The Statistical Turn in Trapped-Ion Noise Diagnostics
From Heating Rates to a Decision Engine
Stewardship: U. Warring Status: Draft · February 2026
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Harbour Architecture Compliance Notice
This document is a Sail (essay + handbook-facing protocol narrative) within the Open-Science Harbour. It is classified Tier 2: an operational synthesis that imports physics from Tier 1 coastlines and boundary conditions from Tier 0, but does not itself constitute a fundamental law or a novel theoretical result.
Authority flows from use, not endorsement. The frameworks referenced herein—macroscopic QED, the Lévy–Khintchine decomposition, translation-covariant quantum Markov semigroups—are Tier 1 imports under local stewardship. Standard-Model electrodynamics and quantum mechanics are external coastlines referenced as constraints. No claim in this Sail exceeds the validity domain stated in §9.
Lock-Key separation applies. The conceptual locks (three operational classes, two-scalar classifier, validation architecture) remain stable; interpretive keys (specific scattering models, mitigation strategies, shot-count estimates) remain free and revisable.
Distribution scales, not authority.
Abstract
Motional decoherence in trapped-ion systems is conventionally characterised by a single scalar—the heating rate nˉ˙. This reporting convention discards statistical information that distinguishes fundamentally different noise mechanisms requiring different mitigation strategies. We present a Tier-2 operational framework—the Decision Engine—that classifies electromagnetically coupled noise into three operational classes (Gaussian diffusion, compound-Poisson jumps, Lévy-type heavy-tailed processes) within the validity domain of harmonic confinement, Markovian bath dynamics, and stationary statistics. Classification proceeds via two dimensionless scalars: the event-count parameter λτ and the Markov-accessibility parameter μ=τ/τc. We provide a numbered decision protocol with explicit gate nodes, shot-count estimates, and a three-pillar validation architecture incorporating mandatory consistency closure. All claims are bounded to single-ion motional decoherence under linear electromagnetic coupling; extensions beyond this domain are flagged as out-of-scope branch conditions.
1. Introduction — The Statistical Turn
1.1 Anomalous heating as bottleneck
The scalability of trapped-ion quantum information processing remains constrained by motional decoherence—the uncontrolled degradation of the harmonic oscillator's quantum state through interaction with electromagnetic environments. For over three decades, the community has characterised this process through a single figure of merit: the mean heating rate nˉ˙, typically extracted from sideband thermometry after variable delay times. Comprehensive reviews have catalogued the dependence of nˉ˙ on frequency, ion–surface distance, and temperature, seeking correlations with surface-science models of patch potentials, adatom fluctuators, and electrode contamination (Brownnutt et al., 2015).
1.2 Why heating-rate-only reporting is epistemically lossy
The reduction of complex environmental dynamics to a single scalar discards the vast majority of statistical information carried by the noise signal. Two physically distinct noise processes—one consisting of many weak, continuous fluctuations, the other of rare but violent momentum kicks—can produce identical values of nˉ˙ whilst requiring orthogonal mitigation strategies. The heating rate is a second-moment quantity; it is blind to skewness, kurtosis, waiting-time structure, and tail behaviour. Under the implicit assumption of Gaussian statistics, the power spectral density SE(ω) is a sufficient statistic. When that assumption fails—as it must for discrete collision events, charging transients, or heavy-tailed excursions—the spectral approach systematically misdiagnoses the environment.
1.3 Statistical Universality Hypothesis (bounded)
Within the validity domain of single-ion harmonic confinement, linear electromagnetic coupling, Markovian bath dynamics, and stationary noise statistics (§9), we advance the following bounded hypothesis: all electromagnetically coupled noise sources acting on a trapped ion's motional degree of freedom fall into one of three operational classes, distinguished by the statistical character of the environmental current sources $\mathbf{J}(\mathbf{r}, t)$ that drive the fields. These classes—Gaussian (diffusive), compound-Poisson (jump-dominated), and Lévy-type (heavy-tailed)—arise from the Lévy–Khintchine decomposition of infinitely divisible processes and are exhaustive within the stated domain. The classification is operational: it determines which observables are informative and which measurement strategies are optimal, independently of the microscopic identity of the source.
