2022

[1,4]Roberto Rizzato These authors contributed equally to this work. These
authors contributed equally to this work. [1,4]Dominik B. Bucher

1]Technical University of Munich, TUM School of Natural Sciences, Department
of Chemistry, Lichtenbergstraße 4, Garching bei München, 85748, Germany

2]Technical University of Munich, TUM School of Natural Sciences, Department
of Physics, James-Franck-Str. 1, Garching bei München, 85748, Germany

3]Walter Schottky Institute, TUM School of Natural Sciences, Am Coulombwall 4,
Garching bei München, 85748, Germany

4]Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4,
München, D-80799, Germany

5]Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and
Materials Research, Bautzner Landstraße 400, Dresden, 01328, Germany

# Extending the coherence time of spin defects in hBN enables advanced qubit
control and quantum sensing

<EMAIL_ADDRESS>Martin Schalk  Stephan Mohr  Joachim P. Leibold  Jens
C. Hermann  Fleming Bruckmaier  Peirui Ji  Georgy V. Astakhov  Ulrich Kentsch
Manfred Helm  Andreas V. Stier  Jonathan J. Finley<EMAIL_ADDRESS>[  [
[  [  [

###### Abstract

Spin defects in hexagonal Boron Nitride (hBN) attract increasing interest for
quantum technology since they represent optically-addressable qubits in a van
der Waals material. In particular, negatively-charged boron vacancy centers
(${V_{B}}^{-}$) in hBN have shown promise as sensors of temperature, pressure,
and static magnetic fields. However, the short spin coherence time of this
defect currently limits its scope for quantum technology. Here, we apply
dynamical decoupling techniques to suppress magnetic noise and extend the spin
coherence time by nearly two orders of magnitude, approaching the fundamental
$T_{1}$ relaxation limit. Based on this improvement, we demonstrate advanced
spin control and a set of quantum sensing protocols to detect electromagnetic
signals in the MHz range with sub-Hz resolution. This work lays the foundation
for nanoscale sensing using spin defects in an exfoliable material and opens a
promising path to quantum sensors and quantum networks integrated into ultra-
thin structures.

##  1 Main

Optically addressable spin defects in semiconductors are promising systems for
various applications in quantum science and technology, including sensing and
metrologyweber_2010 ; awschalom_quantum_2013 ; degen_quantum_2017 ;
awschalom_2021 . In contrast to other defects typically hosted in 3D
crystalsHeremans_2016 , the recently discovered boron vacancy center
($V_{B}^{-}$) in hexagonal boron nitride (hBN)abdi_color_2018 ;
gottscholl_initialization_2020  is embedded in a van der Waals material which
can be exfoliated down to the limit of a single monolayergeim_van_2013 ;
novoselov_2d_2016 (Figure 1a). Such a unique feature would be advantageous for
a wide range of applications where a minimal spatial separation of the defect
to a specific target is highly desired. For example, in nanoscale quantum
sensing spatial resolution is determined by the proximity of the defect to the
test objectrugar_2013 ; barry_sensitivity_2020 ; tetienne_prospects_2021 , or
for integrated quantum photonic devices van der Waals materials can be readily
exfoliated onto different substrates and used as spin-photon
interfacesawschalom_2021 . Furthermore, the $V_{B}^{-}$ center, incorporated
in ultra-thin hBN foils, allows for easier integration of manipulable qubits
in 2D heterostructures. This possibility opens up unexplored paths for
investigating novel composite materials and phenomena in nanoelectronics,
nanophotonics, and spintronicsnovoselov_2d_2016 ; geim_van_2013 ; miao_2019 ;
lemme_2d_2022 ; duan_2022 . The first protocols for generating $V_{B}^{-}$
centers in hBN have been recently presentedkianinia_generation_2020 ;
gao_femtosecond_2021 , and their spectroscopic characterization has been
accomplished in several studiesgottscholl_initialization_2020 ;
gottscholl_room_2021 ; luxmoore_2022 ; murzakhanov_electronnuclear_2022 ;
yu_excited-state_2022 . Importantly, by detecting the changes in their
optically detected magnetic resonance (ODMR) spectra, the $V_{B}^{-}$ centers
demonstrated to work as sensors for temperature, pressure, and static magnetic
fields, in some cases being competitive with more mature spin defect-based
sensorsgottscholl_spin_2021 . Based on these results, first applications have
been demonstrated for the magnetic and temperature nanoscale imaging of low-
dimensional materialshealey_quantum_2022 ; huang_wide_2022 .

Coherent control of ensembles of $V_{B}^{-}$ centers has recently been
showngottscholl_room_2021 ; gao_high-contrast_2021 ; gao_nuclear_2022 ;
haykal_decoherence_2022 ; luxmoore_2022 , although coherence times on the
order of $\sim 100$ nanoseconds have been reportedhaykal_decoherence_2022 ;
liu_coherent_2022 . These short timescales significantly restrict the utility
of such spin qubits and discourage the development of applications based on
coherent spin manipulation.

In this work, we overcome this limitation and achieve an over 80-fold
extension of the room-temperature $T_{2}$ coherence time of $V_{B}^{-}$
ensembles in hBN. The combination of efficient microwave (MW) delivery and
precise MW control allowed us to perform dynamical decoupling schemes, such as
Carr-Purcell-Meiboom-Gillmeiboom_modified_1958  (CPMG) to efficiently suppress
magnetic noise from the spin bath and increase the $V_{B}^{-}$ coherence.
Furthermore, we generate $V_{B}^{-}$ dressed-states (DS) by applying spinlock
pulses and show that spin coherence can be preserved for a time ($T_{1\rho}$),
which is on the same order as the spin-lattice relaxation time $T_{1}$ and
nearly 2 orders of magnitude longer than the spin-echo $T_{2}$ time. These
improvements enable the detection of radiofrequency (RF) signals with a
frequency resolution far beyond the intrinsic spin defect coherence time. This
work broadens the applicability of $V_{B}^{-}$ defects in hBN, opening up new
opportunities for nanoscale quantum technologies.

##  2 Characterization of $V_{B}^{-}$spin properties

The negatively-charged boron vacancy center (${V_{B}}^{-}$) consists of a
missing boron atom in the hBN lattice surrounded by three equivalent nitrogen
atoms (Figure 1a). Ten electrons occupying six defect orbitals result in a
triplet $S=1$ ground state consisting of the $|m_{s}=0\rangle$ ($|0\rangle$)
and $|m_{s}=\pm 1\rangle$ ($|\pm 1\rangle$) spin-statesabdi_color_2018 ;
ivady_ab_2020 . At zero magnetic field, the $|\pm 1\rangle$ states are
degenerate but separated in energy from the $|0\rangle$ state due to a zero-
field splitting (ZFS) of $D\sim$3.47\text{\,}\mathrm{GHz}$$. The transition
from the ground state to the excited state by green laser illumination (e.g.,
$\lambda=$532\text{\,}\mathrm{nm}$$) is followed by a phonon-assisted
radiative decay with broad photoluminescence (PL) peaking at
$\sim$850\text{\,}\mathrm{nm}$$qian_unveiling ; qian_2022 . A spin-state-
dependent relaxation path through inter-system crossing (ISC) leads to two
important consequences: 1) the ${V_{B}}^{-}$ defects can be optically
initialized into the $|0\rangle$ state under ambient conditions; 2) the
$|0\rangle$ and $|\pm 1\rangle$ states can be distinguished by their spin-
state dependent PL (Figure 1b)reimers_2020 ; ivady_ab_2020 ;
gottscholl_initialization_2020 ; luxmoore_2022 .

