Universality and threshold laws for collision lifetimes at ultralow temperatures

While collision lifetimes are a fundamental property of few-body scattering events, their behavior at ultralow temperatures is not completely understood. Using multichannel quantum-defect theory, we derive a multichannel expression for the Smith lifetime $Q$ matrix that is valid far away from scattering resonances. For collisions mediated by short-range interaction potentials of the form $\ensuremath{-}{C}_{6}/{R}^{6}$, we obtain the average time delay in the incident $s$-wave collision channel in the limit of zero collision energy $E$ as ${Q}_{11}=A/\sqrt{E}+B$, 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 $y$.