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

Chaos An Interdisciplinary Journal of Nonlinear Science

Disk-shaped Bose–Einstein condensates in the presence of an harmonic trap and an optical lattice

Todd Kapitula, P. G. Kevrekidis, D. J. Frantzeskakis

Published Apr 9, 2008• 8 citations
Publisher

Abstract

We study the existence and stability of solutions of the two-dimensional nonlinear Schrodinger equation in the combined presence of a parabolic and a periodic potential. The motivating physical example consists of Bose-Einstein condensates confined in an harmonic (e.g., magnetic) trap and an optical lattice. By connecting the nonlinear problem with the underlying linear spectrum, we examine the bifurcation of nonlinear modes out of the linear ones for both focusing and defocusing nonlinearities. In particular, we find real-valued solutions (such as multipoles) and complex-valued ones (such as vortices). A primary motivation of the present work is to develop "rules of thumb" about what waveforms to expect emerging in the nonlinear problem and about the stability of those modes. As a case example of the latter, we find that among the real-valued solutions, the one with larger norm for a fixed value of the chemical potential is expected to be unstable.

DOI
10.1063/1.2897311

Related papers

Bose-Einstein condensates in the presence of a magnetic trap and optical lattice

Todd Kapitula, P. G. Kevrekidis

In this paper we consider solutions of a nonlinear Schrodinger equation with a parabolic and a periodic potential motivated from the dynamics of Bose-Einstein condensates. Our starting point is the corresponding linear problem which we analyze through regular perturbation and homogenization techniques. We then use Lyapunov-Schmidt theory to establish the persistence and bifurcation of the linear states in the presence of attractive and repulsive nonlinear inter-particle interactions. Stability of such solutions is also examined and a count is given of the potential real, complex and imaginary eigenvalues with negative Krein signature that such solutions may possess. The results are corroborated with numerical computations.

Chaos An Interdisciplinary Journal of Nonlinear Science•Sep 1, 2005•32 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

Bose–Einstein condensates in the presence of a magnetic trap and optical lattice: two-mode approximation

Todd Kapitula, P. G. Kevrekidis

In this paper we study the solutions to a nonlinear Schrödinger equation with harmonic and periodic potentials, motivated from the recent interest in these models as mean-field descriptions of Bose–Einstein condensates. We use a two-mode Galerkin approximation to study the dynamics of the full model. The phase plane and stability analysis of the reduced model yield very good agreement with the findings of the full partial differential equation. A particularly interesting finding of the stability analysis is a spontaneous symmetry breaking through a branching bifurcation, resulting in the stabilization of asymmetric states and the destabilization of symmetric or anti-symmetric ones. We also highlight the important differences between the cases of symmetric potentials and those of weakly asymmetric potentials.

Nonlinearity•Aug 8, 2005•39 citations

Structure and stability of two-dimensional Bose–Einstein condensates under both harmonic and lattice confinement

Kody J. H. Law, P. G. Kevrekidis, Brian P. Anderson +2 more

In this work, we study pancake-shaped Bose-Einstein condensates confined by both a cylindrically symmetric harmonic potential and an optical lattice with equal periodicity in two orthogonal directions. We first identify the spectrum of the underlying two-dimensional linear problem through multiple-scale techniques. Then, we use the results obtained in the linear limit as a starting point for a nonlinear existence and stability analysis of the lowest energy states, emanating from the linear ones, in the nonlinear problem. Two-parameter continuations of these states are performed for increasing nonlinearity and optical lattice strengths, and their instabilities and temporal evolution are investigated. It is found that the ground state as well as one of the excited states are either stable or weakly unstable for both attractive and repulsive interatomic interactions.

arXiv:0803.3240•Journal of Physics B Atomic Molecular and Optical Physics•Sep 22, 2008•9 citations

Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates

G. Herring, Lincoln D. Carr, R. Carretero-González +2 more

Starting from the spectrum of the radially symmetric quantum harmonic oscillator in two dimensions, we create a large set of nonlinear solutions. The relevant three principal branches, with ${n}_{r}=0,1$, and 2 radial nodes, respectively, are systematically continued as a function of the chemical potential and their linear stability is analyzed in detail, in the absence as well as in the presence of topological charge $m$, i.e., vorticity. It is found that for repulsive interatomic interactions only the ground state is linearly stable throughout the parameter range examined. Furthermore, this is true for topological charges $m=0$ or 1; solutions with higher topological charge can be unstable even in that case. All higher excited states are found to be unstable in a wide parametric regime. However, for the focusing (attractive) case the ground state with ${n}_{r}=0$ and $m=0$ can only be stable for a sufficiently low number of atoms. Once again, excited states are found to be generically unstable. For unstable profiles, the dynamical evolution of the corresponding branches is also followed to monitor the temporal development of the instability.

arXiv:0709.2193•Physical Review A•Feb 27, 2008•47 citations

Localized modes of binary mixtures of Bose-Einstein condensates in nonlinear optical lattices

