Spin-orbit coupled spinor Bose-Einstein condensates

Stable three-dimensional vortex solitons of high topological charge in a Rydberg-dressed Bose-Einstein condensate with spin-orbit coupling

Stable vortex solitons (VSs) are objects of great interest for fundamental studies and various applications, including particle trapping, microscopy, data encoding, and matter-wave gyroscopes. However, three-dimensional (3D) VSs with high topological charges, supported by self-attractive nonlinearities, are unstable against fragmentation, which eventually leads to internal blowup (supercritical collapse) of the fragments. Here, we propose a scheme for realizing stable 3D VSs with topological charges up to 5 and 6 in the two components of a binary, Rydberg-dressed Bose-Einstein condensate with spin-orbit coupling (SOC). We show that, if the SOC strength exceeds a critical value, the rotational symmetry of the VSs in the transverse plane gets broken, resulting in a separation of the two components. Nevertheless, the VSs with the broken symmetry remain stable. The VS stability domains are identified in the system's parameter space. Moreover, application of torque to the stable VSs sets them in the state of robust gyroscopic precession.

Observation of Momentum Space Josephson Effects in Weakly Coupled Bose-Einstein Condensates

The momentum space Josephson effect describes the supercurrent flow between weakly coupled Bose-Einstein condensates (BECs) at two discrete momentum states. Here, we experimentally observe this exotic phenomenon using a BEC with Raman-induced spin-orbit coupling, where the tunneling between two local band minima is implemented by the momentum kick of an additional optical lattice. A sudden quench of the Raman detuning induces coherent spin-momentum oscillations of the BEC, which is analogous to the ac Josephson effect. We observe both plasma and regular Josephson oscillations in different parameter regimes. The experimental results agree well with the theoretical model and numerical simulation and showcase the important role of nonlinear interactions. We also show that the measurement of the Josephson plasma frequency gives the Bogoliubov zero quasimomentum gap, which determines the mass of the corresponding pseudo-Goldstone mode, a long-sought phenomenon in particle physics. The observation of momentum space Josephson physics offers an exciting platform for quantum simulation and sensing utilizing momentum states as a synthetic degree.

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.

Matter-wave solitons in an array of spin-orbit-coupled Bose-Einstein condensates

We investigate matter-wave solitons in a binary Bose-Einstein condensate (BEC) with spin-orbit (SO) coupling, loaded in a one-dimensional (1D) deep optical lattice and a three-dimensional anisotropic magnetic trap, which creates an array of elongated sub-BECs with transverse tunneling. We show that the system supports 1D continuous and discrete solitons localized in the longitudinal (along the array) and the transverse (across the array) directions, respectively. In addition, such solitons are always unpolarized in the zero-momentum state but polarized in finite-momentum states. We also show that the system supports stable two-dimensional semidiscrete solitons, including single- and multiple-peaked ones, localized in both the longitudinal and transverse directions. Stability diagrams for single-peaked semidiscrete solitons in different parameter spaces are identified. The results reported here are beneficial not only for understanding the physical property of SO-coupled BECs but also for generating new types of matter-wave solitons.

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.

Interplay between spin-orbit couplings and residual interatomic interactions in the modulational instability of two-component Bose-Einstein condensates

The nonlinear dynamics induced by the modulation instability (MI) of a binary mixture in an atomic Bose-Einstein condensate (BEC) is investigated theoretically under the joint effects of higher-order residual nonlinearities and helicoidal spin-orbit (SO) coupling in a regime of unbalanced chemical potential. The analysis relies on a system of modified coupled Gross-Pitaevskii equations on which the linear stability analysis of plane-wave solutions is performed, from which an expression of the MI gain is obtained. A parametric analysis of regions of instability is carried out, where effects originating from the higher-order interactions and the helicoidal spin-orbit coupling are confronted under different combinations of the signs of the intra- and intercomponent interaction strengths. Direct numerical calculations on the generic model support our analytical predictions and show that the higher-order interspecies interaction and the SO coupling can balance each other suitably for stability to take place. Mainly, it is found that the residual nonlinearity preserves and reinforces the stability of miscible pairs of condensates with SO coupling. Additionally, when a miscible binary mixture of condensates with SO coupling is modulationally unstable, the presence of residual nonlinearity may help soften such instability. Our results finally suggest that MI-induced formation of stable solitons in mixtures of BECs with two-body attraction may be preserved by the residual nonlinearity even though the latter enhances the instability.

