Coverage Analysis of Rydberg Atom Quantum Receiver Arrays: A Stochastic Geometry Approach

Abstract: Rydberg atomic quantum receivers (RAQRs) offer quantum-limited sensitivity and broadband tunability. It is not obvious whether this device-level advantage also improves network reliability, since in dense deployments, aggregate interference can push the atomic transducer out of its small-signal regime. This paper addresses the question by embedding the RAQR front end into a stochastic geometry (SG) coverage analysis. Starting with the atomic master equation and balanced coherent optical detection, we derive a third-order complex baseband model that retains both the linear gain and the leading cubic nonlinearity. A Bussgang decomposition converts the per-element nonlinear response into an equivalent linear gain plus a distance-dependent distortion noise. Using this equivalent model, we derive the post maximal-ratio combining (MRC) SINR and obtain tractable expressions for the conditional and spatially averaged coverage probabilities. The analytical results show that RAQRs outperform conventional receivers in sparse deployments. However, when the base station (BS) density becomes large, nonlinear distortion reduces this advantage and may make RAQRs perform worse. Simulation results validate the analytical expressions and confirm that the central design tradeoff is between linear gain and cubic nonlinearity.

• The paper combines a third-order Rydberg atom receiver model, Bussgang linearization, and stochastic geometry to predict uplink coverage within about 1.5 percentage points of simulation results.
• RAQRs outperform conventional receivers in sparse networks, achieving 0.82 versus 0.61 coverage at 5 dB, but cubic distortion reverses the ranking in dense networks, where coverage falls to 0.47 versus 0.87.
• Receiver design should optimize the nonlinearity ratio |c₃/c₁| alongside small-signal gain, since detuning, atomic species, and array size determine whether quantum-limited noise or interference distortion dominates.

Motivation and problem statement

Rydberg atomic quantum receivers (RAQRs) transduce incident RF fields through the response of highly excited atoms, read out optically rather than through an electronic front end. Experimental demonstrations report sensitivities on the order of tens of nV/cm/√Hz and sub-nV/cm minimum detectable fields, and prior work has established equivalent baseband models for superheterodyne RAQRs in single-link settings. However, these models are first-order (linear) and are validated only under weak-field conditions. The paper's central observation is that this weak-field assumption fails precisely where network-level analysis matters: in a dense cellular deployment, aggregate interference from nearby active users drives large excursions of the Rabi frequency around the local-oscillator operating point, pushing the atomic transducer into gain compression. The paper therefore asks whether the device-level sensitivity advantage of RAQRs survives at the network level, and answers it by combining a third-order baseband model with stochastic geometry (SG) coverage analysis.

Third-order baseband signal model

The receiver is modeled as a four-level ladder system driven by probe, coupling, and LO fields, with the desired RF signal offset from the LO by $f_\delta$. Solving the Lindblad master equation in steady state yields the probe coherence $\rho_{21}$ as a rational function of $\Omega_{\mathrm{RF}}$, whose nonlinearity propagates through the susceptibility to the balanced coherent optical detection (BCOD) output voltage. Expanding the detector voltage $V^{(B)}$ to third order around $\Omega_\ell$ and retaining only terms that survive baseband filtering (even orders produce DC and $2f_\delta$ components; the cubic term contributes to the fundamental with effective coefficient $1/8$), the per-element complex baseband output is

$v_m = c_1 u_m + c_3 |u_m|^2 u_m + w_m,$

where $c_1$ recovers the small-signal gain of the existing first-order model and $c_3$ captures the leading nonlinearity. A key structural result is that the linearity region of the element is governed entirely by the ratio $\rho_{21}$0: for tolerance $\rho_{21}$1, the linear approximation holds within $\rho_{21}$2. Because $\rho_{21}$3 tends to grow with $\rho_{21}$4 at typical operating points, maximizing small-signal gain alone does not enlarge the linear region — an observation that foreshadows the paper's main design tradeoff.

Bussgang linearization and coverage analysis

Conditioned on the serving distance $\rho_{21}$5, the per-element input is approximated as circularly symmetric complex Gaussian with power $\rho_{21}$6, derived via Campbell's theorem over the interfering-user PPP. Applying the Bussgang decomposition to $\rho_{21}$7 gives an effective gain $\rho_{21}$8 and distortion variance $\rho_{21}$9. Two consequences follow directly: distortion scales cubically with input power (hence cubically in BS density), so it is negligible in sparse networks but dominant in dense ones; and the resulting post-MRC SINR takes the same form as the classical linear-receiver expression, with the conventional thermal noise replaced by a distance-dependent effective noise $\Omega_{\mathrm{RF}}$0.

