High-precision measurements of the $^{87}$Rb $D$-line tune-out wavelength

We report a measurement of a light wavelength at which the ac electric polarizability equals zero for $^{87}$Rb atoms in the $F=2$ ground hyperfine state. The experiment uses a condensate interferometer to find this “tune-out” wavelength for the scalar polarizability, which lies at 790.032388(32) nm. Our result can be used to determine the ratio of matrix elements $| \langle 5P_{3/2} \| d \| 5S_{1/2} \rangle / \langle 5P_{1/2} \| d \| 5S_{1/2} \rangle |^2 = 1.99221(3)$, a 100-fold improvement over previous experimental values. We discuss techniques for accurate determination and control of light polarization as well as progress on measurements of the vector polarizability between the $D1$ and $D2$ spectral lines. Measurements of tune-out wavelengths and the vector polarizability between multiple lines allows separation of individual contributions to the polarizability from higher-lying states and the core up to ratios of matrix elements. Accurate knowledge of these ratios should serve useful as a theoretical benchmark and in atomic parity violation experiments.