Raman laser induced self-organization with topology in a dipolar condensate

We investigate the ground states of a dipolar Bose-Einstein condensate (BEC) subject to Raman laser induced spin-orbit coupling with mean-field theory. Owing to the interplay between spin-orbit coupling and atom-atom interactions, the BEC presents remarkable self-organization behavior and thus hosts various exotic phases including vortex with discrete rotational symmetry, stripe with spin helix, and chiral lattices with C₄ symmetry. The peculiar chiral self-organized array of square lattice, which spontaneously breaks both U(1) and rotational symmetries, is observed when the contact interaction is considerable in comparison with the spin-orbit coupling. Moreover, we show that the Raman-induced spin-orbit coupling plays a crucial role in forming rich topological spin textures of the chiral self-organized phases by introducing a channel for atoms to turn on spin flipping between two components. The self-organization phenomena predicted here feature topology owing to spin-orbit coupling. In addition, we find long-lived metastable self-organized arrays with C₆ symmetry in the case of strong spin-orbit coupling. We also present a proposal to observe these predicted phases in ultracold atomic dipolar gases with laser-induced spin-orbit coupling, which may stimulate broad theoretical as well as experimental interest.

Effective potentials in a rotating spin-orbit-coupled spin-1 spinor condensate

We theoretically study the stationary-state vortex lattice configurations of rotating spin-orbit (SO)- and coherently-coupled spin-1 Bose-Einstein condensates (BECs) trapped in quasi-two-dimensional harmonic potentials. The combined effects of rotation, SO and coherent couplings are analyzed systematically from the single-particle perspective. Through the single-particle Hamiltonian, which is exactly solvable for one-dimensional coupling, we illustrate that a boson in these rotating SO- and coherently-coupled condensates are subjected to effective toroidal, symmetric double-well, or asymmetric double-well potentials under specific coupling and rotation strengths. In the presence of mean-field interactions, using the coupled Gross-Pitaevskii formalism at moderate to high rotation frequencies, the analytically obtained effective potential minima and the numerically obtained coarse-grained density maxima position are in excellent agreement. On rapid rotation, we further find that the spin-expectation per particle of an antiferromagnetic spin-1 BEC approaches unity indicating a similarity in the response with ferromagnetic SO-coupled condensates.

The quantum dynamics of two-component Bose-Einstein condensate: anSp(4,R)symmetry approach

The compact groups such asSU(n) andSO(n) groups have been heavily studied and applied in the study of quantum many body systems. However, the non-compact groups such as the real symplectic groups are less touched. In this paper, it is revealed that the quantum dynamics of two-component Bose-Einstein condensate can be described by a non-compact real symplectic groupSp(4,R). With this group, an explicit form of the wavefunction in any time of the evolution can be given, meanwhile, this whole time evolution can be shown to correspond to a trajectory in a six-dimensional manifold. By introducing a polar coordinate, we can visualize this six-dimensional manifold in 2d unit disk and reveal the relation between the behavior of the trajectory in this manifold and the eigenenergies of the Hamiltonian. Furthermore, the time evolution of expectation value of a physical observable such as number operator is proven closely related to the behavior of the trajectory in this manifold.

Fledgling Quantum Spin Hall Effect in Pseudo Gap Phase of Bi2212

We studied the emergence of the quantum spin Hall (QSH) states for the pseudo-gap (PG) phase of Bi2212 bilayer system, assumed to be D-density wave (DDW) ordered, starting with a strong Rashba spin-orbit coupling (SOC) armed, and the time reversal symmetry (TRS) complaint Bloch Hamiltonian. The presence of strong SOC gives rise to non-trivial, spin-momentum locked spin texture tunable by electric field. The emergence of quantum anomalous Hall effect with TRS broken Chiral DDW Hamiltonian of Das Sarma et al. is found to be possible.

