Effective-range description of a Bose gas under strong one- or two-dimensional confinement

Abstract

We point out that theories describing s-wave collisions of bosonic atoms confined in one-or two-dimensional geometries can be extended to much tighter confinements than previously thought. This is achieved by replacing the scattering length by an energy-dependent scattering length which was already introduced for the calculation of energy levels under 3D confinement. This replacement accurately predicts the position of confinementinduced resonances in strongly confined geometries.Many experiments investigating the properties of cold atomic gases and Bose-Einstein condensates are now performed in tightly confining traps, such as tight optical lattices, leading to systems of reduced dimensionality [1,2,3,4]. There are many uses for such confinements. In spectroscopic measurements, they eliminate unwanted Doppler and recoil effects [5,6]. They can also be used to create tunable analogs of condensed matter systems, and give the possibility to investigate remarkable many-body regimes in low dimensions such as the Tonks-Girardeau gas [7,8,9,10]. The theory of s-wave atomic collisions in strongly confined systems has been established in Refs.[7] and [11] for 2D and 1D confinement, respectively. Both predict a confinement-induced resonance of the effective 1D or 2D interaction strength. These predictions rely on a description of the atomic interaction in terms of the scattering length only. However, in 3D confined systems, it was shown that a more refined description is needed for very tight confinement [12,13]. Similarly, in 2D confined systems, numerical calculations in Ref. [14] showed that the scattering length description of Ref.[7] may be insufficient. In this paper, we present an accurate analytical description for scattering in 1D and 2D geometries based on the findings of Refs. [13,15].We consider a gas of bosonic atoms in an optical lattice and assume that there is little tunelling between the lattice cells, so that each cell is independent. The atoms in a cell are confined by a trapping potential which will be assumed harmonic (which is true near the centre of the cell). Let us consider a pair of atoms in such a cell. For a harmonic potential, the centre-of-mass motion decouples from the relative motion and the stationary Schrödinger equation for the relative motion wave function ψ( r) reads:Here, r = (x, y, z) is the relative coordinate with separation r, µ the reduced mass, U (r) the isotropic atom-atom interaction potential, V ( r) the trapping potential, and E the relative energy.For 2D confinement (tube or wave guide geometry), the atoms are strongly confined in the xy directions and (almost) free to move in the z direction, therefore we set:where ρ = ρ and ρ is the projection of r on the xy plane. For 1D confinement (pancake geometry), the atoms are strongly confined in the z direction and (almost) free to move in the xy directions:Here, ω is the trapping frequency at the centre of the cell and we define σ = /(2µω) as the typical length scale associated to the trap in the confined directions.Any scatteri...

Ultracold-atom collisions in atomic waveguides: A two-channel analysis

“….. Equation (46) permits to recover the results deduced by using the zero-range potential approach [18,[21][22][23].…”

“…This issue has been already the subject to many theoretical studies [4,5,[18][19][20][21][22][23] where a zero-range potential approach was used and much more sophisticated multichannel studies have been performed in Refs. [24][25][26].…”

Confinement-induced resonances in ultracold atom-ion systems

“…The difference was not ascribable to numerical imprecision, but rather because of the non-zero colliding energy. In a subsequent study [51], it has been showed that by using the formula (5) with the replacement a 3D → a 3D (k) in the so-called effective-range approximation [52,53] …”

Confinement-induced resonances in anharmonic waveguides

“…A direct generalization of Olshanii's theory to anisotropic transverse confinement shows that there is only one harmonic CIR (HCIR) resonance, no matter how large the transverse anisotropy [19]. For large anisotropy, this theory crosses over smoothly to the case of a quasi-2D trap, where a single HCIR occurs with a negative S-wave scattering length, a 3D < 0 [20][21][22].…”