This hypothesis does not claim universality across all quantum systems, all coupling mechanisms, or all decoherence channels. It is restricted to motional decoherence of a single trapped ion under electromagnetic coupling within the Markov-accessible zone defined below.
1.4 Deliverable: Decision Engine and validation separation
The primary deliverable of this Sail is the Decision Engine (§7)—a procedural protocol that converts abstract statistical classification into a linear sequence of experimental decisions. The secondary deliverable is the Validation Architecture (§8), which enforces rigorous separation between inference validity (did we identify the correct statistical class?) and physical predictability (does the identified class match the microscopic model?), joined by a mandatory consistency closure test.
2. Tier Map and Reader Contract
Tier Map
T0 — Boundary Conditions
Harmonic confinement; linear coupling $H_\text{int} = -qxE_x$; Markovian bath ($\mu \gg 1$); stationarity; single-ion restriction
Define the validity domain. When any T0 condition fails, the Decision Engine degrades (see below).
External constraints; not revisable within this Sail.
T1 — Physics Imports
Macroscopic QED / Green tensor $\mathbf{G}(\mathbf{r}_0, \mathbf{r}; \omega)$; Lévy–Khintchine decomposition; translation-covariant Markov semigroups (Holevo, Vacchini); quantum linear Boltzmann equation
Provide the theoretical foundation. Imported, not derived here.
Coastlines under broad endorsement (QED) or local stewardship (Holevo–Vacchini structure theorems).
T2 — Decision Engine
Two-scalar classifier ($\lambda\tau$, $\mu$); three operational classes; decision protocol; validation architecture; shot-count estimates
The operational synthesis—this Sail's contribution.
Local stewardship; revisable as experiments refine estimates.
Degradation Protocol
When a Tier-0 boundary condition fails, the Decision Engine does not produce undefined output; it degrades gracefully:
Markov breakdown ($\mu \lesssim 1$)
Bath correlation time $\tau_c$ comparable to or exceeding interrogation time $\tau$
Downgrade to screening heuristic only. Flag output as "Markov-inaccessible; classification provisional." Open non-Markov branch: recommend memory-kernel analysis or fractional-Brownian-motion extension.
Anharmonic confinement
$\lvert\beta\rvert x_0 \gtrsim m\omega_t^2$ or large motional excitation
Downgrade dephasing predictions; heating classification may remain valid if coupling remains approximately linear. Flag enhanced dephasing channel.
Non-stationarity detected
Drift in calibration interleaves exceeds tolerance
Suspend classification. Require drift subtraction and re-entry.
Multi-ion correlation
More than one ion in shared potential
Engine scope exceeded. Flag as "multi-mode extension required."
Reader Contract
To the reader. This document is a Tier-2 operational guide. It synthesises established physics (Tier 1) into a practical classification protocol, bounded by explicit assumptions (Tier 0). It does not introduce new fundamental physics, claim exhaustive coverage beyond the stated domain, or guarantee that the Decision Engine will correctly classify noise under conditions violating the boundary conditions listed above.
When the Engine says "Gaussian," it means: within the Markov-accessible zone, under harmonic confinement, with stationary statistics, the data are consistent with the Gaussian operational class at the stated confidence level. The Engine's output is a provisional classification subject to the three-pillar validation of §8.
The Decision Engine is a tool for structuring experimental enquiry, not a substitute for physical judgement. Proceed accordingly.
3. Universal Mediation Principle (T1 Import)
The interaction between a trapped ion at equilibrium position $\mathbf{r}_0$ and its electromagnetic environment is governed by
$$H_\text{int} = -qx , E_x(\mathbf{r}_0, t),$$
where $E_x$ is the $x$-component of the electric field. The field is not an independent degree of freedom; it is generated by stochastic current densities $\mathbf{J}(\mathbf{r}, t)$ distributed throughout the environment—electrodes, dielectrics, vacuum chamber walls, background gas—and mediated by the electromagnetic Green tensor:
$$E_x(\mathbf{r}0, \omega) = i\mu_0 \omega \int \mathrm{d}^3 r ; G{x\alpha}(\mathbf{r}0, \mathbf{r}; \omega) , J\alpha(\mathbf{r}, \omega).$$
The force spectrum entering the master equation is
$$S_F(\omega) = q^2 \int \mathrm{d}^3 r , \mathrm{d}^3 r' ; G_{x\alpha}(\mathbf{r}0, \mathbf{r}; \omega) , S{JJ}^{\alpha\beta}(\mathbf{r}, \mathbf{r}'; \omega) , G_{x\beta}^*(\mathbf{r}_0, \mathbf{r}'; \omega).$$
This factorisation is the Universal Mediation Principle (within the linear-coupling domain): the problem separates into a deterministic filter ($\mathbf{G}$, encoding trap geometry and material response) and a stochastic driver ($\mathbf{J}$, encoding the temporal dynamics of the source). The geometry does not change the statistical class of the noise—only its magnitude and spectral colour. A Gaussian source produces Gaussian fields at the ion; a Poissonian source produces Poissonian fields. Linear response preserves the character of the distribution.
