Phase transitions share the universal feature of enhanced fluctuations near the transition point. Here we show that density fluctuations reveal how a Bose-Einstein condensate of dipolar atoms spontaneously breaks its translation symmetry and enters the supersolid state of matter -- a phase that combines superfluidity with crystalline order. We report on the first direct in situ measurement of density fluctuations across the superfluid-supersolid phase transition. This allows us to introduce a general and straightforward way to extract the static structure factor, estimate the spectrum of elementary excitations and image the dominant fluctuation patterns. We observe a strong response in the static structure factor and infer a distinct roton minimum in the dispersion relation. Furthermore, we show that the characteristic fluctuations correspond to elementary excitations such as the roton modes, which have been theoretically predicted to be dominant at the quantum critical point, and that the supersolid state supports both superfluid as well as crystal phonons.
We theoretically investigate the spectrum of elementary excitations of a trapped dipolar quantum gas across the BEC-supersolid phase transition. Our calculations reveal the existence of distinct Higgs and Nambu-Goldstone modes that emerge from the softening roton modes of the dipolar BEC at the phase transition point. On the supersolid side of the transition, the energy of the Higgs mode increases rapidly, leading to a strong coupling to higher-lying modes. Our study highlights how the symmetry-breaking nature of the supersolid state translates to finite-size systems.
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 …
We theoretically investigate the ground states and the spectrum of elementary excitations across the superfluid to droplet crystallization transition of an oblate dipolar Bose-Einstein condensate. We systematically identify regimes where spontaneous rotational symmetry breaking leads to the emergence of a supersolid phase with characteristic collective excitations, such as the Higgs amplitude mode. Furthermore, we study the dynamics across the transition and show how these supersolids can be realized with standard protocols in state-of-the-art experiments.
Supersolidity -- a quantum-mechanical phenomenon characterized by the presence of both superfluidity and crystalline order -- was initially envisioned in the context of bulk solid helium, as a possible answer to the question of whether a solid could have superfluid properties. While supersolidity has not been observed in solid helium (despite much effort), ultracold atomic gases have provided a fundamentally new approach, recently enabling the observation and study of supersolids with dipolar atoms. However, unlike the proposed phenomena in helium, these gaseous systems have so far only shown …
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 selfbound 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 o…
We study the spectrum of elementary excitations of a dipolar Bose gas in a three-dimensional anisotropic trap across the superfluid-supersolid phase transition. Theoretically, we show that, when entering the supersolid phase, two distinct excitation branches appear, respectively associated with dominantly crystal and superfluid excitations. These results confirm infinite-system predictions, showing that finite-size effects play only a small qualitative role, and connect the two branches to the simultaneous occurrence of crystal and superfluid orders. Experimentally, we probe compressional exci…
A supersolid is a fascinating phase of matter, combining the global phase coherence of a superfluid with hallmarks of solids, e.g. a spontaneous breaking of the translational symmetry. Recently, states with such counter-intuitive properties have been realized in experiments using ultracold quantum gases with strong dipolar interactions. Here, we investigate the response of a supersolid state to phase excitations which shatter the global phase coherence. After the creation of those excitations, we observe a rapid re-establishment of a global phase coherence, suggesting the presence of a superfl…
We numerically study the many-body physics of molecular Bose-Einstein condensates with strong dipole-dipole interactions. We observe the formation of self-bound droplets, and explore phase diagrams that feature a variety of exotic supersolid states. In all of these cases, the large and tunable molecular dipole moments enable the study of unexplored regimes and phenomena, including liquid-like density saturation and universal stability scaling laws for droplets, as well as pattern formation and the limits of droplet supersolidity. We discuss a realistic experimental approach to realize both the…
We investigate the novel density distributions acquired by a dipolar Bose-Einstein condensed gas confined in a box potential, with special focus on the effects of supersolidity. Differently from the case of harmonic trapping, the ground state density reveals a strong depletion in the bulk region and an accumulation of atoms near the walls, well separated from the bulk, as a consequence of the competition between the attractive and the repulsive nature of the dipolar force. In a quasi two-dimensional geometry characterized by cylindrical box trapping, we observe the emergence of a ring-like con…