All experiments presented in this work were conducted under ambient conditions
on $V_{B}^{-}$ ensembles obtained by He+ implantation of hBN flakes
($\sim$100\text{\,}\mathrm{nm}$$-thick, details in the Methods section). As
depicted schematically in Figure 1c, the flakes are directly transferred onto
a gold MW microstripline that is used for $V_{B}^{-}$ spin manipulation. A
microscope for spatially resolved ODMR measurements has been built that can
address defined areas of the sample with a laser spot size of
$\sim$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ diameter (see Figure 1d and
Experimental setup in Methods).

As a first characterization experiment, we perform ODMR at a bias magnetic
field $B_{0}\sim$8\text{\,}\mathrm{mT}$$. The spectrum shows a broad resonance
corresponding to the $|0\rangle\rightarrow|-1\rangle$ transition (Figure 1e),
revealing characteristic features due to the strong hyperfine (HF) coupling of
the $V_{B}^{-}$ electronic spin with the three equivalent 14N nuclei. The HF
lines can be fitted with seven Gaussian functions, separated by
$\sim$44\text{\,}\mathrm{MHz}$$gottscholl_initialization_2020 ;
gottscholl_room_2021 ; haykal_decoherence_2022 ; gao_nuclear_2022 . Their
lineshape indicate the inhomogeneously broadened nature of the spectrum, where
the electronic spins experience a broad distribution of local magnetic fields
due to the intricate HF structure. From this first observation, one can expect
that fast decoherence might dramatically affect the $V_{B}^{-}$ electronic
spins. To perform coherent control of the ${V_{B}}^{-}$ centers, we run Rabi
experiments with the MW frequency at the central peak of the ODMR spectrum and
monitor the fluorescence contrast while sweeping the MW pulse duration
$t_{p}$. Figure 1f shows Rabi oscillations for different MW amplitudes. We
observe a $6-7\%$ fluorescence contrast at maximum amplitude with a
$\pi$-pulse duration of $t_{p}^{(\pi)}=$7.5\text{\,}\mathrm{ns}$$,
corresponding to a Rabi frequency of $\nu_{R}\sim$67\text{\,}\mathrm{MHz}$$.
The maximization of the Rabi frequency is crucial for two reasons: 1) it
allows for short pulses that are necessary for an efficient spin manipulation,
especially in the presence of fast spin-dephasing, 2) it allows us to drive a
large bandwidth of the ODMR spectrum. Fourier transformation of our
rectangular $\pi/2$-pulses would give an excitation bandwidth of approximately
$1/t_{p}^{(\pi/2)}=$250\text{\,}\mathrm{MHz}$$ for a
$\sim$4\text{\,}\mathrm{ns}$$ duration. This has the positive effect of
increasing the observable contrast whilst reducing detrimental spectral
diffusion effects.

To characterize the $V_{B}^{-}$ relaxation properties, we measure the spin-
lattice relaxation time $T_{1}$ using a protocol consisting of two
$5\text{\,}\mathrm{\SIUnitSymbolMicro s}$-long laser pulses for initialization
and readout separated by the sweep time $\tau$. A time constant
$T_{1}\sim$6\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ is extracted from a mono-
exponential fit of the resulting contrast decay (see Figure 1g). We continue
to measure the native coherence time $T_{2}$ using the spin-echo sequence
depicted in Figure 1h. Here, the fluorescence contrast is detected while
sweeping the free precession time $\tau$. By fitting the resulting signal
decay with a stretched exponential functionbar-gill_suppression_2012 ; bar-
gill_solid-state_2013  (see Pulse sequences, normalizations and fittings in
Methods), a time constant $T_{2}\sim$90\text{\,}\mathrm{ns}$$ is extracted,
consistent with recent literaturehaykal_decoherence_2022 ; liu_coherent_2022 .
Such a coherence time is short, especially compared to other ensembles of spin
defects in 3D-host materials, where typical $T_{2}$ times lie in the vicinity
of the microsecond timescales at room temperaturebarry_sensitivity_2020 .
Interestingly, a clear oscillation appears superimposed on the spin-echo
decay. This modulation is tentatively assigned to the interaction of the
$V_{B}^{-}$ electronic spin with the three 14N nuclei since the frequency of
$\sim$45\text{\,}\mathrm{MHz}$$, fit by a cosine function (see Methods and
Supplementary Note 1), matches the HF couplinggottscholl_initialization_2020 .
An improvement of the coherence time is a critical step for utilizing
$V_{B}^{-}$ in hBN in advanced quantum technologies, where qubit coherent
manipulation and quantum information storage/retrieval are each essential
preconditions.

##  3 Extension of the $V_{B}^{-}$ coherence

Dynamical decoupling (DD) techniques are traditional tools of nuclear magnetic
resonance spectroscopyCarr_Purcell_1954 ; meiboom_modified_1958  and have been
extensively applied in the past years to prolong the coherence of spin defects
in solid-state materialsryan_robust_2010 ; de_lange_universal_2010 ;
naydenov_dynamical_2011 ; pham_enhanced_2012 ; abobeih_one-second_2018 ;
barry_sensitivity_2020 ; anderson_five-second_nodate . Here, we apply this
approach to $V_{B}^{-}$ ensembles in hBN to improve their short $T_{2}$ times
and unlock possibilities based on advanced spin manipulation.

A significant source of $V_{B}^{-}$ decoherence is likely to be found in spin
‘flip-flops’ from the bath spins (nuclei or paramagnetic impurities)
surrounding the defectshaykal_decoherence_2022 ; liu_coherent_2022 . These
processes cause random magnetic field fluctuations felt by the electronic spin
on a time scale set by the average interactions involvedsousa2009electron ;
pham_enhanced_2012 ; bar-gill_suppression_2012 ; ye_spin_2019 . We show that
the CPMG dynamical decoupling protocolCarr_Purcell_1954  can be applied to
decouple the $V_{B}^{-}$ spins from magnetic noise. This is done by applying
resonant MW $\pi$-pulses, following the scheme
$(\pi/2)_{x}[-\tau-(\pi)_{y}-\tau]_{N}$ (see inset in Figure 2a), that have
the effect to periodically re-phase the $V_{B}^{-}$ superpositions and sustain
their coherence over longer timescales. Then, the coherence is mapped to the
spin populations via a last $(\pi/2)_{x}$ pulse for the final optical readout.
If $\tau$ is shorter than the spin bath fluctuation correlation time $t_{c}$,
the magnetic noise would appear to be time-independentsousa2009electron ; bar-
gill_suppression_2012 , and the train of $\pi$-pulses can effectively cancel
it out.

