# Absorption spectroscopy and Stokes polarimetry in a 87Rb vapour in the Voigt
geometry with a 1.5 T external magnetic field

Francisco S. Ponciano-Ojeda<EMAIL_ADDRESS>Joint
Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham
University, South Road, Durham, DH1 3LE, United Kingdom  Fraser D. Logue
Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham
University, South Road, Durham, DH1 3LE, United Kingdom  Ifan G. Hughes  Joint
Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham
University, South Road, Durham, DH1 3LE, United Kingdom

(August 28, 2024)

###### Abstract

This paper provides details of a spectroscopic investigation of a thermal 87Rb
atomic vapour. The experiment was conducted with an external magnetic field of
1.5 T in the Voigt geometry. Very good quantitative agreement between
experimental data and theory is found for all four Stokes parameters–with RMS
errors of $\sim 1.5$% in all cases. From the fits to our experimental data a
value for the magnetic field strength is extracted, along with the angle
between the magnetic field and the polarisation of the light. The effects of
the cell window birefringence on the optical rotation signals are
characterised. This allows us to carry out precise measurements at a high
field strength and arbitrary geometries, allowing further development of
possible areas of application for atomic magnetometers.

##  I Introduction

Over the past decades atomic spectroscopy has cemented itself as one of the
cornerstones of precision measurements. From helping define the standards for
time-keeping Terrien (1968); Campbell and Phillips (2011); Ludlow  _et al._
(2015) to measurement of electromagnetic fields Osterwalder and Merkt (1999);
Thiele  _et al._ (2015); Sedlacek  _et al._ (2013); Kumar  _et al._ (2017);
Horsley  _et al._ (2015); Horsley and Treutlein (2016), spectroscopic
techniques have become key tools in high precision systems. In recent years,
together with the push for new technologies in other areas, this has seen the
field of atom-based sensors burgeon. Of particular interest are magnetic field
sensors Budker and Romalis (2007) which rely on the interaction of the nuclear
and electronic spins with the external field and well-known spectroscopic
signals. Applications of such atomic sensors span a variety of disciplines,
ranging from explosives detection Lee  _et al._ (2006); microfluidics Xu  _et
al._ (2008); medical imaging of soft tissues Bison  _et al._ (2009); Alem  _et
al._ (2015); Sander  _et al._ (2012); Boto  _et al._ (2017); gyroscopes Donley
_et al._ (2009); and measurements on spin-active solid-state systems Arnold
_et al._ (2017). In order to improve and sustain the development of these
technologies, atomic spectroscopy in the presence of external magnetic fields
has become an area of wide interest Sargsyan  _et al._ (2015a); Arimondo  _et
al._ (2016); Whiting  _et al._ (2018); Si  _et al._ (2020); Nyakang’o  _et
al._ (2020). This in turn has given rise to a variety of experiments and
optical devices Weller  _et al._ (2012a); Abel  _et al._ (2009); Kiefer  _et
al._ (2014); Zentile  _et al._ (2015a, b); Rotondaro  _et al._ (2015);
Keaveney  _et al._ (2016, 2018a) that help demonstrate the depth of
understanding of the physics involved in said interactions, furthering the
reach and possible applications of this area. Nevertheless, this in-depth
understanding of the interactions between atoms and an externally applied
magnetic field has primarily occurred for fields up to $\sim 1$ T.

The use of spectroscopic techniques in atomic systems has received less
attention at the higher end ($>1$ T) of the range of field strengths for
different reasons. There is the matter of most methods used to obtain large
fields being destructive in their nature, thus complicating the experimental
reproducibility. In such experiments, the Zeeman splitting of energy levels in
alkali-metal atoms is used to observe field strengths of the order of
tens/hundreds of Tesla via spectroscopic techniques Gomez  _et al._ (2014);
Garn  _et al._ (1966); Banasek  _et al._ (2016). Non-destructive techniques
for producing these large fields also exist, and this has enabled work in
large pulsed magnetic fields up to $58$ T with similar alkali-atom systems
Ciampini  _et al._ (2017); George  _et al._ (2017). Aside from the large
Zeeman splitting produced at such large magnetic fields, there are other
changes in the atoms that in turn allow additional effects to be observed. Of
these changes the most relevant is the decoupling of the nuclear and
electronic spins in the atom, with the external magnetic field now being a
common axis for the precession of both. This is known as the hyperfine
Paschen-Back regime Umfer  _et al._ (1992); Windholz (1985); Windholz and
Musso (1988); Olsen  _et al._ (2011); Weller  _et al._ (2012b); Zentile  _et
al._ (2014); Sargsyan  _et al._ (2017) and, in contrast to the experiments at
lower fields ($<10$ mT) – where it is easily and directly measured Budker and
Romalis (2007) – the Larmor precession frequency is not of interest as it can
be too high for conventional electronic systems. Rather, the characteristic
absorption spectrum of the atomic system is a more straightforward way of
obtaining information about the magnetic field.

Work has previously been done in atomic vapours of both low and high atomic
densities Siddons  _et al._ (2008); Weller  _et al._ (2011); Lange  _et al._
(2020), and other relevant magneto-optic effects have been the subject of
extensive studies Siddons  _et al._ (2009a, b, 2010); Weller  _et al._
(2012c). This atom-light interaction whilst in the presence of an external
magnetic field has also been used in other systems such as Whiting  _et al._
(2015, 2016, 2017, 2018). For the majority of said work the $\vec{k}$ vector
of the laser beam used is parallel to the direction of the external magnetic
field $\vec{B}$–the Faraday geometry. This gives rise to characteristic
absorption spectra that are highly symmetric Weller  _et al._ (2012b);
Sargsyan  _et al._ (2014a) and reflect only the magnitude of the magnetic
field Zentile  _et al._ (2014). However, there has been some work Sargsyan
_et al._ (2015b); Keaveney  _et al._ (2019) that makes use of the less studied
Voigt configuration where, rather than being parallel, the $\vec{k}$ vector of
the light and the direction of the magnetic field $\vec{B}$ are perpendicular.
The change in geometry results in a change in the atomic transitions permitted
by the selection rules Sargsyan  _et al._ (2014b); Hakhumyan  _et al._ (2012)
and thus allows for information not only on the magnitude but also on the
direction of the magnetic field to be experimentally observed.

In this work we provide a detailed spectroscopic analysis of measurements
taken in the Voigt geometry with an atomic vapour at a magnetic field strength
of $1.5$ T. These measurements include optical rotation signals, expressed in
terms of the Stokes parameters. Our theoretical model ElecSus describes the
atomic susceptibility Zentile  _et al._ (2015c); Keaveney  _et al._ (2018b)
and provides a means of fitting the experimental data, from which values of
the magnetic field strength are extracted and compared against expected
values. From the fit we also extract the relative orientation of the magnetic
field to a given known input light polarisation. Finally, we demonstrate a
method to measure the birefringence of the cell windows by looking at small
changes in the expected optical rotation signals produced by the atomic medium
in the presence of a strong external magnetic field.

The text continues in the following manner: section II gives a brief summary
of the theory behind the model used in analysing our experimental data; the
experimental methods to obtain these data are presented in section III; a
discussion and comparison of our model and the experimental spectra appear in
section IV. The paper finishes with further elaboration upon the analysis of
our experimental data in section V where the optical rotation signals are used
to extract information regarding the birefringence introduced by the cell
windows.

