A stripe phase with supersolid properties in spin–orbit-coupled Bose–Einstein condensates

Excitations of a Binary Dipolar Supersolid

We predict a rich excitation spectrum of a binary dipolar supersolid in a linear crystal geometry, where the ground state consists of two partially immiscible components with alternating, interlocking domains. We identify three Goldstone branches, each with first-sound, second-sound, or spin-sound character. In analogy with a diatomic crystal, the resulting lattice has a two-domain primitive basis and we find that the crystal (first-sound-like) branch is split into optical and acoustic phonons. We also find a spin-Higgs branch that is associated with the supersolid modulation amplitude.

Measurement of the superfluid fraction of a supersolid by Josephson effect

A new class of superfluids and superconductors with spatially periodic modulation of the superfluid density is arising1-12. It might be related to the supersolid phase of matter, in which the spontaneous breaking of gauge and translational symmetries leads to a spatially modulated macroscopic wavefunction13-16. This relation was recognized only in some cases1,2,5-9 and there is the need for a universal property quantifying the differences between supersolids and ordinary matter, such as the superfluid fraction, which measures the reduction in superfluid stiffness resulting from the spatial modulation16-18. The superfluid fraction was introduced long ago16, but it has not yet been assessed experimentally. Here we demonstrate an innovative method to measure the superfluid fraction based on the Josephson effect, a ubiquitous phenomenon associated with the presence of a physical barrier between two superfluids or superconductors19, which might also be expected for supersolids20, owing to the spatial modulation. We demonstrate that individual cells of a supersolid can sustain Josephson oscillations and we show that, from the current-phase dynamics, we can derive directly the superfluid fraction. Our study of a cold-atom dipolar supersolid7 reveals a relatively large sub-unity superfluid fraction that makes realistic the study of previously unknown phenomena such as partially quantized vortices and supercurrents16-18. Our results open a new direction of research that may unify the description of all supersolid-like systems.

Optical spin hall effect in exciton-polariton condensates in lead halide perovskite microcavities

An exciton-polariton condensate is a hybrid light-matter state in the quantum fluid phase. The photonic component endows it with characters of spin, as represented by circular polarization. Spin-polarization can form stochastically for quasi-equilibrium exciton-polariton condensates at parallel momentum vector k|| ∼ 0 from bifurcation or deterministically for propagating condensates at k|| > 0 from the optical spin-Hall effect (OSHE). Here, we report deterministic spin-polarization in exciton-polariton condensates at k|| ∼ 0 in microcavities containing methylammonium lead bromide perovskite (CH3NH3PbBr3) single crystals under non-resonant and linearly polarized excitation. We observe two energetically split condensates with opposite circular polarizations and attribute this observation to the presence of strong birefringence, which introduces a large OSHE at k|| ∼ 0 and pins the condensates in a particular spin state. Such spin-polarized exciton-polariton condensates may serve not only as circularly polarized laser sources but also as effective alternatives to ultracold atom Bose-Einstein condensates in quantum simulators of many-body spin-orbit coupling processes.

Topological spin-orbit-coupled fermions beyond rotating wave approximation

The realization of spin-orbit-coupled ultracold gases has driven a wide range of research and is typically based on the rotating wave approximation (RWA). By neglecting the counter-rotating terms, RWA characterizes a single near-resonant spin-orbit (SO) coupling in a two-level system. Here, we propose and experimentally realize a new scheme for achieving a pair of two-dimensional (2D) SO couplings for ultracold fermions beyond RWA. This work not only realizes the first anomalous Floquet topological Fermi gas beyond RWA, but also significantly improves the lifetime of the 2D-SO-coupled Fermi gas. Based on pump-probe quench measurements, we observe a deterministic phase relation between two sets of SO couplings, which is characteristic of our beyond-RWA scheme and enables the two SO couplings to be simultaneously tuned to the optimum 2D configurations. We observe intriguing band topology by measuring two-ring band-inversion surfaces, quantitatively consistent with a Floquet topological Fermi gas in the regime of high Chern numbers. Our study can open an avenue to explore exotic SO physics and anomalous topological states based on long-lived SO-coupled ultracold fermions.

Giant magnetocaloric effect in spin supersolid candidate Na2BaCo(PO4)2

Supersolid, an exotic quantum state of matter that consists of particles forming an incompressible solid structure while simultaneously showing superfluidity of zero viscosity1, is one of the long-standing pursuits in fundamental research2,3. Although the initial report of 4He supersolid turned out to be an artefact4, this intriguing quantum matter has inspired enthusiastic investigations into ultracold quantum gases5-8. Nevertheless, the realization of supersolidity in condensed matter remains elusive. Here we find evidence for a quantum magnetic analogue of supersolid-the spin supersolid-in the recently synthesized triangular-lattice antiferromagnet Na2BaCo(PO4)2 (ref. 9). Notably, a giant magnetocaloric effect related to the spin supersolidity is observed in the demagnetization cooling process, manifesting itself as two prominent valley-like regimes, with the lowest temperature attaining below 100 mK. Not only is there an experimentally determined series of critical fields but the demagnetization cooling profile also shows excellent agreement with the theoretical simulations with an easy-axis Heisenberg model. Neutron diffractions also successfully locate the proposed spin supersolid phases by revealing the coexistence of three-sublattice spin solid order and interlayer incommensurability indicative of the spin superfluidity. Thus, our results reveal a strong entropic effect of the spin supersolid phase in a frustrated quantum magnet and open up a viable and promising avenue for applications in sub-kelvin refrigeration, especially in the context of persistent concerns about helium shortages10,11.

Glitches in Rotating Supersolids

Glitches, spin-up events in neutron stars, are of prime interest, as they reveal properties of nuclear matter at subnuclear densities. We numerically investigate the glitch mechanism due to vortex unpinning using analogies between neutron stars and dipolar supersolids. We explore the vortex and crystal dynamics during a glitch and its dependence on the supersolid quality, providing a tool to study glitches from different radial depths of a neutron star. Benchmarking our theory against neutron-star observations, our work will open a new avenue for the quantum simulation of stellar objects from Earth.

