Physics-Aware Power Optimization for Rydberg Quantum Arrays via Barankin Bound Minimization

• THz attenuation and negligible initial-access beamforming gain create an acute link-budget deficit, requiring beam alignment in an extremely low-SNR regime.Source: §I · p. 1
• Fixed-gain, fixed-noise array models miss how EIT conversion gain, photon shot noise, and power broadening jointly depend on probe and coupling laser Rabi frequencies.Source: §I · p. 1
• The CRB captures local error variance but misses finite-separation spatial ambiguities that cause threshold breakdown, making it overly optimistic at low SNR.Source: §I · p. 1
• The proposed framework minimizes a stochastic BRB, deriving a closed-form multipoint matrix with arcsine-prior calibration and mapping EIT readout physics into array covariance.Source: §I · pp. 1–2

• The model constructs the channel, array geometry, and baseband stochastic model to link array covariance to quantum parameters.Source: §II · p. 2
• Narrowband training pilots support a narrowband array model; the compact aperture motivates a far-field, rank-one, LoS-dominant channel despite strong THz attenuation.Source: §II · p. 2
• A four-level atomic ladder uses probe and coupling lasers, with a resonant THz field inducing Autler-Townes splitting in the EIT spectrum.Source: §II · p. 2
• A fixed optical-LO incidence angle creates a distributed phase shift across cells, which affects the RAQ-ULA baseband phase.Source: §II · p. 2
• Independent photodetector readouts yield spatially uncorrelated noise, while quantum conversion gain and noise variance depend nonlinearly on laser Rabi frequencies.Source: §II · p. 3

• The stochastic CRB provides a local benchmark, while the multipoint BRB targets spatial ambiguities and threshold breakdown in THz initial access.Source: §III · p. 3
• The BRB offers a nonasymptotic error characterization by including finite-separation uncertainty beyond local differential analysis.Source: §III · p. 3
• The spatial covariance determinant is σ^2(M−1)(σ^2 + M|β0|^2Ps).Source: §III · p. 3

A. Stochastic CRB

• For the zero-mean CSCG observation model, the stochastic CRB follows from the Slepian–Bangs formula.Source: A. Stochastic CRB · p. 3
• The CRB's Fisher information depends on the local steering-vector derivative projected orthogonally to the steering vector.Source: A. Stochastic CRB · p. 3
• The ULA local Fisher metric scales as O(M^3), but ignores finite-separation sidelobe ambiguities and can be overly optimistic in attenuated THz scenarios.Source: A. Stochastic CRB · p. 3

B. Stochastic Multipoint Barankin Bound

• Test points are placed near finite-aperture correlation ambiguity peaks, including null-to-null beamwidth intervals and sidelobe crests.The selection strategy relies on the far-field Dirichlet kernel; near-field spherical-wave effects require a two-dimensional angle-distance grid.Source: B. Stochastic Multipoint Barankin Bound · p. 3
• The stochastic Barankin matrix has a closed-form expression when every retained test-point-pair matrix Mlr is positive definite.The low-SNR integrability condition ensures the defining integral is finite.Source: B. Stochastic Multipoint Barankin Bound · pp. 3–4
• In RAQ-ULAs, the gain and noise in the bound depend on laser Rabi frequencies, making the bound a power-optimization criterion rather than only an evaluation metric.The quantum conversion gain and photon shot noise are tied to the microscopic EIT readout dynamics.Source: B. Stochastic Multipoint Barankin Bound · p. 4
• An arcsine-transformed prior calibrates the BRB to the bounded DOA domain and sets its pure-noise uncertainty saturation to Varc = π^2/4 − 2.The prior follows from treating spatial frequency in the sin θ domain as uninformative.Source: B. Stochastic Multipoint Barankin Bound · p. 4
• The BRB captures spatial-ambiguity effects through finite-separation likelihood ratios, unlike the local CRB.As incident SNR decreases, the prior-calibrated BRB approaches the bounded-domain saturation level Varc.Source: B. Stochastic Multipoint Barankin Bound · p. 4

IV. PHYSICS-AWARE POWER OPTIMIZATION

• Microscopic quantum dynamics are mapped to macroscopic array parameters so laser powers jointly shape the array covariance and ambiguity-sensitive BRB.The allocation objective is nonconvex and minimizes the prior-calibrated BRB.Source: §IV · p. 4
• The optimization uses an MM framework to minimize the resulting nonconvex BRB objective.Source: §IV · p. 4
• The design criterion is global BRB performance beyond the local CRB.Source: §IV · p. 4

