Beyond sensitivity: mechanism-resolved error budgets for designing quantum sensors

Motivation

• The framework addresses applications requiring simultaneous measures because magnetometry platforms serve distinct demands for accuracy, low-frequency sensitivity, robustness, and imaging trade-offs.Current designs often rely on expert judgment about the dominant limiter and tune sensitivity empirically while assuming other metrics follow.Source: §1 · pp. 1–2

Contribution

• A single device-grounded open-system solve computes sensitivity, accuracy, and robustness and attributes each metric to interacting physical mechanisms using Shapley values.The framework extends to other figures of merit such as dynamic range and uses the tangent master equation to co-propagate the state and its field derivative.Source: §1 · p. 3

Validation and computation

• The method is validated against measured nitrogen-vacancy diamond and cesium optically pumped magnetometer devices, with GPU batching making full coalition attribution affordable.It combines a common open quantum system model, mechanism-level Shapley attribution, and validation across two sensing platforms.Source: §1 · p. 3

Main result

• Spin dephasing limits sensitivity, the thermal ground-state shift limits accuracy, and optical leakage limits robustness under specified experimental conditions.Thus, equal sensitivity does not imply equal accuracy or calibration stability because different metrics trace to different mechanisms.Source: §1 · p. 3

Design transfer

• By linking each metric to measurable intermediates, the model predicts the gain from addressing its dominant mechanism and transfers beyond magnetometry to other optically read out open quantum sensors.Examples include optical clocks, atom interferometers, and Rydberg electrometers.Source: §1 · p. 3

2.1 A per-mechanism, multi-metric attribution model

• A single tangent master-equation solve co-propagates the state and field derivative, while Shapley values assign interacting mechanisms to each metric.Optical leakage is modeled explicitly as residual readout-channel pump power, capturing contrast loss and photon-baseline shifts that phenomenological dephasing cannot represent.Source: §2.1 · pp. 4, 6
• Central finite differencing agrees to better than 10^-4, and the engine recovers the field to σb = 32 nT, whereas C = 0.17 reaches only 87 nT.The readout contrast is a prediction computed from the ten-level model rather than an input, and the engine reproduces coherence–density literature across nearly three decades.Source: §2.1 · p. 6

2.2 Metric-dependent attribution

• 89% of sensitivity attribution comes from dephasing, while thermal shift contributes an apparent 282 nT accuracy bias and optical leakage dominates robustness.The metric limiters therefore differ: dephasing controls sensitivity, the temperature-driven ground-state shift controls accuracy, and optical leakage controls robustness.Source: Figure 1 · p. 5, Figure 3 · p. 8, §2.2 · pp. 8–9
• Dephasing dominates at about 89% across short and intermediate τ, then slips to 83% at the longest τ as optical leakage costs a rising fraction of sensitivity.The thermal channel and T1 never exceed 2% because neither reshapes the fringe contrast that sets ηB.Source: §2.2 · pp. 8–9
• At long interrogation times, calibration over-correction cancels the thermal bias near τ ≈2.5 µs, whereas intermediate times approach the full 282 nT net bias.Leakage contributes negatively at short times and partially masks the thermal term alongside the shot-noise floor.Source: §2.2 · p. 9
• Coherence dominates sensitivity, temperature dominates accuracy, and leakage drift dominates robustness, with this ordering stable across 500 shot-noise realizations and the stated parameter ranges.Pure dephasing lowers sensitivity without biasing the recovered field, whereas thermal shifts and dissipative leakage bias accuracy; the thermal channel corresponds to ∂T Dgs/γe ≈2.8 µT/K, or 1 nT↔0.35 mK.Source: §2.2 · p. 9
• 57%: sensitivity is dephasing-limited across this fraction of the design plane, while robustness is leakage-drift-limited over 78%.Accuracy is temperature-limited below ε ≈3 × 10^-5 and leakage-limited above, with bias falling to ∼7 nT on the cancellation ridge.Source: §2.2 · p. 10
• 8 to 1500 nT: accuracy bias spans this range along a single iso-ηB locus at 100 pT, showing that sensitivity tuning does not ensure accuracy.No single interrogation time optimizes all three metrics; robustness is best at the shortest τ, where sensitivity is worst.Source: §2.2 · pp. 9–10

