Atom-Atom Scattering under Cylindrical Harmonic Confinement: Numerical and Analytic Studies of the Confinement Induced Resonance

Abstract

M. Olshanii [Phys. Rev. Lett. 81, 938 (1998)] recently solved the atom-atom scattering problem with a pseudopotential interaction in the presence of transverse harmonic confinement, i.e. within an 'atom waveguide', deriving an effective one-dimensional coupling constant that diverged at a "confinement induced resonance" (CIR). Here, we report numerical results for finite range potentials that corroborate this resonance. In addition, we now present a physical interpretation of this effect as a novel type of Feshbach resonance in which the transverse modes of the waveguide assume the roles of 'open' and 'closed' scattering channels. [1,2,3,4,5,6,7,8]. One goal of such experiments is to reach the 'single-mode' or quasi-1D regime, where only the ground state of transverse motion is significantly populated at thermal equilibrium. This regime is of great practical interest due to the potential for ultra-sensitive rotation and gravitational gradient detection with guided single-mode atom interferometers. In addition to such applications, reaching the quasi-1D regime is of significant theoretical interest as the 1D delta-interacting boson gas represents one of the few known fully integrable quantum field theories. In a finite system with infinitely strong (hard-core) deltafunction interactions, boson many-body states in 1D have been shown to correspond via a one-to-one mapping with the highly-correlated states of the corresponding noninteracting Fermi gas [9]. The properties of this TonksGirardeau gas have been a topic of significant current theoretical interest [9,10,11,12,13,14,15] in anticipation of future atom-waveguide experiments. Additionally, the homogeneous 1D Bose with arbitrary-strength delta-function interactions, known as the Lieb Liniger model, is also a fully integrable system [16].To make the connection between experiments in tightly confining waveguides and theoretical models in 1D, it is necessary to know the relationship between the effective 1D coupling constant, g 1D , and the 3D scattering length, a. This problem was first addressed rigorously in [22], where it was predicted that a 'confinement induced resonance' (CIR) modifies the effective interaction, resulting in an effective 1D coupling strength which can be tuned from −∞ to +∞ by varying the transverse width of the * Electronic address: thbergeman@notes.cc.sunysb.edu † Electronic address: mmoore@cfa.harvard.edu ‡ Electronic address: olshanii@phys4adm.usc.edu waveguide, a ⊥ over a small range in the vicinity of the resonance at a ⊥ = Ca, where C = 1.4603 . . .. Hence this resonance clearly has significant implications for atomwaveguide experiments. Until now, however, there has been no convincing physical explanation for the effect, thus raising questions concerning its appearance in systems with finite-range interactions.The primary goal of this Letter, therefore, is to present numerical calculations of scattering in the presence of a cylindrical harmonic potential using finite-range atomatom potentials, confirming the existence of the CIR. ...

Effective interactions, Fermi–Bose duality, and ground states of ultracold atomic vapors in tight de Broglie waveguides

“…Notice that at a s = |ζ(1/2)| −1 a ⊥ the coupling constant diverges, signifying the so-called "confinement induced resonance" (CIR). The significance of this resonance has been confirmed in ab initio two-body numerical calculations with finite-range realistic interatomic potentials [20] and in many-body Monte-Carlo simulations [29].…”

“…Hence the whole range of 1D coupling constants from −∞ to +∞ is experimentally achievable by tuning a s over a narrow range in the neighborhood of the 1D resonance. It was shown recently [20] that this is a 1D Feshbach resonance between ground and excited transverse vibrational manifolds. It was shown in [20] and in Sec.…”

“…implying the existence of a confinement-induced resonance CIR [16,20] of the coupling constant as a s is tuned via a 3D Feshbach resonance [32] past the resonance point a s /a ⊥ = |ζ(1/2)| −1 = 0.6848 . .…”

“…We assume herein that both µ < ω ⊥ and k B T < ω ⊥ as in [8,18,19]. Nevertheless, virtually excited transverse modes renormalize the effective 1D coupling constant via a confinement-induced resonance, as first shown for bosons [16,20] and recently for spinpolarized fermionic vapors [21]. At the very low densities of ultracold atomic vapors, the 3D interatomic interactions are usually adequately described by the s-wave scattering length and a corresponding 3D zero-range pseudopotential, the input for the original derivation [16] of an effective 1D interaction between waveguide-confined spinless bosonic atoms.…”

“…A more detailed and comprehensive theory has been developed recently [22]. In the simplest case of spinless bosons the resultant effective 1D interaction is of the form g B 1D δ(z) where g B 1D is an explicit and nontrivial function [16,20,22] of the 3D s-wave scattering length, z = z 1 − z 2 , and z 1 and z 2 are 1D coordinates of the interacting atoms measured along the longitudinal axis of the waveguide. This derivation [16,22] will be described in Sec.…”