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Divalent Rydberg atoms in optical lattices: intensity landscape and magic trapping

We develop a theoretical understanding of trapping divalent Rydberg atoms in optical lattices. Because the size of the Rydberg electron cloud can be comparable to the scale of spatial variations of laser intensity, we pay special attention to averaging optical fields over the atomic wavefunctions. Optical potential is proportional to the ac Stark polarizability. We find that in the independent particle approximation for the valence electrons, this polarizability breaks into two contributions: the singly ionized core polarizability and the contribution from the Rydberg electron. Unlike the usually employed free electron polarizability, the Rydberg contribution depends both on laser intensity profile and the rotational symmetry of the total electronic wavefunction. We focus on the $J=0$ Rydberg states of Sr and evaluate the dynamic polarizabilities of the 5s$n$s($^1S_0$) and 5s$n$p($^3P_0$) Rydberg states. We specifically choose Sr atom for its optical lattice clock applications. We find that there are several magic wavelengths in the infrared region of the spectrum at which the differential Stark shift between the clock states (5s$^2$($^1S_0$) and 5s5p($^3P_0$)) and the $J=0$ Rydberg states, 5s$n$s($^1S_0$) and 5s$n$p($^3P_0$), vanishes. We tabulate these wavelengths as a function of the principal quantum number $n$ of the Rydberg electron. We find that because the contribution to the total polarizability from the Rydberg electron vanishes at short wavelengths, magic wavelengths below $\sim$1000 nm are ``universal" as they do not depend on the principal quantum number $n$.