##  II Theoretical model

In order to fit our experimental data we use ElecSus. The theory behind this
software has been treated in detail in refs. Zentile  _et al._ (2015c);
Keaveney  _et al._ (2018b). For this work we summarise this theory and
highlight the points of most relevance. We begin by assuming an atom-light
system operating in the weak-probe regime Smith and Hughes (2004); Sherlock
and Hughes (2009), although recent work Lange  _et al._ (2020) provides
methods to generalise this into a strong-probe regime. The base for the model
lies in using the complex electric susceptibility of the atomic medium,
$\chi(\Delta)$, as a function of the optical frequency detuning
$\hbar\Delta=(\hbar\omega_{\rm{laser}}-\hbar\omega_{0})$ near the alkali
D-line resonances. Here we define $\omega_{\rm{laser}}$ as the angular
frequency of the laser and $\omega_{0}$ as the angular frequency of the atomic
transition. We also note that experimentally it is more common to use linear
detuning, $\Delta/2\pi$. Using a matrix representation we construct the atomic
states for our system in the uncoupled $m_{L},m_{S},m_{I}$ basis. We can then
account for the internal energy levels that come about as a result of the fine
and hyperfine structure to which we add terms relevant in describing
interactions with external magnetic fields, via the Zeeman effect. With this
representation of the Hamiltonian for the atomic system, we proceed to
calculate the eigenvalues, which correspond to the transition energies, and
absolute line strengths from the dipole matrix elements. As a final step, we
incorporate the Doppler effect in our model via a Voigt profile (convolution
of the Lorentzian homogeneous linewidth and the Gaussian from the Maxwellian
velocity distribution) for each of the atomic transitions as defined by their
energy and strength. In this manner, the total susceptibility of the medium is
calculated as the sum of the individual susceptibilities for all dipole-
allowed transitions, each with an effective detuning in the scope of a global
linear detuning. At this point it is convenient to split the susceptibility
into separate components for $\sigma^{\pm}$ and $\pi$ transitions.

For a given total susceptibility the propagation of light through the atomic
medium is done by solving the corresponding wave equation and finding the two
propagation eigenmodes. Each eigenmode is expressed in terms of the components
of the susceptibility, which allow the light to couple to the transitions in
distinct ways. One can also find a relation between the eigenmodes and the
complex refractive indices of the medium. The exact coupling of the eigenmodes
to the transitions depends on the geometry of the atom-light system. In our
experiment the system is set up in the Voigt geometry as shown in figure 1,
which consists of setting the externally applied magnetic field vector
$\vec{B}$ perpendicular to the light wavevector $\vec{k}$. We further
constrain the geometry of the system by taking the electric field vector of
our laser beam $\vec{E}$ along the Cartesian $x$-axis, which allows us to
consider our beam as a linearly polarised plane-wave propagating along the
$z$-axis and polarised in the $x-y$ plane. In this manner, for the Voigt
geometry we take the external magnetic field vector in the $x-y$ plane, with
an angle $\phi_{B}$ defined as the direction of said vector relative to the
$x$-axis ($\vec{E}$). This angle also gives a simple way of describing the
relative coupling of the light to the atomic electron transitions as a
consequence of the projections of $\vec{B}$ onto $\vec{E}$, which in turn
change the angular-momentum algebra. For a projection of $\vec{B}$ parallel to
the polarisation of the light (i.e. $\vec{E}$) $\pi$ transitions with $\Delta
m_{J}=0$ are driven, while for a projection of $\vec{B}$ perpendicular to the
polarisation of the light the $\sigma^{\pm}$ transitions, with $\Delta
m_{J}=\pm 1$, are driven Adams and Hughes (2019). Circular polarisation of
left- or right-handedness in the Voigt geometry can be treated by assuming a
linear combination of the $x$ and $y$ components of linearly polarised light
as there is no immediate effect of the relative phase between the two. In this
particular case, the relative phase $\phi=(2n-1)\pi/4$, with $n$ an integer,
is merely reflected in the strength of each component, which is equal for the
$x$ and $y$ components. A corollary of the above is that the system exhibits
equivalent solutions for values of $\phi_{B}$ modulo $\pi$, and this in turn
results in the inability to determine the absolute direction of the magnetic
field with just one measurement.

Figure 1: General geometry for experiments involving an externally applied
magnetic field and light; for the Voigt geometry where $\vec{k}\perp\vec{B}$,
$\theta_{B}=\pi/2$. The light is assumed to be linearly polarised in the $x-y$
plane, with the electric field vector $\vec{E}$ taken here to be vertically
polarised along the $x$-axis. The external magnetic field $\vec{B}$ makes an
angle $\phi_{B}$ with respect to $\vec{E}$, which when projected onto the
$x-y$ plane gives components parallel and perpendicular to $\vec{E}$. The
former drives $\pi$ transitions and the latter $\sigma^{\pm}$ transitions.

As mentioned above, the propagation through the atomic medium of each of the
eigenmodes has an associated complex refractive index. In general, the medium
has two unique refractive indices, which in light of them being complex, gives
rise to dichroism and birefringence. To take these effects into account, the
electric field of the light is propagated through the atomic medium. The input
electric field is first transformed into the coordinate system of the medium’s
eigenbasis, which corresponds to the Cartesian basis in the Voigt geometry,
with the orthogonal components coinciding with the parallel and perpendicular
directions to the external magnetic field vector. Once the field has been
written in this new basis it is propagated using each index, $n_{i}$, over a
distance $L$ in the medium by using the multiplicative factor
$e^{{\rm{i}}n_{i}kL}$. The resulting electric field is then transformed back
into the laboratory coordinate system. At this stage the output electric field
can be conveniently analysed  via Stokes polarimetry Schaefer  _et al._
(2007); Weller  _et al._ (2012c). This formalism provides a set of parameters
which are easily obtained with standard laboratory equipment and measurements
of the intensity of light in different sets of orthogonal polarisation bases.
The Stokes parameter $S_{0}$ is independent of the measurement basis, and for
an incident beam of intensity $I_{0}$ is defined as

$\displaystyle
S_{0}\equiv\frac{I_{x}+I_{y}}{I_{0}}=\frac{I_{\nearrow}+I_{\searrow}}{I_{0}}=\frac{I_{\rm{RCP}}+I_{\rm{LCP}}}{I_{0}}.$
(1)

Physically, $S_{0}$ represents the normalised total transmitted intensity. The
remaining Stokes parameters, $S_{1},\,S_{2}\,\&\,S_{3}$,

$\displaystyle S_{1}\equiv$
$\displaystyle\displaystyle\frac{I_{x}-I_{y}}{I_{0}},$ (2) $\displaystyle
S_{2}\equiv$
$\displaystyle\displaystyle\frac{I_{\nearrow}-I_{\searrow}}{I_{0}},$ (3)
$\displaystyle S_{3}\equiv$
$\displaystyle\displaystyle\frac{I_{\rm{RCP}}-I_{\rm{LCP}}}{I_{0}},$ (4)

give information on the optical rotation of light generated by the atomic
medium by looking at the differences between the orthogonal polarisation
components in the horizontal ($I_{x},I_{y}$), diagonal
($I_{\nearrow},I_{\searrow}$) and circular bases
($I_{\rm{RCP}},I_{\rm{LCP}}$), respectively.

