1]Department of Engineering, University of Exeter, EX4 4QF, UK

2]Hitachi Cambridge Laboratory, Hitachi Europe Ltd., CB3 0HE,UK

*]corresponding author: Charlie J. Patrickson<EMAIL_ADDRESS>
# Continuous drive heterodyne microwave sensing with spin qubits in hexagonal
boron nitride

Charlie J. Patrickson  Valentin Haemmerli  Shi Guo  Andrew J. Ramsay  Isaac J.
Luxmoore  [  [  [

###### Abstract

Quantum sensors that use solid state spin defects have emerged as effective
probes of weak alternating magnetic signals. By recording the phase of a
signal relative to an external clock, these devices can resolve signal
frequencies to a precision orders of magnitude longer than the spin state
lifetime. However, these quantum heterodyne protocols suffer from sub-optimal
sensitivity, as they are currently limited to pulsed spin control techniques,
which are susceptible to cumulative pulse-area errors, or single continuous
drives which offer no protection of the spin coherence. Here, we present a
control scheme based on a continuous microwave drive that extends spin
coherence towards the effective $T_{2}\approx\frac{1}{2}T_{1}$ limit and can
resolve the frequency, amplitude and phase of GHz magnetic fields. The scheme
is demonstrated using an ensemble of boron vacancies in hexagonal boron
nitride, and achieves an amplitude sensitivity of $\eta\approx
3-5\>\mathrm{\mu T\sqrt{Hz}}$ and phase sensitivity of $\eta_{\phi}\approx
0.076\>\mathrm{rads\sqrt{Hz}}$. By repeatedly referencing the phase of a
resonant signal against the coherent continuous microwave drive in a quantum
heterodyne demonstration, we measure a GHz signal with a resolution $<$1 Hz
over a 10 s measurement. Achieving this level of performance in a two-
dimensional material platform could have broad applications, from probing
nanoscale condensed matter systems to integration into heterostructures for
quantum networking.

## Introduction

Electron spins confined to crystalline point defects have attracted
significant interest for magnetic field sensing in ambient conditions. To
date, progress in the field has been dominated by the nitrogen vacancy (NV)
center in diamond, with sustained efforts in material engineering and
measurement protocols producing notable applications in materials science[1,
2], biosensing[3, 4] and nanoscale NMR [5]. For the detection of AC fields,
outstanding amplitude sensitivities in the $\mathrm{pT/\sqrt{Hz}}$ range[6, 7]
have been demonstrated by using large ensembles of NV-centers embedded in bulk
material. However, achieving comparable performance in small nanodiamonds,
where the sensor can be placed in close proximity to a signal source, has
remained a challenge [8, 9, 10]. This has motivated growing interest in
alternative materials, with two-dimensional foils of hexagonal boron nitride
showing particular promise[11, 12]. The system hosts multiple optically active
defects capable of coherent spin control, most notably the boron vacancy[13],
and recently discovered single carbon-related defects of ambiguous
structure[14, 15].

Two important metrics in the characterisation of spin-based sensors are the
frequency resolution and the sensitivity to low amplitude signals. Effective
sensing protocols enhance both properties, which is often achieved by
extending the spin state lifetime using dynamical decoupling. For NV centres,
this has lead to coherence times approaching an effective limit of
$T_{2}\approx\frac{1}{2}T_{1}$ [16], with the frequency resolution and
sensitivity similarly bounded. However, by exploiting the spin’s sensitivity
to signal phase, pulsed quantum heterodyne measurements have achieved
frequency resolutions far beyond the limit imposed by $T_{1}$ for signal
frequencies $<$ 50 MHz[17, 18, 19, 20]. The technique builds up a stroboscopic
image of a signal waveform by recording the instantaneous signal phase against
an external clock, which can have a coherence many orders of magnitude better
than $T_{1}$. To access GHz frequencies, other heterodyne approaches have used
a pulsed microwave (MW) drive that dresses the electron spin transition with
additional energy levels, which form a Mollow triplet in the absorption
spectrum[21, 22]. However pulsed methods are sub-optimal, as the sensitivity
suffers from pulse-area errors[23, 24] and requires high microwave powers[25].
Alternatively, single continuous drives have been used to replicate the phase
response, [26, 22], but the sensitivity of these schemes is limited by
coherence times $<<T_{2}$.

In this work, we present a continuous drive sensing protocol that can detect
the phase, frequency and amplitude of an AC magnetic field, whilst also
achieving coherence times approaching $T_{2}\approx\frac{1}{2}T_{1}$. The
scheme uses a single, continuous phase modulated microwave field, often
referred to as continuous concatenated dynamical decoupling (CCDD)[27, 28], to
drive the spin along two different axes, at two different frequencies. A
signal that is resonant with these rotations will cause the spin to deviate
from the CCDD driven trajectory, with a path that depends on the signal phase.
Information about the new trajectory, and therefore signal phase, is revealed
by projecting the spin onto the z-axis through optical readout. We demonstrate
the scheme using a $V_{B}^{-}$ ensemble in hexagonal boron nitride, where it
successfully suppresses the effects of magnetic noise from the host III-V
nuclear spin bath. In comparison to other CCDD sensing schemes[29], we show
that the microwave drive parameters can be used to switch the spin response
between phase and amplitude detection. The device’s amplitude and phase
sensitivity are quantified as, $\eta\approx 3-5~{}\mathrm{\mu T\sqrt{Hz}}$ and
$\eta_{\phi}\approx 0.076~{}\mathrm{rads\sqrt{Hz}}$, respectively. Finally, we
use the scheme to repeatedly record signal phase in a quantum heterodyne
protocol, achieving a frequency resolution of 0.118 Hz at $\sim$2.31 GHz, over
a total measurement time of 10 seconds. The scheme provides comprehensive
characterisation of AC signals, without the accumulation of errors found in
pulsed sequences, and is equally applicable to trapped atom and ion systems,
and other solid state defects.

