Degenerate quantum gases with spin-orbit coupling: a review

Optical spin hall effect in exciton-polariton condensates in lead halide perovskite microcavities

An exciton-polariton condensate is a hybrid light-matter state in the quantum fluid phase. The photonic component endows it with characters of spin, as represented by circular polarization. Spin-polarization can form stochastically for quasi-equilibrium exciton-polariton condensates at parallel momentum vector k|| ∼ 0 from bifurcation or deterministically for propagating condensates at k|| > 0 from the optical spin-Hall effect (OSHE). Here, we report deterministic spin-polarization in exciton-polariton condensates at k|| ∼ 0 in microcavities containing methylammonium lead bromide perovskite (CH3NH3PbBr3) single crystals under non-resonant and linearly polarized excitation. We observe two energetically split condensates with opposite circular polarizations and attribute this observation to the presence of strong birefringence, which introduces a large OSHE at k|| ∼ 0 and pins the condensates in a particular spin state. Such spin-polarized exciton-polariton condensates may serve not only as circularly polarized laser sources but also as effective alternatives to ultracold atom Bose-Einstein condensates in quantum simulators of many-body spin-orbit coupling processes.

Topological spin-orbit-coupled fermions beyond rotating wave approximation

The realization of spin-orbit-coupled ultracold gases has driven a wide range of research and is typically based on the rotating wave approximation (RWA). By neglecting the counter-rotating terms, RWA characterizes a single near-resonant spin-orbit (SO) coupling in a two-level system. Here, we propose and experimentally realize a new scheme for achieving a pair of two-dimensional (2D) SO couplings for ultracold fermions beyond RWA. This work not only realizes the first anomalous Floquet topological Fermi gas beyond RWA, but also significantly improves the lifetime of the 2D-SO-coupled Fermi gas. Based on pump-probe quench measurements, we observe a deterministic phase relation between two sets of SO couplings, which is characteristic of our beyond-RWA scheme and enables the two SO couplings to be simultaneously tuned to the optimum 2D configurations. We observe intriguing band topology by measuring two-ring band-inversion surfaces, quantitatively consistent with a Floquet topological Fermi gas in the regime of high Chern numbers. Our study can open an avenue to explore exotic SO physics and anomalous topological states based on long-lived SO-coupled ultracold fermions.

Field-Induced Non-BEC Transitions in Frustrated Magnets

Frustrated spin systems have traditionally proven challenging to understand, owing to a scarcity of controlled methods for their analyses. By contrast, under strong magnetic fields, certain aspects of spin systems admit simpler and universal description in terms of hardcore bosons. The bosonic formalism is anchored by the phenomenon of Bose-Einstein condensation (BEC), which has helped explain the behaviors of a wide range of magnetic compounds under applied magnetic fields. Here, we focus on the interplay between frustration and externally applied magnetic field to identify instances where the BEC paradigm is no longer applicable. As a representative example, we consider the antiferromagnetic J_{1}-J_{2}-J_{3} model on the square lattice in the presence of a uniform external magnetic field, and demonstrate that the frustration-driven suppression of the Néel order leads to a Lifshitz transition for the hardcore bosons. In the vicinity of the Lifshitz point, the physics becomes unmoored from the BEC paradigm, and the behavior of the system, both at and below the saturation field, is controlled by a Lifshitz multicritical point. We obtain the resultant universal scaling behaviors, and provide strong evidence for the existence of a frustration and magnetic-field driven correlated bosonic liquid state along the entire phase boundary separating the Néel phase from other magnetically ordered states.

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Constrained Motions and Slow Dynamics in One-Dimensional Bosons with Double-Well Dispersion

We demonstrate slow dynamics and constrained motion of domain walls in one-dimensional (1D) interacting bosons with double-well dispersion. In the symmetry-broken regime, the domain-wall motion is "fractonlike"-a single domain wall cannot move freely, while two nearby domain walls can move collectively. Consequently, we find an Ohmic-like linear response and a vanishing superfluid stiffness, which are atypical for a Bose condensate in a 1D translation invariant closed quantum system. Near Lifshitz quantum critical point, we obtain superfluid stiffness ρ_{s}∼T and sound velocity v_{s}∼T^{1/2}, showing similar unconventional low-temperature slow dynamics to the symmetry-broken regime. Particularly, the superfluid stiffness suggests an order by disorder effect as ρ_{s} increases with temperature. Our results pave the way for studying fractons in ultracold atom experiments.

