Spin-1 Spinor Bose-Einstein Condensate Research Articles

Spin-orbit-coupled spin-1 Bose-Einstein condensates in a toroidal trap: Even-petal-number necklacelike state and persistent flow

Spin-orbit coupling has novel spin-flip symmetries, a spin-1 spinor Bose-Einstein condensate owns meaningful interactions, and a toroidal trap is topologically nontrivial. We incorporate the three together and study the ground-state phase diagram in a Rashba spin-orbit-coupled spin-1 Bose-Einstein condensate with a toroidal trap. The spin-flip symmetries give rise to two different interesting phases: persistent flows with a unit phase winding difference between three components, and necklace states with even petal-number. The existing parameter regimes and properties of these phases are characterized by two-dimension numerical calculations and an azimuthal analytical one-dimension model.

Multi-fold binary Darboux transformation and mixed solitons of a three-component Gross-Pitaevskii system in the spinor Bose-Einstein condensate

The Bose-Einstein-condensation applications give rise to the superfluidity in the liquid helium and superconductivity in the metals. In this paper, we work on a three-component Gross-Pitaevskii system, which describes the matter waves in an spin-1 spinor Bose-Einstein condensate. We construct a multi-fold binary Darboux transformation with the zero seed solutions to describe the three vertical spin projection of the spin-1 spinor BEC, which is different from all the existing Darboux-type ones for the same system, and derive three types of the exponential-and-rational mixed soliton solutions associated with two conjugate complex eigenvalues. For such mixed solitons, we give their asymptotic expressions, indicating that they consist of the Ieda-Miyakawa-Wadati (IMW)-polar-state or IMW-ferromagnetic solitons but possess the time-dependent velocities. Asymptotically and graphically, interaction mechanisms between the mixed and exponential solitons are classified in six cases, and we exhibit the inelastic and elastic interactions through calculating the modifications of the polarization matrices and phase shifts for the two interacting solitons. We find that both the IMW-polar-state solitons, including the mixed and exponential solitons, can not alter the other soliton’s intensity distribution during the interaction, while the mixed or exponential soliton in the IMW-ferromagnetic state does.

OpenMP solver for rotating spin-1 spin–orbit- and Rabi-coupled Bose–Einstein condensates