Figure 2: Schematic of the most relevant components of the optical setup used
for acquiring experimental data. Light from a distributed feedback (DFB) laser
passes through an optical isolator (OI) and is split into two components.
Approximately half of the light is used for a reference optical system used to
calibrate the laser scan in a zero magnetic field environment; the system
consists of a room temperature 75 mm natural abundance Rb reference cell and a
Fabry-Pérot etalon made with two mirrors (M). The rest of the light is sent to
the experiment cell via an acousto-optic modulator (AOM) that can be used to
keep the power constant Truong  _et al._ (2012) and polarising beamsplitter
(PBS) cubes and half-/quarter-wave plates ($\lambda/2\,\&\,\lambda/4$,
respectively) to ensure the input light is in the desired polarisation state.
The 1 mm long experiment cell contains isotopically enriched 87Rb ($99\%$
purity), and is placed in a cylindrical magnet designed to give a primarily
axial field of 1.62 T at its centre Trénec  _et al._ (2011). A polarisation-
sensitive detection scheme is used after the experiment cell, consisting of a
PBS, retarder wave plate and two balanced photodiodes (PD), that allow a
voltage signal with information on the optical rotation and absorption of the
medium to be measured. Plano-convex lenses (L) are used to resize the beam at
different points in order to avoid significant clipping. Figure 3: Internal
copper heater block used in the experiments in the Voigt geometry. Shown here
is the main copper block, with an angled slot for the $1$ mm cell to avoid
back-reflection from the cell windows and two spaces for $5$ mm right-angle
prisms. Light enters and exits the heater block through two optical access
holes (seen in the upper-left) and passes through the cell perpendicular to
its original direction of incidence. The larger hole (bottom-centre) houses a
resistive heater element that serves to heat the entire block in order to
raise the temperature of the vapour in the interior of the cell to the desired
point. This block is covered by a copper lid and housed in a custom PTFE
cylinder that preserves the optical access and cable feed-throughs necessary
to carry out experiments when the whole assembly is located inside the
permanent magnet. Figure 4: Spectroscopy of the Rb D2 line for a system with
an external magnetic field of 1.5 T arranged in the Voigt geometry. The
bottom-left of the figure shows the evolution of the ground and excited state
energy levels in 87Rb, corresponding to the 5$S_{1/2}$ and 5$P_{3/2}$ atomic
energy levels, respectively, with an increasing magnetic field strength of up
to 1.5 T. In the lower right of the figure, the ground-state levels involved
in the atomic transitions at 1.5 T, including the admixtures due to remnant
effects of the hyperfine interaction, are given in the $|m_{J},m_{I}\rangle$
basis. The upper panels show the calculated spectra, as functions of linear
detuning, for $\pi$ (olive) transitions ($\vec{E}\parallel\vec{B}$), and
$\sigma^{+}$ (blue), $\sigma^{-}$ (purple) transitions
($\vec{E}\perp\vec{B}$). The coloured arrows in the energy level diagram point
from the initial to the final states involved in the transition, with
overlapping arrows showing the presence of weaker transitions due to the
admixtures present in the system.

##  III Experimental setup

Experimental work for this paper was carried out using the optical setup
described in figure 2. The laser used is a tuneable distributed feedback (DFB)
laser, with a central wavelength of 780 nm and a quoted linewidth of $<2$ MHz.
The frequency is tuned by changing the temperature of the laser chip, which
allows a frequency mode-hop free tuning range over many hundreds of GHz. A
typical scan of this size takes $\sim 2$ seconds and is limited by the
response time of the temperature control circuit of the laser. The emitted
light from the laser is split using polarisation optics after an optical
isolator to maintain high polarisation purity and manage the beam intensity.
Some of the light is taken to a secondary optical setup for calibration of the
laser frequency scan. A signal from a 75 mm natural abundance Rb reference
cell provides an absolute frequency reference, whereas a transmission signal
from a Fabry-Pérot etalon is used to linearise the scan. These two signals are
then processed as outlined in ref. Keaveney (2014) to obtain spectra as a
function of linear detuning. The remainder of the light is then passed through
several polarising elements that allow us to obtain a well-defined linear
polarisation to use in our experiment cell. A circular polarisation can be
obtained by replacing the half-wave retarder plate ($\lambda/2$) for a
quarter-wave retarder plate ($\lambda/4$) before the light goes through the
experiment cell.

The experiments are carried out in a cuboidal vapour cell with a cavity length
of 1 mm containing isotopically enriched $99\%$ purity 87Rb. In order to
provide a large enough atomic density and optical depth for observing the
desired phenomena the cell must be heated. Microfabricated cells, such as this
one, have previously been found to exhibit birefringent properties when heated
to their optimal operating temperatures Weller (2013); Sargsyan  _et al._
(2006); Jahier  _et al._ (2000); Steffen  _et al._ (2013). Details on the
fabrication of this cell can be found in Knappe  _et al._ (2005). The cell and
beam-steering optics (uncoated right-angle prisms) are placed in a custom-made
copper holder which also houses an internal heater, as seen in figure 3.
During operation the temperature is set by applying a voltage to the heater to
raise the temperature of the copper surrounding the cell. The stability of the
holder is maintained passively by fixing the voltage and allowing
thermalisation of the metal block with the surroundings. The copper bed is
enclosed in a secondary cylinder made of PTFE that acts as a shield and is
only in weak thermal contact with it in order to ensure there are no sharp
fluctuations in the temperature of the system. Due to the reduced footprint of
the system imposed by the central bore size of the magnet it was not possible
to include a suitable temperature sensor in this mount.

The external magnetic field is generated by a cylindrical permanent magnet,
with a central bore of diameter $22$ mm along its axis, designed using the
“magic sphere” configuration described in Trénec  _et al._ (2011). The maximum
value of the field produced by the magnet is 1.62 T at its centre, with the
field strength quickly falling radially outwards to the ends of the magnet.
The cylindrical PTFE assembly housing the copper heater block with the
experiment cell sits inside the magnet’s bore. This particular design allows
for a well-characterised field along the axis of the magnet, which in turn
ensures field homogeneity across the length of the experiment cell, while
maintaining the field on the outside of the magnet to below hundreds of mT.
Further details on the construction and characterisation of the magnet can be
found in ref. Trénec  _et al._ (2011).

The experiment is conducted in the weak-probe regime Sherlock and Hughes
(2009) to avoid optical pumping. Therefore the optical power of the
interrogating laser beam (i.e. the probe beam) is maintained below the
saturation intensity. For a beam waist ($1/e^{2}$) of approximately $0.7$ mm
this means keeping the power close to  1 $\rm{\mu}$W. A probe beam of these
characteristics gives an effective spatial resolution roughly equal to the
volume of atoms interrogated by the laser beam, which can be considered to be
a cylinder whose length is that of the experiment vapour cell ($1$ mm) and
whose radius is the beam waist. It is worth nothing that the length of the
cylindrical heater block assembly previously described ($28$ mm) is smaller
than the length of the magnet ($152$ mm) and this generates difficulties in
aligning the heater block axis relative to the laboratory frame of reference
once inside the magnet. As such, the heater block is slightly rolled about the
axis of the magnet ($z$) which in turn results in a relative orientation of
the $x,y$ axes of the atom and the laboratory $x,y$ axes that can be described
by an effective offset in the angle $\phi_{B}$.

##  IV Analysis of atomic absorption spectra at high fields

Figure 4 is created using the theoretical model described in section II and
shows the evolution of the atomic energy levels as a function of magnetic
field strength. In addition, the predicted absorption spectra in a 1.5 T
external magnetic field are depicted. In the lower panel the ground- and
excited-state energy level diagrams show, respectively, the states in the
5$S_{1/2}$ and 5$P_{3/2}$ manifolds at 1.5 T as well as the initial and final
states involved in each of the allowed transitions. Seen in the top two panels
are the calculated absorption for an isotopically enriched vapour cell ($99\%$
purity 87Rb) at $100^{\circ}$C, separated into $\pi$ (upper panel) and
$\sigma^{\pm}$ (lower panel) transitions. The position of the vertical lines
serves to indicate the atomic resonances in terms of the linear frequency. The
colour of these lines corresponds to different types of transitions: $\pi$
transitions are olive green, $\sigma^{+}$ transitions are blue and
$\sigma^{-}$ transitions are purple. For the ground state, the decompositions
in the $m_{J},m_{I}$ basis are given to the right of the corresponding
sublevel. The arrows, with colours again corresponding to the type of
transition as previously mentioned, are semi-transparent to evidence the fact
there are still some overlapping transitions due to a small remnant admixture
in the state decomposition due to hyperfine interactions. This can also be
seen in the respective detunings of the transitions on the top two panels,
where $\pi$ and $\sigma^{\pm}$ have frequencies that coincide between them.