Dynamical self-trapping of two-dimensional binary solitons in cross-combined linear and nonlinear optical lattices

Dynamical and self-trapping properties of two-dimensional (2D) binary mixtures of Bose-Einstein condensates in cross-combined lattices, consisting of a one-dimensional (1D) linear optical lattice (LOL) in the x direction for the first component and a 1D nonlinear optical lattice (NOL) in the y direction for the second component, are analytically and numerically investigated. The existence and stability of 2D binary matter wave solitons in these settings are demonstrated both by variational analysis and by direct numerical integration of the coupled Gross-Pitaevskii equations. We find that in the absence of the NOL, binary solitons, stabilized by the action of the 1D LOL and by the attractive intercomponent interaction, can freely move in the y direction. In the presence of the NOL, we find, quite remarkably, the existence of threshold curves in the parameter space separating regions where solitons can move from regions where the solitons become dynamically self-trapped. The mechanism underlying the dynamical self-trapping phenomenon (DSTP) is qualitatively understood in terms of a dynamical barrier induced by the NOL, similar to the Peirls-Nabarro barrier of solitons in discrete lattices. DSTP is numerically demonstrated for binary solitons that are put in motion both by phase imprinting and by the action of external potentials applied in the y direction. In the latter case, we show that the trapping action of the NOL allows one to maintain a 2D binary soliton at rest in a nonequilibrium position of a parabolic trap or to prevent it from falling under the action of gravity. Possible applications of the results are also briefly discussed.

Tuning Photon-Mediated Interactions in a Multimode Cavity: From Supersolid to Insulating Droplets Hosting Phononic Excitations

Ultracold atoms trapped in laser-generated optical lattices serve as a versatile platform for quantum simulations. However, as these lattices are infinitely stiff, they do not allow to emulate phonon degrees of freedom. This restriction can be lifted in emerged optical lattices inside multimode cavities. Motivated by recent experimental progress in multimode cavity QED, we propose a scheme to implement and study supersolid and droplet states with phononlike lattice excitations by coupling a Bose gas to many longitudinal modes of a ring cavity. The interplay between contact collisional and tunable-range cavity-mediated interactions leads to a rich phase diagram, which includes elastic supersolid as well as insulating droplet phases exhibiting roton-type mode softening for a continuous range of momenta across the superradiant phase transition. The nontrivial dynamic response of the system to a local density perturbation further proves the existence of phononlike modes.

Emergence of a Spin Microemulsion in Spin-Orbit Coupled Bose-Einstein Condensates

We report the first numerical prediction of a "spin microemulsion"-a phase with undulating spin domains resembling classical bicontinuous oil-water-surfactant emulsions-in two-dimensional systems of spinor Bose-Einstein condensates with isotropic Rashba spin-orbit coupling. Using field-theoretic numerical simulations, we investigated the melting of a low-temperature stripe phase with supersolid character and find that the stripes lose their superfluidity at elevated temperature and undergo a Kosterlitz-Thouless-like transition into a spin microemulsion. Momentum distribution calculations highlight a thermally broadened occupation of the Rashba circle of low-energy states with macroscopic and isotropic occupation around the ring. We provide a finite-temperature phase diagram that positions the emulsion as an intermediate, structured isotropic phase with residual quantum character before transitioning at higher temperature into a structureless normal fluid.

Photon blockade with a trapped Λ-type three-level atom in asymmetrical cavity

We propose a scheme to manipulate strong and nonreciprocal photon blockades in asymmetrical Fabry-Perot cavity with a Λ-type three-level atom. Utilizing the mechanisms of both conventional and unconventional blockade, the strong photon blockade is achieved by the anharmonic eigenenergy spectrum brought by Λ-type atom and the destructive quantum interference effect induced by a microwave field. By optimizing the system parameters, the manipulation of strong photon blockade over a wide range of cavity detuning can be realized. Using spatial symmetry breaking introduced by the asymmetry of cavity, the direction-dependent nonreciprocal photon blockade can be achieved, and the nonreciprocity can reach the maximum at optimal cavity detuning. In particular, manipulating the occurring position of nonreciprocal photon blockade can be implemented by simply adjusting the cavity detuning. Our scheme provides feasible access for generating high-quality nonreciprocal single-photon sources.

Observation of Anisotropic Superfluid Density in an Artificial Crystal

We experimentally and theoretically investigate the anisotropic speed of sound of an atomic superfluid (SF) Bose-Einstein condensate in a 1D optical lattice. Because the speed of sound derives from the SF density, this implies that the SF density is itself anisotropic. We find that the speed of sound is decreased by the optical lattice, and the SF density is concomitantly reduced. This reduction is accompanied by the appearance of a zero entropy normal fluid in the purely Bose condensed phase. The reduction in SF density-first predicted [A. J. Leggett, Phys. Rev. Lett. 25, 1543 (1970).PRLTAO0031-900710.1103/PhysRevLett.25.1543] in the context of supersolidity-results from the coexistence of superfluidity and density modulations, but is agnostic about the origin of the modulations. We additionally measure the moment of inertia of the system in a scissors mode experiment, demonstrating the existence of rotational flow. As such we shed light on some supersolid properties using imposed, rather than spontaneously formed, density order.

Time Translation Symmetry Breaking in an Isolated Spin-Orbit-Coupled Fluid of Light

We study the interplay between intrinsic spin-orbit coupling and nonlinear photon-photon interactions in a nonparaxial, elliptically polarized fluid of light propagating in a bulk Kerr medium. We find that in situations where the nonlinear interactions induce birefringence, i.e., a polarization-dependent nonlinear refractive index, their interplay with spin-orbit coupling results in an interference between the two polarization components of the fluid traveling at different wave vectors, which entails the breaking of translation symmetry along the propagation direction. This phenomenon leads to a Floquet band structure in the Bogoliubov spectrum of the fluid, and to characteristic oscillations of its intensity correlations. We characterize these oscillations in detail and point out their exponential growth at large propagation distances, revealing the presence of parametric resonances.

