Interaction-induced two-photon edge states in an extended Hubbard model realized in a cavity array

We study theoretically two-photon states in a periodic array of coupled cavities with both on-site and nonlocal Kerr-type nonlinearities. In the absence of nonlinearity the structure is topologically trivial and possesses no edge states. The interplay of two nonlinear interaction mechanisms described by the extended Hubbard model facilitates the formation of edge states of bound photon pairs. Numerical and exact analytical results for the two-photon wave functions are presented. Our findings thus shed light onto the edge states of composite particles and their localization properties.

Figure 1

(a) Array of identical coupled resonators described by the extended Hubbard model. (b) Illustration of the equivalent two-dimensional problem.

Figure 2

(a) Calculated doublon dispersion and the continuum of quasi-independent photon states for $U/J=7,W/J=3$. The dashed line indicates the energy of the doublon edge state. (b)–(d) Probability distributions calculated for a finite chain composed of a $N=21$ resonator: (b) doublon $U$ band, (c) doublon $W$ band, and (d) doubly degenerate doublon edge state.

Figure 3

(a) and (b) Doublon dispersion and the continuum of quasi-independent photon states. (c) and (d) Enlarged fragment of the doublon dispersion curve in the region of collapse. (e) and (f) Real and imaginary parts of the wave-number $\varkappa$ describing photons colocalization in the region of collapse. Calculations are performed for (a), (c), and (e) $U/J=1,W/J=3$ and (b), (d), and (f) $U/J=3,W/J=1$.

Figure 4

Doublon tunneling pathways and qualitative explanation of the doublon edge states emergence.

Figure 5

Phase diagrams of (a) stable doublon branches and (b) doublon edge states. (a) Parameter domains corresponding to the two stable doublon branches are highlighted in blue. (b) Parameter domains corresponding to the two doublon edge states are shown in green.

Figure 6

(a) and (b) Energy bands of quasi-independent photons (green) and doublons (red) versus $W/J$ for the fixed value of $U/J=6$. The dashed line indicates the energy of the doublon edge state. Panel (b) shows the relative positions of the doublon $W$ band and the doublon edge state in more detail. Point $T$ indicates the transition from negative to positive doublon mass. (c)–(e) Dispersion of the doublon $W$ band for the fixed value of $U/J=6$ and (c) $W/J=2.1$; (d) $W/J=2.37$; (e) $W/J=2.7$.