Figure 5: Experimental data (blue circles; colour online) expressed in terms
of the linear detuning of the laser scan for a horizontally polarised input
beam with the corresponding fit (purple line; colour online) using our ElecSus
model, with residuals shown (bottom panel). Data and the theoretical fit are
in very good agreement (RMS error of 1.2%). For this spectrum the free
parameters in the fit are: $\phi_{\rm{B}}$, the angle of the magnetic field
with respect to the $x$-axis, $B$, the magnetic field strength and $T$, the
temperature of the atoms. Average values of $\phi_{\rm{B}}=(0.4491\pm 0.0007)$
rad, $B=(1.52\pm 0.08)$ T and $T=(108.94\pm 0.04)^{\circ}$C are obtained from
fitting five spectra. All other parameters for the system are fixed as
follows: $\theta_{\rm{B}}=\pi/2$, $\Gamma_{\rm{Buff}}=350$ MHz and
$\delta_{\rm{shift}}=50$ MHz. Figure 6: Experimental data (blue circles;
colour online) expressed in terms of the linear detuning of the laser scan for
a vertically polarised input beam with the corresponding fit (purple line;
coloured online) using our ElecSus model, with residuals shown (bottom panel).
Data and the theoretical fit are in very good agreement (RMS error of 1.2%).
For this spectrum the free parameters in the fit are: $\phi_{\rm{B}}$, the
angle of the magnetic field with respect to the $x$-axis, $B$, the magnetic
field strength and $T$, the temperature of the atoms. Average values of
$\phi_{\rm{B}}=(2.0082\pm 0.0007)$ rad, $B=(1.52\pm 0.07)$ T and $T=(110.23\pm
0.03)$C are obtained from fitting five spectra. All other parameters for the
system are fixed as follows: $\theta_{\rm{B}}=\pi/2$, $\Gamma_{\rm{Buff}}=350$
MHz and $\delta_{\rm{shift}}=50$ MHz.

At a field strength of 1.5 T, the 5$P_{3/2}$ states strongly decouple into the
$m_{J},m_{I}$ basis and result in four multiplets organised by
$m_{J}=3/2,1/2,-1/2,-3/2$ projections. Furthermore, at this magnetic field
strength the ground state has a much more complete decoupling than that
observed at lower field strengths Keaveney  _et al._ (2019); Zentile  _et al._
(2014); Whiting  _et al._ (2015, 2016, 2017, 2018). As such, we can treat the
system as being completely in the Hyperfine Paschen-Back (HPB) regime. This
means that the initial hyperfine ground states are now split into two distinct
groups, corresponding to the projections $m_{J}=\pm 1/2$, with the
$m_{I}=3/2,1/2,-1/2,-3/2$ states clearly defined despite the Doppler-
broadening in the vapour. This gives us well-defined multiplets of four
‘strong’ transitions
($|m_{J},m_{I}\rangle\rightarrow|m_{J}^{\prime},m_{I}\rangle$, with
$m_{J}^{\prime}=m_{J}\;(\pi),m_{J}+1\;(\sigma^{+}),m_{J}-1\;(\sigma^{-})$). It
is worth adding that at this point there are still some ‘weak’ transitions
present due to the ground states not being pure eigenstates in the
$m_{J},m_{I}$ basis. This is evident in the small ($<1\%$) admixture of the
opposite $m_{J}$ state in the state decomposition on the bottom right of fig.
4 (more details can be found in ref. Zentile  _et al._ (2014)).

Figures 5 and 6 show experimental data, averaged over five spectra, that have
been fitted using the theory obtained with ElecSus. From the fit, the RMS
error between data and theory is calculated for figure 5(6) to be 1.2%. The
residuals R, multiplied by a factor of 100 for ease of viewing, are also
shown. The absence of obvious structure in the residuals, and the large
multiplier required for the residuals, are indicators of a very good fit
Hughes and Hase (2010). The fit is carried out with three free parameters:
$\phi_{\rm{B}}$, the angle between the magnetic field and the direction of
polarisation of the laser beam (see figure 1), taken to be linear along the
$x$($y$)-axis; $B$, the magnitude of the magnetic field the atoms are exposed
to and $T$, the temperature of the atoms in the experiment cell. Other
significant experimental parameters, such as those relating to the effects due
to buffer gas in the cell, such as the amount of inhomogeneous broadening
$\Gamma_{\rm{Buff}}$ caused by collisions and a shift in the frequency of the
transitions $\delta_{\rm{shift}}$, are kept fixed. The values for these fixed
parameters are obtained a priori by fitting other spectra similar to those
averaged and shown in figures 5 and 6. Of the remaining parameters in the fit,
the field angle $\theta_{\rm{B}}$ is fixed by the geometry of the experimental
setup in the Voigt geometry ($\theta_{\rm{B}}=\pi/2$). We attribute the
significant buffer gas broadening in the spectra to He atoms trapped in the
cell, after the cell was exposed to a He environment in previous experiments
and note that this does not generate important shifts in any of the resonance
lines. Using the literature values of the broadening coefficient for He
Rotondaro and Perram (1997), we extract a pressure of $\sim 18$ torr ($\sim
24$ mbar) for the amount of said buffer gas in our experiment cell. Note that
while the time necessary to acquire a spectrum is on the order of a second,
the time needed to analyse the data and generate a fit is on the order of
minutes due to the complexity of the parameter space.

For the spectrum shown in figure 5 we obtain a value of
$\phi_{\rm{B}}=(0.4491\pm 0.0007)$ radians ($(25.74\pm 0.04)^{\circ}$).
Similarly, for the fit shown in figure 6 we obtain a value of
$\phi_{\rm{B}}=(2.0081\pm 0.0007)$ radians ($(115.05\pm 0.04)^{\circ}$). Both
of these values differ from their corresponding expected values by $\approx
0.45$ radians ($\sim 25^{\circ}$), which we take as a systematic error due to
the orientation of the cell heater block inside the bore of the cylindrical
magnet used in the experiment. As a result, there is excitation of both $\pi$
and $\sigma^{\pm}$ transitions in both spectra shown due to the presence of
parallel and perpendicular components of $\vec{B}$ along the direction of
polarisation of the light. In this case, the difference in strength between
the transitions is given as a simple factor of $\cos^{2}(\phi_{\rm{B}})$ for
the parallel component and $\sin^{2}(\phi_{\rm{B}})$ for the perpendicular
component. According to the difference in the expected value of
$\phi_{\rm{B}}$ obtained from the fits, this results in an approximate $4:1$
ratio in the strengths of the lines; this is clearly visible in both figure 5
and 6.

Similarly, for both spectra shown in figures 5 and 6 we obtain a value
$B=1.52$ T. The uncertainties in these values, $80$ mT and $70$ mT,
respectively for the two spectra, can be mainly attributed to linearity of the
laser scan in our experiment. The DFB laser used in our experiment allows for
a large mode-hop-free scan ($\sim 150$ GHz) at the expense of a non-linearity
that is introduced as the frequency is changed. It is this non-linearity,
along with other systematic errors in the linearisation and calibration of the
frequency axis, that is the primary source of uncertainty in our measurements.
In future work we plan to design the experiment so that this non-linearity can
be reduced in order to improve the precision in our measurement of the
magnetic field strength.

Furthering the ideas and work presented in Keaveney  _et al._ (2019), we
propose this as an atomic technique for measuring large magnetic fields and
their relative orientation. With the system completely in the hyperfine
Paschen-Back regime, the Zeeman shift presented in all the resonance line
positions allows for a better determination of the magnetic field strength. In
addition to this, the relative strength between sets of transitions, due to
different coupling strengths, is better observed. This in turn allows a more
precise determination of the relative orientation of the direction of
polarisation (i.e. of the electric field vector) of the light with respect to
the external magnetic field. Thus, the present system and technique lead to a
natural application of atomic-based spectroscopy in vector magnetometry.

