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

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Geometrically confined three-dimensional Bose–Einstein condensates

Robert A. Van Gorder

Published Jul 1, 2025• 1 citation
Publisher

Abstract

Bose–Einstein condensates (BECs) comprise a state of matter in which separate atoms or subatomic particles coalesce into a single macroscopic wave function when cooled to near absolute zero. BECs are confined within traps when carrying out experiments, and in the present paper, we model BECs confined within three-dimensional infinite potential wells corresponding to impenetrable boxes with prescribed geometric properties. To do this, we study the Gross–Pitaevskii equation (GPE) governing the macroscopic wave function within generic three-dimensional finite boxes, first describing analytical and numerical approaches for solving the ground state macroscopic wave function, and then employing these methods to understand how the geometry of a confining box influences properties of the BEC wave function. We show that the chemical potential of the BEC is sensitive to the size and shape of the box, with the chemical potential lowest for a spherical box and comparatively large for boxes with more extreme aspect ratios. Compared to commonly employed harmonic traps, geometrically confined BECs exhibit larger variations of the chemical potential with the self-interaction parameter, yet geometrically confined attractive BECs may be more robust against collapse. In particular, we show that the collapse threshold for an attractive BEC can be modified by the specific manner of geometric confinement.

DOI
10.1098/rspa.2024.0871

Related papers

Geometrically confined single-particle quantum systems

Robert A. Van Gorder

Spatial confinement of quantum systems has been used to model atoms under high pressure, quantum dots, and Bose–Einstein condensates. Spherical boxes are commonly employed, yet they are energy-minimizing, and other box geometries will be more relevant for some applications. We study generic single-particle quantum systems comprising a central physical potential confined within a three-dimensional infinite potential well corresponding to an impenetrable box having prescribed geometric properties; we refer to this as geometric confinement. There is a marked distortion of the wavefunction and energy spectrum, relative to the standard unconfined system, when a confining box is made small enough. We employ asymptotic analysis in combination with numerical simulations to better understand this behaviour, determining lengthscales on which each of the physical potential and confining box respectively dominate the structure of quantum states, with Padé approximants employed to connect asymptotic scalings between these two regimes. This combination of methods permits us to determine, in a fairly general manner, how and to what extent box size and shape modify a given single-particle quantum system held under geometric confinement. We provide a variety of examples of single-particle central potentials and confining geometries to illustrate our methods and motivate the theoretical findings.

Annals of Physics•Jun 10, 2026•0 citations

Theoretical Studies of dilute Bose-Einstein condensates in a double-well potential

E. Sakellari

In this Thesis we apply the Gross-Pitaevskii equation (GPE) to describe properties of a dilute, near zero temperature Bose gas for various confining geometries. We start by reviewing some basic information about the density, the chemical potential and elementary excitations of a dilute atomic condensate confined in a single harmonic trap for a Bose condensate with repulsive and attractive interactions and we also discuss the stability in the case of attractive interactions. We extend our study to a one and three dimensional double-well trap. We investigate the eigenenergy levels and show that the nonlinearity leads to triangular structures which appear either in the ground or excited states for the case of a Bose condensate with attractive or repulsive interactions respectively. We apply the eigenenergy level picture to analyse Josephson effects induced when the barrier IS moved at a constant velocity across the trapping potential or by the application of a time-dependent potential gradient. The GPE simulations are compared to the predictions of a nonlinear two state model. Above a critical velocity there is a transition to a superposition of ground and excited states which leads to sudden changes in the population difference. The direction of Josephson flow depends critically on the initial state of the system and we discuss the feasibility of experimental control of the atomic flow using phase-imprinting. The stability of a low temperature Bose-Einstein condensate with attract interactions in one and three dimensional double-well potentiate is discussed. The condensate is shown to collapse at a critical potential gradient which corresponds to a critical number of atoms in one of the two wells. Finally we investigate the stability and tunnelling effects in a multi-well system.

•Jan 1, 2004•0 citations

Bose-Einstein Condensation of Confined Atomic Gases at Ultra Low Temperatures

Mohammed Serhan

In this work I solve the Gross-Pitaevskii equation describing an atomic gas confined in an isotropic harmonic trap by introducing a variational wavefunction of Gaussian type. The chemical potential of the system is calculated and the solutions are discussed in the weakly and strongly interacting regimes. For the attractive system with negative scattering length the maximum number of atoms that can be put in the condensate without collapse begins is calculated.

