Sound propagation in striped supersolid cold gases at zero temperature
Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Università di Trento（特伦托大学物理系）；CNR NANOTEC, Institute of Nanotechnology（意大利国家研究委员会纳米技术研究所）；INFN, Sezione di Lecce（意大利国家核物理研究院莱切分部）；Trento Institute for Fundamental Physics and Applications, INFN（特伦托基础物理与应用研究所）
Elena Poli, Giovanni I. Martone, Sandro Stringari, Alessio Recati
We present a unified hydrodynamic approach for the sound propagation in the stripe phases realized in ultracold dipolar gas and spin-orbit-coupled BEC platforms at zero temperature. Despite the deep difference of the two platforms at a microscopic level, a similar hydrodynamic description can be formulated at a macroscopic level. The main difference between the two platforms is the lack of Galilean invariance in the spin-orbit case, resulting in a different identification of the normal (nonsuperfluid) component of the density, which leads to new terms in the equation for the current. In both cases the spectrum comprises two sounds, reflecting the spontaneous breaking of the U(1) and translational symmetries. Both sounds exhibit an anisotropic behavior. A comparison with the first and second sounds of the smectic-A liquid crystal is also presented.
We analyze the role played by quantum fluctuations on a Raman Spin-Orbit Coupled system in the stripe phase. We show that beyond mean-field effects stabilize the collapse predicted by mean-field theory and induce the emergence of two phases: a gas and a liquid, which also show spatial periodicity along a privileged direction. We show that the energetically favored phase is determined by the Raman coupling and the spin-dependent scattering lengths. We obtain the ground-state solution of the finite system by solving the extended Gross-Pitaevskii equation and find self-bound, droplet-like solutio…
We investigate the quench dynamics of the dipolar bosons in two dimensional optical lattice of square geometry using the time dependent Gutzwiller method. The system exhibits different density orders like the checkerboard and the striped pattern, depending upon the polarization angle of the dipoles. We quench the hopping parameter across the striped density wave (SDW) to striped supersolid (SSS) phase transition, and obtain the scaling laws for the correlation length and topological vortex density, as function of the quench rate. The results are reminiscent of the Kibble-Zurek mechanism (KZM).…
The striped state of ultracold bosons with Rashba spin-orbit coupling in a homogeneous infinite system has, as we show, a constant particle flow, which in a finite-size system would accumulate particles at the boundaries; it is thus not a physical steady state of the system. We propose, as a variational ansatz, a condensate wave function for a weakly trapped system which behaves similarly to the striped state near the center, but does not have particle flow at the boundaries. This state has a line of unquantized coreless vortices. We show, by minimizing the total energy, that our modified stri…
We show anisotropy of the dipole interaction between magnetic atoms or polar molecules can stabilize new quantum phases in an optical lattice. Using a well controlled numerical method based on the tensor network algorithm, we calculate phase diagram of the resultant effective Hamiltonian in a two-dimensional square lattice - an anisotropic Hubbard model of hard-core bosons with attractive interaction in one direction and repulsive interaction in the other direction. Besides the conventional superfluid and the Mott insulator states, we find the striped and the checkerboard charge density wave s…
We analyze the Bose-Hubbard model with a three-body hardcore constraint by mapping the system to a theory of two coupled bosonic degrees of freedom. We find striking features that could be observable in experiments, including a quantum Ising critical point on the transition from atomic to dimer superfluidity at unit filling, and a continuous supersolid phase for strongly bound dimers.
