# Modelling Rabi oscillations for widefield radiofrequency imaging in
nitrogen-vacancy centers in diamond

Simone Magaletti1, Ludovic Mayer1, Jean-François Roch2, Thierry Debuisschert1
1Thales Research and Technology, 1 Avenue Augustin Fresnel, Palaiseau Cedex,
91767, France 2Université Paris-Saclay, CNRS, ENS Paris-Saclay,
CentraleSupelec, LuMIn, Gif-sur-Yvette, 91190, France
<EMAIL_ADDRESS>
###### Abstract

In this paper we study the dynamics of an ensemble of nitrogen-vacancy centers
in diamond when its photoluminescence is detected by means of a widefield
imaging system. We develop a seven-level model and use it to simulate the
widefield detection of nitrogen-vacancy centers Rabi oscillations. The
simulation results are compared with experimental measurements showing a good
agreement. In particular, we use the model to explain the asymmetric shape of
the detected Rabi oscillations due to an incomplete repolarization of the
nitrogen-vacancy center during the pulse sequence implemented for the
detection of Rabi oscillations.

##  1 Introduction

The fast spreading of miniaturized RF-based applications requires the
development of tools that are able to map the near-field radiofrequency (RF)
radiation generated by electromagnetic devices to analyze and monitor their
performance. Function and failure analysis [1] as well as material [2, 3] and
device [4] characterization are some of the several applications that require
the measurement of the spatial distribution of a RF field, with both high
spatial resolution and high sensitivity. Recently, quantum sensors based on
vapor cells [5], ultracold atoms [6] and negatively charged nitrogen-vacancy
centers (NV) in diamond [7], have been used to realize the near-field spatial
mapping of the RF radiation. Their working principle relies on measuring Rabi
oscillations to retrieve the amplitude of the electric or magnetic field
exciting the system. Since Rabi frequency is directly related to the amplitude
of the electromagnetic field by physical constants known with high accuracy,
this technique does not require any calibration procedure [8, 9]. Here we
discuss the dynamics of an ensemble of NV centers in diamond when a widefield
imaging system is used for the detection of Rabi oscillations and we point out
the main differences with the detection method using a confocal optical
microscopy set-up operated in photon counting regime with an avalanche
photodiode (APD). Furthermore, we develop a model that, taking into account
the camera gating time, simulates the NV center dynamics and the
photoluminescence (PL) acquisition process, resulting in a versatile tool for
applications based on widefield imaging.

##  2 Widefield detection of nitrogen-vacancy center Rabi oscillations

Figure 1: (a) NV center in the diamond lattice. The blue sphere represents the
nitrogen atom, the white sphere the associated carbon vacancy and the black
spheres are the carbon atoms. (b) NV center energy levels. (c) NV center
transition rates [10, 11]. $k_{ij}$ corresponds to the transition rate from
level i to level j, according to the level numbering defined in (b).

The negatively charged nitrogen-vacancy center is a spin-1 diamond defect
composed of a nitrogen atom and a carbon vacancy in two adjacent lattice
positions [12] (figure 1a). The nitrogen-vacancy axis (NV axis) defines the
intrinsic spin quantification axis in absence of static magnetic field. The
negatively charged NV center energy level structure is depicted in figure 1b.
The ground state (G.S.) and the excited state (E.S.) are spin triplets while
the metastable state (M.S.) is a spin singlet. Spin states are labelled as
$\ket{m_{s}}$, where $m_{s}$ is the spin projection quantum number along the
NV axis. When no static magnetic field is applied, the ground state energy
difference between the $\ket{0}$ and the two degenerate $\ket{\pm 1}$ spin
sublevels corresponds to a Bohr frequency of 2.87 GHz. All sublevels are
equally populated at room temperature. Under optical pumping, two decay paths
exist: a radiative spin-conserving decay, which is responsible of the PL, and
a non-radiative intersystem crossing through the metastable singlet state. The
latter is more probable when the NV center is in $\ket{\pm 1}$ (figure 1c) and
allows both the optical polarization of the NV center in $\ket{0}$ and the
optical readout of the spin state by monitoring the PL emission rate of the
system [13]. The resonance frequencies of the two ground state transitions
($\ket{0}\rightarrow\ket{\pm 1}$) allowed by the spin selection rules ($\Delta
m_{s}=\pm 1$) can be controlled by a static magnetic field through the Zeeman
effect. Moreover, owing to its long coherence time (T2 in the millisecond
range) at room temperature, the NV center spin can be coherently manipulated
by a RF field making possible the optical detection of Rabi oscillations [14].