Quartet Superfluid in Two-Dimensional Mass-Imbalanced Fermi Mixtures

Quartet superfluid (QSF) is a distinct type of fermion superfluidity that exhibits high-order correlation beyond the conventional BCS pairing paradigm. In this Letter, we report the emergent QSF in 2D mass-imbalanced Fermi mixtures with two-body contact interactions. This is facilitated by the formation of a quartet bound state in vacuum that consists of a light atom and three heavy fermions. For an optimized heavy-light number ratio 3:1, we identify QSF as the ground state in a considerable parameter regime of mass imbalance and 2D coupling strength. Its unique high-order correlation can be manifested in the momentum-space crystallization of a pairing field and density distribution of heavy fermions. Our results can be readily detected in Fermi-Fermi mixtures nowadays realized in cold atoms laboratories, and meanwhile shed light on exotic superfluidity in a broad context of mass-imbalanced fermion mixtures.

Integrator for general spin-s Gross-Pitaevskii systems

We provide an algorithm, i-SPin 2, for evolving general spin-s Gross-Pitaevskii or nonlinear Schrödinger systems carrying a variety of interactions, where the 2s+1 components of the "spinor" field represent the different spin-multiplicity states. We consider many nonrelativistic interactions up to quartic order in the Schrödinger field (both short and long range, and spin-dependent and spin-independent interactions), including explicit spin-orbit couplings. The algorithm allows for spatially varying external and/or self-generated vector potentials that couple to the spin density of the field. Our work can be used for scenarios ranging from laboratory systems such as spinor Bose-Einstein condensates (BECs), to cosmological or astrophysical systems such as self-interacting bosonic dark matter. As examples, we provide results for two different setups of spin-1 BECs that employ a varying magnetic field and spin-orbit coupling, respectively, and also collisions of spin-1 solitons in dark matter. Our symplectic algorithm is second-order accurate in time, and is extensible to the known higher-order-accurate methods.

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.

Dipolar physics: a review of experiments with magnetic quantum gases

Since the achievement of quantum degeneracy in gases of chromium atoms in 2004, the experimental investigation of ultracold gases made of highly magnetic atoms has blossomed. The field has yielded the observation of many unprecedented phenomena, in particular those in which long-range and anisotropic dipole-dipole interactions (DDIs) play a crucial role. In this review, we aim to present the aspects of the magnetic quantum-gas platform that make it unique for exploring ultracold and quantum physics as well as to give a thorough overview of experimental achievements. Highly magnetic atoms distinguish themselves by the fact that their electronic ground-state configuration possesses a large electronic total angular momentum. This results in a large magnetic moment and a rich electronic transition spectrum. Such transitions are useful for cooling, trapping, and manipulating these atoms. The complex atomic structure and large dipolar moments of these atoms also lead to a dense spectrum of resonances in their two-body scattering behaviour. These resonances can be used to control the interatomic interactions and, in particular, the relative importance of contact over dipolar interactions. These features provide exquisite control knobs for exploring the few- and many-body physics of dipolar quantum gases. The study of dipolar effects in magnetic quantum gases has covered various few-body phenomena that are based on elastic and inelastic anisotropic scattering. Various many-body effects have also been demonstrated. These affect both the shape, stability, dynamics, and excitations of fully polarised repulsive Bose or Fermi gases. Beyond the mean-field instability, strong dipolar interactions competing with slightly weaker contact interactions between magnetic bosons yield new quantum-stabilised states, among which are self-bound droplets, droplet assemblies, and supersolids. Dipolar interactions also deeply affect the physics of atomic gases with an internal degree of freedom as these interactions intrinsically couple spin and atomic motion. Finally, long-range dipolar interactions can stabilise strongly correlated excited states of 1D gases and also impact the physics of lattice-confined systems, both at the spin-polarised level (Hubbard models with off-site interactions) and at the spinful level (XYZ models). In the present manuscript, we aim to provide an extensive overview of the various related experimental achievements up to the present.

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.

