Atomic Clock Measurements of Quantum Scattering Phase Shifts Spanning Feshbach Resonances at Ultralow Fields

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 $1\mu \mathrm{K}$. Our low spread in collision energy yields phase variations of order $\pm \pi /2$ 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.

Figure 1

(a) Atoms in a coherent superposition of the cesium clock states collide with atoms in a target state. The clock atom wave packet scatters as a spherically outgoing $s$-wave (pink shell) and continues unscattered (violet cloud). The clock faces indicate the differential scattering phase shift of the clock coherence. (b) Transition probability for scattered (solid blue) and unscattered (dashed grey) clock atoms. Each data point represents a single fountain launch with target atoms in $|3,-3⟩$, a mean collision energy of $E_c=798\mathrm{nK}$, and $B=80\mathrm{mG}$. (c) After the first Ramsey pulse, the clock atoms prepared with dash-dot blue velocity distribution collide with the target atoms with the dotted purple distribution. The collisions redistribute the clock atom velocities (solid green). (d) Subtracting the initial clock velocity distribution from the distribution with scattering shows the net redistribution, which shifts the initial velocity class towards $v=0$. The mean collision energy can be tuned by changing the initial selected and final detected velocity.

Figure 2

(a)–(g) Magnetic field dependence of the differential phase shift ${\mathrm{\Phi }}_j$ for target atoms in the $|4,m_F\ne 0⟩$ and $|3,m_F\ne 0⟩$ states; negative $B$ corresponds to the opposite sign of $m_F$. The scattering phase shifts vary rapidly with magnetic field through a series of Feshbach resonances. The blue circles (red diamonds) are experimental results for mean collision energies of 616–656 nK (746–798 nK) and the curves are corresponding energy-averaged results from coupled-channel calculations on the best-fit potential [23]. (h) Comparison of measured (red diamonds) and theoretical values of ${\mathrm{\Phi }}_{3,\pm 3}$. All are shown after subtracting ${\mathrm{\Phi }}_{3,\pm 3}$ from the best-fit potential [23]. The best-fit potential and experimental results differ by $\approx 0.1\mathrm{rad}$ throughout the range and their variations through the Feshbach resonances agree very well. The previous best potential (solid red line) [5] gives much larger deviations from experiment through the resonances. The six dashed curves indicate the uncertainty of the best-fit potential. Their differences from the best-fit potential are small compared to the $\approx 0.1\mathrm{rad}$ offset of the experimental results.