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

We develop a theoretical understanding of the trapping of 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 wave functions. The 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 the laser intensity profile and on the rotational symmetry of the total electronic wave function. We focus on the $J=0$ Rydberg states of Sr and evaluate the dynamic polarizabilities of the $5sns$(${}^1{S}_0$) and $5snp$(${}^3{P}_0$) Rydberg states. We specifically chose the 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 [$5s^2$(${}^1{S}_0$) and $5s5p$(${}^3{P}_0$)] and the $J=0$ Rydberg states [$5sns$(${}^1{S}_0$) and $5snp$(${}^3{P}_0$)] 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 $n$.

Figure 1

Polarizability of the Sr${}^{+}$ ion in the 5$s$ ground state (solid orange), and the landscaping polarizabilities ${\alpha }_{ns}^{\mathrm{lsc},J=0}\left(\lambda \right)$ for the Rydberg electron in the $110s$ (dashed maroon) and $130s$ (dot-dashed blue) states of the Sr atom.

Figure 2

Dynamic polarizabilities for the 5$s$${}^2$(${}^1{S}_0$) (dashed red) ground and the 5$s$5$p$(${}^3{P}_0$) excited clock states (solid purple) of Sr. The magic wavelength where the two polarizabilities are equal is marked by an open circle at 814 nm.

Figure 3

Total polarizabilities for the 5$sns$(${}^1{S}_0$) (upper panel) and 5$snp$(${}^3{P}_0$) (lower panel) Rydberg states of Sr for $n=50$ (dashed brown), 100 (orange), 160 (green), and 180 (blue) plotted with the ground-state 5$s$${}^2$(${}^1{S}_0$) polarizability (solid red). The similarity between the polarizabilities of the 5$sns$(${}^1{S}_0$) and 5$snp$(${}^3{P}_0$) Rydberg states stems from the restriction to the $J=0$ term in the expansion of the ${\left(A_{VG}\right)}^2$ operator, which is dictated by the total $J$ of the Rydberg state.

Figure 4

Total polarizabilities for the $5sns$(${}^1{S}_0$) (upper panel) and $5snp$(${}^3{P}_0$) (lower panel) Rydberg states of Sr for $n=50$ (dashed brown), 100 (orange), 160 (green), and 180 (blue) plotted with the upper clock state $5s5p$(${}^3{P}_0$) (solid purple) polarizability. Two special points at which the $5s5p$(${}^3{P}_0$) state polarizability matches those of the Rydberg states in the high-$n$ limit are marked by open circles. These “universal” magic wavelengths are at 596 and 1362 nm.