2.3 The error budget of a real device

• The NV−/NVT charge fraction decreases monotonically with optical intensity, from 0.59 to 0.085 on the 0.3 ppm diamond and from 0.73 to 0.25 on the 4.5 ppm diamond.Source: §2.3 · p. 10
• P⋆≈2.2 × 10^6 W/cm2: sensitivity turns over at this interior optimum on the 0.3 ppm diamond, while the 4.5 ppm optimum lies at the edge of the measured range.At the lowest intensity on the 4.5 ppm diamond, ∼125 photons per shot create a 240-fold photon deficit that outweighs its fourfold contrast advantage, making sensitivity roughly four times worse.Source: §2.3 · pp. 11–12, Figure 4 · p. 11
• 282 nT: the thermal Dgs(T) shift at a 0.1 K stability specification sets the accuracy floor, which remains nearly flat across almost three decades of intensity.On the 0.3 ppm diamond, the sensitivity budget shifts from a 32% shot-noise share at the lowest intensity to a 63% charge-state share at the highest, with charge-state dilution dominating at P⋆.Source: §2.3 · p. 12
• On the real NV device, thermal Dgs bias remains one to two orders of magnitude above the shot floor across three decades of intensity, keeping NRMSE flat.Thus optical-intensity tuning reshapes sensitivity without materially changing accuracy, whose limiter is thermal bias.Source: §2.3 · p. 12
• The framework prescribes pushing optical intensity toward P⋆ when sensitivity governs, but directing accuracy improvements toward temperature stabilization because thermal bias sets accuracy.The same control knob therefore must be tuned against the metric required by the application.Source: §2.3 · p. 12

2.4 Cross-platform provisioning and design by specification

• ~5 mm localization requires rejecting ambient field noise; error rises to ~60 mm at raw single-channel ambient, while realistic 1% gain spread contributes only a ~0.37 mm floor.The localization analysis is limited by field noise rather than calibration, with the 5 mm target reached after common-mode rejection.Source: Figure 5 · p. 13, §2.4 · p. 14
• The cesium OPM’s intrinsic sensitivity floor lies roughly three orders of magnitude below measured field noise, so sensitivity is not the application’s binding constraint.The design margin indicates that the atomic layer could be traded for cost, robustness, or a simpler source before becoming the bottleneck.Source: Figure 5 · p. 13, §2.4 · p. 14

• 8 to 1500 nT accuracy bias occurs at identical nominal sensitivity, while dephasing, thermal ground-state shift, and optical leakage limit sensitivity, accuracy, and robustness, respectively.The bias assumes ΔT = 0.1 K and spans a factor of approximately 200, so no single figure of merit certifies a design.Source: §3 · pp. 14–15
• The interaction-aware Shapley decomposition resolves nonlinear, dissipative sensor mechanisms into signed, per-metric budgets rather than assuming their contributions simply add.It complements scalar coherence-time, Cramér–Rao, and closed-form sensitivity summaries by decomposing sensitivity, accuracy, and robustness into their physical causes.Source: §3 · p. 15
• Measured device data make the model a digital twin that maps each fabricated sensor to its attributed budget and predicts the gain from removing each limiter.The outputs connect an accuracy target to temperature stability and an optical-leakage robustness target to a low-leakage acousto-optic or integrated-photonics switch.Source: §3 · p. 15
• Differentiable metric trade-offs could support specification-driven gradient design, mechanism-based control or learning, and physically accurate synthetic training data with ground-truth labels.The framework identifies mechanisms and target levers but does not perform the redesign or demonstrate the resulting metric improvement.Source: §3 · p. 15
• Attribution distinguishes design-time fixes for fabrication-frozen parameters from online correction of adjustable controls, but experimental validation of predicted metric gains remains necessary.The runtime extension includes closed-loop, self-adapting calibration, while the paper stops short of carrying out the implied redesign.Source: §3 · p. 15
• The workflow transfers from an NV-diamond ensemble to a cesium optically pumped magnetometer array and is framed for open quantum sensors read out by photon counting.Future reliability requires extending Gaussian noise models to non-Gaussian processes and improving identifiability with richer readout channels.Source: §3 · pp. 15–16

Open-system model of the NV center

• The NV center is modeled as a ten-level open system containing ground and excited spin manifolds, orbital structure, and a metastable singlet.Thirty-one jump operators represent optical pumping, radiative decay, intersystem crossing, singlet decay, phonon processes, and relaxation.Source: Open-system model of the NV center · p. 16

