Supersolid Research Articles

Non-equilibrium dynamics of ultracold lattice bosons inside a cavity

We study the non-equilibrium quench dynamics crossing a continuous phase transition between the charge density wave (CDW) and supersolid (SS) phases of a bosonic lattice gas with cavity-mediated interactions. When changing the hopping amplitude in the Hamiltonian as a function of time, we investigate the scaling behavior of the correlation length and vortex density with respect to the quench time and find that there is a threshold of the quench rate separating two distinct scaling regimes. When slowly varying the system below that threshold, we find a power-law scaling as predicted by the Kibble–Zurek mechanism (KZM). While considering fast quench above that threshold, a deviation from the KZM prediction occurs, manifested by a saturation of the defect density. We further show that such distinct scaling behaviors during different dynamic procedures can be understood through comparing the relaxation time and the quench rate.

Magnetic supersolid phases of two-dimensional extended Bose–Hubbard model with spin–orbit coupling

We investigate the quantum phases and phase transitions for spin–orbit coupled two-species bosons with nearest-neighbor (NN) interaction in a two-dimensional square lattice using inhomogeneous dynamical Guztwiller mean-field method. Under the effect of spin–orbit coupling and NN interaction, we uncover a rich variety of different magnetic supersolid (SS) phases. In the presence of intraspecies NN interaction, the phase diagram exhibits the phase-twisted double-checkerboard SS (PT-DCSS) and phase-striped double-checkerboard SS (PS-DCSS) phases. For both intra- and interspecies NN interactions, apart from the phase-twisted lattice SS (PT-LSS) and phase-striped lattice SS (PS-LSS) phases, some nontrivial SS phases with interesting properties occur. More importantly, we find that the emergences of these nontrivial SS phases are dependent of the interspecies on-site interaction. To further characterize the SS phases, we also discuss the spin-dependent momentum distributions and magnetic textures. The magnetic textures, such as antiferromagnetic, spiral and stripe orders are shown. Finally, we give the fully analytical insights into the numerical results.

Temporal Bistability in the Dissipative Dicke‐Bose‐Hubbard System

AbstractA driven‐dissipative system is considered, consisting of an atomic Bose‐Einstein condensates loaded into a 2D Hubbard lattice and coupled to a single mode of an optical cavity. Due to the interplay between strong, repulsive atomic interaction and the atom‐cavity coupling, the system exhibits several phases of atoms and photons including the atomic superfluid (SF) and supersolid (SS). The dynamical behavior of the system, where dissipation is included by means of Lindblad master equation formalism. Due to the discontinuous nature of the Dicke transition for strong atomic repulsion, extended co‐existence region of different phases are found. The resulting switching dynamics are investigated, particularly between the coexisting SF and SS phases, which eventually becomes damped by the dissipation.

Quantum phases of Bose–Bose mixtures in an optical lattice with artificial magnetic flux

We investigate quantum phases of the Bose–Bose mixtures in a two dimensional (2D) square lattice under artificial magnetic flux. Our analysis employs the inhomogeneous Dynamical Gutzwiller Mean-Field Theory (DGMFT) to investigate ground state within the two-component Bose–Hubbard model. The results based on various inter-component interactions and magnetic flux. In the case of weak inter-component interaction (wherein the inter-component interactions remain subordinate to intra-component interactions, symbolically expressed as |Uab|≤U), we observe a multitude of captivating quantum phases within the Bose–Bose mixtures. These include the emergence of the jagged density-wave supersolid(SS) phase, the latticed-like density-wave SS phase, and counterflow superfluid (CFSF) phase. Notably, the introduction of artificial magnetic flux expands the range of existence for CFSF phase. In the case of strong inter-component interaction(wherein the inter-component interactions surpass intra-component interactions, symbolically expressed as Uab>U), we delve deeper into the classification of the spatially separated phase, aptly termed the phase separated supersolid (P-SS).

Two-dimensional miscible-immiscible supersolid and droplet crystal state in a homonuclear dipolar bosonic mixture

The recent realization of a binary dipolar Bose-Einstein condensate [A. Trautmann et al., Phys. Rev. Lett. 121, 213601 (2018)] presents new exciting aspects for studying quantum droplets and supersolids in a binary mixture. Motivated by this experiment, we study the ground-state phases and dynamics of a Dy-Dy mixture. The dipolar bosonic mixture exhibits qualitatively novel and rich physics. Relying on the three-dimensional numerical simulations in the extended Gross-Pitaevskii framework, we unravel the ground-state phase diagrams and characterize their different possible phases. The emergent phases include single-droplet (SD), multiple-droplets (MD), doubly supersolid (SS), and superfluid (SF) states in both miscible and immiscible phases. Intriguing mixed ground states are observed for an imbalanced binary mixture, including a combination of SS-SF, SS-MD, and SS-SS phases. We also explore the dynamics across the phase boundaries by linear quenches of interspecies and intraspecies scattering lengths. During these dynamical processes, we observe an abrupt change in phase which initially results in some instability in the system and forms some metastable states in the intermediate timescale. However, the state produced in long-time evolution is similar to our predicted ground state. Although we demonstrate the possible results for a Dy-Dy mixture and for a specific parameter range of intraspecies and interspecies scattering lengths, our results are in general valid for other dipolar mixtures and may become an important benchmark for future experimental scenarios.