Intrinsic Anomalous Hall Effect in a Bosonic Chiral Superfluid

The anomalous Hall effect has had a profound influence on the understanding of many electronic topological materials but is much less studied in their bosonic counterparts. We predict that an intrinsic anomalous Hall effect exists in a recently realized bosonic chiral superfluid, a p-orbital Bose-Einstein condensate in a 2D hexagonal boron nitride optical lattice [Wang et al., Nature (London) 596, 227 (2021)NATUAS0028-083610.1038/s41586-021-03702-0]. We evaluate the frequency-dependent Hall conductivity within a multi-orbital Bose-Hubbard model that accurately captures the real experimental system. We find that in the high frequency limit, the Hall conductivity is determined by finite loop current correlations on the s-orbital residing sublattice, the latter a defining feature of the system's chirality. In the opposite limit, the dc Hall conductivity can trace its origin back to the noninteracting band Berry curvature at the condensation momentum, although the contribution from atomic interactions can be significant. We discuss available experimental probes to observe this intrinsic anomalous Hall effect at both zero and finite frequencies.

Wave Packet Dynamics in Synthetic Non-Abelian Gauge Fields

It is generally admitted that in quantum mechanics, the electromagnetic potentials have physical interpretations otherwise absent in classical physics as illustrated by the Aharonov-Bohm effect. In 1984, Berry interpreted this effect as a geometrical phase factor. The same year, Wilczek and Zee generalized the concept of Berry phases to degenerate levels and showed that a non-Abelian gauge field arises in these systems. In sharp contrast with the Abelian case, spatially uniform non-Abelian gauge fields can induce particle noninertial motion. We explore this intriguing phenomenon with a degenerated Fermionic atomic gas subject to a two-dimensional synthetic SU(2) non-Abelian gauge field. We reveal the spin Hall nature of the noninertial dynamic as well as its anisotropy in amplitude and frequency due to the spin texture of the system. We finally draw the similarities and differences of the observed wave packet dynamic and the celebrated Zitterbewegung effect of the relativistic Dirac equation.

Realization of a Floquet-Engineered Moat Band for Ultracold Atoms

We experimentally engineer a moatlike dispersion in a system of weakly interacting bosons. By periodically modulating the amplitude of a checkerboard optical lattice, the two lowest isolated bands are hybridized such that the single particle energy displays a continuum of nearly degenerate minima that lie along a circle in reciprocal space. The moatlike structure is confirmed by observing a zero group velocity at nonzero quasimomentum and we directly observe the effect of the modified dispersion on the trajectory of the center of mass position of the condensate. We measure the lifetime of condensates loaded into different moat bands at different quasimomenta and compare to theoretical predictions based on a linear stability analysis of Bogoliubov excitations. We find that the condensate decay increases rapidly as the quasimomentum is decreased below the radius of the moat minimum, and argue that such dynamical instability is characteristic of moatlike dispersions, including spin-orbit coupled systems. The ground state of strongly interacting bosons in such degenerate energy landscapes is expected to be highly correlated, and our work represents a step toward realizing fractional quantum Hall-like states of bosons in an optical lattice.

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.

Localization and spin dynamics of spin-orbit-coupled Bose-Einstein condensates in deep optical lattices

We analytically and numerically discuss the dynamics of two pseudospin components Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in deep optical lattices. Rich localized phenomena, such as breathers, solitons, self-trapping, and diffusion, are revealed and strongly depend on the strength of the atomic interaction, SOC, Raman detuning, and the spin polarization (i.e., the initial population difference of atoms between the two pseudospin components of BECs). The critical conditions for the transition of localized states are derived analytically. Based on the critical conditions, the detailed dynamical phase diagram describing the different dynamical regimes is derived. When the Raman detuning satisfies a critical condition, localized states with a fixed initial spin polarization can be observed. When the critical condition is not satisfied, we use two quenching methods, i.e., suddenly and linearly quenching Raman detuning from the soliton or breather state, to discuss the spin dynamics, phase transition, and wave packet dynamics by numerical simulation. The sudden quenching results in a damped oscillation of spin polarization and transforms the system to a new polarized state. Interestingly, the linear quenching of Raman detuning induces a controllable phase transition from an unpolarized phase to an expected polarized phase, while the soliton or breather dynamics is maintained.

Localization on a Synthetic Hall Cylinder

By engineering laser-atom interactions, both Hall ribbons and Hall cylinders as fundamental theoretical tools in condensed matter physics have recently been synthesized in laboratories. Here, we show that turning a synthetic Hall ribbon into a synthetic Hall cylinder could naturally lead to localization. Unlike a Hall ribbon, a Hall cylinder hosts an intrinsic lattice, which arises due to the periodic boundary condition in the azimuthal direction, in addition to the external periodic potential imposed by extra lasers. When these two lattices are incommensurate, localization may occur on a synthetic Hall cylinder. Near the localization-delocalization transitions, physical observables strongly depend on the axial magnetic flux, providing us a sensitive means to probe either the transition or the axial flux using one another. In the irrational limit, physical observables are no longer affected by the axial flux, signifying a scheme to suppress decoherence induced by fluctuations of the axial flux.

