Two-dimensional Gross-Pitaevskii Equation Research Articles

Coherence measurements of polaritons in thermal equilibrium reveal a power law for two-dimensional condensates.

We have created a spatially homogeneous polariton condensate in thermal equilibrium, up to very high condensate fraction. Under these conditions, we have measured the coherence as a function of momentum and determined the total coherent fraction of this boson system from very low density up to density well above the condensation transition. These measurements reveal a consistent power law for the coherent fraction as a function of the total density over nearly three orders of its magnitude. The same power law is seen in numerical simulations solving the two-dimensional Gross-Pitaevskii equation for the equilibrium coherence.

Two-dimensional vector solitons in Bose-Einstein-condensate mixtures

We derive two decoupled KP-I equations from the system of two-dimensional (2D) Gross-Pitaevskii equations for a two-component Bose-Einstein condensate (BEC), using the multiple-scale expansion method. We produce asymptotic analytical vector-soliton solutions, viz., dark-dark (DD) and dark-antidark (DAD) one-soliton and two-soliton states, by tuning coupling constants and norms of species, and address their evolution numerically under the action of the harmonic-oscillator (HO) trap, in the local-density approximation. We find that shallow single-line DD and DAD solitons are stable, while single-lump DAD solutions (weakly localized truly-2D states) split and lead to nucleation of two half-vortices. We also find that the BEC mixture placed in the HO trap admits stable asymptotic multi-soliton solutions, e.g., two-line DD and DAD solitons and two-lump DD ones, which were not reported before. In particular, the two-lump solutions describe inelastic collisions between the lumps. The analysis developed in this work may also be applied to systems with spin-orbit coupling and gauge fields, which have been realized in atomic BEC.

Solitary Vortex Dynamics of 2D Bose-Einstein Condensates with Higher-Order Nonlinear Interactions

For study of Continuous matter waves in Bose-Einstein condensates in nonlinear and quantum atom optics, the two-dimensional Gross-Pitaevskii equation (GPE) is chosen as the reliable model for studying the dynamics of vortices in the framework of mean-field theory. In related problems in several recent studies showing that higher-order interrelationships are an indispensable component of the GPE even at the mean-field level, by numerically estimating the vortex dynamics variables. In this paper, derive the vortex soliton solutions using the variational method and investigate the effect of higher-order nonlinear corrections on the behavior of the vortex dynamics, which are shown to have an important impact on the vortex dynamics behavior.

Stokes drift and impurity transport in a quantum fluid

Stokes drift is a classical fluid effect in which traveling waves transfer momentum to tracers of the fluid, resulting in a nonzero drift velocity in the direction of the incoming wave; this effect is the driving mechanism allowing particles, i.e., impurities, to be transported by the flow. In a classical (viscous) fluid this happens usually due to the presence of viscous drag forces; because of the eventual absence of viscosity in quantum fluids, impurities are driven by inertial effects and pressure gradients only. We present theoretical predictions of a Stokes drift analogous in quantum fluids finding that, at the leading order, the drift direction and amplitude depend on the initial impurity position with respect to the wave phase, and at the second order, our theoretical model recovers the classical Stokes drift but with a coefficient that depends on the relative particle-fluid density ratio. Our theoretical predictions are obtained for classical impurities using multitime analytical asymptotic expansions. Numerical simulations of a two-dimensional Gross-Pitaevskii equation coupled with a classical impurity corroborate our findings. Our findings are experimentally testable, for instance, using fluids of light obtained in photorefractive crystals.Received 15 December 2022Revised 19 April 2023Accepted 17 May 2023DOI:https://doi.org/10.1103/PhysRevA.107.L061303Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasFluid-particle interactionsPhysical SystemsQuantum fluids & solidsSuperfluidsUltracold gasesTechniquesPerturbative methodsNonlinear DynamicsFluid DynamicsAtomic, Molecular & Optical

Modulation instability of two-dimensional Bose-Einstein condensates with helicoidal and a mixture of Rashba-Dresselhaus spin-orbit couplings

The modulational instability (MI) process is exclusively studied in a two-component Bose-Einstein condensate (BEC) which includes Rashba-Dresselhaus (RD) spin-orbit (SO) and helicoidal SO couplings. A generalized set of two-dimensional (2D) Gross-Pitaevskii (GP) equations is derived. The tunability of the helicoidal gauge potential is exploited to address BECs dynamics in a square lattice. The MI growth rate is derived, and parametric analyses of MI show the dependence of the instability on interatomic interaction strengths, the RD SO coupling, and helicoidal SO coupling, which combines the gauge amplitude and the helicoidal gauge potential. Direct numerical simulations are carried out to confirm the analytical predictions. Trains of solitons are obtained, and their behaviors are debated when the RD SO parameters are varied under different combinations between the gauge amplitude and the helicoidal gauge potential. The latter gives a potential way to manipulate the trapping capacities of the proposed BEC model.