What changes / what doesn't
Statistical class (Gaussian / Poisson / Lévy)
Source current statistics $\mathbf{J}(\mathbf{r}, t)$
No (within linear coupling)
Spectral shape of $S_F(\omega)$
Both $\mathbf{G}$ and $S_{JJ}$
Yes — $\mathbf{G}$ filters spectral content
Overall noise magnitude
Both $\mathbf{G}$ and $S_{JJ}$
Yes — geometric coupling efficiency
Heating/dephasing balance
Spectral weight near $\omega_t$ vs. near $0$
Yes — $\mathbf{G}$ redistributes spectral weight
Directional anisotropy
Spatial structure of $\mathbf{G}$
Yes — extended trajectories break isotropy
Implication for diagnostics. The trap can be treated as a known filter. Diagnostic effort focuses on analysing the output statistics of the ion's motional state to infer the input statistics of the environment.
4. Regime Classifier
Classification proceeds via two dimensionless scalars that gate all subsequent analysis.
4.1 The event-count parameter $\lambda\tau$
$$\lambda\tau = \text{(characteristic event rate)} \times \text{(interrogation time per shot)}.$$
For impulsive processes, $\lambda$ is the Poisson arrival rate. For continuous fields, $\lambda^{-1}$ represents an effective correlation time, so $\lambda\tau \sim \tau/\tau_c$ counts independent fluctuation intervals. The parameter organises temporal statistics:
$\lambda\tau \gg 1$ (diffusive regime): Many events per shot. Central Limit Theorem applies. Statistics appear Gaussian regardless of single-event character.
$\lambda\tau \sim 1$ (intermediate regime): Zero to a few events per shot. Non-Gaussian cumulants become measurable. This is typically the most diagnostic regime.
$\lambda\tau \ll 1$ (sparse regime): Most shots show no event. Rare shots show large jumps. Waiting-time distributions become the primary observable.
Crucially, $\lambda\tau$ characterises event statistics, not spectral content. A Gaussian source with $1/f$ spectrum ($\lambda\tau \gg 1$) can produce strong dephasing but weak heating; a Poisson collision process ($\lambda\tau \ll 1$) with large momentum transfer produces primarily heating. The framework treats both aspects independently.
4.2 The Markov-accessibility parameter $\mu$
$$\mu = \frac{\tau}{\tau_c},$$
where $\tau_c$ is the bath correlation time—the timescale over which environmental correlations decay. The parameter gates the validity of the Markov approximation:
$\mu \gg 1$ (Markov-accessible zone): The interrogation time spans many bath correlation times. The memoryless approximation holds. The Lévy–Khintchine generator is valid. The Decision Engine operates normally.
$\mu \lesssim 1$ (Markov-inaccessible zone): Environmental memory persists across the measurement window. The generator formalism breaks down. The Decision Engine degrades to a screening heuristic. Non-Markov branches (memory-kernel methods, fractional Brownian motion) must be pursued separately.
4.3 Joint gating logic
Both scalars must be evaluated before classification proceeds. The $\mu$ gate is applied first: if $\mu \lesssim 1$, the Engine halts normal classification and flags the output as provisional. Only when $\mu \gg 1$ does $\lambda\tau$ determine which observables to prioritise.
Failure mode. If only $\lambda\tau$ is evaluated without $\mu$, a source with slow environmental memory ($\tau_c \sim \tau$) could be misclassified as sparse-Poisson ($\lambda\tau \ll 1$) when it is in fact non-Markovian. The $\mu$ gate prevents this diagnostic error.
