Magic polarization for cancellation of light shifts in two-photon optical clocks

We find a simple solution to the problem of probe laser light shifts in two-photon optical atomic clocks. We show that there exists a magic polarization at which the light shifts of the two atomic states involved in the clock transition are identical. We calculate the differential polarizability as a function of laser polarization for two-photon optical clocks based on neutral calcium and strontium, estimate the magic polarization angle for these clocks, and determine the extent to which probe laser light shifts can be suppressed. We show that the light shift and the two-photon excitation rate can be independently controlled using the probe laser polarization.

Figure 1

Level structures of (a) ${}^{40}\mathrm{Ca}$ and (b) ${}^{88}\mathrm{Sr}$. The main cooling line for calcium (strontium) atoms is the ${}^{1}S_0\text{−}{}^{1}P_1$ cycling transition at 423 nm (461 nm), and the ${}^{1}S_0\text{−}{}^{1}D_2$ clock transition involves two photons at 915 nm (993 nm).

Figure 2

Dynamic polarizabilities of (a) strontium and (b) calcium two-photon clock states at the magic polarization angle ${\theta }_0$. The vertical dotted gray line indicates the frequency of the clock transition, at which the polarizabilities of the ground and excited states can be made equal.

Figure 3

Light shifts for calcium and strontium due to the probe laser at two-photon resonance, as a function of linear polarization angle. $\theta =0^{\circ }$ denotes $z$-polarized light, $\theta ={90}^{\circ }$ denotes $x$-polarized light.

Figure 4

Differential polarizability between the clock states in atomic units for (a) calcium and (b) strontium, plotted on a Poincaré sphere. On the sphere, the polarizability is cylindrically symmetric around the axis connecting $x$ polarization to $z$ polarization. The magic polarization angle corresponds to the band with zero differential polarizability.

Figure 5

Two-photon Rabi frequencies for strontium and calcium clock transitions as a function of probe laser polarization angle. Rabi frequencies are shown for three values of the angle $\varphi$, the phase difference between $\widehat{x}$ and $\widehat{z}$ components which describes the degree of circular polarization. The plot shows ${\left|\mathrm{\Omega }\right|}^2$ normalized to its value at $\theta =0$. The value of ${\mathrm{\Omega }}_0=4.85\left(22.1\right)\mathrm{Hz}/(\mathrm{W}/{\mathrm{cm}}^2)$ for calcium (strontium). The magic polarizations for calcium and strontium are indicated with a dashed vertical line.