Dynamical-Freezing-Enhanced AC Magnetometry

• Dynamical-freezing-enhanced AC magnetometry is a technique that leverages sharply tuned dynamical states to arrest magnetic relaxation, resulting in steep susceptibility slopes and narrow detection passbands.
• It employs diverse mechanisms, including exchange-biased thin films, Floquet-driven NV-spin ensembles, and control-engineered decoupling, to extend coherent sensing times well beyond native limits.
• This strategy offers strong frequency selectivity and enhanced sensitivity for detecting AC fields while managing trade-offs between bandwidth narrowing and loss suppression.

Dynamical-freezing-enhanced AC magnetometry is a family of AC-field sensing strategies that exploits a sharply tuned dynamical regime in which magnetic relaxation is arrested, strongly slowed, or projected into a protected subspace. Depending on the platform, the relevant freezing mechanism is a magnetic freezing transition detected through complex transverse susceptibility in an exchange-biased thin film, an emergent Floquet conservation law in a periodically driven interacting spin ensemble, or a control-engineered suppression of detuning noise, amplitude inhomogeneity, and low-frequency bath fluctuations under continuous or concatenated dynamical decoupling. In each case, the sensing advantage is obtained not from static magnetization alone, but from the combination of steep susceptibility slopes, loss resonances, narrow-band filter functions, or sensing times extended far beyond the native coherence scale [(Urazhdin et al., 2018); (Lu et al., 30 Jul 2025); (Kim et al., 2022); (Kitamura et al., 23 Oct 2025); (Farfurnik et al., 2017); (Hirose et al., 2012)].

1. Conceptual basis and scope

In the cited literature, “dynamical freezing” does not denote a single microscopic mechanism. In a thin CoO/Permalloy bilayer it refers to an abrupt variation of the frequency-dependent imaginary part of the AC susceptibility near the exchange-bias blocking temperature, consistent with a magnetic freezing transition inferred from time-domain studies of magnetic aging. In an interacting NV-spin ensemble under periodic driving it refers to a breakdown of thermalization caused by an emergent conservation law at special Floquet detunings. In continuous and concatenated dynamical-decoupling protocols it refers to the suppression of slow noise, detuning spread, and drive-amplitude inhomogeneity by control engineering rather than to a thermodynamic transition [(Urazhdin et al., 2018); (Lu et al., 30 Jul 2025); (Kim et al., 2022); (Kitamura et al., 23 Oct 2025); (Farfurnik et al., 2017); (Hirose et al., 2012)].

A common operational structure nevertheless recurs. The sensor is biased so that the target AC field perturbs a response function with a large derivative, or so that the signal couples linearly inside a subspace protected against dominant decoherence channels. In the thin-film susceptibility setting this appears as a narrow $\chi''$ peak and an enhanced $\partial \chi'/\partial H$ near the blocking transition. In Floquet and dressed-state NV protocols it appears as a narrow resonance in detuning space or in the filter function, together with sensing times that exceed the bare $T_2$ or $T_2^*$ by large factors. This suggests a unifying description: dynamical-freezing-enhanced magnetometry uses a dynamical bottleneck to convert small AC perturbations into either unusually large susceptibility changes or unusually persistent coherent signals.

A related misconception is to equate freezing with generic decoupling. The available results are narrower. The enhancement is typically frequency selective, often tunable, and usually accompanied by a bandwidth penalty. Long-lived response is therefore not synonymous with wideband response; in several platforms the gain in sensitivity is explicitly obtained by narrowing the effective passband (Lu et al., 30 Jul 2025, Kitamura et al., 23 Oct 2025, Gupta et al., 2018).