Tangent solve and pulses

• Field derivatives co-propagate with the density matrix, producing the exact spin-readout derivative from one open-system solve for an ideal Ramsey sequence.The sensed field enters through the effective detuning, with instantaneous π/2 pulses represented in the coherent initial state.Source: Tangent solve and pulses · p. 17

Optical readout and photon statistics

• Optical contrast is computed from a separate first-principles ten-level optical-cycle solve and depends on temperature, strain, and laser power without fitted readout parameters.For the Fig. 2c budget, ⟨C⟩=0.46, Ceff=0.137, and Ndet = 4.04 × 10^8 detected photons per shot.Source: Optical readout and photon statistics · p. 17

Charge-state and saturation measurements

• Measured NV− charge fractions and fluorescence saturation are incorporated through the power-dependent effective contrast Ceff(P) = Coptical(P) fNV−(P) and photon budget Ndet(P).Measurements used two diamond growths spanning 0.3 and 4.5 ppm under 532 nm excitation.Source: Charge-state and saturation measurements · pp. 17–18

Stochastic terms and frequency response

• Time-dependent noise is synthesized from power spectral densities and can enter as Zeeman detuning or drive noise, while deterministic main-text results omit these traces.Frequency response ηB(f) is obtained by driving the model at frequency f and summing the two Fisher-information quadratures.Source: Stochastic terms and frequency response · p. 18

Estimation and attribution

• Shapley attribution assigns signed, order-independent mechanism shares that sum exactly to each metric, unlike leave-one-out differences when error processes interact.A source that improves a metric receives a negative contribution; Shapley averaging removes activation-order dependence.Source: Estimation and attribution · pp. 18–19
• Sensitivity attributions agree with leave-one-out estimates within a few percent, whereas accuracy bias leaves approximately 90% in the residual without Shapley partitioning.The accuracy interaction is physical: thermal shifts and leakage combine through nonlinear calibration inversion.Source: Estimation and attribution · p. 19
• Shapley attribution identifies each metric’s dominant mechanism consistently across realizations and parameter draws, while accounting for signed accuracy contributions and uncertainty.Accuracy shares use Poisson-readout resampling; sensitivity and robustness uncertainties are assessed parametrically. Leave-one-out accounting can deviate from Shapley shares by up to ∼20% of ηB.Source: Estimation and attribution · pp. 19–20

Numerical implementation

• Both master equations are integrated in batched form across spatial or per-pixel realizations, noise realizations, parameter sweeps, and per-mechanism dissipative-channel toggles; when dynamics close on the {|0⟩, |−1⟩} subspace, two-level reduction reproduces the ten-level result.Source: Numerical implementation · p. 20

GPU acceleration

• GPU batching evaluates attribution and design-space sweeps as single propagations, running 50–100× faster than the NumPy CPU reference.The implementation uses eight NVIDIA H100 GPUs with the cuQuantum cuDensityMat backend and supports up to 4096 noise realizations in benchmark sweeps.Source: GPU acceleration · p. 20

Scalar optically-pumped magnetometer

• The cesium magnetometer instantiation uses a Breit–Rabi Hamiltonian, three mean-field relaxation channels, tangent Fisher-information sensitivity, and noise-based robustness and localization budgets.Sensitivity budgets split ηB into spin-projection and photon-readout shot-noise terms; localization design varies effective per-sensor field noise and calibration error.Source: Scalar optically-pumped magnetometer · p. 21

Magnetocardiography recording

• The magnetocardiogram uses a 26-channel unshielded scalar magnetometer array, averaging ∼340 cardiac cycles before fitting the QRS-peak current-dipole field.The recording was obtained from an author-participant, with array acquisition described in Ref..Source: Magnetocardiography recording · p. 21

• The human recording was a single, non-invasive, non-clinical technical measurement obtained from an author-participant and not used for diagnosis, treatment, or clinical decision-making.Source: Ethics declarations · p. 22
• The participant provided written consent for the recording, manuscript publication, and resulting processed MCG data.Source: Ethics declarations · p. 22
• Raw identifiable data will not be shared; processed MCG data may be available from the corresponding author on reasonable request, subject to privacy and data-governance requirements.Source: Ethics declarations · p. 22

