Hyperpolarizability Effects in a Sr Optical Lattice Clock

We report the observation of a higher-order frequency shift due to the trapping field in a ${}^{87}\mathrm{Sr}$ optical lattice clock. We show that, at the magic wavelength of the lattice, where the first-order term cancels, the higher-order shift will not constitute a limitation to the fractional accuracy of the clock at a level of ${10}^{-18}$. This result is achieved by operating the clock at very high trapping intensity up to $400\mathrm{kW}/{\mathrm{cm}}^2$ and by a specific study of the effect of the two two-photon transitions near the magic wavelength.

Figure 1

Energy levels of Sr relevant to this Letter. The clock transition at 698 nm couples the two lowest energy states of the atom.

Figure 2

Motional spectrum of the atoms in the optical lattice. Each point corresponds to a single measurement of 1 s total duration. Black curve: Optimal operating conditions (probe pulse of 10 ms duration and $2\mu \mathrm{W}$ power). The inset is a zoom on the carrier. Grey curve: The first-order longitudinal motional sidebands are enhanced by applying long probe pulses of 200 ms at the maximum available power at 698 nm (2 mW).

Figure 3

(a) Typical experimental data used to derive the first- and second-order frequency shift coefficients plotted in (b) and (c). Shown here is the frequency shift of the clock transition vs lattice depth for 5 different lattice wavelengths: 813.406 (four lowest points in the graph), 813.413, 813.428, 813.436, and 813.444 nm (four highest points). Also shown is a linear fit of each of these five sets of data. (b) First-order frequency shift vs ${\lambda }_L$ scaled back to $U_0=E_r$. Also plotted is a linear fit of these data (${\chi }^2=1.1$). The inset shows measurements of ${\lambda }_m$. ■: this work. ▲: Ref. 7. ●: Ref. 9. (c) Higher-order frequency shift vs ${\lambda }_L$ scaled back to $U_0=E_r$. All these points are compatible with zero. Their average is $-4(36)\times {10}^{-7}\mathrm{Hz}$ (${\chi }^2=1.02$). The arrow on the ${\lambda }_L$ axis corresponds to the ${}^{3}P_0\to {}^{1}P_1$ transition at 813.360 nm. The vertical dotted line is at the magic wavelength 813.428 nm.

Figure 4

(a) Higher-order frequency shift around the ${}^{3}P_0\to {}^{3}F_2$ transition at 818.570 nm scaled back to $U_0=E_r$. The five vertical arrows on the wavelength axis correspond to the hyperfine substates of $5s4f{}^{3}F_2$ ($F=13/2$ to $F=5/2$ from left to right). (b) Atomic frequency shift vs trapping depth for two lattice wavelengths on both sides of the two-photon transition to substate $F=13/2$. The linear light shift has been removed for clarity. The bold line is a fit of the data with a parabola.

Figure 5

Higher-order frequency shift at the magic wavelength scaled back to $U_0=E_r$. The value of the open square is the weighted average of the seven other points (${\chi }^2=1.8$).