2. Exchange-biased CoO/Permalloy bilayers

The thin-film realization is based on variable-temperature, variable-frequency transverse magneto-optical susceptibility measurements in a CoO/Permalloy bilayer. A small in-plane DC bias field $H_{dc}$ saturates the Permalloy layer while the CoO antiferromagnet remains essentially insensitive to $H_{dc}$. A perpendicular AC field,

$h_{ac}(t)=H_{ac}\cos \omega t,$

is applied through a ferrite-cored coil, and the oscillation of the Permalloy magnetization is read out by the transverse magneto-optic Kerr effect. Lock-in detection of the first harmonic yields the complex transverse susceptibility

$$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$

with temperature tunable from $80\,\mathrm{K}$ to $325\,\mathrm{K}$ and frequency tunable from approximately $\partial \chi'/\partial H$0 to approximately $\partial \chi'/\partial H$1 (Urazhdin et al., 2018).

In the frozen-antiferromagnet limit, and in the absence of antiferromagnetic losses, the transverse susceptibility reduces to the textbook form of a uniformly magnetized ferromagnet with effective anisotropy field $\partial \chi'/\partial H$2,

$\partial \chi'/\partial H$3

If the antiferromagnetic exchange field relaxes with a single Debye-like time scale $\partial \chi'/\partial H$4, then

$\partial \chi'/\partial H$5

with thermally activated dynamics

$\partial \chi'/\partial H$6

The corresponding real and imaginary parts are

$\partial \chi'/\partial H$7

$\partial \chi'/\partial H$8

so that the loss peak obeys the condition $\partial \chi'/\partial H$9 (Urazhdin et al., 2018).

Experimentally, below the Néel temperature $T_2$0 a finite $T_2$1 emerges and grows rapidly on cooling toward the exchange-bias blocking temperature $T_2$2–$T_2$3. At $T_2$4, with $T_2$5 and $T_2$6, $T_2$7 exhibits a clear maximum near $T_2$8; above and below $T_2$9, $T_2^*$0 falls toward zero in quantitative agreement with a Debye peak. When temperature is swept at $T_2^*$1 or $T_2^*$2, $T_2^*$3 shows a narrow peak of full width $T_2^*$4 centered at $T_2^*$5, whereas $T_2^*$6 shows only a modest step-like change (Urazhdin et al., 2018).

The magnetometric implication is explicit. Near $T_2^*$7, the slope $T_2^*$8 increases by approximately $T_2^*$9 on crossing through $H_{dc}$0, implying that a small field change $H_{dc}$1 produces a larger $H_{dc}$2 and therefore an enhanced field resolution. The loss peak also yields an effective quality factor

$H_{dc}$3

which defines a narrow low-noise detection band around $H_{dc}$4. Below $H_{dc}$5 the antiferromagnet is essentially frozen on the measurement time scale, giving a flat, frequency-independent $H_{dc}$6 and negligible $H_{dc}$7; above $H_{dc}$8 the response again has lower slope. The enhanced regime is therefore confined to a narrow freezing window. Using $H_{dc}$9 and $H_{dc}$0 gives $H_{dc}$1–$H_{dc}$2, or approximately $H_{dc}$3–$H_{dc}$4 (Urazhdin et al., 2018).

The same study also established that the directional asymmetry of the hysteresis loop is associated mainly with the difference in reversal mechanisms between the two reversed states of magnetization stabilized by the exchange-induced uniaxial anisotropy, and that this exchange-induced uniaxial anisotropy is much larger than the exchange-induced unidirectional anisotropy of the ferromagnet. The measurement approach was presented as enabling precise characterization of the dynamical and static characteristics of thin-film magnetic heterostructures with possible applications in reconfigurable magnonic and neuromorphic circuits (Urazhdin et al., 2018).

3. Floquet dynamical freezing in interacting NV-spin ensembles

A quantum implementation was demonstrated in an ensemble of approximately $H_{dc}$5 interacting NV-center electron spins in diamond. In the rotating frame, the periodically driven system is described by

$H_{dc}$6

with

$H_{dc}$7

Here $H_{dc}$8 are dipolar couplings, $H_{dc}$9 is residual on-site disorder, and $h_{ac}(t)=H_{ac}\cos \omega t,$0 alternates between $h_{ac}(t)=H_{ac}\cos \omega t,$1 and $h_{ac}(t)=H_{ac}\cos \omega t,$2 every half-period $h_{ac}(t)=H_{ac}\cos \omega t,$3. The experiment operated in the regime

