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Fundamental CRB-Rate Tradeoff of Rydberg Atomic Receivers for Integrated Sensing and Communication

Yin Zhang, Jiayi Zhang, Bokai Xu, Jieao Zhu, Bo Ai

arXiv:2610.08696v1eess.SP

TL;DR

RAR-based ISAC has a communication–sensing tradeoff shaped not only by signaling and resource allocation but also by shared atomic response. This paper models a multi-branch Rydberg receiver and finds that coupling strength and LO allocation jointly shape bandwidth, conversion gain, and the achievable CRB-rate tradeoff.

  • Problem

    In RAR-based ISAC, the communication–sensing tradeoff remains insufficiently understood because both functions share atomic response, adding receiver behavior to classical resource-allocation and signaling tradeoffs.

    Source: §I · pp. 1–2

  • Method

    The paper models a multi-branch RAR by deriving its dynamic response from the LO-biased five-level Lindblad master equation and formulating a rate-constrained joint optimization over coupling strength and LO allocation.

    Source: §I · p. 2

  • Results

    Simulations show joint tuning balances bandwidth broadening and IBGI; at default parameters, RAR-based ISAC achieves smaller sensing CRBs than time-sharing RF-ISAC, while RF-ISAC reaches a higher communication-rate endpoint.

    Source: §I · p. 2

  • Takeaways & Limitations

    The achievable CRB-rate region must account for coupling-laser bandwidth and atomic-response effects alongside LO-branch interactions.

    Source: §I · p. 2, §VI · p. 14

Abstract

from arXiv · show

Integrated sensing and communication (ISAC) is a key technology for next generation wireless networks, requiring highly sensitive, compact receiver front ends with multiband coverage. Rydberg atomic receivers (RARs) provide a promising architecture that satisfies these requirements owing to their high sensitivity and continuous frequency tuning capability. Unlike in classical ISAC systems utilizing orthogonal resource allocation, executing concurrent multi band tasks within a single atomic ensemble relies on shared energy state transitions. Owing to this shared structure, the communication and sensing tradeoff in RAR-based ISAC introduces an atomic-response tradeoff beyond the classical subspace and deterministic-random tradeoffs. This paper studies a multi-branch RAR (MB-RAR) architecture for joint communication and radar sensing. By solving the five level Lindblad master equation at the local oscillator (LO) biased operating point and linearizing the communication and sensing radio frequency (RF) signals as small perturbations, we obtain the double-sideband dynamic signal response of the MB-RAR. The model reveals two intrinsic constraints. First, increasing the coupling laser strength broadens the response bandwidth but can weaken the atomic response. Second, strengthening one branch's LO can suppress the other branch's conversion gain through the shared Rydberg state, even when the latter's LO bias remains fixed, which is termed the inter band gain inhibition (IBGI). Based on these mechanisms, we formulate the achievable Cram\'er--Rao bound (CRB)-rate region and develop a rate-constrained joint search over coupling strength and LO power allocation. Numerical results validate the proposed response model and show that joint optimization provides a more complete characterization of the achievable CRB-rate region.

I. INTRODUCTION

RAR-based ISAC adds receiver-side atomic response to the communication–sensing tradeoff, because both functions share Rydberg states. The paper models a multi-branch receiver and characterizes how its coupling laser and LO branches shape performance.

