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 $n$th probe harmonic carries the factor $e<sup>{inΦ_S}$. 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.

• The paper demonstrates LO-free recovery of an RF signal’s phase and amplitude by using a static DC field to Stark-mix a near-degenerate Rydberg pair and close a phase-sensitive optical loop.
• The exact harmonic phase law makes the first demodulated probe harmonic rotate directly with the RF phase, while amplitude estimation requires an injective response map and follows RMSE laws set by harmonic SNR and local sensitivity.
• The analysis shows that phase and amplitude have different optimal bias angles—approximately 0.49 and 0.60 radians, with a balanced value near 0.56 radians—and that bias nonuniformity sharply reduces coherent gain and phase performance.

Overview and problem statement

Rydberg-atom receivers based on ladder electromagnetically induced transparency (EIT) are well established for RF field measurement, but recovering the carrier phase of a single received RF tone has historically required an auxiliary reference: either an RF local oscillator (LO) injected into the atoms (as in atomic-mixer, superheterodyne, and phase-modulation schemes) or additional time-varying optical/RF reference structure. The paper by Katkov and Zlatanov proposes a third route: a static DC bias field that Stark-mixes a near-degenerate upper Rydberg pair, activating an otherwise absent optical pathway and closing a phase-sensitive loop within a receiver driven only by the probe field, the coupling field, and the received signal itself. "LO-free" here means no auxiliary RF LO inside the vapor cell; ordinary synchronous demodulation outside the cell is still assumed.

The received signal is $A_S\cos(\omega_S t+\Phi_S)$ with unknown $A_S$ and $\Phi_S$ at a known or separately tracked carrier $\omega_S$. The paper's contributions are: (i) the DC-Stark loop-closure mechanism; (ii) an exact harmonic phase law for the periodic steady state under uniform bias; (iii) phase and amplitude estimators from a demodulated probe harmonic; and (iv) high-SNR RMSE laws, mixing-angle design criteria, and a quasistatic treatment of bias nonuniformity.

Reduced model and loop-closure mechanism

The constructive model uses four states $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$ with probe on $\ket{1}\leftrightarrow\ket{2}$, coupling on $\ket{2}\leftrightarrow\ket{3}$, and the received RF coupling the bare pair $\ket{3}\leftrightarrow\ket{4}$. A static bias $E_z$ mixes only the near-degenerate upper pair, producing Stark states with mixing angle

$$\tan(2\theta)=\beta=\frac{2E_z|\mu_{34}^{z}|}{\hbar\Delta_{34}},$$

and effective couplings $A_S$0, $A_S$1, and $A_S$2. The leg $A_S$3 vanishes at $A_S$4 and is the bias-enabled pathway that closes the loop. A design trade-off is visible immediately: increasing $A_S$5 strengthens the loop-closing leg but weakens the effective RF coupling through $A_S$6 — this competition generates the distinct phase- and amplitude-optimal angles derived later. The authors are explicit that the trigonometric forms belong to this minimal isolated-pair model; the general theory requires only well-defined effective couplings.

After a rotating-wave approximation and a gauge transformation placing the factor $A_S$7 on the bias-enabled leg, the reduced Hamiltonian takes Floquet form with Fourier components only at $A_S$8 and $A_S$9. The construction neglects additional nearby Stark states, magnetic-sublevel structure, Doppler averaging, transport, and the diagonal oscillatory term from the transformed signal dipole; these omissions shift quantitative response maps but do not alter the structural phase result, provided the phase dependence remains confined to $\Phi_S$0 and the dissipator stays time independent.

Exact harmonic phase law

Because the unknown phase enters the Hamiltonian only through $\Phi_S$1, changing $\Phi_S$2 is equivalent to a time shift of the drive. Under uniqueness of the periodic steady state (PSS), each Fourier coefficient of the density operator obeys

$\Phi_S$3

This law is exact — detunings, linewidths, and dissipation modify the coefficients $\Phi_S$4 but not the multiplicative factor. For the probe coherence, harmonic magnitudes are independent of $\Phi_S$5 while phases shift linearly with slope $\Phi_S$6, so the full PSS can be computed once at $\Phi_S$7 via nearest-neighbor-coupled harmonic-balance equations (a block-tridiagonal Liouville-space system after truncation to $\Phi_S$8).

Estimators and high-SNR error laws

Phase is recovered from the argument of a demodulated harmonic relative to a calibrated reference:

$\Phi_S$9

The first harmonic ($\omega_S$0) is emphasized as branch-unambiguous; higher harmonics carry intrinsic modulo-$\omega_S$1 ambiguity. Amplitude recovery inverts the response map $\omega_S$2, which must be injective on the operating branch — otherwise inversion is ambiguous without prior information or a second observable.

Under additive circular complex Gaussian noise on the demodulated phasor, linearization yields two distinct laws:

$\omega_S$3

where $\omega_S$4 is the logarithmic sensitivity. Phase accuracy depends only on harmonic SNR; amplitude accuracy additionally depends on the local response slope, packaged into an effective amplitude SNR $\omega_S$5. These are local, high-SNR approximations that neglect phase wrapping, inverse-map ambiguity, and curvature effects.