Path integral molecular dynamics for thermodynamics and Green's function of ultracold spinor bosons

Most recently, the path integral molecular dynamics has been successfully used to consider the thermodynamics of single-component identical bosons and fermions. In this work, the path integral molecular dynamics is developed to simulate the thermodynamics, Green's function and momentum distribution of two-component bosons in three dimensions. As an example of our general method, we consider the thermodynamics of up to sixteen bosons in a three-dimensional harmonic trap. For noninteracting spinor bosons, our simulation shows a bump in the heat capacity. As the repulsive interaction strength increases, however, we find the gradual disappearance of the bump in the heat capacity. We believe this simulation result can be tested by ultracold spinor bosons with optical lattices and magnetic-field Feshbach resonance to tune the inter-particle interaction. We also calculate Green's function and momentum distribution of spinor bosons. Our work facilitates the exact numerical simulation of spinor bosons, whose property is one of the major problems in ultracold Bose gases.

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.

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.

Charge-4e Superconductivity from Nematic Superconductors in Two and Three Dimensions

Charge-4e superconductivity as a novel phase of matter remains elusive so far. Here, we show that charge-4e phase can arise as a vestigial order above the nematic superconducting transition temperature in time-reversal-invariant nematic superconductors. On the one hand, the nontrivial topological defect-nematic vortex-is energetically favored over the superconducting phase vortex when the nematic stiffness is less than the superfluid stiffness; consequently the charge-4e phase emerges by proliferation of nematic vortices upon increasing temperatures. On the other hand, the Ginzburg-Landau theory of the nematic superconductors has two distinct decoupling channels to either charge-4e orders or nematic orders; by analyzing the competition between the effective mass of the charge-4e order and the cubic potential of the nematic order, we find a sizable regime where the charge-4e order is favored. These two analyses consistently show that nematic superconductors can provide a promising route to realize charge-4e phases, which may apply to candidate nematic superconductors such as PbTaSe_{2} and twisted bilayer graphene.

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.

Electronic Floquet gyro-liquid crystal

Floquet engineering uses coherent time-periodic drives to realize designer band structures on-demand, thus yielding a versatile approach for inducing a wide range of exotic quantum many-body phenomena. Here we show how this approach can be used to induce non-equilibrium correlated states with spontaneously broken symmetry in lightly doped semiconductors. In the presence of a resonant driving field, the system spontaneously develops quantum liquid crystalline order featuring strong anisotropy whose directionality rotates as a function of time. The phase transition occurs in the steady state of the system achieved due to the interplay between the coherent external drive, electron-electron interactions, and dissipative processes arising from the coupling to phonons and the electromagnetic environment. We obtain the phase diagram of the system using numerical calculations that match predictions obtained from a phenomenological treatment and discuss the conditions on the system and the external drive under which spontaneous symmetry breaking occurs. Our results demonstrate that coherent driving can be used to induce non-equilibrium quantum phases of matter with dynamical broken symmetry.

Moiré pattern of interference dislocations in condensate of indirect excitons

Interference patterns provide direct measurement of coherent propagation of matter waves in quantum systems. Superfluidity in Bose-Einstein condensates of excitons can enable long-range ballistic exciton propagation and can lead to emerging long-scale interference patterns. Indirect excitons (IXs) are formed by electrons and holes in separated layers. The theory predicts that the reduced IX recombination enables IX superfluid propagation over macroscopic distances. Here, we present dislocation-like phase singularities in interference patterns produced by condensate of IXs. We analyze how exciton vortices and skyrmions should appear in the interference experiments and show that the observed interference dislocations are not associated with these phase defects. We show that the observed interference dislocations originate from the moiré effect in combined interference patterns of propagating condensate matter waves. The interference dislocations are formed by the IX matter waves ballistically propagating over macroscopic distances. The long-range ballistic IX propagation is the evidence for IX condensate superfluidity.

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.

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.

Stripe and supersolid phases of spin-orbit coupled spin-2 Bose-Einstein condensates in an optical lattice

We study the ground-state phases of two-dimensional spin-orbit coupled spin-2 Bose-Einstein condensates in a one-dimensional spin-dependent optical lattice. Due to the competition among optical lattice, spin-orbit coupling and spin-exchange interaction, the exotic ground-state phases are found, i.e. three types of the stripe phases and three types of the supersolid phases. The spin-exchange interaction can adjust the direction of the stripe in the stripe phase and generate various vortex lattice structures in the supersolid phase, which shows that the spin-exchange interaction plays an important role in the formation of the stripe and supersolid phases of spin-orbit coupled spin-2 Bose-Einstein condensates in an optical lattice.