In Figure 2a, we show the signal intensity of the spin-echo obtained after $N$
MW $\pi$-pulses upon increasing the delay $\tau$ between them. We plot the
decays against the total pulse sequence time $t_{s}=2N\tau$. This results in
multiple decay curves, showing how fast decoherence occurs depending on the
number of $\pi$-pulses utilized for noise suppression. Then, we extract the
characteristic $T_{2}$ times by fitting every curve (see Methods). We observe
a factor $\sim 50$ increase in the coherence time
($T_{2}^{(1000)}\sim$4.2\text{\,}\mathrm{\SIUnitSymbolMicro s}$$) by applying
up to 1000 $\pi$-pulses, with respect to the spin-echo with a single
$\pi$-pulse ($T_{2}\sim$90\text{\,}\mathrm{ns}$$). The enhanced coherence
times closely approach the spin-lattice relaxation time, which is the
theoretical limitslichter2013principles . We note that dynamical decoupling
does not affect the strong modulation in the spin-echo $T_{2}$ measurement. We
also note that the finite duration of the MW pulses sets the shortest pulse
sequence time $t_{s}$. Moreover, particularly for large $N$, we expect
contributions from pulse errors and $T_{1}$ relaxationnaydenov_dynamical_2011
; pham_enhanced_2012 .

In Figure 2b, we show the increase of the $T_{2}$ times versus the number of
$\pi$-pulses $N$. The plot depicts a sub-linear dependence that can be fit
with a simple power function $f(N)=a\times N^{s}$ where $s\sim 0.60$, which is
in good agreement with the theoretical dependence of $T_{2}\propto N^{2/3}$
expected for a Lorentzian spin bath, in the limit of long spin bath
correlation times $t_{c}\gg\tau$sousa2009electron ; bar-gill_suppression_2012
. These results demonstrate that dynamical decoupling works effectively for
our sample, similar to the situation encountered for nitrogen-vacancy(NV)
ensembles in diamond in the presence of high-density paramagnetic impurities
(50-100 ppm) and natural abundance 13C nuclear spinsbar-gill_suppression_2012
; pham_enhanced_2012  or for $V$Si- in 4H-SiCsimin_2017 .

We also note that a factor of two increase in coherence time by a CPMG
protocol was shown by conventional electron paramagnetic resonance (EPR)
spectroscopy, although under very different experimental conditions (magnetic
field of $\sim$3\text{\,}\mathrm{T}$$ and cryogenic temperature of
$\sim$50\text{\,}\mathrm{K}$$)murzakhanov_electronnuclear_2022 .

An alternative approach that preserves coherence relies on generating
$V_{B}^{-}$ dressed spin states using spinlock pulse
sequencesschweiger_principles_2001 ; london_detecting_2013 . After optical
initialization of the defects in the $|$0$\rangle$ state, a $(\pi/2)_{x}$
pulse generates their coherent superposition which is then locked along the
$y$-axis of the Bloch sphere by a spinlock pulse. We measure the spin-lattice
relaxation time in the electron-spin rotating frame ($T_{1\rho}$) with the
experiment depicted in Figure 2c. Here, the $V_{B}^{-}$ fluorescence contrast
is monitored while increasing the spinlock pulse duration. We demonstrate that
the $V_{B}^{-}$ dressed-states and, therefore, the electronic spin coherence
can survive $\sim$80x longer than in the case of the spin-echo $T_{2}$ (Figure
1h). The decay of the spinlocked coherence can be fit by a bi-exponential
function (see Methods) and reaches the $\sim T_{1}$ relaxation time.

##  4 Sensing RF signals with $V_{B}^{-}$ defects in hBN

With the improved coherence time, the application of advanced quantum sensing
protocols is now possible. In particular, we explore oscillating (AC) magnetic
field sensing by alternatively using pulsed (CPMG-like) or continuous
(spinlock) dynamical decoupling techniques. We test the response of the
$V_{B}^{-}$-based sensor to an oscillating magnetic field in the
radiofrequency range of the form $B_{RF}(t)=b_{RF}$cos$(2\pi\nu_{RF}t+\phi)$,
where $b_{RF}$ is the amplitude, $\nu_{RF}$ the frequency and $\phi$ the phase
of the RF signal.

As a first example, we use a pulsed dynamical decoupling sequence (XY8-N),
depicted in Figure 3a. During the total free evolution time $t_{s}$, the
oscillating magnetic field induces the accumulation of a relative phase
$\theta$ to the $V_{B}^{-}$ coherence ($|\psi\rangle=(|0\rangle\pm
e^{i\theta}|1\rangle)/\sqrt{2}$). The DD sequence then acts as a narrow-band
RF filter and the $V_{B}^{-}$ superposition accumulates a maximal phase
$\theta(t_{s})=(2/\pi)\gamma b_{RF}t_{s}$abe_2016  if the condition
$\tau=1/(4\nu_{RF})$ is matched, leading to a dip in the fluorescence
intensity. In the experiment depicted in Figure 3b, we study the sensor’s
spectral response by sweeping $\nu_{RF}$ while keeping $\tau$ at a defined
value. Contrast dips appear at the expected RF frequencies for three different
values of $\tau$. These dips can be fitted by the characteristic sinc-squared
functiondegen_quantum_2017 (see Methods). In Figure 3c, we show the dependence
of the detected signal on the number of $\pi$-pulses. As anticipated, the
XY8-N sequence acts as a band-pass filter whose spectral width narrows down by
increasing Ndegen_quantum_2017 . However, simultaneously, the signal-to-noise
ratio (SNR) decreases due to increasing decoherence. Furthermore, due to the
short coherence time $T_{2}$, the pulsed DD sensing protocol fails at
detecting frequencies $\nu_{RF}\lesssim
1/T_{2}\lesssim$10\text{\,}\mathrm{MHz}$$.