$h_{ac}(t)=H_{ac}\cos \omega t,$4

The leading Floquet Hamiltonian contains a transverse term whose coefficient vanishes when

$h_{ac}(t)=H_{ac}\cos \omega t,$5

At these detunings, $h_{ac}(t)=H_{ac}\cos \omega t,$6 becomes approximately conserved up to corrections of order $h_{ac}(t)=H_{ac}\cos \omega t,$7, the Hilbert space fractures into disconnected $h_{ac}(t)=H_{ac}\cos \omega t,$8 sectors, and $h_{ac}(t)=H_{ac}\cos \omega t,$9 freezes over many Floquet periods (Lu et al., 30 Jul 2025).

The experimental platform used a dense NV ensemble in bulk diamond with two-level encoding $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$0 and $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$1. A static field $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$2 selected one NV orientation and yielded a transition frequency of approximately $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$3. The dipolar couplings were given by $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$4, and the mean interaction-limited coherence time under a standard PDD sequence was $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$5. A Floquet block of duration $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$6 was built from two mirrored half-periods with detunings $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$7 and $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$8, continuous transverse drive $$\chi_T(\omega,T)\equiv \frac{M_T}{H_{ac}}=\chi'(\omega,T)-i\chi''(\omega,T),$$9, and trains of fast $80\,\mathrm{K}$0 pulses of duration $80\,\mathrm{K}$1 and inter-pulse spacing $80\,\mathrm{K}$2 (Lu et al., 30 Jul 2025).

At freezing detunings $80\,\mathrm{K}$3, the measured stroboscopic magnetization $80\,\mathrm{K}$4 remained constant for up to $80\,\mathrm{K}$5 Floquet cycles, or $80\,\mathrm{K}$6, more than an order of magnitude longer than $80\,\mathrm{K}$7. Away from the freezing points, $80\,\mathrm{K}$8 thermalized rapidly to zero within $80\,\mathrm{K}$9. Intra-period measurements at time steps of $325\,\mathrm{K}$0 resolved coherent micromotion in all three spin components. Fourier spectra exhibited peaks at multiples of $325\,\mathrm{K}$1, with $325\,\mathrm{K}$2 and $325\,\mathrm{K}$3 at $325\,\mathrm{K}$4 and $325\,\mathrm{K}$5 additionally at $325\,\mathrm{K}$6 and $325\,\mathrm{K}$7, in agreement with the analytic kick operator and with DTWA simulations (Lu et al., 30 Jul 2025).

The magnetometric protocol exploited the sharp detuning dependence of $325\,\mathrm{K}$8. During each Floquet period $325\,\mathrm{K}$9, two additional inverting $\partial \chi'/\partial H$00 pulses were inserted at $\partial \chi'/\partial H$01 and $\partial \chi'/\partial H$02 to rectify the AC field, while a PDD train with $\partial \chi'/\partial H$03 generated a narrow-band filter centered at

$\partial \chi'/\partial H$04

For an applied phase-locked field $\partial \chi'/\partial H$05, the effective additional detuning is

$\partial \chi'/\partial H$06

and, to leading order,

$\partial \chi'/\partial H$07

The magnetic sensitivity was defined as

$\partial \chi'/\partial H$08

The best dynamical-freezing protocol reached $\partial \chi'/\partial H$09 at sensing times $\partial \chi'/\partial H$10, whereas the optimal PDD protocol gave approximately $\partial \chi'/\partial H$11 near $\partial \chi'/\partial H$12, corresponding to a $\partial \chi'/\partial H$13 improvement (Lu et al., 30 Jul 2025).

The improvement is not uniform over all regimes. In the short-time regime $\partial \chi'/\partial H$14, PDD can have a steeper slope at $\partial \chi'/\partial H$15 and therefore slightly better sensitivity, whereas for $\partial \chi'/\partial H$16 the freezing protocol is superior because $\partial \chi'/\partial H$17 remains nearly constant until systematic pulse errors accumulate. The same work identified pulse errors, microwave inhomogeneity, and the narrow-band nature of the PDD filter as the principal constraints on further extension (Lu et al., 30 Jul 2025).

