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

Vladislav Katkov, Nikola Zlatanov

Abstract

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.

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

Abstract

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 th probe harmonic carries the factor . 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.

Paper Structure

This paper contains 59 sections, 1 theorem, 159 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Reduced receiver model and estimation target
  3. Fields, basis states, and conventions
  4. DC-Stark-induced effective couplings
  5. Uniform-bias benchmark and estimation target
  6. Rotating-frame Hamiltonian under uniform bias
  7. DC Stark mixing of the upper bare pair
  8. Effective Hamiltonian in the co-rotating frame
  9. Validity and scope of the reduced Hamiltonian
  10. Periodic steady state and exact harmonic phase law
  11. Driven open-system model
  12. Exact harmonic phase law
  13. Harmonic-balance equations
  14. Optical readout and recovery of signal phase and amplitude
  15. Observable and calibrated harmonic readout
  16. ...and 44 more sections

Key Result

Proposition 1

Assume that: (i) the Hamiltonian depends on the unknown signal phase only through the factors $e^{\pm i(\omega_S t+\Phi_S)}$ in (eq:Heff_main), (ii) the dissipator $\mathcal{D}[\rho]$ in (eq:master_eq) is time independent, and (iii) the driven system admits a unique periodic steady state for each $\ for every integer $n$.

Figures (9)

  • Figure 1: Conceptual mechanism of the proposed receiver. Left: without a DC bias, the standard ladder-EIT configuration plus the received RF coupling does not supply the additional upper optical pathway needed for a phase-sensitive loop. Right: a static DC bias Stark-mixes the upper pair, activates a bias-enabled optical leg, and closes a loop within a receiver driven only by the probe, coupling, and received RF fields. The resulting periodic probe response carries the phase of the received signal in its harmonics.
  • Figure 2: Reduced receiver model in the Stark basis. A DC bias mixes the upper bare pair so that the coupling tone drives both $\ket{2}\leftrightarrow\ket{3^{(S)}}$ and the bias-enabled leg $\ket{2}\leftrightarrow\ket{4^{(S)}}$, while the received RF field couples the Stark states. In the minimal constructive model, $\Omega_{23}=\Omega_c\cos\theta$, $\Omega_{24}=\Omega_c\sin\theta$, and $\Omega_{34}=\Omega_S\cos(2\theta)$.
  • Figure 3: Numerical confirmation of the exact first-harmonic phase law at the balanced nominal operating point. The DC-Stark-enabled loop generates a nonzero first harmonic, and varying $\Phi_S$ rotates that harmonic rigidly in the complex plane. The residual plot shows that the reference-normalized first-harmonic phase follows the received-signal phase with numerically negligible deviation.
  • Figure 4: First-harmonic response map and local logarithmic sensitivity at the nominal operating point. Panel (a) shows the injective branch of $m(\Omega_S)=|P_{21}^{(1)}(\Omega_S)|$ together with the chosen design level $\Omega_{S,0}=0.12$. Panel (b) shows $s(\Omega_S)=d\ln m/d\ln\Omega_S$ on the same branch. Amplitude recovery requires both injectivity of the response map and a sufficiently large local slope.
  • Figure 5: Uniform-bias design landscape at the nominal design level $\Omega_{S,0}$. Panel (a) compares the full first-harmonic phase metric $M_{\phi}(\theta)=m(\Omega_{S,0},\theta)$ with the scaled perturbative proxy $f(\theta)$, showing that the full-model phase optimum is shifted away from the perturbative seed. Panel (b) overlays the normalized phase metric $M_{\phi}$, amplitude metric $M_A$, and a representative equal-weight joint utility derived from $J(\theta)$. The crossing of the normalized phase and amplitude curves indicates the balanced operating point $\theta_{\mathrm{bal}}^{\star}$, which lies between the separate phase- and amplitude-optimal angles.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Proposition 1: Exact harmonic phase law
  • Remark 1