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

This work develops a microscopic account of sequence-dependent decoherence
time 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.