Figure 3d displays an alternative approach for sensing RF signals based on
rotating-frame magnetometry(loretz_radio-frequency_2013 ;
hirose_continuous_2012 ; wang_nanoscale_2021 ). It exploits the $V_{B}^{-}$
coherence locked on the transversal plane by a spinlock pulse of duration
$t_{SL}$. Matching the $V_{B}^{-}$ Rabi frequency with the sensing frequency
$\nu_{R}=\nu_{RF}$ drives transitions between $V_{B}^{-}$ dressed-states
$|\pm\rangle=(|\ 0\rangle\pm ie^{i\theta}|\ 1\rangle)/\sqrt{2}$. This has the
consequence of inducing a Rabi evolution in the rotating frame that occurs
with an oscillation rate set by the RF amplitude $b_{RF}$. This dressed-state
dynamics determines the accumulation of a relative phase
$\theta(t_{SL})=\frac{1}{2}\gamma b_{RF}t_{SL}$, required for sensing (see
Supplementary Note 2). In Supplementary Note 3, we demonstrate coherent
control of the $V_{B}^{-}$ dressed-states and probe their evolution during the
spinlock pulsejeschke_coherent_1999 ; loretz_radio-frequency_2013 ;
wang_nanoscale_2021 .

By sweeping the frequency of the RF and performing the spinlock experiments
with different MW amplitudes (see Supplementary Note 4), we observe dips
similar to the data in Figure 3b. However, the broad ODMR line and the MW
field inhomogeneities throughout the $V_{B}^{-}$ ensemble lead to a
distribution of effective Rabi frequencies as discussed in our previous
workrizzato_polarization_2022  which result in broad dips that can be fitted
as a sum of Lorentzian functions (Figure 3b). We observe that the sensing
frequency bandwidth depends on the spinlock MW amplitude allowing access to
frequencies $\nu_{RF}<$10\text{\,}\mathrm{MHz}$$ (see Figure 3e). In addition,
we study the spectral response of the sensor with different spinlock durations
$t_{SL}$ (Figure 3e). We observe a gradual increase of the SNR until an
optimal duration of $0.5-1$\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ after
which the signal starts to degrade, approaching the $T_{1}{\rho}$ and $T_{1}$
time limits. Line-broadening and a gradual shift to higher frequencies are
also observed, likely due to heating caused by the long MW spinlock pulse.
These results demonstrate how advanced quantum sensing protocols can be
applied using spin defects in 2D materials and for qubit-control in 2D quantum
registersliu_2d_2019 ; tadokoro_designs_2021 ; taminiau_2019 ; abobeih_one-
second_2018 ; gao_nuclear_2022 .

##  5 Sensing of RF signals with arbitrary frequency resolution

The frequency resolution in dynamical decoupling sequences is typically
restricted by the duration of the pulse sequence and thus by the coherence
time. This is in particular a limitation for $V_{B}^{-}$ defects in hBN, but
it can be overcome by applying a sensing scheme called coherently averaged
synchronized readout (CASR)glenn_high-resolution_2018 ; boss_quantum_2017 ;
schmitt_submillihertz_2017 .

Figure 3g shows the pulse scheme, which consists of a train of DD sub-
sequences (in our case XY8-2), which are synchronized with the RF signal. If
$\nu_{RF}\neq 1/4\tau$, consecutive phase shifts of the RF signal with respect
to the DD sequences will occur, resulting in a downmixed fluorescence contrast
oscillating at the difference-frequency $\Delta\nu=\nu_{DD}-\nu_{RF}$
($1000\text{\,}\mathrm{Hz}$ in our case). This scheme enables arbitrary
frequency resolution since it is no longer limited by the intrinsinc coherence
$T_{2}$ time of the solid-state spin system. We run the protocol for a total
measurement time $t_{m}=$2\text{\,}\mathrm{s}$$ (Figure 3h), resulting in a
time-dependent measurement that can be Fourier transformed to give a peak with
sub-Hertz linewidth (Figure 3h). Importantly, this method can render our
ultra-thin quantum sensor capable of sensing oscillatory magnetic fields with
a high-resolution, with possible applications in magnetic resonance
spectroscopyallert_advances_2022 .

##  6 Conclusion

This work addresses the short spin coherence of the hBN spin defects, a severe
limitation for their application in quantum technology. Utilizing dynamical
decoupling protocols, we achieve a $\sim 50$ times extension of the coherence
with respect to the single spin-echo $T_{2}$, approaching the limit of the
longitudinal relaxation time $T_{1}$. Furthermore, by generating $V_{B}^{-}$
dressed-states with spinlock pulses, we demonstrate an extension of the
coherence up to $\sim$7.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, overcoming
the spin-echo $T_{2}$ by nearly two orders of magnitude. The improved
coherence times enable us to demonstrate RF signal detection in several
complementary experiments. In particular, we demonstrate that despite the
intrinsically short $V_{B}^{-}$coherence, sensing RF frequencies with sub-Hz
frequency resolution is possible.

The presented work paves the way to establish nanoscale magnetic resonance
sensing and imaging using spin-based quantum sensors embedded in a van der
Waals material. Thanks to the possible miniaturization and integration in 2D
heterostructures, these systems will render magnetic resonance techniques
functional for exploring emergent phenomena in low-dimensional quantum
materials and devices. Finally, the improved spin control and the intriguing
nuclear-spin environment surrounding the defects open a promising path to the
realization of multi-qubit registers for quantum sensors and quantum networks
integrated into ultra-thin structures.

##  7 Methods

###  7.1 Experimental setup

Initialization of the $V_{B}^{-}$ensemble is realized with a
$532\text{\,}\mathrm{nm}$ laser (Opus 532, Novanta photonics) at a power of
$\sim$100\text{\,}\mathrm{mW}$$ (CW). Laser pulses are timed by an acousto-
optic modulator (3250-220, Gooch and Housego) with typical pulse durations of
$5\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The laser light is reflected by a
dichroic mirror (DMLP650, Thorlabs) after which it is focused on the hBN flake
by an objective (CFI Plan Apochromat VC 20x, NIKON) with a numerical aperture
of NA=0.75. Photoluminescence (PL) is collected by the same objective and
alternatively focused by a tube lens on: 1) an Avalanche photodiode (APD)
(A-Cube-S3000-10, Laser Components) for the spectroscopic path; 2) a camera
(a2A3840-45ucBAS, Basler) for the imaging of the sample. The excitation green
light and possible unwanted fluorescence from other defects are filtered out
using a long-pass filter with a cut-on wavelength of 736 nm (Brightline
736/128, Semrock). The output voltage of the APD is digitized with a data
acquisition unit (USB-6221 DAQ, National Instruments). An arbitrary waveform
generator (AWG) up to 2.5GS/s (AT-AWG-GS2500, Active Technology) is used to
syncronize the experiment and generate rectangular arbitrary-phased RF pulses
for $V_{B}^{-}$ spin control. For up-conversion of the AWG RF frequency
(typically 250 MHz) to the MW frequency required for $V_{B}^{-}$ driving,
mixing with a local oscillator generator (SG384, Stanford Research Systems) is
realized by means of an IQ mixer (MMIQ0218LXPC 2030, Marki). The amplified
microwave pulses (ZHL-16W-43-S+, Mini-Circuits) are delivered by a gold strip-
line to the hBN sample. A permanent magnet underneath the sample holder is
utilized for applying the magnetic field of $\sim 8$ mT. The ESR frequency is
used to determine the magnetic field strength $B_{0}$ as well as the
$V_{B}^{-}$0,-1 resonance frequency f$V_{B}^{-}$. The radiowaves signals used
to demonstrate AC magnetometry are produced by an RF waveform generator
(DG1022z, Rigol) connected to a 30 W amplifier (LZY-22+, Minicircuits). A
small wire loop placed in the proximity of the sample was used for the RF
delivery.