Critical Temperature in the BCS-BEC Crossover with Spin-Orbit Coupling

We review the study of the superfluid phase transition in a system of fermions whose interaction can be tuned continuously along the crossover from Bardeen–Cooper–Schrieffer (BCS) superconducting phase to a Bose–Einstein condensate (BEC), also in the presence of a spin–orbit coupling. Below a critical temperature the system is characterized by an order parameter. Generally a mean field approximation cannot reproduce the correct behavior of the critical temperature Tc over the whole crossover. We analyze the crucial role of quantum fluctuations beyond the mean-field approach useful to find Tc along the crossover in the presence of a spin–orbit coupling, within a path integral approach. A formal and detailed derivation for the set of equations useful to derive Tc is performed in the presence of Rashba, Dresselhaus and Zeeman couplings. In particular in the case of only Rashba coupling, for which the spin–orbit effects are more relevant, the two-body bound state exists for any value of the interaction, namely in the full crossover. As a result the effective masses of the emerging bosonic excitations are finite also in the BCS regime.

Generating Giant Vortex in a Fermi Superfluid via Spin-Orbital-Angular-Momentum Coupling

Spin-orbital-angular-momentum (SOAM) coupling has been realized in recent experiments of Bose-Einstein condensates [Chen et al., Phys. Rev. Lett. 121, 113204 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.113204 and Zhang et al., Phys. Rev. Lett. 122, 110402 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.110402], where the orbital angular momentum imprinted upon bosons leads to quantized vortices. For fermions, such an exotic synthetic gauge field can provide fertile ground for fascinating pairing schemes and rich superfluid phases, which are yet to be explored. Here we demonstrate how SOAM coupling stabilizes vortices in Fermi superfluids through a unique mechanism that can be viewed as the angular analog to that of the spin-orbit-coupling-induced Fulde-Ferrell state under a Fermi surface deformation. Remarkably, the vortex size is comparable with the beam waist of Raman lasers generating the SOAM coupling, which is typically much larger than previously observed vortices in Fermi superfluids. With tunable size and core structure, these giant vortex states provide unprecedented experimental access to topological defects in Fermi superfluids.

Quantum Smectic Gauge Theory

We present a gauge theory formulation of a two-dimensional quantum smectic and its relatives, motivated by their realizations in correlated quantum matter. The description gives a unified treatment of phonons and topological defects, respectively, encoded in a pair of coupled gauge fields and corresponding charges. The charges exhibit subdimensional constrained quantum dynamics and anomalously slow highly anisotropic diffusion of disclinations inside a smectic. This approach gives a transparent description of a multistage quantum melting transition of a two-dimensional commensurate crystal (through an incommensurate crystal-a supersolid) into a quantum smectic, which subsequently melts into a quantum nematic and isotropic superfluids, all in terms of a sequence of Higgs transitions.

Spin-Nematic Vortex States in Cold Atoms

The (pseudo)spin degrees of freedom greatly enriches the physics of cold atoms. This is particularly so for systems with high spins (i.e., spin quantum number larger than 1/2). For example, one can construct not only the rank-1 spin vector, but also the rank-2 spin tensor in high spin systems. Here we propose a simple scheme to couple the spin tensor and the center-of-mass orbital angular momentum in a spin-1 cold atom system and show that this leads to a new quantum phase of the matter: the spin-nematic vortex state that features vorticity in an SU(2) spin-nematic tensor subspace. Under proper conditions, such states are characterized by quantized topological numbers. Our work opens up new avenues of research in topological quantum matter with high spins.

Non-Abelian generalizations of the Hofstadter model: spin-orbit-coupled butterfly pairs

The Hofstadter model, well known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum Hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment, we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin-orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and symmetric gauges, the corresponding non-Abelian generalizations exhibit distinct spectra due to the non-commutativity of the gauge fields. We derive the genuine (necessary and sufficient) non-Abelian condition for the two models from the commutativity of their arbitrary loop operators. At zero energy, the models are gapless and host Weyl and Dirac points protected by internal and crystalline symmetries. Double (8-fold), triple (12-fold), and quadrupole (16-fold) Dirac points also emerge, especially under equal hopping phases of the non-Abelian potentials. At other fillings, the gapped phases of the models give rise to topological insulators. We conclude by discussing possible schemes for experimental realization of the models on photonic platforms.