5. Three Operational Classes (Bounded)
Within the Markov-accessible zone ($\mu \gg 1$), under harmonic confinement and stationary statistics, all electromagnetically coupled noise falls into three operational classes distinguished by the Lévy–Khintchine decomposition of the master-equation generator:
$$\dot{\rho} = -i[H_0, \rho] + \mathcal{D}_G[\rho] + \mathcal{J}_P[\rho],$$
where $\mathcal{D}_G$ is the Gaussian diffusion superoperator and $\mathcal{J}_P$ is the compound-Poisson jump superoperator. Heavy-tailed Lévy processes extend $\mathcal{J}_P$ to divergent-variance jump-size distributions.
5.1 Gaussian (Diffusive) — $\lambda\tau \gg 1$
Physical picture. Many weak, independent fluctuations aggregate via the Central Limit Theorem. Sources include Johnson–Nyquist noise in electrodes, control-field pickup, dense patch-potential fluctuations, and high-flux photon scattering.
Signature → Measurement → Inference → Mitigation lever
Signature
Linear heating $\bar{n}(t) = \bar{n}_0 + \dot{\bar{n}},t$; thermal phonon distribution $P(n) \propto \bar{n}^n/(\bar{n}+1)^{n+1}$; vanishing higher-order cumulants $\kappa_3 \approx 0$, $\kappa_4 \approx 0$; Allan variance $\sigma_A^2(\tau) \propto \tau^{-1}$.
Measurement
Power spectral density $S_E(\omega)$ via sideband heating-rate scans at multiple frequencies; sideband asymmetry mapping; Ramsey coherence decay envelopes.
Inference
$\dot{\bar{n}} \approx q^2 S_E(\omega_t)/(4m\hbar\omega_t)$. Spectral shape (white, $1/f^\alpha$, Lorentzian) identifies source class. Dephasing separately from low-frequency $S_{\delta\omega}(\Omega)$.
Mitigation lever (handbook-level strategy)
Reduce source amplitude (lower temperature, lower electrode resistance); spectral filtering (dynamical decoupling for dephasing); geometric optimisation (increase ion–surface distance to exploit $\mathbf{G}$ scaling).
Domain note. The PSD is a sufficient statistic only within this class. Applying spectral methods to non-Gaussian noise produces systematically biased estimates.
5.2 Compound-Poisson (Jump-Dominated) — $\lambda\tau \sim 1$ to $\ll 1$
Physical picture. Discrete, independent events deliver momentum kicks $\Delta p$ drawn from a distribution $\nu(\Delta p)$ with finite variance. Each kick implements the unitary displacement $U_{\Delta p} = \exp[-i\Delta p , x/\hbar]$, simultaneously changing the ion's energy and phase. Sources include background-gas collisions (Langevin, hard-sphere, resonant charge exchange), single-photon recoil at low flux, and bistable fluctuators (telegraph noise).
Signature → Measurement → Inference → Mitigation lever
Signature
Non-zero skewness $\kappa_3/\kappa_2^{3/2} \sim 0.3\text{–}1$; excess kurtosis; non-thermal $P(n)$ (displaced/coherent-state character after single kicks); exponential waiting-time distribution $p(\Delta t) = \lambda e^{-\lambda \Delta t}$; Allan variance rollover near $\tau \sim 1/\lambda$; correlated energy and phase fluctuations.
Measurement
Higher-order cumulants from $\sim 10^4$ stroboscopic shots; waiting-time histograms from time-stamped single-shot readout ($\sim 10^3$ events); jump-size distribution $P(\Delta n)$ from $\sim 10^5$ shots; Allan variance over extended traces (hours).
Inference
Event rate $\lambda$ from waiting-time exponential fit or Allan-variance rollover. Jump-size distribution $\nu(\Delta n)$ mapped to scattering cross-section $\mathrm{d}\sigma/\mathrm{d}\Omega$ via phonon transition matrix elements $P_{n \to n'}(\Delta p)$. Species identification from $\nu(\Delta p)$ shape: forward-peaked (Langevin), isotropic (hard-sphere), state-dependent (charge exchange).