Superconductivity, Charge Density Wave, and Supersolidity in Flat Bands with a Tunable Quantum Metric

Predicting the fate of an interacting system in the limit where the electronic bandwidth is quenched is often highly nontrivial. The complex interplay between interactions and quantum fluctuations driven by the band geometry can drive competition between various ground states, such as charge density wave order and superconductivity. In this work, we study an electronic model of topologically trivial flat bands with a continuously tunable Fubini-Study metric in the presence of on-site attraction and nearest-neighbor repulsion, using numerically exact quantum Monte Carlo simulations. By varying the electron filling and the minimal spatial extent of the localized flat-band Wannier wave functions, we obtain a number of intertwined orders. These include a phase with coexisting charge density wave order and superconductivity, i.e., a supersolid. In spite of the nonperturbative nature of the problem, we identify an analytically tractable limit associated with a "small" spatial extent of the Wannier functions and derive a low-energy effective Hamiltonian that can well describe our numerical results. We also provide unambiguous evidence for the violation of any putative lower bound on the zero-temperature superfluid stiffness in geometrically nontrivial flat bands.

Superfluid Fraction in an Interacting Spatially Modulated Bose-Einstein Condensate

At zero temperature, a Galilean-invariant Bose fluid is expected to be fully superfluid. Here we investigate theoretically and experimentally the quenching of the superfluid density of a dilute Bose-Einstein condensate due to the breaking of translational (and thus Galilean) invariance by an external 1D periodic potential. Both Leggett's bound fixed by the knowledge of the total density and the anisotropy of the sound velocity provide a consistent determination of the superfluid fraction. The use of a large-period lattice emphasizes the important role of two-body interactions on superfluidity.

Dynamics of Stripe Patterns in Supersolid Spin-Orbit-Coupled Bose Gases

Despite ground-breaking observations of supersolidity in spin-orbit-coupled Bose-Einstein condensates, until now the dynamics of the emerging spatially periodic density modulations has been vastly unexplored. Here, we demonstrate the nonrigidity of the density stripes in such a supersolid condensate and explore their dynamic behavior subject to spin perturbations. We show both analytically in infinite systems and numerically in the presence of a harmonic trap how spin waves affect the supersolid's density profile in the form of crystal waves, inducing oscillations of the periodicity as well as the orientation of the fringes. Both these features are well within reach of present-day experiments. Our results show that this system is a paradigmatic supersolid, featuring superfluidity in conjunction with a fully dynamic crystalline structure.

Dipolar physics: a review of experiments with magnetic quantum gases

Since the achievement of quantum degeneracy in gases of chromium atoms in 2004, the experimental investigation of ultracold gases made of highly magnetic atoms has blossomed. The field has yielded the observation of many unprecedented phenomena, in particular those in which long-range and anisotropic dipole-dipole interactions (DDIs) play a crucial role. In this review, we aim to present the aspects of the magnetic quantum-gas platform that make it unique for exploring ultracold and quantum physics as well as to give a thorough overview of experimental achievements. Highly magnetic atoms distinguish themselves by the fact that their electronic ground-state configuration possesses a large electronic total angular momentum. This results in a large magnetic moment and a rich electronic transition spectrum. Such transitions are useful for cooling, trapping, and manipulating these atoms. The complex atomic structure and large dipolar moments of these atoms also lead to a dense spectrum of resonances in their two-body scattering behaviour. These resonances can be used to control the interatomic interactions and, in particular, the relative importance of contact over dipolar interactions. These features provide exquisite control knobs for exploring the few- and many-body physics of dipolar quantum gases. The study of dipolar effects in magnetic quantum gases has covered various few-body phenomena that are based on elastic and inelastic anisotropic scattering. Various many-body effects have also been demonstrated. These affect both the shape, stability, dynamics, and excitations of fully polarised repulsive Bose or Fermi gases. Beyond the mean-field instability, strong dipolar interactions competing with slightly weaker contact interactions between magnetic bosons yield new quantum-stabilised states, among which are self-bound droplets, droplet assemblies, and supersolids. Dipolar interactions also deeply affect the physics of atomic gases with an internal degree of freedom as these interactions intrinsically couple spin and atomic motion. Finally, long-range dipolar interactions can stabilise strongly correlated excited states of 1D gases and also impact the physics of lattice-confined systems, both at the spin-polarised level (Hubbard models with off-site interactions) and at the spinful level (XYZ models). In the present manuscript, we aim to provide an extensive overview of the various related experimental achievements up to the present.

Manipulating polariton condensates by Rashba-Dresselhaus coupling at room temperature

Spin-orbit coupling plays an important role in the spin Hall effect and topological insulators. Bose-Einstein condensates with spin-orbit coupling show remarkable quantum phase transition. In this work we control an exciton polariton condensate – a macroscopically coherent state of hybrid light and matter excitations – by virtue of the Rashba-Dresselhaus (RD) spin-orbit coupling. This is achieved in a liquid-crystal filled microcavity where CsPbBr₃ perovskite microplates act as the gain material at room temperature. Specifically, we realize an artificial gauge field acting on the CsPbBr₃ exciton polariton condensate, splitting the condensate fractions with opposite spins in both momentum and real space. Besides the ground states, higher-order discrete polariton modes can also be split by the RD effect. Our work paves the way to manipulate exciton polariton condensates with a synthetic gauge field based on the RD spin-orbit coupling at room temperature.

Engineered spin-orbit coupling can induce novel quantum phases in a Bose-Einstein condensate, however such demonstrations have been limited to cold atom systems. Here the authors realize a exciton-polarion condensate with tunable spin-orbit coupling in a liquid crystal microcavity at room temperature.

Two-Dimensional Supersolid Formation in Dipolar Condensates

Dipolar condensates have recently been coaxed to form the long-sought supersolid phase. While one-dimensional supersolids may be prepared by triggering a roton instability, we find that such a procedure in two dimensions (2D) leads to a loss of both global phase coherence and crystalline order. Unlike in 1D, the 2D roton modes have little in common with the supersolid configuration. We develop a finite-temperature stochastic Gross-Pitaevskii theory that includes beyond-mean-field effects to explore the formation process in 2D and find that evaporative cooling directly into the supersolid phase-hence bypassing the first-order roton instability-can produce a robust supersolid in a circular trap. Importantly, the resulting supersolid is stable at the final nonzero temperature. We then experimentally produce a 2D supersolid in a near-circular trap through such an evaporative procedure. Our work provides insight into the process of supersolid formation in 2D and defines a realistic path to the formation of large two-dimensional supersolid arrays.

