This work develops a microscopic account of sequence-dependent decoherencetime ratios in dynamical-decoupling (DD) spectroscopy. We show that a nontrivial ratio between coherence times measured under XY-8 and CPMG does not arise generically from standard Gaussian pure-dephasing theory: in the white-noise limit, Parseval’s theorem enforces R ≡ τXY/τCPMG = 1, so any deviation from unity must originate from spectral structure rather than total filter power alone. We then consider a minimal noncommuting-noise extension in which sequence-dependent channel redistribution shifts one transverse contribution sampled by XY-8 from ωeff to ωeff/2, while CPMG continues to sample the relevant channels at ωeff. In the narrow-band regime this yields the microscopic ratio law R = 2S(ωeff)/[S(ωeff) + S(ωeff/2)]. For power-law spectra S(ω) ωα, the ratio becomes R = 2/(1 + 2 α), giving R = 1 for white∝⁻ noise (α = 0), R = 4/3 for an Ohmic bath (α = 1), and R = 2/3 for 1/f noise (α = −1). The value R = 4/3 is therefore not universal but emerges as the special signature of an Ohmic spectrum. These results identify the XY-8/CPMG coherence-time ratio as a spectral-slope-sensitive observable and suggest a direct diagnostic strategy: measuring R in the narrow-band regime directly constrains the local bath exponent governing the sampled frequency window. This establishes a distinct theoretical direction from the effective benchmark treatment of Paper A and from the time-discontinuous channel-accessibility mechanism of Paper B, while remaining directly testable on NV-center platforms. V2: This version clarifies the phenomenological status of the XY-8/CPMG ratio law and reorganizes the manuscript so that the sequence-dependent spectral-slope interpretation is self-contained. Parts of the earlier Paper A framework have been re-examined and absorbed into the present paper in a revised form: rather than treating R=4/3R=4/3R=4/3 as a protocol-universal value, the ratio is now interpreted as a bath-dependent observable determined by the local spectral structure, with R=4/3R=4/3R=4/3 identified as the ideal Ohmic point. The revised version also makes the assumptions explicit (narrow-band regime, equal-weight transverse channels, and common exponential-decay form), strengthens the white-noise no-go benchmark, refines the inverse-problem discussion, and clarifies that the half-frequency XY-8 contribution is introduced as a minimal phenomenological assumption illustrated by a toy toggling-frame model rather than a complete microscopic derivation.
This work develops a microscopic account of sequence-dependent decoherencetime ratios in dynamical-decoupling (DD) spectroscopy. We show that a nontrivial ratio between coherence times measured under XY-8 and CPMG does not arise generically from standard Gaussian pure-dephasing theory: in the white-noise limit, Parseval’s theorem enforces R ≡ τXY/τCPMG = 1, so any deviation from unity must originate from spectral structure rather than total filter power alone. We then consider a minimal noncommuting-noise extension in which sequence-dependent channel redistribution shifts one transverse contribution sampled by XY-8 from ωeff to ωeff/2, while CPMG continues to sample the relevant channels at ωeff. In the narrow-band regime this yields the microscopic ratio law R = 2S(ωeff)/[S(ωeff) + S(ωeff/2)]. For power-law spectra S(ω) ωα, the ratio becomes R = 2/(1 + 2 α), giving R = 1 for white∝⁻ noise (α = 0), R = 4/3 for an Ohmic bath (α = 1), and R = 2/3 for 1/f noise (α = −1). The value R = 4/3 is therefore not universal but emerges as the special signature of an Ohmic spectrum. These results identify the XY-8/CPMG coherence-time ratio as a spectral-slope-sensitive observable and suggest a direct diagnostic strategy: measuring R in the narrow-band regime directly constrains the local bath exponent governing the sampled frequency window. This establishes a distinct theoretical direction from the effective benchmark treatment of Paper A and from the time-discontinuous channel-accessibility mechanism of Paper B, while remaining directly testable on NV-center platforms. V2: This version clarifies the phenomenological status of the XY-8/CPMG ratio law and reorganizes the manuscript so that the sequence-dependent spectral-slope interpretation is self-contained. Parts of the earlier Paper A framework have been re-examined and absorbed into the present paper in a revised form: rather than treating R=4/3R=4/3R=4/3 as a protocol-universal value, the ratio is now interpreted as a bath-dependent observable determined by the local spectral structure, with R=4/3R=4/3R=4/3 identified as the ideal Ohmic point. The revised version also makes the assumptions explicit (narrow-band regime, equal-weight transverse channels, and common exponential-decay form), strengthens the white-noise no-go benchmark, refines the inverse-problem discussion, and clarifies that the half-frequency XY-8 contribution is introduced as a minimal phenomenological assumption illustrated by a toy toggling-frame model rather than a complete microscopic derivation.
