We discuss the novel frequency shift due to inhomogeneous excitations, particularly in the context of optical frequency
lattice clocks. We analyze in detail the frequency shifts due to scattering-induced tunneling. These shifts have terms that
are linear and non-linear in the atomic density. We show that they are small at moderate to high lattice depths.
Kurt Gibble, Kurt Gibble,
"Frequency shifts of colliding fermions in optical lattice clocks", Proc. SPIE 8132, Time and Frequency Metrology III, 813209 (12 September 2011); doi: 10.1117/12.893949; https://doi.org/10.1117/12.893949