###  7.2 Sample preparation

The hexagonal boron nitride van der Waals flakes were cleaved and tape-
exfoliated starting from hBN seed crystals (2D semiconductors) on a silicon
wafer with a $70\text{\,}\mathrm{nm}$ top oxide layer. We then implanted the
exfoliated samples at the ion beam facility (Helmoltz-Zentrum Dresden-
Rossdorf, HZDR) with a helium ion fluence of $10^{14}$ions/cm2 at an energy of
$3$ keV. Once the photoluminescence spectrum of the boron vacancies was
verified, we transferred the boron vacancy containing hexagonal boron nitride
with a standard dry transfer method on top of a gold strip-line evaporated on
a sapphire substrate. The gold strip-line was connected to a printed circuit
board using several bond-wires in parallel to improve impedance matching and
enabling high power microwave delivery for short Rabi pulses.

###  7.3 Pulse sequences, normalizations and fittings

####  7.3.1 Characterization of the $V_{B}^{-}$ spin defects

ODMR measurement. For all experiments throughout this work,
$5\text{\,}\mathrm{\SIUnitSymbolMicro s}$-long laser pulses were used for
initialization/readout. The ODMR measurement displayed in Figure 1e) was
performed using a $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$-long MW pulse at
$\sim$ 1mW power for the driving of the $V_{B}^{-}$ spin-populations. The
fluorescence contrast was monitored by increasing the MW frequency. For
normalization and noise cancellation purposes, a second reference sequence was
applied right after the first one, differing only from the MW being
offbucher_2019 . The single data points result from dividing the fluorescence
readouts of the two sequences. 10,000 averages for each data point and further
600 averages of the full frequency sweep were recorded. After baseline
subtraction, the lineshape was fitted with seven Gaussian functions of the
form $a[$exp$(-$ln$(2)(f-f_{0})^{2}/$LW${}^{2})]$ whose amplitude $a$ and
frequency ($f_{0}$) were used as fitting parameters whereas the half-width-
half-maximum LW was kept at a constant value of $22\text{\,}\mathrm{MHz}$.

Rabi nutation experiment. Rabi oscillations were recorded by setting the MW
frequency in the center of the ODMR spectrum and sweeping the MW pulse
duration $t_{p}$ in stepwise increments. Different MW power (-5, -14 and
-19dBm) were used corresponding to the different datasets in Figure 1f). For
each point, 10,000 averages were acquired following the same normalization and
noise cancellation procedure used for the ODMR experiments. The Rabi
oscillations were fitted using the following function:
$1-c/2+c/2($cos$(2\pi\nu_{R}t+\Phi))[a\times$exp$(-bt)+m\times$exp$(-nt)]$
where $c$ is a term for coherence normalization, $\nu_{R}$ the Rabi frequency,
$\Phi$ a phase term and the bi-exponential factor accounts for the damping of
the oscillation.

$\mathbf{T_{1}}$ measurement. The $T_{1}$ time (see Figure 1g) was measured by
sweeping the time $\tau$ between initialization and readout pulses in a range
from $10\text{\,}\mathrm{ns}$ to $\sim$30\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$. For noise cancellation purposes, a second reference sequence was applied
right after the first one, differing only for a MW $\pi$-pulse inserted after
the first laser pulse. The datapoints were then obtained by dividing the
readouts of the two consecutive sequences. Every point was averaged 100,000
times and the whole time sweep averaged 4 times. The $T_{1}$ decay curve was
fitted with a simple mono-exponential function of the form
$a($exp$(-\tau/T_{1}))$ where $T_{1}=5.84\pm 0.05$\mathrm{\SIUnitSymbolMicro
s}$$.

$\mathbf{T_{2}}$ measurement. A spin-echo sequence was set up following the
scheme [$(\pi/2)_{x}$ \- $\tau$ \- $(\pi)_{y}$ \- $\tau$ \- $(\pi/2)_{x}$]
where $\tau$ is swept from $10\text{\,}\mathrm{ns}$ to
$\sim$200\text{\,}\mathrm{ns}$$. Referencing for noise cancellation was
achieved by alternating the last MW-pulse of the spin-echo sequence from
$\pi/2$ to $3/2\pi$bucher_2019 . Every point was averaged 100,000 times and
the whole time sweep averaged 5 times. The decay was fitted with the function:
$a($exp$(-2\tau/T_{2})^{c})+b($exp$(-d\tau))$cos$(2\pi f\tau)$ where the first
stretched exponential yields the time constant $T_{2}=88.5\pm
0.7$\mathrm{ns}$$ with $c=1.37\pm 0.01$. To guide the eye in the figures, we
fitted the oscillation superimposed on the decay with a cosine function with
$f=$44.7\text{\,}\mathrm{MHz}$$, multiplied by an exponential term which
reproduces the damping of the observed oscillation with $d=1.2\times 10^{7}$
s-1.

####  7.3.2 Extension of coherence

CPMG experiments. We applied the same scheme utilized for the $T_{2}$
measurement, that is monitoring the spin-echo signal while increasing the free
evolution time $\tau$. Multiple decoherence curves were acquired for pulse
sequences with growing number $N$ of $\pi$-pulses, up to $N=1000$. The data
were normalized according to the following procedure. First, for all datasets
the time axis was multiplied by a scaling factor $S=2N$ accounting for the
real time elapsed between the first and the last $\pi/2$ pulse. Then, the
decay curve corresponding to the single $\pi$-pulse experiment (spin-echo) was
fitted by the function:
$a($exp$(-t_{s}/T_{2})^{c})+b($exp$(-dt_{s}))$cos$(2\pi ft_{s}/S)$ (details
see above). The point of maximum contrast, namely maximum intensity of the
spin-echo, when no decoherence has occurred yet, has been extrapolated from
the fit in correspondence to the time $t=0$ of the decay. Then, the contrast
of all datasets has been normalized to this value and then fitted with the
same function, keeping only the amplitudes $a$ and $b$ locked. The fitted
$T_{2}$ values and relative exponents $c$ are reported in the table below.