Dynamical-decoupling spectroscopy using solid-state spin qubits provides a powerful probe of environmental noise across wide frequency ranges. Here we identify a previously unrecognized observable fixed-point behavior in decoherence dynamics arising from additive decoherence channels sampled under narrow-band spectral filtering. We show that the ratio of decoherence times between two experimentally distinguishable configurations converges to a parameter-independent plateau value R ≈ 4/3 over an intermediate range of pulse numbers. This fixed point is independent of microscopic noise models and admits a natural geometric interpretation in terms of decoherence channel structure in the influence-functional description of open quantum systems. A central consequence is a strong experimental prediction: the existence of the plateau requires a band-limited spectral distortion, such that reconstructed noise spectra satisfy S_XY(ω)/S_CPMG(ω) ≈ 4/3 within a finite frequency window while approaching unity outside it. Numerical simulations demonstrate robustness of the effect across realistic noise environments without parameter fine tuning. We further propose an experimentally accessible protocol based solely on standard dynamical-decoupling measurements on a single qubit, enabling direct falsification of the predicted behavior. More broadly, these results indicate that measurement filtering can generate emergent invariant structures in open quantum-system dynamics, suggesting a new organizing principle for interpreting decoherence spectroscopy across quantum sensing platforms. Version 2 (v2) update: This version clarifies the logical structure of the paper by separating the existence of the measurement-induced plateau from the specific benchmark value R=4/3R=4/3R=4/3. The plateau mechanism is presented as a consequence of narrow-band spectral cancellation, while the value R=4/3R=4/3R=4/3 is treated more conservatively as a minimal effective-weight assignment consistent with pulse-sequence symmetry and the numerical results. The revised manuscript also makes the falsifiability structure more explicit and narrows the scope of the numerical claims: the simulations support the existence and robustness of the plateau, but are not presented as a unique microscopic derivation of the effective channel weights. Note (updated): This record corresponds to an earlier exploratory version of the idea. Parts of the theoretical interpretation presented here were not fully established and have been refined in subsequent work. In particular, the universal interpretation of ratio plateaus should be treated with caution. Readers are encouraged to consult the later work (see related records) for a more robust and conservative formulation of the effective description.
This Zenodo record provides the digitized data points and derived summary table used in our cross-platform reanalysis of dynamical-decoupling (DD) protocol performance via the dimensionless ratio eta(tau) = T_alt(tau) / T_CPMG(tau), where T denotes the experiment-specific decay-time metric (e.g., R^{-1}, T2, or an exponential lifetime). Datasets included: Solid-state NMR (Álvarez & Suter, 2011): digitized eta(tau) vs pulse delay tau. NV ensemble (Farfurnik et al., 2015): digitized T2 vs pulse number N for CPMG, XY8, and KDD-XY8, plus the clean reference table; an effective-axis mapping tau_eff = k/N (k = 51.75 ms) is used for tau-like visualization. Optical spin-wave memory (Ma et al., 2021): digitized (t, ln A(t)) points used to extract lifetimes T = -1/slope for CPMG (tau = 50, 100, 1000 ms) and KDDx (tau = 50, 100, 1000 ms). We use these data to test the spectral-RG scaling relation alpha(gamma) = gamma/(1+gamma) in the narrow-band sampling regime, and to quantify protocol-dependent performance through eta(tau) and windowed geometric-mean summaries (Supplementary Table S1 in the accompanying manuscript). Contact: Hiroyuki Shioiri (ORCID 0009-0000-2928-5685).