##  V Sensitivity of optical rotation signals to the birefringence of cell
windows

At the high magnetic field strength used for this work, optical rotation
phenomena can provide additional information about the medium through which
the light propagates Ejlli  _et al._ (2020). In the case of an atomic medium,
measuring this optical rotation via the Stokes parameters, as mentioned in
section II, proves to be of natural interest for understanding the
interactions between the atoms and the external magnetic field. In this case,
we have experimentally measured the dichroism and birefringence of the atomic
medium in the three bases corresponding to the $S_{1},S_{2}\textrm{ and
}S_{3}$ parameters as shown in figures 7, 8 and 9, respectively.

Figure 7: Experimental data (blue circles) taken as a function of linear
detuning for a linearly polarised input beam with the corresponding fit
(purple line) of the $S_{1}$ parameter taking window birefringence into
account, with residuals R shown. Data and the theoretical fit are in very good
agreement (RMS error of $\sim$2%). In this case the fixed parameters are
$\Gamma_{\rm{Buff}}=350$ MHz, $\delta_{\rm{shift}}=50$ MHz and
$\theta_{\rm{B}}=\pi/2$, while the fit allows $T,B,\phi_{\rm{B}}$ to float;
also included in the fit are the parameters to take into account the
birefringence effects of the cell windows ($\theta_{\rm{BR}},\phi_{\rm{BR}}$).
A fit without the effect of the cell window birefringence (broken line) is
included for comparison. Figure 8: Experimental data (blue circles) taken as a
function of linear detuning for a linearly polarised input beam with the
corresponding fit (purple line) of the $S_{2}$ parameter taking window
birefringence into account, with residuals R shown. Data and the theoretical
fit are in very good agreement (RMS error of $\sim$3%). In this case the fixed
parameters are $\Gamma_{\rm{Buff}}=350$ MHz, $\delta_{\rm{shift}}=50$ MHz and
$\theta_{\rm{B}}=\pi/2$, while the fit allows $T,B,\phi_{\rm{B}}$ to float;
also included in the fit are the parameters to take into account the
birefringence effects of the cell windows ($\theta_{\rm{BR}},\phi_{\rm{BR}}$).
A fit without the effect of the cell window birefringence (broken line) is
included for comparison. Figure 9: Experimental data (blue circles) taken as a
function of linear detuning for a linearly polarised input beam with the
corresponding fit (purple line) of the $S_{3}$ parameter taking window
birefringence into account, with residuals R shown. Data and the theoretical
fit are in very good agreement (RMS error of $\sim$2%). In this case the fixed
parameters are $\Gamma_{\rm{Buff}}=350$ MHz, $\delta_{\rm{shift}}=50$ MHz and
$\theta_{\rm{B}}=\pi/2$, while the fit allows $T,B,\phi_{\rm{B}}$ to float;
also included in the fit are the parameters to take into account the
birefringence effects of the cell windows ($\theta_{\rm{BR}},\phi_{\rm{BR}}$).
A fit without the effect of the cell window birefringence (broken line) is
included for comparison.

In order to carry out these measurements a set of polarising optics
(PBS$+\lambda/2$, $\lambda/4$) and two photodiodes was set up, as seen in
figure 2, to measure the light transmitted through the experiment cell in
terms of the linear detuning of the laser scan. This allows for two orthogonal
polarisation components to be recorded simultaneously and then be processed
into the corresponding Stokes parameter for the basis in question. We take the
definitions of the Stokes parameters as used in references Zentile  _et al._
(2015c); Keaveney  _et al._ (2018b). Figure 7 shows the $S_{1}$ parameter,
taken as the difference between the orthogonal linear polarisations (i.e.
horizontal and vertical) (eq. 2). Figure 8 shows the $S_{2}$ parameter,
defined as the difference between the linear polarisations in a basis rotated
by $\pi/4$, giving diagonal components in a Cartesian basis (eq. 3). Lastly,
figure 9 shows the last Stokes parameter, $S_{3}$, as the difference between
orthogonal circular polarisations in the helicity basis, (i.e. left-hand and
right-hand circular) (eq. 4). In these cases, given a well-defined input
polarisation, the transmitted light gives information regarding the linear and
circular birefringence of the atomic medium.

Using our theoretical model, _ElecSus_ , we proceed to fit the data to each of
the three Stokes parameters mentioned above. Very good agreement Hughes and
Hase (2010) between the model (solid purple curve) and data (blue circles) is
evident in figures 7, 8 and 9. Despite this, there are still slight
discrepancies between the experimental data and the fit. We can try and remove
some of these errors by taking into account the birefringence of the windows
of our experiment cell. To do this, we include in our model the effects of two
thin, birefringent windows interacting with the electric field of our laser
beam twice: once before the light enters the atomic medium and once when the
light has passed through the atomic medium and exits the cell. We carry out
these calculations by using the Jones matrix formalism Keaveney  _et al._
(2018b), so that in this case the transmitted electric field $E_{\rm{out}}$ in
our experiment can be written as

$\displaystyle
E_{\rm{out}}=\mathcal{M}_{\theta_{\rm{BR}},\phi_{\rm{BR}}}\times\mathcal{J}_{\rm{atoms}}\times\mathcal{M}_{\theta_{\rm{BR}},\phi_{\rm{BR}}}\times
E_{\rm{in}},$

where $E_{\rm{in}}$ is the incident electric field,
$\mathcal{M}_{\theta_{\rm{BR}},\phi_{\rm{BR}}}$ is the Jones matrix
representing the birefringent window of the cell and
$\mathcal{J}_{\rm{atoms}}$ is the Jones matrix representing the dichroic and
birefringent atomic medium. The matrix
$\mathcal{M}_{\theta_{\rm{BR}},\phi_{\rm{BR}}}$ has been included twice to
account for the entry and exit windows of the experiment cell. This output
electric field can be multiplied by the appropriate Jones’ matrices to give
the desired polarisation components to process into the form of the different
Stokes’ parameters.

From our fits we can see that the birefringence due to the cell windows is
considerably small. In this case the cell windows have a thickness of
$300\leavevmode\nobreak\ \rm{\mu m}$ each Knappe  _et al._ (2005). Using the
literature value for the Verdet coefficient of glass Carr  _et al._ (2020) at
the wavelength of the Rb D2 line ($\sim 780$ nm), each cell window induces a
rotation of $\approx 0.05^{\circ}$. Here we make an initial assumption that
both of these windows are identical in their birefringent properties. Figures
7, 8 and 9 show very good agreement between the data (blue circles) and the
model considering birefringence (broken green curve) Hughes and Hase (2010).
The residuals R shown in these figures correspond to the results of the fits
that include the birefringence of both cell windows. We obtain average values
of $(0.96\pm 0.16)$ radians ($(55\pm 9)^{\circ}$) for the angle
$\theta_{\rm{BR}}$ and of $(0.06\pm 0.03)$ radians ($(3\pm 2)^{\circ}$) for
the angle $\phi_{\rm{BR}}$. It is worth noting that these values correspond to
a fit of the effect _both_ cell windows have on the electric field transmitted
through the cell. Comparing the fits to the experimental data with and without
the birefringence effects from the cell windows we can see that this effect is
particularly evident in the $S_{3}$ parameter, as seen in figure 9. Due to the
definition of said parameter (eq. 4) we can proceed to say that the cell
windows have a predominantly circular birefringence. The discrepancy between
the rotation induced in the cell due to the field ($\approx 0.1^{\circ}$) and
the measured value for $\phi_{\rm{BR}}$ is assumed to be caused by the heating
of the windows as well as mechanical stresses from the fabrication process and
the optical setup. In particular, we exploit the sensitivity of the atomic
system to optical rotation, in this case in the basis of orthogonal circular
polarisation states, to obtain a signal that enhances these effects so that
they are clearly visible. As such, this experimental system provides a tool to
characterise these birefringence effects due to vapour cell windows in order
to reduce systematic errors in future measurements.