Two-component dipolar condensates are now experimentally producible, and we theoretically investigate the nature of supersolidity in this system. We predict the existence of a binary supersolid state in which the two components form a series of alternating domains, producing an immiscible double supersolid. Remarkably, we find that a dipolar component can even induce supersolidity in a nondipolar component. In stark contrast to single-component dipolar supersolids, alternating-domain supersolids do not require quantum stabilization, and the number of crystal sites is not strictly limited by th…
Angular oscillations can provide a useful probe of the superfluid properties of a system. Such measurements have recently been applied to dipolar supersolids, which exhibit both density modulation and phase coherence, and for which robust probes of superfluidity are particularly interesting. So far, these investigations have been confined to linear droplet arrays. Here, we explore angular oscillations in systems with 2D structure, which in principle have greater sensitivity to superfluidity. Surprisingly, in both experiment and simulation, we find that the frequency of angular oscillations rem…
A supersolid is a counter-intuitive state of matter that combines the frictionless flow of a superfluid with the crystal-like periodic density modulation of a solid. Since the first prediction in the 1950s, experimental efforts to realize this state have focussed mainly on Helium, where supersolidity remains elusive. Recently, supersolidity has also been studied intensively in ultracold quantum gases, and some of its defining properties have been induced in spin-orbit coupled Bose-Einstein condensates (BECs) and BECs coupled to two crossed optical cavities. However, the periodicity of the crys…
We study theoretically and experimentally the emergence of supersolid properties in a dipolar Bose-Einstein condensate. The theory reveals a ground state phase diagram with three distinct regimes - a regular Bose-Einstein condensate, incoherent and coherent arrays of quantum droplets. In the latter the droplets are connected by a finite superfluid density, which leads - in addition to the periodic density modulation - to a robust phase coherence throughout the whole system. We further theoretically demonstrate that we are able to dynamically approach the ground state in our experiment and that…
Experiments on dipolar Bose-Einstein condensates have recently reported the observation of supersolidity. Although quantized vortices constitute a key probe of superfluidity, their observability in dipolar supersolids is largely prevented by the strong density depletion caused by the formation of droplets. We present a novel approach to the nucleation of vortices and their observation, based on the quenching of the s-wave scattering length across the superfluid-supersolid transition. Starting from a slowly rotating, vortex-free, configuration in the superfluid phase, we predict vortex nucleati…
Breakthrough experiments have newly explored the fascinating physics of dipolar quantum droplets and supersolids. The recent realization of dipolar mixtures opens further intriguing possibilities. We show that under rather general conditions, the presence of a second component catalyzes droplet nucleation and supersolidity in an otherwise unmodulated condensate. Droplet catalyzation in miscible mixtures, which may occur even for a surprisingly small impurity doping, results from a local roton instability triggered by the doping-dependent modification of the effective dipolar strength. The cata…
We present the theoretical mean-field zero-temperature phase diagram of a Bose-Einstein condensate (BEC) with dipolar interactions loaded into an optical lattice with a staggered flux. Apart from uniform superfluid, checkerboard supersolid and striped supersolid phases, we identify several supersolid phases with staggered vortices, which can be seen as combinations of supersolid phases found in earlier work on dipolar BECs and a staggered-vortex phase found for bosons in optical lattices with staggered flux. By allowing for different phases and densities on each of the four sites of the elemen…
Recent breakthrough experiments on dipolar condensates have reported the creation of supersolids, including two-dimensional arrays of quantum droplets. Droplet arrays are, however, not the only possible non-trivial density arrangement resulting from the interplay of mean-field instability and quantum stabilization. Several other possible density patterns may occur in trapped condensates at higher densities, including the so-called honeycomb supersolid, a phase that exists, as it is also the case of a triangular droplet supersolid, in the thermodynamic limit. We show that compared to droplet su…
We calculate the mean-field phase diagram of a zero-temperature, binary Bose mixture on a square optical lattice, where one species possesses a non-negligible dipole moment. Remarkably, this system exhibits supersolidity for anomalously weak dipolar interaction strengths, which are readily accessible with current experimental capabilities. The supersolid phases are robust, in that they occupy large regions in the parameter space. Further, we identify a first-order quantum phase transition between supersolid and superfluid phases. Our results demonstrate the rich features of the dipolar Bose mi…