Finite temperature phases and excitations of bosons on a square lattice: a cluster mean field study

We study the finite temperature phases and collective excitations of hardcore as well as softcore bosons on a square lattice with nearest and next nearest neighbor interactions, focusing on the formation of various types of supersolid (SS) phases and their stability under thermal fluctuations. The interplay between the on-site, nearest, and next nearest neighbor interactions leads to various density ordering and structural transitions, which we have plotted out. Thermodynamic properties and phase diagrams are obtained by cluster mean field theory at finite temperatures, which includes quantum effects systematically, and they are compared with the single-site mean field (MF) results. We investigate the melting process of the SS phase to normal fluid (NF), which can occur in at least two steps due to the presence of two competing orders in the SS. A tetra-critical point exists at finite temperature and exhibits intriguing behavior, which is analyzed for different regimes of interactions. The phase diagrams reveal the different pathways of the thermal transition of SSs to the NF phase, for different interaction regimes, which can be accessible by thermal quench protocols used in recent experiments. We show how the phases and the transitions between them can be identified from the characteristic features of the excitation spectrum. We analyze the appearance of a low-energy gapped mode apart from the gapless sound mode in the SS phase, which is analogous to the gapped mode recently studied for dipolar SS phases. Finally, we discuss the relevance of the results of the present work in the context of ongoing experiments on ultracold atomic gases and newly observed SS phases.

Vortex structures in a rotating Rydberg-dressed Bose-Einstein condensate with the Lee-Huang-Yang correction

We numerically investigate the vortex structures in a fast rotating two-dimensional Rydberg-dressed Bose-Einstein condensate (BEC) taking into account the Lee-Huang-Yang (LHY) quantum correction. In a rotating BEC of reduced dimensionality, it is shown that there is room to tune the LHY coupling against the short- and long-range couplings. In the absence of the LHY effect, hexagonal, square, striped, and honeycomb lattices can exhibit upon increasing the long-range interaction, which can be realized in the context of superfluid (SF)-supersolid (SS) transition. In the presence of the LHY effect, competition between LHY and long-range terms results in rich lattice structures. In particular, clustering of multiple vortices can occur due to the trapping effect of the SS triangles or grids. Estimates of number of vortices in the clustering are provided.

Phases and collective modes of bosons in a triangular lattice at finite temperature: A cluster mean field study

Motivated by the realization of Bose-Einstein condensates (BEC) in non-cubic lattices, in this work we study the phases and collective excitation of bosons with nearest neighbor interaction in a triangular lattice at finite temperature, using mean field (MF) and cluster mean field (CMF) theory. We compute the finite temperature phase diagram both for hardcore and softcore bosons, as well analyze the effect of correlation arising due to lattice frustration and interaction systematically using CMF method. A semi-analytic estimate of the transition temperatures between different phases are derived within the framework of MF Landau theory, particularly for hardcore bosons. Apart from the usual phases such as density waves (DW) and superfluid (SF), we also characterize different supersolids (SS). These phases and their transitions at finite temperature are identified from the collective modes. The low lying excitations, particularly Goldstone and Higgs modes of the supersolid can be detected in the ongoing cold atom experiments.

Self-gravitating Λ-media

We address the question whether a medium featuring p+ρ =0, dubbed Λ-medium, has to be necessarily a cosmological constant. By using effective field theory, we show that this is not the case for a class of media comprising perfect fluids, solids and special super solids, providing an explicit construction. The low energy excitations are non trivial and lensing, the growth of large scale structures can be used to clearly distinguish Λ-media from a cosmological constant.

Out-of-equilibrium dynamics of multiple second-order quantum phase transitions in an extended Bose-Hubbard model: Superfluid, supersolid, and density wave

In this paper, we study the dynamics of the Bose-Hubbard model with the nearest-neighbor repulsion by using time-dependent Gutzwiller methods. Near the unit filling, the phase diagram of the model contains density wave (DW), supersolid (SS) and superfluid (SF). The three phases are separated by two second-order phase transitions. We study "slow-quench" dynamics by varying the hopping parameter in the Hamiltonian as a function of time. In the phase transitions from the DW to SS and from the DW to SF, we focus on how the SF order forms and study scaling laws of the SF correlation length, vortex density, etc. The results are compared with the Kibble-Zurek scaling. On the other hand from the SF to DW, we study how the DW order evolves with generation of the domain walls and vortices. Measurement of first-order SF coherence reveals interesting behavior in the DW regime.

