Differential nuclear-spin-dependent light shifts and state mixing of Rydberg atoms

In this paper we present a detailed analysis of the nuclear-spin manifolds associated with the $ns$ Rydberg levels of $^{87}\mathrm{Rb}$ atoms that interact with both magnetic and optical lattice fields. Eigenvalues and eigenkets for the Rydberg manifold are obtained and used to study the dynamics of phase-matched emission following illumination of an ensemble of cold atoms with excitation and readout laser pulses. By comparing the measured emission signal to predictions of a model that accounts for the quantized motion of atoms in a one-dimensional optical lattice potential, we are able to extract the Rydberg hyperfine and light shift contributions to the observed modulation frequencies. In this way the hyperfine splitting of Rydberg $ns$ levels is measured for $n$ in the range of 30 to 65. Our results should be relevant for realizations of high-fidelity Rydberg qubits confined in optical potentials.