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…
In this paper, we study Bose-Einstein condensation of Rydberg-dressed atoms considering finite range interactions. We use Hartree-Fock-Bogoliubov approximation based on Mean-Field approach. Moreover, within this approximation modified by the finite-range character of the two-body interaction we shall obtain analytical expressions for thermodynamic quantities of Rydberg-dressed Bose gas. The imaginary part of the quasiparticle spectrum of a BEC signals the instability of the roton mode with respect to the formation of supersolid state. Our theory predicts a second-order quantum phase transition…
Supersolid is a long-sought exotic phase of matter, which is characterized by the coexistence of a diagonal long-range order of solid and an off-diagonal long-range order of superfluid. Possible candidates to realize such a phase have been previously considered, including hard-core bosons with long-range interaction and soft-core bosons. Here we demonstrate that an ultracold atomic condensate of hard-core bosons with contact interaction can establish a supersolid phase when simultaneously subjected to spin-orbit coupling and a spin-dependent periodic potential. This supersolid phase is accompa…
Chirality represents a kind of symmetry breaking characterized by the noncoincidence of an object with its mirror image and has been attracting intense attention in a broad range of scientific areas. The recent realization of spin-orbit coupling in ultracold atomic gases provides a new perspective to study quantum states with chirality. In this Letter, we demonstrate that the combined effects of spin-orbit coupling and interatomic soft-core long-range interaction can induce an exotic supersolid phase in which the chiral symmetry is broken with spontaneous emergence of circulating particle curr…
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 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.
One of the major challenges in realizing a strongly interacting lattice gas using Rydberg states is the occurrence of avalanche loss processes. As these are directly proportional to the total Rydberg fraction, the commonly suggested solution is using far off-resonantly excited Rydberg states. We instead propose the realization of a correlated bosonic lattice gas at near-resonant excitation, where the total Rydberg fraction in the bulk is low due to the strong, interaction-driven effective detuning. Using real-space dynamical mean-field theory we show that its reduced effect at the boundary of …
We present results of computer simulations at low temperature of a two-dimensional system of dipolar bosons, with dipole moments aligned at an arbitrary angle with respect to the direction perpendicular to the plane. The phase diagram includes a homogeneous superfluid phase, as well as triangular and striped crystalline phases, as the particle density and the tilt angle are varied. In the striped solid, no phase coherence among stripes and consequently no ``supersolid" phase is found, in disagreement with recent theoretical predictions.
A major obstacle for the experimental realization of a supersolid phase with cold atomic gases in an optical lattice is the weakness of the nearest-neighbor interactions achievable via magnetic dipole-dipole interactions. In this letter, we show that using a large filling of atoms within each well the characteristic energy scales are strongly enhanced. Within this regime, the system is well described by the rotor model, and the qualitative behavior of the phase diagram derives from mean-field theory. We find a stable supersolid phase for realistic parameters with chromium atoms.
We investigate a cold atomic mixture of spinless bosons and fermions in two-dimensional optical lattices. In the presence of a nested Fermi surface, the bosons may develop a fascinating supersolid behavior characterized by a finite superfluid density as well as a spatial density wave order. Focusing on the triangular lattice geometry and combining a general Landau-Ginzburg-Wilson approach with microscopically derived mean-field theory, we find an exotic supersolid phase at a fermionic band-filling of $n_f = 3/4$ with a Kagome-type crystalline order. We also address the case of anisotropic hopp…
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…
Supersolidity - a coexistence of superfluidity and crystalline or amorphous density variations - has been vividly debated ever since its conjecture. While the initial focus was on helium-4, recent experiments uncovered supersolidity in ultra-cold dipolar quantum gases. Here, we propose a new self-bound supersolid phase in a binary mixture of Bose gases with short-range interactions, making use of the non-trivial properties of spin-orbit coupling. We find that a first-order phase transition from a self-bound supersolid stripe phase to a zero-minimum droplet state of the Bose gas occurs as a fun…
We investigate the possibility of realizing supersolid quantum phases in bosonic Rydberg-excited quantum lattice gases in the presence of non-unitary processes, by simulating the dynamical evolution starting from initial preparation in non-dissipative equilibrium states. Within Gutzwiller theory, we first analyze the many-body ground-state of a bosonic Rydberg-excited quantum gas in a two dimensional optical lattice for variable atomic hopping rates and Rabi detunings. Furthermore, we perform time evolution of different supersolid phases using the Lindblad-master equation. With the inclusion o…
