Rydberg-atom-based radio-frequency sensors: amplitude-regime sensing

Rydberg atom-based radio frequency electromagnetic field sensors are drawing wide-spread interest because of their unique properties, such as small size, dielectric construction, and self-calibration. These photonic sensors use lasers to prepare atoms and read out the atomic response to a radio frequency electromagnetic field based on electromagnetically induced transparency, or related phenomena. Much of the theoretical work has focused on the Autler-Townes splitting induced by the radio frequency wave. The amplitude regime, where the change in transmission observed on resonance is measured to determine electric field strength, has received less attention. In this paper, we deliver analytic expressions that are useful for calculating the absorption coefficient in the amplitude regime. Our main goal is to describe the analytic expressions for the absorption coefficient and demonstrate their validity over a large range of the interesting parameter space. The effect of the thermal motion of the atoms is explicitly addressed. The analytic formulas for the absorption coefficient for different types of Doppler broadening are compared to estimate the sensitivity under conditions where it is limited by the laser shot noise. Residual Doppler shifts are shown to limit sensitivity. The expressions, approximations and descriptions presented in the paper are important for understanding the absorption of Rydberg atom-based sensors in the amplitude regime. This provides insight into the physics of multi-level interference phenomena.