A Bose gas under the effect of Raman Spin-Orbit Coupling (SOC) is analyzed using the Discrete Spin T-moves Diffusion Monte Carlo method. By computing the energy as well as the static structure factor and the superfluid fraction of the system, the emergence of an energetically favorable supersolid stripe state is observed, which is in agreement with recent observations. A significant enhancement of the stability of the stripe phase with respect to the mean-field prediction is observed when the strength of the inter-atomic correlations is increased. We also quantify and characterize the degree of superfluidity of the stripes and show that this quantity is mostly determined by the ratio between the Raman coupling and the square of the momentum difference between the pair of SOC inducing laser beams.
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…
Traveling supersolid stripe patterns in spin-orbit-coupled Bose-Einstein condensates
CNR NANOTEC, Institute of Nanotechnology（意大利国家研究委员会纳米技术研究所）；Russian Quantum Center（俄罗斯量子中心）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；University of Amsterdam（阿姆斯特丹大学）
By means of quantum Monte Carlo simulations we study phase diagrams of dipolar bosons in a square optical lattice. The dipoles in the system are parallel to each other and their orientation can be fixed in any direction of the three-dimensional space. Starting from experimentally tunable parameters like scattering length and dipolar interaction strength, we derive the parameters entering the effective Hamiltonian. Depending on the direction of the dipoles, various types of supersolids (e.g. checkerboard, stripe) and solids (checkerboard, stripe, diagonal stripe, and an incompressible phase) ca…
The phase diagram of a spin-orbit-coupled two-component Bose gas includes a supersolid stripe phase, which is featuring density modulations along the direction of the spin-orbit coupling. This phase has been recently found experimentally [J.~Li \textit{et al.}, Nature (London) \textbf{543}, 91 (2017)]. In the present work we characterize the superfluid behavior of the stripe phase by calculating the drag force acting on a moving impurity. Because of the gapless band structure of the excitation spectrum, the Landau critical velocity vanishes if the motion is not strictly parallel to the stripes…
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…
Supersolidity is an intriguing concept. It combines the property of superfluid flow with the long-range spatial periodicity of solids, two properties which are often mutually exclusive. The original discussion of quantum crystals and supersolidity focuses on solid Helium-4 where it was predicted that vacancies could form dilute weakly interacting Bose-Einstein condensates. In this system, direct observation of supersolidity has been elusive. The concept of supersolidity was then generalized to include other superfluid systems which break the translational symmetry of space. One of such systems…
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…
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）
We use microscopic Bogoliubov-de Gennes formalism to explore the ground-state phase diagram of the single-band attractive Hofstadter-Hubbard model on a square lattice. We show that the interplay between the Hofstadter butterfly and superfluidity breaks spatial symmetry, and gives rise to stripe-ordered superfluid and supersolid phases in large parameter spaces. We also discuss the effects of a trapping potential and comment on the viability of observing stripe-ordered phases with cold Fermi gases.
Strong, long-range interactions present a unique challenge for the theoretical investigation of quantum many-body lattice models, due to the generation of large numbers of competing states at low energy. Here, we investigate a class of extended bosonic Hubbard models with off-site terms interpolating between short and infinite range, thus allowing for an exact numerical solution for all interaction strengths. We predict a novel type of stripe crystal at strong coupling. Most interestingly, for intermediate interaction strengths we demonstrate that the stripes can turn superfluid, thus leading …
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 b…
We study the ground state phase diagram of a two dimensional mixed-spin system of coupled alternating spin-1 and $1/2$ chains with a stripe supersolid phase. Utilizing different analytical and numerical approaches such as mean field approximation, cluster mean field theory and linear spin wave theory, we demonstrate that our system displays a rich ground state phase diagram including novel stripe supersolid, solids with different fillings and super-counterfluid phases, in addition to a stripe solid with half filling, superfluid and Mott insulating phases. In order to find a minimal mixed-spin …
We analyse the coexistence of superfluid and density wave (stripe) order in a quasi-two-dimensional gas of dipolar fermions aligned by an external field. Remarkably, the anisotropic nature of the dipolar interaction allows for such a coexistence in a large region of the zero temperature phase diagram. In this region, the repulsive part of the interaction drives the stripe formation and the attractive part induces the pairing, resulting in a supersolid with $p$-wave Cooper pairs aligned along the stripes. From a momentum space perspective, the stability of the supersolid phase is due to the fac…
