Hyperpolarizability and operational magic wavelength in an optical lattice clock

One of the largest systematic frequency shifts in optical lattice clocks arises from light shifts due to the trapping lattice. At the 1x10\textasciicircum -18 level we find that light shift effects beyond the dominant electric dipole coupling (hyperpolarizability, magnetic dipole, and electric quadrupole) become relevant. Including finite temperature effects, we observe a simple linear $+$ quadratic scaling of clock frequency shift with trap depth. For any choice of trap depth, we may tune our trapping laser to a corresponding ``operational magic wavelength'' where the clock shift is insensitive to uncontrolled changes in trap depth. We further explore atomic temperature-dependent effects by implementing a third stage of quenched sideband laser cooling on the ultra-narrow $^{\mathrm{1}}$S$_{\mathrm{0}}$ to $^{\mathrm{3}}$P$_{\mathrm{0}}$ clock transition. This cooling allows us to achieve near unit occupation of the ground band of our 1-D optical lattice (corresponding to temperatures in the 100's of nK) within a few 10's of ms. This cooling implies increased atomic confinement, which we use as a lever arm to explore previously unobservable magnetic dipole and electric quadrupole effects.