4. Control-engineered freezing by continuous and concatenated decoupling

A second major lineage of freezing-enhanced AC magnetometry arises from continuous and concatenated dynamical decoupling. The central idea is to transform the sensor into a dressed or doubly dressed basis in which the target-field coupling survives at a slow rate while detuning noise, drive-amplitude errors, or spin-bath couplings are strongly suppressed. In the continuous-driving analysis of rotary-echo and spin-lock magnetometry, strong resonant driving shifts the qubit into a toggling frame where slow longitudinal noise averages to zero, while the AC signal is admitted through a narrow passband. Constant drive produces a filter centered around $\partial \chi'/\partial H$18, and the $\partial \chi'/\partial H$19 rotary-echo protocol produces a dominant passband at $\partial \chi'/\partial H$20; under realistic NV-center conditions, rotary echo with $\partial \chi'/\partial H$21–$\partial \chi'/\partial H$22 and moderate drive-amplitude noise of approximately $\partial \chi'/\partial H$23 can outperform both pulsed CPMG and simple spin lock by up to approximately $\partial \chi'/\partial H$24 in sensitivity (Hirose et al., 2012).

The low-frequency implementation based on a single NV qubit and double dressing used concatenated continuous dynamical decoupling with a microwave drive and a secondary RF drive. In the dressed basis, hyperfine couplings and slow qubit-energy noise acquire the suppression factor

$\partial \chi'/\partial H$25

so that, in the ideal limit of large detunings, the qubit becomes effectively decoupled from the spin bath and low-frequency noise. The reported coherence times were $\partial \chi'/\partial H$26 (bare) $\partial \chi'/\partial H$27, $\partial \chi'/\partial H$28 (single-dressed) $\partial \chi'/\partial H$29, and $\partial \chi'/\partial H$30 (double-dressed) $\partial \chi'/\partial H$31, corresponding to approximately $\partial \chi'/\partial H$32 and $\partial \chi'/\partial H$33 extensions. The same protocol suppressed spin-bath noise below $\partial \chi'/\partial H$34 by as much as $\partial \chi'/\partial H$35 and reached signal-to-noise ratios up to $\partial \chi'/\partial H$36 for coherent sub-MHz AC signals (Kim et al., 2022).

At higher frequencies, an ensemble implementation used concatenated continuous dynamical decoupling to freeze out spatially inhomogeneous dynamics in a large NV ensemble. In the rotating frame at $\partial \chi'/\partial H$37, a first resonant drive with $\partial \chi'/\partial H$38 created the first dressed manifold, and a second drive with $\partial \chi'/\partial H$39 created the second dressed manifold. The residual static detuning $\partial \chi'/\partial H$40 was suppressed to $\partial \chi'/\partial H$41, and first-drive amplitude errors were suppressed in the second-dressed frame. Experimentally, direct Rabi oscillations decayed with $\partial \chi'/\partial H$42, whereas the two-drive dressed coherence reached $\partial \chi'/\partial H$43 and, under a weak target drive, $\partial \chi'/\partial H$44. The lower amplitude cutoff for detectable GHz-range AC fields was reduced from approximately $\partial \chi'/\partial H$45 in direct Rabi sensing to approximately $\partial \chi'/\partial H$46 under CCDD, the dynamic range was extended from roughly $\partial \chi'/\partial H$47 to roughly $\partial \chi'/\partial H$48, and the reported sensitivity was approximately $\partial \chi'/\partial H$49 (Kitamura et al., 23 Oct 2025).

Pulsed concatenation provides a complementary route. In NV-ensemble AC magnetometry with XY-type pulse trains, the relevant phase accumulation is governed by the filter function

$\partial \chi'/\partial H$50

and the sensitivity obeys

$\partial \chi'/\partial H$51

For concatenated XY$\partial \chi'/\partial H$52 sequences, the modulation function generates filter zeros at low frequencies up to order $\partial \chi'/\partial H$53, strongly suppressing slow noise. Experimentally, Farfurnik and collaborators found that concatenation enhanced sensitivity only when many pulses were required and pulse imperfections became critical: at approximately $\partial \chi'/\partial H$54, conventional XY8 gave $\partial \chi'/\partial H$55, whereas second-level concatenated XY8 with $\partial \chi'/\partial H$56 pulses recovered contrast and improved the sensitivity to approximately $\partial \chi'/\partial H$57 (Farfurnik et al., 2017).

