Phase- and photon-number-dependent NOON-state localization in flat-band lattices

Flat-band lattices supporting compact localized states provide a versatile platform for exploring unconventional transport phenomena in photonic, ultracold atomic, electronic, and other systems. Here, we investigate the transport of path-entangled multi-photon NOON states in a flat-band rhombic lattice and observe intriguing localization-delocalization features that depend on both the phase and photon number of the NOON states. To experimentally emulate photon number correlations, we develop an intensity correlation measurement protocol using coherent laser light with tunable relative phases. We first apply this protocol to show spatial bunching and anti-bunching of two-photon NOON states in a one-dimensional waveguide lattice. In the rhombic lattice, we show that for an even (odd) photon number $N$, localization occurs at $0 \, (π)$ phase of the NOON state with a probability of $2^{1-N}$, which is demonstrated up to eight photons. Our results open an exciting route towards understanding the dynamics of correlated photons in complex photonic networks.