We consider two-dimensional (2D) localized modes in the second-harmonic-generating (χ(2)) system with the harmonic-oscillator (HO) trapping potential. In addition to its realization in optics, the system describes the mean-field dynamics of mixed atomic-molecular Bose–Einstein condensates (BECs). The existence and stability of various modes is determined by their total power, N, topological charge, m/2 [m is the intrinsic vorticity of the second-harmonic (SH) field], and χ(2) mismatch, q. The analysis is carried out in a numerical form and, in parallel, by means of the variational approximation (VA), which produces results that agree well with numerical findings. Below a certain power threshold, N≤Nc(m)(q), all trapped modes are of the single-color type, represented by the SH component only, while the fundamental frequency (FF) one is absent. In contrast with the usual situation, where such modes are always unstable, we demonstrate that they are stable, for m=0, 1, 2 (the mode with m=1 may be formally considered as a semivortex with topological charge m/2=1/2), at N≤Nc(m)(q), and unstable above this threshold. On the other hand, Nc(m)(q)≡0 at q≥qmax (in our notation, qmax=1); hence the single-color modes are unstable in the latter case. At N=Nc(m), the modes with m=0 and m=2 undergo a pitchfork bifurcation, which gives rise to two-color states, which remain completely stable for m=0. The two-color vortices with m=2 (topological charge 1) have an upper stability border, N=Nc2(q). Above the border, they exhibit periodic splittings and recombinations, while keeping their vorticity. The semivortex does not bifurcate; at N=Nc(m=1), it exhibits quasi-chaotic oscillations and a rotating “groove” resembling a screw-edge dislocation induced by the semi-integer vorticity.
We consider two-dimensional (2D) localized modes in the second-harmonic-generating (χ(2)) system with the harmonic-oscillator (HO) trapping potential. In addition to its realization in optics, the system describes the mean-field dynamics of mixed atomic-molecular Bose–Einstein condensates (BECs). The existence and stability of various modes is determined by their total power, N, topological charge, m/2 [m is the intrinsic vorticity of the second-harmonic (SH) field], and χ(2) mismatch, q. The analysis is carried out in a numerical form and, in parallel, by means of the variational approximation (VA), which produces results that agree well with numerical findings. Below a certain power threshold, N≤Nc(m)(q), all trapped modes are of the single-color type, represented by the SH component only, while the fundamental frequency (FF) one is absent. In contrast with the usual situation, where such modes are always unstable, we demonstrate that they are stable, for m=0, 1, 2 (the mode with m=1 may be formally considered as a semivortex with topological charge m/2=1/2), at N≤Nc(m)(q), and unstable above this threshold. On the other hand, Nc(m)(q)≡0 at q≥qmax (in our notation, qmax=1); hence the single-color modes are unstable in the latter case. At N=Nc(m), the modes with m=0 and m=2 undergo a pitchfork bifurcation, which gives rise to two-color states, which remain completely stable for m=0. The two-color vortices with m=2 (topological charge 1) have an upper stability border, N=Nc2(q). Above the border, they exhibit periodic splittings and recombinations, while keeping their vorticity. The semivortex does not bifurcate; at N=Nc(m=1), it exhibits quasi-chaotic oscillations and a rotating “groove” resembling a screw-edge dislocation induced by the semi-integer vorticity.
We consider two-dimensional (2D) localized vortical modes in the three-wave system with the quadratic ( χ ( 2 ) ) nonlinearity, alias nondegenerate second-harmonic (SH)-generating system, guided by the isotropic harmonic-oscillator (alias parabolic) confining potential. In addition to the straightforward realization in optics, the system models mixed atomic-molecular Bose–Einstein condensates. The main issue is stability of the vortex modes, which is investigated through computation of instability growth rates for eigenmodes of small perturbations, and by means of direct simulations. The threshold of parametric instability for single-color beams, represented solely by the SH with zero vorticity, is found in an analytical form with the help of the variational approximation. Trapped states with vorticities ( + 1 , − 1 , 0 ) in the two fundamental-frequency components and the SH one (the so-called hidden-vorticity modes) are completely unstable. Also unstable are semi-vortices , with component vorticities ( 1 , 0 , 1 ) . However, full vortices, with charges ( 1 , 1 , 2 ) , have a well-defined stability region. Unstable full vortices feature regions of robust dynamical behavior, where they periodically split and recombine, keeping their vortical content.