5. Sensitivity enhancement, bandwidth selection, and frequency filtering

Across these realizations, the gain mechanism can be classified by whether it acts primarily through response slope, through coherence-time extension, or through spectral filtering. In the CoO/Permalloy bilayer, the decisive quantities are the sharp increase in $\partial \chi'/\partial H$58 near $\partial \chi'/\partial H$59 and the narrow Debye-like $\partial \chi'/\partial H$60 peak at $\partial \chi'/\partial H$61; the quality factor is $\partial \chi'/\partial H$62 and the characteristic bandwidth is of order $\partial \chi'/\partial H$63, tunable by temperature (Urazhdin et al., 2018). In the Floquet NV ensemble, the decisive quantity is the sharp dependence of $\partial \chi'/\partial H$64 on detuning near the freezing points, together with sensing times extending far beyond the interaction-limited $\partial \chi'/\partial H$65 (Lu et al., 30 Jul 2025). In double-dressed and CCDD schemes, the main resource is the extension of the dressed-state coherence time while restricting the accepted signal band to the dressed splitting (Kim et al., 2022, Kitamura et al., 23 Oct 2025).

The bandwidth penalty is explicit in several platforms. In the optical-fiber AC magnetometer based on Terbium doping and built-in half-wave plates, the waveplate spacing is chosen as

$\partial \chi'/\partial H$66

so that the pulse sequence is synchronous with the zero-crossings of the AC field. On resonance, the net Faraday rotation is $\partial \chi'/\partial H$67, giving a minimum detectable field proportional to $\partial \chi'/\partial H$68, but the detectable-frequency bandwidth scales as

$\partial \chi'/\partial H$69

Thus longer fibers improve raw sensitivity while narrowing the detection band. Numerical simulations showed that, for a $\partial \chi'/\partial H$70 fiber and waveplate placement errors up to approximately $\partial \chi'/\partial H$71 of the nominal spacing, the final state fidelity remained above $\partial \chi'/\partial H$72 when $\partial \chi'/\partial H$73 (Gupta et al., 2018).

A closely related susceptibility-based example is square artificial spin ice. There the mean relaxation time follows the Vogel-Fulcher-Tammann form

$\partial \chi'/\partial H$74

and the freezing temperature extracted from the susceptibility peak satisfies

$\partial \chi'/\partial H$75

As the inter-island gap increased and the dipolar coupling weakened, the fitted freezing temperature shifted downward from $\partial \chi'/\partial H$76 to $\partial \chi'/\partial H$77 and then to $\partial \chi'/\partial H$78. Around each $\partial \chi'/\partial H$79, $\partial \chi'/\partial H$80 displayed a single symmetric peak with full width at half maximum of approximately $\partial \chi'/\partial H$81–$\partial \chi'/\partial H$82, $\partial \chi'/\partial H$83 after normalization, and baseline Kerr-rotation noise of order $\partial \chi'/\partial H$84 that could be driven below $\partial \chi'/\partial H$85 by longer lock-in integration at lower frequency, yielding signal-to-noise ratios exceeding $\partial \chi'/\partial H$86. The steep slopes $\partial \chi'/\partial H$87 near $\partial \chi'/\partial H$88, reaching up to approximately $\partial \chi'/\partial H$89 in the strongest-coupled array, were identified as the effective gain element for sensitive detection of small temperature or field perturbations (Pohlit et al., 2020).