Sinc Collocation Numerical Methods for Solving Two-Dimensional Gross-Pitaevskii Equations with Non-Homogeneous Dirichlet Boundary Conditions

Sinc Collocation Numerical Methods for Solving Two-Dimensional Gross-Pitaevskii Equations with Non-Homogeneous Dirichlet Boundary Conditions

Dynamical Mechanisms of Vortex Pinning in Superfluid Thin Films.

We characterize the mechanisms of vortex pinning in a superfluid thin film described by the two-dimensional Gross-Pitaevskii equation. We consider a vortex "scattering experiment" whereby a single vortex in a superfluid flow interacts with a circular, uniform pinning potential. By an analogy with linear dielectrics, we develop an analytical hydrodynamic approximation that predicts vortex trajectories, the vortex fixed point and the unpinning velocity. We then solve the Gross-Pitaevskii equation to validate this model, and build a phase portrait of vortex pinning. We identify two different dynamical pinning mechanisms marked by distinctive phonon emission signatures: one enabled by acoustic radiation and another mediated by vortex dipoles nucleated within the pin. Relative to obstacle size, we find that pinning potentials on the order of the healing length are more effective for vortex capture. Our results could be useful in mitigating the negative effects of drag due to vortices in superfluid channels, in analogy to maximizing supercurrents in type-II superconductors.

Exact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensation

IntroductionThe Gross-Pitaevskii equation is a class of the nonlinear Schrödinger equation, whose exact solution, especially soliton solution, is proposed for understanding and studying Bose-Einstein condensate and some nonlinear phenomena occurring in the intersection field of Bose-Einstein condensate with some other fields. It is an important subject to investigate their exact solutions. ObjectivesWe give multi-soliton of a two-dimensional Gross-Pitaevskii system which contains the time-varying trapping potential with a few interactions of multi-soliton. Through analytical and graphical analysis, we obtain one-, two- and three-soliton which are affected by the strength of atomic interaction. The asymptotic expression of two-soliton embodies the properties of solitons. We can give some interactions of solitons of different structures including parabolic soliton, line-soliton and dromion-like structure. MethodsBy constructing an appropriate Hirota bilinear form, the multi-soliton solution of the system is obtained. The soliton elastic interaction is analyzed via asymptotic analysis. ResultsThe results in this paper theoretically provide the analytical bright soliton solution in the two-dimensional Bose-Einstein condensation model and their interesting interaction. To our best knowledge, the discussion and results in this work are new and important in different fields. ConclusionsThe study enriches the existing nonlinear phenomena of the Gross-Pitaevskii model in Bose-Einstein condensation, and prove that the Hirota bilinear method and asymptotic analysis method are powerful and effective techniques in physical sciences and engineering for analyzing nonlinear mathematical-physical equations and their solutions. These provide a valuable basis and reference for the controllability of bright soliton phenomenon in experiments for high-dimensional Bose-Einstein condensation.

Effects of Higher Order Interaction on Vortex Formation in Bose-Einstein Condensates

We consider theoretically the formation of vortex in rotating Bose-Einstein condensates (BECs) with higher order interaction (HOI). Our results are obtained from the twodimensional Gross-Pitaevskii equation. As the first step, for the certain number vortices, we discuss the ground state properties and show that the critical rotation frequency for HOI is smaller than those without HOI. As the increasing of HOI strength, the critical rotation frequency decreases. In addition, we verify that the Feynman rule is meet well. Moreover, we study the vortex dynamics.Numerical results indicate that the angular momentum remains almost unchanged irrespective of the HOI strength. The time taken for the nucleation of vortices pays less for strong HOI. These results suggest that the HOI is favorable to rotate the condensate, and this mechanism is useful to control the vortex number in BECs.

