LO-Free Phase and Amplitude Recovery of an RF Signal with a DC-Stark-Enabled Rydberg Receiver

We present a theoretical framework for recovering the amplitude and carrier phase of a single received RF field with a Rydberg-atom receiver, without injecting an RF local oscillator (LO) into the atoms. The key enabling mechanism is a static DC bias applied to the vapor cell: by Stark-mixing a near-degenerate Rydberg pair, the bias activates an otherwise absent upper optical pathway and closes a phase-sensitive loop within a receiver driven only by the standard probe/coupling pair and the received RF field. For a spatially uniform bias, we derive an effective four-level rotating-frame Hamiltonian of Floquet form and show that the periodic steady state obeys an exact harmonic phase law, so that the $n$th probe harmonic carries the factor $e^{inΦ_S}$. This yields direct estimators for the signal phase and amplitude from a demodulated probe harmonic, with amplitude recovery obtained by inverting an injective harmonic response map. In the high-SNR regime, we derive explicit RMSE laws and use them to identify distinct phase-optimal and amplitude-optimal bias-controlled mixing angles, together with a weighted joint-design criterion and a balanced compromise angle that equalizes the fractional phase and amplitude penalties. We then extend the analysis to nonuniform DC bias through quasistatic spatial averaging and show that bias inhomogeneity reduces coherent gain for phase readout while also reshaping the amplitude-response slope. Numerical examples validate the phase law, illustrate response-map inversion and mixing-angle trade-offs, and quantify the penalties induced by bias nonuniformity. The results establish a minimal route to coherent Rydberg reception of a single RF signal without an auxiliary RF LO in the atoms.