Double-layer Bose-Einstein condensates: A quantum phase transition in the transverse direction, and reduction to two dimensions

We revisit the problem of the reduction of the three-dimensional (3D) dynamics of Bose-Einstein condensates, under the action of strong confinement in one direction (z), to a 2D mean-field equation. We address this problem for the confining potential with a singular term, viz., V_{z}(z)=2z^{2}+ζ^{2}/z^{2}, with constant ζ. A quantum phase transition is induced by the latter term, between the ground state (GS) of the harmonic oscillator and the 3D condensate split in two parallel noninteracting layers, which is a manifestation of the "superselection" effect. A realization of the respective physical setting is proposed, making use of resonant coupling to an optical field, with the resonance detuning modulated along z. The reduction of the full 3D Gross-Pitaevskii equation (GPE) to the 2D nonpolynomial Schrödinger equation (NPSE) is based on the factorized ansatz, with the z -dependent multiplier represented by an exact GS solution of the 1D Schrödinger equation with potential V_{z}(z). For both repulsive and attractive signs of the nonlinearity, the 2D NPSE produces GS and vortex states, that are virtually indistinguishable from the respective numerical solutions provided by full 3D GPE. In the case of the self-attraction, the threshold for the onset of the collapse, predicted by the 2D NPSE, is also virtually identical to its counterpart obtained from the 3D equation. In the same case, stability and instability of vortices with topological charge S=1, 2, and 3 are considered in detail. Thus, the procedure of the spatial-dimension reduction, 3D → 2D, produces very accurate results, and it may be used in other settings.

Stability and quantum escape dynamics of spin-orbit-coupled Bose-Einstein condensates in the shallow trap

The Bose-Einstein condensates in a finite depth potential well provide an ideal platform to study the quantum escape dynamics. In this paper, the ground state, tunneling, and diffusion dynamics of the spin-orbit coupling (SOC) of Bose-Einstein condensates with two pseudospin components in a shallow trap are studied analytically and numerically. The phase transition between the plane-wave phase and zero-momentum phase of the ground state is obtained. Furthermore, the stability of the ground state is discussed, and the stability diagram in the parameter space is provided. The bound state (in which condensates are stably trapped in the potential well), the quasibound state (in which condensates tunnel through the well), and the unstable state (in which diffusion occurs) are revealed. We find that the finite depth potential well has an important effect on the phase transition of the ground state, and, interestingly, SOC can stabilize the system against the diffusion and manipulate the tunneling and diffusion dynamics. In particular, spatial anisotropic tunneling and diffusion dynamics of the two pseudospin components induced by SOC in quasibound and unstable states are observed. We provide an effective model and method to study and control the quantum tunneling and diffusion dynamics.

Nematic-Orbit Coupling and Nematic Density Waves in Spin-1 Condensates

We propose the creation of artificial nematic-orbit coupling in spin-1 Bose-Einstein condensates, in analogy with spin-orbit coupling. Using a suitably designed microwave chip, the quadratic Zeeman shift, normally uniform in space, can be made to be spatiotemporally varying, leading to a coupling between spatial and nematic degrees of freedom. A phase diagram is explored where three quantum phases with the nematic order emerge: easy axis, easy plane with single-well structure, and easy plane with double-well structure in momentum space. By including spin-dependent and spin-independent interactions, we also obtain the low energy excitation spectra in these three phases. Last, we show that the nematic-orbit coupling leads to a periodic nematic density modulation in relation to the period λ_{T} of the cosinusoidal quadratic Zeeman term. Our results point to the rich possibilities for manipulation of tensorial degrees of freedom in ultracold gases without requiring Raman lasers, and therefore, obviating light-scattering induced heating.

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.

Cavity-Mediated Unconventional Pairing in Ultracold Fermionic Atoms

We investigate long-range pairing interactions between ultracold fermionic atoms confined in an optical lattice which are mediated by the coupling to a cavity. In the absence of other perturbations, we find three degenerate pairing symmetries for a two-dimensional square lattice. By tuning a weak local atomic interaction via a Feshbach resonance or by tuning a weak magnetic field, the superfluid system can be driven from a topologically trivial s wave to topologically ordered, chiral superfluids containing Majorana edge states. Our work points out a novel path towards the creation of exotic superfluid states by exploiting the competition between long-range and short-range interactions.

