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.
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