Ground state of spin-orbit coupled rotating ferromagnetic Bose-Einstein condensate in toroidal trap

Realization of spinor Bose-Einstein condensate in an optical trap has made it possible to create a variety of topological nontrivial structures, due to the vector character of the order parameter. Recently, artificial spin-orbit coupling in the spinor Bose-Einstein condensate, owing to coupling between the spin and the center-of-mass motion of the atom, provides an unprecedented opportunity to search for novel quantum states. As is well known, the potential well in the Bose-Einstein condensate is adjustable. The toroidal trap is an important model potential because of its simplicity and richness in physics. In particular, the spinor Bose-Einstein condensate under the toroidal trap has brought an ideal platform for studying fascinating properties of a superfluid, such as persistent flow and symmetry-breaking localization. For the case of the spin-orbit-coupled Bose-Einstein condensate, the previous studies of the toroidal trap mainly focused on the two-component or antiferromagnetic case. However, in the presence of a toroidal trap, there remains an open question whether the combined effects of the spin-orbit coupling and rotation can produce previously unknown types of topological excitations in the ferromagnetic Bose-Einstein condensate. In this work, by using quasi two-dimensional Gross-Pitaevskii equations, we study the ground state structure of spin-orbit coupled rotating ferromagnetic Bose-Einstein condensate in the toroidal trap. We concentrate on the effects of the spin-orbit coupling and the rotation on the ground states. The numerical results show that in the presence of a toroidal trap, the ground state structure is displayed as half-skyrmion chain with circular distribution. Adjusting the strength of spin-orbit coupling not only changes the number of half-skyrmion in the system, but also controls the symmetry of half-skyrmion with circular distribution. As the rotation frequency increases, the system undergoes the transitions from the plane wave to the half-skyrmion chain with circular distribution, and eventually developing the half-skyrmion phase of triangular lattice. Next, we examine the effect of spin-independent interaction on spin-orbit coupled rotating spinor Bose-Einstein condensate. As the spin-independent interaction increases, the topological defects in the condensate increase due to the variation of the local magnetic order. We also discuss the influence of well shape on the ground state structure. These topological structures can be detected via the time-of-flight absorption imaging technique. The spin-orbit coupled spinor Bose-Einstein condensate in the toroidal trap is an important quantum platform, which not only opens up a new avenue for exploring the exotic topological structures, but also is crucial for realizing the transitions among different ground states. This work paves the way for futureexploring the topological defects and the corresponding dynamical stability in quantum system subjected to the toroidal trap.

Asymmetric solitary vortex evolution in two-dimensional Bose-Einstein condensate in harmonic trap

We studied the evolution of an asymmetric solitary vortex in a two-dimensional Bose-Einstein condensate starting from a quasi-one dimensional state with external harmonic trapping potential modulation. We derived the analytical solution of the asymmetric solitary vortex based on the two-dimensional Gross-Pitaevskii equation model and variational method. Furthermore, we identified the oscillation mode between the extreme asymmetric state and circular symmetric state for the derived solitary vortex. The obtained theoretical results can be used as guides for relevant experimental studies of asymmetric solitary vortices under similar physical conditions.

Dynamics of interacting dark soliton stripes

In the present work we examine the statics and dynamics of multiple parallel dark soliton stripes in a two-dimensional Bose-Einstein condensate. Our principal goal is to study the effect of the interaction between the stripes on the transverse instability of the individual stripes. We use a recently developed adiabatic invariant formulation to derive a quasi-analytical prediction for the stripe equilibrium position and for the Bogoliubov-de Gennes spectrum of excitations of stationary stripes. The cases of two-, three- and four-stripe states are studied in detail. We subsequently test our predictions against numerical simulations of the full two-dimensional Gross-Pitaevskii equation. We find that the number of unstable eigenmodes increases as the number of stripes increases due to (unstable) relative motions between the stripes. Their corresponding growth rates do not significantly change, although for large chemical potentials, the larger the stripe number, the larger the maximal instability growth rate. The instability induced dynamics of multiple stripe states and their decay into vortices are also investigated.

Collective excitations and tunneling dynamics in long bosonic Josephson junctions

We investigate the low-energy dynamics of two coupled anisotropic Bose-Einstein condensates forming a long Josephson junction. The theoretical study is performed in the framework of the two-dimensional Gross-Pitaevskii equation and the Bogoliubov-de Gennes formalism. We analyze the excitation spectrum of the coupled Bose condensates and show how low-energy excitations of the condensates lead to multiple-frequency oscillations of the atomic populations in the two wells. This analysis generalizes the standard bosnic Josephson euqation approach. We also develop a one-dimensional hydrodynamic model of the coupled condensates, that is capable to reproduce the excitation spectrum and population dynamics of the system.

