Quantum scattering in quasi-one-dimensional cylindrical confinement

Abstract

Finite size effects alter not only the energy levels of small systems, but can also lead to new effective interactions within these systems. Here the problem of low energy quantum scattering by a spherically symmetric short range potential in the presence of a general cylindrical confinement is investigated. A Green's function formalism is developed which accounts for the full 3D nature of the scattering potential by incorporating all phase-shifts and their couplings. This quasi-1D geometry gives rise to scattering resonances and weakly localized states, whose binding energies and wavefunctions can be systematically calculated. Possible applications include e.g. impurity scattering in ballistic quasi-1D quantum wires in mesoscopic systems and in atomic matter wave guides. In the particular case of parabolic confinement, the present formalism can also be applied to pair collision processes such as two-body interactions. Weakly bound pairs and quasi-molecules induced by the confinement and having zero or higher orbital angular momentum can be predicted, such as p-and d-wave pairings.

Energy-dependentℓ-wave confinement-induced resonances

“…a van der Waals tail. This approach provides a generalization of the works of Granger et al [18] and Kim et al [38] incorporating however all the higher partial waves and contributions from all the closed channels. Furthermore, going beyond the previous studies we derive the connection of the physical K-matrix with all the relevant scattering observables obtaining thus the full scattering wave function.…”

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

“…The length a0 is the Bohr radius, B0 is the magnetic field value at resonance, a bg is the background scattering length and R is defined in Eq. (20).…”

“…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].…”

One-dimensional ultracold atomic gases: Impact of the effective range on integrability

“…Using the standard method, one uses in the Schrödinger equation the δ source terms which are related to the two-body singularities of the wave function [44] and correspond to the expressions of the Λ potential in Eqs. (28) and (29). In our case, one obtains Eq.…”

“…(2) can be expressed as a function of 3D scattering parameters in the homogeneous space [25][26][27][28][29][30][31][32]. In what follows, we consider only positive values of the effective range parameter b η > 0, an assumption justified in the limit of narrow resonances [20,21].…”