Four mixing-angle criteria follow for $\omega_S$6: the phase-optimal angle maximizes the first-harmonic magnitude; the amplitude-optimal angle maximizes $\omega_S$7; a weighted joint-design angle minimizes a weighted sum of squared errors; and a weight-free balanced angle minimizes the maximum fractional degradation relative to the individual optima. In the perturbative regime all angular dependence collapses to $\omega_S$8, giving the seed rule $\omega_S$9 ($\{\ket{1},\ket{2},\ket{3},\ket{4}\}$0) — explicitly flagged as an initialization heuristic, not a design theorem.

Nonuniform bias

For $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$1 treated quasistatically, every local first-harmonic response carries the same factor $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$2, so any linear spatial average preserves the exact phase law while altering the complex coefficient. Two mechanisms act: spatial variation of the local mixing angle, and inhomogeneous RF detuning via the bias-dependent dressed splitting. The coherent-gain factor $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$3 reduces the effective phase SNR as $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$4, so the phase-RMSE penalty is exactly $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$5. Amplitude recovery is affected more subtly: averaging reshapes the effective response-map slope $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$6, and moderate nonuniformity can reduce coherent gain while simultaneously steepening the local branch — so amplitude behavior is not determined by $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$7 alone.

Numerical results

Production calculations truncate harmonics at $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$8 (validated against $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$9: relative error below $\ket{1}\leftrightarrow\ket{2}$0 at the nominal point by $\ket{1}\leftrightarrow\ket{2}$1, below $\ket{1}\leftrightarrow\ket{2}$2 at a stress point by $\ket{1}\leftrightarrow\ket{2}$3), with direct BDF time-domain integration confirming the Floquet reconstruction. At the nominal point ($\ket{1}\leftrightarrow\ket{2}$4, $\ket{1}\leftrightarrow\ket{2}$5, $\ket{1}\leftrightarrow\ket{2}$6, $\ket{1}\leftrightarrow\ket{2}$7, balanced angle $\ket{1}\leftrightarrow\ket{2}$8 rad), varying $\ket{1}\leftrightarrow\ket{2}$9 over $\ket{2}\leftrightarrow\ket{3}$0 rotates the first-harmonic phasor rigidly, with residuals at machine precision — a direct confirmation of the exact phase law. At $\ket{2}\leftrightarrow\ket{3}$1 the first harmonic collapses to zero, consistent with the absence of loop closure.

Key numerical findings:

Quantity	Value
Full-model phase optimum $\ket{2}\leftrightarrow\ket{3}$2	≈ 0.49 rad
Amplitude optimum $\ket{2}\leftrightarrow\ket{3}$3	≈ 0.60 rad
Balanced angle $\ket{2}\leftrightarrow\ket{3}$4	≈ 0.56 rad
Perturbative seed $\ket{2}\leftrightarrow\ket{3}$5	≈ 0.393 rad
Local sensitivity at nominal point	$\ket{2}\leftrightarrow\ket{3}$6
Sensitivity under nonuniformity	$\ket{2}\leftrightarrow\ket{3}$7
Coherent gain $\ket{2}\leftrightarrow\ket{3}$8 for $\ket{2}\leftrightarrow\ket{3}$9	0.32, 0.22, 0.10

Two results deserve emphasis. First, the full-model phase optimum lies clearly right of the perturbative seed, confirming that $\ket{3}\leftrightarrow\ket{4}$0 is genuinely non-optimal outside the weak-coupling regime — the authors state plainly that it should be used only for initialization. Second, Monte-Carlo RMSE curves ($\ket{3}\leftrightarrow\ket{4}$1 trials per point) match the high-SNR laws closely; under nonuniform bias, phase curves collapse when replotted against $\ket{3}\leftrightarrow\ket{4}$2, and amplitude curves collapse only after the additional slope correction $\ket{3}\leftrightarrow\ket{4}$3, validating the two-factor amplitude theory. Notably, the coherent-gain values are severe — even 1% relative nonuniformity costs roughly a factor of three in coherent gain in this parameter set — underscoring the authors' own conclusion that bias homogeneity and calibration are part of the receiver design problem, not secondary corrections.

Limitations and open questions

The framework is theoretical throughout, built on a deliberately minimal four-level model. It relies on the isolated-pair Stark approximation, neglects bias-induced mixing with lower ladder states and magnetic-sublevel structure, omits Doppler and transit effects, treats nonuniformity as static rather than fluctuating, discards the diagonal oscillatory dipole term, and assumes the carrier frequency is known or tracked — it is not a theory of blind carrier acquisition. Harmonic magnitudes, response maps, prefactors, and optimal angles are model dependent and will shift under manifold-level descriptions. No species-specific level selection or experimental realization is provided; the coherent-gain losses reported under even modest nonuniformity raise the concrete open question of how closely a practical vapor-cell geometry with realistic electrode fields can approach the uniform-bias benchmark.

Conclusion

The paper establishes that a static DC bias, by Stark-mixing a near-degenerate Rydberg pair, suffices to close a phase-sensitive reception loop for a single RF tone without any RF LO in the atoms. The resulting periodic steady state obeys an exact harmonic phase law, enabling direct phase estimation and injective-map amplitude recovery, with explicit high-SNR RMSE laws and four distinct bias-controlled operating-point criteria. Spatial bias nonuniformity preserves the phase law but degrades performance through coherent-gain loss and slope reshaping, quantified by collapse laws confirmed numerically. The work provides a minimal analytical foundation whose translation to specific atomic species and device geometries remains the principal outstanding step.