##  VI Conclusions

In this work we have presented a spectroscopy-based technique using a thermal
alkali-metal vapour that allows absolute magnetic field strength and
orientation of the field with respect to the light polarisation to be
measured. An isotopically enriched sample of 87Rb was used; however the
principles behind this technique can be easily applied to other alkali-metal
atoms. Very good agreement was found between the theoretical model used and
our detailed spectroscopic measurements of the Stokes parameters of the
medium. We have used polarisation-sensitive detection in order to better
constrain the polarisation angle measured, as well as measure the
birefringence effects due to the vapour cell windows. Using this technique it
is also possible to envisage a precise spectroscopy setup for atomic
magnetometry in large ($>1$ T) fields. Furthermore, the work here presented
opens up new areas of research using atomic vapours, such as measurement of
fundamental constants via precision thermometry using the ground-state
populations.

##  VII Acknowledgements

The authors would like to thank Jacques Vigué for kindly providing the
permanent magnet used, James Keaveney for his contributions and insightful
discussions, Danielle Pizzey for her comments and suggestions on the
manuscript, Stephen Lishman and members of the mechanical workshop for their
aid in fabricating the mechanical components for the experiment. The authors
acknowledge funding from EPSRC (Grant EP/R002061/1) and Durham University.
FSPO gratefully acknowledges a Durham Doctoral Scholarship. The data presented
in this paper are available from 111DOI://10.15128/r1kk91fk595.

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# The quantum dynamic range of room temperature spin imaging

Martin Schalk<EMAIL_ADDRESS>Walter Schottky Institut, Physics
Department and MCQST, Technische Universität München, 85748 Garching, Germany
Riccardo Silvioli  Walter Schottky Institut, Physics Department and MCQST,
Technische Universität München, 85748 Garching, Germany  Karina Houska  Walter
Schottky Institut, Physics Department and MCQST, Technische Universität
München, 85748 Garching, Germany  Niels van Venrooy  Walter Schottky Institut,
Physics Department and MCQST, Technische Universität München, 85748 Garching,
Germany  Katrin Schneider  Walter Schottky Institut, Physics Department and
MCQST, Technische Universität München, 85748 Garching, Germany  Nathan P.
Wilson  Walter Schottky Institut, Physics Department and MCQST, Technische
Universität München, 85748 Garching, Germany  Jan Luxa  Chemistry Department,
University of Chemistry and Technology Prague, 16628 Prague, Czech Republic
Zdeněk Sofer  Chemistry Department, University of Chemistry and Technology
Prague, 16628 Prague, Czech Republic  Dominik Bucher  Chemistry Department and
MCQST, Technische Universität München, 85748 Garching , Germany  Andreas V.
Stier  Walter Schottky Institut, Physics Department and MCQST, Technische
Universität München, 85748 Garching, Germany  Jonathan J. Finley
<EMAIL_ADDRESS>Walter Schottky Institut, Physics Department and MCQST,
Technische Universität München, 85748 Garching, Germany

###### Abstract

Magnetic resonance imaging of spin systems combines scientific applications in
medicine, chemistry and physics. Here, we investigate the pixel-wise coherent
quantum dynamics of spins consisting of a 40 by 40 micron sized region of
interest implanted with nitrogen vacancy centers (NV) coupled to a nano-
magnetic flake of CrTe2. CrTe2 is an in-plane van der Waals ferromagnet, which
we can probe quantitatively by the NV electron’s spin signal even at room
temperature. First, we combine the nano-scale sample shapes measured by atomic
force microscope with the magnetic resonance imaging data. We then map out the
coherent dynamics of the colour centers coupled to the van der Waals
ferromagnet using pixel-wise coherent Rabi and Ramsey imaging of the NV sensor
layer. Next, we fit the pixel-wise solution of the Hamiltonian to the quantum
sensor data. Combining data and model, we can explore the detuning range of
the spin oscillation with a quantum dynamic range of over
$\left|\Delta_{max}\right|=$60\text{\,}\mathrm{MHz}$$ in the Ramsey
interferometry mode. Finally, we show the effect of the CrTe2 sample on the
coherence of the NV sensor layer and measure a 70 times increase in the
maximum frequency of the quantum oscillation going from the Rabi to the Ramsey
imaging mode.

###### keywords:

American Chemical Society, LaTeX

NV, 2D,

##  1 Introduction

Spinful materials, biological reactions and active electronic devices
essentially induce magnetic fields that can be imaged on the micron to nano-
scale using quantum sensors1, 2, 3. Image-processed camera read-out and
homogeneous quantum control are key challenges in the context of quantum
sensing for characterizing and improving smart sensors, coherent memories,
bio-molecular and spin devices under ambient conditions. Here, we image a
layer of nitrogen vacancy centers as quantum sensors coupled to a magnetic van
der Waals material to investigate nano-magnetism and the strongly detuned
dynamic range of coherent spin oscillations with a wide field of view. We
define the quantum dynamic range of the quantum sensor as the maximum detuning
range we can resolve by means of a pixel-wise coherent spin oscillation driven
by single frequency pulses. Complementary to the common low quantum dynamic
range approaches of optimizing the sensitivity at a single frequency4, 5 and
three level atom sensing (3LS) schemes such as spins driven by two frequency
tones6, 7, 8, 9, we here explore the full quantum dynamic range of room
temperature spin imaging in a strongly detuned magnetic environment. Magnetic
van der Waals materials10 like CrTe2 offer opportunities to investigate
magnetic phase transitions, fingerprints of quantum spin phases and are the
material of choice to explore the atomic limit of scaling spintronic
memories11. While single spin scanning techniques have been used to image the
static magnetization12, 13, 14 and spatially resolved domain walls15, 16 of 2D
magnets, the wide-field imaging mode17 has several benefits: First, the NV
ensemble signal is bright and homogeneous and consequently offers higher
combined spin sensitivity than using single spins18.

Figure 1:  Quantitative nano-magnetic imaging with a wide-field of view a) Van
der Waals magnet CrTe Raman modes (b) Atomic force microscope height profile,
inset horizontal height cut of the $100\text{\,}\mathrm{nm}$ thick CrTe2
magnetic flake c) NV spin directions in the diamond lattice used for 3D vector
magnetometry d) Coherent wide-field camera setup for direct spin imaging e)
Magnetic flux density of CrTe2 imaged as a quadrupol by the NV electron spin
resonance detuning, the colormap depicts the $B_{z}$ component; $B_{x}$ and
$B_{y}$ are shown as field lines in black.

Second, the camera acquires pixel-wise spin information in parallel and thus
opens up easy access to imaging science and even real-time image processing on
micro-second (camera limited) time-scales19. Third, the sensor signal is
robust and does not bleach or change during subsequent measurements due to the
solid-state integration of the quantum sensor layer into the protecting
diamond lattice. The point-wise coupling approach in scanning mode
measurements13, 12 induces oscillations of the the fork with respect to the
sample and microwave field leading to more involved stability and fork
oscillation calibration requirements20. Most importantly, the fixed coupling
constants and distances make homogeneous coherent control of the whole NV
layer as a fully quantum-enabled sensor with high dynamic range feasible as we
outline in this letter.