Formation of Matter-Wave Droplet Lattices in Multi-Color Periodic Confinements

In the paper, we introduce a new model that addresses the generation of quantum droplets (QDs) in the binary Bose–Einstein condensate (BEC) mixture with mutually symmetric spinor components loaded in multi-color optical lattices (MOLs) of commensurate wavelengths and tunable intensities. The considered MOL confinement is the combination of the four-color optical lattice with an exponential periodic trap, which includes the complete set of the Fourier harmonics. Employing the one-dimensional (1D) extended Gross–Pitäevskii equation (eGPE), we calculate the exact analytical form of the wavefunction, MF/BMF nonlinearities, and MOL trap parameters. Utilizing the exact solutions, the formation of supersolid-like spatially periodic matter-wave droplet lattices and superlattices is illustrated under the space-periodic nonlinearity management. The precise positioning of the density maxima/minima of the droplet patterns at the center of the trap and tunable Anderson-like localization are observed by tuning the symmetry and amplitude of the considered MOL trap. The stability of the obtained solution is confirmed using the Vakhitov–Kolokolov (VK) criterion.

Supersolid gap soliton in a Bose-Einstein condensate and optical ring cavity coupling system

The system of a transversely pumped Bose-Einstein condensate (BEC) coupled to a lossy ring cavity can favor a supersolid steady state. Here we find the existence of supersolid gap soliton in such a driven-dissipative system. By numerically solving the mean-field atom-cavity field coupling equations, gap solitons of a few different families have been identified. Their dynamical properties, including stability, propagation, and soliton collision, are also studied. Due to the feedback atom-intracavity field interaction, these supersolid gap solitons show numerous new features compared with the usual BEC gap solitons in static optical lattices.

Discovery of quantum phases in the Shastry-Sutherland compound SrCu2(BO3)2 under extreme conditions of field and pressure

The 2-dimensional layered oxide material SrCu2(BO3)2, long studied as a realization of the Shastry-Sutherland spin topology, exhibits a range of intriguing physics as a function of both hydrostatic pressure and magnetic field, with a still debated intermediate plaquette phase appearing at approximately 20 kbar and a possible deconfined critical point at higher pressure. Here, we employ a tunnel diode oscillator (TDO) technique to probe the behavior in the combined extreme conditions of high pressure, high magnetic field, and low temperature. We reveal an extensive phase space consisting of multiple magnetic analogs of the elusive supersolid phase and a magnetization plateau. In particular, a 10 × 2 supersolid and a 1/5 plateau, identified by infinite Projected Entangled Pair States (iPEPS) calculations, are found to rely on the presence of both magnetic and non-magnetic particles in the sea of dimer singlets. These states are best understood as descendants of the full-plaquette phase, the leading candidate for the intermediate phase of SrCu2(BO3)2.

Spin-orbit coupling in buckled monolayer nitrogene

Buckled monolayer nitrogene has been recently predicted to be stable above the room temperature. The low atomic number of nitrogen atom suggests, that spin–orbit coupling in nitrogene is weak, similar to graphene or silicene. We employ first principles calculations and perform a systematic study of the intrinsic and extrinsic spin–orbit coupling in this material. We calculate the spin mixing parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^2$$\end{document}b2, reflecting the strength of the intrinsic spin–orbit coupling and find, that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^2$$\end{document}b2 is relatively small, on the order of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-6}$$\end{document}10-6. It also displays a weak anisotropy, opposite for electrons and holes. To study extrinsic effects of spin–orbit coupling we apply a transverse electric field enabling spin–orbit fields \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}Ω. We find, that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}Ω are on the order of a single \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document}μeV in the valence band, and tens to a hundred of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document}μeV in the conduction band, depending on the applied electric field. Similar to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^2$$\end{document}b2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}Ω is also anisotropic, in particular for the conduction electrons.

Prediction of a Supersolid Phase in High-Pressure Deuterium

Supersolid is a mysterious and puzzling state of matter whose possible existence has stirred a vigorous debate among physicists for over 60 years. Its elusive nature stems from the coexistence of two seemingly contradicting properties, long-range order and superfluidity. We report computational evidence of a supersolid phase of deuterium under high pressure (p>800  GPa) and low temperature (T<1.0  K). In our simulations, that are based on bosonic path integral molecular dynamics, we observe a highly concerted exchange of atoms while the system preserves its crystalline order. The exchange processes are favored by the soft core interactions between deuterium atoms that form a densely packed metallic solid. At the zero temperature limit, Bose-Einstein condensation is observed as the permutation probability of N deuterium atoms approaches 1/N with a finite superfluid fraction. Our study provides concrete evidence for the existence of a supersolid phase in high-pressure deuterium and could provide insights on the future investigation of supersolid phases in real materials.

Crystallization of bosonic quantum Hall states in a rotating quantum gas

The dominance of interactions over kinetic energy lies at the heart of strongly correlated quantum matter, from fractional quantum Hall liquids¹, to atoms in optical lattices² and twisted bilayer graphene³. Crystalline phases often compete with correlated quantum liquids, and transitions between them occur when the energy cost of forming a density wave approaches zero. A prime example occurs for electrons in high-strength magnetic fields, where the instability of quantum Hall liquids towards a Wigner crystal^4-9 is heralded by a roton-like softening of density modulations at the magnetic length^7,10-12. Remarkably, interacting bosons in a gauge field are also expected to form analogous liquid and crystalline states^13-21. However, combining interactions with strong synthetic magnetic fields has been a challenge for experiments on bosonic quantum gases^18,21. Here we study the purely interaction-driven dynamics of a Landau gauge Bose-Einstein condensate²² in and near the lowest Landau level. We observe a spontaneous crystallization driven by condensation of magneto-rotons^7,10, excitations visible as density modulations at the magnetic length. Increasing the cloud density smoothly connects this behaviour to a quantum version of the Kelvin-Helmholtz hydrodynamic instability, driven by the sheared internal flow profile of the rapidly rotating condensate. At long times the condensate self-organizes into a persistent array of droplets separated by vortex streets, which are stabilized by a balance of interactions and effective magnetic forces.

