A method for performing a precision measurement of the Rydberg constant, R∞, using cold circular Rydberg atoms is proposed. These states have long lifetimes, as well as negligible quantumelectrodynamics (QED) and no nuclear-overlap corrections. Due to these advantages, the measurement can help solve the "proton radius puzzle" [Bernauer, Pohl, Sci. Am. 310, 32 (2014)]. The atoms are trapped using a Rydberg-atom optical lattice, and transitions are driven using a recentlydemonstrated lattice-modulation technique to perform Doppler-free spectroscopy. The circular-state transition frequency yields R∞. Laser wavelengths and beam geometries are selected such that the lattice-induced transition shift is minimized. The selected transitions have no first-order Zeeman and Stark corrections, leaving only manageable second-order Zeeman and Stark shifts. For Rb, the projected relative uncertainty of R∞ in a measurement under the presence of the Earth's gravity is 10 −11 , with the main contribution coming from the residual lattice shift. This could be reduced in a future micro-gravity implementation. The next-important systematic arises from the Rb + polarizability (relative-uncertainty contribution of ≈ 3 × 10 −12 ).

We present measurements of the hyperfine structure splittings of nS 1/2 Rydberg states of 85 Rb for n = 43, 44, 45 and 46. From the splittings, the hyperfine coupling constant, AHFS, is determined to be 15.372 (80) GHz. This result is an order-of-magnitude improvement from previous measurements. We study and account for systematic uncertainty sources, such as unwanted electric and magnetic fields, dipolar Rydberg-Rydberg interactions, and AC shifts. Initial evidence for hyperfine-mixed Rydberg pair states is found.

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