We present OpenMP version of a Fortran program for solving the Gross–Pitaevskii equation for a harmonically trapped three-component rotating spin-1 spinor Bose–Einstein condensate (BEC) in two spatial dimensions with or without spin–orbit (SO) and Rabi couplings. The program uses either Rashba or Dresselhaus SO coupling. We use the split-step Crank–Nicolson discretization scheme for imaginary- and real-time propagation to calculate stationary states and BEC dynamics, respectively. New version program summaryProgram title: BEC-GP-SPINOR-ROT-OMP, a program package containing programs spin-SO-rot-imre2d-omp.f90, with util.f90.CPC Library link to program files:https://doi.org/10.17632/j3wr4wn946.2Licensing provisions: Apache License 2.0Programming language: OpenMP Fortran 90/95. The program is tested with the GNU, Intel, PGI, and Oracle (former Sun) compilers.Supplementary material: File Supp.pdf gives additional details about the new program version and the underlying physical system.Journal Reference of previous version: Comput. Phys. Commun. 259 (2021) 107657.Does the new version supersede the previous version?: Only partially. The program spin-SO-rot-imre2d-omp.f90 supersedes spin-SO-imre2d-omp.f90, while the one-dimensional program is not part of this package.Nature of problem: The present Open Multi-Processing (OpenMP) Fortran program solves the time-dependent nonlinear partial differential Gross–Pitaevskii (GP) equation for a trapped rotating spinor Bose–Einstein condensate (BEC) in two spatial dimensions.Solution method: We employ the split-step Crank–Nicolson scheme to discretize the time-dependent GP equation in space and time. The discretized equation is then solved by imaginary- or real-time propagation, employing adequately small space and time steps, to yield the solution of stationary and non-stationary problems, respectively.Reason for new version: The BEC is a special form of matter called superfluid. A hallmark of superfluidity is the formation of quantized vortices in a rotating BEC. The present program can be used to study the generation of quantized vortices in a rotating spin-1 trapped BEC and hence should be of general interest to researchers from various fields.Summary of revisions: Previously we published Fortran [1] and C [2] programs for solving the mean-field GP equation for a BEC, which are now enjoying widespread use. Later we extended these programs to the more complex scenario of dipolar BECs [3], spin-1 spinor BECs [4], and of rotating BECs [5]. The OpenMP [6, 7] and CUDA/MPI [8, 9, 10] versions of these programs, designed to make these faster and more efficient in multi-core computers, are also available. In this paper we present Fortran 90/95 program for solving the GP equation of a two-dimensional (2D) rotating spin-1 spinor BEC with Rashba [11] and Dresselhaus [12] spin–orbit (SO) coupling and Rabi coupling, involving a modification over the same for a spin-1 spinor BEC [4]. A new input parameter OMEG, which represents the angular velocity of rotation Ω of the spin-1 spinor BEC, has been introduced in the program, following Ref. [5]. Besides this new parameter, the execution of the present program follows the same procedure as the 2D program of Ref. [4]. All other input parameters in the two programs are identical and the reader is advised to consult that reference for further details. For some values of input parameters the quantized vortices of a rotating BEC could be arranged in the form of a lattice with a certain spatial symmetry, e.g., triangular or square lattice [5]. In our numerical study, we established recently such a symmetric lattice structure for a Rashba SO-coupled rotating spin-1 BEC in the simplest case, without the Rabi coupling [13]. A Dresselhaus SO-coupled rotating spin-1 BEC should also lead to identical structure, provided the sign of the angular velocity of rotation is changed. For the sake of completeness, in the supplementary material related to this article that can be found online at URL we provide the corresponding GP equations for a rotating spin-1 BEC with some instructive numerical examples. The program package BEC-GP-SPINOR-ROT-OMP contains the programs spin-SO-rot-imre2d-omp.f90 and util.f90 in the directory src, as well as the files makefile and README.md. The makefile allows automated compilation of the program using different supported compilers (GNU, Intel, PGI, Oracle) by a simple make command, as in Ref. [4]. The file README.md contains instructions on how to compile and run the programs. The directory output contains examples of matching outputs of imaginary- and real-time propagation programs in sub-directories with a generic name rotxgamyferro or rotxgamyantiferro, where x denotes the value of the angular velocity of rotation Ω and y denotes the strength of the SO coupling γ for ferromagnetic (c0=482,c2=15) and antiferromagnetic (c0=669,c2=−3.1) cases. The results in imaginary-time sub-directories rot.3gam.5ferro and rot.3gam.5antiferro are calculated using the respective converged imaginary-time wave functions with zero angular velocity. The real-time sub-directories rot.3gam.5ferro and rot.3gam.5polar contain real-time results calculated using the respective converged imaginary-time wave functions as inputs. These sub-directories also contain gnuplot programs fig*.gnu which can be used to generate fig*.eps figure files of component densities.

Vortex-lattice formation in a spin–orbit coupled rotating spin-1 condensate

We study the vortex-lattice formation in a rotating Rashba spin–orbit (SO) coupled quasi-two-dimensional (quasi-2D) hyper-fine spin-1 spinor Bose–Einstein condensate (BEC) in the x–y plane using a numerical solution of the underlying mean-field Gross–Pitaevskii equation. In this case, the non-rotating Rashba SO-coupled spinor BEC can have topological excitation in the form of vortices of different angular momenta in the three components, e.g. the (0, +1, +2)- and (−1, 0, +1)-type states in ferromagnetic and anti-ferromagnetic spinor BEC: the numbers in the parenthesis denote the intrinsic angular momentum of the vortex states of the three components with the negative sign denoting an anti-vortex. The presence of these states with intrinsic vorticity breaks the symmetry between rotation with vorticity along the z and −z axes and thus generates a rich variety of vortex-lattice and anti-vortex-lattice states in a rotating quasi-2D spin-1 spinor ferromagnetic and anti-ferromagnetic BEC, not possible in a scalar BEC. For weak SO coupling, we find two types of symmetries of these states − hexagonal and ‘square’. The hexagonal (square) symmetry state has vortices arranged in closed concentric orbits with a maximum of 6, 12, 18… (8, 12, 16…) vortices in successive orbits. Of these two symmetries, the square vortex-lattice state is found to have the smaller energy.