Figure 1:  Overview of the sample and protocol. (a) Darkfield microscope image
of the device. The CCDD microwave drive,
$H_{c}(\Omega,\epsilon_{m},\theta_{m})$ and signal field $H_{s}(\phi_{s})$ are
applied to the $V_{B}^{-}$ spin ensemble via the co-planar waveguide. (b)
Simplified model of the $V_{B}^{-}$ optical and spin states, with spin
dependent decay rates, $\gamma_{0},\gamma_{1},\kappa_{0}$ and $\kappa_{1}$.
(c) The CCDD drive field resonantly addresses the two-level system
corresponding to the $|m_{s}=0\rangle$ to $|m_{s}=-1\rangle$ spin transition,
producing concatenated dressed states defined by the drive amplitudes $\Omega$
and $\epsilon_{m}$, and drive phase $\theta_{m}$. A resonant signal $H_{s}$
will drive an additional transition between these states. (d) Signal driven
$V_{B}^{-}$ spin evolution in a doubly rotating reference frame. The frame is
selected so that the drive field, $H_{c}(\Omega,\epsilon_{m},\theta_{m})$
(turquoise arrow), reduces to a DC component in the YZ plane, with the polar
angle determined by the phase of the drive, $\theta_{m}$. A resonant signal
field applied along the x-axis exerts a torque on the spin vector (black
arrow). The z-component of the subsequent spin trajectory is insensitive to
signal phase if $\theta_{m}=0$ (top), but is sensitive if
$\theta_{m}=\frac{\pi}{2}$ (bottom). This can be detected via optical readout,
which projects the spin onto the z-axis. To demonstrate this the spin
trajectory is plotted under two different signal phases, $\phi_{s}=0$ (blue)
and $\pi/2$ (green).

## Methods

### Experimental Setup

The device, shown in Fig. 1(a) is constructed of a sapphire substrate,
patterned with a gold co-planar waveguide (CPW), which delivers the microwave
drive fields to an ion-irradiated hBN flake on top. A 488nm laser is modulated
by an acousto-optic modulator, and focused through an NA=0.55 microscope
objective for optical excitation. The same objective collects
photoluminescence, which is detected using a single photon avalanche detector
(SPAD). The microwave drive and signal waveforms are generated using an
arbitrary waveform generator (AWG). All results were recorded at room
temperature in air. (see methods for further experimental details).

The boron vacancy hosts two unpaired electrons, producing a spin-1 triplet
ground state aligned along the principal crystal axis. A simplified
representation of this system and the relevant optical states are presented in
Fig. 1(b). The $m_{s}=0$ and $m_{s}=\pm 1$ spin states are separated by a zero
field splitting (ZFS) of $D\approx 3.5~{}\mathrm{GHz}$, and a strain field of
$E\approx 59~{}\mathrm{MHz}$[30]. We use off resonant optical pumping to
prepare the spin into the $m_{s}=0$ state. The same optical excitation
provides readout of the spin state, where the emitted photoluminescence (PL)
is brighter for the $m_{s}=0$ than the $m_{s}=\pm 1$ states. A DC-magnetic
field applied along the c-axis of the hBN, $B_{z}\approx 207~{}\mathrm{mT}$,
produces a Zeeman shift of $\omega_{\pm 1}=\pm\gamma_{e}B_{z}\approx\pm
5.74~{}\mathrm{GHz}$. The frequency of the $m_{s}=0\Leftrightarrow m_{s}=-1$
transition used in these experiments is then
$\omega_{0}=D-E-\omega_{-1}=2.32~{}\mathrm{GHz}$.

The hBN is of natural isotopic composition, with approximately
$99.6\%~{}\mathrm{{}^{14}N}$ nuclei, which has a nuclear spin $I=1$. The boron
vacancy couples to the three nearest nitrogen nuclei[31], with a strong
hyperfine interaction (HFI) of $47~{}\mathrm{MHz}$. Fluctuations in the
nuclear spin bath [32] result in short $T_{2}^{*}$ coherence times, typically
below $100~{}\mathrm{ns}$. We have shown previously that a strong CCDD
microwave drive of $100~{}\mathrm{MHz}$ can mitigate the effects of this
inhomogeneous noise and extend the coherence time to the few microsecond
range[30].

### Phase detection using a double microwave drive

The central idea of CCDD is to use a pair of microwave drives to isolate the
spin from magnetic noise. The first drive protects against low frequency phase
noise. If applied in quadrature, the second drive counteracts fluctuations in
the first drive [27]. When used in this way, a resonantly driven spin vector
remains co-aligned with the second drive, creating a protected eigenstate.
However, if exposed to a resonant signal field, effectively a third drive, the
spin vector will deviate. The spin-projection along the z-axis is then
detected optically and forms the basis of CCDD sensing schemes [29, 33, 34].

Whilst the signal phase influences the resulting spin trajectory, it has
little impact on the z-projection (see Fig. 1 (d)) and is therefore difficult
to detect with optical readout techniques used in typical CCDD sensing
schemes[29, 33, 34]. To sense the signal phase, we apply the second drive in
phase with the first drive, perpendicular to the spin vector, so that it
modulates the Rabi frequency. This reduces the protection provided by the
microwave drive [30]. Instead, coherence stabilization is now provided by the
signal field (Fig. 2(a)). The signal driven spin trajectory is again dependent
on signal phase, but crucially, so is the projection along the z-axis (Fig. 1
(d)).

The dynamics are determined by the system Hamiltonian. Choosing to resonantly
drive the $m_{s}=0\Leftrightarrow m_{s}=-1$ transition we approximate a two-
level system, $H_{0}=\frac{1}{2}\omega_{0}\sigma_{z}$. The CCDD drive field,
$H_{C}$, acts along the x-axis and in general, the signal field $H_{s}$ can be
applied along an arbitrary axis,

$\displaystyle\begin{gathered}H=H_{0}+H_{C}+H_{s},\\\
H_{C}=\Omega\cos{(\omega_{0}t-\frac{2\epsilon_{m}}{\Omega}\sin{(\omega_{m}t-\theta_{m})}})\sigma_{x},\\\
H_{s}=(g_{x}\sigma_{x}+g_{y}\sigma_{y}+g_{z}\sigma_{z})\cos(\omega_{s}t+\phi_{s})\end{gathered}$
(3)

where $\omega_{0}$ is the energy gap of the two level system, $\Omega$ and
$\epsilon_{m}$ describe the amplitudes of the first and second CCDD drive
fields, respectively. The dynamics of the optically detected spin z-component
are sensitive to the signal amplitude $g_{i}$, the signal frequency
$\omega_{s}$ and, if the drive phase is $\theta_{m}=\frac{\pi}{2}$, the signal
phase, $\phi_{s}$.