Mitigation lever (handbook-level strategy)
Reduce event rate (improve vacuum, eliminate charging sources); time-domain filtering (erasure conversion—detect kick, discard corrupted shot); increase gate speed to operate between kicks ($\tau_\text{gate} \ll 1/\lambda$).
Domain note. The compound-Poisson class requires finite variance of $\nu(\Delta p)$. When variance diverges, the process enters the Lévy-type class.
5.3 Lévy-Type (Heavy-Tailed) — Power-Law $\nu(\Delta p)$
Physical picture. Jump-size distributions with power-law tails $\nu(\Delta p) \propto |\Delta p|^{-1-\alpha}$ for $0 < \alpha < 2$ produce divergent variance and anomalous diffusion $\langle(\Delta x)^2\rangle \propto t^{2/\alpha}$. Candidate mechanisms include Coulomb scattering on charged dust grains (long-range $1/r^2$ potential), micro-discharge events, cosmic-ray impacts, and possibly self-organised-critical surface-patch avalanches.
Signature → Measurement → Inference → Mitigation lever
Signature
Power-law tails in $P(\Delta n) \propto
Measurement
Large-$N$ single-shot datasets ($\sim 10^5\text{–}10^6$ shots); complementary cumulative distribution function (CCDF) on log–log scale; tail-exponent estimation (Hill estimator or maximum-likelihood stable-law fit); i.i.d. vs. clustered test (autocorrelation of extreme events).
Inference
Stability parameter $\alpha$ from tail fit. Feasibility gate applies (see §7 and §8.4): tail-exponent inference requires $N_\text{ev} \gtrsim 10^3\text{–}10^4$ events in the tail, implying acquisition time $T_\text{acq} \sim N_\text{ev}/(\lambda d)$ where $d$ is the tail-exceedance fraction. Beyond the stability horizon ($T_\text{acq}$ exceeds practical drift-free measurement windows), downgrade to detection-only: report "heavy tails detected" without quantitative $\alpha$.
Mitigation lever (handbook-level strategy)
Averaging is counterproductive (rare catastrophic events contaminate the mean). Use robust statistics (median, trimmed mean); threshold-based rejection of extreme excursions; erasure-conversion protocols in quantum error correction; reduce defect density (a single Lévy-type defect can dominate even in otherwise clean systems).
Critical distinction: Lévy-type heavy tails do not automatically imply long memory. A Lévy process can be i.i.d. (memoryless jumps with heavy-tailed sizes) or clustered (correlated inter-event times). The i.i.d. vs. clustered distinction must be tested explicitly via autocorrelation of exceedances. Clustering would indicate a non-Markov component and trigger the $\mu$-gate degradation.
6. Decision Engine
The following numbered protocol constitutes the Decision Engine. Each node references a measurable quantity and an order-of-magnitude shot count. Gate nodes are marked with ⊳.
Pre-requisite. Calibrate the stroboscopic measurement chain: ground-state preparation fidelity, readout fidelity, and baseline technical noise floor. Establish interleaved calibration shots for drift monitoring.
Shot-count reference (order-of-magnitude):
Detect $\kappa_3 \neq 0$ at $3\sigma$
$\sim 10^4$ shots
Reconstruct $\nu(\Delta n)$ (5 bins)
$\sim 10^5$ shots
Test heavy-tail exponent $\alpha$
$\sim 10^6$ events
Allan-variance rollover near $1/\lambda$
Hours of continuous trace
Sideband asymmetry frequency map
$\sim 10^2$ frequency scans
7. Validation Architecture
Pillar 1 — Classification Power (Inference Validity)
Question: Given finite data, can we reliably distinguish the three classes?
Method: Monte Carlo confusion matrix. Generate synthetic datasets for each class (Gaussian, compound-Poisson, Lévy-type) with parameters matching the experimental regime (including realistic readout fidelity, drift contamination, and finite $N$). Feed synthetic data through the Decision Engine. Compute $C_{ij} = P(\text{predicted class } i \mid \text{true class } j)$.
Requirement: Diagonal elements $C_{ii} > 0.90$ at the achieved shot count $N$. If off-diagonal confusion is high (e.g., Lévy frequently misclassified as Poisson), the data volume is insufficient. Action: acquire more shots or accept reduced classification confidence with explicit error bars.
