Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1501.00854 · Journal of Optics

Stabilization of three-wave vortex beams in the waveguide

A. Gammal, Boris A. Malomed

Published Mar 27, 2015• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2040-8978/17/4/045503

Related papers

Stabilizing single- and two-color vortex beams in quadratic media by a trapping potential

Hidetsugu Sakaguchi, Boris A. Malomed

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.

arXiv:1208.4191•Journal of the Optical Society of America B•Sep 13, 2012•22 citations

Hidden vorticity in binary Bose-Einstein condensates

Marijana Brtka, A. Gammal, Boris A. Malomed

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.

Physical Review A•Nov 10, 2010•62 citations

Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction

DUMITRU BOGDAN MIHALACHE, Dumitru Mazilu, Boris A. Malomed +1 more

We perform accurate investigation of stability of localized vortices in an effectively two-dimensional (``pancake-shaped'') trapped Bose-Einstein condensate with negative scattering length. The analysis combines computation of the stability eigenvalues and direct simulations. The states with vorticity $S=1$ are stable in a third of their existence region, $0<N<(1∕3){N}_{\mathrm{max}}^{(S=1)}$, where $N$ is the number of atoms, and ${N}_{\mathrm{max}}^{(S=1)}$ is the corresponding collapse threshold. Stable vortices easily self-trap from arbitrary initial configurations with embedded vorticity. In an adjacent interval, $(1∕3){N}_{\mathrm{max}}^{(S=1)}<N<0.43{N}_{\mathrm{max}}^{(S=1)}$, the unstable vortex periodically splits in twofragments and recombines. At $N>0.43{N}_{\mathrm{max}}^{(S=1)}$, the fragments do not recombine, as each one collapses by itself. The results are compared with those in the full three-dimensional (3D) Gross-Pitaevskii equation. In a moderately anisotropic 3D configuration, with the aspect ratio $\sqrt{10}$, the stability interval of the $S=1$ vortices occupies $\ensuremath{\approx}40%$ of their existence region, hence the two-dimensional (2D) limit provides for a reasonable approximation in this case. For the isotropic 3D configuration, the stability interval expands to 65% of the existence domain. Overall, the vorticity heightens the actual collapse threshold by a factor of up to $2$. All vortices with $S\ensuremath{\geqslant}2$ are unstable.

arXiv:cond-mat/0603793•Physical Review A•Apr 26, 2006•120 citations

Three-dimensional Bose—Einstein condensate vortex solitons under optical lattice and harmonic confinements

Ying Wang, Feng-De Zong, Feng-Bo Li

We predict three-dimensional vortex solitons in a Bose—Einstein condensate under a complex potential, which is the combination of a two-dimensional parabolic trap along the transverse radial direction and a one-dimensional optical-lattice potential along the z axis direction. The vortex solitons are built in the form of a layer-chain structure made of several fundamental vortices along the optical-lattice direction. This has not been reported before in the three-dimensional Bose—Einstein condensate. By using a combination of the energy density functional method with direct numerical simulation, we find three-dimensional vortex solitons with topological charges χ = 1, χ = 2, and χ = 3. Moreover, the macroscopic quantum tunneling and chirp phenomena of the vortex solitons are shown in the evolution. Therein, the occurrence of macroscopic quantum tunneling provides the possibility for the experimental realization of quantum tunneling. Specifically, we successfully manipulate the vortex solitons along the optical lattice direction. The stability limits for dragging the vortex solitons from an initial fixed position to a prescribed location are further pursued.

Chinese Physics B•Mar 1, 2013•3 citations

Elliptic vortices in optical waveguides and self-attractive Bose-Einstein condensates

Fangwei Ye, Liangwei Dong, Boris A. Malomed +2 more

We investigate vortex solitons in models representing “pancake-shaped” Bose-Einstein condensates (BECs) with attractive interatomic interactions and an anisotropic harmonic trapping potential, or polarization-preserving waveguides, such as a photonic-crystal or radial-bandgap fiber with an elliptically deformed core. Elliptic-vortex states bifurcate from dipole modes. In spite of strongly asymmetric shapes, the elliptic vortices with topological charge 1 are stable in a wide region in their existence domain. The dependence of the stability region on the eccentricity of the elliptic trap is reported. Input circular wave functions with embedded vorticity slowly relax into stationary elliptic vortices, featuring persistent rotation in the course of the evolution. All higher-order vortices are unstable.