Inverse Geometry and Identifiability of Protocol Ratios in Dynamical-Decoupling Spectroscopy This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Building on reduced observation geometry, it asks the inverse question: given an observed protocol ratio, what spectral information is uniquely reconstructible, what is only weakly constrained, and what is fundamentally unrecoverable? The analysis is formulated in terms of the Jacobian of the reduced observation map and the associated Fisher information matrix. The paper shows that the inverse problem separates into three qualitatively distinct sectors. In the regular sector, the ratio map is locally invertible and reconstruction accuracy is controlled by a Cramér–Rao bound. For the pure power-law family S(ω)=Aω^α, the ratio R(α)=2/(1+2^{-α}) is strictly monotone, admits a closed-form inverse, and yields an explicit conditioning profile through the slope ∂_αR. In the degenerate sector, exemplified by the white-noise family S(ω)=A, the protocol ratio is exactly pinned to R=1 for all amplitudes. The Fisher information for the amplitude direction vanishes identically, so amplitude reconstruction from ratio data alone is impossible. This gives an exact no-go theorem for identifiability from protocol ratios in the white-noise case. In the branched sector, non-injectivity of the inverse map produces intrinsic ambiguity. Using the mixed family S(ω; α1, α2, w)=wAω^{α1}+(1−w)Aω^{α2}, the paper shows both global non-injectivity and local fold branching. Along an explicit symmetric slice, the ratio develops a fold at Rc=4/3, so nearby observed values admit two distinct inverse branches. These results establish protocol ratios as inverse-geometric probes with sharply defined domains of identifiability and intrinsic limits of spectral reconstruction. The framework clarifies the difference between unique inference, exact information collapse, and multi-valued inversion, and provides a basis for future extensions to multi-ratio, mixed-spectrum, and frequency-resolved inverse design. V2:This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Interpreting the reduced observation map through Fisher information, we show that the inverse problem separates into three sectors: regular (locally invertible with Cramér–Rao bounds), degenerate (exact collapse with vanishing Fisher information), and branched (non-injective with multiple solutions). Using power-law and mixed spectral families, we derive explicit inversion formulas, a white-noise no-go theorem, and fold-induced branching. The framework establishes protocol ratios as inverse-geometric probes with sharply defined limits of spectral reconstruction.
This work formulates dynamical-decoupling (DD) spectroscopy as a reduced observation problem. Rather than treating measured coherence responses as direct probes of the full environmental noise spectrum, the paper introduces a protocol-dependent map from parametrized spectral families to a small set of observables and analyzes its local Jacobian structure. The main result is a geometric framework in which spectral directions are classified into observable, weakly observable, and compressed sectors according to the singular-value structure and effective rank of the reduced map. Within this framework, plateau behavior and fixed protocol ratios arise when the reduced image becomes effectively low-dimensional, while abrupt transitions are interpreted as switching between response sectors with different local observability structure. A running power-law example is used to illustrate how amplitude and slope deformations are transmitted differently into protocol observables, and how protocol ratios can become locally insensitive without implying full spectral universality. The analysis emphasizes that DD protocols access environmental structure only through compressed, protocol-shaped images of the underlying spectrum. This provides a unifying interpretation of several familiar DD phenomena, including plateau regions, fixed-ratio structures, and sharp protocol-level transitions, in terms of reduced observation geometry rather than direct spectral reconstruction. v2 :This work formulates dynamical-decoupling spectroscopy as a reduced observation map from parameterized noise spectra to protocol-level observables. Using the Jacobian of this map, we classify spectral directions into observable, weakly observable, and compressed sectors via an effective-rank criterion. Plateau behavior and fixed protocol ratios are shown to arise from geometric compression of the reduced image, combined with a proportional-locking condition, rather than from full spectral universality. Abrupt transitions are interpreted as switching between response sectors with different local observability structures. The framework provides a unified geometric interpretation of sensitivity, compression, and invariance in DD-based noise spectroscopy.