Roton-Induced Bose Polaron in the Presence of Synthetic Spin-Orbit Coupling

We predict the existence of a roton-induced Bose polaron for an impurity immersed in a three-dimensional Bose-Einstein condensate with Raman-laser-induced spin-orbit coupling, where the condensate is in a finite-momentum plane-wave state with an intriguing roton minimum in its excitation spectrum. This novel polaron is formed by dressing the impurity with roton excitations, instead of phonon excitations as in a conventional (i.e., phonon-induced) Bose polaron, and acquires a significant center-of-mass momentum and highly anisotropic effective mass. We find that the roton-induced polaron evolves from a phonon-induced polaron, as the interaction between impurity and atoms increases across a Feshbach resonance. The evolution is not smooth, and a first-order phase transition from a phonon- to roton-induced polaron is observed at a critical interaction strength.

Supersolid Stripe Crystal from Finite-Range Interactions on a Lattice

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 to a self-assembled array of quasi-one-dimensional superfluids. These bosonic superstripes turn into an isotropic supersolid with decreasing the interaction strength. The mechanism for stripe formation is based on cluster self-assembling in the corresponding classical ground state, reminiscent of classical soft-matter models of polymers, different from recently proposed mechanisms for cold gases of alkali or dipolar magnetic atoms.

Stable Multiring and Rotating Solitons in Two-Dimensional Spin-Orbit-Coupled Bose-Einstein Condensates with a Radially Periodic Potential

We consider two-dimensional spin-orbit-coupled atomic Bose-Einstein condensate in a radially periodic potential. The system supports different types of stable self-sustained states including radially symmetric vorticity-carrying modes with different topological charges in two spinor components that may have multiring profiles and at the same time remain remarkably stable for repulsive interactions. Solitons of the second type show persistent rotation with constant angular frequency. They can be stable for both repulsive and attractive interatomic interactions. Because of the inequivalence between clockwise and counterclockwise rotation directions introduced by spin-orbit coupling, the properties of such solitons strongly differ for positive and negative rotation frequencies. The collision of solitons located in the same or different rings is accompanied by a change of the rotation frequency that depends on the phase difference between colliding solitons.

Spin current generation and relaxation in a quenched spin-orbit-coupled Bose-Einstein condensate

Understanding the effects of spin-orbit coupling (SOC) and many-body interactions on spin transport is important in condensed matter physics and spintronics. This topic has been intensively studied for spin carriers such as electrons but barely explored for charge-neutral bosonic quasiparticles (including their condensates), which hold promises for coherent spin transport over macroscopic distances. Here, we explore the effects of synthetic SOC (induced by optical Raman coupling) and atomic interactions on the spin transport in an atomic Bose-Einstein condensate (BEC), where the spin-dipole mode (SDM, actuated by quenching the Raman coupling) of two interacting spin components constitutes an alternating spin current. We experimentally observe that SOC significantly enhances the SDM damping while reducing the thermalization (the reduction of the condensate fraction). We also observe generation of BEC collective excitations such as shape oscillations. Our theory reveals that the SOC-modified interference, immiscibility, and interaction between the spin components can play crucial roles in spin transport.

Spin-orbit coupling is interesting for fundamental understanding of spin transport and quench dynamics. Here the authors demonstrate spin-current generation and its relaxation in spin-orbit-coupled Bose-Einstein condensates of Rb atoms in different spin states.

Beliaev Damping of a Spin-Orbit-Coupled Bose-Einstein Condensate

Beliaev damping provides a fundamental mechanism for dissipation of quasiparticles. Previous research has shown that the two-component internal degrees of freedom has no nontrivial effect on Beliaev damping. Here we provide the first example where the spinor nature of Bose gases can manifest itself in the Beliaev damping by way of spin-obit coupling. We identify novel features of the Beliaev decay rate due to spin-orbit coupling; in particular, it shows an explicit dependence on the spin-density interaction and diverges at the interaction-modified phase boundary between the zero-momentum and plane wave phases. This represents a manifestation of the effect of spin-orbit coupling in the beyond-mean-field regime, which by breaking Galilean invariance couples excitations in the density and spin channels. We further show that the measurement of the Beliaev damping rate is experimentally feasible through the measurement of spin polarizability susceptibility, which has been already achieved in spin-orbit-coupled Bose gases.

