"Spatially modulated ordered states, such as crystals, liquid crystals, and antiferromagnets, are ubiquitous in nature but relatively difficult to realize in ultracold atomic systems. In the present work, we present a scheme for generating controllable cavity-mediated interactions between atoms, and show that these interactions give rise to a crystallization transition in the case of a transversely pumped optical cavity. We focus on the case of multimode cavities, in which the interactions are relatively local and the range of possible ordered configurations (and consequently of low-energy fluctuations) is large; as we show, the crystallization transition for a Bose-Einstein condensate in a multimode cavity is driven first-order by fluctuations, through the Brazovskii effect. The ordered state to which this crystallization transition gives rise is a ""supersolid,"" possessing both superfluid and solid order. We address the crystallization transition and the properties of the ordered state, discuss the experimental feasibility of observing these, and finally show how ordering in layered systems of atoms is geometrically frustrated. We then introduce a more straightforward realization of frustrated cavity-mediated interactions, viz. systems of randomly-positioned spins in multimode cavities. We show by means of a mapping to a variant of the Hopfield associative-memory model that such systems exhibit a spin-glass phase. Finally, we consider a different ultracold-atomic setting---that of spin-orbit-coupled Bose gases---in which the Brazovskii effect has a profound influence on the low-temperature phases, leading to a universal preference for stripe-like ordering at zero temperature and bosonic pair condensation at nonzero temperatures."
"Spatially modulated ordered states, such as crystals, liquid crystals, and antiferromagnets, are ubiquitous in nature but relatively difficult to realize in ultracold atomic systems. In the present work, we present a scheme for generating controllable cavity-mediated interactions between atoms, and show that these interactions give rise to a crystallization transition in the case of a transversely pumped optical cavity. We focus on the case of multimode cavities, in which the interactions are relatively local and the range of possible ordered configurations (and consequently of low-energy fluctuations) is large; as we show, the crystallization transition for a Bose-Einstein condensate in a multimode cavity is driven first-order by fluctuations, through the Brazovskii effect. The ordered state to which this crystallization transition gives rise is a ""supersolid,"" possessing both superfluid and solid order. We address the crystallization transition and the properties of the ordered state, discuss the experimental feasibility of observing these, and finally show how ordering in layered systems of atoms is geometrically frustrated. We then introduce a more straightforward realization of frustrated cavity-mediated interactions, viz. systems of randomly-positioned spins in multimode cavities. We show by means of a mapping to a variant of the Hopfield associative-memory model that such systems exhibit a spin-glass phase. Finally, we consider a different ultracold-atomic setting---that of spin-orbit-coupled Bose gases---in which the Brazovskii effect has a profound influence on the low-temperature phases, leading to a universal preference for stripe-like ordering at zero temperature and bosonic pair condensation at nonzero temperatures."
The self-organization of a Bose-Einstein condensate (BEC) in a transversely pumped optical cavity is a process akin to crystallization: when pumped by a laser of sufficient intensity, the coupled matter and light fields evolve, spontaneously, into a spatially modulated pattern, or crystal, whose lattice structure is dictated by the geometry of the cavity.In cavities having multiple degenerate modes, the quasicontinuum of possible lattice arrangements, and the continuous symmetry breaking associated with the adoption of a particular lattice arrangement, give rise to phenomena such as phonons, defects, and frustration, which have hitherto been unexplored in ultracold atomic settings involving neutral atoms.The present work develops a nonequilibrium field-theoretic approach to explore the self-organization of a BEC in a pumped, lossy optical cavity.We find that the transition is well described, in the regime of primary interest, by an effective equilibrium theory.At nonzero temperatures, the self-organization occurs via a fluctuation-driven first-order phase transition of the Brazovskii class; this transition persists to zero temperature and crosses over into a quantum phase transition.We make further use of our field-theoretic description to investigate the role of nonequilibrium fluctuations in the self-organization transition, as well as to explore the nucleation of ordered-phase droplets, the nature and energetics of topological defects, supersolidity in the ordered phase, and the possibility of frustration controlled by the cavity geometry.In addition, we discuss the range of experimental parameters for which we expect the phenomena described here to be observable, along with possible schemes for detecting ordering and fluctuations via either atomic correlations or the correlations of the light emitted from the cavity.