We interpret this interaction by considering a doubly rotating reference
frame, first with respect to $\frac{1}{2}\omega_{0}\sigma_{z}$, and then
$\frac{1}{2}\omega_{m}\sigma_{x}^{\prime}$ (primes denote the reference
frame). This is a rotating frame that tracks the Rabi oscillation driven by
$\Omega$. We focus on $g_{x}$, taking $g_{y}=g_{z}=0$, as each signal vector
drives similar spin dynamics but for different sensor resonances and signal
phases (see Supplementary Note 3). Following the method in ref. 34, for the
case $\Omega=\omega_{m}$ we find,

$\displaystyle\begin{gathered}H^{{}^{\prime\prime}}=H_{C}^{{}^{\prime\prime}}+H_{s}^{{}^{\prime\prime}}=\frac{\epsilon_{m}}{2}[\sin{(\theta_{m})}\sigma_{y}^{{}^{\prime\prime}}+\cos{(\theta_{m})}\sigma_{z}^{{}^{\prime\prime}}]+\frac{g_{x}}{2}(\sigma_{x}^{\prime\prime}\cos((\omega_{s}-\omega_{0})t+\phi_{s}))\end{gathered}$
(5)

where we have applied the rotating wave approximation and assumed
$\epsilon_{m}<<\Omega$. The CCDD drive reduces to a time-independent magnetic
field, $\epsilon_{m}$, which points in the $YZ^{{}^{\prime\prime}}$ plane at a
polar angle described by the phase of the drive, $\theta_{m}$. This is
illustrated in Fig. 1(d). A benefit of using this approach is that the CCDD
dressed spin states produce six tuneable transitions centred on the electron
spin resonance. This means that our device can select signal frequencies
across a $\sim 300~{}\mathrm{MHz}$ range at a fixed DC field[34] (also see
Supplementary Note 2). In Eq. 5 we omit five of the resonances and consider
only the $\omega_{s}=\omega_{0}-\epsilon_{m}$ resonance, as the sensor couples
more strongly to the signal field here[35, 34]. For other sensor resonances,
including off-resonant signals, see Supplementary Note 2.

$H^{{}^{\prime\prime}}$ sets the spin trajectory via the Heisenberg equation,
$\dot{\sigma^{{}^{\prime\prime}}}=i[H^{{}^{\prime\prime}},\sigma^{{}^{\prime\prime}}]$.
To provide a more intuitive interpretation, we re-express this as the rotation
$\vec{\dot{\sigma}}^{{}^{\prime\prime}}=\vec{H}^{{}^{\prime\prime}}(t)\times\vec{\sigma}^{{}^{\prime\prime}}$.
To make the device sensitive to signal phase, we cast $\epsilon_{m}$ along the
$y^{{}^{\prime\prime}}$-axis by choosing $\theta_{m}=\frac{\pi}{2}$. This
causes the spin vector to rotate in the $XZ^{{}^{\prime\prime}}$ plane at the
frequency $\epsilon_{m}$. Mixing the rotating spin with a signal oscillating
at the frequency $\epsilon_{m}$ in the rotating frame will produce a constant
torque on the spin vector, causing it to rotate. Crucially, the coupling
strength depends on the signal phase. To illustrate this interaction, in Fig.
1(d) we plot the trajectories of two different spin vectors exposed to signals
of phase $\phi_{s}=0$ (blue), and $\phi_{s}=\frac{\pi}{2}$ (green) (see
methods). Evolution occurs in the same doubly rotating frame described above.
The torque produced by the signal is largest for $\phi_{s}=0$, causing a rapid
divergence of the spin vector from its original trajectory, which lay in the
$XZ^{{}^{\prime\prime}}$ plane. The coupling strength is minimised for
$\phi_{s}=\frac{\pi}{2}$. The cumulative impact on $\dot{\sigma}$ can be seen
after $\epsilon_{m}t=4\pi$, where the z-projection of the spin vector is
drastically different for the two cases.

Figure 2: $\bm{V_{B}^{-}}$ spin dynamics driven by a CCDD field,
$\bm{H_{c}(\theta_{m})}$, and a signal, $\bm{H_{s}(\phi_{s})}$. The signal is
resonant with the $\omega_{s}=\omega_{0}-\epsilon_{m}=2.31~{}\mathrm{GHz}$
sensor transition. (a) Phase-sensitive detection Rabi measurements with the
drive phase $\theta_{m}=\frac{\pi}{2}$, plotted without (turquoise) and with a
signal applied for signal phases of $\phi_{s}=0$ (blue) and
$\phi_{s}=\frac{\pi}{2}$ (green). The inset highlights both the extended spin
coherence in the presence of the signal, and the different sensor responses
for the two signal phases. (b) Experimental sequence used to record the plots
shown in (a) and (c)-(g). The contrast is recorded as $\mathrm{C_{\Delta
T}}=\mathrm{(P_{T}-P_{T+\Delta T})/P_{T+\Delta T}}$, where the reference
readout is measured with $\Delta T=5\>\mathrm{ns}=\frac{\pi}{\omega_{m}}$ to
cancel the effects of $T_{1}$ decay [36]. (c) Fourier transforms of (a). The
signal enhances the FFT amplitude for signal phases of $\phi_{s}=0$, and
creates additional Fourier components for $\phi_{s}=\frac{\pi}{2}$, where the
spectrum is offset for clarity. (d) Equivalent of (c) for signal phases of
$0<\phi_{s}<2\pi$, showing a smooth transition between two response regimes
defined by $\phi_{s}=n\pi$ and $\phi_{s}=\frac{n\pi}{2}$. Dashed white lines
correspond to spectra shown in (c). (e) Phase insensitive detection Rabi
measurement where the drive phase $\theta_{m}=0$. (f) Fourier transform of
(e). (g) Fourier response for signal phases $0<\phi_{s}<2\pi$. Dashed white
lines correspond to spectra shown in (f).

## Results

### Phase sensitive microwave detection

Figure 3: Benchmarking of the sensor response to signal amplitude and phase.
Contrast as a function of signal amplitude for $\theta_{m}=\frac{\pi}{2}$ in
(a) and $\theta_{m}=0$ in (b), plotted for signal phases of $\phi_{s}=0$
(blue), $\frac{\pi}{2}$ (green). The inset of (c) describes the experimental
sequence used for (a)-(d), with a fixed pulsewidth of $T_{MW}=950\mathrm{ns}$,
and the contrast calculated as $\mathrm{C_{0}}=\mathrm{(P_{T}-P_{0})/P_{0}}$.
(c) The change in contrast with and without a signal applied as a function of
signal amplitude. Linear fits provide ${\max\lvert\frac{\partial(\Delta
C)}{\partial B}\rvert}$ for the sensitivity calculation in Eq. 6. (d) Phase
sensitivity as a function of signal amplitude. A value for
${\max\lvert\frac{\partial(\Delta C)}{\partial\phi_{s}}\rvert}$ was required
for each phase sensitivity calculation. The measurement used for a signal
amplitude of 0.9 MHz is provided in the inset as an example.