A standard detection scheme for NV center Rabi oscillations is depicted in
figure 2a. A laser pulse with a duration of a few µs polarizes the system in
$\ket{0}$. After a waiting time of a few hundreds of ns, necessary for the
depletion of the metastable state, the system is excited by a RF pulse of
duration $\tau$ on resonance with one of the two NV center ground state spin
transitions. The cycle is then repeated sweeping $\tau$. At the beginning of
each laser pulse an APD integrates the PL over a duration $T$ ($N$signal) in
order to readout the NV center spin state and, at the end of the pulse, when
the system is polarized in $\ket{0}$, the APD is opened again to take a
reference signal ($N$reference). The optically detected magnetic resonance
(ODMR) contrast is defined as:

$C=\frac{N_{\text{reference}}-N_{\text{signal}}}{N_{\text{reference}}}$ (1)

Owing to the time dependence of the PL (figure 2b), the contrast is a function
of time. For long integration time, the contrast tends to zero due to the
repolarization of the NV centers in $\ket{0}$ [15]. The contrast is maximized
by optimizing the opening time of the APD, usually some hundreds of ns [16].
This measurement scheme was used in Ref. [17] to image the RF field generated
by a micrometer-scale RF stripline by means of a scanning diamond tip with an
NV center at its apex.

Figure 2: NV center Rabi oscillations. (a) Pulse sequence for the detection of
NV centers Rabi oscillations by means of an APD. (b) NV center PL dynamics
when the NV center is initially in $\ket{0}$ (blue) or in $\ket{\pm 1}$
(orange). In green, time dependence of the related contrast (eq. 1). (c) Pulse
sequence for the detection of NV centers Rabi oscillations by means of a
camera. (d) Experimental contrast for different RF pulse durations. The fit is
realized using eq. 2 and we obtain the following parameters:
$a_{\text{R}}=0.0206\pm 0.0002$, $b_{\text{R}}=1.08\pm 0.08$ µs and
$c_{\text{R}}\leavevmode\nobreak\ =\leavevmode\nobreak\ (12.0\pm 0.2)\cdot
10^{6}$ rad$\cdot$ s-1. $d_{\text{R}}$ is a phase term which accounts for the
fact that, for technical reason, the acquisition started at 0.5 µs instead of
0 µs. (e-f) Experimentally measured contrast as a function of the RF pulse
duration, for different laser powers (e) and different laser pulse durations
(f). Asymmetric Rabi oscillations are observed for low power or short duration
laser pulses.

RF field imaging can also be realized using an NV center ensemble in bulk
diamond and a widefield imaging system to avoid any scanning procedure,
leading to a multiplex advantage in the acquisition time of the RF magnetic
field map [18]. In this case, an alternative pulse scheme is needed to account
for the much larger gating time of a camera compared to an APD (figure 2c).
For a single camera exposure, several cycles consisting of a laser pulse, a
waiting time for the relaxation of the metastable state and a RF pulse of
duration $\tau$ are repeated. The same sequence is then applied with the RF
field off to take a reference image. For each RF pulse duration, the ODMR
contrast is defined following eq. 1. We realized the experiment using a laser
pulse duration of 1 µs, a waiting time of 400 ns and we swept the RF pulse
duration from 500 ns to 4 µs (Appendix A). The contrast evolution for
different values of $\tau$ at a given position in the diamond is plotted in
figure 2d, where we recognize the expected oscillating behavior due to Rabi
oscillations. Data are fitted using the function [17]:

$f(\tau)=a_{\text{R}}\cdot\left(1-e^{-\tau/b_{\text{R}}}\cdot\cos\left[c_{\text{R}}\cdot\tau+d_{\text{R}}\right]\right)$
(2)

where $a$R, $b$R, $c$R, $d$R are respectively the amplitude, the decay time,
the angular frequency and the phase of the Rabi oscillations [19]. Equation 2
partially fits data that exhibit an asymmetric shape. We repeated the
measurement for different laser powers (figure 2e) and laser pulse durations
(figure 2f) showing that the increase of these two parameters leads to more
symmetric but less contrasted oscillations between the spin states. The
decrease in contrast is directly related to the repolarization of the NV
centers in $\ket{0}$, as already described in figure 2b. In fact, differently
from the previously described scheme based on APDs, in this case the detector
is opened during the entire laser pulse sequence. As a consequence, the spin
readout and the NV center repolarization occur simultaneously and cannot be
addressed separately. Therefore a compromise has to be found between either a
full repolarization of the NV centers or a high ODMR contrast. To have a
better understanding of these mechanisms as well as of the asymmetric shape of
the Rabi oscillations, we model the dynamics of an ensemble of NV centers when
a widefield imaging system is employed for the detection of the Rabi
oscillations.

##  3 Simulation

The pulse sequence implemented in our experiment consists of three steps:

  * •

The application of the laser pulse;

  * •

The depletion of the metastable state;

  * •

The application of the RF pulse.

For the first two steps, the NV center can be modelled as a seven-level system
and its dynamics is described by rate equations that neglect the spin
coherence terms [20, 21]. On the contrary, the last step only addresses one of
the two ground state spin transitions ($\ket{0}\rightarrow\ket{-1}$ in our
case) and can be described by the Bloch equations for a two level system [22].
Merging the two approaches [23], we can write the temporal evolution of the
density matrix of the system ($\rho$) as:

$\frac{d\rho}{dt}=\frac{1}{\imath\hbar}\left[H,\rho\right]+\left\\{\frac{d\rho}{dt}\right\\}_{\text{{incoh}}}$
(3)

In this equation, $H$ is the rotating-wave approximation of the Hamiltonian
describing the interaction between the NV center and a RF field on resonance
with the ground state spin transition $\ket{0}\rightarrow\ket{-1}$ [16]. It
results:

$H=\frac{\hbar\Omega_{R}}{2}\left(\ket{0}\bra{-1}\rme^{i\omega_{0}t}+\ket{-1}\bra{0}\rme^{-i\omega_{0}t}\right)$
(4)

where 0 is the angular frequency of the spin resonance and R is the Rabi
angular frequency. The last term $\\{\frac{d\rho}{dt}\\}_{\text{{incoh}}}$
accounts for the optical pumping and the relaxation due to interaction with
the environment. We then obtain the following set of equations:

$\displaystyle\frac{dn_{1}}{dt}$
$\displaystyle=-n_{1}W_{p}+n_{4}k_{41}+n_{7}k_{71}+\Omega_{R}\cdot n_{c}$ (5a)
$\displaystyle\frac{dn_{2}}{dt}$
$\displaystyle=-n_{2}W_{p}+n_{5}k_{52}+n_{7}k_{72}-\Omega_{R}\cdot n_{c}$ (5b)
$\displaystyle\frac{dn_{3}}{dt}$
$\displaystyle=-n_{3}W_{p}+n_{6}k_{63}+n_{7}k_{73}$ (5c)
$\displaystyle\frac{dn_{4}}{dt}$
$\displaystyle=n_{1}W_{p}-n_{4}k_{41}-n_{4}k_{47}$ (5d)
$\displaystyle\frac{dn_{5}}{dt}$
$\displaystyle=n_{2}W_{p}-n_{5}k_{52}-n_{5}k_{57}$ (5e)
$\displaystyle\frac{dn_{6}}{dt}$
$\displaystyle=n_{3}W_{p}-n_{6}k_{63}-n_{6}k_{67}$ (5f) $\displaystyle 1$
$\displaystyle=n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}$ (5g)
$\displaystyle\frac{dn_{c}}{dt}$ $\displaystyle=-\Gamma_{2}\cdot
n_{c}+\frac{\Omega_{R}}{2}\cdot(n_{2}-n_{1})$ (5h)

where $n_{i}=\rho_{ii}$ is the NV center population of the level $i$ (figure
2b) and $n_{c}=\Im[\rho_{21}\text{e}^{i\omega_{0}t}]$ represents the coherence
between level 1 and 2. $W$p is the laser pumping parameter and depends on the
laser intensity (Appendix B), and $\Gamma$2 is the spin decoherence rate. The
spin-lattice relaxation, which has a characteristic T1 time of approximately a
few ms, is neglected since it is three order of magnitude longer than the
characteristic times of the time evolution in the previous equations. The pump
excites a vibronic sideband which decays quickly via phonon emission to levels
4, 5 and 6. This allows us to neglect the down-transition rates due to the
pump light. The NV center PL is proportional to the excited state population
defined as:

$n_{\text{E}}(t)=n_{4}(t)+n_{5}(t)+n_{6}(t)$ (6)

Figure 3: Simulation results. (a) Time evolution of the NV center population.
The plot at the bottom is a zoom of the one at the top. Green areas correspond
to the laser pulse, light blue areas to the metastable state depletion, pink
areas to the RF pulse. $n$2 and $n$3 as well as $n$5 and $n$6 are often
superimposed. (b) Simulation of the detected Rabi oscillations when the laser
pulse duration is 10 µs and 1 µs. (c) Rabi oscillations for different
saturation parameters and laser pulse duration equal to 1 µs. (d) Rabi
oscillations for different laser pulse durations and a saturation parameter
equal to 0.1.

To account for the several pulse cycles in a single camera exposure, the three
steps are iterated several times until a stationary behavior is reached
(figure 3a). For each step, the rate equations are solved using as initial
condition the final condition of the previous step. The initial condition of
the first step of the first iteration is the thermal equilibrium state at room
temperature, that is: $n\leavevmode\nobreak\ =\leavevmode\nobreak\
[n\textsubscript{1};\leavevmode\nobreak\
n\textsubscript{2};\leavevmode\nobreak\
n\textsubscript{3};\leavevmode\nobreak\
n\textsubscript{4};\leavevmode\nobreak\
n\textsubscript{5};\leavevmode\nobreak\
n\textsubscript{6};\leavevmode\nobreak\
n\textsubscript{7};\leavevmode\nobreak\
n\textsubscript{c}]\leavevmode\nobreak\ =\leavevmode\nobreak\
[1/3;\leavevmode\nobreak\ 1/3;\leavevmode\nobreak\ 1/3;\leavevmode\nobreak\
0;\leavevmode\nobreak\ 0;\leavevmode\nobreak\ 0;\leavevmode\nobreak\
0;\leavevmode\nobreak\ 0]$. During the laser pulse we set $\Omega$R = 0,
because the RF field is off, and $W$p = $1.9\leavevmode\nobreak\
\times\leavevmode\nobreak\ 10^{6}$ s-1. This value is chosen according to our
experimental condition and corresponds to NV centers pumped at a laser power
equal to 10% of the saturation power (Appendix B). The spin decoherence rate
induced by the optical pumping process reads as [16]:

$\Gamma_{\text{c}}=\Gamma_{\text{c}}^{\infty}\frac{s}{s+1}$ (7)

where $\Gamma_{\text{c}}^{\infty}=8\times 10^{7}$ s-1 is the inverse of the
excited state lifetime ($\approx$ 13 ns); $s$ is the PL saturation parameter
defined as the ratio between the laser power used in the experiment and the
saturation power. We set $s$ equal to 0.1 (Appendix B) and $\Gamma$2 equal to
$\Gamma$c. During the depletion of the metastable state, the laser and the RF
driving field are off ($W$p = $\Omega$R = 0) and the spin coherence relaxation
rate is related to the NV center T2* time. We set
$Γ_{2}=\frac{1}{\text{T}_{2}^{*}}=5\times 10^{5}\leavevmode\nobreak\ s^{-1}$
assuming T2* = 2 µs, which is the typical value for optical grade diamond
[24]. Finally, during the RF pulse, we set: $\Omega$R =
1.5$\leavevmode\nobreak\ \times\leavevmode\nobreak\ $107 rad $\cdot$s-1, $W$p
= 0 and $\Gamma$2 = 5$\leavevmode\nobreak\ \times\leavevmode\nobreak\ $105
s-1. The simulation is then performed for different RF pulse durations. The
reference signal is obtained by setting $\Omega$R = 0 during all the
simulation.