Table 1: Fitted $T_{2}$ time constants and corresponding stretch-exponents $c$ for the data shown in Figures 2a,b. N ($\pi$-pulses) |  $T_{2}$ [$\mathrm{ns}$] | $c$
---|---|---
1 | $90\pm 2$ | $1.425\pm 0.026$
4 | $220\pm 4$ | $1.403\pm 0.025$
16 | $500\pm 5$ | $1.435\pm 0.016$
64 | $940\pm 11$ | $1.299\pm 0.015$
300 | $1759\pm 32$ | $1.309\pm 0.024$
600 | $2776\pm 35$ | $1.265\pm 0.016$
800 | $3434\pm 30$ | $1.257\pm 0.011$
1000 | $4204\pm 44$ | $1.310\pm 0.014$

$\mathbf{T_{1\rho}}$ measurement. $T_{1\rho}$ was measured using a pulse
sequence $[(\pi/2)_{x}-d-($spinlock$)_{y}-d-(\pi/2)_{x}]$ and monitoring the
resulting fluorescence contrast while applying step-by-step increments of the
spinlock pulse duration $t_{SL}$. $3.5\text{\,}\mathrm{ns}$-long $\pi/2$
pulses were used and the delay times $d$ kept to a value as short as possible
($\sim 1-2$ ns), in order to avoid significant dephasing during this time. The
spinlock amplitude was kept at 10% of the MW amplitude utilized for the
$\pi/2$-pulses. 100,000 averages were taken for each datapoint. The resulting
curve fits a bi-exponential decay of the form:
$a($exp$(-t_{SL}/T_{1a}))+b($exp$(-t_{SL}/T_{1b}))$, with $a=0.504\pm 0.007$,
$T_{1a}=1.38\pm 0.03$ $\mathrm{\SIUnitSymbolMicro s}$, $b=0.554\pm 0.003$,
$T_{1b}=7.52\pm 0.06$ $\mathrm{\SIUnitSymbolMicro s}$.

####  7.3.3 AC-sensing

XY-N protocol. AC field sensing was performed by means of a XY8-2 sequence
following the scheme: $(\pi/2)_{x}[[-\tau-(\pi)_{\phi}-\tau]_{8}]_{2}$, with
the following $\pi$ pulses phase-scheme: $\phi=[x-y-x-y-y-x-y-x]_{2}$. $\pi/2$
and $\pi$-pulse durations of respectively $3.5\text{\,}\mathrm{ns}$ and
$7\text{\,}\mathrm{ns}$ have been used. The fluorescence contrast was
monitored while the RF frequency $\nu_{RF}$ was swept in a range of a few MHz
around the condition for the sensing frequency matching: $\nu_{RF}=1/4\tau$.
Referencing for noise cancellation was achieved by alternating the last MW-
pulse of the spin-echo sequence from $\pi/2$ to $3/2\pi$. Every point was
averaged 100,000 times and the whole sweep averaged 25 times. The observed
dips were first baseline-corrected by subtracting a line fitting the baseline
in the spectral region without signal and then fitted with the following
function: $a/2[$sin$(2\pi\tau N(\nu-\nu_{RF}))/2\pi\tau N(\nu-\nu_{RF})]^{2}$
leaving all 4 parameters $a$, $\tau$, $N$ and $\nu_{RF}$ free to fit. For
instance, for the XY8-2 experiment set to $\tau=13$ ns, $a=-1\times 10^{-3}$,
$\tau=0.013$, $N=16.1$ and $\nu_{RF}=19.8$. The same procedure has been
applied for the data in Figure 3c.

Spinlock protocol. For sensing, the same spinlock sequence described above for
the measurement of the $T_{1\rho}$ was utilized. A $\pi/2$-pulse of
$3.5\text{\,}\mathrm{ns}$ and a spinlock duration
$t_{SL}=$0.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ were used. The
fluorescence contrast was monitored while the RF frequency $\nu_{RF}$ was
swept in a range of a few MHz around the matching condition:
$\nu_{R}=\nu_{RF}$. Five different values of Rabi frequency were used, setting
the spinlock MW to the 2%, 3%, 5%, 9%, 11% of the MW amplitude utilized for
the $\pi/2$-pulses. The Rabi oscillations corresponding to the latter two
spinlock amplitudes were recorded for a check (Supplementary Note 4). The
observed dips were first baseline-corrected by subtracting a line fitting the
spectral region without signal and then fitted with several Lorentzian
functions: $a/(1+(\nu-\nu_{RF})^{2}/$LW${}^{2})$ whose sum reproduces the
overall lineshape. In these fits only the linewidth of the Lorentzians was
kept locked to $1/t_{SL}=$2\text{\,}\mathrm{MHz}$$. Same pulse sequence and
same background subtraction was applied in Figure 3f. In this case the fitting
was performed by simple Gaussian functions.

CASR protocol. We applied a pulse sequence which was synchronized with the
sensing RF signal and consisted of concatenated XY8-2 sub-sequences repeated
so that the timing between them was an integer number of the RF period. For
each XY8 sequence, we used $5\text{\,}\mathrm{\SIUnitSymbolMicro s}$ long
laser pulses and $\pi$-pulses of $8\text{\,}\mathrm{ns}$ for
initialization/readout and MW manipulation, respectively. We sensed an RF
frequency $\nu_{RF}\sim$18\text{\,}\mathrm{MHz}$$ by setting the interpulse
delay $\tau=$14\text{\,}\mathrm{ns}$$. To get a relative frequency
$\Delta\nu=$1000\text{\,}\mathrm{Hz}$$, the RF was shifted to
$\nu_{RF}=$18.001\text{\,}\mathrm{MHz}$$. A total measurement time $t_{m}$ of
2 seconds was utilized. The detected time trace was then Fourier transformed
and the magnitude plotted in Figure 3i. The resulting peak in the frequency
domain was fitted with a modified Lorentzian function to determine the full
width half-maximum (FWHM). The signal was the result of 1000 averages.