F. Kh. Abdullaev, A. Gammal, Mario Salerno +1 more

The properties of the localized states of a two-component Bose-Einstein condensate confined in a nonlinear periodic potential (nonlinear optical lattice) are investigated. We discuss the existence of different types of solitons and study their stability by means of analytical and numerical approaches. The symmetry properties of the localized states with respect to nonlinear optical lattices are also investigated. We show that nonlinear optical lattices allow the existence of bright soliton modes with equal symmetry in both components and bright localized modes of mixed symmetry type, as well as dark-bright bound states and bright modes on periodic backgrounds. In spite of the quasi-one-dimensional nature of the problem, the fundamental symmetric localized modes undergo a delocalizing transition when the strength of the nonlinear optical lattice is varied. This transition is associated with the existence of an unstable solution, which exhibits a shrinking (decaying) behavior for slightly overcritical (undercritical) variations in the number of atoms.

arXiv:0710.0607•Physical Review A•Feb 12, 2008•56 citations
We study the existence and stability of solutions of the two-dimensional nonlinear Schrodinger equation in the combined presence of a parabolic and a periodic potential. The motivating physical example consists of Bose-Einstein condensates confined in an harmonic (e.g., magnetic) trap and an optical lattice. By connecting the nonlinear problem with the underlying linear spectrum, we examine the bifurcation of nonlinear modes out of the linear ones for both focusing and defocusing nonlinearities. In particular, we find real-valued solutions (such as multipoles) and complex-valued ones (such as vortices). A primary motivation of the present work is to develop "rules of thumb" about what waveforms to expect emerging in the nonlinear problem and about the stability of those modes. As a case example of the latter, we find that among the real-valued solutions, the one with larger norm for a fixed value of the chemical potential is expected to be unstable.
In this paper we consider solutions of a nonlinear Schrodinger equation with a parabolic and a periodic potential motivated from the dynamics of Bose-Einstein condensates. Our starting point is the corresponding linear problem which we analyze through regular perturbation and homogenization techniques. We then use Lyapunov-Schmidt theory to establish the persistence and bifurcation of the linear states in the presence of attractive and repulsive nonlinear inter-particle interactions. Stability of such solutions is also examined and a count is given of the potential real, complex and imaginary eigenvalues with negative Krein signature that such solutions may possess. The results are corroborated with numerical computations.
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.
In this paper we study the solutions to a nonlinear Schrödinger equation with harmonic and periodic potentials, motivated from the recent interest in these models as mean-field descriptions of Bose–Einstein condensates. We use a two-mode Galerkin approximation to study the dynamics of the full model. The phase plane and stability analysis of the reduced model yield very good agreement with the findings of the full partial differential equation. A particularly interesting finding of the stability analysis is a spontaneous symmetry breaking through a branching bifurcation, resulting in the stabilization of asymmetric states and the destabilization of symmetric or anti-symmetric ones. We also highlight the important differences between the cases of symmetric potentials and those of weakly asymmetric potentials.
In this work, we study pancake-shaped Bose-Einstein condensates confined by both a cylindrically symmetric harmonic potential and an optical lattice with equal periodicity in two orthogonal directions. We first identify the spectrum of the underlying two-dimensional linear problem through multiple-scale techniques. Then, we use the results obtained in the linear limit as a starting point for a nonlinear existence and stability analysis of the lowest energy states, emanating from the linear ones, in the nonlinear problem. Two-parameter continuations of these states are performed for increasing nonlinearity and optical lattice strengths, and their instabilities and temporal evolution are investigated. It is found that the ground state as well as one of the excited states are either stable or weakly unstable for both attractive and repulsive interatomic interactions.
Starting from the spectrum of the radially symmetric quantum harmonic oscillator in two dimensions, we create a large set of nonlinear solutions. The relevant three principal branches, with nr=0,1{n}_{r}=0,1nr​=0,1, and 2 radial nodes, respectively, are systematically continued as a function of the chemical potential and their linear stability is analyzed in detail, in the absence as well as in the presence of topological charge mmm, i.e., vorticity. It is found that for repulsive interatomic interactions only the ground state is linearly stable throughout the parameter range examined. Furthermore, this is true for topological charges m=0m=0m=0 or 1; solutions with higher topological charge can be unstable even in that case. All higher excited states are found to be unstable in a wide parametric regime. However, for the focusing (attractive) case the ground state with nr=0{n}_{r}=0nr​=0 and m=0m=0m=0 can only be stable for a sufficiently low number of atoms. Once again, excited states are found to be generically unstable. For unstable profiles, the dynamical evolution of the corresponding branches is also followed to monitor the temporal development of the instability.
The properties of the localized states of a two-component Bose-Einstein condensate confined in a nonlinear periodic potential (nonlinear optical lattice) are investigated. We discuss the existence of different types of solitons and study their stability by means of analytical and numerical approaches. The symmetry properties of the localized states with respect to nonlinear optical lattices are also investigated. We show that nonlinear optical lattices allow the existence of bright soliton modes with equal symmetry in both components and bright localized modes of mixed symmetry type, as well as dark-bright bound states and bright modes on periodic backgrounds. In spite of the quasi-one-dimensional nature of the problem, the fundamental symmetric localized modes undergo a delocalizing transition when the strength of the nonlinear optical lattice is varied. This transition is associated with the existence of an unstable solution, which exhibits a shrinking (decaying) behavior for slightly overcritical (undercritical) variations in the number of atoms.