##  2 Results and discussion

Figure 1 schematically portrays the coherent wide-field imaging experiment.
The synthesized bulk crystals show the Raman $E_{2g}$ and $A_{1g}$ modes of
CrTe2, verifying the crystallography of the in-plane room temperature
ferromagnet14, as measured by Raman spectroscopy in figure 1 (a). The shape
and height of the exfoliated van der Waals flake is mapped out by an atomic
force microscope height profile in figure 1 (b). CrTe2 transferred on top of
the diamond sensor induces a magnetic field, which is commonly probed by
electron spin resonance (ESR) spectroscopy21, 22. The NV ensemble’s electron
spin resonances are linearly magnetic bias field dependent and therefore
naturally calibrated via23, 24:

$\begin{split}\mathcal{H}_{NV,i}&=\gamma_{el}\mathbf{B}\cdot\mathbf{S}+\mathbf{S}\cdot\mathbf{D}\cdot\mathbf{S}\\\
&\approx\gamma_{el}B_{i}\left(\text{sin}\text{\hskip
1.69998pt}{\theta}_{B}\cdot\text{cos}\text{\hskip
1.69998pt}{\phi}_{B}\cdot{\hat{S}}_{x}+\text{sin}\text{\hskip
1.69998pt}{\theta}_{B}\cdot\text{sin}\text{\hskip
1.69998pt}{\phi}_{B}\cdot{\hat{S}}_{y}+\text{cos}\text{\hskip
1.69998pt}{\phi}_{B}\cdot{\hat{S}}_{z}\right)+\frac{D}{h}\hat{S}_{z}^{2},\\\
\end{split}$ (1)

where $\mathbf{S}$ (S = 1) is the spin operator of the NV electron spin, the
index $i$ stands for the eight possible spin orientations (four NV directions
in diamond times two spin orientations),
$\gamma_{el}=$28\text{\,}\mathrm{GHz}\text{\,}{\mathrm{T}}^{-1}$$ is the NV
electron’s gyromagnetic ratio and the offset
$D/h\approx$2.87\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ is the zero field
splitting. We use three $i=(1,2,3)$ NV spin orientations depicted in orange,
green, and blue in figure 1 (c) to calculate a three-dimensional magnetic
field map. Here, each color map represents the measured electron spin
resonance frequency shift $\Delta f_{NV,i}\propto\gamma_{el}{\Delta B_{i}}$.
The spin resonance shifts of the three different NV directions directly relate
to a 3D cartesian lab coordinate system ($B_{x},B_{y},B_{z}$), where the
directional expansion in equation 1 is determined by the diamond crystal
orientation (100) and vector projections.23, 24, 25, 26

Figure 2: Dynamic range of a pixel-wise detuned Rabi oscillation a)
Homogeneous wide-field Rabi spin driving on the bare sensor with Rabi data in
direct comparison to the numerical rotating wave approximation model b) Bloch
sphere representation of the measured Rabi oscillation c) Measurement of the
detuned Rabi oscillation showing a symmetric increase of the Rabi frequency
around zero detuning, quantum dynamic range indicated by dashed lines and
arrows, sample effect by dark blue arrows d) Single camera pixel Rabi
oscillation measurement and fit (red dashed line) e) Full Rabi decay map, red
square indicates pixel (40,20) f) Rabi frequency image with dynamical spin
speed-up induced by the CrTe2 magnet.

The 3D magnetic field plot - as color plot for $B_{z}$ and black field lines
for the $B_{x}$ and $B_{y}$ components - in figure 1 (e) shows a quadrupolar
magnetic field distribution $\mathbf{B}$. Using the NV ensemble spin
resonance, we can thus directly image the magnetic flux distribution by the
in-plane magnetization of the nanoscopic CrTe2 sample14, 27 at a distance of
$z=$100\text{\,}\mathrm{n}\mathrm{m}$$. We combine the shape of the flake as a
mask for clarity in the upcoming measurements, since the CrTe2 sample darkens
the NV center photoluminescence signal out in the imaging path. Additional to
spin-mixture induced frequency shifts in continuous wave spin resonance
spectroscopy28, the magnetic field distribution of our sample provides a bias
field to the NV spin dynamics. A triggered camera read-out combined with drive
pulses observe spin dynamics of the sensor NV layer coupled to the CrTe2.
Thus, we can in particular use the coherent dynamics of the NV layer as a tool
to probe nano-magnetic samples like CrTe2 as follows. The wide-field quantum
sensor dynamics $S_{NV,\square}(t)$ is influenced by the three main components
drive $\omega_{d}$, detuning $\Delta$ and decay $\xi_{env}$. Briefly, the
driven NV ensemble resonance can be described as an ensemble of atoms, a
quantum two-level system (2LS) with spin-1, 29 and spin transition frequency
$\omega_{s}$ using the rotating wave approximation:

$\mathcal{H}_{RWA}=\frac{\Delta}{2}S_{z}+2\eta S_{x},$ (2)

where $\Delta=\omega_{s,\square}-\omega_{d}$ is the detuning of the spin
resonance with respect to the drive frequency $\omega_{d}$. As shown in
equation 1 and figure 1 (c), the spin transition frequency
$\omega_{s,\square}=\frac{f_{NV}}{2\pi}\approx D/h+\gamma_{el}\cdot B_{z}$
depends for every pixel on the zero field splitting and the local magnetic
bias field. We initialize the complete sensor plane with a green laser pulse
in the ground state and apply a resonant microwave drive
$\mathcal{H}_{d}=\gamma_{el}b_{1}\cos{\left(\omega_{s}t\right)}S_{x}=2\eta
S_{x}$ with strength $\eta$ to the NV ensemble. With that, we can perform
pixel-wise Rabi oscillations with the effective Rabi frequency
$\Omega_{\square}=\sqrt{\Delta^{2}+4\eta^{2}}$. We use a microwave split-ring
resonator antenna30 to apply a homogeneous drive field so that the drive
strength $\eta\approx\mathsf{const.}$ $\forall\square$ is constant or
continuously defined for all pixels. Figure 2 (a) shows the bare sensor Rabi
oscillation and its coherence, where all pixels are averaged into one curve
proving the wide-field homogeneity and displaying the good control parameter
range for the rotating wave approximation shown on the Bloch sphere 2 (b). The
orange line shows the numerical solution to the rotating wave approximation
model $\mathcal{H}_{RWA}$ fitting well to the wide-field data. Figure 2c then
shows the measured detuned Rabi oscillation following closely the solution
$S_{NV,\square}(t)=\sqrt{\frac{\Delta^{2}\sin^{2}{\left(\frac{\Omega
t}{2}\right)}}{\Omega^{2}}+\cos^{2}{\left(\frac{\Omega t}{2}\right)}}$ with a
dynamic range of $\left|\Delta_{Rabi}\right|=$15\text{\,}\mathrm{MHz}$$. We
emphasize, that if both the strength $\eta$ of the microwave field drive and
the detuned spin frequency $\omega_{s,\square}$ influence the pixel’s Rabi
frequency28, 31, it is difficult to distinguish between sensor or sample-
related frequency shifts induced by either inhomogeneous control or magnetic
bias. The Rabi imaging in figure 2 shows that the Rabi oscillations (f) get
faster and also decay faster (e) for higher magnetic field detuning induced by
the CrTe2 $\Delta_{\ce{CrTe2}}>0$ as indicated by the blue arrows in (c). The
sensor has a slight radial background stemming from the ring resonator’s
microwave field profile which can be fitted by the plane
$$7.94\text{\,}\mathrm{kHz}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$\cdot
X+$6.50\text{\,}\mathrm{kHz}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$\cdot Y+$13.382\,50\text{\,}\mathrm{MHz}$=Z$ around the mean Rabi
frequency $\Omega=$13.3825\text{\,}\mathrm{MHz}$$. In the decay map in figure
2 (e) we can clearly identify the magnetic CrTe2 van der Waals flake by faster
relaxation and smaller additional surface background effects which can
originate from charged surface contamination or strain, which change the T1
spin-lattice relaxation time32. The pixel-wise Rabi imaging (see for instance
pixel 40,20 in (a)) with homogenous wide-field control can to this end be used
as a naturally calibrated sensor for temperature, strain, microwave and
magnetic gradient sensing. Subsequently, we investigate Ramsey interferometry
and the free induction decay influenced by CrTe2.

