Realization of Bose-Einstein Condensates in Lower Dimensions

Abstract

Bose-Einstein condensates of sodium atoms have been prepared in optical and magnetic traps in which the energy-level spacing in one or two dimensions exceeds the interaction energy between atoms, realizing condensates of lower dimensionality. The cross-over into two-dimensional and onedimensional condensates was observed by a change in aspect ratio and saturation of the release energy when the number of trapped atoms was reduced.New physics can be explored when the hierarchy of physical parameters changes. This is evident in dilute gases, where the onset of Bose-Einstein condensation occurs when the thermal deBroglie wavelength becomes longer than the average distance between atoms. Dilutegas condensates of density n in axially-symmetric traps are characterized by four length scales: Their radius R ⊥ , their axial half-length R z , the scattering length a which parameterizes the strength of the two-body interaction, and the healing length ξ = (4πna) −1/2 . In almost all experiments on Bose-Einstein condensates, both the radius and length are determined by the interaction between the atoms and thus, R ⊥ , R z ≫ ξ ≫ a. In this regime, a BEC is three-dimensional and is well-described by the socalled Thomas-Fermi approximation [1]. A qualitatively different behavior of a BEC is expected when the healing length is larger than either R ⊥ or R z since then the condensate becomes restricted to one or two dimensions, respectively. New phenomena that may be observed in this regime are for example quasi-condensates [2-4] and a Tonk's gas of impenetrable bosons [4][5][6].In this Letter, we report the experimental realization of cigar-shaped one-dimensional condensates with R z > ξ > R ⊥ and disk-shaped two-dimensional condensates with R ⊥ > ξ > R z . The cross-over from 3D to 1D or 2D was explored by reducing the number of atoms in condensates which were trapped in highly elongated magnetic traps (1D) and disk-shaped optical traps (2D) and measuring the release energy. In harmonic traps, lower dimensionality is reached when µ 3D = 4π 2 a n/m < ω t . Here, ω t is the trapping frequency in the tightly confining dimension(s) and µ 3D is the interaction energy of a weakly interacting BEC, which in 3D corresponds to the chemical potential. Other experiments in which the interaction energy was comparable to the level spacing of the confining potential include condensates in onedimensional optical lattices [8] and the cross-over to an ideal-gas (zero-D) condensate [7], both at relatively low numbers of condensate atoms.Naturally, the number of interacting atoms in a lowerdimensional condensate is limited. The peak interaction energy of a 3D condensate of N atoms with mass m is given by1/2 are the oscillator lengths of the harmonic potential. The cross-over to 1D and 2D, defined by µ 3D = ω t or equivalently ξ = l t occurs if the number of condensate atoms becomeswhere we have used the scattering length (a = 2.75 nm) and mass of 23 Na atoms to derive the numerical factor. Our traps feature extreme aspect ratios resulting in N 1D > ...

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

“…Such lower dimensional BECs have been studied theoretically [92][93][94][95][96][97][98] (see also Ref. [51] for a rigorous mathematical analysis) and have been realized experimentally in optical and magnetic traps [99], in optical lattice potentials [100][101][102][103] and surface microtraps [78,79].…”

“…[51] for a rigorous mathematical analysis) and have been realized experimentally in optical and magnetic traps [99], in optical lattice potentials [100][101][102][103] and surface microtraps [78,79].…”

Coherence properties of the two-dimensional Bose-Einstein condensate

“…In the limitñ ≡ 0 the correlation function (9) is given by the Heaviside function Θ(1 − r/r con ). However, there is a small contribution from the quantum depletion of the condensate that smoothes the sharp corner as g (1) drops to zero, causing a loss of numerical accuracy as we divide two very similar small numbers in (12).…”

“…Bose-Einstein condensation (BEC) in (quasi-) twodimensional systems has only recently been obtained in the laboratory [1,2]. Thus, many properties have yet to be explored both experimentally and theoretically.…”

Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

“…Recent advances in the experimental physics of Bose-Einstein condensates [14,15,6] have led to the experimental realisation of a one-dimensional Bose gas of ultra-cold alkali atoms. One expects [39] that the microscopic forces are such that there is an effective one-body confining harmonic potential acting on each atom individually, and an effective infinitely short range contact potential acting between neighbouring atoms.…”