Spin waves and Collisional Frequency Shifts of a Trapped-Atom Clock

We excite spin-waves with spatially inhomogeneous pulses and study the resulting frequency shifts of a chip-scale atomic clock of trapped $^{87}$Rb. The density-dependent frequency shifts of the hyperfine transition simulate the s-wave collisional frequency shifts of fermions, including those of optical lattice clocks. As the spin polarizations oscillate in the trap, the frequency shift reverses and it depends on the area of the second Ramsey pulse, exhibiting a predicted beyond mean-field frequency shift. Numerical and analytic models illustrate the observed behaviors.