To probe the CCDD driven spin dynamics in the presence of a resonant signal,
we perform Rabi experiments with the pulse sequence shown in Fig. 2(b). The
sequence samples the z-component of the spin trajectory after simultaneously
applying a control and signal microwave pulse of duration, $T_{MW}$
($T_{MW}+\Delta T$), with optical readout measurements $\mathrm{P_{T}}$
($\mathrm{P_{T+\Delta T}}$). These values are then used to calculate a
normalised contrast, $\mathrm{C_{\Delta T}}=\mathrm{(P_{T}-P_{T+\Delta
T})/P_{T+\Delta T}}$, where $\Delta T=5\mathrm{ns}=\frac{\pi}{\omega_{m}}$ to
nullify the effects of $T_{1}$ decay. In Fig. 2(a) we use the sequence to
illustrate the spin response when the drive phase $\theta_{m}=\frac{\pi}{2}$.
Note that without a signal (turquoise) the ensemble decoheres within $\sim
1000$ ns due to unprotected fluctuations in the CCDD drive amplitude $\Omega$
[30]. However, in our scheme this is corrected by the signal field, which
behaves as an additional decoupling drive. This improves the devices
sensitivity by extending spin coherence. This is shown in Fig. 2(a) for a
signal amplitude of $g_{x}$ = 2 MHz, frequency
$\omega_{s}=\omega_{0}-\epsilon_{m}$ = 2.31 GHz, and signal phases of
$\phi_{s}=0$ (blue) and $\phi_{s}=\frac{\pi}{2}$ (green). The inset depicts
the marked difference in spin response for the two phases, and can be
visualised by the Fourier transforms in Fig. 2(c) where we find a pair of
nested Mollow triplets. The central frequency is produced by the CCDD drive at
$\Omega=100\,\mathrm{MHz}$. The sidebands at $\Omega\pm\epsilon_{m}=$ 90 and
110 MHz are produced by mixing with the second CCDD drive. The splitting of
the central peak and sidebands is caused by the signal field
$g_{x}/2=1\,\mathrm{MHz}$ [34]. These dynamics reflect the trajectories
plotted in Fig. 1(d), however they are superimposed onto the frequency
components of the rotating reference frame, introducing additional complexity
in the spin response. Note that the Fourier frequencies are independent of
(dependent on) signal amplitude, $g_{x}$, for signal phases of $\phi_{s}=n\pi$
($\phi_{s}=\frac{n\pi}{2}$), where n is an integer (see Supplementary Note 4).
To highlight the distinction with CCDD sensing schemes that omit drive phase,
in Fig. 2(e) we plot Rabi measurements with the drive phase $\theta_{m}=0$.
Here the plots are qualitatively the same, and the Fourier transforms in Fig.
2(f) and (g) confirm that the sensor response is independent of signal phase.

To operate the device as a sensor we use the pulse sequence depicted in Fig.
3(b). The contrast $\mathrm{C_{0}}=\mathrm{(P_{T}-P_{0})/P_{0}}$ is recorded
at a single pulsewidth of $T_{MW}=950~{}\mathrm{ns}$, which corresponds to a
peak in the CCDD Rabi oscillation [37]. The reference readout $P_{0}$ is
collected with the CCDD MW drive, $H_{c}(\theta_{m})=0$, and signal,
$H_{s}(\phi_{s})=0$. Turning the CCDD drive off is useful in the case of an
unknown continuous signal as the dressed spin states are removed, providing a
robust reference measurement that is unresponsive to the signal field. In Fig.
3(a) we record the contrast $\mathrm{C_{0}}$ as a function of signal amplitude
for signal phases of $\phi_{s}=0,\frac{\pi}{2}$. The diverging responses
demonstrates the sensitivity to signal amplitude and phase. In Fig. 3(b) we
again compare to the case of $\theta_{m}=0$, where the response is sensitive
to signal amplitude, but not signal phase.

To benchmark the sensors performance, we calculate the amplitude sensitivity,
$\eta$,

$\displaystyle\eta=\frac{\mathrm{S}(t_{m})}{\max\lvert\frac{\partial(\Delta
C_{0})}{\partial g_{x}}\rvert}\sqrt{t_{m}}$ (6)

where $g_{x}$ is the signal amplitude, $\Delta
C_{0}=|(C_{0}(g_{x})-C_{0}(g_{x}=0)|$ describes the change in contrast due to
the signal field, S$(t_{m})$ is the standard deviation in $\Delta C_{0}$, and
$t_{m}$ is the measurement time. To calculate
${\max\lvert\frac{\partial(\Delta C_{0})}{\partial g}\rvert}$ we plot the
change in contrast, $\Delta C_{0}$ as a function of signal amplitude, $g_{x}$,
for different drive and signal phases in Fig. 3(c). Each data point is
averaged over 10 measurements to provide an estimate of the standard
deviation, $S(t_{m})$.

For $\theta_{m}=\frac{\pi}{2}$, this gives sensitivities of $5.1~{}\mathrm{\mu
T/\sqrt{Hz}}$ and $3.4~{}\mathrm{\mu T/\sqrt{Hz}}$ for signal phases of
$\phi_{s}=0$ and $\phi_{s}=\frac{\pi}{2}$, respectively. These values are
comparable to other AC magnetometry schemes using $V_{B}^{-}$ ensembles [38,
34], and as a reference for our device we find $\eta=2.5\mathrm{\mu
T/\sqrt{Hz}}$ in the case of zero drive phase, $\theta_{m}=0$. This suggests
that the protocol is not only able to resolve signal phase, but does so with
only a modest decrease in amplitude sensitivity.