Figure 3a shows the time evolution of the different level populations when the
laser pulse duration is set to 10 µs and both the waiting time for the
depletion of the metastable state and the RF pulse duration are set to 1 µs.
During the laser pulse (green areas on figure 3a) we can observe the
polarization of the system in the level 1 as well as an increase of population
both in the metastable and in the excited states. For this pumping parameter,
the system is polarized after a few µs (Appendix B). Turning off the laser
(light blue areas on figure 3a) we observe the faster decay of the excited
states and the slower decay of the metastable state, leading to the increase
of the ground states population. Finally, when the RF field is on resonance
with the ground state $\ket{0}\rightarrow\ket{-1}$ transition (pink areas on
figure 3a), the populations of these two levels, as well as the spin coherence
term nc start oscillating. Note that the amplitude of the Rabi oscillations
depends on the population difference between $\ket{0}$ and $\ket{-1}$, and is
thus directly related to the degree of polarization of the system. Turning the
laser on, we restart the pulse sequence. After some iterations, small compared
to the total number of iterations during a single exposure of the camera, the
system reaches a steady state behaviour. We evaluate the PL intensity (eq. 6)
and the ODMR contrast (eq. 1) integrating over the duration of the laser pulse
of the last iteration. Figure 3b shows the contrast versus the RF pulse
duration for two different laser pulse durations: 10 µs, sufficient to fully
repolarize the system at the end of the pulse, and 1 µs, too short to
completely repolarize the system. The fits are realized using eq. 2 and we
obtain the same Rabi frequency for the two curves, despite the asymmetric
shape. The lower contrast for the longest laser pulse is due to the
repolarization of the system, as already shown in figure 2b. Figure 3c-d show
the evolution of the ODMR contrast for different pumping parameters and laser
pulse durations respectively. In agreement with the experimental data (figure
2d-f), we observe asymmetric and highly contrasted Rabi oscillations both for
short laser pulses and low pumping parameters.

We further investigate the asymmetry in the Rabi oscillations looking at NV
center dynamics when the RF pulse duration corresponds to a -pulse or a
2-pulse for two different laser pulse durations: 10 µs (figure 4a-b) and 1 µs
(figure 4c-d). A long laser pulse (10 µs) allows a complete repolarization of
the NV center after each RF pulse. The polarization is thus independent of the
initial state and of the RF pulse itself (figure 4a-b). A short laser pulse
(figure 4c-d) leads to an incomplete polarization of the NV centers that
depends on the RF pulse duration itself. For example, after a 2-pulse (figure
4d) the system is easier to fully repolarize than after a -pulse (figure 4c).
As a consequence, a short laser pulse may allow a full repolarization in the
former case but not in the latter. Since the amplitude of Rabi oscillations
(associated to the parameter $a$R in eq. 2) depends on the polarization of the
system before each RF pulse, it is independent of the RF pulse duration $\tau$
when the laser pulse is able to completely repolarize the NV centers; whereas
it is a function of $\tau$, causing the asymmetric shape observed in the
detected signal (figure 2d) in the other case.

Figure 4: NV center ground state $\ket{0}$ and $\ket{-1}$ population during
the last iteration of the simulation, when the system has reached the steady-
state. (a) Laser pulse duration 10 µs, RF pulse duration corresponding to a -
pulse. (b) Laser pulse duration 10 µs, RF pulse duration corresponding to a
2-pulse. (c) Laser pulse duration 1 µs, RF pulse duration corresponding to a
-pulse.(d) Laser pulse duration 1 µs, RF pulse duration corresponding to a
2-pulse.

##  4 Conclusion

We modelled the widefield detection of NV center Rabi oscillations and used
this model to investigate the results obtained in our experiment. Although the
time response of the camera is much lower than the Rabi oscillations period,
we are nevertheless able to observe those ones using a proper pulse sequence.
In particular, we showed that the asymmetric shape of the detected Rabi
oscillations is due to an incomplete repolarization of the NV centers at the
end of the laser pulse when this latter is too short. More generally, the
model allows studying and visualizing the behavior of an ensemble of NV center
taking into account the specificities of the widefield detection procedure,
namely a collection of the fluorescence over the whole duration of the pump
laser pulse and a gating-time of the camera much longer than the
characteristic time of the NV centers Rabi oscillations. It results in a
powerful and versatile tool to improve widefield Rabi oscillations detection
and thus the imaging of RF field. Combining the coherent control of the spin
with the multiplexing advantages of the widefield imaging opens the way to new
applications based on NV centers in diamond.