Experimental realization of a non-magnetic one-way spin switch

Controlling magnetism through non-magnetic means is highly desirable for future electronic devices, as such means typically have ultra-low power requirements and can provide coherent control. In recent years, great experimental progress has been made in the field of electrical manipulation of magnetism in numerous material systems. These studies generally do not consider the directionality of the applied non-magnetic potentials and/or magnetism switching. Here, we theoretically conceive and experimentally demonstrate a non-magnetic one-way spin switch device using a spin-orbit coupled Bose-Einstein condensate subjected to a moving spin-independent repulsive dipole potential. The physical foundation of this unidirectional device is based on the breakdown of Galilean invariance in the presence of spin-orbit coupling. Such a one-way spin switch opens an avenue for designing quantum devices with unique functionalities and may facilitate further experimental investigations of other one-way spintronic and atomtronic devices.

Tunable spinful matter wave valve

We investigate the transport problem that a spinful matter wave is incident on a strong localized spin-orbit-coupled Bose-Einstein condensate in optical lattices, where the localization is admitted by atom interaction only existing at one particular site, and the spin-orbit coupling arouse spatial rotation of the spin texture. We find that tuning the spin orientation of the localized Bose-Einstein condensate can lead to spin-nonreciprocal/spin-reciprocal transport, meaning the transport properties are dependent on/independent of the spin orientation of incident waves. In the former case, we obtain the conditions to achieve transparency, beam-splitting, and blockade of the incident wave with a given spin orientation, and furthermore the ones to perfectly isolate incident waves of different spin orientation, while in the latter, we obtain the condition to maximize the conversion of different spin states. The result may be useful to develop a novel spinful matter wave valve that integrates spin switcher, beam-splitter, isolator, and converter. The method can also be applied to other real systems, e.g., realizing perfect isolation of spin states in magnetism, which is otherwise rather difficult.

Ultra-low noise magnetic field for quantum gases

A ultralow noise magnetic field is essential for many branches of scientific research. Examples include experiments conducted on ultracold atoms, quantum simulations, and precision measurements. In ultracold atom experiments specifically, a bias magnetic field will often serve as a quantization axis and be applied for Zeeman splitting. As atomic states are usually sensitive to magnetic fields, a magnetic field characterized by ultralow noise as well as high stability is typically required for experimentation. For this study, a bias magnetic field is successfully stabilized at 14.5 G, with the root mean square value of the noise reduced to 18.5 μG (1.28 ppm) by placing μ-metal magnetic shields together with a dynamical feedback circuit. Long-time instability is also regulated consistently below 7 μG. The level of noise exhibited in the bias magnetic field is further confirmed by evaluating the coherence time of a Bose-Einstein condensate characterized by Rabi oscillation. It is concluded that this approach can be applied to other physical systems as well.

Ground-State Phase Diagram of a Spin-Orbital-Angular-Momentum Coupled Bose-Einstein Condensate

By inducing a Raman transition using a pair of Gaussian and Laguerre-Gaussian laser beams, we realize a ^{87}Rb condensate whose orbital angular momentum (OAM) and its internal spin states are coupled. By varying the detuning and the coupling strength of the Raman transition, we experimentally map out the ground-state phase diagram of the system for the first time. The transitions between different phases feature a discontinuous jump of the OAM and the spin polarization, and hence are of first order. We demonstrate the hysteresis loop associated with such first-order phase transitions. The role of interatomic interaction is also elucidated. Our work paves the way to explore exotic quantum phases in the spin-orbital-angular-momentum coupled quantum gases.

Nonlinear Management of Topological Solitons in a Spin-Orbit-Coupled System

We consider possibilities to control dynamics of solitons of two types, maintained by the combination of cubic attraction and spin-orbit coupling (SOC) in a two-component system, namely, semi-dipoles (SDs) and mixed modes (MMs), by making the relative strength of the cross-attraction, γ , a function of time periodically oscillating around the critical value, γ = 1 , which is an SD/MM stability boundary in the static system. The structure of SDs is represented by the combination of a fundamental soliton in one component and localized dipole mode in the other, while MMs combine fundamental and dipole terms in each component. Systematic numerical analysis reveals a finite bistability region for the SDs and MMs around γ = 1 , which does not exist in the absence of the periodic temporal modulation (“management”), as well as emergence of specific instability troughs and stability tongues for the solitons of both types, which may be explained as manifestations of resonances between the time-periodic modulation and intrinsic modes of the solitons. The system can be implemented in Bose-Einstein condensates (BECs), and emulated in nonlinear optical waveguides.

