Exciton-polariton systems can sustain macroscopic quantum states (MQSs) under a periodic potential\nmodulation. In this paper, we investigate the structure of these states in acoustic square lattices by\nprobing their wave functions in real and momentum spaces using spectral tomography.Weshow that\nthe polariton MQSs, when excited by a Gaussian laser beam, self-organize in a concentric structure,\nconsisting of a single, two-dimensional gap-soliton (GS) state surrounded by one dimensional (1D)\nMQSs with lower energy. The latter form at hyperbolical points of the modulated polariton dispersion.\nWhile the size of the GS tends to saturate with increasing particle density, the emission region of the\nsurrounding 1D states increases. The existence of these MQSs in acoustic lattices is quantitatively\nsupported by a theoretical model based on the variational solution of the Grossââ?¬â??Pitaevskii equation.\nThe formation of the 1D states in a ring around the central GS is attributed to the energy gradient in\nthis region, which reduces the overall symmetry of the lattice. The results broaden the experimental\nunderstanding of self-localized polariton states, which may prove relevant for functionalities\nexploiting solitonic objects.
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