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:cond-mat/0305273 · Journal of Physics B Atomic Molecular and Optical Physics

Transition from resonances to bound states in nonlinear systems: application to Bose–Einstein condensates

Nimrod Moiseyev, Lincoln D. Carr, Boris A. Malomed, Y. B. Band

Published Apr 24, 2004• 53 citations
arXivHTMLPDF

Abstract

It is shown using the Gross–Pitaevskii equation that resonance states of Bose–Einstein condensates with attractive interactions can be stabilized into true bound states. A semiclassical variational approximation and an independent quantum variational numerical method are used to calculate the energies (chemical potentials) and linewidths of resonances of the time-independent Gross–Pitaevskii equation; both methods produce similar results. Borders between the regimes of resonances, bound states and, in two and three dimensions, collapse are identified.

DOI
10.1088/0953-4075/37/9/l02

Related papers

Geometric resonances in Bose–Einstein condensates with two- and three-body interactions

Hamid Al-Jibbouri, Ivana Vasić, Antun Balaž +1 more

We investigate geometric resonances in Bose-Einstein condensates by solving the underlying time-dependent Gross-Pitaevskii equation for systems with two- and three-body interactions in an axially-symmetric harmonic trap. To this end, we use a recently developed analytical method [Phys. Rev. A 84, 013618 (2011)], based on both a perturbative expansion and a Poincaré-Lindstedt analysis of a Gaussian variational approach, as well as a detailed numerical study of a set of ordinary differential equations for variational parameters. By changing the anisotropy of the confining potential, we numerically observe and analytically describe strong nonlinear effects: shifts in the frequencies and mode coupling of collective modes, as well as resonances. Furthermore, we discuss in detail the stability of a Bose-Einstein condensate in the presence of an attractive two-body interaction and a repulsive three-body interaction. In particular, we show that a small repulsive three-body interaction is able to significantly extend the stability region of the condensate.

arXiv:1208.0991•Journal of Physics B Atomic Molecular and Optical Physics•Mar 7, 2013•39 citations

ENERGY AND CHEMICAL POTENTIAL ASYMPTOTICS FOR THE GROUND STATE OF BOSE-EINSTEIN CONDENSATES IN THE SEMICLASSICAL REGIME

Weizhu Bao, Fong Yin Lim, Yanzhi Zhang

Asymptotic approximations for the energy and chemical potential of the ground state in Bose-Einstein condensates are presented in the semiclassical regime with several typical trap-ping potentials. As preparatory steps, we begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE), review several typical external trapping potentials, scale the 3D GPE and show how to reduce it to 1D and 2D GPEs in certain limiting trapping frequency regime. For the 1D box potential, we derive asymptotic approximations up to o(1) in term of the scaled interacting pa-rameter d for energy and chemical potential of the ground and all excited states in both weakly interacting regime, i.e. d! 0 and strongly repulsive interacting regime, i.e. d! 1, respec-tively. For the 1D harmonic oscillator, double well and optical lattice potentials, as well as a more general external potential in high dimensions, we get asymptotic approximations up to o(1) in term of the scaled interacting parameter d for the energy and chemical potential of the ground state in semiclassical regime, i.e. d!1. Our extensive numerical results conrm all our asymp-totic approximations, provide convergence rate and suggest several very interesting conclusions.

•Jan 1, 2007•17 citations

Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. II. Applications

Stefan Rau, Jörg Main, Holger Cartarius +2 more

Bose-Einstein condensates with an attractive $1/r$ interaction and with dipole-dipole interaction are investigated in the framework of the Gaussian variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A 82, 023610 (2010)]. We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation, or even superior in that coupled Gaussians are capable of producing both stable and unstable states of the Gross-Pitaevskii equation and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. As an alternative to numerical solutions of the Bogoliubov--de Gennes equations, the stability of the stationary condensate wave functions is investigated by analyzing the stability properties of the dynamical equations of motion for the Gaussian variational parameters in the local vicinity of the stationary fixed points. For blood-cell-shaped dipolar condensates it is shown that on the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.