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.

Topological bands for ultracold atoms

There have been significant recent advances in realizing band structures with geometrical and topological features in experiments on cold atomic gases. This review summarizes these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. Descriptions are given of the different methods that have been used to generate these novel band structures for cold atoms and of the physical observables that have allowed their characterization. The focus is on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. Also described is how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, descriptions are given of the forms of interparticle interactions that emerge when atoms are subjected to these energy bands and of some of the many-body phases that may be sought in future experiments.

Rotating Atomic Quantum Gases with Light-Induced Azimuthal Gauge Potentials and the Observation of the Hess-Fairbank Effect

We demonstrate synthetic azimuthal gauge potentials for Bose-Einstein condensates from engineering atom-light couplings. The gauge potential is created by adiabatically loading the condensate into the lowest energy Raman-dressed state, achieving a coreless vortex state. The azimuthal gauge potentials act as effective rotations and are tunable by the Raman coupling and detuning. We characterize the spin textures of the dressed states, in agreements with the theory. The lowest energy dressed state is stable with a 4.5-s half-atom-number-fraction lifetime. In addition, we exploit the azimuthal gauge potential to demonstrate the Hess-Fairbank effect, the analogue of Meissner effect in superconductors. The atoms in the absolute ground state has a zero quasiangular momentum and transits into a polar-core vortex when the synthetic magnetic flux is tuned to exceed a critical value. Our demonstration serves as a paradigm to create topological excitations by tailoring atom-light interactions where both types of SO(3) vortices in the |⟨F[over →]⟩|=1 manifold, coreless vortices and polar-core vortices, are created in our experiment. The gauge field in the stationary Hamiltonian opens a path to investigating rotation properties of atomic superfluids under thermal equilibrium.

Collective dipole oscillations of a spin-orbit coupled Fermi gas

The collective dipole mode is induced and measured in a spin-orbit (SO) coupled degenerate Fermi gas of ¹⁷³Yb atoms. Using a differential optical Stark shift, we split the degeneracy of three hyperfine states in the ground manifold, and independently couple consecutive spin states with the equal Raman transitions. A relatively long-lived spin-orbit-coupled Fermi gas, readily being realized with a narrow optical transition, allows to explore a single-minimum dispersion where three minima of spin-1 system merge into and to monitor collective dipole modes of fermions in the strong coupling regime. The measured oscillation frequency of the dipole mode is compared with the semi-classical calculation in the single-particle regime. Our work should pave the way towards the characterization of spin-orbit-coupled fermions with large spin s >\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}$$\end{document}12 in the strong coupling regime.

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.

Highly Controllable and Robust 2D Spin-Orbit Coupling for Quantum Gases

We report the realization of a robust and highly controllable two-dimensional (2D) spin-orbit (SO) coupling with a topological nontrivial band structure. By applying a retro-reflected 2D optical lattice, phase tunable Raman couplings are formed into the antisymmetric Raman lattice structure, and generate the 2D SO coupling with precise inversion and C_{4} symmetries, leading to considerably enlarged topological regions. The lifetime of the 2D SO coupled Bose-Einstein condensate reaches several seconds, which enables exploring fine-tuning interaction effects. These essential advantages of the present new realization open the door to explore exotic quantum many-body effects and nonequilibrium dynamics with novel topology.

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.

Spin-Orbital-Angular-Momentum Coupled Bose-Einstein Condensates

We demonstrate coupling between the atomic spin- and orbital-angular momentum (OAM) of the atom's center-of-mass motion in a Bose-Einstein condensate (BEC). The coupling is induced by Raman-dressing lasers with a Laguerre-Gaussian beam and creates coreless vortices in an F=1 ^{87}Rb spinor BEC. We observe correlations between spin and OAM in the dressed state and characterize the spin texture; the result is in good agreement with the theory. In the presence of the Raman field, our dressed state is stable for 0.1 s or longer, and it decays due to collision-induced relaxation. As we turn off the Raman beams, the vortex cores in the bare spin |m_{F}=1⟩ and |-1⟩ split. These spin-OAM coupled systems with the Raman-dressing approach have great potential for exploring new topological textures and quantum states.