We consider a two-dimensional (2D) model of a rotating attractive Bose-Einstein condensate (BEC), trapped in an external potential. First, a harmonic potential with the critical strength is considered, which generates quasisolitons at the lowest Landau level (LLL). We describe a family of the LLL quasisolitons using both numerical method and a variational approximation (VA), which are in good agreement with each other. We demonstrate that kicking the LLL mode or applying a ramp potential sets it in the Larmor (cyclotron) motion that can also be accurately modeled by the VA. Collisions between two such moving modes may be elastic or inelastic depending on their total norm. If an additional confining potential is applied along with the ramp, it creates a stationary edge state. Applying a kick to the edge state in the direction of the ramp gives rise to a skipping motion in the perpendicular direction. These regimes may be interpreted as the Hall effect for the quasisolitons. Next, we consider the condensate trapped in an axisymmetric quartic potential. Three species of localized states and their stability regions are identified, viz., vortices with arbitrary topological charge m, ``crescents'' (mixed-vorticity states), and strongly localized center-of-mass (c.m.) states, alias quasisolitons, shifted off the rotation pivot. These results are similar to those reported before for the model with a combined quadratic-quartic trap. Stable pairs of c.m. states set at diametrically opposite points are found, too. We present a VA which provides for an accurate description of vortices with all values of m, and of the c.m. states. We also demonstrate that kicking them in the azimuthal direction sets the quasisolitons in epitrochoidal motion (which is also accurately predicted by the VA), collisions between them being elastic.
It is commonly known that two-dimensional mean-field models of optical and matter waves with cubic self-attraction cannot produce stable solitons in free space because of the occurrence of collapse in the same setting. By means of numerical analysis and variational approximation, we demonstrate that the two-component model of the Bose-Einstein condensate with the spin-orbit Rashba coupling and cubic attractive interactions gives rise to solitary-vortex complexes of two types: semivortices (SVs, with a vortex in one component and a fundamental soliton in the other), and mixed modes (MMs, with topological charges 0 and ±1 mixed in both components). These two-dimensional composite modes can be created using the trapping harmonic-oscillator (HO) potential, but remain stable in free space, if the trap is gradually removed. The SVs and MMs realize the ground state of the system, provided that the self-attraction in the two components is, respectively, stronger or weaker than the cross attraction between them. The SVs and MMs which are not the ground states are subject to a drift instability. In free space (in the absence of the HO trap), modes of both types degenerate into unstable Townes solitons when their norms attain the respective critical values, while there is no lower existence threshold for the stable modes. Moving free-space stable solitons are also found in the present non-Galilean-invariant system, up to a critical velocity. Collisions between two moving solitons lead to their merger into a single one.
We consider a binary Bose-Einstein condensate (BEC) described by a system of two-dimensional (2D) Gross-Pitaevskii equations with the harmonic-oscillator trapping potential. The intraspecies interactions are attractive, while the interaction between the species may have either sign. The same model applies to the copropagation of bimodal beams in photonic-crystal fibers. We consider a family of trapped hidden-vorticity (HV) modes in the form of bound states of two components with opposite vorticities ${S}_{1,2}=\ifmmode\pm\else\textpm\fi{}1$, the total angular momentum being zero. A challenging problem is the stability of the HV modes. By means of a linear-stability analysis and direct simulations, stability domains are identified in a relevant parameter plane. In direct simulations, stable HV modes feature robustness against large perturbations, while unstable ones split into fragments whose number is identical to the azimuthal index of the fastest growing perturbation eigenmode. Conditions allowing for the creation of the HV modes in the experiment are discussed too. For comparison, a similar but simpler problem is studied in an analytical form, viz., the modulational instability of an HV state in a one-dimensional (1D) system with periodic boundary conditions (this system models a counterflow in a binary BEC mixture loaded into a toroidal trap or a bimodal optical beam coupled into a cylindrical shell). We demonstrate that the stabilization of the 1D HV modes is impossible, which stresses the significance of the stabilization of the HV modes in the 2D setting.