Spin-orbit-coupled exciton-polariton condensates in lead halide perovskites

Spin-orbit coupling (SOC) is responsible for a range of spintronic and topological processes in condensed matter. Here, we show photonic analogs of SOCs in exciton-polaritons and their condensates in microcavities composed of birefringent lead halide perovskite single crystals. The presence of crystalline anisotropy coupled with splitting in the optical cavity of the transverse electric and transverse magnetic modes gives rise to a non-Abelian gauge field, which can be described by the Rashba-Dresselhaus Hamiltonian near the degenerate points of the two polarization modes. With increasing density, the exciton-polaritons with pseudospin textures undergo phase transitions to competing condensates with orthogonal polarizations. Unlike their pure photonic counterparts, these exciton-polaritons and condensates inherit nonlinearity from their excitonic components and may serve as quantum simulators of many-body SOC processes.

Localization and spin dynamics of spin-orbit-coupled Bose-Einstein condensates in deep optical lattices

We analytically and numerically discuss the dynamics of two pseudospin components Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in deep optical lattices. Rich localized phenomena, such as breathers, solitons, self-trapping, and diffusion, are revealed and strongly depend on the strength of the atomic interaction, SOC, Raman detuning, and the spin polarization (i.e., the initial population difference of atoms between the two pseudospin components of BECs). The critical conditions for the transition of localized states are derived analytically. Based on the critical conditions, the detailed dynamical phase diagram describing the different dynamical regimes is derived. When the Raman detuning satisfies a critical condition, localized states with a fixed initial spin polarization can be observed. When the critical condition is not satisfied, we use two quenching methods, i.e., suddenly and linearly quenching Raman detuning from the soliton or breather state, to discuss the spin dynamics, phase transition, and wave packet dynamics by numerical simulation. The sudden quenching results in a damped oscillation of spin polarization and transforms the system to a new polarized state. Interestingly, the linear quenching of Raman detuning induces a controllable phase transition from an unpolarized phase to an expected polarized phase, while the soliton or breather dynamics is maintained.

Spatially Modulated Superfluid State in Two-Dimensional ^{4}He Films

The second layer of ^{4}He films adsorbed on a graphite substrate is an excellent experimental platform to study the interplay between superfluid and structural orders. Here, we report a rigid two-frequency torsional oscillator study on the second layer as a function of temperature and ^{4}He atomic density. For the first time, we show experimentally that the superfluid density is independent of frequency, which can be interpreted as unequivocal evidence of genuine superfluidity. The phase diagram established in this work reveals that a superfluid phase coexists with hexatic density-wave correlation and a registered solid phase. This suggests the second layer as a candidate for hosting two exotic quantum ground states: the spatially modulated superfluid and supersolid phases resulting from the interplay between superfluid and structural orders.

Symbiotic solitons in quasi-one- and quasi-two-dimensional spin-1 condensates

We study the formation of spin-1 symbiotic spinor solitons in a quasi-one- (quasi-1D) and quasi-two-dimensional (quasi-2D) hyperfine spin F=1 ferromagnetic Bose-Einstein condensate (BEC). The symbiotic solitons necessarily have a repulsive intraspecies interaction and are bound due to an attractive interspecies interaction. Due to a collapse instability in higher dimensions, an additional spin-orbit coupling is necessary to stabilize a quasi-2D symbiotic spinor soliton. Although a quasi-1D symbiotic soliton has a simple Gaussian-type density distribution, novel spatial periodic structure in density is found in quasi-2D symbiotic SO-coupled spinor solitons. For a weak SO coupling, the quasi-2D solitons are of the (-1,0,+1) or (+1,0,-1) type with intrinsic vorticity and multiring structure, for Rashba or Dresselhaus SO coupling, respectively, where the numbers in the parentheses are angular momenta projections in spin components F_{z}=+1,0,-1, respectively. For a strong SO coupling, stripe and superlattice solitons, respectively, with a stripe and square-lattice modulation in density, are found in addition to the multiring solitons. The stationary states were obtained by imaginary-time propagation of a mean-field model; dynamical stability of the solitons was established by real-time propagation over a long period of time. The possibility of the creation of such a soliton by removing the trap of a confined spin-1 BEC in a laboratory is also demonstrated.

Exciting the Goldstone Modes of a Supersolid Spin-Orbit-Coupled Bose Gas

Supersolidity is deeply connected with the emergence of Goldstone modes, reflecting the spontaneous breaking of both phase and translational symmetry. Here, we propose accessible signatures of these modes in harmonically trapped spin-orbit-coupled Bose-Einstein condensates, where supersolidity appears in the form of stripes. By suddenly changing the trapping frequency, an axial breathing oscillation is generated, whose behavior changes drastically at the critical Raman coupling. Above the transition, a single mode of hybridized density and spin nature is excited, while below it, we predict a beating effect signaling the excitation of a Goldstone spin-dipole mode. We further provide evidence for the Goldstone mode associated with the translational motion of stripes. Our results open up new perspectives for probing supersolid properties in experimentally relevant configurations with both symmetric as well as highly asymmetric intraspecies interactions.

Spatial order in a two-dimensional spin-orbit-coupled spin-1/2 condensate: superlattice, multi-ring and stripe formation

We demonstrate the formation of stable spatially-ordered states in auniformand alsotrappedquasi-two-dimensional (quasi-2D) Rashba or Dresselhaus spin-orbit (SO) coupled pseudo spin-1/2 Bose-Einstein condensate using the mean-field Gross-Pitaevskii equation. For weak SO coupling, one can have a circularly-symmetric (0, +1)- or (0, -1)-type multi-ring state with intrinsic vorticity, for Rashba or Dresselhaus SO coupling, respectively, where the numbers in the parentheses denote the net angular momentum projection in the two components, in addition to a circularly-asymmetric degenerate state with zero net angular momentum projection. For intermediate SO couplings, in addition to the above two types, one can also have states with stripe pattern in component densities with no periodic modulation in total density. The stripe state continues to exist for large SO coupling. In addition, a new spatially-periodic state appears in the uniform system: asuperlatticestate, possessing some properties of asupersolid, with a square-lattice pattern in component densities and also in total density. In a trapped system the superlattice state is slightly different with multi-ring pattern in component density and a square-lattice pattern in total density. For an equal mixture of Rashba and Dresselhaus SO couplings, in both uniform and trapped systems, only stripe states are found for all strengths of SO couplings. In a uniform system all these states are quasi-2D solitonic states.