Spin-1 spin–orbit- and Rabi-coupled Bose–Einstein condensate solver

We present OpenMP versions of FORTRAN programs for solving the Gross–Pitaevskii equation for a harmonically trapped three-component spin-1 spinor Bose–Einstein condensate (BEC) in one (1D) and two (2D) spatial dimensions with or without spin–orbit (SO) and Rabi couplings. Several different forms of SO coupling are included in the programs. We use the split-step Crank–Nicolson discretization for imaginary- and real-time propagation to calculate stationary states and BEC dynamics, respectively. The imaginary-time propagation programs calculate the lowest-energy stationary state. The real-time propagation programs can be used to study the dynamics. The simulation input parameters are provided at the beginning of each program. The programs propagate the condensate wave function and calculate several relevant physical quantities. Outputs of the programs include the wave function, energy, root-mean-square sizes, different density profiles (linear density for the 1D program, linear and surface densities for the 2D program). The imaginary- or real-time propagation can start with an analytic wave function or a pre-calculated numerical wave function. The imaginary-time propagation usually starts with an analytic wave function, while the real-time propagation is often initiated with the previously calculated converged imaginary-time wave function. Program summaryProgram title: BEC-GP-SPINOR, consisting of: BEC-GP-SPINOR-OMP package, containing programs spin-SO-imre1d-omp.f90 and spin-SO-imre2d-omp.f90, with util.f90.CPC Library link to program files:https://doi.org/10.17632/j3wr4wn946.1Licensing provisions: Apache License 2.0Programming language: OpenMP FORTRAN. The FORTRAN programs are tested with the GNU, Intel, PGI, and Oracle compiler.Nature of problem: The present Open Multi-Processing (OpenMP) FORTRAN programs solve the time-dependent nonlinear partial differential Gross–Pitaevskii (GP) equation for a trapped spinor Bose–Einstein condensate, with or without spin–orbit coupling, in one and two spatial dimensions.Solution method: We employ the split-step Crank–Nicolson rule to discretize the time-dependent GP equation in space and time. The discretized equation is then solved by imaginary- or real-time propagation, employing adequately small space and time steps, to yield the solution of stationary and non-stationary problems, respectively.

Manifold mixing in the temporal evolution of a spin-1 spinor Bose-Einstein condensate.

Hydrodynamics of the spin-1 Bose-Einstein condensate is investigated without restriction of its allowed manifold to either the ferromagnetic or the antiferromagnetic phase. Mixing of the two manifolds is found to be a generic feature in the temporal evolution of the condensate, regardless of the sign and strength of the spin-dependent interaction. Hydrodynamic theory is developed based on a new representation of the spin-1 condensate wave function as a linear combination of the well-known wave functions specific to antiferromagnetic and ferromagnetic manifolds only. Dynamical constraints unique to each submanifold are derived for the first time, demonstrating that efforts to write down hydrodynamic theory in one specific manifold are generally invalid. Certain exceptions, such as a uniform spiral state in the antiferromagnetic manifold, are shown to sustain dynamical evolution within the same manifold over time.

Spin turbulence in spinor Bose-Einstein condensates

We summarize the recent theoretical and numerical works on spin turbulence (ST) in spin-1 spinor Bose-Einstein condensates. When the system is excited from the ground state, it goes through hydrodynamic instability to ST in which the spin density vector has various disordered direction. The properties of ST depend on whether the spin-dependent interaction is ferromagnetic or antiferromagnetic. ST has some characteristics different from other kinds of turbulence in quantum fluids. Firstly, the spectrum of the spin-dependent interaction energy exhibits the characteristic power law different from the usual Kolmogorov -5/3 law. Secondly, ST can show the spin-glass-like behavior; the spin density vectors are spatially random but temporally frozen.