The phase diagram of a Bose-Einstein condensate with Raman-induced spin-orbit coupling includes a stripe phase with supersolid features. In this work we develop a perturbation approach to study the ground state and the Bogoliubov modes of this phase, holding for small values of the Raman coupling. We obtain analytical predictions for the most relevant observables (including the periodicity of stripes, sound velocities, compressibility, and magnetic susceptibility) which are in excellent agreement with the exact (non perturbative) numerical results, obtained for significantly large values of th…
We demonstrate spontaneous generation of spatially-periodic supersolid-like super-lattice and stripe solitons in Rashba spin-orbit (SO) coupled spin-1 and spin-2 quasi-two-dimensional nonmagnetic Bose-Einstein condensates (BECs). The solitons in a weakly SO-coupled spin-1 BEC are circularly-symmetric of $(-1, 0, +1)$ and $(0, +1, +2)$ types and have inherent vorticity; the numbers in the parentheses are the winding numbers in hyper-spin components $+1, 0, -1$, respectively. The circularly-symmetric solitons in an SO-coupled spin-2 BEC are of types $(-2, -1, 0, +1, +2)$ and $(-1, 0, +1, +2, +3)…
We developed functional integral formulation for the stripe phase of a spinor Bose-Einstein condensates with Rashba spin-orbit coupling. The excitation spectrum is found to exhibit double gapless band structures, identified to be two Goldstone modes resulting from spontaneously broken internal gauge symmetry and translational invariance symmetry. The sound velocities display anisotropic behaviors with the lower branch vanishes in the direction perpendicular to the stripe in the x-y plane. At the transition point between the plane wave phase and the stripe phase, physical quantities such as flu…
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 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 study the quantum-droplet state in a 3-dimensional (3D) Bose gas in the presence of 1D spin-orbit-coupling and Raman coupling, especially the stripe phase with density modulation, by numerically computing the ground state energy including the mean-field energy and Lee-HuangYang correction. In this droplet state, the stripe can exist in a wider range of Raman coupling, compared with the BEC-gas state. More intriguingly, both spin-orbit-coupling and Raman coupling strengths can be used to tune the droplet density.
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…
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…
We study the many-body phases of bosonic atoms with $N$ internal states confined to a 1D optical lattice under the influence of a synthetic magnetic field and strong repulsive interactions. The $N$ internal states of the atoms are coupled via Raman transitions creating the synthetic magnetic field in the space of internal spin states corresponding to recent experimental realisations. We focus on the case of strong $\mbox{SU}(N)$ invariant local density-density interactions in which each site of the 1D lattice is at most singly occupied, and strong Raman coupling, in distinction to previous wor…
Within mean field Gross-Pitaevskii framework, ultra cold atomic condensates with long range interaction is predicted to have a supersolid like ground state beyond a critical interaction strength. Such mean field supersolid like ground state has periodically modulated superfluid density which implies the coexistence of superfluid and crystalline order. Ultra cold atomic system in such mean field ground state can be subjected to artificial gauge field created either through rotation or by introducing space dependent coupling among hyperfine states of the atoms using Raman lasers. Starting from t…
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…
Recently, stripe phases in spin-orbit coupled Bose-Einstein condensates (BECs) have attracted much attention since they are identified as supersolid phases. In this paper, we exploit experimentally reachable parameters and show theoretically that annular stripe phases with large stripe spacing and high stripe contrast can be achieved in spin-orbital-angular-momentum coupled (SOAMC) BECs. In addition to using Gross-Pitaevskii numerical simulations, we develop a variational ansatz that captures the essential interaction effects to first order, which are not present in the ansatz employed in prev…
We study supersolid-like crystalline structures emerging in the stationary states of a quasi-two-dimensional spin-orbit (SO)-coupled spin-2 condensate in the ferromagnetic, cyclic, and antiferro-magnetic phases by solving a mean-field model.Interplay of different strengths of SO coupling and interatomic interactions gives rise to a variety of non-trivial density patterns in the emergent solutions. For small SO-coupling strengths $γ$ ($γ\approx 0.5$), the ground state is an axisymmetric multi-ring soliton for polar, cyclic and weakly-ferromagnetic interactions, whereas for stronger-ferromagneti…
