Universality and threshold laws for collision lifetimes at ultralow temperatures

TL;DR

This paper derives a universal expression for collision lifetimes at ultralow temperatures, revealing threshold laws and the dominance of elastic scattering, with implications for understanding few-body quantum collision dynamics.

Contribution

It introduces a general formula for the Smith lifetime matrix at ultralow energies using multichannel quantum defect theory, extending previous models.

Findings

Collision lifetime scales as 1/√E at zero energy.

Elastic scattering dominates the collision lifetime.

Inelastic loss shortens the collision lifetime with increasing loss parameter.

Abstract

While collision lifetimes are a fundamental property of few-body scattering events, their behavior at ultralow temperatures is not completely understood. We derive a general expression for the Smith lifetime Q-matrix using multichannel quantum defect theory, which allows us to obtain the average time delay in the incident $s$-wave collision channel in the limit of zero collision energy $E$, $Q_{11}=A/\sqrt{E} + B$, where $A$ and $B$ are constants. We show that the time delay is dominated by elastic scattering, and contains an additional multichannel contribution independent of the two-body scattering length. We also obtain the expressions for the collision lifetime using the universal model of Idziaszek and Julienne [Phys. Rev. Lett. 104, 113202 (2010)]. The lifetime acquires an imaginary part in the presence of inelastic loss due to the lack of unitarity of the S-matrix, and shortens with increasing the short-range loss parameter $y$.