Raman-dressed spin-1 spin-orbit-coupled quantum gas

The recently realized spin-orbit-coupled quantum gases [Lin et al., Nature (London) 471, 83 (2011); Wang et al., Phys. Rev. Lett. 109, 095301 (2012); Cheuk et al., Phys. Rev. Lett. 109, 095302 (2012)] mark a breakthrough in the cold atom community. In these experiments, two hyperfine states are selected from a hyperfine manifold to mimic a pseudospin-1/2 spin-orbit-coupled system by the method of Raman dressing, which is applicable to both bosonic and fermionic gases. In this paper, we show that the method used in these experiments can be generalized to create any large pseudospin spin-orbit-coupled gas if more hyperfine states are coupled equally by the Raman lasers. As an example, we study, in detail, a quantum gas with three hyperfine states coupled by the Raman lasers and show, when the state-dependent energy shifts of the three states are comparable, triple-degenerate minima will appear at the bottom of the band dispersions, thus, realizing a spin-1 spin-orbit-coupled quantum gas. A novel feature of this three-minima regime is that there can be two different kinds of stripe phases with different wavelengths, which has an interesting connection to the ferromagnetic and polar phases of spin-1 spinor Bose-Einstein condensates without spin-orbit coupling.

Spin turbulence with small spin magnitude in spin-1 spinor Bose-Einstein condensates

We theoretically and numerically study spin turbulence (ST) with small spin magnitude in spin-1 spinor Bose-Einstein condensates by using the spin-1 spinor Gross-Pitaevskii (GP)equations. This kind of ST is realized in two cases: (i) with antiferromagnetic interaction and (ii) with ferromagnetic interaction under a static magnetic field. The ST with small spin magnitude can exhibit two characteristic power laws in the spectrum of the spin-dependent interaction energy: -1 and -7/3 power laws in the low- and high-wave-number regions, respectively. These power laws are derived from a Kolmogorov-type dimensional scaling analysis for the equations of motion of the spin vector and nematic tensor. To confirm these power laws, we perform a numerical calculation of the spin-1 spinor GP equations in a two-dimensional uniform system. In case (i), the -7/3 power law appears in the high-wave-number region, but the spectrum in the low-wave-number region deviates from the -1 power law. In contrast, both -1 and -7/3 power laws are found to clearly appear in case (ii).

Spin Turbulence in Spin-1 Spinor Bose-Einstein Condensate with Antiferromagnetic Interaction

We theoretically and numerically study turbulence in spin-1 spinor Bose-Einstein condensates (BECs) with antiferromagnetic (AF) interaction by using the spinor Gross-Pitaevskii equations. In this system, the dynamical instability is caused by the counterflow, leading to the disturbed distribution of the spin density vector, which we call spin turbulence (ST). Previously, we investigated the ST in spin-1 spinor BEC with ferromagnetic interaction, finding that the spectrum of spin-dependent interaction energy exhibited the −7/3 power law. In this study, we investigate the ST in the spin-1 spinor BEC with AF interaction, pointing out the −1 and −7/3 power laws in the low and high wave number regions, respectively. These power laws can be obtained by the Kolmogorov-type dimensional scaling analysis for the equations of spin vector and nematic tensor. However, our numerical calculation finds the −7/3 power law in the high wave number region, while the spectrum in the low wave number region deviates from the −1 power law.

Anomalous Tunneling of Spin Wave in Polar State of Spin-1 BEC

We investigate tunneling properties of collective spin-wave excitations in the polar state of a spin-1 spinor Bose-Einstein condensate. Within the mean-field theory at T = 0, we show that when the condensate is in the critical supercurrent state, the spin wave mode exhibits perfect transmission through a nonmagnetic potential barrier in the low energy limit, unless the strength of a spin-independent interaction co equals that of a spin-dependent interaction co Such an anomalous tunneling behavior is absent in the case of a magnetic barrier. We also clarify a scaling law of the transmission probability as a function of the mode energy.