Energy-level splitting for weakly interacting bosons in a harmonic trap

We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap. When the interactions are turned off, the energy levels are equidistant and highly degenerate. At linear order in the coupling parameter, these degenerate levels split, and we study the patterns of this splitting. It turns out that the problem is mathematically identical to diagonalizing the quantum resonant system of the two-dimensional Gross-Pitaevskii equation, whose classical counterpart has been studied previously in the mathematical literature on turbulence. Our purpose is to explore the implications of the symmetries and energy bounds of this resonant system, previously studied for the classical case, for the quantum level splitting. Simplifications in computing the splitting spectrum numerically result from exploiting the symmetries. The highest energy state emanating from each unperturbed level is explicitly described by our analytics. We furthermore discuss the energy-level spacing distributions in the spirit of quantum chaos theory. After separating the eigenvalues into blocks with respect to the known conservation laws, we observe the Wigner-Dyson statistics within specific large blocks, which leaves little room for further integrable structures in the problem beyond the symmetries that are already explicitly known.

Solvable Cubic Resonant Systems

Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross-Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross-Pitaevskii equation, but not for any other cases.

Two infinite families of resonant solutions for the Gross-Pitaevskii equation

We consider the two-dimensional Gross-Pitaevskii equation describing a Bose-Einstein condensate in an isotropic harmonic trap. In the small coupling regime, this equation is accurately approximated over long times by the corresponding nonlinear resonant system whose structure is determined by the fully resonant spectrum of the linearized problem. We focus on two types of consistent truncations of this resonant system: first, to sets of modes of fixed angular momentum, and second, to excited Landau levels. Each of these truncations admits a set of explicit analytic solutions with initial conditions parametrized by three complex numbers. Viewed in position space, the fixed angular momentum solutions describe modulated oscillations of dark rings, while the excited Landau level solutions describe modulated precession of small arrays of vortices and antivortices. We place our findings in the context of similar results for other spatially confined nonlinear Hamiltonian systems in recent literature.

Vortex Dynamics of Rotating Dipolar Bose-Einstein Condensates in Synthetic Magnetic Field

In the framework of time-dependent two-dimensional Gross-Pitaevskii equation, we investigate the dynamics of vortex formation in rotating dipolar Bose-Einstein condensates in synthetic magnetic field (SMF) and compare with rotating frame (RF) method. The formation of vortices are calculated, considering effects of the rotational frequency, dipole strength, tilting angle and trap ratio. The results we found are that in SMF, the formation of steady state vortices is much slower than that in RF, and it is more difficult to add large angular momentum to the condensates than to do so in RF.

Particles and fields in superfluids: Insights from the two-dimensional Gross-Pitaevskii equation

We carry out extensive direct numerical simulations to investigate the interaction of active particles and fields in the two-dimensional Gross-Pitaevskii superfluid, in both simple and turbulent flows. The particles are active in the sense that they affect the superfluid even as they are affected by it. We tune the mass of the particles, which is an important control parameter. At the one-particle level, we show how light, neutral, and heavy particles move in the superfluid, when a constant external force acts on them; in particular, beyond a critical velocity, at which a vortex-antivortex pair is emitted, particle motion can be periodic or chaotic. We demonstrate that the interaction of a particle with vortices leads to dynamics that depends sensitively on the particle characteristics. We also demonstrate that assemblies of particles and vortices can have rich, and often turbulent, spatiotemporal evolution. In particular, we consider the dynamics of the following illustrative initial configurations: (a) one particle placed in front of a translating vortex-antivortex pair; (b) two particles placed in front of a translating vortex-antivortex pair; (c) a single particle moving in the presence of counter-rotating vortex clusters; (d) four particles in the presence of counter-rotating vortex clusters. We compare our work with earlier studies and examine its implications for recent experimental studies in superfluid helium and Bose-Einstein condensates.