Two-dimensional supersolidity in a dipolar quantum gas

Supersolid states simultaneously feature properties typically associated with a solid and with a superfluid. Like a solid, they possess crystalline order, manifesting as a periodic modulation of the particle density; but unlike a typical solid, they also have superfluid properties, resulting from coherent particle delocalization across the system. Such states were initially envisioned in the context of bulk solid helium, as a possible answer to the question of whether a solid could have superfluid properties^1-5. Although supersolidity has not been observed in solid helium (despite much effort)⁶, ultracold atomic gases provide an alternative approach, recently enabling the observation and study of supersolids with dipolar atoms^7-16. However, unlike the proposed phenomena in helium, these gaseous systems have so far only shown supersolidity along a single direction. Here we demonstrate the extension of supersolid properties into two dimensions by preparing a supersolid quantum gas of dysprosium atoms on both sides of a structural phase transition similar to those occurring in ionic chains^17-20, quantum wires^21,22 and theoretically in chains of individual dipolar particles^23,24. This opens the possibility of studying rich excitation properties^25-28, including vortex formation^29-31, and ground-state phases with varied geometrical structure^7,32 in a highly flexible and controllable system.

Stripe phases in WSe2/WS2 moiré superlattices

Stripe phases, in which the rotational symmetry of charge density is spontaneously broken, occur in many strongly correlated systems with competing interactions^1-11. However, identifying and studying such stripe phases remains challenging. Here we uncover stripe phases in WSe₂/WS₂ moiré superlattices by combining optical anisotropy and electronic compressibility measurements. We find strong electronic anisotropy over a large doping range peaked at 1/2 filling of the moiré superlattice. The 1/2 state is incompressible and assigned to an insulating stripe crystal phase. Wide-field imaging reveals domain configurations with a preferential alignment along the high-symmetry axes of the moiré superlattice. Away from 1/2 filling, we observe additional stripe crystals at commensurate filling 1/4, 2/5 and 3/5, and compressible electronic liquid crystal states at incommensurate fillings. Our results demonstrate that two-dimensional semiconductor moiré superlattices are a highly tunable platform from which to study the stripe phases and their interplay with other symmetry breaking ground states.

Self-Organization in Cold Atoms Mediated by Diffractive Coupling

This article discusses self-organization in cold atoms via light-mediated interactions induced by feedback from a single retro-reflecting mirror. Diffractive dephasing between the pump beam and the spontaneous sidebands selects the lattice period. Spontaneous breaking of the rotational and translational symmetry occur in the 2D plane transverse to the pump. We elucidate how diffractive ripples couple sites on the self-induced atomic lattice. The nonlinear phase shift of the atomic cloud imprinted onto the optical beam is the parameter determining coupling strength. The interaction can be tailored to operate either on external degrees of freedom leading to atomic crystallization for thermal atoms and supersolids for a quantum degenerate gas, or on internal degrees of freedom like populations of the excited state or Zeeman sublevels. Using the light polarization degrees of freedom on the Poincaré sphere (helicity and polarization direction), specific irreducible tensor components of the atomic Zeeman states can be coupled leading to spontaneous magnetic ordering of states of dipolar and quadrupolar nature. The requirements for critical interaction strength are compared for the different situations. Connections and extensions to longitudinally pumped cavities, counterpropagating beam schemes and the CARL instability are discussed.

Birth, Life, and Death of a Dipolar Supersolid

In the short time since the first observation of supersolid states of ultracold dipolar atoms, substantial progress has been made in understanding the zero-temperature phase diagram and low-energy excitations of these systems. Less is known, however, about their finite-temperature properties, particularly relevant for supersolids formed by cooling through direct evaporation. Here, we explore this realm by characterizing the evaporative formation and subsequent decay of a dipolar supersolid by combining high-resolution in-trap imaging with time-of-flight observables. As our atomic system cools toward quantum degeneracy, it first undergoes a transition from thermal gas to a crystalline state with the appearance of periodic density modulation. This is followed by a transition to a supersolid state with the emergence of long-range phase coherence. Further, we explore the role of temperature in the development of the modulated state.

Supersolid-like states in a two-dimensional trapped spin-orbit-coupled spin-1 condensate

We study supersolid-like states in a quasi-two-dimensional trapped Rashba and Dresselhaus spin-orbit (SO) coupled spin-1 condensate. For small strengths of SO couplingγ(γ⪅ 0.75), in the ferromagnetic phase, circularly-symmetric (0, ±1, ±2)- and (∓1, 0, ±1)-type states are formed where the numbers in the parentheses denote the angular momentum of the vortex at the center of the components and where the upper (lower) sign correspond to Rashba (Dresselhaus) coupling; in the antiferromagnetic phase, only (∓1, 0, ±1)-type states are formed. For large strengths of SO coupling, supersolid-like superlattice and superstripe states are formed in the ferromagnetic phase. In the antiferromagnetic phase, for large strengths of SO coupling, supersolid-like superstripe and multi-ring states are formed. For an equal mixture of Rashba and Dresselhaus SO couplings, only a superstripe state is found. All these states are found to be dynamically stable and hence accessible in an experiment and will enhance the fundamental understanding of crystallization onto radially periodic states in solids.

Exotic Vortex States with Discrete Rotational Symmetry in Atomic Fermi Gases with Spin-Orbital-Angular-Momentum Coupling

We investigate the superfluidity of a two-component Fermi gas with spin-orbital-angular-momentum coupling (SOAMC). Because of the intricate interplay of SOAMC, two-photon detuning and atom-atom interaction, a family of vortex ground states emerges in a broad parameter regime of the phase diagram, in contrast to the usual case where an external rotation or magnetic field is generally required. More strikingly, an unprecedented vortex state, which breaks the continuous rotational symmetry to a discrete one spontaneously, is predicted to occur. The underlying physics are elucidated and verified by numerical simulations. The unique density distributions of the predicted vortex states enable a direct observation in experiment.