Spin Turbulence Made by the Oscillating Magnetic Field in a Spin-1 Spinor Bose-Einstein Condensate

We numerically study spin turbulence made by the oscillating magnetic field in a two-dimensional homogeneous spin-1 spinor Bose-Einstein condensate. We confine ourselves to the case of the ferromagnetic interaction, where the ground state is ferromagnetic. By the oscillating magnetic field along some direction makes the system unstable. At first, the spin density vector revolves in the plane perpendicular to the magnetic field, exciting long-wavelength modes. Secondly, appear some vector components along the magnetic field. Finally, the system becomes spin turbulence. In terms of the energy spectrum of the spin-dependent interaction energy, the peak appears first at low wave number region. Gradually, the peak shifts from low to high wave number region. Eventually, the spectrum exhibits the −7/3 power law. The −7/3 power law is confirmed by the scaling analysis using the hydrodynamic equation of the spinor Bose-Einstein condensate.

Beliaev theory of spinor Bose–Einstein condensates

By generalizing the Green’s function approach developed by Beliaev [S.T. Beliaev, Sov. Phys. JETP 7 (1958) 299; S.T. Beliaev, Sov. Phys. JETP 7 (1958) 289], we study effects of quantum fluctuations on the energy spectra of spin-1 spinor Bose–Einstein condensates, in particular, of a 87Rb condensate in the presence of an external magnetic field. We find that due to quantum fluctuations, the effective mass of magnons, which characterizes the quadratic dispersion relation of spin-wave excitations, increases compared with its mean-field value. The enhancement factor turns out to be the same for two distinct quantum phases: the ferromagnetic and polar phases, and it is a function of only the gas parameter. The lifetime of magnons in a spin-1 87Rb spinor condensate is shown to be much longer than that of phonons due to the difference in their dispersion relations. We propose a scheme to measure the effective mass of magnons in a spinor Bose gas by utilizing the effect of magnons’ nonlinear dispersion relation on the time evolution of the distribution of transverse magnetization. This type of measurement can be applied, for example, to precision magnetometry.

Spin Turbulence and the −7/3 Power Law in a Trapped Spin-1 Spinor Bose-Einstein Condensate

We numerically study the spin turbulence in a two-dimensional trapped spin-1 spinor Bose-Einstein condensate, focusing on the energy spectrum. The spin turbulence in the trapped system is generated through the instability of the initial helical structure of the spin density vector. The spectrum of the spin-dependent interaction energy in the ferromagnetic case becomes to exhibit the −7/3 power law as the spin turbulence develops, which was confirmed in a uniform system by the counterflow instability. Hence the spin turbulence with the −7/3 energy spectrum is robust independently of how the turbulence is generated.

Excitation transport through a domain wall in a Bose-Einstein condensate

We investigate the tunneling properties of collective excitations through a domain wall in the ferromagnetic phase of a spin-1 spinor Bose-Einstein condensate. Within the mean-field theory at $T=0$, we show that the transverse spin wave undergoes perfect reflection in the low-energy limit. This reflection property differs considerably from that of a domain wall in a Heisenberg ferromagnet where spin-wave excitations exhibit perfect transmission at arbitrary energy. When the Bogoliubov mode is scattered from this domain wall soliton, the transmission and reflection coefficients exhibit pronounced nonmonotonicity. In particular, we find perfect reflection of the Bogoliubov mode at energies where bound states appear. This is in stark contrast to the perfect transmission of the Bogoliubov mode with arbitrary energy through a dark soliton in a scalar Bose-Einstein condensate.

