Near-threshold scaling of resonant inelastic collisions at ultralow temperatures

We show that the cross sections for a broad range of resonant inelastic processes accompanied by excitation exchange (such as spin exchange, F\"orster resonance, or angular momentum exchange) exhibit a near-threshold scaling ${E}^{\mathrm{\ensuremath{\Delta}}{m}_{12}}$, where $E$ is the collision energy, $\mathrm{\ensuremath{\Delta}}{m}_{12}={m}_{1}^{\ensuremath{'}}+{m}_{2}^{\ensuremath{'}}\ensuremath{-}{m}_{1}\ensuremath{-}{m}_{2}$, and ${m}_{i}$ and ${m}_{i}^{\ensuremath{'}}$ are the initial and final angular momentum projections of the colliding species ($i=1,2$). In particular, the inelastic cross sections for $\mathrm{\ensuremath{\Delta}}{m}_{12}=0$ transitions display an unconventional ${E}^{0}$ scaling similar to that of elastic cross sections, and their rates vanish as ${T}^{\mathrm{\ensuremath{\Delta}}{m}_{12}+1/2}$. For collisions dominated by even partial waves (such as those of identical bosons in the same internal state) the scaling is modified to ${\ensuremath{\sigma}}_{\text{inel}}\ensuremath{\propto}{E}^{\mathrm{\ensuremath{\Delta}}{m}_{12}+1}$ if $\mathrm{\ensuremath{\Delta}}{m}_{12}$ is odd. We present accurate quantum scattering calculations that illustrate these modified threshold laws for resonant spin exchange in ultracold $\text{Rb}+\text{Rb}$ and ${\mathrm{O}}_{2}+{\mathrm{O}}_{2}$ collisions. Our results illustrate that the well-known ${k}^{\ensuremath{-}1}$ threshold scaling of inelastic cross sections only applies to exothermic, rather than resonant, inelastic processes.

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