Supplementary Information

• The supplement develops and validates the open-quantum-system model, tangent master equation, first-principles inputs, metric definitions, and Shapley attribution.Supplementary Notes 1–4 provide the model, closed-form validation, rate and contrast inputs, and metric-attribution framework.Source: Supplementary Information · p. 27
• The supplement includes numerical methods, parameter tables, and validation that support the deterministic single-point and real-device results reported in the main text.Its overview frames these materials as the derivational and numerical foundation for the paper’s main-text results.Source: Supplementary Information · p. 27
• The supplement adds a time-dependent-noise model, multi-GPU implementation, and per-pixel spatial output for large-scale analysis.These developments are exercised only in the supplementary results, unlike the deterministic single-point results in the main text.Source: Supplementary Information · p. 27
• The supplementary material expands the cross-platform optically pumped magnetometer analysis and describes experimental extraction of charge-state fraction from photoluminescence spectra.The second sensor platform is developed and exercised in the Supplementary Information, while real-device results are reported in the main text.Source: Supplementary Information · p. 27

1.1 NV Center Hamiltonian

• The 10-level NV− Hamiltonian combines intrinsic electronic structure, spatiotemporal controls and sensed fields, and time-dependent noise while accounting for temperature-dependent rates, laser leakage, and strain.The implementation uses ground, excited, and shelving states; interaction-picture and rotating-wave treatments retain relevant control and noise terms in the final 10 × 10 model.Source: §1.1 · p. 27

1.2 Lindblad Master Equation

• The digital twin solves a Lindblad master equation for density-matrix trajectories, combining time-dependent noise, control protocols, external fields, and material-dependent energy levels to predict sensitivity and accuracy.Dissipative processes are represented by time- and spatially dependent rates and corresponding non-unitary jump operators in the interaction frame.Source: §1.2 · p. 31
• The Lindblad model treats ground-state channels consistently across interaction and Schrödinger pictures, while excited-manifold cross terms are removed by secular orbital averaging above approximately 80 K.Global phases cancel inside the dissipator; the dissipation-rate inputs and attribution operating point are tabulated separately.Source: §1.2 · p. 32, Table S3 · p. 33

1.3 Tangent Master Equation

• The tangent master equation directly propagates parameter derivatives with the density matrix, avoiding repeated perturbed simulations and supplying derivatives for Fisher information and sensitivity bounds.The construction preserves the density matrix’s physical properties while tangent operators remain generically indefinite.Source: §1.3 · p. 34

Tangent master equation for magnetic-field sensing (θ = bsense)

• For magnetic-field sensing, b_sense enters explicitly through the ground-manifold detuning; fixed collapse operators and b-independent rates make only the Hamiltonian contribute to its tangent equation.The initial tangent is zero when NV initialization is field-independent, and the resulting measurement derivative feeds Fisher information for Ramsey or echo sequences.Source: Tangent master equation for… · pp. 34–35

1.4 Poisson Statistics and Photon Shot Noise

• Photon counting over independent emitters yields a binomial-to-Poisson Fisher-information framework that evaluates photon-shot-noise and time-dependent-noise impacts on sensitivity.The same treatment applies to the optically pumped magnetometer using transmitted-probe rotation and Larmor frequency, mapped to field through the gyromagnetic ratio.Source: §1.4 · p. 36

2.1 Numerical checks of the assembled model

• The model reaches ∼0.69 ms=0 polarization from several initial states within ∼2 µs under optical pumping, reproducing the NV− spin-initialization mechanism.Spin-selective intersystem crossing through the shelving singlet drives repolarization and sets the readout contrast.Source: §2.1 · p. 36, Figure S1 · p. 37
• The end-to-end digital twin recovers ˆB=1.91 ± 0.21 µT for a true 2.0 µT field, with realistic uncertainty ∼4.0 times the ideal projection Cramér–Rao bound.The gap reflects finite contrast, 1/f dephasing, and photon shot noise in the simulated readout.Source: §2.1 · pp. 37–38, Figure S2 · p. 38

2.2 Tangent master equation versus finite difference

• The tangent derivative agrees with central finite differences to better than 10−4 and reproduces closed-form Fisher limits to ∼5 × 10−5 without integration-grid drift.Under Markovian dephasing, the Fisher information also tracks the analytic expression with agreement ∼3 × 10−3.Source: §2.2 · p. 38, Figure S3 · p. 39
• Finite differences reach ∼10−12 only near (∆b)⋆≈5 × 10−11 T, whereas the tangent solve reaches the roundoff floor ∼2.2 × 10−16 without step tuning.Truncation and cancellation make the finite-difference optimum problem-dependent and require retuning ∆b at each operating point.Source: §2.2 · pp. 39–40