Vortex-lattice formation in a spin-orbit coupled rotating spin-1 condensate

We study the vortex-lattice formation in a rotating Rashba spin-orbit (SO) coupled quasi-two-dimensional (quasi-2D) hyper-fine spin-1 spinor Bose-Einstein condensate (BEC) in the x-y plane using a numerical solution of the underlying mean-field Gross-Pitaevskii equation. In this case, the non-rotating Rashba SO-coupled spinor BEC can have topological excitation in the form of vortices of different angular momenta in the three components, e.g. the (0, +1, +2)- and (-1, 0, +1)-type states in ferromagnetic and anti-ferromagnetic spinor BEC: the numbers in the parenthesis denote the intrinsic angular momentum of the vortex states of the three components with the negative sign denoting an anti-vortex. The presence of these states with intrinsic vorticity breaks the symmetry between rotation with vorticity along the z and -z axes and thus generates a rich variety of vortex-lattice and anti-vortex-lattice states in a rotating quasi-2D spin-1 spinor ferromagnetic and anti-ferromagnetic BEC, not possible in a scalar BEC. For weak SO coupling, we find two types of symmetries of these states - hexagonal and 'square'. The hexagonal (square) symmetry state has vortices arranged in closed concentric orbits with a maximum of 6, 12, 18… (8, 12, 16…) vortices in successive orbits. Of these two symmetries, the square vortex-lattice state is found to have the smaller energy.

Ground-state phase and superfluidity of tunable spin-orbit-coupled Bose-Einstein condensates

We theoretically study the ground-state phases and superfluidity of tunable spin-orbit-coupled Bose-Einstein condensates (BECs) under the periodic driving of Raman coupling. An effective time-independent Floquet Hamiltonian is proposed by using a high-frequency approximation, and we find single-particle dispersion, spin-orbit-coupling, and asymmetrical nonlinear two-body interaction can be modulated effectively by the periodic driving. The critical Raman coupling characterizing the phase transition and relevant physical quantities in three different phases (the stripe phase, plane-wave phase, and zero momentum phase) are obtained analytically. Our results indicate that the boundary of ground-state phases can be controlled and the system will undergo three different phase transitions by adjusting the external driving. Interestingly, we find the contrast of the stripe density can be enhanced by the periodic driving in the stripe phase. We also study the superfluidity of tunable spin-orbit-coupled BECs and find the dynamical instability can be tuned by the periodic driving of Raman coupling. Furthermore, the sound velocity of the ground-state and superfluidity state can be controlled effectively by tuning the periodic driving strength. Our results indicate that the periodic driving of Raman coupling provides a powerful tool to manipulate the ground-state phase transition and dynamical instability of spin-orbit-coupled BECs.

New states of matter with fine-tuned interactions: quantum droplets and dipolar supersolids

Quantum fluctuations can stabilize Bose-Einstein condensates (BEC) against the mean-field collapse. Stabilization of the condensate has been observed in quantum degenerate Bose-Bose mixtures and dipolar BECs. The fine-tuning of the interatomic interactions can lead to the emergence of two new states of matter: liquid-like self-bound quantum droplets and supersolid crystals formed from these droplets. We review the properties of these exotic states of matter and summarize the experimental progress made using dipolar quantum gases and Bose-Bose mixtures. We conclude with an outline of important open questions that could be addressed in the future.

Ideal Weyl semimetal with 3D spin-orbit coupled ultracold quantum gas

There is an immense effort in search for various types of Weyl semimetals, of which the most fundamental phase consists of the minimal number of i.e. two Weyl points, but is hard to engineer in solids. Here we demonstrate how such fundamental Weyl semimetal can be realized in a maneuverable optical Raman lattice, with which the three-dimensional (3D) spin-orbit (SO) coupling is synthesised for ultracold atoms. In addition, a new novel Weyl phase with coexisting Weyl nodal points and nodal ring is also predicted here, and is shown to be protected by nontrivial linking numbers. We further propose feasible techniques to precisely resolve 3D Weyl band topology through 2D equilibrium and dynamical measurements. This work leads to the first realization of the most fundamental Weyl semimetal band and the 3D SO coupling for ultracold quantum gases, which are respectively the significant issues in the condensed matter and ultracold atom physics.

Spin-Nematic Vortex States in Cold Atoms

The (pseudo)spin degrees of freedom greatly enriches the physics of cold atoms. This is particularly so for systems with high spins (i.e., spin quantum number larger than 1/2). For example, one can construct not only the rank-1 spin vector, but also the rank-2 spin tensor in high spin systems. Here we propose a simple scheme to couple the spin tensor and the center-of-mass orbital angular momentum in a spin-1 cold atom system and show that this leads to a new quantum phase of the matter: the spin-nematic vortex state that features vorticity in an SU(2) spin-nematic tensor subspace. Under proper conditions, such states are characterized by quantized topological numbers. Our work opens up new avenues of research in topological quantum matter with high spins.

Bénard-von Kármán vortex street in a spin-orbit-coupled Bose-Einstein condensate

The dynamics of pseudo-spin-1/2 Bose-Einstein condensates with weak spin-orbit coupling through a moving obstacle potential are studied numerically. Four types of wakes are observed and the phase diagrams are determined for different spin-orbit coupling strengths. The conditions to form Bénard-von Kármán vortex street are rather rigorous, and we investigate in detail the dynamical characteristics of the vortex streets. The two point vortices in a pair rotate around their center, and the angular velocity and their distance oscillate periodically. The oscillation intensifies with increasing spin-orbit coupling strengths, and it makes part of the vortex pairs dissociate into separate vortices or combine into single ones and destroys the vortex street in the end. The width b of the street and the distance l between two consecutive vortex pairs of the same circulation are determined by the potential radius and its moving velocity, respectively. The b/l ratios are independent of the spin-orbit coupling strength and fall in the range 0.19-0.27, which is a little smaller than the stability criterion 0.28 for classical fluids. Proper b/l ratios are necessary to form Bénard-von Kármán vortex street, but the spin-orbit coupling strength affects the stability of the street patterns. Finally, we propose a protocol to experimentally realize the vortex street in ^{87}Rb spin-orbit-coupling Bose-Einstein condensates.

