Floquet analysis of coherence in periodically driven diamond NV ensemble systems

Abstract: High-density nitrogen-vacancy (NV) ensembles are promising platforms for solid-state quantum sensing, but their performance is limited by dipolar interactions and inhomogeneous dephasing. Periodic decoupling sequences such as Waugh-Huber-Haeberlen (WAHUHA) can extend the observed stroboscopic decay time. However, it remains unclear that a longer effective dephasing time yield improved magnetic-field sensitivity. Here, we show that WAHUHA control increases the effective inhomogeneous dephasing time of a dense NV ensemble from $T_2^\ast$ of 0.9 $μ$s to $T_{2,eff}^\ast$ of 31 $μ$s, while producing little improvement in dc magnetic-field sensitivity. Using detuning-resolved stroboscopic spectroscopy and finite-pulse Floquet analysis, we show that the long-lived signal arises from phase wrapping and quasi-energy branch folding of the one-cycle unitary. These effects reshape the stroboscopic spectrum and suppress the detuning-to-phase transduction slope, $dΦ/dΔ$, which governs the dc magnetic-field response. Our results demonstrate that, under periodic driving, an extended effective dephasing time does not necessarily translate into enhanced dc sensitivity and establish finite-pulse Floquet analysis as a practical framework for evaluating coherence in spin ensembles.

• The paper demonstrates that periodic WAHUHA driving extends the effective coherence time in diamond NV ensembles, yet this does not improve DC magnetometry because of phase wrapping effects.
• Stroboscopic finite-pulse Floquet analysis is used to rigorously model phase evolution and quantify the impact of control imperfections on magnetic sensitivity.
• The work highlights that optimizing quantum sensors requires co-engineering both the coherence envelope and phase transduction, not merely prolonging coherence times.

Floquet Analysis of Coherence in Periodically Driven Diamond NV Ensemble Systems

Introduction

The study rigorously investigates the stroboscopic coherence and magnetic sensitivity of dense diamond nitrogen-vacancy (NV) ensembles subjected to periodic WAHUHA decoupling. NV centers in diamond are robust solid-state systems for quantum sensing, but at high densities, their performance is fundamentally limited by strong homonuclear dipolar interactions and environmental inhomogeneities. The work evaluates whether increasing effective coherence times via periodic driving sequences, specifically the WAHUHA protocol, translates into genuine improvements in DC magnetometry, utilizing finite-pulse Floquet theory as a central analytic tool.

Periodic Decoupling, Control Protocols, and Floquet Framework

High-density NV ensembles experience rapid dephasing from both NV-NV dipolar interactions and coupling to residual paramagnetic impurities such as P1 centers. Conventional dynamical decoupling sequences such as CPMG or XY sequences, though effective for extending coherence, are incompatible with DC magnetometry—these refocus the signal from both environmental noise and the DC field of interest, nullifying the desired sensor response. WAHUHA, originating from solid-state NMR, is distinct in that it symmetrically rotates the quantization axis through [100], [010], and [001] over the cycle, averaging dipolar interactions while retaining a finite response to DC fields along [111]. In the ideal-pulse limit, the DC response is attenuated by a theoretical factor of $1/3$.

However, in experimental regimes where only stroboscopic measurements after each full period are viable and pulses are of finite width, average Hamiltonian theory yields an incomplete description. Instead, the one-cycle Floquet unitary precisely governs the observed signal. The full driven system is described within the framework of finite-pulse Floquet theory, capturing phase evolution accrued not only during free evolution but also within pulses. This approach elucidates the true relationship between extended effective coherence (here denoted $T_{2,\mathrm{eff}}$) and DC sensitivity in periodically driven open quantum systems.