##  8 Figures

Figure 1: Characterization of the $V_{B}^{-}$ defect’s spin properties. a)
Idealized structure of $V_{B}^{-}$ spin defects (red arrows) in hBN layers.
Nitrogen atoms are displayed in blue, boron in grey. The number of the spin
defects is only for illustrative purpose and does not reflect the actual
density. b) Simplified energy levels displaying the ground and excited triplet
states, the optical excitation transitions (green arrows), the fluorescence
pathways (red arrows) and the non-radiative pathways through ISC to the S=0
singlet state (yellow arrows) which allows for spin-state dependent optical
detection. c) Microscope image of the hBN flake positioned on the gold MW
stripline. d) Image of the sample from the ODMR setup with the laser spot
(blue, false color) for the initialization/interrogation of the spin
ensembles. In the inset the Gaussian profile of the laser spot is shown. e)
ODMR spectrum of the $|0\rangle\rightarrow|-1\rangle$ transition. The data
points are fitted with single Gaussian functions (dotted lines) that are
summed up to give the overall spectral lineshape (solid-blue line). f) Rabi
oscillations for four different MW amplitudes. Inset, left: pulse sequence for
the Rabi nutation experiment. Right: dependence of the Rabi frequency
$\nu_{R}$versus the square root of microwave power P. g) Semi-log plot of the
spin-lattice relaxation time decay ($T_{1}$) measured by the recovery pulse
sequence (inset). A $T_{1}$ time of $\sim$6\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$ is extracted from the mono-exponential fit. h) Semi-log plot of the
coherence time $T_{2}$ measured with the depicted spin-echo sequence.
$T_{2}=$90\text{\,}\mathrm{ns}$$ is obtained from a stretched exponential fit.
A strong signal modulation is superimposed to the same $T_{2}$ decay curve
with a frequency of $\sim$45\text{\,}\mathrm{MHz}$$. Figure 2: Extension of
coherence for $V_{B}^{-}$ in hBN. a) Measured decoherence curves (semi-log
plot) for the $V_{B}^{-}$ spins under the effect of different CPMG decoupling
pulse sequences with increasing number of $\pi$-pulses N. Coloured dots and
solid lines are data points and fits, respectively. In black, the $T_{1}$
measurement of Figure 1g is shown with the corresponding fit as dotted line.
Bottom, left: Dynamical decoupling (CPMG) pulse sequence. Top right: enlarged
plot area for better visualization of the data points in the dotted red
rectangle.b) Plot of the $T_{2}$ values extracted from fitting the curves in
(a) vs the number of $\pi$-pulses $N$ utilized in each CPMG experiment. c)
Rotating frame spin-lattice relaxation time ($T_{1\rho}$) measured with the
depicted spinlock sequence.  Figure 3: RF-sensing with $V_{B}^{-}$ in hBN. a)
Pulsed-DD protocol used for sensing RF signals. An XY8-2 pulse sequence was
utilized for the experiments in (b),(c) keeping $\tau$ fixed while sweeping
the RF frequency. b) Spectral response of the $V_{B}^{-}$ sensor as
fluorescence contrast dips occurring at matching of the interpulse delay
$\tau$ with the RF signal period according to: $\tau=1/(4\nu_{RF})$. The data
points (dots) are best fitted with a (sinc)2 function. c) Dependence of the
lineshape on the number of pulses for XY8-1,2,3,4 sequences setting
$\tau=$15\text{\,}\mathrm{ns}$$. d) Spinlock-based protocol for RF-sensing.
The spinlock pulse duration was kept fixed
($t_{SL}=$0.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$$) and the RF frequency
was swept. e) Spectral response as fluorescence contrast dips occurring at
matching of the spinlock amplitude with the RF frequency according to
$\nu_{R}=\nu_{RF}$. The labels next to each dip indicate decreasing values of
$\nu_{R}$ which are normalized on the maximum spinlock amplitude used,
corresponding to $\nu_{R}^{(1)}\sim$18\text{\,}\mathrm{MHz}$$. f) Dependence
of the lineshape on the spinlock duration used for the experiment with
spinlock amplitude set to $\nu_{R}^{(1)}$. g) Coherently-Averaged Syncronized
Readout (CASR) protocol. XY8-2 subsequences are synchronized to the sensing
frequency for 2 seconds. h)Time-domain CASR signal. Each detected point
corresponds to an optical readout represented by the red circles in (g). In
the inset, a zoom-in of the signal in the first $4\text{\,}\mathrm{ms}$ shows
the oscillations at $\Delta\nu=$1000\text{\,}\mathrm{Hz}$$. i)Fourier
transformation of the signal in (h) yields a sharp peak at the relative
frequency $\Delta\nu$. A Lorentzian linefit of the peak (red line) results in
a $0.9\text{\,}\mathrm{Hz}$ linewidth.

## Acknowledgment

R.R. and D.B.B. would like to thank Prof. Dr. Steffen Glaser for stimulating
discussions. R.R. thanks Nick Neuling and Giovanna Salvitti for their help in
the lab. This study was funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) - 412351169 within the Emmy Noether program. R.R.
acknowledges support from the DFG Walter Benjamin Programme (Project RI
3319/1-1). J.J.F. and D.B.B. acknowledges support from the DFG under Germany’s
Excellence Strategy—EXC 2089/1—390776260 and the EXC-2111 390814868. Support
by the Ion Beam Center (IBC) at HZDR is gratefully acknowledged.

## Additional Information

Author Contributions R.R. M.S. and D.B.B. conceived the idea of AC sensing
with $V_{B}^{-}$ in hBN. R.R. and D.B.B. designed the research. R.R. carried
out the experiments. M.S. and P.J. prepared the sample and fabricated the
microstructure for MW delivery. R.R. M.S. and S.M. built the ODMR setup. F.B.
programmed the pulse sequences, J.C.H. and J.P.L. helped in the optimization
of the experimental setup, J.C.H. contributed with the theoretical
derivations. G.V.A., U.K. and M.H. were responsible for the ion implantation
for generation of the $V_{B}^{-}$ centers in the hBN substrate. A.V.S. and
J.J.F. advised on several aspects of theory and experiments. R.R., M.S. and
D.B.B. analyzed the data. R.R. and D.B.B. wrote the manuscript with inputs
from all authors. All authors reviewed the manuscript and suggested
improvements. All correspondence and request for materials should be addressed
to<EMAIL_ADDRESS>or D.B.B. (dominik.bucher@tum.de).

Competing interests. All authors declare that they have no competing
interests.

Data and materials availability. All data needed to evaluate the conclusions
in the paper and/or the Supplementary Materials. Additional data related to
this paper may be requested from the authors.

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## Supplementary Information

### Supplementary Note 1: ESEEM Modulation

In the $T_{2}$ measurement reported in Figure 1h and in the CPMG experiments
in Figure 2a, we observe a strong modulation of the time-domain electron spin
signal reminiscent of the electron spin envelope modulation (ESEEM) observed
in other systems when the central electronic spin strongly interacts with
proximate nuclei, such as 14N and 13C. Fitting this oscillation with a
function $F(t)\propto$cos$(2\pi ft)$ yields a frequency of
$f\sim$45\text{\,}\mathrm{MHz}$$, matching the hyperfine coupling of the three
equivalent 14N surrounding the defect. A similar effect can also be predicted
by using a well-known EPR calculation toolboxstoll_easyspin_2006  and
simulating a two-pulse ESEEM experiment using the following parameters: spin
S=1;g-factor=2;zero-field-splitting: D=3.47 GHz,E=60 MHz. Three 14N have been
considered in the spin system, coupled to the electronic spin with the
following hyperfine tensors A1=[80 57 47], A2=[46 91 48], A3=[80 57
47]ivady_ab_2020 ; liu_coherent_2022 . Further parameters: ext.field = 8 mT,
MWfreq=3.2 GHz, excit. bandwidth=250 MHz, Exp.Sequence=2pESEEM, resolution
dt=0.001$\mu$s. Relaxation parameters: $T_{1}$=6$\mu$s, $T_{2}$=80 ns.

Figure S1: Simulation of a two-pulse ESEEM experiment. A two-pulse ESEEM
experiment is simulated with the Easyspin function ”saffron”.