Minimum shot counts for $>90%$ diagonal (indicative):
Gaussian vs. Poisson: $N \sim 10^3\text{–}10^4$
Poisson vs. Lévy tail: $N \sim 10^5\text{–}10^6$ (tail events)
Recommended tests: Anderson–Darling (sensitive to tail deviations) over Kolmogorov–Smirnov.
Pillar 2 — Physical Predictability (Model Correctness)
Question: Does the identified statistical class align with the physics predicted by the Green tensor and known source models?
Method: Compare inferred parameters against independent physical predictions:
If Class I (Gaussian) identified: heating rate should exhibit predicted geometric scaling ($d^{-4}$ for patch potentials, $d^{-2}$ for Johnson noise). Temperature dependence should match ($\dot{\bar{n}} \propto T$ for Johnson).
If Class II (Poisson) identified: event rate $\lambda$ should scale linearly with background pressure (for gas collisions) or laser intensity (for photon scattering). Pressure-independence falsifies the gas-collision assignment.
If Class III (Lévy-type) identified: candidate mechanism must physically support power-law momentum transfer (Coulomb scattering, not Langevin).
Cross-validation protocol: Deliberately perturb one physical parameter (e.g., worsen vacuum pressure by $10\times$) and verify that the Engine's inferred $\lambda$ responds as predicted. If it does not, the physical assignment is wrong even if the statistical classification is correct.
Green tensor benchmarking: Compare numerically computed $\mathbf{G}$ against measured heating-rate geometric scalings. Discrepancies indicate geometry-modelling errors that propagate into all quantitative predictions.
Pillar 3 — Consistency Closure (Mandatory)
Question: Do Pillars 1 and 2, taken together, produce a self-consistent picture?
Closure equation. Define the total predicted noise budget:
$$\sigma_\text{tot}^2 = \sigma_\text{Gaussian}^2 + \sigma_\text{Poisson}^2 + \sigma_\text{Lévy}^2,$$
where each term is computed from the inferred parameters of the respective class (diffusion coefficient, event rate $\times$ mean kick variance, tail contribution up to truncation). The closure test requires:
$$\left| \frac{\sigma_\text{tot}^2 - \sigma_\text{measured}^2}{\sigma_\text{measured}^2} \right| < \epsilon_\text{closure},$$
with $\epsilon_\text{closure}$ set by the combined uncertainty budget (statistical + systematic).
Failure actions:
If $\sigma_\text{tot}^2 < \sigma_\text{measured}^2$: a noise source is missing. Search for additional mechanisms.
If $\sigma_\text{tot}^2 > \sigma_\text{measured}^2$: parameters are overestimated or classes are double-counted. Review inference assumptions.
If closure fails after iterative refinement: archive as Divergent (per Council-3 Resolution Clause). Report the discrepancy explicitly; do not force agreement.
Pillar-3 is mandatory. Pillars 1 and 2 are necessary but not sufficient. Without closure, a statistically valid and physically plausible classification can still produce an inconsistent noise budget—indicating that the decomposition is incomplete or that the validity domain has been exceeded.
7.4 Feasibility Gate for Lévy Exponent Inference
Estimating the stability parameter $\alpha$ from tail data requires sufficient events in the power-law regime. The acquisition timescale is
$$T_\text{acq} \sim \frac{N_\text{ev}}{\lambda , d},$$
where $N_\text{ev} \sim 10^3\text{–}10^4$ is the minimum number of tail events, $\lambda$ is the total event rate, and $d$ is the fraction of events exceeding the tail threshold.
Stability horizon: The maximum drift-free measurement window achievable with interleaved calibration (typically hours to a day, depending on laboratory stability).
If $T_\text{acq} <$ stability horizon: proceed with $\alpha$ estimation.
If $T_\text{acq} >$ stability horizon: downgrade to detection-only. Report that heavy tails are detected but $\alpha$ is not reliably estimable at current event rates. Do not publish a point estimate of $\alpha$ without the requisite statistical power.
8. Limitations and Extensions (Strictly Bounded)
The Decision Engine operates within the validity domain specified in §2 (Tier Map). The following extensions lie outside the current scope; branch conditions are defined to prevent silent extrapolation.
