Higher-order effects on the precision of clocks of neutral atoms in optical lattices

The recent progress in designing optical lattice clocks with fractional uncertainties below ${10}^{-17}$ requires unprecedented precision in estimating the role of higher-order effects of atom-lattice interactions. In this paper, we present results of systematic theoretical evaluations of the multipole, nonlinear, and anharmonic effects on the optical-lattice-based clocks of alkaline-earth-like atoms. Modifications of the model-potential approach are introduced to minimize discrepancies of theoretical evaluations from the most reliable experimental data. Dipole polarizabilities, hyperpolarizabilities, and multipolar polarizabilities for neutral Ca, Sr, Yb, Zn, Cd, and Hg atoms are calculated in the modified approach.

Figure 1

The wavelength dependence of the hyperpolarizability of the clock transition in Sr atoms for linear (solid curves) and circular (dashed curves) polarization of the lattice-laser wave. The vertical dash-dotted lines indicate positions of two-photon resonances on the $5s7p\left({}^{3}P_2\right)$ state at ${\lambda }_1=795.5\mathrm{nm}$, on the $5s7p\left({}^{3}P_0\right)$ state at ${\lambda }_2=797\mathrm{nm}$ (this resonance does not appear for circular polarization), and on the $5s4f\left({}^{3}F_2\right)$ state at ${\lambda }_3=818.6\mathrm{nm}$. The solid vertical line indicates the MWL at 813.43 nm, where the corresponding hyperpolarizabilities are $\mathrm{\Delta }{\beta }^l=-200$ and $\mathrm{\Delta }{\beta }^c=-311\mu \mathrm{Hz}/{\left(\mathrm{kW}/\mathrm{c}{\mathrm{m}}^2\right)}^2$.

Figure 2

Real (thick curves) and imaginary (thin, almost horizontal lines) parts of hyperpolarizabilities $\mathrm{\Delta }\beta$ in Cd atoms, as functions of the wavelength $\lambda$ of the circularly (dashed) and linearly (solid curves) polarized lattice laser, in the region between two-photon resonances on the $5s6s\left({}^{1}S_0\right)$ single-electron excited state at 375.163 nm for the ground-state hyperpolarizability and single-photon resonance on the $5s6s\left({}^{3}S_1\right)$ state at 467.95 nm for the hyperpolarizability of the excited clock state $5s5p\left({}^{3}P_0\right)$. Corresponding values at the magic wavelength of ${\lambda }_m=414.4\mathrm{nm}$ (the dash-dotted vertical line) are $\mathrm{\Delta }{\beta }^l=-5.47+2.02i$ and $\mathrm{\Delta }{\beta }^c=19.5+3.01i\mu \mathrm{Hz}/{\left(\mathrm{kW}/\mathrm{c}{\mathrm{m}}^2\right)}^2$. The imaginary parts of the hyperpolarizabilities determine the rate of the two-photon ionization of the upper clock state.

Figure 3

Real (thick) and imaginary (thin, almost horizontal lines) parts of differences of the clock-state hyperpolarizabilities in Hg atoms, as functions of the wavelength of the lattice-laser circularly (dashed) and linearly (solid lines) polarized radiation, in the region between two-photon resonance on the singlet single-electron excited state $6s7s\left({}^{1}S_0\right)$ at 312.851 nm for the ground-state hyperpolarizability and single-photon resonance on the triplet state $6s7s\left({}^{3}S_1\right)$ at 404.770 nm for the hyperpolarizability of the excited clock state $6s6p\left({}^{3}P_0\right)$. Corresponding values at the magic wavelength of ${\lambda }_m=362.57\mathrm{nm}$ (marked by the dashed vertical line) are $\mathrm{\Delta }{\beta }^l=-2.67+0.82i$ and $\mathrm{\Delta }{\beta }^c=0.94+1.21i\mu \mathrm{Hz}/{\left(\mathrm{kW}/\mathrm{c}{\mathrm{m}}^2\right)}^2$.