Quantum Spin Dynamics in a Normal Bose Gas with Spin-Orbit Coupling

In this Letter, we investigate spin dynamics of a two-component Bose gas with spin-orbit coupling realized in cold atom experiments. We derive coupled hydrodynamic equations for number and spin densities as well as their associated currents. Specializing to the quasi-one-dimensional situation, we obtain analytic solutions of the spin helix structure and its dynamics in both adiabatic and diabatic regimes. In the adiabatic regime, the transverse spin decays parabolically in the short-time limit and exponentially in the long-time limit, depending on initial polarization. In contrast, in the diabatic regime, transverse spin density and current oscillate in a way similar to the charge-current oscillation in an undamped LC circuit. The effects of Rabi coupling on the short-time spin dynamics is also discussed. Finally, using realistic experimental parameters for ^{87}Rb, we show that the timescales for spin dynamics is of the order of milliseconds to a few seconds and can be observed experimentally.

Electromagnetically induced transparency in a spin-orbit coupled Bose-Einstein condensate

The artificial field can be generated by properly arranging pulsed magnetic fields interacting with a Bose-Einstein condensate (BEC), which can be widely used to simulate the phenomena of traditional condensed matter physics, such as spin-orbit (SO) coupling and the neutral atom spin Hall effect. The introduction of SO coupling in a BEC will alter its optical properties. Eletromagnetically induced transparency (EIT) is a powerful tool that can change and detect the properties of an atomic medium in a nondestructive way. It is important and interesting to study EIT properties and to investigate the effects of SO coupling on EIT. In this paper, we investigate EIT in a SO-coupled BEC. Not only is the transparency existing, but the real and imaginary parts of the susceptibility have an additional red frequency shift, which is linearly proportional to the strength of the SO coupling. By using this unconventional, sensitive EIT spectrum, SO coupling can be detected and its strength can be accurately measured according to the frequency shift.

Ground-state phases of spin-orbit coupled spin-1 Bose-Einstein condensate in a plane quadrupole field

We study the ground-state phases of two-dimensional spin-orbit coupled spin-1 Bose-Einstein condensate loaded in a plane quadrupole field. In the absence of rotation, for the fixed spin-orbit coupling strength, the ordinary stripe phase is found when the strength of the magnetic field gradient is small. As the strength of magnetic field gradient enhances, the system realizes the phases with three layer vortices along the radial direction. The number of vortices in the second layer is successively increased and the vortices in the outermost layer disappear when the strength of magnetic field gradient surpass the critical value. For the large strength of magnetic field gradient, the system only has the inner layer vortices. The magnetic field inhibits the region of vortices. For the fixed magnetic field gradient strength, the vortices of the system elongate along the radial direction and form a series of vortex lines, the number of the vortex line increases as the strength of spin-orbit coupling enhances. By adding the rotation, for the fixed strengths of spin-orbit coupling and magnetic field gradient, the number of second layer vortices also successively increases as the rotational frequency increases. The number of vortices in the certain layer of the ground-state density can be regularly changed under the effects of the magnetic field and spin-orbit coupling.

Chiral Supersolid in Spin-Orbit-Coupled Bose Gases with Soft-Core Long-Range Interactions

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 current. This implies that a finite angular momentum can be generated with neither rotation nor effective magnetic field. The direction of the angular momentum can be altered by adjusting the strength of spin-orbit coupling or interatomic interaction. The predicted chiral supersolid phase can be experimentally observed in Rydberg-dressed Bose-Einstein condensates with spin-orbit coupling.

Searching for Supersolidity in Ultracold Atomic Bose Condensates with Rashba Spin-Orbit Coupling

We developed a functional integral formulation for the stripe phase of 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 behavior with the lower branch vanishing 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 fluctuation correction to the ground-state energy and quantum depletion of the condensates exhibit discontinuity, characteristic of the first-order phase transition. Despite strong quantum fluctuations induced by Rashba spin-orbit coupling, we show that the supersolid phase is stable against quantum depletion. Finally, we extend our formulation to finite temperatures to account for interactions between excitations.