Figure 3: Pixel-wise coherent Ramsey interferometry and detuning a) Ramsey
fringes of the whole sensor with detuned microwave drive exploring the full
quantum dynamic range marked by dashed lines b) Simulation of the Ramsey
fringes using a two-level quantum interference model, the orange line shows
the experimental MW drive set-point and the red (blue) arrows indicate the
magnetic detuning induced by the sample in positive (negative) direction c)
Single pixel Ramsey oscillation data blue and red-dashed trend line fit d) Map
of the pixel-wise Ramsey frequency detuned by the CrTe2 van der Waals magnet.

The free induction decay of the ensemble is then detected as Ramsey fringes
approaching the ideal form (at strong drive $\eta\gg\Delta$):

$S_{NV,\square}(\tau)=\sum_{n}\langle\sigma_{z,n}\rangle\propto\left(\frac{\cos{\left(\Delta\tau\right)}}{2}+\frac{1}{2}\right)\cdot
e^{-\tau/T_{2}^{*}}=\cos^{2}(\Delta\cdot\tau/2)\cdot e^{-\tau/T_{2}^{*}}.$ (3)

Figure 3 (a) shows the measured quantum interference of the whole sensor chip
under detuning. The model in 3 (b) of the Ramsey interferometry assuming a two
level quantum interference corresponds remarkably well with no free parameters
to the measurement even in the case that all spins in all pixels evolve and
are measured in parallel. In the following, we drive the spin resonance
$f_{res}=$2.3636\text{\,}\mathrm{GHz}$$ with drive frequency
$f_{drive}=$2.3836\text{\,}\mathrm{GHz}$$ at detuning
$\Delta=$20\text{\,}\mathrm{MHz}$$ as marked in figure 3 (b). The single pixel
(40,20) data in figure 3 (c) shows that a Ramsey oscillation can even be
resolved on a single camera pixel without binning. The Ramsey frequency is
then plotted in figure 3 (d) showing a strong quadrupolar detuning of the
Ramsey frequency influence by the CrTe2 sample. The sample speeds up the
Ramsey oscillation in the positive magnetically detuned part and decreases
Ramsey frequency at the sample edges of negative magnetic field. Here, the
sample shifts in positive and negative directions and the sign is given by the
orientation of the rotating frame defined by the Z spin ensemble direction and
the X drive axis. Ramsey interferometry is sensitive to smaller magnetic field
changes and detunings since the change in frequency for the Ramsey measurement
$\Delta f_{Ramsey,max}\approx$35\text{\,}\mathrm{MHz}$$ is nearly two orders
of magnitude higher than the Rabi frequency change $\Delta
f_{Rabi,max}\approx$0.5\text{\,}\mathrm{MHz}$$ by directly quantitatively
comparing the colorbar of the detuned Rabi and Ramsey interferometry
measurements on the same sample with fixed maximum drive power $\eta$. The
optimal sensor set-point is centered around the maximum rate of change under
detuning $\max\left(\frac{\partial S_{NV,\square}(t)}{\partial\Delta}\right)$,
which then equivalently corresponds to the most sensitive quantum dynamic
range. The quantum dynamics trend of the increased Ramsey frequency range is
also in accordance with Ramsey limited sensing outperforming continuous wave
or Rabi protocols33. The quantum dynamic range is over
$\left|\Delta_{max}\right|=$60\text{\,}\mathrm{MHz}$$ for Ramsey and around
$\left|\Delta_{Rabi}\right|=$15\text{\,}\mathrm{MHz}$$ for the Rabi imaging
mode as marked in figure 3 (a) respectively 2 (c).

Technique | Dynamics | Drive | Sensitivity | Dynamic Range | Ref.
---|---|---|---|---|---
Spin Resonance | spin mixture111no full coherent oscillation but statistical spin phase mixture | broad-band | $<$\mathrm{k}\mathrm{H}\mathrm{z}$$ | $>$10\text{\,}\mathrm{GHz}$$ | 21, 22
Relaxometry | spin decay | single f | $<$\mathrm{k}\mathrm{H}\mathrm{z}$$ | $\mathcal{O}($1\text{\,}\mathrm{GHz}$)$ | 34
Rabi | 2LS, driven | single f | $<$\mathrm{k}\mathrm{H}\mathrm{z}$$ | $\mathcal{O}($10\text{\,}\mathrm{MHz}$)$ | 35
Ramsey | 2LS, free | single f | $<$\mathrm{H}\mathrm{z}$$ | $\mathcal{O}($100\text{\,}\mathrm{MHz}$)$ | 7, 8
Homodyne | 2LS, synced | single f |  $<$1\text{\,}\mathrm{mHz}$$222determined spot-wise, not by wide-field imaging, divide by $\gamma$ for magnetic units in $\mathrm{T}$ | $<$100\text{\,}\mathrm{kHz}$$ | 5, 36, 37
Heterodyne | 3LS | two f/chirp333smooth transition between multilevel chirp and broad-band spin resonance spectroscopy38 | $<$100\text{\,}\mathrm{mHz}$$ | $>$100\text{\,}\mathrm{MHz}$$ | 9
|  |  |  |  |
Table 1: Design aspects and quantum dynamics of spin imaging techniques.

Table 1 summarizes the design aspects of quantum dynamic imaging showing that
the detection of fast quantum oscillations in a strongly detuned magnetic
environment requires a high dynamic sensing range. Therefore, CrTe2 can be
imaged by spin resonance, Rabi and Ramsey characterized in this letter, where
Ramsey provides the best compromise in terms of drive-free, sample induced
quantum oscillations, a high dynamic range and sufficient sensitivity. Future
homodyne31, 37 or heterodyne9 sensing protocols together with optimized light
collection39 are clear ways to approach the wide-field sensitivity limit for
fast cameras40, but will also either reduce the accessible dynamic range or
conceal quantum dynamics in multi-level, incoherent excitations6, 7, 8. Here,
we emphasize the importance of homogenous wide-field quantum control and
coherent, collective spin dynamics investigating a strongly detuned spin
environment induced by the CrTe2 van der Waals magnet. Most quantum phase or
entanglement-based sensing schemes then equally require homogeneity and
quantum-transduced frequency stability41, 42 over the whole sensor plane as an
essential prerequisite. We conclude that the stable control of the pixel-wise
quantum dynamics and a high dynamic detuning range are important design and
characterization aspects for practical quantum imaging which must not be
omitted next to the optimization of the single spot resolution or the highest
sensitivity at low quantum dynamic range.

##  3 Experimental

The spin imaging setup consists of: Green laser: DPSS Spectra Physics 532nm at
P=0.2W, Acousto-optic modulator: G&H 3350-199, Dichroic mirror: Thorlabs
DMLP550, Pulse generator: Swabian instrument pulsestreamer, Time of flight
camera: Basler acA 1920-155um, Objective: Leica 100x NA 0.75 HC PL FLUOTAR,
Microwave generator: Windfreak SynthHD PRO, Microwave antenna: Split ring
resonator antenna with a homogenous Bz field of
$1\text{\,}\mathrm{m}\mathrm{m}$ radius centered around
$2.8\text{\,}\mathrm{GHz}$, (design files are available in the supplementary
information). Sample preparation: element6 electronic grade diamond,
implantation Innovion with ^15N fluence
$2\text{\times}{10}^{12}\text{\,}{\mathrm{cm}}^{-2}$ at energy
$20\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$, diamond is excited from the side
in a total internal reflection geometry (cf. supplementary).

All authors gratefully acknowledge the German Science Foundation (DFG) for
financial support, as well as the clusters of excellence MCQST (EXS-2111) and
e-conversion (EXS-2089). J. J. Finley acknowledge the state of Bavaria via the
One Munich Strategy and Munich Quantum Valley. We greatfully acknowledge
support and discussions to the sample preparation with Bucher lab at TUM
chemistry department and Sofer lab at UCT Prague.

The theoretical models are based on the two-level systems in rotating wave
approximation and Ramsey spectroscopy using the quantum optics package43. All
experimental procedures, the RF design files, theoretical models, CrTe2
synthesis and characterization data for the samples and measurements are
available online or upon request.

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