arXiv:1007.3182•Physical Review A•Aug 13, 2010•33 citations

13 Geometric Resonances in Bose-Einstein Condensates with Two- and Three-Body Interactions

Hamid Al-Jibbouri

We investigate geometric resonances in Bose-Einstein condensates by solving the underlying time-dependent Gross-Pitaevskii (GP) equation for systems with two-and three-body interactions in an axially-symmetric harmonic trap.To this end, we use a recently developed analytical method [Vidanović I et al 2011 Phys.Rev. A 84 013618], based on both a perturbative expansion and a Poincaré-Lindstedt analysis of a Gaussian variational approach, as well as a detailed numerical study of a set of ordinary differential equations for variational parameters.By changing the anisotropy of the confining potential, we numerically observe and analytically describe strong nonlinear effects: shifts in the frequencies and mode coupling of collective modes, as well as resonances.Furthermore, we discuss in detail the stability of a Bose-Einstein condensate in the presence of an attractive two-body interaction and a repulsive threebody interaction.In particular, we show that a small repulsive three-body interaction is able to significantly extend the stability region of the condensate.

•Aug 16, 2016•40 citations

Dynamics of Bose–Einstein Condensates Subject to the Pöschl–Teller Potential through Numerical and Variational Solutions of the Gross–Pitaevskii Equation

Lucas Carvalho Pereira, Valter Aragão do Nascimento

We present for the first time an approach about Bose-Einstein condensates made up of atoms with attractive interatomic interactions confined to the Pöschl-Teller hyperbolic potential. In this paper, we consider a Bose-Einstein condensate confined in a cigar-shaped, and it was modeled by the mean field equation known as the Gross-Pitaevskii equation. An analytical (variational method) and numerical (two-step Crank-Nicolson) approach is proposed to study the proposed model of interatomic interaction. The solutions of the one-dimensional Gross-Pitaevskii equation obtained in this paper confirmed, from a theoretical point of view, the possibility of the Pöschl-Teller potential to confine Bose-Einstein condensates. The chemical potential as a function of the depth of the Pöschl-Teller potential showed a behavior very similar to the cases of Bose-Einstein condensates and superfluid Fermi gases in optical lattices and optical superlattices. The results presented in this paper can open the way for several applications in atomic and molecular physics, solid state physics, condensed matter physics, and material sciences.

Materials•May 13, 2020•7 citations

STABILITY OF BOSE–EINSTEIN CONDENSATES CONFINED IN TRAPS

Takeya Tsurumi, Hirofumi Morise, Miki Wadati

Bose–Einstein condensation has been realized as dilute atomic vapors. This achievement has generated immense interest in this field. This article review of recent theoretical research into the properties of trapped dilute-gas Bose–Einstein condensates. Among these properties, stability of Bose–Einstein condensates confined in traps is mainly discussed. Static properties of the ground state are investigated by using the variational method. The analysis is extended to the stability of two-component condensates. Time-development of the condensate is well-described by the Gross–Pitaevskii equation which is known in nonlinear physics as the no nlinear Schrödinger equation. For the case that the inter-atomic potential is effectively attractive, a singularity of the solution emerges in a finite time. This phenomenon which we call collapse explains the upper bound for the number of atoms in such condensates under traps.