Coverage probabilities are obtained using a closed-form Laplace transform of the aggregate interference for path-loss exponent $\Omega_{\mathrm{RF}}$1, combined with a Kolmogorov–Smirnov-optimal approximation of the Gamma CCDF that reduces exactly to the standard exponential case for $\Omega_{\mathrm{RF}}$2. The largest absolute deviation between analysis and Monte Carlo simulation across all validated curves is about 0.015 (1.5 percentage points), confirming the tractability–accuracy tradeoff.

The benchmark comparison isolates the tradeoff cleanly: both receivers share the same interference Laplace transform and differ only in the noise exponent. At short serving distances, Bussgang distortion inflates $\Omega_{\mathrm{RF}}$3 above thermal noise, penalizing the RAQR; at large distances, distortion vanishes and the RAQR's intrinsic noise floor — approximately $\Omega_{\mathrm{RF}}$4 of the conventional noise power under typical parameters — yields a coverage advantage that grows exponentially with threshold and path loss.

Operating regimes and numerical findings

The simulations quantify when each regime dominates:

• Sparse networks: At $\Omega_{\mathrm{RF}}$5 (ISD ≈ 316 m), the optimized $\Omega_{\mathrm{RF}}$6Cs RAQR achieves coverage of 0.82 at $\Omega_{\mathrm{RF}}$7 dB versus 0.61 for the conventional receiver.
• Dense networks: At $\Omega_{\mathrm{RF}}$8 (ISD ≈ 32 m), the ranking reverses: Cs drops to 0.47 while the conventional receiver reaches 0.87. Crossover densities lie near $\Omega_{\mathrm{RF}}$9–$V^{(B)}$0 depending on threshold.
• Detuning choice: The $V^{(B)}$1-optimal detunings give $V^{(B)}$2 but $V^{(B)}$3; the resonant point gives $V^{(B)}$4 but $V^{(B)}$5. In dense networks the resonant configuration outperforms the gain-optimized one, demonstrating that detuning optimization should target linearity rather than small-signal gain.
• Atomic species: Optimized $V^{(B)}$6Rb offers $V^{(B)}$7 lower linear gain than Cs but a $V^{(B)}$8 smaller nonlinearity ratio ($V^{(B)}$9 vs. $\Omega_\ell$0). In dense networks Rb attains 0.73 coverage versus Cs's 0.47, though both remain below the conventional receiver.
• Array scaling: Increasing $\Omega_\ell$1 narrows the gap but does not eliminate it for high thresholds: at $\Omega_\ell$2 dB parity is reached near $\Omega_\ell$3, whereas at $\Omega_\ell$4 dB the conventional receiver still leads by about 5.3% at $\Omega_\ell$5.

An additional extension models coupling-induced receive correlation in conventional arrays via an exponential correlation matrix. Correlation increases the variance of the MRC channel gain and raises the interference factor, degrading conventional-array coverage, while the RAQR array — idealized as coupling-free — retains full array gain. Under $\Omega_\ell$6, conventional-array outage remains several times larger than the RAQR's even at the largest simulated $\Omega_\ell$7.

Limitations and open questions

The analysis rests on several assumptions stated by the authors. The Gaussian approximation of the conditional input distribution relies on a CLT argument over many interferers and may degrade in sparse networks with few dominant interferers. The Bussgang parameters are computed from the conditional second moment rather than instantaneous PPP realizations, and the Gamma-CCDF approximation introduces residual error beyond $\Omega_\ell$8. In the correlation extension, the random interference projection scale $\Omega_\ell$9 is replaced by its deterministic surrogate $2f_\delta$0, which preserves the mean but discards fluctuation effects. The RAQR array is idealized as perfectly uncoupled ($2f_\delta$1); whether real vapor-cell arrays exhibit inter-element coupling or correlated laser noise across elements is not modeled. Finally, the framework assumes perfect CSI and narrowband transmission; wideband signals, imperfect CSI, and higher-order nonlinearities remain unaddressed.

Conclusion

This paper embeds a third-order RAQR baseband model into a stochastic-geometry uplink coverage analysis via Bussgang linearization, producing tractable SINR and coverage expressions validated against Monte Carlo simulation within roughly 1.5 percentage points. The results establish that the RAQR advantage is regime-dependent: quantum-limited noise dominates in sparse networks, while cubic distortion — scaling as the sixth power of input power — can make RAQRs underperform conventional receivers in dense deployments. The ratio $2f_\delta$2, controllable through detuning and species selection, emerges as the governing design parameter, indicating that receiver optimization for network deployment must balance small-signal sensitivity against linearity rather than maximize gain alone.