Rotation-Symmetry-Enforced Coupling of Spin and Angular Momentum for p-Orbital Bosons

Intrinsic spin angular-momentum coupling of an electron has a relativistic quantum origin with the coupling arising from charged orbits, which does not carry over to charge-neutral atoms. Here, we propose a mechanism of spontaneous generation of spin angular-momentum coupling with spinor atomic bosons loaded into p-orbital bands of a two-dimensional optical lattice. This spin angular-momentum coupling originates from many-body correlations and spontaneous symmetry breaking in a superfluid, with the key ingredients attributed to spin-channel quantum fluctuations and an approximate rotation symmetry. The resultant spin angular-momentum intertwined superfluid has Dirac excitations. In the presence of a chemical potential difference for adjacent sites, it provides a bosonic analogue of a symmetry-protected-topological insulator. Through a dynamical mean-field calculation, this novel superfluid is found to be a generic low-temperature phase, and it gives way to Mott localization only at strong interactions and even-integer fillings. We show the temperature to reach this order is accessible with present experiments.

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.

Exotic complexes in one-dimensional Bose-Einstein condensates with spin-orbit coupling

By means of the F-expansion method and intensive numerical simulations, the existence of three families of nonlinear matter waves including Jacobi elliptic functions, solitons, and triangular periodic functions, is demonstrated for spin-orbit coupled Bose-Einstein condensates with a linear potential. In addition, several complexes are obtained by taking two distinct solutions of each family or two distinct families. These solutions sustain different types of two-body interactions in the condensate that can be repulsive, attractive, or attractive and repulsive. Whereas the spin-orbit coupling destabilized these nonlinear matter waves, the linear potential leads to a stabilization. The numerical results are in excellent agreement with our analytical findings and it can be expected that the proposed robust solutions should be observable for experimentally relevant conditions.

Hyper- and hybrid nonlocality

The controlled generation and identification of quantum correlations, usually encoded in either qubits or continuous degrees of freedom, builds the foundation of quantum information science. Recently, more sophisticated approaches, involving a combination of two distinct degrees of freedom, have been proposed to improve on the traditional strategies. Hyperentanglement describes simultaneous entanglement in more than one distinct degree of freedom, whereas hybrid entanglement refers to entanglement shared between a discrete and a continuous degree of freedom. In this work we propose a scheme that allows us to combine the two approaches, and to extend them to the strongest form of quantum correlations. Specifically, we show how two identical, initially separated particles can be manipulated to produce Bell nonlocality among their spins, among their momenta, as well as across their spins and momenta. We discuss possible experimental realizations with atomic and photonic systems.

Realizing Fulde-Ferrell Superfluids via a Dark-State Control of Feshbach Resonances

We propose that the long-sought Fulde-Ferrell superfluidity with nonzero momentum pairing can be realized in ultracold two-component Fermi gases of ^{40}K or ^{6}Li atoms by optically tuning their magnetic Feshbach resonances via the creation of a closed-channel dark state with a Doppler-shifted Stark effect. In this scheme, two counterpropagating optical fields are applied to couple two molecular states in the closed channel to an excited molecular state, leading to a significant violation of Galilean invariance in the dark-state regime and hence to the possibility of Fulde-Ferrell superfluidity. We develop a field theoretical formulation for both two-body and many-body problems and predict that the Fulde-Ferrell state has remarkable properties, such as anisotropic single-particle dispersion relation, suppressed superfluid density at zero temperature, anisotropic sound velocity, and rotonic collective mode. The latter two features can be experimentally probed using Bragg spectroscopy, providing a smoking-gun proof of Fulde-Ferrell superfluidity.

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.

Cold molecules: Progress in quantum engineering of chemistry and quantum matter

Cooling atoms to ultralow temperatures has produced a wealth of opportunities in fundamental physics, precision metrology, and quantum science. The more recent application of sophisticated cooling techniques to molecules, which has been more challenging to implement owing to the complexity of molecular structures, has now opened the door to the longstanding goal of precisely controlling molecular internal and external degrees of freedom and the resulting interaction processes. This line of research can leverage fundamental insights into how molecules interact and evolve to enable the control of reaction chemistry and the design and realization of a range of advanced quantum materials.

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.