Clock Spectroscopy of Interacting Fermions in a Harmonic Trap

We investigate the dynamics during Ramsey interrogation of interacting fermions in a harmonic trap. We consider the effect of both s-wave and p-wave collisions during the dynamics, including processes that change the spatial modes of particles. Prior theoretical treatments utilize the so-called spin model\footnote{K. Gibble, Phys. Rev. Lett. 103, 113202 (2009)}$^,$\footnote{A.M. Rey, A.V. Gorshov, C. Rubbo, Phys. Rev. Lett. 109, 260402 (2009)}$^,$\footnote{Z. Yu and C. J. Pethick, Phys. Rev. Lett. 104, 010801 (2010)} which includes processes that change the internal states of atoms, but leave the vibrational modes unchanged. We discuss how the inclusion of these mode-changing processes modifies the predicted density-dependent frequency shifts and contrast of Ramsey fringes, both of which are relevant for the precision of optical lattice clocks using fermionic alkaline-earth atoms. We also discuss how the frequency shifts and contrast depend on the pulse areas, interaction strength, and temperature - and how these dependences are affected by the inclusion of mode-changing collisions in the calculations.