2.3 Benchmark table

• Table S4 documents reproducible validation checks against independent references for thermal apparent-field bias, readout contrast, Cramér–Rao sensitivity, and GPU/CPU parity.Each headline quantity used in the main text is traceable to a documented check at the stated tolerance.Source: §2.3 · p. 40, Table S4 · p. 41

Numerical Implementations and Benchmarks

• Sensitivity is computed from total Fisher information accumulated over repeated sensing cycles, with the Cramér–Rao bound determining the reported ηB.The simulation propagates ρI(t) and Gb(t), computes p0(b) and ∂bp0(b), assembles F1(b), and reports ηB.Source: Numerical Implementations and Benchmarks · p. 41

3.1 Temperature-Dependent Phonon Processes

• The operating-range rate includes the two-phonon Raman term of Eq. (S22) up to cutoff ℏΩ = 0.168 eV and scales as T^5 at low reduced temperature.Same-direction two-phonon channels contribute below 1% for the strains considered.Source: §3.1 · pp. 42–43

3.2 Optical readout: contrast and photon budget

• The open-system readout model computes effective contrast Ceff and detected-photon budget Ndet rather than assuming them, incorporating temperature, strain, charge state, and collection efficiency.A tabulated contrast surface replaces per-point optical solves; interpolation error stays below 2% in the operating region and yields a ∼10^6× speed-up over 10^4 direct solves.Source: §3.2 · p. 43, Figure S5 · p. 44

attribution

• Sensitivity, accuracy, and robustness are functionals of one open-system solve, evaluated at nominal parameters except that robustness measures sensitivity stability under parameter drift.The solve uses the field-dependent density matrix and its field tangent; sensitivity and accuracy are read at θ0, while robustness probes drift about θ0.Source: attribution · p. 44

4.3 Per-mechanism attribution by Shapley values

• Shapley attribution assigns each mechanism a unique, order-independent share of metric degradation, using mechanism-specific off operations and metric-specific value functions.The thermal off operation is carrier recalibration rather than removal of a physical channel; non-additivity captures mechanism interaction, while thermal recalibration leaves the readout contrast constant across coalitions.Source: attribution · p. 44, §4.3 · p. 47

4.1 The three metrics as functionals of one solve

• Sensitivity uses Fisher-information-limited resolution, accuracy separates deterministic calibration bias from shot-noise precision, and robustness quantifies drift-induced sensitivity inflation.R = 1 denotes a stable operating point, while R > 1 represents fractional sensitivity inflation; the linearized budget assumes uncorrelated parameter drifts, with correlated drifts adding cross terms and non-additivity.Source: §4.1 · pp. 45–46

4.2 Unbiased and biased Cram´er–Rao bounds: precision versus accuracy

• The biased Cramér–Rao bound separates achievable precision, set by Fisher information and ηB, from the accuracy floor, set by the independent bias β.Thus, minimizing ηB or maximizing Fisher information does not guarantee a small β, because the two metrics are bounded by different quantities and mechanisms.Source: §4.2 · p. 47

4.3 Per-mechanism attribution by Shapley values

• The accuracy budget requires order-averaged Shapley attribution because dephasing lowers visibility and nonlinearly changes the thermal apparent-field bias, whereas sensitivity is effectively additive.Thermal alone gives 282 nT, dephasing reduces this to 189 nT, and leakage brings the full-device value to 110 nT; the dephasing–thermal cross term is −91 nT.Source: §4.3 · pp. 49–50
• Accuracy non-additivity is dominated by pairwise terms but includes a −2.1 nT higher-order remainder, about 2% of the total, while the sensitivity remainder is essentially zero.A single leave-one-out ordering leaves 89% of the accuracy bias in a non-additive residual, whereas Shapley returns a unique, order-independent partition.Source: §4.3 · p. 50
• −91 nT from dephasing–thermal interaction accounts for 83% of accuracy degradation, while sensitivity interactions remain at most 4% of total and LOO agrees with Shapley within a few percent.The full coalition values underlying these decompositions are listed in Supplementary Table S5, and the interaction indices and LOO residuals in Supplementary Table S6.Source: §4.3 · p. 49, Table S5 · p. 50, Table S6 · p. 51