Recent studies on supersolidity in a single-component Bose-Einstein condensate (BEC) have relied on the Lee-Huang-Yang (LHY) correction for stabilization of self-bound droplets, which however involves a high density inside the droplets, limiting the lifetime of the supersolid. Here we propose a two-component mixture of dipolar and nondipolar BECs, such as an $^{166}$Er-$^{87}$Rb mixture, to create and stabilize a supersolid without the LHY correction, which can suppress the atomic loss and may allow observation of the long-time dynamics of the supersolid. In such a system, supersolidity can be…
Dipolar Bose-Einstein condensates represent a powerful platform for the exploration of quantum many-body phenomena arising from long-range interactions. A series of recent experiments has demonstrated the formation of supersolid states of matter. Subsequent theoretical works have shown that quantum fluctuations can affect the underlying phase transition and may lead to the emergence of supersolids with various lattice structures in dipolar condensates. In this work we explore the signatures of such different geometries in confined finite condensates. In addition to previously found triangular …
We study superfluidity of supersolid phases of dipolar Bose gases in two-dimensional optical lattices. We perform linear stability analyses for the corresponding dipolar Bose-Hubbard model in the hardcore boson limit to show that a supersolid can have stable superflow until the flow velocity reaches a certain critical value. The critical velocity for the supersolid is found to be significantly smaller than that for a conventional superfluid phase. We propose that the critical velocity can be used as a signature to identify the superfluidity of the supersolid phase in experiment.
We investigate a system of dipolar bosons in an optical lattice with local two and three-body interactions. Using the mean-field theory approach, we obtain the ground state phase diagram of the extended Bose-Hubbard (EBH) model with both repulsive and attractive three-body interac- tions. We show that the additional three-body on-site interaction has strong effects on the phase diagram especially on the supersolid phase. Positive values of the three-body interaction lead to the enhancement of the gapped phases at densities larger than unity by reducing the supersolid region. However, a small a…
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…
Gases of doubly-dipolar particles, with both magnetic and electric dipole moments, offer intriguing novel possibilities. We show that the interplay between doubly-dipolar interactions, quantum stabilization, and external confinement results in a rich ground-state physics of supersolids and incoherent droplet arrays in doubly-dipolar condensates. Our study reveals novel possibilities for engineering quantum droplets and droplet supersolids, including supersolid-supersolid transitions and the realization of supersolid arrays of pancake droplets.
A key manifestation of superfluidity in liquids and gases is a reduction of the moment of inertia under slow rotations. Non-classical rotational effects have been searched for a long time also for the elusive supersolid phase of matter, in which superfluidity coexists with a lattice structure. Here we show that the recently discovered supersolid phase in dipolar quantum gases features a reduced moment of inertia. We study a peculiar rotational oscillation mode in a harmonic potential, the scissors mode, already employed for superfluids. From the measured moment of inertia, we infer a superflui…
Supersolid phases as a result of a coexistence of superfluid and density ordered checkerboard phases are predicted to appear in ultracold Fermi molecules confined in a bilayer array of two-dimensional square optical lattices. We demonstrate the existence of these phases within the inhomogeneous mean-field approach. In particular, we show that tuning the interlayer separation distance at a fixed value of the chemical potential produces different fractions of superfluid, density ordered, and supersolid phases.
We investigate topological supersolidity of dipolar Fermi gases in a spin-dependent 2D optical lattice. Numerical results show that the topological supersolid states can be synthesized via the combination of topological superfluid states with the stripe order, where the topological superfluid states generated with dipolar interaction possess the $Δ_{x}+iΔ_{y}$ order, and it is of D class topological classification. By adjusting the ratio between hopping amplitude $t_{x}/t_{y}$ and interaction strength $U$ with dipole orientation $ϕ\approx \fracπ{4}$, the system will undergo phase transitions a…
We explore spatial symmetry breaking of a dipolar Bose Einstein condensate in the thermodynamic limit and reveal a critical point in the phase diagram at which crystallization occurs via a second order phase transition. This behavior is traced back to the significant effects of quantum fluctuations in dipolar condensates, which moreover stabilize a new supersolid phase, namely a regular honeycomb pattern with maximal modulational contrast and near-perfect superfluidity.
