Resonant energy transfer Rbns+Rbns+2hν→Rbnp1/2+Rbnp3/2 in a frozen Rydberg gas

We have observed the process $\mathrm{Rb}ns+\mathrm{Rb}\phantom{\rule{4pt}{0ex}}ns+2h\ensuremath{\nu}\ensuremath{\rightarrow}\mathrm{Rb}\phantom{\rule{4pt}{0ex}}n{p}_{1/2}+\mathrm{Rb}\phantom{\rule{4pt}{0ex}}n{p}_{3/2}$ from $n=34$ to $n=40$ in a frozen gas of Rb Rydberg atoms. It is resonant when the microwave frequency is halfway between the $ns\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}n{p}_{1/2}$ and $ns\ensuremath{\rightarrow}n{p}_{3/2}$ frequencies, which range from 57 to 106 GHz. The process cannot occur in isolated atoms, nor can it occur if the magnetic quantum numbers are unchanged, an implicit assumption of one-dimensional models. A Floquet-Forster model shows that the coupling between the initial and final states involves the absorption of two microwave photons and the dipole-dipole interaction, which leads to a coupling proportional to the product of the density, the microwave field squared, and ${n}^{*14}$, where ${n}^{*}$ is the effective quantum number of the $n{p}_{3/2}$ state. We have experimentally verified these dependences. The observed resonances are asymmetric, with a low-frequency tail, which we attribute to the van der Waals shift of the final $n{p}_{1/2}n{p}_{3/2}$ state due to its dipole-dipole interaction with the nearby $ns(n+1)s$ state. While the van der Waals shift is negligible for most of the atoms in the Rydberg gas, it is not for the pairs of close atoms which undergo this transition.