Non-Markovian environments ($\mu \lesssim 1$). When bath correlation times exceed the interrogation window—as for $1/f$ patch-potential fluctuations with very low corner frequencies—the Lévy–Khintchine generator is not applicable. Branch: memory-kernel methods, fractional Brownian motion frameworks, or whitening of known spectral drifts before re-entry into the Engine.
Multi-ion correlated dynamics. When multiple ions share a common potential, the Green tensor becomes a matrix coupling spatial modes. Common-mode noise (typically Class I technical noise) and differential-mode noise (potentially Class II/III surface effects) can in principle be separated, but the single-ion restriction of this Sail does not cover the inference protocol for correlated multi-mode decoherence. Branch: extend $\mathbf{G}$ to multi-mode form; modify cumulant analysis for joint distributions.
Feedback and active stabilisation. Closed-loop systems modify the effective noise statistics seen by the ion. The open-loop classification of this Sail applies to the uncompensated noise; the effect of feedback on the residual noise class is a separate analysis.
Non-electromagnetic decoherence. Magnetic-field noise coupling to spin-motion entangled states, gravitational perturbations, and other non-electromagnetic channels are outside the universal-mediation framework. The Engine does not classify such sources.
Heavy-tailed Lévy models (quantitative). A full quantitative treatment of $\alpha < 2$ stable laws—including finite-size effects, truncated tails, and the crossover from compound-Poisson to genuinely heavy-tailed behaviour—requires dedicated theoretical and numerical work beyond the scope of this Sail. The current treatment provides a detection and feasibility framework; quantitative Lévy modelling is deferred to future coastline-level work.
All extensions are flagged as out of scope; branch conditions are defined.
9. Conclusion — What Changes for Labs
Reporting standard
The Decision Engine implies a richer reporting standard than the conventional $\dot{\bar{n}}$. We suggest that trapped-ion noise characterisation reports include, at minimum:
The heating rate $\dot{\bar{n}}$ (for backward compatibility).
The regime parameters $\lambda\tau$ and $\mu$ at the measurement conditions used.
The identified operational class (Gaussian / compound-Poisson / Lévy-type / mixed / indeterminate) with confidence level.
Higher-order cumulants $\kappa_3$, $\kappa_4$ (even if consistent with zero—the null result is informative).
The full phonon distribution $P(n|\tau)$ or sufficient statistics thereof.
Validation status: which pillars passed, whether closure was achieved.
A trap with low $\dot{\bar{n}}$ but Lévy-type statistics is, for quantum error correction purposes, potentially worse than a trap with higher $\dot{\bar{n}}$ but confirmed Gaussian statistics—because uncorrectable extreme events defeat standard error correction codes whilst Gaussian noise can be managed by conventional filtering and averaging.
Protocol reference
This Sail is designed to serve as a protocol reference: the Decision Engine (§7) provides the operational flowchart; the validation architecture (§8) provides the quality assurance; the glossary (Appendix A) provides the shared vocabulary. Together, they define a reproducible methodology for statistical noise characterisation that can be adopted across laboratories without requiring each group to independently derive the underlying theory.
Appendix A — Glossary
Event rate
$\lambda$
Characteristic frequency of environmental disturbances (Hz). For collisions: $\lambda = n_g \sigma v$. For continuous noise: $\lambda \sim 1/\tau_c$.
Interrogation time
$\tau$
Duration of the idle/noise-evolution period in a single stroboscopic shot (s). A controlled experimental parameter.
Bath correlation time
$\tau_c$
Timescale over which environmental correlations decay. For Markovian baths, $\tau_c \ll \omega_t^{-1}$.
Event-count parameter
$\lambda\tau$
Dimensionless product counting expected independent noise events per shot. Organises temporal statistics: diffusive ($\gg 1$), intermediate ($\sim 1$), sparse ($\ll 1$).
Markov-accessibility parameter
$\mu = \tau/\tau_c$
Dimensionless ratio gating the validity of the Markov approximation. $\mu \gg 1$: Markov-accessible (Engine valid). $\mu \lesssim 1$: Markov-inaccessible (Engine degrades).
Event
—
A single discrete disturbance (collision, charging transient, photon recoil) producing a resolvable change in the ion's motional state. For continuous noise, an "event" is understood as one independent fluctuation interval of duration $\sim \tau_c$.
Jump
—
The instantaneous change in phonon number $\Delta n$ resulting from a single event. Related to momentum transfer $\Delta p$ via the displacement operator $U_{\Delta p}$ and phonon transition amplitudes.