Superfluid-Quasicrystal in a Bose-Einstein Condensate

A quasicrystal is a class of ordered structures defying conventional classification of solid crystals and may carry classically forbidden (e.g., fivefold) 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 fivefold rotational symmetry can be realized as the ground state of a Bose-Einstein condensate within a practical experimental scheme. There exists a rich phase diagram consisting of various superfluid-quasicrystal, supersolid, and plane-wave phases. Our scheme can be generalized for generating other higher-order (e.g., sevenfold) quasicrystal states, and provides a platform for investigating such new exotic quantum matter.

The phase diagram and stability of trapped D-dimensional spin-orbit coupled Bose-Einstein condensate

By variational analysis and direct numerical simulation, we study the phase transition and stability of a trapped D-dimensional Bose-Einstein condensate with spin-orbit coupling. The complete phase and stability diagrams of the system are presented in full parameter space, while the collapse dynamics induced by the mean-filed attraction and the mechanism for stabilizing the collapse by spin-orbit coupling are illustrated explicitly. Particularly, a full and deep understanding of the dependence of phase transition and stability mechanism on geometric dimensionality and external trap potential is revealed. It is shown that the spin-orbit coupling can modify the dispersion relations, which can balance the mean-filed attractive interaction and result in a spin polarized or overlapped state to stabilize the collapse, then changes the collapsing threshold dependent on the geometric dimensionality and external trap potential. Moreover, from 2D to 3D system, the mean-field attraction for inducing the collapse is reduced and the collapse speed is enhanced, namely, the collapse can be more easily stabilized in 2D system. That is, the collapse can be manipulated by adjusting the spin-orbit coupling, Raman coupling, geometric dimensionality and the external trap potential, which can provide a possible way for elaborating the collapse dynamics experimentally.

Spin-Tensor-Momentum-Coupled Bose-Einstein Condensates

The recent experimental realization of spin-orbit coupling for ultracold atomic gases provides a powerful platform for exploring many interesting quantum phenomena. In these studies, spin represents the spin vector (spin 1/2 or spin 1) and orbit represents the linear momentum. Here we propose a scheme to realize a new type of spin-tensor-momentum coupling (STMC) in spin-1 ultracold atomic gases. We study the ground state properties of interacting Bose-Einstein condensates with STMC and find interesting new types of stripe superfluid phases and multicritical points for phase transitions. Furthermore, STMC makes it possible to study quantum states with dynamical stripe orders that display density modulation with a long tunable period and high visibility, paving the way for the direct experimental observation of a new dynamical supersolidlike state. Our scheme for generating STMC can be generalized to other systems and may open the door for exploring novel quantum physics and device applications.

Strong coupling phases of the spin-orbit-coupled spin-1 Bose-Hubbard chain: odd integer Mott lobes and helical magnetic phases

We study the odd integer filled Mott phases of a spin-1 Bose-Hubbard chain and determine their fate in the presence of a Raman induced spin-orbit coupling which has been achieved in ultracold atomic gases; this system is described by a quantum spin-1 chain with a spiral magnetic field. The spiral magnetic field initially induces helical order with either ferromagnetic or dimer order parameters, giving rise to a spiral paramagnet at large field. The spiral ferromagnet-to-paramagnet phase transition is in a novel universality class, with critical exponents associated with the divergence of the correlation length ν ≈ 2 / 3 and the order parameter susceptibility γ ≈ 1 / 2 . We solve the effective spin model exactly using the density matrix renormalization group, and compare with both a large-S classical solution and a phenomenological Landau theory. We discuss how these exotic bosonic magnetic phases can be produced and probed in ultracold atomic experiments in optical lattices.

Diffused Vorticity and Moment of Inertia of a Spin-Orbit Coupled Bose-Einstein Condensate

By developing the hydrodynamic theory of spinor superfluids, we calculate the moment of inertia of a harmonically trapped Bose-Einstein condensate with spin-orbit coupling. We show that the velocity field associated with the rotation of the fluid exhibits diffused vorticity, in contrast to the irrotational behavior characterizing a superfluid. Both Raman-induced and Rashba spin-orbit couplings are considered. In the first case the moment of inertia takes the rigid value at the transition between the plane wave and the single minimum phase, while in the latter case the rigid value is achieved in the limit of isotropic Rashba coupling. A procedure to generate the rigid rotation of the fluid and to measure the moment of inertia is proposed. The quenching of the quantum of circulation h/m, caused by Raman-induced spin-orbit coupling in a toroidal geometry, is also discussed.