The inset in Fig. 3(d) shows an analogous measurement for calculating the
phase sensitivity $\eta_{\phi}$, with the drive phase
$\theta_{m}=\frac{\pi}{2}$. For a fixed value of $T_{MW}$ the contrast $\Delta
C_{0}$ has a strong dependence on signal amplitude, as it effects both the
spin coherence and the Fourier components of the spin response. To
characterise this relationship, we plot the phase sensitivity, $\eta_{\phi}$,
as a function of signal amplitude, $g_{x}$, in the main panel of Fig. 3(d). A
minimum in the phase sensitivity of $\eta_{\phi}=0.076\mathrm{rads/\sqrt{Hz}}$
is reached for signal amplitudes of $g_{x}\approx 35\mathrm{\mu T}$. At the
cost of amplitude sensitivity, this minimum could be improved by optimising
$T_{MW}$ to reduce the total measurement time, $t_{m}$ (see Supplementary Note
5). Note that the signal amplitude can also be measured independent of signal
phase by choosing $\theta_{m}=0$[34].

Figure 4: Quantum heterodyne measurements using a continuous microwave drive.
(a) Schematic of the experimental sequence. A coherent microwave drive
(turquoise) is used across sequential measurements, each lasting $T_{MW}$, to
record the phase of a continuous, coherent signal (blue). The sensor output
(bottom panel) records the relative detuning. (b) Experimental photon time
trace, giving an excerpt of the sensor output analogous to the bottom panel of
(a). The total measurement time $t_{m}$ was 10 seconds, with an average of 1.8
photons collected per measurement. (c) FFT of the autocorrelation taken from
(b). The inset shows a Gaussian fit providing a SNR of 235, with the FWHM
giving a frequency resolution of 0.118 Hz. (d) FFT SNR as a function of total
measurement time, $t_{m}$. Taking the FFT of the autocorrelated data improves
the SNR scaling from $\sqrt{t_{m}}$ to $t_{m}$.

###  0.1 CCDD heterodyne sensing

In the measurements of Fig. 3 the sequence repetition time,
$T_{rep}=5~{}\mathrm{\mu s}$, and the phase measurement is averaged over
$10^{5}$ measurements (total measurement time, $t_{m}=1~{}\mathrm{s}$). The
signal phase is set to be the same at the beginning of each measurement
period, as would be the case for an interferometric measurement of a signal
exactly resonant with $\omega_{0}+\epsilon_{m}$. However, the phase
sensitivity also allows this method to be adapted for quantum heterodyne
sensing [18, 17], where the instantaneous phase information is recorded for
each individual measurement. Fig. 4(a) provides an overview of the CCDD
quantum heterodyne sensing technique. The CCDD drive field acts as the local
clock and provides a stable coherent phase reference, and is switched on for a
time $T_{MW}$ during each sensing sequence. The instantaneous phase difference
between the continuously applied signal and the local clock provided by the
CCDD drive is encoded in the PL intensity originating from the subsequent
laser pulse. In this way, the relative phase is tracked as a function of time,
thereby sampling the beat frequency between signal and clock. Taking the
Fourier transform reveals the detuning of the signal from the clock, with a
frequency resolution determined by the total number of measurements multiplied
by $T_{rep}$.

Fig. 4(b) shows the experimental realisation of the CCDD quantum heterodyne
method, where an excerpt of a spin dependent PL time trace is plotted. To
record the data, a continuous signal of frequency $2310.008~{}\mathrm{MHz}$
was generated by the AWG and applied via the CPW. The
$\omega_{0}+\epsilon_{m}=2310~{}\mathrm{MHz}$ resonance provided the $H_{c}$
clock frequency. An average of 1.8 photons were recorded per readout sequence.
The time sequence data is recorded for a total measurement time,
$t_{m}=10~{}\mathrm{s}$. The data is autocorrelated, then fast Fourier
transformed [22], as plotted in Fig. 4(c). The dominant Fourier component
$\omega_{f}$ corresponds to the signal demodulated by the clock at
$16~{}\mathrm{kHz}$, and has a full-width half maximum of 0.118 Hz. Note that
the peak appears at twice the true detuned frequency of
$\delta=\omega_{s}-\omega_{0}-\epsilon_{m}=8~{}\mathrm{kHz}$. This is because
the sensor response cycles twice for signal phases between $0\geq\phi_{s}\geq
2\pi$, as can be seen in the inset of Fig. 3(d). Comparing the peak height to
the standard deviation of the baseline gives a signal to noise ratio, SNR=235.

For a fixed set of CCDD drive parameters, the device can detect detuned
signals within a 100 kHz range of the sensor resonance. This is demonstrated
in Fig. 4(d), where we plot the peak frequency from the autocorrelation FFT as
a function of signal detuning. This range is determined by the measurement
repetition rate, $1/T_{rep}=400~{}\mathrm{kHz}$, which sets the Nyquist
frequency of the PL time trace. This limit could be extended by reducing
$T_{MW}$ from $950~{}\mathrm{ns}$ to 10s of $\mathrm{ns}$ (see Supplementary
Note 5) or by shifting the sensor resonance with the $H_{c}$ drive amplitude,
$\epsilon_{m}$ [34]. Although 4(c) shows that the sign of the detuning is not
implicit from a single measurement, this can also be deduced using
$\epsilon_{m}$. Finally, in Fig. 4(d) we characterise how the SNR scales with
measurement time, confirming that $\mathrm{SNR}\propto t_{m}$ (green) with
autocorrelation, compared to a shot noise limited
$\mathrm{SNR}\propto\sqrt{t_{m}}$ without (blue)[22].

## Discussion

To conclude, we have demonstrated a phase modulated CCDD sensing protocol that
extends the coherence time of a $V_{B}^{-}$ ensemble into the $\mathrm{\mu s}$
regime by suppressing the effect of magnetic noise from the nuclear spin bath,
whilst providing tuneable control over the phase, frequency and amplitude
response of the sensor. Benchmarking our sensor, we measure amplitude and
phase sensitivities of $\eta\approx 3-5\>\mathrm{\mu T\sqrt{Hz}}$ and
$\eta_{\phi}\approx 0.076\>\mathrm{rads\sqrt{Hz}}$, respectively. Finally, the
sensing protocol is embedded in a quantum heterodyne measurement, where we
measure a frequency resolution below 1 Hz for a signal frequency of $\sim$2.31
GHz.