arXiv:cond-mat/9912470•International Journal of Modern Physics B•Mar 20, 2000•31 citations
It is shown using the Gross–Pitaevskii equation that resonance states of Bose–Einstein condensates with attractive interactions can be stabilized into true bound states. A semiclassical variational approximation and an independent quantum variational numerical method are used to calculate the energies (chemical potentials) and linewidths of resonances of the time-independent Gross–Pitaevskii equation; both methods produce similar results. Borders between the regimes of resonances, bound states and, in two and three dimensions, collapse are identified.
We investigate geometric resonances in Bose-Einstein condensates by solving the underlying time-dependent Gross-Pitaevskii equation for systems with two- and three-body interactions in an axially-symmetric harmonic trap. To this end, we use a recently developed analytical method [Phys. Rev. A 84, 013618 (2011)], based on both a perturbative expansion and a Poincaré-Lindstedt analysis of a Gaussian variational approach, as well as a detailed numerical study of a set of ordinary differential equations for variational parameters. By changing the anisotropy of the confining potential, we numerically observe and analytically describe strong nonlinear effects: shifts in the frequencies and mode coupling of collective modes, as well as resonances. Furthermore, we discuss in detail the stability of a Bose-Einstein condensate in the presence of an attractive two-body interaction and a repulsive three-body interaction. In particular, we show that a small repulsive three-body interaction is able to significantly extend the stability region of the condensate.
Asymptotic approximations for the energy and chemical potential of the ground state in Bose-Einstein condensates are presented in the semiclassical regime with several typical trap-ping potentials. As preparatory steps, we begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE), review several typical external trapping potentials, scale the 3D GPE and show how to reduce it to 1D and 2D GPEs in certain limiting trapping frequency regime. For the 1D box potential, we derive asymptotic approximations up to o(1) in term of the scaled interacting pa-rameter d for energy and chemical potential of the ground and all excited states in both weakly interacting regime, i.e. d! 0 and strongly repulsive interacting regime, i.e. d! 1, respec-tively. For the 1D harmonic oscillator, double well and optical lattice potentials, as well as a more general external potential in high dimensions, we get asymptotic approximations up to o(1) in term of the scaled interacting parameter d for the energy and chemical potential of the ground state in semiclassical regime, i.e. d!1. Our extensive numerical results conrm all our asymp-totic approximations, provide convergence rate and suggest several very interesting conclusions.
Bose-Einstein condensates with an attractive 1/r1/r1/r interaction and with dipole-dipole interaction are investigated in the framework of the Gaussian variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A 82, 023610 (2010)]. We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation, or even superior in that coupled Gaussians are capable of producing both stable and unstable states of the Gross-Pitaevskii equation and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. As an alternative to numerical solutions of the Bogoliubov--de Gennes equations, the stability of the stationary condensate wave functions is investigated by analyzing the stability properties of the dynamical equations of motion for the Gaussian variational parameters in the local vicinity of the stationary fixed points. For blood-cell-shaped dipolar condensates it is shown that on the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.
We investigate geometric resonances in Bose-Einstein condensates by solving the underlying time-dependent Gross-Pitaevskii (GP) equation for systems with two-and three-body interactions in an axially-symmetric harmonic trap.To this end, we use a recently developed analytical method [Vidanović I et al 2011 Phys.Rev. A 84 013618], based on both a perturbative expansion and a Poincaré-Lindstedt analysis of a Gaussian variational approach, as well as a detailed numerical study of a set of ordinary differential equations for variational parameters.By changing the anisotropy of the confining potential, we numerically observe and analytically describe strong nonlinear effects: shifts in the frequencies and mode coupling of collective modes, as well as resonances.Furthermore, we discuss in detail the stability of a Bose-Einstein condensate in the presence of an attractive two-body interaction and a repulsive threebody interaction.In particular, we show that a small repulsive three-body interaction is able to significantly extend the stability region of the condensate.
We present for the first time an approach about Bose-Einstein condensates made up of atoms with attractive interatomic interactions confined to the Pöschl-Teller hyperbolic potential. In this paper, we consider a Bose-Einstein condensate confined in a cigar-shaped, and it was modeled by the mean field equation known as the Gross-Pitaevskii equation. An analytical (variational method) and numerical (two-step Crank-Nicolson) approach is proposed to study the proposed model of interatomic interaction. The solutions of the one-dimensional Gross-Pitaevskii equation obtained in this paper confirmed, from a theoretical point of view, the possibility of the Pöschl-Teller potential to confine Bose-Einstein condensates. The chemical potential as a function of the depth of the Pöschl-Teller potential showed a behavior very similar to the cases of Bose-Einstein condensates and superfluid Fermi gases in optical lattices and optical superlattices. The results presented in this paper can open the way for several applications in atomic and molecular physics, solid state physics, condensed matter physics, and material sciences.
Bose–Einstein condensation has been realized as dilute atomic vapors. This achievement has generated immense interest in this field. This article review of recent theoretical research into the properties of trapped dilute-gas Bose–Einstein condensates. Among these properties, stability of Bose–Einstein condensates confined in traps is mainly discussed. Static properties of the ground state are investigated by using the variational method. The analysis is extended to the stability of two-component condensates. Time-development of the condensate is well-described by the Gross–Pitaevskii equation which is known in nonlinear physics as the no nlinear Schrödinger equation. For the case that the inter-atomic potential is effectively attractive, a singularity of the solution emerges in a finite time. This phenomenon which we call collapse explains the upper bound for the number of atoms in such condensates under traps.