Experimental Observation of a Topological Band Gap Opening in Ultracold Fermi Gases with Two-Dimensional Spin-Orbit Coupling

The recent experimental realization of synthetic spin-orbit coupling (SOC) opens a new avenue for exploring novel quantum states with ultracold atoms. However, in experiments for generating two-dimensional SOC (e.g., Rashba type), a perpendicular Zeeman field, which opens a band gap at the Dirac point and induces many topological phenomena, is still lacking. Here, we theoretically propose and experimentally realize a simple scheme for generating two-dimensional SOC and a perpendicular Zeeman field simultaneously in ultracold Fermi gases by tuning the polarization of three Raman lasers that couple three hyperfine ground states of atoms. The resulting band gap opening at the Dirac point is probed using spin injection radio-frequency spectroscopy. Our observation may pave the way for exploring topological transport and topological superfluids with exotic Majorana and Weyl fermion excitations in ultracold atoms.

Ground states of a Bose-Einstein Condensate in a one-dimensional laser-assisted optical lattice

We study the ground-state behavior of a Bose-Einstein Condensate (BEC) in a Raman-laser-assisted one-dimensional (1D) optical lattice potential forming a multilayer system. We find that, such system can be described by an effective model with spin-orbit coupling (SOC) of pseudospin (N-1)/2, where N is the number of layers. Due to the intricate interplay between atomic interactions, SOC and laser-assisted tunnelings, the ground-state phase diagrams generally consist of three phases-a stripe, a plane wave and a normal phase with zero-momentum, touching at a quantum tricritical point. More important, even though the single-particle states only minimize at zero-momentum for odd N, the many-body ground states may still develop finite momenta. The underlying mechanisms are elucidated. Our results provide an alternative way to realize an effective spin-orbit coupling of Bose gas with the Raman-laser-assisted optical lattice, and would also be beneficial to the studies on SOC effects in spinor Bose systems with large spin.

Spin-Orbit Coupling and Spin Textures in Optical Superlattices

We propose and demonstrate a new approach for realizing spin-orbit coupling with ultracold atoms. We use orbital levels in a double-well potential as pseudospin states. Two-photon Raman transitions between left and right wells induce spin-orbit coupling. This scheme does not require near resonant light, features adjustable interactions by shaping the double-well potential, and does not depend on special properties of the atoms. A pseudospinor Bose-Einstein condensate spontaneously acquires an antiferromagnetic pseudospin texture, which breaks the lattice symmetry similar to a supersolid.

Tricriticalities and Quantum Phases in Spin-Orbit-Coupled Spin-1 Bose Gases

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 detail, and the emergence of tricritical points, which are the direct consequence of the spin-dependent interactions, is explicitly discussed.

Vortex solitons in two-dimensional spin-orbit coupled Bose-Einstein condensates: Effects of the Rashba-Dresselhaus coupling and Zeeman splitting

We present an analysis of two-dimensional (2D) matter-wave solitons, governed by the pseudospinor system of Gross-Pitaevskii equations with self- and cross attraction, which includes the spin-orbit coupling (SOC) in the general Rashba-Dresselhaus form, and, separately, the Rashba coupling and the Zeeman splitting. Families of semivortex (SV) and mixed-mode (MM) solitons are constructed, which exist and are stable in free space, as the SOC terms prevent the onset of the critical collapse and create the otherwise missing ground states in the form of the solitons. The Dresselhaus SOC produces a destructive effect on the vortex solitons, while the Zeeman term tends to convert the MM states into the SV ones, which eventually suffer delocalization. Existence domains and stability boundaries are identified for the soliton families. For physically relevant parameters of the SOC system, the number of atoms in the 2D solitons is limited by ∼1.5×10^{4}. The results are obtained by means of combined analytical and numerical methods.