Our quantum heterodyne protocol employs a continuous concatenated dynamic
decoupling sequence, in contrast to previous implementations using pulsed
sequences[17, 18, 19, 20]. The majority of these schemes are limited to signal
frequencies below $\sim 50~{}\mathrm{MHz}$ because they require the electron
spin flip rate to be comparable to the signal frequency. This requires
complex, high power pulse sequences susceptible to accumulated pulse-area
errors which can impair the sensitivity[24, 10, 23, 25]. The well established
CASR protocol[19] was recently demonstrated using a $V_{B}^{-}$ ensemble in
hBN for the first time[38], where the authors achieved a frequency resolution
of 0.9 Hz for an 18 MHz signal sampled for 2000 $\mathrm{s}$, compared with
0.118 Hz resolution for a 10 s integration in this work. Note that, whilst the
results presented here focused on GHz signals, the same protocol and
experimental setup is also capable of detecting signals in the 10-150 MHz
range [34].

Newer pulsed protocols capable of detecting GHz signals have been proposed[39]
and demonstrated[22] using NV centres in diamond, however these still suffer
from the aforementioned complexity and power constraints. A continuous scheme
is also presented in [22], however, the single drive does not extend the
coherence time, limiting the sensitivity. Other continuous heterodyne schemes
take advantage of the natural $\mathrm{\mu s}$ coherence times inherent to NV
centres in isotopically purified diamond[26, 6, 40]. However these are not an
option with $V_{B}^{-}$ ensembles, where the natural spin state lifetime is
$<100\>\mathrm{ns}$ [36, 31]. In contrast, our scheme is able to extend
$V_{B}^{-}$ coherence times into the $\mathrm{\mu s}$ regime, whilst remaining
sensitive to signal phase, frequency and amplitude. It is important to
acknowledge the significance of achieving this in a two-dimensional material,
which could enable nanoscale sensor-source distances unaffected by dangling
bonds and surface imperfections in the host material.

This system can be applied as an effective probe of other low dimensional
condensed matter systems that present AC magnetic fields[1, 41]. For instance,
the out of plane DC field used in this work would support forward volume spin
wave modes in ferromagnetic thin films[42, 43]. These spin waves are a
promising platform for next generation data transfer[42], as they avoid ohmic
heat loss and support GHz to THz fields. For example, spin waves resonant with
our device could be optically excited[44, 45] at one end of the CPW. As these
spin waves propagate with $\leq 100~{}\mathrm{\mu m}$ wavelengths[46], the
phase dependent sensor response could be used with time correlated laser
scanning confocal microscopy to image the spin waveform along the length of
the CPW, potentially providing new insight into spin wave dispersion
relations[47, 44]. This sensing scheme could also aid in the development of
microwave circuitry, to probe failure modes or ohmic heating in regions of
high current loads[48, 49, 50]. More broadly, combining CCDD schemes with
bright single spin defects[14], and using local nuclear spins as ancilla
qubits, presents a promising platform for future quantum sensing endeavours.

##  1 Methods

###  1.1 Experimental

PL is excited using a 488 nm diode laser pulsed by an acousto-optic modulator.
An objective lens (N.A.=0.55) is used to focus the laser to a diffraction-
limited spot on the hBN flake and at the center of the co-planar waveguide.
Photoluminescence from the boron vacancy ensemble is collected with the same
objective, filtered by a 750 nm long pass filter and recorded on a single
photon avalanche diode (SPAD). The microwave control and signal waveforms are
generated using an arbitrary waveform generator, amplified and applied via a
circulator to the CPW. The other end is terminated 50 ohms. The optical and
microwave excitation are synchronised with a digital pattern generator and the
photon counts recorded with time-tagging electronics. All experiments applied
a constant phase offset of 0.07$\pi$ to the drive field, as this produced the
optimum sensor response to signal phase in the device.

The maximum waveform length is limited by the AWG memory to approximately 2.5
ms. For simplicity, we choose a waveform length of 1 ms and ensure the
frequency of all control and signal components is an integer multiple of 1
kHz. The 1 ms sequence is repeated resulting in signals with greater coherence
than dictated by the memory constraints [[22]]. The optical and microwave
control pulses are applied to the sample, along with the constant signal field
for a total measurement time, $t_{m}$. All detected photons are time-tagged
and saved. In post-processing a time gate is applied to keep only photons that
arrive within a time $t_{gate}=~{}\mathrm{350ns}$ of the beginning of each
optical pulse (repetition rate of 400 kHz). These photons are then binned
according to the time that has elapsed since the beginning of the measurement,
with an example shown in Fig. 4(b). We fast Fourier transform the
autocorrelation of this data to generate a spectrum of the signal, down-
converted by the frequency of the double-dressed resonance,
$\omega_{0}-\epsilon_{m}$.

###  1.2 Model

The Bloch-vector dynamics plotted in Fig. 1(d) were numerically calculated
using the Heisenberg equation
$\dot{\sigma^{{}^{\prime}}}=i[H_{C}^{{}^{\prime}},\sigma^{{}^{\prime}}]$,
which is re-expressed as a rotation,
$\vec{\dot{\sigma}}=\vec{H}^{{}^{\prime\prime}}(t)\times\vec{\sigma}$, where
$\vec{\dot{\sigma}}$ is the time derivative of the spin vector,
$\vec{H}^{{}^{\prime\prime}}(t)$ is the interaction picture Hamiltonian as
defined in the main text, and $\vec{\sigma}$ is the spin vector. For each time
step used in the model, a single effective field is calculated by first
summing the vector components in $\vec{H}^{{}^{\prime\prime}}(t)$. This forms
an axis rotation for the spin vector, with the angle of rotation determined by
the magnitude of the summed vector components. The process is repeated
iteratively for all time steps to produce the plotted trajectories.

Acknowledgments This work was supported by the Engineering and Physical
Sciences Research Council [Grant numbers EP/S001557/1 and EP/L015331/1],
Partnership Resource Funding from the Quantum Computing and Simulation Hub
[EP/T001062/1] and an Engineering and Physical Sciences Research Council iCASE
in partnership with Oxford Instruments Plasma Technology. Ion implantation was
performed by Keith Heasman and Julian Fletcher at the University of Surrey Ion
Beam Centre. For the purpose of open access, the author has applied a
‘Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising’.