##  Appendix A Experimental set-up

The diamond is an optical grade CVD sample (Element six) with two {100} main
faces and four {110} facets. It is pumped by a 532 nm laser modulated by an
AOM in a double pass configuration. Total losses on the pumping branch are
approximately 3dB. The PL is collected from a {110} facet through an imaging
system composed of a microscope objective (20$\times$, 0.33 NA), a polarizer,
an optical filter (FF01-697-75 nm band-pass filter, Semrock) to suppress the
laser backscattered light and the PL of the neutrally charged nitrogen vacancy
center, and a CMOS camera (UI-5240CP-M, IDS). Two neodymium magnets generate
an homogeneous static magnetic field in proximity of the NV centers imaged by
the camera. The magnetic field is oriented along one of the two NV center
families laying on the {110} plane. The magnetic field is set approximately at
51 mT, near the excited state level anticrossing, to optically polarize the NV
center nitrogen nuclear spin [25]. In this condition, the hyperfine structure
of the NV center, which results from the interaction between the NV center
electron spin and the nitrogen nuclear spin, and which for 14N consists in
three hyperfine peaks, reduces to a single peak, making the NV center almost
equivalent to a two-level system[16]. The RF field is generated by a RF
generator, pulsed through a RF switch and brought in proximity of the NV
centers through a loop antenna. The RF switch and the AOM generator are
synchronized by means of a delay generator.

For each RF pulse duration, several images are acquired both with the RF field
on and off to improve the signal-to-noise ratio of the measurement.

Figure 5: Experimental set-up. On the left of the dashed line the imaging
branch of the experimental set-up. On the right of the dashed line the RF and
the optical pumping branches. The violet top right inset shows the view from
the camera of the diamond area. The magnetic field is aligned along one of the
two NV center families laying in the diamond {110} plane

##  Appendix B NV center dynamics

Figure 6: Simulation of the NV centers dynamics. (a) Steady-state solution of
NV center excited state for different value of $W$p (b) Time evolution of the
metastable state population after the laser is turned off for different values
of the saturation parameter $s$ (c) Time evolution of the ground state
polarization when the laser is turned on for different values of the
saturation parameter s.

We solve eq. 5 in the stationary state condition for different values of $W$p.
In figure 6a we plot the excited state population ($n$E=$n$4+$n$5+$n$6,
proportional to the emitted PL) for different values of $W$p. We fit the curve
by a saturation law:

$n_{\text{E}}(W_{\text{p}})=a_{\text{p}}\cdot\frac{W_{\text{p}}}{W_{\text{p}}+W_{\text{p}}^{\text{Sat}}}$
(8)

where $W$pSat is the value of $W$p for which the PL saturates and ap is a
constant of proportionality. Inserting the typical values of the NV transition
rates [10, 11], we found $W$pSat = 1.9$\leavevmode\nobreak\
\times\leavevmode\nobreak\ $107 s-1. Thus, we can easily evaluate the
saturation intensity ($I$Sat) as:

$I^{\text{Sat}}=\frac{W_{\text{p}}^{\text{Sat}}\cdot c\cdot
h}{\sigma\cdot\lambda}$ (9)

where $h$ is the Plank constant, $c$ is the speed of light, is the laser
wavelength and = 3 $\times$ 10-21 m2 is the NV center cross section [26]. We
obtain $I$Sat = 2.3 mW$\cdot$µm-2. Considering our experimental set-up, the
laser is focused on the diamond with a beam waist $w_{0}$ = 18 µm. The
saturation power can be calculated using the formula:

$P^{\text{Sat}}=\frac{\pi w_{0}^{2}}{2}\cdot I^{\text{Sat}}$ (10)

We obtain $P$Sat = 1.2 W. We define the saturation parameter ($s$) as:
$s=\frac{P}{P^{\text{Sat}}}$ where $P$ is the laser power used in the
experiment, in our case in the range [75; 250] mW. Therefore $s$ is in the
range [0.06; 0.2]. The values of WP corresponding to our experiment are thus
in the range [106; 4 $\times$ 106] s-1.

Figure 6b shows the time evolution of the metastable state population when the
laser is turned on for different values of the saturation parameter. We can
observe that the typical metastable depletion time is around 400 ns, the value
used for the experiment.