We study the ground state phase diagram and the quantum phase transitions in spin-1 Bose gases with Raman induced spin-orbit coupling. In addition to the Bose-Einstein condensates with uniform density, three types of stripe condensation phases that simultaneously break the U(1) symmetry and the translation symmetry are identified. The transitions between these phases are investigated, and the occurrences of the various tricritical points are predicted. The excitation spectra in the plane-wave phase and the zero-momentum phase show rich roton-maxon structures, and their instabilities indicate t…
The recent realization of synthetic spin-orbit coupling represents an outstanding achievement in the physics of ultracold quantum gases. In this review we explore the properties of a spin-orbit-coupled Bose-Einstein condensate with equal Rashba and Dresselhaus strengths. This system presents a rich phase diagram, which exhibits a tricritical point separating a zero-momentum phase, a spin-polarized plane-wave phase, and a stripe phase. In the stripe phase translational invariance is spontaneously broken, in analogy with supersolids. Spin-orbit coupling also strongly affects the dynamics of the …
International Center for Quantum Materials and School of Physics, Peking University（北京大学量子材料科学与工程研究院和物理学院）；Hefei National Laboratory（合肥国家实验室）；Department of Physics and Astronomy and IQ Initiative, University of Pittsburgh（匹兹堡大学物理与天文学系及IQ倡议）；International Quantum Academy（国际量子学院）
Quasicrystal is a class of ordered structures defying conventional classification of solid crystals and may carry classically forbidden (e.g., 5-fold) rotational symmetries. In view of long-sought supersolids, a natural question is whether a superfluid can spontaneously form quasicrystalline order that is not possessed by the underlying Hamiltonian, forming "superfluid-quasicrystals". Here we show that a superfluid-quasicrystal stripe state with the minimal 5-fold rotational symmetry can be realized as the ground state of a Bose-Einstein condensate within a practical experimental scheme. There…
We report on a theoretical study of a ring-shaped Bose-Einstein condensate with Raman-induced spin-orbital-angular-momentum coupling. We analyze the structure of the ground-state of the system depending different physical parameters and reveal a peculiar fine structure within the stripe phase on the phase diagram. We demonstrate the existence of the predicted stripe sub-phases within a wide range of physical parameters, their traceability through physical observables, and compare the results with several commonly used variational approximations. We also show that predicted sub-phases of the st…
We measure the collective excitation spectrum of a spin-orbit coupled Bose-Einstein condensate using Bragg spectroscopy. The spin-orbit coupling is generated by Raman dressing of atomic hyperfine states. When the Raman detuning is reduced, mode softening at a finite momentum is revealed, which provides insight towards a supersolid-like phase transition. We find that for the parameters of our system, this softening stops at a finite excitation gap and is symmetric under a sign change of the Raman detuning. Finally, using a moving barrier that is swept through the BEC, we also show the effect of…
Significant experimental progress has been made recently for observing long-sought supersolid-like states in Bose-Einstein condensates, where spatial translational symmetry is spontaneously broken by anisotropic interactions to form a stripe order. Meanwhile, the superfluid stripe ground state was also observed by applying a weak optical lattice that forces the symmetry breaking. Despite of the similarity of the ground states, here we show that these two symmetry breaking mechanisms can be distinguished by their collective excitation spectra. In contrast to gapless Goldstone modes of the \text…
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.…
We study two-component Bose gases with Raman induced spin-orbit coupling via a perturbation approach at finite temperature. For weak coupling, free energy is expanded in terms of Raman coupling strength up to the second order, where the coefficient (referred to as Raman susceptibility) is determined according to linear response theory. The equation of state for the stripe phase and the plane-wave phase are obtained in Popov approximation, and the first order transition between these two phases is investigated. As temperature increases, we find the phase boundary bends toward the stripe phase s…
Soon after its theoretical prediction, striped-density states in the presence of synthetic spin-orbit coupling were realized in Bose-Einstein condensates of ultracold neutral atoms [J.-R. Li et al., Nature \textbf{543}, 91 (2017)]. The achievement opens avenues to explore the interplay of superfluidity and crystalline order in the search for supersolid features and materials. The system considered is essentially made of two linearly coupled Bose-Einstein condensates, that is a pseudo-spin-$1/2$ system, subject to a spin-dependent gauge field $σ_z \hbar k_\ell$. Under these conditions the strip…