4.4 Uncertainty quantification of the attribution shares

• Across swept assumptions, the dominant mechanism of each metric is unchanged; leakage remains the leading robustness contributor when the leakage-drift budget satisfies σε ≳ 1.8 × 10−6.At the adopted σε = 10−5 assumption, the leakage share of Rlin is approximately 96%.Source: §4.4 · p. 51
• Accuracy is shot-noise-limited at low photon number but saturates at the thermal Dgs bias once σb averages below it, becoming nearly flat across the operating range.The shot-noise precision floor scales as σb ∝1/(Ceff N), while the thermal Dgs bias is 282 nT.Source: §4.4 · p. 51, Figure S6 · p. 52
• Across five two-parameter design planes, sensitivity is dominated by dephasing, robustness by optical leakage, and accuracy by the thermal Dgs shift over most of the design space.Optical leakage takes over for accuracy only in high-leakage, low-thermal-drift corners, showing that metric-dependent limiter inversion is not specific to one design.Source: §4.4 · p. 52

Noise Models

• Across every design plane, dephasing limits sensitivity, optical leakage limits robustness, and the thermal ground-state shift limits accuracy.Figure S7 shows these dominant-limiter boundaries across the tested parameter planes and metrics.Source: Noise Models · p. 53

5.1 Noise sources

• The model includes field, phase, amplitude, and optical-rate stochastic channels plus photon shot noise, inserted into Hamiltonian or dissipative-rate terms at their physical locations.Phase noise affects effective detuning through its derivative, amplitude noise scales the control envelope, field noise shifts the Zeeman term, and optical noise modifies collapse-operator rates.Source: Noise Models · p. 52, §5.1 · p. 54

5.2 From PSDs to Time-Domain Noise Traces and Hamiltonian Insertion

• The band-limited synthesis preserves the target 1/f slope at −1.00 across the sensing band, unlike naive decimation, which aliases it toward −0.64 and inflates variance.The delivered variance remains independent of the oversampling factor under band-limited synthesis, as shown in Figure S8.Source: §5.2 · p. 54, Figure S8 · p. 55

5.3 Sensitivity under time-dependent noise

• Quasi-static 1/f2 and 1/f field noise produce the deepest, most sharply rising sensitivity curves, whereas a ∼2 MHz Ornstein–Uhlenbeck process gives the shallowest.Sensitivity is computed from 4096 independent realizations by co-propagating the density matrix and field derivative under stochastic detuning.Source: §5.3 · p. 56, Figure S9 · p. 57

Simulation Methods

• The simulations propagate NV dynamics and tangent-space equations under noise realizations synthesized from Section 5 protocols.The resulting observables support readout probabilities, field tangents, and Fisher-information calculations.Source: Simulation Methods · p. 57, §6.1 · p. 58

6.1 Multi-GPU CUDA-Q Backend

• The backend uses cuDensityMat mixed-state simulation with MPI-distributed batches spanning spatial points, noise realizations, and pulse-sequence indices.Each GPU evolves its shard of a batch sized B = |r| × |ξ| × |s| through an MGMN communicator.Source: §6.1 · p. 57
• The Liouvillian includes Hamiltonian commutators and dissipators, while time-dependent drives, noise traces, and detuning enter through call-time parameters.Fixed-step RK4 advances the density matrix; tangent propagation reuses the operator action with the additional source term ∂θL ρ.Source: §6.1 · p. 58
• A 64 × 64 synthetic grid with 64 noise realizations per pixel and Nt=1500 steps produced GPU-scale, per-pixel metric-dependent attribution in one run.The backend generated six maps across eight GPUs, with each pixel carrying distinct material parameters and independent open-system dynamics.Source: §6.1 · p. 59, Figure S10 · p. 60

model and cross-platform test

• The cross-platform test applies the same open-system, Fisher-sensitivity, and design-margin construction to cesium-133 vapor and a real magnetocardiography recording.It tests whether metric-dependent, device-grounded provisioning transfers from the original sensing platform.Source: model and cross-platform test · p. 61

7.1 Cesium-133 ground-state model and Breit–Rabi spectrum

• The cesium ground-state model uses the exact Breit–Rabi spectrum, reproduces its eigenvalues to a few parts in 10^6 across the geomagnetic range, and identifies quadratic Zeeman curvature as the source of heading dependence.The stretched F = 4 manifold has γ ≈ 3.50 Hz/nT in the low-field regime.Source: §7.1 · p. 62

7.2 Mean-field spin-exchange master equation and readout

• The OPM dynamics use a mean-field master equation in which spin-exchange collisions couple each atom to the ensemble spin, producing a nonlinear dissipator alongside spin destruction and optical pumping.The density matrix is read out through free-induction decay on two quadratures, recovering amplitude and phase independently.Source: §7.2 · pp. 63–64

