  The interaction between light and an atom causes perturbations in the atom's
energy levels, known as the light-shift. These light-shifts are a key source of
inaccuracy in atomic clocks, and can also deteriorate their precision. We
present a study of light-shifts and associated dynamic polarizabilities for a
two-photon atomic clock based on the $5S_{1/2}$-$5D_{5/2}$ transition in
rubidium-87 over the range 770 nm to 800 nm. We determine experimental and
theoretical values for a magic wavelength in this range and the electric dipole
(E1) matrix element for the $5P_{3/2}$-$5D_{5/2}$ transition. We find a magic
wavelength of 776.179(5) nm (experimental) and 776.21 nm (theoretical) in the
vicinity of the $5P_{3/2}$-$5D_{5/2}$ resonance, and the corresponding reduced
E1 matrix element 1.80(6) $ea_0$ (experimental) and 1.96(15) $ea_0$
(theoretical). These values resolve a previous discrepancy between theory and
experiment.