Dynamic Onset of Feynman Relation in the Phonon Regime

The Feynman relation, a much celebrated condensed matter physics gemstone for more than 70 years, predicts that the density excitation spectrum and structure factor of a condensed Bosonic system in the phonon regime drops linear and continuously to zero. Until now, this widely accepted monotonic excitation energy drop as the function of reduced quasi-momentum has never been challenged in a spin-preserving process. We show rigorously that in a light-matter wave-mixing process in a Bosonic quantum gas, an optical-dipole potential arising from the internally-generated field can profoundly alter the Feynman relation and result in a new dynamic relation that exhibits an astonishing non-Feynman-like onset and cut-off in the excitation spectrum of the ground state energy of spin-preserving processes. This is the first time that a nonlinear optical process is shown to actively and significantly alter the density excitation response of a quantum gas. Indeed, this dynamic relation with a non-Feynman onset and cut-off has no correspondence in either nonlinear optics of a normal gas or a phonon-based condensed matter Bogoliubov theory.

Magnetic phases of spin-1 spin-orbit-coupled Bose gases

Phases of matter are characterized by order parameters describing the type and degree of order in a system. Here we experimentally explore the magnetic phases present in a near-zero temperature spin-1 spin-orbit-coupled atomic Bose gas and the quantum phase transitions between these phases. We observe ferromagnetic and unpolarized phases, which are stabilized by spin-orbit coupling's explicit locking between spin and motion. These phases are separated by a critical curve containing both first- and second-order transitions joined at a tricritical point. The first-order transition, with observed width as small as h × 4 Hz, gives rise to long-lived metastable states. These measurements are all in agreement with theory.

Tunable atomic spin-orbit coupling synthesized with a modulating gradient magnetic field

We report the observation of synthesized spin-orbit coupling (SOC) for ultracold spin-1 ⁸⁷Rb atoms. Different from earlier experiments where a one dimensional (1D) atomic SOC of pseudo-spin-1/2 is synthesized with Raman laser fields, the scheme we demonstrate employs a gradient magnetic field (GMF) and ground-state atoms, thus is immune to atomic spontaneous emission. The strength of SOC we realize can be tuned by changing the modulation amplitude of the GMF, and the effect of the SOC is confirmed through the studies of: 1) the collective dipole oscillation of an atomic condensate in a harmonic trap after the synthesized SOC is abruptly turned on; and 2) the minimum energy state at a finite adiabatically adjusted momentum when SOC strength is slowly ramped up. The condensate coherence is found to remain very good after driven by modulating GMFs. Our scheme presents an alternative means for studying interacting many-body systems with synthesized SOC.

A two-dimensional algebraic quantum liquid produced by an atomic simulator of the quantum Lifshitz model

Bosons have a natural instinct to condense at zero temperature. It is a long-standing challenge to create a high-dimensional quantum liquid that does not exhibit long-range order at the ground state, as either extreme experimental parameters or sophisticated designs of microscopic Hamiltonians are required for suppressing the condensation. Here we show that synthetic gauge fields for ultracold atoms, using either the Raman scheme or shaken lattices, provide physicists a simple and practical scheme to produce a two-dimensional algebraic quantum liquid at the ground state. This quantum liquid arises at a critical Lifshitz point, where a two-dimensional quartic dispersion emerges in the momentum space, and many fundamental properties of two-dimensional bosons are changed in its proximity. Such an ideal simulator of the quantum Lifshitz model allows experimentalists to directly visualize and explore the deconfinement transition of topological excitations, an intriguing phenomenon that is difficult to access in other systems.

Degenerate quantum gases with spin-orbit coupling: a review

This review focuses on recent developments in synthetic spin-orbit (SO) coupling in ultracold atomic gases. Two types of SO coupling are discussed. One is Raman process induced coupling between spin and motion along one of the spatial directions and the other is Rashba SO coupling. We emphasize their common features in both single-particle and two-body physics and the consequences of both in many-body physics. For instance, single particle ground state degeneracy leads to novel features of superfluidity and a richer phase diagram; increased low-energy density-of-state enhances interaction effects; the absence of Galilean invariance and spin-momentum locking gives rise to intriguing behaviours of superfluid critical velocity and novel quantum dynamics; and the mixing of two-body singlet and triplet states yields a novel fermion pairing structure and topological superfluids. With these examples, we show that investigating SO coupling in cold atom systems can, enrich our understanding of basic phenomena such as superfluidity, provide a good platform for simulating condensed matter states such as topological superfluids and more importantly, result in novel quantum systems such as SO coupled unitary Fermi gas and high spin quantum gases. Finally we also point out major challenges and some possible future directions.