Applied Physics Research•Sep 23, 2017•0 citations

Two-species Bose–Einstein condensate in an optical lattice: analytical approximate formulae

R. Cipolatti, Leonardo Villegas-Lelovsky, Ming-Chiang Chung +1 more

Abstract Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose–Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross–Pitaevskii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom–atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.

arXiv:1508.04341•Journal of Physics A Mathematical and Theoretical•Feb 22, 2016•5 citations

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

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
Bose–Einstein condensates (BECs) comprise a state of matter in which separate atoms or subatomic particles coalesce into a single macroscopic wave function when cooled to near absolute zero. BECs are confined within traps when carrying out experiments, and in the present paper, we model BECs confined within three-dimensional infinite potential wells corresponding to impenetrable boxes with prescribed geometric properties. To do this, we study the Gross–Pitaevskii equation (GPE) governing the macroscopic wave function within generic three-dimensional finite boxes, first describing analytical and numerical approaches for solving the ground state macroscopic wave function, and then employing these methods to understand how the geometry of a confining box influences properties of the BEC wave function. We show that the chemical potential of the BEC is sensitive to the size and shape of the box, with the chemical potential lowest for a spherical box and comparatively large for boxes with more extreme aspect ratios. Compared to commonly employed harmonic traps, geometrically confined BECs exhibit larger variations of the chemical potential with the self-interaction parameter, yet geometrically confined attractive BECs may be more robust against collapse. In particular, we show that the collapse threshold for an attractive BEC can be modified by the specific manner of geometric confinement.
Spatial confinement of quantum systems has been used to model atoms under high pressure, quantum dots, and Bose–Einstein condensates. Spherical boxes are commonly employed, yet they are energy-minimizing, and other box geometries will be more relevant for some applications. We study generic single-particle quantum systems comprising a central physical potential confined within a three-dimensional infinite potential well corresponding to an impenetrable box having prescribed geometric properties; we refer to this as geometric confinement. There is a marked distortion of the wavefunction and energy spectrum, relative to the standard unconfined system, when a confining box is made small enough. We employ asymptotic analysis in combination with numerical simulations to better understand this behaviour, determining lengthscales on which each of the physical potential and confining box respectively dominate the structure of quantum states, with Padé approximants employed to connect asymptotic scalings between these two regimes. This combination of methods permits us to determine, in a fairly general manner, how and to what extent box size and shape modify a given single-particle quantum system held under geometric confinement. We provide a variety of examples of single-particle central potentials and confining geometries to illustrate our methods and motivate the theoretical findings.
In this Thesis we apply the Gross-Pitaevskii equation (GPE) to describe properties of a dilute, near zero temperature Bose gas for various confining geometries. We start by reviewing some basic information about the density, the chemical potential and elementary excitations of a dilute atomic condensate confined in a single harmonic trap for a Bose condensate with repulsive and attractive interactions and we also discuss the stability in the case of attractive interactions. We extend our study to a one and three dimensional double-well trap. We investigate the eigenenergy levels and show that the nonlinearity leads to triangular structures which appear either in the ground or excited states for the case of a Bose condensate with attractive or repulsive interactions respectively. We apply the eigenenergy level picture to analyse Josephson effects induced when the barrier IS moved at a constant velocity across the trapping potential or by the application of a time-dependent potential gradient. The GPE simulations are compared to the predictions of a nonlinear two state model. Above a critical velocity there is a transition to a superposition of ground and excited states which leads to sudden changes in the population difference. The direction of Josephson flow depends critically on the initial state of the system and we discuss the feasibility of experimental control of the atomic flow using phase-imprinting. The stability of a low temperature Bose-Einstein condensate with attract interactions in one and three dimensional double-well potentiate is discussed. The condensate is shown to collapse at a critical potential gradient which corresponds to a critical number of atoms in one of the two wells. Finally we investigate the stability and tunnelling effects in a multi-well system.
Abstract Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose–Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross–Pitaevskii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom–atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.
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.
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.