A. Quantum Macroscopic Mapping and CRB Baseline

• Probe intensity trades optical response against power broadening and photon shot noise in the EIT readout.An excessively strong probe perturbs atomic coherence and narrows the transparency window, while increasing the photon shot-noise floor.Source: A. Quantum Macroscopic Mapping and CRB… · p. 4
• The CRB-based SNR metric has a unique maximizing probe intensity for each fixed coupling intensity.The stationary point is the unique maximizer because the metric tends to zero at both zero and infinite probe intensity.Source: A. Quantum Macroscopic Mapping and CRB… · p. 4
• The CRB-optimized allocation is an analytical local-SNR baseline because it ignores finite-separation ambiguity errors that can cause premature threshold breakdown at very low RF power.Source: A. Quantum Macroscopic Mapping and CRB… · p. 5

B. BRB Minimization via Lipschitz-MM Framework

• The objective minimizes the prior-calibrated BRB over laser powers subject to EIT operating constraints and the hardware power budget.A positive lower bound excludes degenerate zero-power points.Source: B. BRB Minimization via Lipschitz-MM… · p. 5
• A Lipschitz-based quadratic surrogate majorizes the smooth BRB objective over the feasible set and is tight at the current iterate.Its minimizer has a projected-gradient form.Source: B. BRB Minimization via Lipschitz-MM… · p. 5
• Backtracking increases the surrogate parameter until majorization holds, ensuring monotonic objective decrease and convergence to a feasible stationary point.Source: B. BRB Minimization via Lipschitz-MM… · p. 5

C. Computational Complexity and Engineering Feasibility

• The MM framework optimizes only a two-dimensional vector, with constant per-iteration complexity for projection onto the feasible polygon.The remaining cost is evaluating the BRB objective and gradient over selected test points.Source: C. Computational Complexity and… · p. 5
• Retaining at most six prominent sidelobe test points keeps the Barankin matrix dimension low and decoupled from array size.Diagonal regularization and truncated SVD provide numerical stability and constant matrix-inversion complexity.Source: C. Computational Complexity and… · p. 5
• The algorithm typically converges within 10 iterations, supporting low-latency power allocation within the THz channel coherence interval.The section presents this alongside the truncated test-point strategy for large arrays.Source: C. Computational Complexity and… · p. 5

V. SIMULATION RESULTS

• The evaluation uses an RAQ-ULA with M = 10 elements and N = 20 snapshots, stabilizing Barankin-matrix inversion with truncated SVD at tolerance 10^-10.The empirical MLE uses a dense spatial grid to reduce quantization errors.Source: §V · p. 5
• The 1/η penalty has a U-shaped dependence on probe Rabi frequency, and its analytical optimum matches the numerical minimizer.The interior operating point balances conversion-gain enhancement, power broadening, and photon shot noise.Source: §V · pp. 5–6, Figure 2 · p. 6
• Around 4 dB, the stochastic BRB captures threshold breakdown and better matches empirical MLE behavior, while the stochastic CRB is overly optimistic at low SNR.In the pure-noise region, the MLE approaches the bounded-manifold saturation level Varc.Source: §V · p. 6, Figure 3 · p. 6
• From three initializations, the backtracking MM algorithm decreases the BRB objective monotonically and converges to a similar stationary value within 10 iterations.The starts include points near the hardware boundary, under imbalanced EIT, and at low power.Source: §V · p. 6
• Equal-power allocation breaks down around 7 dB, while maximum-power allocation can delay breakdown but excessive probe power broadens the atomic resonance and degrades EIT readout.The proposed allocation balances the photon shot-noise floor against the power-broadening penalty by minimizing the prior-calibrated BRB.Source: §V · p. 6
• The proposed allocation reduces photon shot noise variance by over 70% versus the saturated probe regime and achieves lower RMSE than constrained SNR maximization around 0–5 dB.The solution is approximately (x*, y*) = (5.86, 94.14) MHz^2, compared with the maximum-power boundary allocation (33.33, 66.67) MHz^2.Source: §V · p. 6

VI. CONCLUSION

• A backtracking MM algorithm minimizes the prior-calibrated BRB under physical laser-power constraints; Fig. 4 depicts its monotonic convergence.The framework uses a closed-form stochastic multipoint BRB with arcsine-prior calibration and an EIT readout mapping.Source: §VI · p. 6, Figure 4 · p. 7
• Simulations demonstrate improved threshold-breakdown prediction and an enlarged reliable operating region under severe THz attenuation.The conclusion also reports monotonic objective decrease.Source: §VI · p. 6
• Future directions include integrating data-transmission constraints for ISAC and extending the allocation methodology to distributed architectures.The current framework focuses on spatial sensing.Source: §VI · p. 7