7.3 Tangent solve for the OPM

• The field tangent is co-propagated with the density matrix, including the Frechet derivative of nonlinear spin-exchange feedback, which distinguishes the OPM solve from the linear NV Lindbladian.The tangent directly provides ∂B⟨Fy(t)⟩ without a finite-difference reference.Source: §7.3 · p. 64

7.4 Cram´er–Rao sensitivity

• The OPM sensitivity is more than four orders of magnitude below unshielded ambient noise across the cardiac band, so the quantum layer never limits measurement.The conclusion holds under both assumed readout-noise and spin-projection-noise models; a 1 ms window would inflate ηB by approximately 2.4×.Source: §7.4 · p. 65

7.5 Heading-error accuracy bias

• Heading-dependent nonlinear Zeeman, nuclear-Zeeman, and light-shift effects set a geometry-dependent accuracy floor without entering sensitivity, while cardiac beat averaging suppresses their near-DC offsets.The nonlinear Zeeman contribution dominates; the array-level heading bias is small after R-peak-aligned averaging.Source: §7.5 · p. 65

7.6 Common-mode rejection by principal-component analysis

• Common-mode rejection removes the dominant spatially correlated ambient field by principal-component analysis; retaining one component is conservative because additional components remove genuine spatial structure.The cardiac-band residual power spectral density defines the rejected floor, with k=1 suppressing it substantially.Source: §7.6 · p. 66

7.7 Equivalent-current-dipole localization

• The application-level localization metric models the cardiac source as an equivalent current dipole with moment m at position r0 using an exact point-dipole lead.This establishes the source model for evaluating cardiac-field localization accuracy.Source: §7.7 · p. 66
• 0.37 mm localization error at 1% gain match is about 14× below the 5 mm clinical specification, making array-level field noise—not atomic sensitivity—the limiting factor.The error rises to approximately 60 mm at raw single-channel ambient noise and scales with cross-sensor gain mismatch.Source: §7.7 · p. 66, Figure S11 · p. 67

7.8 From the model to the Fig. 5 panels

• The cesium model and the single unshielded recording generate every Fig. 5 quantity through the same core routines, including cardiac waveforms and the ∼7 cm dipole fit.The R-peak-aligned average resolves P, QRS, and T waves, while the equivalent-current-dipole model reproduces the measured 26-sensor QRS field.Source: §7.8 · pp. 67–68
• 265 pT residual accuracy spread remains after shared heading-error bias calibrates out across sensors clustered within 1–2° of θ ≈90°.Near θ ≈90°, the bias varies with slope |dβ/dθ| ≈0.12 nT/deg and is nearly common-mode.Source: §7.8 · p. 68, Figure S12 · p. 68
• 13× common-mode rejection reduces integrated ambient noise from 58.7 to 4.4 pT/ while removing at most 0.01% of the QRS amplitude.The leading spatial mode carries 99.9% of array variance and is nearly orthogonal to the cardiac dipole pattern.Source: §7.8 · p. 67, Figure S12 · p. 68
• 4.0 fT/Hz sensitivity is obtained at an optimal sensing window of approximately 4.9 ms from the modeled free-induction decay and field tangent.The same solve supplies frequency-resolved sensitivity and a Larmor-versus-Breit–Rabi check.Source: §7.8 · p. 67, Figure S12 · p. 68

luminescence spectra

• Only NV− carries spin contrast, so its fraction controls the effective readout contrast, sensitivity, and photon budgets.The effective contrast is Ceff = Coptical fNV−, and the measured charge fraction versus optical power supplies the Fig. 4 analysis.Source: luminescence spectra · p. 69

8.1 Experimental setup and samples

• A 532 nm confocal measurement varied excitation power from 0.1 to 50 mW, while fixed 355 nm excitation provided a predominantly NV0 reference from two diamond substrates.The setup used a 532 nm excitation laser and a 355 nm reference laser held at 3 mW [9].Source: §8.1 · p. 69

8.2 Spectral features

• The charge-state zero-phonon lines occur at 575 nm for NV0 and 637 nm for NV−, with broad sidebands extending roughly 100 nm to longer wavelengths.The 532 nm-excited diamond Raman line at 572.6 nm overlaps the rising edge of the NV0 zero-phonon line and is unresolved separately.Source: §8.2 · p. 69

