Low energy universality and scaling of Van der Waals forces

TL;DR

This paper demonstrates that the effective range in ultra-cold atom scattering with Van der Waals forces follows a universal relation dependent on dispersion coefficients, extending beyond naive estimates.

Contribution

It derives a universal low energy theorem linking the effective range to the scattering length and dispersion coefficients, validated against experimental data.

Findings

Effective range formula matches experimental data for many atom pairs.

Universality is primarily governed by the C6 dispersion coefficient.

Scaling behavior extends beyond naive dimensional analysis.

Abstract

At long distances interactions between neutral ground state atoms can be described by the Van der Waals potential V(r) =-C6/r^6-C8/r^8 - ... . In the ultra-cold regime atom-atom scattering is dominated by s-waves phase shifts given by an effective range expansion p cot d0 (p) = -1/a0 + r0 p^2/2 + ... in terms of the scattering length a0 and the effective range r0. We show that while for these potentials the scattering length cannot be predicted, the effective range is given by the universal low energy theorem r0 = A + B/a0+ C/a0^2 where A,B and C depend on the dispersion coefficients Cn and the reduced di-atom mass. We confront this formula to about a hundred determinations of r0 and a0 and show why the result is dominated by the leading dispersion coefficient C6. Universality and scaling extends much beyond naive dimensional analysis estimates.