I. INTRODUCTION

  • RARs use Rydberg transitions to map incident RF fields to optical responses, supporting broad frequency coverage and concurrent multiband reception in a common ensemble.The paper motivates RARs as an alternative to conventional front ends with antenna and shared-component constraints.Source: §I · p. 1
  • Classical ISAC tradeoffs concern resource allocation and signaling, while RAR-based ISAC also depends on the shared atomic response to communication and sensing signals.Shared Rydberg coherences and microwave-induced transitions couple the two functions.Source: §I · p. 2
  • The paper develops an MB-RAR analytical framework in which separate LO branches address microwave transitions but remain coupled through the shared Rydberg state and common EIT channel.The resulting tradeoff depends on receiver-side atomic response as well as signaling and resource allocation.Source: §I · p. 2
  • The response model solves the five-level Lindblad master equation at an LO-biased operating point and linearizes communication and sensing RF signals as small perturbations.The branch atomic response links microscopic density-matrix dynamics to communication and sensing observation models.Source: §I · p. 2
  • Joint tuning balances response-bandwidth broadening and inter-band gain inhibition; at default parameters, RAR-based ISAC has smaller sensing CRBs than time-sharing RF-ISAC, which attains a higher rate endpoint.The comparison is attributed to the RF-ISAC baseline's larger receive SNR.Source: §I · p. 2

II. ISAC SYSTEM MODELING

  • The uplink system uses task-oriented atomic-response branches for communication and sensing, with an MB-RAR array receiving the signals through heterodyne detection driven by a multitone LO.The RF model includes a static point target and neglects Doppler shift for AoA estimation; the LO is assumed spatially uniform across vapor cells.Source: §II · p. 2, A. Radio Frequency Signal Model · p. 3

B. Double-Sideband Quantum Response of MB-RAR

The MB-RAR response model treats communication and sensing signals as small perturbations of one LO-biased atomic ensemble. It retains both sidebands and branch inputs in the shared optical-coherence response, with Doppler averaging and time-domain calibration used to evaluate that response.

B. Double-Sideband Quantum Response of MB-RAR

  • The five-level MB-RAR uses two resolvable microwave branches coupled through a shared Rydberg state and optical EIT pathway.The communication and sensing transitions connect the shared state to distinct upper Rydberg states, with their respective LO beats forming the intermediate frequencies.Source: B. Double-Sideband Quantum Response of… · p. 3
  • Both LO biases establish one shared atomic operating point, and first-order RF perturbations from both branches contribute to its optical-coherence response.The model obtains the response from the five-level Lindblad dynamics and evaluates the upper- and conjugated lower-sideband inputs at a common output frequency.Source: B. Double-Sideband Quantum Response of… · p. 4
  • The two sideband magnitudes nearly coincide at low IF but separate as IF increases, and optical detuning changes their relative magnitudes and spectral features.The direct and conjugate perturbations drive different coherence pathways; resonant excitation does not guarantee identical responses across the receive band.Source: B. Double-Sideband Quantum Response of… · p. 5
  • The Doppler-averaged resolvent kernels match time-domain Liouvillian calibration markers across the tested IF range, supporting the two-source linearization.The I and Q perturbations have unequal component magnitudes because they couple differently to the biased optical coherence.Source: B. Double-Sideband Quantum Response of… · pp. 5–6, Figure 4 · p. 5
  • The quantum transconductance has local maxima near Ωp = 0.8Γ2 and Ωp = 5.57Γ2, with near-cancellation around Ωp ≈ 1.59Γ2.The higher maximum requires stronger probe excitation; subsequent simulations use 0.8Γ2, retaining appreciable conversion with moderate probe power and about 98.8% ensemble-averaged ground-state occupation.Source: B. Double-Sideband Quantum Response of… · p. 6

C. Quantum Transconductance

The shared optical coherence gives rise to a susceptibility and photocurrent response to weak branch inputs. The quantum transconductance represents the input-to-readout conversion for the scalar I-input readout.

C. Quantum Transconductance

  • The shared optical susceptibility includes contributions from all weak inputs to the Doppler-averaged coherence.The resulting photocurrent is evaluated using Beer–Lambert readout after averaging the coherence over atomic velocities.Source: C. Quantum Transconductance · p. 6
  • The scalar I-input transconductance measures branch input-to-readout conversion, not a separate density matrix or detector.It is defined at the branch’s operating intermediate frequency.Source: C. Quantum Transconductance · p. 6

D. Noise Model

The noise model combines photon shot, quantum projection, and thermal noise at the processed readout. It treats the sources as mutually uncorrelated and approximates the bandlimited residual as complex Gaussian.