Ground-state phases of spin-orbit coupled spin-1 Bose-Einstein condensate in an optical lattice

We study the ground-state phases of two-dimensional (2D) spin-orbit coupled spin-1 Bose-Einstein condensates (BEC) in a one-dimensional (1D) optical lattice with polar and ferromagnetic interactions, respectively. Three types of the ground-state phases are found in polar and ferromagnetic BEC, respectively, i.e., the polar superfluid (PSF) phase, the polar stripe (PST) phase and the polar supersolid (PSS) phase in polar BEC and the ferromagnetic superfluid (FSF) phase, the ferromagnetic stripe (FST) phase and the ferromagnetic supersolid (FSS) phase in ferromagnetic BEC. Main different points between polar and ferromagnetic BECs are the stripe (ST) phase in polar BEC and the supersolid (SS) phase in ferromagnetic BEC for large spin-orbit coupling (SOC). We also investigate the ground-state phase diagrams of spin-orbit coupled spin-1 BEC in a 1D optical lattice with spin-dependent periodic disorder. The range of the SS phase increases and the range of the ST phase decreases with increasing the strength of the spin-dependent periodic disorder in both polar and ferromagnetic BECs, which shows the SS phase is obtained easily in spin-orbit coupled spin-1 BEC in a 1D optical lattice with spin-dependent periodic disorder.

Turbulence in a matter-wave supersolid

Quantum turbulence associated with wave and vortex dynamics is numerically investigated for a two-dimensional trapped atomic Rydberg-dressed Bose-Einstein condensate (BEC). When the coupling constant of the soft-core interaction is over a critical value, the superfluid (SF) system can transition into a hexagonal supersolid (SS) state. Based on the Gross-Pitaevskii equation approach, we have discovered a new characteristic k−13/3 scaling law for wave turbulence in the SS state, that coexists with the waveaction k−1/3 and energy k−1 cascades commonly existing in a SF BEC. The new k−13/3 scaling law implies that the SS system exhibits a negative, minus-one power energy dispersion (E ~ k−1) at the wavevector consistent with the radius of the SS droplet. For vortex turbulence, in addition to the presence of the Kolmogorov energy k−5/3 and Saffman enstrophy k−4 cascades, it is found that large amount of independent vortices and antivortices pinned to the interior of the oscillating SS results in a strong k−1 scaling at the wavevector consistent with the SS lattice constant.

Two supersolid phases in hard-core extended Bose–Hubbard model

The effect of the next-nearest-neighbor (nnn) tunneling on the hard-core extended Bose–Hubbard model on square lattices is investigated. By means of the cluster mean-field theory, the ground-state phase diagrams are determined. When a modest nnn tunneling is introduced, depending on its sign, two distinct supersolid (SS) states with checkerboard crystal structures are found away from half-filing. The characters of various phase transitions out of these two SS states are discussed. In particular, for the case with kinetic frustration, the existence of a half SS phase possessing both solid and unconventional superfluid orders is established. Our work hence sheds light on the search of this interesting SS phase in real ultracold lattice gases with frustrated tunnelings.

Supersolid phases in Bose gases with three-dimensional spin–orbit coupling

We investigate the ground state phases of the two-component Bose–Einstein condensates with three-dimensional Weyl-type spin–orbit coupling (SOC) loaded in a cubic optical lattice. The single-particle lowest energy band exhibits one, two or four pairs of degenerate ground states depending on the configuration of the SOC. We find two types of the supersolid (SS) phases with the isotropic SOC, and the SS phases are signaled by the momentum peaks distribution. We also discuss the spin textures of the SS phases, the spin vectors consist alternatively of two sets of continuous textures at the even and the odd lattice sites.

Supersolid phases in the magnetic fields

We study the ground state phases of the ultracold atomic condensates loaded in a two-dimensional optical lattice with the magnetic fields. Apart from uniform superfluid (SF) phase, four types of supersolid (SS) phases in the presence of the uniform magnetic fluxes and two types of SS phases in the presence of the staggered magnetic fluxes are found. For the system without magnetic flux, except for a certain unit phase factor ϕx(y)=π, the magnetic field has no effect on the system.

