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arXiv:1609.04679 · Physical review. E

Dimensional reduction in Bose-Einstein condensed clouds of atoms confined in tight potentials of any geometry and any interaction strength

Patrik Sandin, Magnus Ögren, Mårten Gulliksson, J. Smyrnakis, Manolis Magiropoulos, G. M. Kavoulakis

Published Jan 23, 2017• 15 citations
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Abstract

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.

DOI
10.1103/physreve.95.012142

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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.
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Within the formalism of the Gross-Pitaevskii equation, we derive effective one- and two-dimensional equations for cigar- and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. These are based on an ansatz for the condensate wavefunction whose width in the tightly-confined direction/s is treated variationally. The equations constitute a coupled partial differential for the low-dimensional wavefunction and algebraic equations for the width parameters. This approach accurately predicts the ground state densities of cigar-shaped and pancake-shaped dipolar Bose-Einstein condensates, and gives strong agreement with the three-dimensional results, even as the trapping is relaxed away from the strict quasi-one- and quasi-two-dimensional regimes. This approach offers a significant improvement over the standard one- and two-dimensional reduction.
Motivated by recent experiments on Bose-Einstein condensed atoms which rotate in annular and/or toroidal traps, we study the effect of the finiteness of the atom number NNN on the states of lowest energy for a fixed expectation value of the angular momentum, under periodic boundary conditions. To attack this problem, we develop a general strategy, considering a linear superposition of the eigenstates of the many-body Hamiltonian, with amplitudes that we extract from the mean-field approximation. This many-body state breaks the symmetry of the Hamiltonian; it has the same energy to leading order in NNN as the mean-field state and the corresponding eigenstate of the Hamiltonian, however, it has a lower energy to subleading order in NNN and thus it is energetically favorable.
We study binary mixtures of Bose–Einstein condensates confined in a two-well potential within the framework of the Gross–Pitaevskii equation. We re-examine both the single component and the binary mixture cases for such a potential. We investigate the most usual dimensional reductions used to solve the Gross–Pitaevskii equations, including the one proposed by Reatto and collaborators. To this end, we compare numerical simulations of the 1D reductions with the full 3D numerical solutions of the Gross–Pitaevskii equation. Our analysis considers an experimentally feasible binary mixture of an F = 1 spinor condensate where two of its Zeeman manifolds are populated.
Abstract By exact numerical solutions of the Gross–Pitaevskii (GP) equation in 3D, we assess the validity of 1D and 2D approximations in the study of Bose–Einstein condensates confined in harmonic trap potentials. Typically, these approximations are performed when one or more of the harmonic frequencies are much greater than the remaining ones, using arguments based on the adiabatic evolution of the initial approximated state. Deviations from the 3D solution are evaluated as a function of both the effective interaction strength and the ratio between the trap frequencies that define the reduced dimension where the condensate is confined. The observables analyzed are both of stationary and dynamical character, namely, the chemical potential, the wave function profiles, and the time evolution of the approximated 1D and 2D stationary states, considered as initial states in the 3D GP equation. Our study, besides setting quantitative limits on approximations previously developed, should be useful in actual experimental studies where quasi-1D and quasi-2D conditions are assumed. From a qualitative perspective, 1D and 2D approximations certainly become valid when the anisotropy is large, but in addition the interaction strength needs to be above a certain threshold.