These examples illustrate a common design rule. The protocol is most effective when the imposed modulation is synchronous with the target field and when the system is operated at the steep side of a dynamical crossover. In susceptibility platforms the synchrony condition is $\partial \chi'/\partial H$90; in pulsed DD it is typically $\partial \chi'/\partial H$91; in the optical implementation it is set by the propagation time between waveplates; in Floquet freezing it is encoded in the resonance condition $\partial \chi'/\partial H$92 (Urazhdin et al., 2018, Farfurnik et al., 2017, Gupta et al., 2018, Lu et al., 30 Jul 2025).

6. Applications, limitations, and research directions

The most immediate applications are in platforms where reconfigurability, frequency selectivity, or history dependence are themselves useful resources. In the CoO/Permalloy system, lithographically defining elements of varying thickness can tune $\partial \chi'/\partial H$93 across an array, enabling spatial or temporal filters with controlled loss peaks. The localized enhancement of $\partial \chi'/\partial H$94 near $\partial \chi'/\partial H$95 was proposed as a tunable thermally programmable damping element in magnonic waveguides, while the aging and memory effects of the putative antiferromagnetic glass were proposed for neuromorphic sensing elements whose effective susceptibility can be set by a thermal write pulse and read out as an AC signal amplitude. The same platform was proposed for practical field sensors mounted on micromachined bridges whose local temperature is modulated around $\partial \chi'/\partial H$96, allowing switching into and out of the high-sensitivity state and lock-in detection of faint AC magnetic signals such as biomagnetic or NMR micro-signals (Urazhdin et al., 2018).

In NV-based sensing, the broader implication is transferability to other interacting spin platforms. The Floquet-freezing study stated that the mechanism relies only on global periodic driving and generic spin-spin couplings and therefore should be directly transferable to P1 centers, donor spins in Si, superconducting qubit arrays, trapped-ion crystals, and cold-atom ensembles in cavity QED. The same work also emphasized that scalability to higher densities may provide further sensitivity gains if dipolar broadening can be mitigated by improved decoupling (Lu et al., 30 Jul 2025). The high-frequency CCDD study similarly framed the second-dressed manifold as a route to weak-signal sensing in large, spatially inhomogeneous ensembles, but identified several practical costs: microwave power consumption and heating, technical complexity associated with two phase-locked microwave sources and composite initialization pulses, reduced readout contrast from partial dressing of hyperfine sidebands, and bandwidth narrowing to a detection line of order $\partial \chi'/\partial H$97 (Kitamura et al., 23 Oct 2025).

Important limits recur across the literature. Narrow-band detection is often intrinsic rather than incidental. In the Floquet NV implementation the PDD filter restricts the detectable bandwidth, and systematic pulse errors eventually cap the useful sensing time (Lu et al., 30 Jul 2025). In the concatenated XY study, cooling to $\partial \chi'/\partial H$98 improved sensitivity only for low-frequency fields of order $\partial \chi'/\partial H$99, where the total experiment time approached $T_2$00 at a small number of applied pulses; at higher frequencies, concatenation helped only when more than approximately $T_2$01 pulses were needed and pulse imperfections dominated (Farfurnik et al., 2017). In the CoO/Permalloy bilayer, the enhanced slope exists only in the narrow freezing window around $T_2$02; above and below that window the susceptibility is either less steep or nearly lossless and frequency independent (Urazhdin et al., 2018).

A further conceptual boundary is that dynamical freezing is not equivalent to universal non-ergodicity. The Floquet NV study explicitly contrasted it with integrability, many-body localization, scars, and Hilbert-space fragmentation, identifying it instead as a distinct breakdown of thermalization through emergent conservation laws (Lu et al., 30 Jul 2025). By contrast, the thin-film and artificial-spin-ice cases are susceptibility signatures of glass-like or collective freezing phenomena, and the continuous-driving cases are deliberate control constructions that “freeze” unwanted dynamics while retaining signal coupling [(Urazhdin et al., 2018); (Pohlit et al., 2020); (Hirose et al., 2012)]. The term therefore spans several mechanisms, but the magnetometric content is consistent: a suitably engineered or naturally occurring freezing condition can turn an otherwise broad, lossy, or rapidly dephasing magnetic response into a sharply amplified and frequency-selective AC sensor.