Bénard-von Kármán vortex street in dipolar Bose-Einstein condensate trapped by square-like potential

Bénard-von Kármán vortex street in dipolar Bose-Einstein Condensate (BEC) trapped by a square-like potential is investigated numerically. In the frame of mean-field theory, the nonlinear dynamic of the dipolar BEC can be described by the so-called two-dimensional Gross-Pitaevskii (GP) equation with long-range interaction. In this paper, we only consider the case that all the dipoles are polarized along the <i>z</i>-axis, which is perpendicular to the plane of disc-shaped BEC. Firstly, the stationary state of the BEC is obtained by the imaginary-time propagation approach. Secondly, the nonlinear dynamic of the BEC, when a moving Gaussian potential exists in such a system, is numerically investigated by the time-splitting Fourier spectral method, in which the stationary state obtained before is set to be the initial state. The results show that when the velocity of the cylindrical obstacle potential reaches a critical value, which depends on interaction strength and the shape of the potential, the vortex-antivortex pairs will be generated alternately in the super-flow behind the obstacle potential. However, in general, such a vortex-antivortex pair structure is dynamically unstable. When the velocity of the obstacle potential increases to a certain value and for a suitable potential width, a stable vortex structure called Bénard-von Kármán vortex street will be formed. While this phenomenon emerges, the vortices in pairs created by the obstacle potential have the same circulation. The pairs with opposite circulations are alternately released from the moving obstacle potential. For larger potential width and velocity, the shedding pattern becomes irregular. We also numerically investigate the effects of the dipole interaction strength, the width and the velocity of the obstacle potential on the vortex structures arising in the wake flow. As a result, the phase graph is presented by lots of numerical calculations for a group of given physical parameters. Thirdly, the drag force on the obstacle potential is also calculated and the mechanical mechanism of vortex pair is analyzed. Finally, we discuss how to find the phenomenon of Bénard-von Kármán vortex street in dipolar BEC experimentally.

Ground state of spin-orbit coupled rotating two-component Bose-Einstein condensate in gradient magnetic field

Two-component Bose-Einstein condensate offers an ideal platform for investigating many intriguing topological defects due to the interplay between intraspecies and interspecies interactions. The recent realization of spin-orbit coupling in two-component Bose-Einstein condensate, owing to coupling between the spin and the centre-of-mass motion of the atom, provides possibly new opportunities to search for novel quantum states. In particular, the gradient magnetic field in the Bose-Einstein condensate has brought a new way to create topologically nontrivial structures including Dirac monopoles and quantum knots. Previous studies of the gradient magnetic field effect in the Bose-Einstein condensate mainly focused on the three-component case. However, it remains unclear how the gradient magnetic field affects the ground state configuration in the rotating two-component Bose-Einstein condensate with spin-orbit coupling. In this work, by using quasi two-dimensional Gross-Pitaevskii equations, we study the ground state structure of a rotating two-component Bose-Einstein condensate with spin-orbit coupling and gradient magnetic field. We concentrate on the effects of the spin-orbit coupling and the gradient magnetic field on the ground state. The numerical results show that increasing the strength of the spin-orbit coupling can induce a phase transition from skyrmion lattice to skyrmion chain in the presence of the gradient magnetic field. Unlike the study of skyrmion in rotating two-component Bose-Einstein condensate with only spin-orbit coupling, the skyrmion chain can occur under the isotropic spin-orbit coupling when the gradient magnetic field is considered. It is worth noting that the skyrmion chain here is arrayed along the diagonal direction. Next we examine the effect of the gradient magnetic field on spin-orbit coupled two-component Bose-Einstein condensate. For the case of weak spin-orbit coupling and the slow rotation, a phase transition from a single plane-wave to half-skyrmion is found through increasing magnetic field gradient strength. For the case of strong spin-orbit coupling and the fast rotation, the nature of the ground state is shown to support the formation of a hidden vortex as the gradient magnetic field is enhanced. These hidden vortices have no visible cores in density distributions but have phase singularities in phase distributions, which are arrayed along the diagonal direction. This result confirms a new method of creating the hidden vortices in the two-component Bose-Einstein condensate. These topological structures can be detected by using the time-of-flight absorption imaging technique. Our results illustrate that the gradient magnetic field not only provides an opportunity to explore the exotic topological structures in spin-orbit coupled spinor Bose-Einstein condensate, but also is crucial for realizing the phase transitions among different ground states. This work paves the way for the future exploring of topological defect and the corresponding dynamical stability in quantum systems subjected to a gradient magnetic field.