## References

  *   * Bertelli et al. [2020] Bertelli, I. et al.: Magnetic resonance imaging of spin-wave transport and interference in a magnetic insulator. Science Advances 6, 3556–3567 (2020) https://doi.org/10.1126/SCIADV.ABD3556
  * Zhou et al. [2020] Zhou, B.B. et al: Spatiotemporal mapping of a photocurrent vortex in monolayer $\mathrm{MoS_{2}}$ using diamond quantum sensors. Physical Review X 10, 011003 (2020) https://doi.org/10.1103/PhysRevX.10.011003
  * Arai et al. [2022] Arai, K. et al.: Millimetre-scale magnetocardiography of living rats with thoracotomy. Communications Physics 5, 200 (2022) https://doi.org/10.1038/s42005-022-00978-0
  * Cao et al. [2020] Cao, Q.-Y. et al.: Protecting quantum spin coherence of nanodiamonds in living cells. Phys. Rev. Applied 13, 024021 (2020) https://doi.org/10.1103/PhysRevApplied.13.024021
  * Staudacher et al. [2013] Staudacher, T. et al.: Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume. Science 339, 561–563 (2013) https://doi.org/10.1126/SCIENCE.1231675
  * Wang et al. [2022] Wang, Z. et al.: Picotesla magnetometry of microwave fields with diamond sensors. Science Advances 8, 8158 (2022) https://doi.org/10.1126/SCIADV.ABQ8158
  * Alsid et al. [2023] Alsid, S.T. et al.: Solid-state microwave magnetometer with picotesla-level sensitivity. Physical Review Applied 19, 054095 (2023) https://doi.org/10.1103/PhysRevApplied.19.054095
  * Doherty et al. [2013] Doherty, M.W., Manson, N.B., Delaney, P., Jelezko, F., Wrachtrup, J., Hollenberg, L.C.L.: The nitrogen-vacancy colour centre in diamond. Physics Reports 528(1), 1–45 (2013) https://doi.org/10.1016/j.physrep.2013.02.001
  * Alkahtani et al. [2019] Alkahtani, M., Lang, J., Naydenov, B., Jelezko, F., Hemmer, P.: Growth of high-purity low-strain fluorescent nanodiamonds. ACS Photonics 6, 1266–1271 (2019) https://doi.org/10.1021/acsphotonics.9b00224
  * Cao et al. [2020] Cao, Q.-Y. et al.: Protecting quantum spin coherence of nanodiamonds in living cells. Physical Review Applied 13, 24021 (2020) https://doi.org/10.1103/PhysRevApplied.13.024021
  * Gottscholl et al. [2021] Gottscholl, A. et al.: Spin defects in hbn as promising temperature, pressure and magnetic field quantum sensors. Nature Communications 12, 4480 (2021) https://doi.org/10.1038/s41467-021-24725-1
  * Vaidya et al. [2023] Vaidya, S., Gao, X., Dikshit, S., Aharonovich, I., Li, T.: Quantum sensing and imaging with spin defects in hexagonal boron nitride. Advances in Physics: X 8(1), 2206049 (2023) https://doi.org/%****␣1._Continuous_heterodyne_scheme.tex␣Line␣525␣****10.1080/23746149.2023.2206049
  * Gottscholl et al. [2020] Gottscholl, A. et al.: Initialization and read-out of intrinsic spin defects in a van der waals crystal at room temperature. Nature Materials 19(5), 540–545 (2020) https://doi.org/10.1038/s41563-020-0619-6
  * Stern et al. [2022] Stern, H.L. et al.: Room-temperature optically detected magnetic resonance of single defects in hexagonal boron nitride. Nature Communications 13, 618 (2022) https://doi.org/10.1038/s41467-022-28169-z
  * Stern et al. [2024] Stern, H.L. et al.: A quantum coherent spin in hexagonal boron nitride at ambient conditions. Nature Materials (2024) https://doi.org/10.1038/s41563-024-01887-z
  * Bar-Gill et al. [2013] Bar-Gill, N., Pham, L.M., Jarmola, A., Budker, D., Walsworth, R.L.: Solid-state electronic spin coherence time approaching one second. Nature communications 4, 1743 (2013) https://doi.org/10.1038/NCOMMS2771
  * Boss et al. [2017] Boss, J.M., Cujia, K.S., Zopes, J., Degen, C.L.: Quantum sensing with arbitrary frequency resolution. Science 356, 837–840 (2017) https://doi.org/10.1126/science.aam7009
  * Schmitt et al. [2017] Schmitt, S. et al.: Submillihertz magnetic spectroscopy performed with a nanoscale quantum sensor. Science 356, 832–837 (2017) https://doi.org/10.1126/science.aam5532
  * Glenn et al. [2018] Glenn, D.R., Bucher, D.B., Lee, J., Lukin, M.D., Park, H., Walsworth, R.L.: High-resolution magnetic resonance spectroscopy using a solid-state spin sensor. Nature 555, 351–354 (2018) https://doi.org/10.1038/nature25781
  * Jiang et al. [2023] Jiang, Z., Cai, H., Cernansky, R., Liu, X., Gao, W.: Quantum sensing of radio-frequency signal with nv centers in sic. Science Advances 9 (2023) https://doi.org/10.1126/sciadv.adg2080
  * Joas et al. [2017] Joas, T., Waeber, A.M., Braunbeck, G., Reinhard, F.: Quantum sensing of weak radio-frequency signals by pulsed mollow absorption spectroscopy. Nature Communications 8, 964 (2017) https://doi.org/10.1038/s41467-017-01158-3
  * Meinel et al. [2021] Meinel, J. et al.: Heterodyne sensing of microwaves with a quantum sensor. Nature Communications 12, 2737 (2021) https://doi.org/10.1038/s41467-021-22714-y
  * Lang et al. [2019] Lang, J.E. et al.: Nonvanishing effect of detuning errors in dynamical-decoupling-based quantum sensing experiments. Physical Review A 99, 012110 (2019) https://doi.org/10.1103/PhysRevA.99.012110
  * Ishikawa et al. [2018] Ishikawa, T., Yoshizawa, A., Mawatari, Y., Watanabe, H., Kashiwaya, S.: Influence of dynamical decoupling sequences with finite-width pulses on quantum sensing for ac magnetometry. Physical Review Applied 10, 54059 (2018) https://doi.org/10.1103/PhysRevApplied.10.054059
  * Aharon et al. [2019] Aharon, N., Schwartz, I., Retzker, A.: Quantum control and sensing of nuclear spins by electron spins under power limitations. Physical Review Letters 122, 120403 (2019) https://doi.org/10.1103/PhysRevLett.122.120403