Markov-accessible
—
A measurement regime in which the Markov approximation holds ($\mu \gg 1$) and the Lévy–Khintchine generator is a valid description of the dynamics.
Closure
—
The Pillar-3 consistency test requiring the sum of inferred noise contributions to match the total measured variance within the uncertainty budget.
Stability horizon
—
The maximum duration over which drift-free measurements can be sustained with interleaved calibration. Sets the upper bound on $T_\text{acq}$ for tail-exponent inference.
Appendix B — Lock-Compliance Checklist
B.1 Tier-0 Boundary Conditions
Box: T0 Boundary Conditions
Harmonic confinement: $U(x) = \frac{1}{2}m\omega_t^2 x^2$ with anharmonic corrections perturbative.
Linear coupling: $H_\text{int} = -qxE_x(\mathbf{r}_0, t)$; ion motion small compared to field-variation scale.
Markovian bath: $\tau_c \ll \omega_t^{-1}$; equivalently $\mu \gg 1$.
Stationary statistics: noise correlations depend only on time differences.
Single-ion restriction: no multi-ion correlated dynamics.
B.2 Degradation Triggers and Branch Actions
$\mu \lesssim 1$
Autocorrelation of noise proxy; $\tau_c$ estimate
Degrade to screening heuristic. Open non-Markov branch.
Non-stationarity
Drift in interleaved calibrations
Halt classification. Subtract drift or shorten acquisition window.
Anharmonic regime
Large motional excitation; measured frequency shifts
Flag enhanced dephasing. Proceed with caution for heating classification.
Multi-ion mode coupling
Correlated heating across ions
Flag scope exceeded. Extend to multi-mode $\mathbf{G}$.
Lévy $T_\text{acq} >$ stability horizon
Feasibility-gate computation
Downgrade to detection-only for tail exponent.
B.3 Claim Bounding List
"all EM noise" (unbounded)
"all electromagnetically coupled noise within the validity domain of §2"
"exhaustive" (unbounded)
"exhaustive within the Markov-accessible zone under harmonic confinement and stationary statistics"
"universal" (unbounded)
"universal within the domain of single-ion motional decoherence under linear EM coupling"
"the Lévy exponent is $\alpha = \ldots$" (without feasibility)
"the Lévy exponent is estimated as $\alpha = \ldots$ subject to the feasibility gate of §7/§8.4"
"Pillars 1 and 2 confirm…" (without closure)
"Pillars 1, 2, and 3 (consistency closure) confirm…"
"determines the regime" ($\lambda\tau$ alone)
"$\lambda\tau$, jointly with the Markov gate $\mu$, determines the accessible regime"
"noise spectroscopy" (implying completeness)
"statistical noise characterisation within the operational framework"
"applies to all quantum systems"
"applies to single-ion motional decoherence; platform generalisations require separate treatment"
Appendix C — Validation Architecture Box
Three-Pillar Validation Summary
1. Classification power
Can we distinguish the three classes at achieved $N$?
Monte Carlo confusion matrix with synthetic data
$C_{ii} > 0.90$ required
Acquire more data or widen confidence interval
2. Physical predictability
Does the class match the physics?
Compare inferred $\lambda$, $\nu(\Delta p)$ against $\mathbf{G}$-tensor predictions and independent physical controls (pressure, temperature, geometry scaling)
Quantitative agreement within uncertainty
Revise physical-model assignment; seek alternative source
3. Consistency closure
Do Pillars 1+2 sum correctly?
Closure equation: $\lvert\sigma_\text{tot}^2 - \sigma_\text{measured}^2\rvert / \sigma_\text{measured}^2 < \epsilon_\text{closure}$
Pass/fail
Missing source ($\sigma_\text{tot}^2$ too small) or double-counting ($\sigma_\text{tot}^2$ too large); iterate or archive as Divergent
Feasibility gate (Lévy tail inference): $T_\text{acq} \sim N_\text{ev}/(\lambda d)$. If $T_\text{acq} >$ stability horizon → downgrade to detection-only; do not publish point estimate of $\alpha$.
Pillar 3 is mandatory. Classification is provisional until closure is verified.
End of Sail v1.0-draft.
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