Strategies for reducing the light shift in atomic clocks

Recent progress in optical lattice clocks requires unprecedented precision in controlling systematic uncertainties at the ${10}^{-18}$ level. Tuning of nonlinear light shifts is shown to reduce lattice-induced clock shift for a wide range of lattice intensity. Based on theoretical multipolar, nonlinear, anharmonic, and higher-order light shifts, we numerically demonstrate possible strategies for Sr, Yb, and Hg clocks to achieve lattice-induced systematic uncertainty below $1\times {10}^{-18}$.

Figure 1

Clock transitions in an optical lattice potential created by counterpropagating laser beams. An atom trapped in $U_{\mathrm{g}}\left(x\right)$ with $\left|x\right|\ll {\lambda }_{\mathrm{m}}/4$ is excited on the ${}^1{S}_0\left(\mathrm{g}\right)\to {}^3{P}_0\left(\mathrm{e}\right)$ clock transition with unperturbed frequency ${\nu }_0$ and lattice-induced clock shift $\Delta {\nu }_{\mathrm{c}}$. Actual laser beams show transverse intensity distribution $I{e}^{-2(y^2+z^2)/r_0^2}$ with beam radius $r_0\gg \lambda$, which weakly confines atoms transversely.

Figure 2

The wavelength-dependent hyperpolarizability for Yb for linear ($\Delta {\beta }^l$, dashed) and circular ($\Delta {\beta }^c$, solid) polarized lattices near the magic wavelength ${\lambda }_{\mathrm{m}}$. The vertical dashed lines indicate two-photon resonances.

Figure 3

Light shift for a Hg clock as a function of laser intensity $I$. The black solid and blue dashed lines correspond to linear-polarized and “magic-elliptical” light with detuning $\delta \nu =0$. With $\delta \nu =-4.66\mathrm{MHz}$ and ${\xi }^{\mathrm{Hg}}=0.75$, the light shift (red solid line) becomes less than $\pm 1$ mHz (green) for the gray shaded region. The red dashed lines indicate the tolerance $\left(0.5\%\right)$ for ${\xi }^{\mathrm{Hg}}$. For linear polarized light $\left(\xi =0\right)$ with $\delta \nu =-2\mathrm{MHz}$, the light shift becomes insensitive to $\Delta I$ around $I\sim 36\mathrm{kW}/{\mathrm{cm}}^2$ (dotted line).

Figure 4

Contour plots of light shifts for (a) Yb and (b) Sr clock transitions as functions of lattice-laser intensity $I$ and detuning $\delta \nu$, for ${\xi }^{\mathrm{Yb}}=0.75$ and ${\xi }^{\mathrm{Sr}}=0$. The red dotted lines show zero light shift. The region bound by red lines corresponds to light shift $|\Delta {\nu }_{\mathrm{c}}|/{\nu }_0\le 1\times {10}^{-18}$. Tuning to the operational magic frequency $\delta \nu$ as indicated by the white dotted lines, a wide range of operational intensities are allowed.