D. Noise Model

  • The three noise sources are modeled as uncorrelated, locally flat-PSD processes over the equivalent noise bandwidth, and the processed residual is approximated as circularly symmetric complex Gaussian.The Gaussian approximation follows from aggregation of many independent microscopic fluctuations after bandlimited baseband processing.Source: D. Noise Model · pp. 6–7
  • Photon shot noise has output PSD SPSN = 2qIPDRL, and its variance grows linearly with detected probe power.The model neglects background light and dark current because the probe beam dominates detected optical power.Source: D. Noise Model · p. 7
  • Quantum projection noise is modeled using the standard-quantum-limit field-sensitivity model and converted to output noise through the branch transconductance.The conversion uses the effective participating atom number and microwave coherence time.Source: D. Noise Model · p. 7
  • Thermal noise combines load-resistor Johnson–Nyquist noise with amplifier-chain excess noise, producing PSD STN = kBTFamp.It is independent of optical power and LO allocation, so it sets a fixed electronic noise floor that dominates at low probe power.Source: D. Noise Model · p. 7

III. QUANTUM TRADEOFF ANALYSIS

The section identifies the coupling-laser tradeoff and inter band gain inhibition as intrinsic tradeoffs in RAR-ISAC.

III. QUANTUM TRADEOFF ANALYSIS

  • The analysis identifies the coupling-laser tradeoff and inter band gain inhibition as two intrinsic RAR-ISAC effects.The accompanying Fig. 6 caption describes steady-state probe-transmission spectra versus detuning for different coupling powers.Source: §III · p. 7, Figure 6 · p. 7
  • Fig. 6 presents steady-state probe-transmission spectra across detuning for different coupling powers.Source: Figure 6 · p. 7

A. Coupling-Laser Tradeoff and Effective Response Bandwidth

The coupling laser affects the communication response bandwidth through the shared EIT channel. The effective bandwidth is measured from the dynamic response, and it increases over the displayed coupling sweep.

A. Coupling-Laser Tradeoff and Effective Response Bandwidth

  • Steady-state transmission features widen and separate as coupling power rises, but the dynamic communication bandwidth additionally depends on RF-to-optical conversion speed.The coupling field changes EIT interference and, at stronger coupling, produces resolved Autler–Townes structure.Source: A. Coupling-Laser Tradeoff and… · p. 7
  • Effective bandwidth Bcom is the smallest frequency at which the normalized dynamic response reaches 1/2 magnitude.It is evaluated numerically over 0–100 MHz at each operating point.Source: A. Coupling-Laser Tradeoff and… · p. 7
  • Fig. 7 compares Doppler-averaged and zero-velocity double-sideband bandwidth responses with a zero-velocity complex-linear reference.The Doppler-averaged double-sideband curve gives Bcom at T = 300 K.Source: A. Coupling-Laser Tradeoff and… · p. 7, Figure 7 · p. 8
  • The Doppler-averaged bandwidth increases over the displayed coupling sweep because stronger coupling lets the common EIT channel follow faster RF perturbations.Doppler averaging combines complex responses with velocity-dependent detunings and reshapes the normalized spectrum.Source: A. Coupling-Laser Tradeoff and… · p. 8, Figure 7 · p. 8

B. Inter Band Gain Inhibition

The two RF branches share a Rydberg state, so their LO biases jointly shape the atomic response and transconductances. This produces inter band gain inhibition and makes sensing gain depend on the joint LO allocation and optical coupling.