  * Staudenmaier et al. [2021] Staudenmaier, N., Schmitt, S., McGuinness, L.P., Jelezko, F.: Phase-sensitive quantum spectroscopy with high-frequency resolution. Physical Review A 104, 020602 (2021) https://doi.org/10.1103/PhysRevA.104.L020602
  * Cai et al. [2012] Cai, J.M. et al.: Robust dynamical decoupling with concatenated continuous driving. New Journal of Physics 14, 113023 (2012) https://doi.org/10.1088/1367-2630/14/11/113023
  * Farfurnik et al. [2017] Farfurnik, D., Aharon, N., Cohen, I., Hovav, Y., Retzker, A., Bar-Gill, N.: Experimental realization of time-dependent phase-modulated continuous dynamical decoupling. Physical Review A 96, 13850 (2017) https://doi.org/10.1103/PhysRevA.96.013850
  * Stark et al. [2017] Stark, A. et al.: Narrow-bandwidth sensing of high-frequency fields with continuous dynamical decoupling. Nature Communications 8, 1105 (2017) https://doi.org/10.1038/s41467-017-01159-2
  * Ramsay et al. [2023] Ramsay, A.J. et al.: Coherence protection of spin qubits in hexagonal boron nitride. Nature Communications 14, 461 (2023) https://doi.org/10.1038/s41467-023-36196-7
  * Haykal et al. [2022] Haykal, A. et al.: Decoherence of $v_{b}^{-}$ spin defects in monoisotopic hexagonal boron nitride. Nature Communications 13, 4347 (2022) https://doi.org/10.1038/s41467-022-31743-0
  * Gong et al. [2023] Gong, R. et al.: Coherent dynamics of strongly interacting electronic spin defects in hexagonal boron nitride. Nature Communications 14, 3299 (2023) https://doi.org/10.1038/s41467-023-39115-y
  * Wang et al. [2021] Wang, G., Liu, Y.X., Cappellaro, P.: Observation of the high-order mollow triplet by quantum mode control with concatenated continuous driving. Physical Review A 103, 022415 (2021) https://doi.org/10.1103/PHYSREVA.103.022415
  * Patrickson et al. [2024] Patrickson, C.J., Baber, S., Gaál, B.B., Ramsay, A.J., Luxmoore, I.J.: High frequency magnetometry with an ensemble of spin qubits in hexagonal boron nitride. npj Quantum Information 10, 5 (2024) https://doi.org/10.1038/s41534-023-00796-4
  * Salhov et al. [2024] Salhov, A., Cao, Q., Cai, J., Retzker, A., Jelezko, F., Genov, G.: Protecting quantum information via destructive interference of correlated noise. Phys. Rev. Lett. 132, 223601 (2024) https://doi.org/10.1103/PhysRevLett.132.223601
  * Baber et al. [2022] Baber, S. et al.: Excited state spectroscopy of boron vacancy defects in hexagonal boron nitride using time-resolved optically detected magnetic resonance. Nano Letters 22, 461–467 (2022) https://doi.org/10.1021/acs.nanolett.1c04366
  * Degen et al. [2017] Degen, C.L., Reinhard, F., Cappellaro, P.: Quantum sensing. Reviews of Modern Physics 89, 035002 (2017) https://doi.org/10.1103/RevModPhys.89.035002
  * Rizzato et al. [2023] Rizzato, R. et al.: Extending the coherence of spin defects in hbn enables advanced qubit control and quantum sensing. Nature Communications 14, 5089 (2023) https://doi.org/10.1038/s41467-023-40473-w
  * Chu et al. [2021] Chu, Y. et al.: Precise spectroscopy of high-frequency oscillating fields with a single-qubit sensor. Physical Review Applied 15, 014031 (2021) https://doi.org/10.1103/PhysRevApplied.15.014031
  * Li et al. [2023] Li, M. et al.: Near-field sensing of microwave magnetic field phase difference enabled by N-V-center spins. Physical Review Applied 19, 054088 (2023) https://doi.org/10.1103/PhysRevApplied.19.054088
  * Lee-Wong et al. [2020] Lee-Wong, E. et al.: Nanoscale detection of magnon excitations with variable wavevectors through a quantum spin sensor. Nano Letters 20, 3284–3290 (2020) https://doi.org/10.1021/acs.nanolett.0c00085
  * Chumak et al. [2015] Chumak, A.V., Vasyuchka, V.I., Serga, A.A., Hillebrands, B.: Magnon spintronics. Nature Physics 11, 453–461 (2015) https://doi.org/10.1038/nphys3347
  * Chen et al. [2019] Chen, J., Wang, C., Liu, C., Tu, S., Bi, L., Yu, H.: Spin wave propagation in ultrathin magnetic insulators with perpendicular magnetic anisotropy. Applied Physics Letters 114, 212401 (2019) https://doi.org/10.1063/1.5093265
  * Hashimoto et al. [2017] Hashimoto, Y. et al.: All-optical observation and reconstruction of spin wave dispersion. Nature Communications 8, 15859 (2017) https://doi.org/10.1038/ncomms15859
  * Au et al. [2013] Au, Y. et al.: Direct excitation of propagating spin waves by focused ultrashort optical pulses. Physical Review Letters 110, 097201 (2013) https://doi.org/10.1103/PhysRevLett.110.097201
  * Bang et al. [2018] Bang, W. et al.: Measurements of long-wavelength spin waves for the magnetic field in the Damon-Eshbach, backward-volume and forward-volume geometries of an yttrium iron garnet film. Journal of Applied Physics 123(12), 123902 (2018) https://doi.org/10.1063/1.5019752
  * Divinskiy et al. [2021] Divinskiy, B. et al.: Dispersionless propagation of ultrashort spin-wave pulses in ultrathin yttrium iron garnet waveguides. Physical Review Applied 16, 024028 (2021) https://doi.org/10.1103/PhysRevApplied.16.024028
  * Böhi et al. [2010] Böhi, P., Riedel, M.F., Hänsch, T.W., Treutlein, P.: Imaging of microwave fields using ultracold atoms. Applied Physics Letters 97, 51101 (2010) https://doi.org/10.1063/1.3470591
  * Garsi et al. [2024] Garsi, M. et al.: Three-dimensional imaging of integrated-circuit activity using quantum defects in diamond. Physical Review Applied 21, 014055 (2024) https://doi.org/10.1103/PhysRevApplied.21.014055
  * Nowodzinski et al. [2015] Nowodzinski, A., Chipaux, M., Toraille, L., Jacques, V., Roch, J.F., Debuisschert, T.: Nitrogen-vacancy centers in diamond for current imaging at the redistributive layer level of integrated circuits. Microelectronics Reliability 55, 1549–1553 (2015) https://doi.org/10.1016/J.MICROREL.2015.06.069