Spin turbulence in a trapped spin-1 spinor Bose-Einstein condensate

We perform a numerical study of spin turbulence in a two-dimensional trapped spin-1 spinor Bose-Einstein condensate, focusing on the energy spectrum. The spin turbulence in the trapped system is generated by instability of the helical structure of the spin density vector in the initial state. Our numerical calculation finds that in the trapped system, the spectrum of the spin-dependent interaction energy in the ferromagnetic case exhibits a $\ensuremath{-}7/3$ power law, which was confirmed in a uniform system by our previous study. The relation between the $\ensuremath{-}7/3$ power law and the motion of the spin density vector is discussed by investigating the orbits of dynamical variables in the spin space.

Counterflow instability and turbulence in a spin-1 spinor Bose-Einstein condensate

We theoretically study counterflow instability and turbulence in a spin-1 spinor Bose-Einstein condensate by the Gross-Pitaevskii equation and the Bogoliubov-de Gennes equation. Our study considers (i) the dynamics induced by the counterflow of two components with different magnetic quantum numbers, which leads to turbulence with spin degrees of freedom, and (ii) the properties of the turbulence. For (i), the behavior of the condensate induced by the counterflow strongly depends on whether the spin-dependent interaction is ferromagnetic or antiferromagnetic, leading to different behaviors for the dispersion relation and the spin density vector, etc. For (ii), we numerically calculate the spectrum of the spin-dependent interaction energy, which also depends on the spin-dependent interaction. The spectrum of the spin-dependent interaction energy in the ferromagnetic case clearly exhibits a $\ensuremath{-}7/3$ power law, which can be explained by scaling analysis. The spectrum in the antiferromagnetic case seems to show some power law too, but the power exponent cannot be estimated as definitely as the ferromagnetic case.

Anomalous tunneling of collective excitations and effects of superflow in the polar phase of a spin-1 spinor Bose-Einstein condensate

We investigate tunneling properties of collective modes in the polar phase of a spin-1 spinor Bose-Einstein condensate. This spinor BEC state has two kinds of gapless modes, i.e., Bogoliubov mode and spin-wave. Within the framework of the mean-field theory at T=0, we show that these Goldstone modes exhibit the perfect transmission in the low-energy limit. Their anomalous tunneling behaviors still hold in the presence of superflow, except in the critical current state. In the critical current state, while the tunneling of Bogoliubov mode is accompanied by finite reflection, the spin-wave still exhibit the perfect transmission, unless the strengths of a spin-dependent and spin-independent interactions take the same value.

Tunneling properties of Bogoliubov mode and spin-wave modes in supercurrent states of a spin-1 ferromagnetic spinor Bose-Einstein condensate

We investigate tunneling properties of collective excitations in the ferromagnetic phase of a spin-1 spinor Bose-Einstein condensate (BEC). In addition to the Bogoliubov mode, this superfluid phase has two spin excitations, namely, the gapless transverse spin wave and the quadrupolar mode with a finite excitation gap. In the mean-field theory at T=0, we examine how these collective modes tunnel through a barrier potential that couples to the local density of particles. In the presence of supercurrent with a finite momentum $q$, while the Bogoliubov mode shows the so-called anomalous tunneling behavior (which is characterized by perfect transmission) in the low energy limit, the transverse spin-wave transmits perfectly only when the momentum $k$ of this mode coincides with $\pm q$. At $k=\pm q$, the wave function of this spin wave has the same form as the condensate wave function in the current carrying state, so that the mechanism of this perfect transmission is found to be the same as tunneling of supercurrent. Using this fact, the perfect transmission of the spin wave is proved for a generic barrier potential. We show that such perfect transmission does not occur in the quadrupolar mode. Further, we consider the effects of potentials breaking U(1) and spin rotation symmetries on the transmission properties of excitations. Our results would be useful for understanding excitation properties of spinor BECs, as well as the anomalous tunneling phenomenon in Bose superfluids.

Complex quantum gases: spinor Bose–Einstein condensates of trapped atomic vapors

We discuss the energy eigenstates, ground and spin mixing dynamics of a spin-1 spinor Bose–Einstein condensate for a dilute atomic vapor confined in an optical trap. Our results go beyond the mean field picture and are developed within a fully quantized framework.