B. Inter Band Gain Inhibition

  • IBGI is the reduction in one branch’s transconductance when the other LO strengthens while the first branch’s own LO bias remains fixed.Both branches couple to the shared Rydberg state, so their LO biases jointly determine the operating point and RF-to-optical response.Source: B. Inter Band Gain Inhibition · p. 8
  • The LO-allocation relation distinguishes direct mixing enhancement from response-per-unit-LO changes, including own-branch broadening and cross-branch effects.The coefficients depend on the operating point and capture changes in steady-state photocurrent and RF-to-optical conversion slope.Source: B. Inter Band Gain Inhibition · p. 8
  • At α = 0.8, simultaneous sensing gain has a 10.73 dB penalty relative to the communication-LO-off reference at the same sensing bias.The communication LO couples the shared state to a damped state, suppressing the optical response to the sensing perturbation.Source: B. Inter Band Gain Inhibition · p. 8, Figure 8 · p. 9
  • The sensing allocation sensitivity separates into direct LO loss, own-branch response recovery, and cross-branch inhibition.The own-branch contribution includes self broadening and population redistribution, while a negative cross-branch coefficient identifies local IBGI.Source: B. Inter Band Gain Inhibition · p. 8
  • Reducing the sensing LO alleviates its own broadening, while the stronger communication LO adds inhibition that largely offsets the resulting response recovery.The simultaneous sensing gain is consequently approximately proportional to √(1−α).Source: B. Inter Band Gain Inhibition · pp. 8–9
  • Switching on the communication LO reverses allocation ordering over lower-to-intermediate coupling and shifts the sensing-gain maximum toward stronger coupling.At stronger coupling, optical dressing weakens both branches’ probe responses, so a smaller relative IBGI penalty does not imply greater sensing transconductance.Source: B. Inter Band Gain Inhibition · p. 9

IV. C-R REGION CHARACTERIZATION

This section defines the communication rate and angle-of-arrival (AoA) Cramér–Rao bound (CRB) used to characterize the achievable communication–sensing tradeoff. It also specifies how the sensing metric extends to multiple separated targets sharing the same receiver operating point.

A. Performance Metrics

  • Communication rate is the effective bandwidth multiplied by log2(1 + communication SNR), with LO allocation affecting rate through branch gain, bandwidth, and readout noise.Source: A. Performance Metrics · p. 9
  • The AoA CRB is expressed using dimensionless sensing SNR and an orthogonal projection, with a reduced form for a uniform linear array.Source: A. Performance Metrics · p. 10
  • For L separated targets at fixed total echo power, each target receives equal power and shares the existing sensing branch and atomic operating point.The evaluation uses the average marginal AoA bound across the targets.Source: A. Performance Metrics · p. 10

B. Physically Feasible Pareto Front

The section characterizes the physically feasible communication–sensing Pareto boundary using a rate-constrained CRB minimization. It establishes the relationship between this ϵ-constraint solution and Pareto-optimal designs, then describes a derivative-free search to trace the boundary.

B. Physically Feasible Pareto Front

  • The design objective is to maximize communication rate while minimizing sensing estimation error over physically feasible operating points.Source: B. Physically Feasible Pareto Front · p. 10
  • For a required rate, the ϵ-constraint formulation first minimizes CRB and then selects the highest-rate design among the minimizers.Source: B. Physically Feasible Pareto Front · p. 10
  • The rate-constrained solution is Pareto optimal, and every Pareto-optimal design solves the formulation at its own achieved rate.Source: B. Physically Feasible Pareto Front · p. 10
  • A derivative-free search is used because rate and CRB require numerical evaluation and rate need not vary monotonically with LO allocation.The two controls are normalized, and accepted updates preserve rate feasibility while reducing CRB.Source: B. Physically Feasible Pareto Front · pp. 10–11
  • Repeating the search over rate requirements traces the boundary, using feasible warm starts and a seed pool to mitigate local minima.The stated complexity is O(N_R N_start I_max |D| C_phys), excluding the fixed cost of computing the two seeds.Source: B. Physically Feasible Pareto Front · p. 11

V. NUMERICAL RESULTS

The numerical-results section evaluates the C-R tradeoff and parameter effects under a specified uplink and point-scatterer sensing setup. It compares joint atomic control with fixed-coupling, time-sharing RAR, and conventional RF-ISAC baselines.