This work solves the inverse design problem for multi‑ratio dynamical‑decoupling (DD) spectroscopy in the presence of a flat noise floor. For spectra of the form S(ω)=Aω−α+B,p=B/A, the asymptotic robustness of reduced inverse reconstruction is governed by the prefactor C(α,{ωk})=σmin(J(1)), the smallest singular value of the leading‑order Jacobian coefficient matrix. We derive an exact analytic expression for the column structure of J(1) and show that its row geometry is controlled by the sensitivity levers λk(α)=logωk+δ(α), which depend logarithmically on frequency. The angular separation of these levers determines the asymptotic robustness, yielding a transparent design principle: maximize lever spread over the accessible frequency range. We map the full design space of three‑frequency protocols within a fixed range [ ωmin,ωmax ] and identify the horizon‑optimal protocol family for each spectral slope α. The optimal geometry concentrates sampling toward the extremes of the accessible range, leaving the center undersampled; the gain over log‑uniform placement exceeds a factor of two for α≥1. We further construct a robust protocol for cases where the target slope is unknown, showing that the lever‑spread principle extends cleanly to broadband optimization. Together with the collapse taxonomy of earlier work, these results close the design side of reduced DD spectroscopy: they identify when floor‑induced information compression is inevitable, and how it can be minimized through protocol geometry. V2: Refined the geometric interpretation of the leading-order Jacobian structure by replacing the previous “lever-spread” formulation with a weighted row-geometry criterion. Corrected the asymptotic lever expressions and row-vector notation, clarified the α-dependent crossover mechanism, and improved the treatment of the α→0 limit. Added extensive numerical verification of the asymptotic scaling law, Jacobian convergence, finite-ρ optimality consistency, and robust broadband protocol optimization.
This paper extends the recoverable‑null and observable‑horizon framework of reduced dynamical‑decoupling (DD) spectroscopy to colored noise floors. Earlier work (Papers Q6–Q7) established that a flat floor B produces a recoverable near‑null: the reduced Jacobian remains full‑rank at every finite floor ratio ρ=B/A, and the weakest reconstruction direction becomes unobservable only when its singular value falls below the measurement noise, defining an observable horizon. Here we replace the flat floor by a shape‑dependent floor ρ Sfloor(ω;ν) and compute the horizon map ρ∗(α;ν) for three canonical spectral families: a flat floor, a 1/f-like floor, and a Lorentzian floor parameterised by a knee frequency ωc. The main findings are: Shape‑dependent horizon deformation: each floor family produces a qualitatively distinct deformation of the horizon map. A flat floor yields a smooth baseline; a 1/f-like floor introduces a sharp but finite minimum at α=1; a Lorentzian floor shifts the horizon‑gain region across the α-axis as ωc varies. 1/f resonance collapse: when the floor exponent matches the signal exponent (α=1), the ln(ωk)-amplified component of the α-column of the leading Jacobian J(1) is suppressed by the prefactor 1−21−α, which vanishes at α=1. This produces a strong recoverable‑null degeneracy and a sharp horizon minimum. Lorentzian knee as a design parameter: the knee frequency ωc determines which α sector benefits from the floor. An intermediate knee near the geometric mean ωminωmax achieves the broadest horizon gain (up to ∼3.5× near α≈0.9). Together with Papers Q6 and Q7, these results complete the three‑part analysis of recoverable‑null geometry under coarse‑grained floors in the DD series. Floor spectral shape emerges as a geometrically active design variable, providing a full toolkit for horizon optimisation in reduced DD spectroscopy under realistic colored noise conditions. v2: Clarified the distinction between recoverable near-nulls and exact structural nulls in reduced DD spectroscopy under colored noise floors. Added the exact 1/f resonance result at α=1, where the observable map loses the logρ direction identically, together with revised horizon geometry, Lorentzian knee-frequency analysis, normalization conventions, convergence checks, and improved figure annotations for structural rank loss.