# Quantum Frequency Mixing using an NV Diamond Microscope

Samuel J. Karlson  MIT Lincoln Laboratory, Lexington, MA 02421, USA
Department of Nuclear Science and Engineering, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Pauli
Kehayias  MIT Lincoln Laboratory, Lexington, MA 02421, USA  Jennifer M.
Schloss  MIT Lincoln Laboratory, Lexington, MA 02421, USA  Andrew C. Maccabe
Current Affiliation: Quantum Science and Engineering Program, Harvard
University, Cambridge, MA 02138, USA  MIT Lincoln Laboratory, Lexington, MA
02421, USA  David F. Phillips  MIT Lincoln Laboratory, Lexington, MA 02421,
USA  Guoqing Wang  Department of Nuclear Science and Engineering,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Research
Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA  Department of Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Paola Cappellaro  Department of Nuclear Science and
Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Research Laboratory of Electronics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Danielle A. Braje<EMAIL_ADDRESS>MIT
Lincoln Laboratory, Lexington, MA 02421, USA

###### Abstract

Wide-field magnetic microscopy using nitrogen-vacancy (NV) centers in diamond
can yield high-quality magnetic images of DC and AC magnetic fields. The
unique combination of micron-scale spatial resolution of scalar or vector
fields at room temperature and parallel camera readout make this an appealing
technique for applications in biology, geology, condensed-matter physics, and
electronics. However, while NV magnetic microscopy has achieved great success
in these areas, historically the accessible frequency range has been limited.
In this paper, we overcome this limitation by implementing the recently
developed technique of quantum frequency mixing. With this approach, we
generate wide-field magnetic images of test structures driven by alternating
currents up to 70 MHz, well outside the reach of DC and Rabi magnetometry
methods. With further improvements, this approach could find utility in
hyperspectral imaging for electronics power spectrum analysis, electronics
diagnostics and troubleshooting, and quantum computing hardware validation.

##  I Introduction

Wide-field magnetic microscopy using nitrogen-vacancy (NV) centers in diamond
has gained attention for its value in diverse applications including geology,
biology, condensed-matter physics, and electronics Levine  _et al._ (2019);
Scholten  _et al._ (2021). Unlike single-pixel scanning magnetic microscopes,
NV-diamond-based microscopes can simultaneously image all pixels in a wide
(mm-scale) field of view (FOV) without moving parts, and can reveal
spatiotemporal dynamics of magnetic fields in samples under study. The
technique offers micron-scale spatial resolution and the potential for
operation over a broad frequency range, and is being applied to a range of
interdisciplinary applications.

Despite the ongoing accomplishments and progress, most wide-field NV magnetic
microscopes have been limited to measuring static or low-frequency ($\lesssim$
1 kHz) magnetic fields, with few exceptions Pham  _et al._ (2011); DeVience
_et al._ (2015); Horsley  _et al._ (2018); Ziem  _et al._ (2019); Briegel  _et
al._ (2024). This is due to the added complexity required for measuring MHz-
frequency and GHz-frequency AC magnetic fields (including pulsed NV
interrogation and gated optical readout) and the greater need for homogeneous
laser, microwave (MW), bias magnetic field, and NV strain profiles across the
FOV. Moreover, these measurements are restricted to specific and often narrow
frequency ranges. For example, AC magnetometry using pulsed dynamical
decoupling is limited by MW $\pi$-pulse durations to typically $<$ 10 MHz
Degen  _et al._ (2017), while Rabi-oscillation AC magnetometry senses in a
narrow frequency range ($\sim$1 MHz) around the NV spin resonance frequency
(usually near $\sim$3 GHz) Alsid  _et al._ (2023). Though varying a strong
bias magnetic field can tune (and extend) the Rabi-oscillation AC magnetometry
frequency range, field uniformity requirements render this method impractical
for macroscopic NV ensembles. As a result, NV AC magnetic imaging
demonstrations often sacrifice frequency range, FOV size, or both.