Stable Nonlinear Modes Sustained by Gauge Fields

We reveal the universal effect of gauge fields on the existence, evolution, and stability of solitons in the spinor multidimensional nonlinear Schrödinger equation. Focusing on the two-dimensional case, we show that when gauge field can be split in a pure gauge and a nonpure gauge generating effective potential, the roles of these components in soliton dynamics are different: the localization characteristics of emerging states are determined by the curvature, while pure gauge affects the stability of the modes. Respectively the solutions can be exactly represented as the envelopes which may depend on the pure gauge implicitly through the effective potential, and modulating stationary carrier-mode states, which are independent of the curvature. Our central finding is that nonzero curvature can lead to the existence of unusual modes, in particular, enabling stable localized self-trapped fundamental and vortex-carrying states in media with constant repulsive interactions without additional external confining potentials and even in the expulsive external traps.

Topological supersolidity of dipolar Fermi gases in a spin-dependent optical lattice

We investigate the topological supersolid states of dipolar Fermi gases trapped in a spin-dependent 2D optical lattice. Our results show that topological supersolid states can be achieved via the combination of topological superfluid states with the stripe order. Different from the general held belief that supersolid state in fermionic system can only survive with simultaneous coexistence of the repulsive and attractive dipolar interaction. We demonstrate that it can be maintained when the dipolar interaction is attractive in both x and y direction. By adjusting the ratio of hopping amplitude between different directions and dipolar interaction strength U, the system will undergo a phase transition among p _x + ip _y superfluid state, p _y -wave superfluid state, and the topological supersolid state. The supersolid state in the attractive environment is proved to be stable by the positive sign of the inverse compressibility. We also design an experimental protocol to realize the staggered next-next-nearest-neighbor hopping via the laser assisted tunneling technique, which is the key to simulate the spin-dependent potential.

Nematic-Orbit Coupling and Nematic Density Waves in Spin-1 Condensates

We propose the creation of artificial nematic-orbit coupling in spin-1 Bose-Einstein condensates, in analogy with spin-orbit coupling. Using a suitably designed microwave chip, the quadratic Zeeman shift, normally uniform in space, can be made to be spatiotemporally varying, leading to a coupling between spatial and nematic degrees of freedom. A phase diagram is explored where three quantum phases with the nematic order emerge: easy axis, easy plane with single-well structure, and easy plane with double-well structure in momentum space. By including spin-dependent and spin-independent interactions, we also obtain the low energy excitation spectra in these three phases. Last, we show that the nematic-orbit coupling leads to a periodic nematic density modulation in relation to the period λ_{T} of the cosinusoidal quadratic Zeeman term. Our results point to the rich possibilities for manipulation of tensorial degrees of freedom in ultracold gases without requiring Raman lasers, and therefore, obviating light-scattering induced heating.

Mixed-spin system with supersolid phases: magnetocaloric effect and thermal properties

Recently, it has been shown that two dimensional frustrated mixed-spin systems with anisotropic exchange interactions display supersolid phases in their ground state phase diagrams even in the absence of long-range interactions. In this paper, using cluster mean field theory, we investigate the effects of thermal fluctuations on the ground state phases of this kind of system and show that various thermal solids and thermal insulators emerge around the ground state solid and Mott insulating phases. We also study the thermodynamic properties and magnetocaloric effect of these systems and demonstrate that at low temperatures, a large cooling rate is seen in the vicinity of the solid-supersolid, solid-superfluid and Mott insulator-superfluid critical points, with the large accumulation of the entropy and the minimums of the isentropes. Our results show the sign change of the magnetocaloric parameter inside the solids and the Mott insulator, which is a characteristic of ordered phases.

Supersolid Properties of a Bose-Einstein Condensate in a Ring Resonator

We investigate the dynamics of a Bose-Einstein condensate interacting with two noninterfering and counterpropagating modes of a ring resonator. Superfluid, supersolid, and dynamic phases are identified experimentally and theoretically. The supersolid phase is obtained for sufficiently equal pump strengths for the two modes. In this regime we observe the emergence of a steady state with crystalline order, which spontaneously breaks the continuous translational symmetry of the system. The supersolidity of this state is demonstrated by the conservation of global phase coherence at the superfluid to supersolid phase transition. Above a critical pump asymmetry the system evolves into a dynamic runaway instability commonly known as collective atomic recoil lasing. We present a phase diagram and characterize the individual phases by comparing theoretical predictions with experimental observations.

Spatial Coherence of Spin-Orbit-Coupled Bose Gases

Spin-orbit-coupled Bose-Einstein condensates (SOBECs) exhibit two new phases of matter, now known as the stripe and plane-wave phases. When two interacting spin components of a SOBEC spatially overlap, density modulations with periodicity given by the spin-orbit coupling strength appear. In equilibrium, these components fully overlap in the miscible stripe phase and overlap only in a domain wall in the immiscible plane-wave phase. Here we probe the density modulation present in any overlapping region with optical Bragg scattering and observe the sudden drop of Bragg scattering as the overlapping region shrinks. Using an atomic analog of the Talbot effect, we demonstrate the existence of long-range coherence between the different spin components in the stripe phase and surprisingly even in the phase-separated plane-wave phase.

Rotating a Supersolid Dipolar Gas

Distinctive features of supersolids show up in their rotational properties. We calculate the moment of inertia of a harmonically trapped dipolar Bose-Einstein condensed gas as a function of the tunable scattering length parameter, providing the transition from the (fully) superfluid to the supersolid phase and eventually to an incoherent crystal of self-bound droplets. The transition from the superfluid to the supersolid phase is characterized by a jump in the moment of inertia, revealing its first order nature. In the case of elongated trapping in the plane of rotation, we show that the moment of inertia determines the value of the frequency of the scissors mode, which is significantly affected by the reduction of superfluidity in the supersolid phase. The case of an in-plane isotropic trapping is instead well suited to study the formation of quantized vortices, which are shown to be characterized, in the supersolid phase, by a sizeable deformed core, caused by the presence of the surrounding density peaks.