The dynamics of a two-dimensional Bose-Einstein condensate mixture, loaded into a dual-core trap, when the beyond-mean-field effects are taken into account, are considered. The effects of quantum fluctuations are described by the Lee-Huang-Yang correction terms in the extended coupled Gross-Pitaevskii equations (GPE). The spatially uniform and inhomogeneous condensate cases are studied. In the first case, the parameter regimes associated with macroscopic quantum tunneling, macroscopic self-trapping, and revival-like localization dynamics are found. The Josephson oscillation frequencies for both the zero-phase and the π -phase modes are derived. As the total atom number is varied, the dynamics exhibit a nontrivial bifurcation structure: along the zero-phase branch, two successive pitchfork bifurcations generate bistability and hysteresis, while the π -phase branch shows a single pitchfork bifurcation. In the second case, the Josephson dynamics for quantum droplets and vortices are investigated. The analytical predictions for the oscillation frequencies of the atomic population between quantum droplets are found, and results are validated by direct numerical simulations of the system of coupled extended GPE. The existence of the Andreev-Bashkin nondissipative drag through simulations of droplet-droplet interactions is shown. The Josephson dynamics of vortex states are studied. It was found that vortices with topological charge S and sufficiently small particle number are typically unstable, breaking up into S + 1 (and occasionally S + 2 ) fundamental, nonvortical fragments, with the time to breakup increasing as the particle number grows. It is also found that unstable asymmetric vortices exhibit the splitting and/or crescentlike instability. For vortices with sufficiently large norms, long-time simulations confirm robust stability against small perturbations. In this stable regime, the properties of Josephson oscillations and Andreev-Bashkin-type entrainment for vortex states with charges S = 1 , 2 , and 3 are investigated.
Abstract A recent analysis has revealed singular but physically relevant localized two-dimensional vortex states with density ∼ r −4/3 at r → 0 and a convergent total norm, which are maintained by the interplay of the potential of the attraction to the center, ∼− r −2 , and a self-repulsive quartic nonlinearity, produced by the Lee-Huang-Yang correction to the mean-field dynamics of Bose–Einstein condensates. In optics, a similar setting, with the density singularity ∼ r −1 , is realized with the help of quintic self-defocusing. Here we present physically relevant antidark singular-vortex states in these systems, existing on top of a flat background. Numerical solutions for them are very accurately approximated by the Thomas–Fermi wave function. Their stability exactly obeys an analytical criterion derived from the consideration of small perturbations. The singular-vortex states exist as well in the case when the effective potential is weakly repulsive. It is demonstrated that the singular vortices can be excited by the input in the form of the ordinary nonsingular vortices, hence the singular modes can be created in the experiment. We also consider regular (dark) vortices maintained by the flat background, under the action of the repulsive central potential ∼+ r −2 . The dark modes with vorticities l = 0 and 1 are completely stable. In the case when the central potential is attractive, but the effective one, which includes the centrifugal term, is repulsive, and, in addition, a weak trapping potential ∼ r 2 is applied, dark vortices with l = 1 feature an intricate pattern of alternating stability and instability regions. Under the action of the instability, states with l = 1 travel along tangled trajectories, which stay in a finite area defined by the trap. The analysis is also reported for dark vortices with l = 2, which feature a complex structure of alternating intervals of stability and instability against splitting. Lastly, simple but novel flat vortices are found at the border between the anidark and dark ones.