  The ability to create and manipulate strongly correlated quantum many-body
states is of central importance to the study of collective phenomena in several
condensed-matter systems. In the last decades, a great amount of work has been
focused on ultracold atoms in optical lattices, which provide a flexible
platform to simulate peculiar phases of matter both for fermionic and bosonic
particles. The recent experimental demonstration of Bose-Einstein condensation
(BEC) of light in dye-filled microcavities has opened the intriguing
possibility to build photonic simulators of solid-state systems, with potential
advantages over their atomic counterpart. A distinctive feature of photon BEC
is the thermo-optical nature of the effective photon-photon interaction, which
is intrinsically nonlocal and can thus induce interactions of arbitrary range.
This offers the opportunity to systematically study the collective behaviour of
many-body systems with tunable interaction range. In this paper, we
theoretically study the effect of nonlocal interactions in photon BEC. We first
present numerical results of BEC in a double-well potential, and then extend
our analysis to a short one-dimensional lattice with open boundaries. By
resorting to a numerical procedure inspired by the Newton-Raphson method, we
simulate the time-independent Gross-Pitaevskii equation and provide evidence of
surface localization induced by nonlocality, where the condensate density is
localized at the boundaries of the potential. Our work paves the way towards
the realization of synthetic matter with photons, where the interplay between
long-range interactions and low dimensionality can lead to the emergence of
unexplored nontrivial collective phenomena.