A. Simulation Setup

  • The simulations model a far-field single-antenna uplink and a point-scatterer target with deterministic echo coefficients, evaluating AoA CRB over K snapshots.Source: A. Simulation Setup · p. 11
  • The Joint Atomic Response MB-RAR jointly optimizes coupling strength and LO allocation against communication-centric, sensing-centric, RAR time-sharing, and conventional RF-ISAC time-sharing baselines.Source: A. Simulation Setup · p. 11
  • Figure 10 shows a CRB-rate region, and its caption specifies that the region concerns the task of single target angle estimation.Source: Figure 10 · p. 11

B. Characterization of the CRB-Rate Pareto Frontier

The numerical results show that joint atomic control characterizes a broader C-R tradeoff, while sensing load and system parameters reshape feasible performance. The conclusion reports smaller sensing CRBs and feasibility under heavier multi-target loads than conventional RF-ISAC, while noting that experimental validation remains future work.

B. Characterization of the CRB-Rate Pareto Frontier

  • The CRB-rate curves trade higher communication rate for larger sensing CRB, and joint control attains higher intermediate-region rates than fixed-coupling MB-RAR.The conventional RF endpoint has the largest rate because its SNR is limited by electrical noise, while RAR rate is limited by conversion efficiency and quantum noise.Source: B. Characterization of the CRB-Rate… · pp. 11–12

C. Communication Rate under Multi-Target Sensing Loads

  • As the number of targets increases at fixed total echo power, feasible schemes lose communication rate, and sensing load can eliminate feasibility before it determines rate.RAR time sharing has the highest rates among the feasible schemes in this comparison; communication-centric MB-RAR has no feasible point under the stated CRB limit.Source: Figure 11 · p. 12, C. Communication Rate under… · p. 12

D. Performance Analysis of Communication and Sensing

D. Performance Analysis of Communication and Sensing / 1 Torr

  • Temperature produces competing effects: stronger atomic response initially improves performance, but probe attenuation can reverse the trend and create an interior CRB minimum.The conventional RF-ISAC baseline varies mildly with temperature and degrades slowly as thermal-noise power spectral density increases.Source: Figure 12 · p. 13, §1 · pp. 13–14

VI. CONCLUSION

  • Joint atomic control yields a more complete C-R region than fixed-coupling operation; RAR has smaller sensing CRBs and remains feasible under heavier multi-target loads than conventional RF-ISAC.Time sharing is preferable in parts of the region where inter-branch competition is strong.Source: §VI · p. 14
  • Future work includes experimental validation of the MB-RAR response model, distributed RAR cooperation, microwave-transition selection, and physics-informed AI for calibration and optimization.Source: §VI · p. 14

APPENDIX A LO SENSITIVITIES AND PROOF OF LEMMA 1

The appendix derives LO sensitivities from changes to the steady-state operating point and branch response. It also proves the angle CRB by eliminating nuisance parameters and evaluating projected steering-vector sensitivity.

APPENDIX A LO SENSITIVITIES AND PROOF OF LEMMA 1

  • The sensitivity equations determine coefficients from the atomic response, and equations (61)–(66) complete the branch-gain derivative evaluation.The appendix also differentiates the susceptibility and branch response.Source: Appendix A · p. 14
  • At a smooth operating point with a unique steady state and decaying trace-zero modes, differentiation determines the operating-point change at each velocity.The trace constraint is part of the system that determines this change.Source: Appendix A · p. 14
  • The LO derivative accounts for changes in both the evolution operator and the steady-state source, while the trace constraint specifies the solution at ω = 0.Source: Appendix A · p. 14

APPENDIX B PROOF OF LEMMA 2

  • A Schur complement eliminates the 2K nuisance parameters to yield the equivalent Fisher information for θ.The snapshots are independent, and the unknown vector includes the real and imaginary parts of each signal and θ.Source: Appendix B · pp. 14–15
  • Substituting the ULA steering-vector derivative into its projection orthogonal to a(θ) completes the CRB(θ) proof.The proof then uses CRB(θ) = J^-1(θ).Source: Appendix B · p. 15
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