2=\gamma$, $J=3\gamma$, $E_1=E_2=\hslash {\omega }_{\text{p}}+0.275\gamma$, and $F_1=0.01\gamma$.

Figure 2

(a) Optimal nonlinearity $U_{\text{opt}}$ and detuning $\Delta E_{\text{opt}}$ plotted as functions of inter-mode coupling strength $J$ normalized to broadening $\gamma$ (${\gamma }_1={\gamma }_2=\gamma$ and $E_1=E_2=E$). Nearly perfect antibunching is obtained for $J>\gamma /\sqrt{2}$. (b) Transition paths leading to the quantum interference responsible for the strong antibunching. One path is the direct excitation from $|10\rangle$ to $|20\rangle$, but it is forbidden by the interference with the other path drawn by dotted arrows.

Figure 3

(a) The time evolution of the second-order correlation function, which oscillates with period $2\pi /J$ as the result of amplitude oscillation between $|01\rangle$ and $|10\rangle$. (b) Equal-time second-order correlation functions plotted as functions of $\Delta E_1=\Delta E_2=\Delta E$ normalized to ${\gamma }_1={\gamma }_2=\gamma$. The spectral width of the antibunching resonance is $\approx 0.3\gamma$. The parameters are $J=3\gamma$, $U_1=U_2=0.0428\gamma$, and $F=0.01\gamma$. $\Delta E=0.275\gamma$ in panel (a).

Figure 4

Equal-time correlation functions plotted as functions of coupling strength $g$ between a cavity and a quantum dot. The solid line represents the results in the system sketched in the inset [Eq. (10)]. The parameters are ${\gamma }_1={\gamma }_2={\gamma }_{\text{ex}}=\gamma$, $J=3\gamma$, $E_1=\hslash {\omega }_{\text{p}}+0.275\gamma$, $E_2-E_1=\gamma$, $E_{\text{ex}}-E_2=2\gamma$, and $F=0.01\gamma$. The dashed line represents the result in the system with one quantum dot and one cavity [Jaynes-Cummings model]. ${\gamma }_1={\gamma }_{\text{ex}}=\gamma$, $E_{\text{ex}}-E_1=2\gamma$, ${\gamma }_1={\gamma }_2=\gamma$, and $\hslash {\omega }_{\text{p}}$ is tuned to the lower one-particle eigenenergy of the Jaynes-Cummings ladder.

Figure 5

(a) Sketch of a triangular lattice of coupled photonic “molecules.” The driven cavities ($i=1$, 2, and 3) are coupled with strength $J_2$. (b) The equal-time second-order correlation functions in each mode (solid line) and between neighbors (dashed line) plotted vs $J_2/\gamma$. The parameters are $J=3\gamma$, $\Delta E=0.450\gamma$, $U=0.0769\gamma$, and $F=0.01\gamma$.