Figure 6c shows the time evolution of the ground state polarization in
$\ket{0}$ when the laser is turned on for different value of s. The time
needed to completely polarize the system increases reducing the laser power.
For a saturation parameter $s$ = 0.1 it is approximately 10 µs. That explains
our choice of using a laser pulse duration of 10 µs for our simulations.

##  Appendix C Map of the RF near field of a loop antenna

Figure 7: Widefield imaging of the RF field emitted by a loop antenna in the
near field region. (a-b) Antenna position. The picture is the full-frame image
acquired by the camera. The laser beam (yellow beam in the picture), the
shadow of the antenna (dark blue semi-circle) and the diamond (light blue
rectangle) are visible. (c-d) Rabi oscillations contrast along the beam
propagation axis (x) for the first (c) and second (d) antenna configuration.
The large horizontal line showing an almost homogeneous contrast at the top of
(d) is the shadow of the antenna. The several thin horizontal lines with low-
contrast are caused by inhomogeneities in the NV center concentration [27].
(e-f) Rabi frequency (R) along the beam propagation axis for the first (e) and
the second (f) antenna position. The RF field (BR) is calculated using the
formula: $B_{\text{R}}=\frac{\sqrt{2}\nu_{\text{R}}}{\gamma}$ [17], where
$\gamma=28$ MHz $\cdot$ mT-1 is the NV center gyromagnetic ratio. The fit (red
line) in (f) is realized using eq. 11.

We mapped the RF signal emitted in near field by the loop antenna implemented
in our set-up (figure 5) along the laser beam propagation axis. We
investigated two configurations. In the first one, NV centers are excited by
the RF field generated inside the wire loop (figure 7a) and, in the second
one, by the one generated outside of it (figure 7b). Rabi oscillations are
driven and detected using the pulse sequence in figure 2c. The laser pulse
duration is set to 2 µs in the first configuration and 5 µs in the second one.
The RF pulse duration is swept from 0.1 µs to 10 µs in steps of 20 ns. Figure
7c-d show the contrast of Rabi oscillations along the beam propagation axis
(x). For each x position, the contrast is averaged along y over the 10 pixels
around the center of the laser beam. We observe that the Rabi frequency
depends on the position and it is the sign that the antenna is emitting an
inhomogeneous RF field. For each position, the Rabi frequency is evaluated by
means of a fast Fourier transform and then plotted in figure 2e-f.

Inside the antenna (figure 2e), the Rabi frequency, and thus the amplitude of
the RF field, is almost homogeneous. Outside the antenna, (figure 2f) the
amplitude of the magnetic field decreases while increasing the distance from
the wire. The fit function employed in figure 2f is:

$f(x)=\frac{a_{\text{W}}}{x+b_{\text{W}}-c_{\text{W}}}$ (11)

where the parameter $a$W accounts for the magnetic field amplitude, $b$W is an
offset radius due to the mismatch between the zero of the x axis and the
center of the wire, and $c$W is the origin of the fit set at pixel number 275.
We obtain $b$W = 57$\pm$3 µm which is comparable to the radius of the wire (50
µm).

## Acknowledgement

We thank Nimba Oshnik and Elke Neu for their support in conceiving the
experimental set-up. This project received funding from the European Union’s
Horizon 2020 research and innovation program under grant agreement No. 820394
(ASTERIQS), the European Union’s Horizon Europe research and innovation
program under grant agreement No. 101080136 (AMADEUS), the Marie Skłodowska-
Curie grant agreement No. 765267 (QuSCo), the QUANTERA grant agreement
ANR-18-QUAN-0008 (MICROSENS), the EIC Pathfinder 2021 program under grant
agreement No. 101046911 (QuMicro) and the EMPIR project 20IND05 (QADeT). We
acknowledge the support of the DGA/AID under grant agreement ANR- 17-ASTR-0020
(ASPEN).

## Author contributions

S.M., L.M., J.-F.R., and T.D. contributed to the design and implementation of
the research, to the simulations and the analysis of the results and to the
writing of the manuscript.

## Conflict of Interest

The authors declare no conflict of interest.

## Data Availability Statement

The data and the code that support the findings of this study are available
from the corresponding author upon reasonable request.

## Reference

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