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arXiv:1507.02910 · Discrete and Continuous Dynamical Systems

Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement

Florian Méhats, Christof Sparber

Published May 1, 2016• 13 citations
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Abstract

We consider the three-dimensional time-dependent Gross-Pitaevskii equation arising in the description of rotating Bose-Einstein condensatesand study the corresponding scaling limit of strongly anisotropic confinement potentials. The resulting effective equations in one ortwo spatial dimensions, respectively, are rigorously obtained as special cases of an averaged three dimensional limit model. In the particularcase where the rotation axis is not parallel to the strongly confining direction the resulting limiting model(s) include a negative, and thus, purelyrepulsive quadratic potential, which is not present in the original equation and which can be seen as an effective centrifugal force counteracting theconfinement.

DOI
10.3934/dcds.2016021

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We consider the three-dimensional time-dependent Gross-Pitaevskii equation arising in the description of rotating Bose-Einstein condensatesand study the corresponding scaling limit of strongly anisotropic confinement potentials. The resulting effective equations in one ortwo spatial dimensions, respectively, are rigorously obtained as special cases of an averaged three dimensional limit model. In the particularcase where the rotation axis is not parallel to the strongly confining direction the resulting limiting model(s) include a negative, and thus, purelyrepulsive quadratic potential, which is not present in the original equation and which can be seen as an effective centrifugal force counteracting theconfinement.
Abstract We study the problem of dimension reduction for the three dimensional Gross–Pitaevskii equation (GPE) describing a Bose–Einstein condensate confined in a strongly anisotropic harmonic trap. Since the gas is assumed to be in a strong interaction regime, we have to analyse two combined singular limits: a semi-classical limit in the transport direction and the strong partial confinement limit in the transversal direction. We prove that both limits commute together and we provide convergence rates. The by-products of this work are approximated models in reduced dimension for the GPE, with a priori estimates of the approximation errors.
We study dimension reduction for the three-dimensional Gross-Pitaevskii equation with a long-range and anisotropic dipole-dipole interaction modeling dipolar Bose-Einstein condensation in a strong interaction regime. The cases of disk shaped condensates (confinement from dimension three to dimension two) and cigar shaped condensates (confinement to dimension one) are analyzed. In both cases, the analysis combines averaging tools and semiclassical techniques. Asymptotic models are derived, with rates of convergence in terms of two small dimensionless parameters characterizing the strength of the confinement and the strength of the interaction between atoms.
Ground states and dynamical properties of a dipolar Bose–Einstein condensate are analyzed based on the Gross–Pitaevskii–Poisson system (GPPS) and its dimension reduction models under an anisotropic confining potential. We begin with the three-dimensional (3D) GPPS and review its quasi-two-dimensional (2D) approximate equations when the trap is strongly confined in the z-direction and quasi-one-dimensional (1D) approximate equations when the trap is strongly confined in the x- and y-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator (−Δ)1/2(-\Delta)^{1/2}(−Δ)1/2 is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.
We consider vortex dynamics in the context of Bose–Einstein condensates (BECs) with a rotating trap, with or without anisotropy. Starting with the Gross–Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.
Motivated by numerous experiments on Bose-Einstein condensed atoms which have been performed in tight trapping potentials of various geometries [elongated and/or toroidal (annular)], we develop a general method which allows us to reduce the corresponding three-dimensional Gross-Pitaevskii equation for the order parameter into an effectively one-dimensional equation, taking into account the interactions (i.e., treating the width of the transverse profile variationally) and the curvature of the trapping potential. As an application of our model we consider atoms which rotate in a toroidal trapping potential. We evaluate the state of lowest energy for a fixed value of the angular momentum within various approximations of the effectively one-dimensional model and compare our results with the full solution of the three-dimensional problem, thus getting evidence for the accuracy of our model.
We present effective reduced equations for the study of a binary Bose-Einstein condensate (BEC), where the confining potentials of the two BEC components have distinct asymmetry so that the components belong to different space dimensions as in a recent experiment [G. Lamporesi et al., Phys. Rev. Lett. 104, 153202 (2010)]. Starting from a binary three-dimensional (3D) Gross-Pitaevskii equation (GPE) and using a Lagrangian variational approach we derive a binary effective nonlinear Schroedinger equation with components in different reduced dimensions, e. g., the first component in one dimension and the second in two dimensions as appropriate to represent a cigar-shaped BEC coupled to a disk-shaped BEC. We demonstrate that the effective reduced binary equation, which depend on the geometry of the system, is quite reliable when compared with the binary 3D GPE and can be efficiently used to perform numerical simulation and analytical calculation for the investigation of static and dynamic properties of a binary BEC in mixed dimensions.