### Supplementary Note 2: Phase accumulation in the spinlock experiment

The rotating-frame Hamiltonian $H_{\text{rot}}$, accounting for a spinlock
pulse of amplitude $\Omega$ around the $y$-axis and an AC magnetic field
signal $B_{RF}(t)=b_{RF}$cos$(2\pi\nu_{RF}t+\phi)$ along the $z$-axis, reads
(in natural units $\hbar=1$):

$H_{\text{rot}}=\frac{1}{2}\Omega\,\sigma_{y}+\frac{1}{2}\gamma
b_{RF}\cos(2\pi\nu_{RF}t+\phi)\,\sigma_{z}\,,$ (1)

with the Pauli matrices $\sigma_{y}$ and $\sigma_{z}$. This equation can be
transferred into a second rotating frame of the Rabi oscillation, where the
first term vanishes. Assuming that the matching condition of
$\Omega=2\pi\nu_{RF}$ is fulfilled and that the rotating wave approximation
for $2\Omega\gg b_{RF}$ is valid, the Hamiltonian $H_{\text{rot,2}}$ in the
second rotating frame becomes:

$H_{\text{rot,2}}=\frac{1}{4}\gamma
b_{RF}\left(\cos(\phi)\sigma_{z}+\sin(\phi)\sigma_{x}\right)\,.$ (2)

Setting the signal phase to $\phi=0$ and implementing the Hamiltonian in a
time evolution operator $U(t)$ leads to the following:

$U(t)=\exp(-\mathrm{i}Ht)=\exp\left(-\frac{\mathrm{i}}{4}\gamma
b_{RF}t\,\sigma_{z}\right)\,.$ (3)

This equation can be connected to the general expression for a rotation
operator

$R_{\theta}=\exp\left(-\frac{\mathrm{i}}{2}\theta\sigma_{z}\right)\,,$ (4)

which describes a rotation of the Bloch vector by an angle of $\theta$ around
the $z$-axis. Comparing equations (3) and (4), gives the phase $\theta$ that
is accumulated during the spinlock pulse on the equatorial plane in the second
rotating frame:

$\theta(t)=\frac{1}{2}\gamma b_{RF}t\,.$ (5)

### Supplementary Note 3: Coherent manipulation of the $V_{B}^{-}$ dressed-
states

In the following experiments the evolution of the $V_{B}^{-}$ dressed-states
during matched spinlock is probed. Here, a spinlock pulse sequence is applied
where the spinlock duration is gradually increased. The RF frequency is kept
at a fixed value, matching the condition $\nu_{RF}=\Omega/{2\pi=18}$ MHz. The
RF phase $\phi$ is kept constant with respect to the pulse sequence.
Furthermore, the last $\pi/2$-pulse has been removed (see Figure S2a) to
directly detect the rotating-frame Rabi evolution as oscillations of the spin
vector’s $z$-projection. The observed signal is modulated at a frequency of
$18\text{\,}\mathrm{MHz}$ in accordance to the matching condition. Moreover,
the phase of these oscillations is sensitive to the RF phase $\phi$ utilized
for driving the dressed-states (see Figure S2b). To, at least qualitatively,
describe the dynamics occurring during the experiment, numerical simulations
have been performed based on the density matrix evolution of a simple spin-1/2
system under the Hamiltonian reported in the equation 1 of the Supplementary
Note 2. The resulting density matrix is then corrected by an exponential term
accounting for dressed-state decoherence using a bi-exponential decay with
time constants of $T_{a}=1.6\times 10^{-7}$ s and $T_{b}=1.5\times 10^{-6}$ s
with amplitudes of 0.95 and 0.05, respectively. These parameters have been
chosen until the oscillation damping in the simulation resembled the
experimental one. The simulation for $\phi=180\deg$ with no relaxation
correction is shown in (Figure S2d) for a comparison. We note that, due to the
RF field amplitude not being strong enough to counteract the fast decoherence,
the detected evolution is likely to be only the very initial part of the
complete Rabi rotation from the $|+\rangle$ to $|-\rangle$ dressed-states
(Figure S2d).

Figure S2: Dressed-state Rabi oscillations a) Spinlock pulse sequence. Each
datapoint correspond to the fluorescence contrast detected immediately after
the spinlock pulse. b) Experiments performed setting different $\phi$ values
resulting in different Rabi phases. c)Numerical simulations of dressed-state
Rabi nutation. d)Overlap of the simulation reported in c) (red line) and the
same numerical simulation with
$t_{SL}\sim$5\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ in the absence of
decoherence (grey).

### Supplementary Note 4: Measurement of Rabi frequencies for spinlock
matching

Spinlock sensing is accomplished by setting the spinlock MW amplitude such
that the corresponding $V_{B}^{-}$ Rabi frequency matches the RF sensing
frequency. In this work, this is practically done by setting the AWG amplitude
to a fraction of the available range, only for the spinlock pulse. For a
check, Rabi experiments have been performed by setting, as the Rabi-pulse
amplitude, the same used for the spinlock pulse. Here, Rabi oscillations
corresponding to the experiments labeled $\nu_{R}^{0.8}$ and $\nu_{R}^{1}$ are
reported. As visible, they yield Rabi frequencies respectively of 15 and 18
MHz, exactly where the dips fall in Figure 3e.

Figure S3: Measurement of Rabi frequencies corresponding to spinlock MW
amplitudes.

### Supplementary Note 5: Spinlock sensing of RF fields by sweeping the
spinlock amplitude

In the main text, the proof-of-principle demonstration of RF sensing by the
spinlock protocol (rotating frame magnetometry) was accomplished by setting
the MW spinlock amplitude at a defined value and sweeping the RF signal
frequency $\nu_{RF}$ until the matching condition for phase accumulation was
met and a dip in the fluorescence contrast was observed. Here, we show the
vice-versa experiment, where the spinlock amplitude $A_{SL}$ is swept until
matching a certain RF frequency. Although the same fundamental physics is
involved, this sensing mode demonstrates the capability of the sensor to seek
an unknown RF signal frequency.

Figure S4: Spinlock sensing by spinlock amplitude sweep. a) Pulse sequence
utilized for the experiment. The RF frequency was kept fixed at a value of
$\nu_{RF}=$14\text{\,}\mathrm{MHz}$$ and the spinlock MW amplitude $A_{SL}$
was swept in a range around $\nu_{RF}$. b) Spinlock amplitude sweep experiment
keeping the RF signal off (red line) and on (blue line). A broad dip appears
centered at a spinlock amplitude value matching the RF frequency. In the inset
the dip is shown after baseline subtraction. The spinlock amplitude,
originally in units of Volt-per-peak (Vpp) was converted to MHz by performing
a calibration matching several RF frequencies at different values of $A_{SL}$,
as done in rizzato_polarization_2022 .