The competition of dipole-dipole and contact interactions leads to exciting new physics in dipolar gases, well-illustrated by the recent observation of quantum droplets and rotons in dipolar condensates. We show that the combination of the roton instability and quantum stabilization leads under proper conditions to a novel regime that presents supersolid properties, due to the coexistence of stripe modulation and phase coherence. In a combined experimental and theoretical analysis, we determine the parameter regime for the formation of coherent stripes, whose lifetime of a few tens of millisec…
We investigate the rotational properties of a dipolar Bose-Einstein condensate trapped in a toroidal geometry. Studying the ground states in the rotating frame and at fixed angular momenta, we observe that the condensate acts in distinctly different ways depending on whether it is in the superfluid or in the supersolid phase. We find that intriguingly, the toroidal dipolar condensate can support a supersolid persistent current which occurs at a local minimum in the ground state energy as a function of angular momentum, where the state has a vortex solution in the superfluid component of the co…
Distintictive 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 on inertia, revealing its first order nature. In the case of elongated trapping in the plane of rotation we show that the the …
We present a theory for the emergence of a supersolid state in a cigar-shaped dipolar quantum Bose gas. Our approach is based on a reduced three-dimensional (3D) theory, where the condensate wavefunction is decomposed into an axial field and a transverse part described variationally. This provides an accurate fully 3D description that is specific to the regime of current experiments and efficient to compute. We apply this theory to understand the phase diagram for a gas in an infinite tube potential. We find that the supersolid transition has continuous and discontinuous regions as the average…
Using a beyond-mean-field model including a Lee-Huang-Yang-type interaction, we demonstrate a supersolid-like spatially-periodic square- and triangular-lattice crystallization of droplets in a polarized dipolar condensate confined by an appropriate three-dimensional (3D) box trap. In this paper we consider a rectangular box (cuboid) trap, a square box (cuboid with two equal sides) trap, a cylindrical box trap and a hexagonal box (hexagonal prism) trap. The droplet lattice is always formed in the $x$-$y$ plane perpendicular to the polarization $z$ direction of dipolar atoms. In contrast to a ha…
We investigate the properties of quantized vortices in a dipolar Bose-Einstein condensed gas by means of a generalised Gross-Pitaevskii equation. The size of the vortex core hugely increases by increasing the weight of the dipolar interaction and approaching the transition to the supersolid phase. The critical angular velocity for the existence of an energetically stable vortex decreases in the supersolid, due to the reduced value of the density in the interdroplet region. The angular momentum per particle associated with the vortex line is shown to be smaller than $\hbar$, reflecting the redu…
We investigate the zero-temperature excitation spectrum of two-dimensional soft-core bosons for a wide range parameters and across the phase transition from a superfluid to a supersolid state. Based on mean field calculations and recent Quantum Monte Carlo results, we demonstrate the applicability of the Bogoliubov-de Gennes equations, even at high interaction strengths where the system forms an insulating cluster crystal. Interestingly, our study reveals that the maximum energy of the longitudinal phonon band in the supersolid phase connects to the maxon energy of the superfluid at the phase …
We study the behavior of the excitation spectrum across the quantum phase transition from a superfluid to a supersolid phase of a dipolar Bose gas confined to a one-dimensional geometry. Including the leading beyond-mean-field effects within an effective Hamiltonian, the analysis is based on Bogoliubov theory with several order parameters accounting for the superfluid as well as solid structure. We find fast convergence of the ground-state energy in the supersolid with the number of order parameters and demonstrate a stable excitation spectrum with two Goldstone modes and an amplitude mode in …
We study by means of first principle Quantum Monte Carlo simulations the ground state phase diagram of a system of dipolar bosons with aligned dipole moments, and with the inclusion of a two-body repulsive potential of varying range. The system is shown to display a supersolid phase in a relatively broad region of the phase diagram, featuring different crystalline patterns depending on the density and on the range of the repulsive part of the interaction (scattering length). The supersolid phase is sandwiched between a classical crystal of parallel filaments and a homogeneous superfluid phase.…