Engineering quantum phases with spontaneously broken symmetries is a major goal of research in different fields. Trapped ultracold Rydberg-excited atoms in optical lattices are a promising platform for realizing quantum phases with broken lattice translational symmetry since they are interacting over distances larger than the lattice constant. Although numerous theoretical works on trapped Rydberg-excited gases have predicted such phases, in particular density wave or supersolid phases, their experimental observation proves to be difficult due to challenges such as scattering processes and the…
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 states with checkerboard crystal structures are found away from half-filing. The characters of various phase transitions out of these two supersolid states are discussed. In particular, for the case with kinetic frustration, the existence of a half supersolid phase possessing both …
The striped phase exhibited by a spin-$1/2$ Bose-Einstein condensate with spin-orbit coupling is characterized by the spontaneous breaking of two continuous symmetries: gauge and translational symmetry. This is a peculiar feature of supersolids and is the consequence of interaction effects. We propose an approach to produce striped configurations with high-contrast fringes, making their experimental detection in atomic gases a realistic perspective. Our approach, whose efficiency is directly confirmed by three-dimensional Gross-Pitaevskii simulations, is based on the space separation of the tw…
Supersolid, a fascinating quantum state of matter, features novel phenomena such as the non-classical rotational inertia and transport anomalies. It is a long standing issue of the coexistence of superfluidity and broken translational symmetry in condensed matter physics. By recent experimental advances to create tunable synthetic spin-orbit coupling in ultracold gases, such highly controllable atomic systems would provide new possibilities to access supersolidity with no counterpart in solids. Here we report that the combination of anharmonicity of trapping potential and spin-orbit coupling w…
By combining theory and experiments, we demonstrate that dipolar quantum gases of both $^{166}$Er and $^{164}$Dy support a state with supersolid properties, where a spontaneous density modulation and a global phase coherence coexist. This paradoxical state occurs in a well defined parameter range, separating the phases of a regular Bose-Einstein condensate and of an insulating droplet array, and is rooted in the roton mode softening, on the one side, and in the stabilization driven by quantum fluctuations, on the other side. Here, we identify the parameter regime for each of the three phases. …
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.
The existence of a paradoxical supersolid phase of matter, possessing the apparently incompatible properties of crystalline order and superfluidity, was predicted 50 years ago. Solid helium was the natural candidate, but there supersolidity has not been observed yet, despite numerous attempts. Ultracold quantum gases have recently shown the appearance of the periodic order typical of a crystal, due to various types of controllable interactions. A crucial feature of a D-dimensional supersolid is the occurrence of up to D+1 gapless excitations reflecting the Goldstone modes associated with the s…
In a recent article [R. Bombin, F. Mazzanti and J. Boronat, Phys. Rev. A100, 063614 (2019)], it is contended that a two-dimensional system of dipolar bosons, with dipole moments aligned at particular angles with respect to the direction perpendicular to the plane of motion, featuring a "striped" crystalline ground state, in turn undergoes a Berezinskii-Kosterlitz-Thouless superfluid transition at low temperature, making it a two-dimensional supersolid. We show here that the results provided therein, obtained by means of Quantum Monte Carlo simulations, do not actually support such a conclusion…
The droplet crystal phase of a symmetric binary mixture of Rydberg-blockaded dipolar Bose gases is studied by computer simulation. At high temperature each droplet comprises on average equal numbers of particles of either component, but the two components demix below the supersolid transition temperature, i.e., droplets mostly consist of particles of one component. Droplets consisting of the same component will also favor clustering. Demixing is driven by quantum tunnelling of particles across droplets over the system, and does not take place in a non-superfluid crystal. This effect should be …
Experiments on quantum degenerate Fermi gases of magnetic atoms and dipolar molecules begin to probe their broken symmetry phases dominated by the long-range, anisotropic dipole-dipole interaction. Several candidate phases including the p-wave superfluid, the stripe density wave, and a supersolid have been proposed theoretically for two-dimensional spinless dipolar Fermi gases. Yet the phase boundaries predicted by different approximations vary greatly, and a definitive phase diagram is still lacking. Here we present a theory that treats all competing many-body instabilities in the particle-pa…
