We consider a spinless ultracold Fermi gas tightly trapped along the axis of an optical resonator and transversely illuminated by a laser closely tuned to a resonator mode. At a certain threshold pump intensity, the homogeneous gas density breaks a Z_{2} symmetry towards a spatially periodic order, which collectively scatters pump photons into the cavity. We show that this known self-ordering transition also occurs for low field seeking fermionic particles when the laser light is blue detuned to an atomic transition. The emergent superradiant optical lattice in this case is homopolar and possesses two distinct dimerizations. Depending on the spontaneously chosen dimerization, the resulting Bloch bands can have a nontrivial topological structure characterized by a nonvanishing Zak phase. In the case where the Fermi momentum is close to half of the cavity-mode wave number, a Peierls-like instability here creates a topological insulator with a gap at the Fermi surface, which hosts a pair of edge states. The topological features of the system can be nondestructively observed via the cavity output: the Zak phase of the bulk coincides with the relative phase between laser and cavity field, while the fingerprint of edge states can be observed as additional broadening in a well-defined frequency window of the cavity spectrum.

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We study spatial spin and density self-ordering of a two-component Bose-Einstein condensate via collective Raman scattering into a linear cavity mode. The onset of the Dicke superradiance phase transition is marked by a simultaneous appearance of a crystalline density order and a spin-wave order. The latter spontaneously breaks the discrete Z2 symmetry between even and odd sites of the cavity optical potential. Moreover, in the superradiant state the continuous U (1) symmetry of the relative phase of the two condensate wavefunctions is explicitly broken by the cavity-induced position-dependent Raman coupling with a zero spatial average. Thus, the spatially-averaged relative condensate phase is locked at either π/2 or −π/2. This continuous symmetry breaking and relative condensate phase locking by a zero-average Raman field can be considered as a generic order-by-disorder process similar to the random-field-induced order in the two-dimensional classical ferromagnetic XY spin model. However, the seed of the random field in our model stems from quantum fluctuations in the cavity field and is a dynamical entity affected by self-ordering. The spectra of elementary excitations exhibit the typical mode softening at the superradiance threshold.Introduction.-Loading Bose-Einstein condensates (BECs) into optical potentials created by dynamic cavity fields has opened a new avenue in ultracold atomic physics [1][2][3][4], paving the way for realization of novel phenomena [5]. Seminal results include the Dicke superradiance phase transition [6][7][8], and quantum phase transitions between superfluid, superradiant Mott insulator, density-wave state, lattice supersolid, and supersolid phase with a broken continuous U (1) symmetry due to the interplay between cavity-mediated long-range interactions and short-range collisional interactions [9][10][11]. On the theoretical side, in addition to studying conventional quantum-optics and self-ordering aspects of coupled quantum-gas-cavity environments [12][13][14][15][16][17][18][19][20], many proposals have been put forward to simulate and realize exotic phenomena for ultracold atoms via coupling to dynamic cavity fields, including synthetic gauge fields [21][22][23][24][25][26], topological states [27][28][29][30], and superconductor-related physics [31].In this Letter we study the Dicke superradiance phase transition for a generalized atomic system with both internal [32][33][34][35][36] and external [6-8] quantized degrees of freedom, i.e., a spinor BEC, coupled to a single mode of a linear cavity (see Fig. 1). The ultracold four-level atoms are transversely illuminated by two sufficiently far red-detuned pump lasers polarized along the cavity axis x so that to induce near resonant two-photon Raman transitions between the lowest two internal atomic states via the same cavity mode with the transverse polarization along z as in Ref. [36]. After adiabatic elimination of the atomic excited states, the system reduces to a twocomponent BEC coupled via a cavity-induced positiondependent ...

Supersolids are characterized by the counterintuitive coexistence of superfluid and crystalline order. Here we study a supersolid phase emerging in the steady state of a driven-dissipative system. We consider a transversely pumped Bose-Einstein condensate trapped along the axis of a ring cavity and coherently coupled to a pair of degenerate counterpropagating cavity modes. Above a threshold pump strength the interference of photons scattered into the two cavity modes results in an emergent superradiant lattice, which spontaneously breaks the continuous translational symmetry towards a periodic atomic pattern. The crystalline steady state inherits the superfluidity of the Bose-Einstein condensate, thus exhibiting genuine properties of a supersolid. A gapless collective Goldstone mode correspondingly appears in the superradiant phase, which can be nondestructively monitored via the relative phase of the two cavity modes on the cavity output. Despite cavity-photon losses the Goldstone mode remains undamped, indicating the robustness of the supersolid phase.

The recent experimental observation of spinor self-ordering of ultracold atoms in optical resonators has set the stage for the exploration of emergent magnetic orders in quantum-gas-cavity systems. Based on this platform, we introduce a generic scheme for the implementation of longrange quantum spin Hamiltonians composed of various types of couplings, including Heisenberg and Dzyaloshinskii-Moriya interactions. Our model is comprised of an effective two-component Bose-Einstein condensate, driven by two classical pump lasers and coupled to a single dynamic mode of a linear cavity in a double Λ scheme. Cavity photons mediate the long-range spin-spin interactions with spatially modulated coupling coefficients, where the latter ones can be tuned by modifying spatial profiles of the pump lasers. As experimentally relevant examples, we demonstrate that by properly choosing the spatial profiles of the pump lasers achiral domain-wall antiferromagnetic and chiral spin-spiral orders emerge beyond critical laser strengths. The transition between these two phases can be observed in a single experimental setup by tuning the reflectivity of a mirror. We also discuss extensions of our scheme for the implementation of other classes of spin Hamiltonians.Introduction.-Quantum magnetism plays a crucial role in many phenomena in condensed matter physics [1], including for instance high-temperature superconductivity [2] and spin liquids [3]. In materials, there exist different forms of interactions between electronic spins. The Heisenberg interaction, originating from the isotropic quantum exchange interaction between electrons, favors ferromagnetic (FM) or antiferromagnetic (AFM) ordering [4]. The more exotic Dzyaloshinskii-Moriya (DM) interaction [5][6][7], stemming from a relativistic antisymmetric exchange interaction, favors chiral states such as spin spiral (SS) and skyrmion [8][9][10][11][12][13][14][15], with potential applications in spintronics [16].