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arXiv:cond-mat/0107001 · Physical Review A

Highly anisotropic Bose-Einstein condensates: Crossover to lower dimensionality

Kunal K. Das

Published Nov 14, 2002• 38 citations
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Abstract

We develop a simple analytical model based on a variational method to explain the properties of trapped cylindrically symmetric Bose-Einstein condensates of varying degrees of anisotropy, well into the regimes of effective one dimension and effective two dimensions. Our results are accurate in regimes where the Thomas-Fermi approximation breaks down and they are shown to be in agreement with recent experimental data. We also present analytic expressions for various ground-state properties and quasiparticle excitations valid for highly anisotropic condensates.

DOI
10.1103/physreva.66.053612

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We develop a simple analytical model based on a variational method to explain the properties of trapped cylindrically symmetric Bose-Einstein condensates of varying degrees of anisotropy, well into the regimes of effective one dimension and effective two dimensions. Our results are accurate in regimes where the Thomas-Fermi approximation breaks down and they are shown to be in agreement with recent experimental data. We also present analytic expressions for various ground-state properties and quasiparticle excitations valid for highly anisotropic condensates.
In this thesis, the properties of the Bose-Einstein condensation (BEC) in low dimensions are reviewed. Three dimensional weakly interacting Bose systems are examined by the variational method. The effects of both the attractive and the repulsive interatomic forces are studied. Thomas-Fermi approximation is applied to find the ground state energy and the chemical potential. The occurrence of the BEC in low dimensional systems, is studied for ideal gases confined by both harmonic and power-law potentials. The properties of BEC in highly anisotropic trap are investigated and the conditions for reduced dimensionality are derived.
We introduce dimensional perturbation techniques to Bose-Einstein condensation of inhomogeneous alkali-metal gases. The perturbation parameter is $\ensuremath{\delta}=1/\ensuremath{\kappa},$ where $\ensuremath{\kappa}$ depends on the effective dimensionality of the condensate and on the angular momentum quantum number. We derive a simple approximation that is more accurate and flexible than the $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ Thomas-Fermi ground-state approximation of the Gross-Pitaevskii equation. The approximation presented here is well suited for calculating properties of states in three dimensions and in low-effective dimensionality, such as vortex states in a highly anisotropic trap.
We investigate D-dimensional atomic Bose-Einstein condensates in a hypercylindrical trap with a vortex core along the z-axis and quantized circulation ℏm\hbar mℏm. We analytically approximate the hypercylindrical Gross-Pitaevskii equation using dimensional perturbation theory with perturbation parameter δ=1/(D+2∣m∣−d)δ=1/(D+2|m|-d)δ=1/(D+2∣m∣−d), \textcolor{black}{where ddd controls the contribution of kinetic energy at zeroth order}. We derive the zeroth-order (δ→0δ\to 0δ→0) semiclassical approximations for the condensate energy, density, chemical potential, and critical vortex rotation speed in arbitrary dimensions. We investigate the effect of trap anisotropy on lower effective dimensionality and compute properties of vortices in higher dimensions motivated by the study of synthetic dimensions and holographic duality, where a higher-dimensional gravitational model corresponds to a lower-dimensional quantum model. In the zeroth-order approximation, we observe crossings between energy levels for different dimensions as a function of interaction strength and anisotropy parameters.