We realize and model a Rydberg-state atom interferometer for measurement of phase and intensity of radio-frequency (RF) electromagnetic waves. A phase reference is supplied to the atoms via a modulated laser beam, enabling atomic measurement of the RF waves phase without an external RF reference wave. The RF and optical fields give rise to closed interferometric loops within the atoms internal Hilbert space. In our experiment, we construct interferometric loops in the state space ${ 6P_{3/2}, 90S_{1/2}, 91S_{1/2}, 90P_{3/2} }$ of cesium and employ them to measure phase and intensity of a 5 GHz RF wave in a room-temperature vapor cell. Electromagnetically induced transparency on the $6S_{1/2}$ to $6P_{3/2}$ transition serves as an all-optical interferometer probe. The RF phase is measured over a range of $pi$, and a sensitivity of 2 mrad is achieved. RF phase and amplitude measurements at sub-millimeter optical spatial resolution are demonstrated.
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