  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 wave's 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.