The self-organization of a Bose-Einstein condensate (BEC) in a transversely pumped optical cavity is a process akin to crystallization: when pumped by a laser of sufficient intensity, the coupled matter and light fields evolve, spontaneously, into a spatially modulated pattern, or crystal, whose lattice structure is dictated by the geometry of the cavity. In cavities having multiple degenerate modes, the quasicontinuum of possible lattice arrangements, and the continuous symmetry breaking associated with the adoption of a particular lattice arrangement, give rise to phenomena such as phonons, defects, and frustration, which have hitherto been unexplored in ultracold atomic settings involving neutral atoms. The present work develops a nonequilibrium field-theoretic approach to explore the self-organization of a BEC in a pumped, lossy optical cavity. We find that the transition is well described, in the regime of primary interest, by an effective equilibrium theory. At nonzero temperatures, the self-organization occurs via a fluctuation-driven first-order phase transition of the Brazovskii class; this transition persists to zero temperature and crosses over into a quantum phase transition. We make further use of our field-theoretic description to investigate the role of nonequilibrium fluctuations in the self-organization transition, as well as to explore the nucleation of ordered-phase droplets, the nature and energetics of topological defects, supersolidity in the ordered phase, and the possibility of frustration controlled by the cavity geometry. In addition, we discuss the range of experimental parameters for which we expect the phenomena described here to be observable, along with possible schemes for detecting ordering and fluctuations via either atomic correlations or the correlations of the light emitted from the cavity.
We show that the effective spin-spin interaction between three-level atoms confined in a multimode optical cavity is long-ranged and sign changing, like the RKKY interaction; therefore, ensembles of such atoms subject to frozen-in positional randomness can realize spin systems having disordered and frustrated interactions. We argue that, whenever the atoms couple to sufficiently many cavity modes, the cavity-mediated interactions give rise to a spin glass. In addition, we show that the quantum dynamics of cavity-confined spin systems is that of a Bose-Hubbard model with strongly disordered hopping but no on-site disorder; this model exhibits a random-singlet glass phase, absent in conventional optical-lattice realizations. We briefly discuss experimental signatures of the realizable phases.
Ultracold bosonic atoms in optical lattices self-organize into a variety of structural and quantum phases when placed into a single-mode cavity and pumped by a laser. Cavity optomechanical effects induce an atom density modulation at the cavity-mode wavelength that competes with the optical lattice arrangement. Simultaneously short-range interactions via particle hopping promote superfluid order such that a variety of structural and quantum coherent phases can occur. We analyze the emerging phase diagram in two dimensions by means of an extended Bose-Hubbard model using a local mean-field approach combined with a superfluid cluster analysis. For commensurate ratios of the cavity and external lattice wavelengths, the Mott insulator-superfluid transition is modified by the appearance of charge density wave and supersolid phases, at which the atomic density supports the buildup of a cavity field. For incommensurate ratios, the optomechanical forces induce the formation of Bose-glass and superglass phases, namely, nonsuperfluid and superfluid phases, respectively, displaying quasiperiodic density modulations, which in addition can exhibit structural and superfluid stripe formation. The onset of such structures is constrained by the on-site interaction and is favorable at fractional densities. Experimental observables are identified and discussed.
We realize a driven-dissipative Ising spin glass using cavity QED in a novel "4/7" multimode geometry. Gases of ultracold atoms trapped within the cavity by optical tweezers serve as effective spins. They are coupled via randomly signed, all-to-all Ising cavity-mediated interactions. Networks of up to n=25 spins are holographically imaged via cavity emission. The system is driven through a frustrated transverse-field Ising transition, and we show that the entropy of the spin glass states depends on the rate at which the transition is crossed. Despite being intrinsically nonequilibrium, the system exhibits phenomena associated with Parisi's theory of equilibrium spin glasses, namely, replica symmetry breaking (RSB) and ultrametric structure. For system sizes up to n=16, we measure the Parisi function q(x), Edwards-Anderson overlap q_{EA}, and ultrametricity K correlator; all indicate a deeply ordered spin glass under RSB. The system can serve as an associative memory and enable aging and rejuvenation studies in driven-dissipative spin glasses at the microscopic level.
We provide strong evidence that the effective spin-spin interaction in a multimodal confocal optical cavity gives rise to a self-induced glassy phase, which emerges exclusively from the peculiar Euclidean correlations and is not related to the presence of disorder as in standard spin glasses. As recently shown, this spin-spin effective interaction is both nonlocal and nontranslational invariant, and randomness in the atoms' positions produces a spin glass phase. Here we consider the simplest feasible disorder-free setting, where atoms form a one-dimensional regular chain and we study the thermodynamics of the resulting effective Ising model. We present extensive results showing that the system has a low-temperature glassy phase. The model depends on the adimensional parameter α=(a/w_{0})^{2}, a being a lattice spacing and w_{0} an interaction length scale. Notably, for rational values of α=p/q, the number of metastable states at low temperature grows exponentially with q and the problem of finding the ground state rapidly becomes computationally intractable, suggesting that the system develops high-energy barriers and ergodicity breaking occurs.