Analytic description of atomic interaction at ultracold temperatures: the case of a single channel

TL;DR

This paper extends analytic descriptions of ultracold atomic interactions to higher angular momenta using quantum-defect theory, introducing a generalized scattering length applicable to all partial waves, enhancing understanding of ultracold scattering phenomena.

Contribution

It generalizes existing quantum-defect theory expansions to higher angular momenta and introduces a new generalized scattering length for all partial waves.

Findings

Analytic description of ultracold shape resonances for all angular momenta.

A well-defined generalized scattering length for $l extgreater=2$.

Foundation for analytic descriptions around Feshbach resonances.

Abstract

We present analytic descriptions of atomic interaction at ultracold temperatures using both single-channel and multichannel quantum-defect theories. In the case of a single channel, addressed in this paper, the expansion of [B. Gao, Phys. Rev. A \textbf{58}, 4222 (1998)] is generalized to higher orders for angular momentum $l\ge 2$ to give a more complete description of ultracold scattering, including an analytic description of ultracold shape resonances of arbitrary $l$. We also introduce a generalized scattering length that is well defined and useful for all partial waves, to replace the traditional definition that fails for $l\ge 2$ due to the long-range van der Waals interaction. The results are used in a companion paper to derive analytic descriptions of atomic interaction around a magnetic Feshbach resonance of arbitrary angular momentum $l$.