Drude weight in hard-core boson systems: Possibility of a finite-temperature ideal conductor

We calculate the Drude weight in the superfluid (SF) and the supersolid (SS) phases of hard core boson (HCB) model using stochastic series expansion (SSE). We demonstrate from our numerical calculations that the normal phase of HCBs in two dimensions can be an ideal conductor with dissipationless transport. In two dimensions, when the ground state is a SF, the superfluid stiffness drops to zero with a Kosterlitz-Thouless type transition at $T_{KT}$. The Drude weight, though is equal to the stiffness below $T_{KT}$, surprisingly stays finite even for temperatures above $T_{KT}$ indicating the non-dissipative transport in the normal state of this system. In contrast to this in a three dimensional SF phase, where the superfluid stiffness goes to zero continuously with a second order phase transition at $T_c$, Drude weight also goes to zero at $T_c$ as expected. We also calculated the Drude weight in a 2D SS phase, where the charge density wave (CDW) order coexists with superfluidity. For the SS phase we studied, superfluidity is lost via Kosterlitz-Thouless transition at $T_{KT}$ and the transition temperature for the CDW order is larger than $T_{KT}$. In striped SS phase where the CDW order breaks the rotational symmetry of the lattice, for $T > T_{KT}$, the system behaves like an ideal conductor along one of the lattice direction while along the other direction it behaves like an insulator. In contrast to this, in star-SS phase, Drude weight along both the lattice directions goes to zero along with the superfluid stiffness and for $T > T_{KT}$ we have a finite temperature phase of a CDW insulator.

Intrinsic-to-extrinsic supersolid transition and fractionally modulated states in a lattice ultracold Bose gas with long-range interaction

We investigate the intrinsic-to-extrinsic supersolid (SS) transition in a lattice ultracold Bose gas with strong long-range interaction. When changing the depth of the periodic lattice potential, the transition is shown to manifest in the ground-state wave function and energy, the change of superfluid fraction ${f}_{s}$, and a roton instability. Near the transition in the extrinsic SS phase, due to the competition between the long-range interaction and the periodic potential, we show that there exist a variety of stable fractionally modulated states (FMSs) upon the change of the effective length of the long-range interaction. Consequence of the transition across different FMSs is discussed.

Hard-Core Bosons on a Two-Dimensional Square Optical Superlattice* *Supported by the National Natural Science Foundation of China under Grant Nos. 10904096, 10604024, 11474025, and the Natural Science Foundation of Beijing under Grant No. 1092009

In this work, we theoretically study hard-core bosons on a two-dimensional square optical superlattice at T = 0. First of all, we present the mean field phase diagram of this model in terms of the chemical potential μ and the alternating potential strength Δ. Besides a superfluid (SF) phase at Δ = 0 and a charge density wave (CDW) phase in the large Δ at half filling, we demonstrate that a supersolid (SS) phase emerges in the moderate Δ. Then, we focus on the μ = 0, e.g., half filling case, using large-S semiclassical spin-wave approximation to study the SS to CDW quantum phase transition. In particular, we calculate the ground-state energy and the superfluid density at the level of 1/S correction. We then compare the spin-wave results with the large scale quantum Monte Carlo (QMC) simulations using the cluster stochastic series expansion (CSSE) algorithm, and find that while the spin wave method is intuitive with clear physical pictures, the quantum critical point is quite different from that of numerical results which is believed to be accurate. We suggest that as simple as it is, this model still exhibits strong quantum fluctuations near the quantum critical point beyond the power of semiclassical spin-wave approach.

Supersolid phase accompanied by a quantum critical point in the intermediate coupling regime of the Holstein model.

We reveal that electron-phonon systems described by the Holstein model on a bipartite lattice exhibit, away from half filling, a supersolid (SS) phase characterized by coexisting charge order (CO) and superconductivity (SC), and an accompanying quantum critical point (QCP). The SS phase, demonstrated by the dynamical mean-field theory with a quantum Monte Carlo impurity solver, emerges in the intermediate-coupling regime, where the peak of the Tc dome is located and the metal-insulator crossover occurs. On the other hand, in the weak- and strong-coupling regimes the CO-SC boundary is of first order with no intervening SS phases. The QCP is associated with the continuous transition from SS to SC and characterized by a reentrant behavior of the SS around it. We further show that the SS-SC transition is hallmarked by diverging charge fluctuations and a kink (peak) in the superfluid density.

Superfluidity and solid order in a two-component Bose gas with dipolar interactions in an optical lattice

In this paper, we study an extended bosonic t-J model in an optical lattice, which describes two-component hard-core bosons with a nearest-neighbor (NN) pseudo-spin interaction, and also inter- and intra-species dipole-dipole interactions (DDI). In particular, we focus on the case in which two component hard-core bosons have anti-parallel polarized dipoles with each other. The global phase diagram is studied by means of the Gutzwiller variational method and also the quantum Monte-Carlo simulations (QMC). The both calculations show that a stripe solid order, besides a checkerboard one, appears as a result of the DDI. By the QMC, we find that two kinds of supersolid (SS) form, checkerboard SS and stripe SS, and we also verify the existence of some exotic phase between the stripe solid and checkerboard SS. Finally by the QMC, we study the t-J-like model, which was experimentally realized recently by A. de Paz et al. [Phys. Rev. Lett. {\bf 111}, 185305 (2013)].