Journal of the Optical Society of America B•Mar 26, 2010•13 citations

Stability of quantized vortices in a Bose-Einstein condensate confined in an optical lattice

Kody J. H. Law, L. Qiao, P. G. Kevrekidis +1 more

We investigate the existence and especially the linear stability of single- and multiple-charge quantized vortex states of nonlinear Schr\"odinger equations in the presence of a periodic and a parabolic potential in two spatial dimensions. The study is motivated by an examination of pancake-shaped Bose-Einstein condensates in the presence of magnetic and optical confinement. A two-parameter space of the condensate's chemical potential versus the periodic potential's strength is scanned for both single- and double-quantized vortex states located at a local minimum or a local maximum of the lattice. Triple-charged vortices are also briefly discussed. Single-charged vortices are found to be stable for cosinusoidal potentials and unstable for sinusoidal ones above a critical strength. Higher-charge vortices are more unstable for both types of potentials, and their dynamical evolution leads to a breakup into single-charged vortices.

arXiv:0803.3251•Physical Review A•May 27, 2008•16 citations
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 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 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.
We perform accurate investigation of stability of localized vortices in an effectively two-dimensional (``pancake-shaped'') trapped Bose-Einstein condensate with negative scattering length. The analysis combines computation of the stability eigenvalues and direct simulations. The states with vorticity S=1S=1S=1 are stable in a third of their existence region, 0<N<(1∕3)Nmax(S=1)0<N<(1∕3){N}_{\mathrm{max}}^{(S=1)}0<N<(1∕3)Nmax(S=1)​, where NNN is the number of atoms, and Nmax(S=1){N}_{\mathrm{max}}^{(S=1)}Nmax(S=1)​ is the corresponding collapse threshold. Stable vortices easily self-trap from arbitrary initial configurations with embedded vorticity. In an adjacent interval, (1∕3)Nmax(S=1)<N<0.43Nmax(S=1)(1∕3){N}_{\mathrm{max}}^{(S=1)}<N<0.43{N}_{\mathrm{max}}^{(S=1)}(1∕3)Nmax(S=1)​<N<0.43Nmax(S=1)​, the unstable vortex periodically splits in twofragments and recombines. At N>0.43Nmax(S=1)N>0.43{N}_{\mathrm{max}}^{(S=1)}N>0.43Nmax(S=1)​, the fragments do not recombine, as each one collapses by itself. The results are compared with those in the full three-dimensional (3D) Gross-Pitaevskii equation. In a moderately anisotropic 3D configuration, with the aspect ratio 10\sqrt{10}10​, the stability interval of the S=1S=1S=1 vortices occupies $\ensuremath{\approx}40%$ of their existence region, hence the two-dimensional (2D) limit provides for a reasonable approximation in this case. For the isotropic 3D configuration, the stability interval expands to 65% of the existence domain. Overall, the vorticity heightens the actual collapse threshold by a factor of up to 222. All vortices with $S\ensuremath{\geqslant}2$ are unstable.
We predict three-dimensional vortex solitons in a Bose—Einstein condensate under a complex potential, which is the combination of a two-dimensional parabolic trap along the transverse radial direction and a one-dimensional optical-lattice potential along the z axis direction. The vortex solitons are built in the form of a layer-chain structure made of several fundamental vortices along the optical-lattice direction. This has not been reported before in the three-dimensional Bose—Einstein condensate. By using a combination of the energy density functional method with direct numerical simulation, we find three-dimensional vortex solitons with topological charges χ = 1, χ = 2, and χ = 3. Moreover, the macroscopic quantum tunneling and chirp phenomena of the vortex solitons are shown in the evolution. Therein, the occurrence of macroscopic quantum tunneling provides the possibility for the experimental realization of quantum tunneling. Specifically, we successfully manipulate the vortex solitons along the optical lattice direction. The stability limits for dragging the vortex solitons from an initial fixed position to a prescribed location are further pursued.
We investigate vortex solitons in models representing “pancake-shaped” Bose-Einstein condensates (BECs) with attractive interatomic interactions and an anisotropic harmonic trapping potential, or polarization-preserving waveguides, such as a photonic-crystal or radial-bandgap fiber with an elliptically deformed core. Elliptic-vortex states bifurcate from dipole modes. In spite of strongly asymmetric shapes, the elliptic vortices with topological charge 1 are stable in a wide region in their existence domain. The dependence of the stability region on the eccentricity of the elliptic trap is reported. Input circular wave functions with embedded vorticity slowly relax into stationary elliptic vortices, featuring persistent rotation in the course of the evolution. All higher-order vortices are unstable.
We investigate the existence and especially the linear stability of single- and multiple-charge quantized vortex states of nonlinear Schr\"odinger equations in the presence of a periodic and a parabolic potential in two spatial dimensions. The study is motivated by an examination of pancake-shaped Bose-Einstein condensates in the presence of magnetic and optical confinement. A two-parameter space of the condensate's chemical potential versus the periodic potential's strength is scanned for both single- and double-quantized vortex states located at a local minimum or a local maximum of the lattice. Triple-charged vortices are also briefly discussed. Single-charged vortices are found to be stable for cosinusoidal potentials and unstable for sinusoidal ones above a critical strength. Higher-charge vortices are more unstable for both types of potentials, and their dynamical evolution leads to a breakup into single-charged vortices.