We study the zero-temperature phase diagram of a spin-orbit-coupled Bose-Einstein condensate of spin $1$, with equally weighted Rashba and Dresselhaus couplings. Depending on the antiferromagnetic or ferromagnetic nature of the interactions, we find three kinds of striped phases with qualitatively different behaviors in the modulations of the density profiles. Phase transitions to the zero-momentum and the plane-wave phases can be induced in experiments by independently varying the Raman coupling strength and the quadratic Zeeman field. The properties of these transitions are investigated in d…
Interfacing unbiased quantum Monte Carlo simulations with state-of-art analytic continuation techniques, we obtain exact numerical results for dynamical density and spin correlations in the attractive Hubbard model, describing a spin-balanced two-dimensional cold Fermi gas on an optical lattice. We focus on half-filling, where on average one fermion occupies each lattice site, and the system displays an intriguing supersolid phase: a superfluid with a checkerboard density modulation. The coexistence of $U(1)$ broken symmetry and the density modulations makes this regime very challenging and in…
In the past few decades, the search for supersolid-like phases has attracted great attention in condensed matter and ultracold atom communities. Here we experimentally demonstrate a route for realizing a superfluid stripe-phase in a spin-orbit coupled Bose-Einstein condensate by employing a weak optical lattice to induce momentum-space hopping between two spin-orbit band minima. We characterize the striped ground state as a function of lattice coupling strength and spin-orbit detuning and find good agreement with mean-field simulations. We observe coherent Rabi oscillations in momentum space b…
Liquid crystals are phases of matter intermediate between crystals and liquids. Whereas classical liquid crystals have been known for a long time and are used in electro-optical displays, much less is known about their quantum counterparts. There is growing evidence that quantum liquid crystals play a central role in many electron systems including high temperature superconductors, but a quantitative understanding is lacking due to disorder and other complications. Here, we analyse the quantum phase diagram of a two-dimensional dipolar gas, which exhibits stripe, nematic and supersolid phases.…
This paper summarizes some of the relevant features exhibited by bi-nary mixtures of Bose-Einstein condensates in the presence of coherent coupling at zero temperature. The coupling, which is experimentally produced by proper photon transitions, can either involve negligible momentum transfer from the electromagnetic radiation (Rabi coupling) or large momentum transfer (Raman coupling) associated with spin-orbit effects.The nature of the quantum phases exhibited by coherently coupled mixtures is discussed in detail, including their paramagnetic, ferromagnetic, and, in the case of spin-orbit co…
We investigate the quantum phases of monodispersed bosonic gases confined to a triangular lattice and interacting via a class of soft-shoulder potentials. The latter correspond to soft-core potentials with an additional hard-core onsite interaction. Using exact quantum Monte Carlo simulations, we show that the low temperature phases for weak and strong interactions following a temperature quench are a homogeneous superfluid and a glass, respectively. The latter is an insulating phase characterized by inhomogeneity in the density distribution and structural disorder. Remarkably, we find that fo…
We theoretically establish the mean-field phase diagram of a homogeneous spin-$1$, spin-orbit coupled Bose gas as a function of the spin-dependent interaction parameter, the Raman coupling strength and the quadratic Zeeman shift. We find that the interplay between spin-orbit coupling and spin-dependent interactions leads to the occurrence of ferromagnetic or ferronematic phases which also break translational symmetry. For weak Raman coupling, increasing attractive spin-dependent interactions (as in $^{87}$Rb or $^7$Li) induces a transition from a uniform to a stripe XY ferromagnet (with no nem…
Several new features arise in the ground-state phase diagram of a spin-1 condensate trapped in an optical trap when the magnetic dipole interaction between the atoms is taken into account along with confinement and spin precession. The boundaries between the regions of ferromagnetic and polar phases move as the dipole strength is varied and the ferromagnetic phases can be modulated. The magnetization of the ferromagnetic phase perpendicular to the field becomes modulated as a helix winding around the magnetic field direction, with a wavelength inversely proportional to the dipole strength. Thi…
We consider imbalanced Fermi gases with strong attractive interactions, for which Cooper-pair formation plays an important role. The two-component mixtures consist either of identical fermionic atoms in two different hyperfine states, or of two different atomic species both occupying only a single hyperfine state. In both cases, the number of atoms for each component is allowed to be different, which leads to a spin imbalance, or spin polarization. Two different atomic species also lead to a mass imbalance. Imbalanced Fermi gases are relevant to condensed-matter physics, nuclear physics and as…
