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Measurement-Induced Fixed Points in Dynamical-Decoupling Noise Spectroscopy

Hiroyuki Shioiri

Published Feb 28, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.18809681

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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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We identify a regime of open quantum system dynamics in which the observed coherence time exhibits a discontinuous transitionarising from selective loss of accessibility ofan effective decoherence channel. Within theGaussian influence-functional formalism, normalization of the Keldysh weight matrix leadsto a parameter-independent coherence-timeratioR≡ τpostτpre = 43 .The result follows entirely from standardopen-system theory and introduces no modification of quantum mechanics. The predicted transition differs fundamentally fromknowndynamical-decoupling fixed points, occurring in time at fixed control parametersand remaining independent of pulse number.Because the framework yields explicit falsification criteria, the predicted ratio providesa direct experimental probe of multi-channeldecoherence structure. The analysis suggeststhat R = 4/3 represents a structural invariant associated with the normalized geometryof the Keldysh representation.
Decoherence is one of the most important obstacles that must be overcome in quantum information processing. It depends on the qubit-environment coupling strength, but also on the spectral composition of the noise generated by the environment. If the spectral density is known, fighting the effect of decoherence can be made more effective. Applying sequences of inversion pulses to the qubit system, we developed a method for dynamical decoupling noise spectroscopy. We generate effective filter functions that probe the environmental spectral density without requiring assumptions about its shape. Comparing different pulse sequences, we recover the complete spectral density function and distinguish different contributions to the overall decoherence.
I present a theory of environmental noise spectroscopy via dynamical decoupling of a qubit at an optimal working point. Considering a sequence of nnn pulses and pure dephasing due to quadratic coupling to Gaussian distributed noise $\ensuremath{\xi}(t)$, I use the linked-cluster (cumulant) expansion to calculate the coherence decay. Solutions allowing for reconstruction of spectral density of noise are given. For noise with correlation time shorter than the time scale on which coherence decays, the noise filtered by the dynamical decoupling procedure can be treated as effectively Gaussian at large nnn, and well-established methods of noise spectroscopy can be used to reconstruct the spectrum of ${\ensuremath{\xi}}^{2}(t)$ noise. On the other hand, for noise of dominant low-frequency character $(1/{f}^{\ensuremath{\beta}}$ noise with $\ensuremath{\beta}>1)$, an infinite-order resummation of the cumulant expansion is necessary, and it leads to an analytical formula for coherence decay having a power-law tail at long times. In this case, the coherence at time ttt depends both on spectral density of $\ensuremath{\xi}(t)$ noise at $\ensuremath{\omega}=n\ensuremath{\pi}/t$, and on the effective low-frequency cutoff of the noise spectrum, which is typically given by the inverse of the data acquisition time. Simulations of decoherence due to purely transverse noise show that the analytical formulas derived in this paper apply in this often encountered case of an optimal working point, provided that the number of pulses is not very large, and that the longitudinal qubit splitting is much larger than the transverse noise amplitude.
Here we explore the prospects of carrying out the single qubit dynamical decoupling spectroscopy of environmental noise characterized by discrete spectral density. We show that the approach to the spectroscopy and spectrum reconstruction has to be adjusted to the resolution of frequency filters produced by the dynamical decoupling pulse sequences. The standard form of spectroscopy is applicable when the filter is unable to resolve discrete spectral lines and only coarse grained approximation of the spectrum is recorded in the qubit's decoherence rate. When the filter's resolution is high enough the qubit probe evolves in a qualitatively different manner and the procedure for recovering spectral density has to be redesigned.