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Universal Scaling of Decoherence under Dynamical Decoupling: A Cross Platform Test of Spectral RG Theory

Hiroyuki Shioiri

Published Mar 4, 2026• 0 citations
Publisher

Abstract

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).

DOI
10.5281/zenodo.18863099

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Shioiri Hiroyuki

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.

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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.

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Apr 29, 2016•4 citations
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).
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.
We investigate the scaling of coherence time T2{T}_{2}T2​ with the number of $\ensuremath{\pi}$ pulses ${n}_{\ensuremath{\pi}}$ in a singlet-triplet spin qubit using Carr-Purcell-Meiboom-Gill (CPMG) and concatenated dynamical decoupling (CDD) pulse sequences. For an even numbers of CPMG pulses, we find a power law ${T}_{2}\ensuremath{\propto}({n}_{\ensuremath{\pi}}{)}^{{\ensuremath{\gamma}}_{e}}$, with ${\ensuremath{\gamma}}_{e}=0.72\ifmmode\pm\else\textpm\fi{}0.01$, essentially independent of the envelope function used to extract T2{T}_{2}T2​. From this surprisingly robust value, a power-law model of the noise spectrum of the environment, $S(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{-}\ensuremath{\beta}}$, yields $\ensuremath{\beta}={\ensuremath{\gamma}}_{e}/(1\ensuremath{-}{\ensuremath{\gamma}}_{e})=2.6\ifmmode\pm\else\textpm\fi{}0.1$. Model values for ${T}_{2}\left({n}_{\ensuremath{\pi}}\right)$ using $\ensuremath{\beta}=2.6$ for CPMG with both even and odd ${n}_{\ensuremath{\pi}}$ up to 32 and CDD orders 3 through 6 compare very well with the experiment.
We study the time dependence of the decoherence factor (DF) of a qubit globally coupled to an environmental spin system (ESS) which is driven across the quantum critical point (QCP) by varying a parameter of its Hamiltonian in time ttt as $1\ensuremath{-}t/\ensuremath{\tau}$ or $\ensuremath{-}t/\ensuremath{\tau}$, to which the qubit is coupled starting at the time $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$; here $\ensuremath{\tau}$ denotes the inverse quenching rate. In the limit of weak coupling we analyze the time evolution of the DF in the vicinity of the QCP (chosen to be at t=0t=0t=0) and define three quantities, namely, the generalized fidelity susceptibility ${\ensuremath{\chi}}_{F}(\ensuremath{\tau})$ (defined right at the QCP), and the decay constants ${\ensuremath{\alpha}}_{1}(\ensuremath{\tau})$ and ${\ensuremath{\alpha}}_{2}(\ensuremath{\tau})$ which dictate the decay of the DF at a small but finite t(>0)t(>0)t(>0). Using a dimensional analysis argument based on the Kibble-Zurek healing length, we show that ${\ensuremath{\chi}}_{F}(\ensuremath{\tau})$ as well as ${\ensuremath{\alpha}}_{1}(\ensuremath{\tau})$ and ${\ensuremath{\alpha}}_{2}(\ensuremath{\tau})$ indeed satisfy universal power-law scaling relations with $\ensuremath{\tau}$ and the exponents are solely determined by the spatial dimensionality of the ESS and the exponents associated with its QCP. Remarkably, using the numerical t-DMRG method, these scaling relations are shown to be valid in both the situations when the ESS is integrable and nonintegrable and also for both linear and nonlinear variation of the parameter. Furthermore, when an integrable ESS is quenched far away from the QCP, there is a predominant Gaussian decay of the DF with a decay constant which also satisfies a universal scaling relation.
This repository contains the simulation and analysis code accompanying the manuscript “Q-Loop-Enabled Deterministic Coherence Management for Portable Quantum Sensors”. It provides a fully reproducible Python/NumPy/QuTiP workflow to evaluate Q-Loop, a continuous phase-modulation protocol for coherence protection in NV-centre magnetometers and related qubit sensors operating under “robotic” composite noise.The code computes filter functions (|F(\omega,\tau)|^2) and decoherence functionals (\chi(\tau)) for free evolution, Q-Loop, and standard pulsed dynamical decoupling sequences (CPMG, XY8, UDD), under both Lorentzian and composite (1/f +) line-rich noise spectra. It generates the main figures in the paper, including: (i) filter-function and noise-weighting plots, and (ii) jitter–flip-angle robustness heatmaps comparing Q-Loop to XY8/CPMG. A Colab-friendly script is included so all figures can be regenerated on a laptop without specialized hardware. The repository also provides example waveform generators (I/Q baseband) for implementing Q-Loop and pulsed sequences on standard AWGs, together with reference parameter sets ((\eta \simeq 3.6, T \simeq 0.97\tau_c)) that reproduce the reported (\sim 5)–(6\times) coherence extension. This archive is intended as a starting point for both numerical studies of coherence management under realistic platform noise and for experimental implementation on ensemble NV magnetometers or other driven qubit sensors.
In this work, we optimize a dynamical decoupling (DD) protocol to improve the spin coherence properties of a dense ensemble of nitrogen-vacancy (NV) centers in diamond. Using liquid nitrogen-based cooling and DD microwave pulses, we increase the transverse coherence time T2 from ∼ 0.7 ms up to ∼ 30 ms. We extend previous work of single-axis (Carr-Purcell-Meiboom-Gill) DD towards the preservation of arbitrary spin states. After performing a detailed analysis of pulse and detuning errors, we compare the performance of various DD protocols. We identify that the concatenated XY8 pulse sequences serves as the optimal control scheme for preserving an arbitrary spin state. Finally, we use the concatenated sequences to demonstrate an immediate improvement of the AC magnetic sensitivity up to a factor of two at 250 kHz. For future work, similar protocols may be used to increase coherence times up to NV-NV interaction time scales, a major step toward the creation of quantum collective NV spin states.