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The self-organization of a Bose-Einstein condensate in a transversely pumpedoptical cavity is a process akin to crystallization: when pumped by a laser ofsufficient intensity, the coupled matter and light fields evolve,spontaneously, into a spatially modulated pattern, or crystal, whose latticestructure is dictated by the geometry of the cavity. In cavities havingmultiple degenerate modes, the quasi-continuum of possible latticearrangements, and the continuous symmetry breaking associated with the adoptionof a particular lattice arrangement, give rise to phenomena such as phonons,defects, and frustration, which have hitherto been unexplored in ultracoldatomic settings involving neutral atoms. The present work develops anonequilibrium field-theoretic approach to explore the self-organization of aBEC in a pumped, lossy optical cavity. We find that the transition is welldescribed, in the regime of primary interest, by an effective equilibriumtheory. At nonzero temperatures, the self-organization occurs via afluctuation-driven first-order phase transition of the Brazovskii class; thistransition persists to zero temperature, and crosses over into a quantum phasetransition of a new universality class. We make further use of ourfield-theoretic description to investigate the role of nonequilibriumfluctuations on the self-organization transition, as well as to explore thenucleation of ordered-phase droplets, the nature and energetics of topologicaldefects, supersolidity in the ordered phase, and the possibility of frustrationcontrolled by the cavity geometry. In addition, we discuss the range ofexperimental parameters for which we expect the phenomena described here to beobservable, along with possible schemes for detecting ordering and fluctuationsvia either atomic correlations or the correlations of the light emitted fromthe cavity.

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

Atom-light crystallization of BECs in multimode cavities: Nonequilibrium  classical and quantum phase transitions, emergent lattices, supersolidity,  and frustration