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We investigate a two-dimensional Bose-Einstein condensate that is opticallydriven via a retro-reflecting mirror, forming a single optical feedback loop.This induces a peculiar type of long-range atomic interaction with highlyoscillatory behavior, and we show here how the sign of the underlyinginteraction potential can be controlled by additional optical elements andexternal fields. This additional tunability enriches the behavior of the systemsubstantially, and gives rise to a surprising range of new ground states of thecondensate. In particular, we find the emergence of self-bound crystals ofquantum droplets with various lattice structures, from simple and familiartriangular arrays to complex superlattice structures and crystals with entirelybroken rotational symmetry. This includes mesoscopic clusters composed of smallnumbers of quantum droplets as well as extended crystalline structures.Importantly, such ordered states are entirely self-bound and stable without anyexternal in-plane confinement, having no counterpart to other quantum-gassettings with long-range atomic interactions.

Self-bound droplet clusters in laser-driven Bose-Einstein condensates