Figure 4

The dependence of the real (a) and imaginary (b) parts of the clock-frequency shift of Cd atoms $\mathrm{\Delta }{\nu }_{\mathrm{cl}}^t(n,\xi ,I)$, trapped in their ground state of vibrations $n=0$, on the intensity $I$ (in $\mathrm{kW}/\mathrm{c}{\mathrm{m}}^2$) of the linear (red solid curves), “magic elliptical” (blue dotted), and circular polarized (black dashed) lattice wave of a “traveling-wave magic frequency.” Imaginary parts of $\mathrm{\Delta }{\nu }_{\mathrm{cl}}(n,\xi ,I)$ determining the clock-transition line broadening are negative values, identical for all strategies of determining the MWL and about one order smaller in magnitude than the real parts [cf. the vertical scales of (a) and (b)].

Figure 5

The dependence of the clock-frequency shift on the intensity $I$ (in $\mathrm{kW}/\mathrm{c}{\mathrm{m}}^2$) of the linearly ($\xi =0$, solid curves), “magic elliptically ($\xi ={\xi }_m$, dotted), and circularly ($\xi =\pm 1$, dashed curves) polarized laser wave (a) at the “motion-insensitive magic frequency” $\mathrm{\Delta }{\nu }_{\mathrm{cl}}^s(n,\xi ,I)$ and (b) at the “$E1$ magic frequency” $\mathrm{\Delta }{\nu }_{\mathrm{cl}}^{E1}(n,\xi ,I)$ (in mHz) in Cd atoms, trapped to their ground-state vibrations ($n=0$). The imaginary parts of $\mathrm{\Delta }{\nu }_{cl}^{s,E1}(n,\xi ,I)$ are given in Fig. 4.

Figure 6

The dependence of the clock-frequency shift $\mathrm{\Delta }{\nu }_{cl}^{E1}(n,\xi ,I)$ (in mHz) in Cd atoms, trapped to their ground-state vibrations ($n=0$), on the intensity $I$ (in $\mathrm{kW}/\mathrm{c}{\mathrm{m}}^2$) of the “magic elliptical” polarized laser wave of operational MWL, red-shifted from the frequency ${\omega }_m^{E1}$ by (a) $\delta =-900$ (lowest solid curve), $\delta =-905$ (middle dotted curve), and $\delta =-910$ kHz (top dash-dotted curve); (b) $\delta =-790$ (lowest solid curve), $\delta =-800$ (middle dotted curve), and $\delta =-810$ kHz (top dash-dotted curve).

Figure 7

Model-potential data for the wavelength (in nanometers) dependence of the lattice-potential depths (in kilohertz at the lattice-laser intensity $I=10\mathrm{kW}/\mathrm{c}{\mathrm{m}}^2$): (a) for Ca atoms in their upper $5s5p^3{P}_0$ (dashed) and lower $5s^2{}^{1}S_0$ (bold curve) clock states; the curves intersect at the MWL ${\lambda }_m^{\mathrm{Ca}}=747\mathrm{nm}$ in a satisfactory agreement with ${\lambda }_m^{\mathrm{Ca}}=735.5\pm 20\mathrm{nm}$ of Ref. [17]; (b) for Cd atoms the curves intersect at the MWL ${\lambda }_m^{\mathrm{Cd}}=414.4\mathrm{nm}$.

Figure 8

Calculated data for the wavelength (in nanometers) dependence of the lattice-potential depths (in kiloHertz at the lattice-laser intensity $I=50\mathrm{kW}/\mathrm{c}{\mathrm{m}}^2$): (a) for Zn atoms in their upper $5s5p^3{P}_0$ (dashed) and lower $5s^2{}^{1}S_0$ (bold curve) clock states; the curves intersect at the MWL ${\lambda }_m^{\mathrm{Zn}}=406.5\mathrm{nm}$; (b) for Hg atoms the curves intersect at ${\lambda }_{\mathrm{theor}}^{\mathrm{Hg}}=364\mathrm{nm}$, rather close to the MWL ${\lambda }_m^{\mathrm{Hg}}=362.57\mathrm{nm}$, determined experimentally in [3, 4].