Sequence-Dependent Spectral-Slope Diagnostics via XY-8/CPMG Coherence-Time Ratios

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.