Experimental Observations and Numerical Analysis

The empirical core of the paper examines the ensemble magnetization decay for both Ramsey (free induction) and WAHUHA-controlled protocols under varying interpulse delays. The Ramsey $T_2^*$ is 0.92 μs, consistent with strong dipolar and static disorder. When WAHUHA is employed with short delay ($t = 20$ ns), the effective dephasing time increases to 1.7 μs, reflecting suppression of NV-NV interactions. Strikingly, with longer delays ($t = 110$ ns), the stroboscopic envelope stretches to $T_{2,\mathrm{eff}} \approx 31$ μs—more than 30 times the Ramsey baseline.

A detuning-resolved Floquet spectroscopy was performed: at short delays, Floquet branches are approximately linear in detuning with a finite gap at the zero-crossing due to pulse imperfections. At long delays, the spectrum displays nonlinear branch folding and phase wrapping, features unique to stroboscopic Floquet sampling. The observed long-lived signal at extended delays is shown to be a direct consequence of quasi-energy branch folding, not true many-body coherence protection. Specifically, the Floquet quasi-energy branches reach the phase-slip boundary and are wrapped into the available Nyquist window under stroboscopic sampling, which generates robust 2T-period oscillatory signals.

Floquet eigenphase calculations, made from the one-cycle propagator stripped of the global phase, identify that "avoided crossing" features and branch offsets in detuning originate from finite pulse effects and control errors, not intrinsic physical resonances.

Disconnection Between Extended Coherence and Sensitivity

A key result is the demonstration that the marked extension of $T_{2,\mathrm{eff}}$ under WAHUHA cannot be equated with improved magnetic sensitivity. The relevant metric for DC magnetometry is the transduction slope $d\phi/d\Delta$ of the Floquet phase with respect to detuning, not simply the dephasing envelope. Under phase wrapping and branch folding, this slope is strongly suppressed even as observable coherence time increases, resulting in no concomitant gain in practical field sensitivity.

Direct mappings of the signal slope with respect to detuning and interpulse delay reveal a fragmented, sharply reduced transduction landscape at long delay, in sharp contrast to the intuitive expectation that longer coherence time should yield superior sensitivity. The long-lived stroboscopic signals in this regime are thus not metrologically advantageous.

Theoretical and Practical Implications

The analysis establishes that in the Floquet-engineered regime probed by periodic decoupling, the maximization of effective coherence time $T_{2,\mathrm{eff}}$ is not a reliable proxy for quantum sensor optimization. Instead, the Floquet transduction slope $d\phi/d\Delta$ must also be considered. Average Hamiltonian theory, while providing asymptotic intuition, is insufficient for determining sensor performance with real finite-width pulses. Finite-pulse Floquet analysis captures both the stroboscopic spectrum and the dynamical suppression of DC sensitivity, providing a more complete theoretical foundation for quantum metrology with periodically driven spin ensembles.

Notably, the period-doubled stroboscopic response observed matches expectations for subharmonic discrete time crystals. However, in this context, the 2T oscillations result from single-particle Floquet spectrum structure, not emergent collective phenomena stabilized by many-body interactions.

For practical quantum sensor design, these results highlight the necessity of co-optimizing both coherence envelope and the structure of phase transduction under the relevant experimental measurement protocol.

Future Prospects

The findings motivate future explorations of advanced Floquet-engineered protocols that may adjust the quasi-energy landscape to avoid phase wrapping-induced suppression, possibly via tailored pulse shaping, nonuniform cycle timing, or measurement sampling schemes. There is also scope for leveraging the detailed detuning-resolved Floquet structure for other quantum control and Hamiltonian engineering applications, including quantum simulation and error correction in dense spin systems.

Conclusion

This work conclusively demonstrates, through rigorous experiment and finite-pulse Floquet analysis, that extended stroboscopic coherence times in periodically driven NV ensembles do not inherently yield improved DC magnetic sensitivity. The formalism deployed clarifies the precise mechanisms—namely, Floquet phase wrapping and branch folding—that decouple observable coherence from metrological utility. Finite-pulse Floquet theory is established as an essential analytic and diagnostic framework for the evaluation and design of quantum control protocols in dense spin ensembles. This paradigm, focusing on phase transduction rather than decay envelopes, will be central to future advances in quantum sensing and metrology.