Probing photon correlations in the dark sites of geometrically frustrated cavity lattices

We explore theoretically the driven-dissipative physics of geometrically frustrated lattices of cavity resonators with relatively weak nonlinearities, i.e., a photon-photon interaction smaller than the loss rate. In such systems, photon modes with zero probability at dark sites are present at the single-particle level due to interference effects. In particular, we study the behavior of a cell with three coupled resonators as well as extended Lieb lattices in one and two dimensions. By considering a partial pumping scheme, with the driving field not applied to the dark sites, we predict that even in the presence of relatively weak photon-photon interactions the nominally dark sites achieve a finite photonic population with strong correlations. We show that this is a consequence of biphoton and multiphoton states that in the absence of frustration would not be visible in the observables.

Figure 1

Sketch of the driven-dissipative one-dimensional Lieb lattice with the partial driving scheme. The elementary unit cell is indicated by the box. Only the $c$ sites are coherently driven and the $b$ sites are dark with respect to the $a$ and $c$ sites. The indicated system parameters are discussed in the text.

Figure 2

(a) Sketch of three coupled driven-dissipative cavities (corresponding to a single unit cell of the Lieb lattice with open boundary conditions) and the corresponding single-particle energy-level structure for the closed system (in units of $J$), with three levels at $E^{\left(1\right)}-{\omega }_c=-\sqrt{2}J$, 0, and $\sqrt{2}J$. (b) Population on the central site $n_b$ (on a logarithmic scale) as a function of the detuning $\mathrm{\Delta }$ (in units of $\gamma$) for $F=\gamma$, $U=0.1\gamma$, and $J=5\gamma$. An anomalous peak is found around $\mathrm{\Delta }=0$, completely missed by the nonequilibrium Gross-Pitaevkii approximation (thin line). (c) Energies of the six two-photon eigenstates (in units of $J$) as a function of the nonlinearity $U$ (in units of $J$). The dotted line is the approximation $U/4$ for small $U/J$ discussed in the text. (d) Probabilities (normalized to the two-photon manifold) of the two-photon eigenstates (3) in the limit $F\to 0$ as a function of the detuning. The dashed line gives the sum of the probabilities for the other four eigenstates. The system parameters are $J=5\gamma$ and $U=0.1\gamma$.

Figure 3

Second-order correlation function $g_b^{\left(2\right)}$ on the central dark site as a function of the system parameters (in units of the loss rate $\gamma$), namely, (a) the detuning $\mathrm{\Delta }$, (b) the driving amplitude $F$, (c) the hopping term $J$, and (d) the on-site photon-photon interaction $U$ [note the logarithmic scales for the vertical axes in (a) and (c) and the horizontal axis in (d)]. While not varied, the fixed parameters are $\mathrm{\Delta }=0$, $F=\gamma$, $U=0.1\gamma$, and $J=5\gamma$. The dotted lines indicate the value $g_b^{\left(2\right)}=1$ for a coherent state. In (a) the dashed line is the analytical result in the limit $F\to 0$ and the solid black line is the result for $g_b^{\left(3\right)}$, which is also strongly peaked around $\mathrm{\Delta }\simeq 0$. In (b) the inset presents the corresponding density on the dark site $n_b$.

Figure 4

Second-order correlation function $g_{i,j}^{\left(2\right)}$ for dark sites as a function of the distance $|i-j|$ between the sites (in terms of unit cells) for a 1D Lieb lattice consisting of 12 unit cells and periodic boundary conditions. The same curves are shown on a logarithmic scale (main figure) and in linear scale (inset). The green triangles are the results for a uniform drive with driving amplitude $F=0.1\gamma$ and hopping strength $J=2\gamma$. The other parameters are the same for all results and are $U=0.3\gamma$ and $\mathrm{\Delta }=0$.

Figure 5

Sketch of the 1D Lieb lattice with open boundary conditions studied here. The numbers $1,2,...,5$ indicate the different dark sites.