We have developed a potential-energy surface for spin-polarized $\mathrm{K}(^{2}S)+{\mathrm{K}}_{2}(^{3}\ensuremath{\Sigma}_{u}^{+})$ collisions and carried out quantum dynamical calculations of vibrational quenching at low and ultralow collision energies for both bosons
39K and
41K and fermions
40K. At collision energies above about
0.1mK the quenching rates are well described by a classical Langevin model, but at lower energies a fully quantal treatment is essential. We find that for the low initial vibrational state considered here
(v=1), the ultracold quenching rates are not substantially suppressed for fermionic atoms. For both bosons and fermions, vibrational quenching is much faster than elastic scattering in the ultralow-temperature regime. This contrasts with the situation found experimentally for molecules formed via Feshbach resonances in very high vibrational states.
We use an atomic fountain clock to measure quantum scattering phase shifts precisely through a series of narrow, low-field Feshbach resonances at average collision energies below 1 μK. Our low spread in collision energy yields phase variations of order ±π/2 for target atoms in several F, m_{F} states. We compare them to a theoretical model and establish the accuracy of the measurements and the theoretical uncertainties from the fitted potential. We find overall excellent agreement, with small statistically significant differences that remain unexplained.