8.3 Spectral decomposition

• The decomposition constructs a pure NV0 line shape from a 355 nm reference and obtains NV− by subtracting the scaled NV0 contribution from 532 nm spectra.The reference spectrum is normalized to unit area, while the NV0 scaling uses the 565–585 nm window where NV− emission is negligible [9].Source: §8.3 · p. 70
• Raman overlap in the 565–585 nm scaling window can bias fNV− slightly low, but the unresolved line leaves a fit residual below 1% of peak intensity.A Raman-free window or explicit Raman basis component would remove this bias.Source: §8.3 · p. 70
• Corrected NV0 and NV− photoluminescence intensities yield the NV−/NVT fraction fNV− used in effective contrast and plotted versus optical power.Non-negative least-squares decomposition achieves a root-mean-square residual below 1% of peak intensity across the dataset.Source: §8.3 · p. 70

Data and code availability

• Charge-state fractions and fluorescence-saturation curves from two diamond growths and three confocal positions set Fig. 4’s power-dependent contrast and photon budget.The growths contained 0.3 and 4.5 ppm nitrogen and used 532 nm excitation.Source: Data and code availability · p. 71
• Per-pixel maps of T2 (Ramsey), T2 (Hahn echo), T1, diagonal stress, and a Ramsey χ2 validity mask support spatially resolved accuracy and sensitivity analysis.The maps were digitized from published characterization figures, with quantization caveats documented in the dataset README.Source: Data and code availability · p. 71
• Imaged sensing and control magnetic-field distributions from a current-carrying wire were sampled on a 64 × 64 grid for four ⟨111⟩ NV orientations.Source: Data and code availability · p. 71
• A single unshielded scalar optically pumped magnetometer recording of a human heartbeat supplied the magnetocardiogram used in Fig. 5, while supporting code and data are available on reasonable request.Source: Data and code availability · p. 71

adaptive, meta-learned control

• A single tangent solve returns the objective gradient, sensor metrics, and Shapley shares, while known parameter couplings make the generalized control vector θ agnostic to physical knob parameterization.The coupling groups control coordinates into mechanism-aligned blocks for targeted updates.Source: adaptive, meta-learned control · p. 72
• The twin generates physically accurate synthetic training data with ground-truth per-mechanism labels, enabling meta-training of a control policy for a device family.Mechanisms can be activated or suppressed in the model, providing supervision unavailable from hardware measurements; batched multi-GPU propagation scales this campaign.Source: adaptive, meta-learned control · p. 72
• Expected adaptation gain grows with measured device variance and saturates with adaptation steps K, with a ceiling A∞ and rate constant β setting diminishing returns.The section reproduces the scaling-law form; its derivation, β, and validity conditions are attributed to Leclerc et al..Source: adaptive, meta-learned control · pp. 72–73
• Mechanism-targeted calibration applies masked gradient descent only to the diagnosed control block, holding inactive parameters fixed and limiting updates by recoverable gain.The online procedure probes the device, estimates task parameters, and applies θ ← θ − η ΠS ∇θL for K steps.Source: adaptive, meta-learned control · pp. 73–74
• Per-mechanism targeting requires an informative readout: scalar measurements can make thermal shifts, dephasing, and optical leakage indistinguishable, whereas time-resolved or spectral data separate them.The twin can rank candidate readouts using the conditioning of their mechanism Fisher matrix; NV and cesium examples show distinct signal-shape separation.Source: adaptive, meta-learned control · p. 73
• The per-mechanism budget distinguishes runtime-adjustable channels from fabrication-fixed properties, allocating effort toward better operation or design; the proposed loop remains un demonstrated.The section emphasizes operational accuracy because slowly varying bias accumulates over time, but states that the closed loop is a direction rather than a demonstration.Source: adaptive, meta-learned control · pp. 73, 74, 76
• End-to-end hardware closure remains open within the controllability and model-fidelity assumptions required by each component.The passage identifies these assumptions as an unresolved condition for closing the loop on hardware.Source: adaptive, meta-learned control · p. 77
• The proposed hardware role depends on satisfying the assumptions required by each component.The supplied passage leaves the specific component assumptions unspecified.Source: adaptive, meta-learned control · p. 77
• The approach reframes hardware from a passive subject of environmental drift toward an active participant in its own behavior.This is presented as the appeal of changing what the hardware is asked to do.Source: adaptive, meta-learned control · p. 77