Universality in s-wave and higher partial wave Feshbach resonances: an illustration with a single atom near two scattering centers

TL;DR

This paper explores the universal properties of atoms near higher partial wave Feshbach resonances, revealing new bound state behaviors and formulas applicable to cold atom systems with two scattering centers.

Contribution

It extends the understanding of universality from s-wave to higher partial wave Feshbach resonances, providing analytical formulas and corrections for bound state energies.

Findings

At Feshbach resonance, $2L+1$ shallow bound states with energies $ ightarrow 1/R^{2L+1}$ for large $R$

Derived simple formulas for bound state energies in terms of a proximity parameter for higher partial waves

Discussed effects of Van der Waals potential on low energy physics

Abstract

It is well-known that cold atoms near s-wave Feshbach resonances have universal properties that are insensitive to the short-range details of the interaction. What is less known is that atoms near higher partial wave Feshbach resonances also have remarkable universal properties. We illustrate this with a single atom interacting resonantly with two fixed static centers. At a Feshbach resonance point with orbital angular momentum $L\ge1$, we find $2L+1$ shallow bound states whose energies behave like $1/R^{2L+1}$ when the distance $R$ between the two centers is large. We then compute corrections to the binding energies due to other parameters in the effective range expansions. For completeness we also compute the binding energies near s-wave Feshbach resonances, taking into account the corrections. Afterwards we turn to the bound states at large but finite scattering volumes. For p-wave and higher partial wave resonances, we derive a simple formula for the energies in terms of a parameter called "proximity parameter". These results are applicable to a free atom interacting resonantly with two atoms that are localized to two lattice sites of an optical lattice, and to one light atom interacting with two heavy ones in free space. Modifications of the low energy physics due to the long range Van der Waals potential are also discussed.