Here we demonstrate a pulsed diamond magnetic microscope with a $1.5\times
1.5$ mm2 FOV size, and image DC and AC magnetic fields up to 70 MHz. To our
knowledge, this is the largest-area pulsed NV magnetic imaging demonstration,
as previous experiments have shown $\sim$0.5 mm2 at best Horsley  _et al._
(2018). To enable wide-frequency sensitivity, we apply the quantum frequency
mixing (QFM) approach recently demonstrated with small NV ensembles Wang  _et
al._ (2022) to our wide-field imager. We validate the technique by imaging AC
magnetic fields from currents in a straight-wire test structure patterned on a
glass slide, where we observe the expected frequency dependence of detected
signals. We then image magnetic fields from a second test structure (an
Archimedean spiral), demonstrating agreement between DC and AC images.
Finally, we explore the QFM magnetometry dynamic range under different signal
frequencies and amplitudes, leading to additional analytical expressions of
the expected behavior. Combining high-performance wide-field NV-diamond
microscopy with the QFM AC magnetometry technique opens the door to new
magnetic microscopy applications, including high-resolution power spectrum
analysis (PSA) imaging of integrated circuits, electronics failure analysis,
validation of $\sim$1-100 MHz electrical components, and even diagnostics of
quantum computing hardware (e.g. ion trap chips) Sutherlin (2015); Gaudestad
_et al._ (2012); Kehayias  _et al._ (2024).

\begin{overpic}[width=433.62pt]{Figure_1.pdf} \centering\put(5.0,57.0){{\Large
a)}} \put(47.0,57.0){{\Large b)}} \put(47.0,27.0){{\Large c)}}
\@add@centering\end{overpic} Figure 1:  (a) Schematic diagram of a QFM
magnetic microscope. A green laser beam illuminates the NV layer. Different
regions of the NV layer (highlighted by blue and red spins) respond to
spatially varying magnetic fields and produce magnetic-field-dependent
fluorescence, which is imaged onto a camera sensor using imaging optics. An
example test structure (an Archimedean spiral), generating a spatially varying
magnetic field, is shown at the sensor plane, and a magnetic map of its
current is imaged at the focal plane. (b) The NV spins act as both quantum
mixers and quantum sensors. In the presence of a signal field (yellow) at
$\omega_{\mathrm{s}}$ and a bias field (teal) at $\omega_{\mathrm{b}}$, a
target field (green) at the sum or difference frequency $\omega_{\mathrm{T}}$
is generated. When $\omega_{\mathrm{b}}$ is chosen so that the sum or
difference frequency is resonant with the NV resonance at $\omega_{0}$,
population oscillations occur between the two spin states. (c) The oscillation
frequency, called the target Rabi frequency $\Omega_{\mathrm{QFM}}$, provides
a measure of the signal field amplitude $\Omega_{\mathrm{sz}}$, given by Eq.
3. Spatial variations in $\Omega_{\mathrm{sz}}$ result in variations in
$\Omega_{\mathrm{QFM}}$ from pixel to pixel, allowing wide-field imaging of AC
signal amplitudes at arbitrary $\omega_{\mathrm{s}}$.

##  II Quantum Frequency Mixing

Frequency mixing is a process that uses a nonlinear response to generate new
frequencies from two periodic input signals. One example classical mixer, the
unbalanced single-diode mixer, uses the nonlinear relationship between input
voltage and response current to multiply two input signals, generating new
frequencies at the sum and difference of the input frequencies. In our work
using NV centers in diamond as a quantum frequency mixer, the NV quantum
system is nonlinear, and we exploit this to multiply two AC magnetic signals
within the quantum system itself Wang  _et al._ (2022). The type of mixing
employed here is heterodyne, and the net effect of our experiment is
implementation of a heterodyne receiver in a quantum system.

When a quantum system is driven with two fields at different frequencies, its
dynamics can be described as a quantum frequency mixer, where the effective
Hamiltonian derivation in the multi-mode Floquet picture allows us to identify
the dynamics to describe a frequency-mixer-based quantum sensing protocol Wang
_et al._ (2022). Here, we summarize the principle of quantum frequency mixing
and its application to this work.

To bring a signal of frequency $\omega_{\mathrm{s}}$ into resonance with an NV
transition frequency $\omega_{0}$, we apply a bias oscillating field with a
frequency $\omega_{\mathrm{b}}$. Here $\omega_{\mathrm{s}}$ and
$\omega_{\mathrm{b}}$ are analogous to the input radio-frequency (RF) and
local oscillator (LO) of a classical mixer. The effective signal generated by
the nonlinear process of the quantum system is at frequency
$\omega_{\mathrm{T}}$, which is analogous to the intermediate frequency (IF)
of a classical mixer.

This process can be described by Floquet theory Leskes  _et al._ (2010);
Scholz  _et al._ (2007). In Floquet analysis, the time-dependent Hamiltonian
in the Hilbert space is converted to a time-independent Hamiltonian in Floquet
space. The combined effect of two Fourier components
$H_{b}e^{i\omega_{\mathrm{b}}t}$ and $H_{s}e^{i\omega_{\mathrm{s}}t}$
associated with the input signal $\omega_{\mathrm{s}}$ and the bias AC field
$\omega_{\mathrm{b}}$, respectively, gives rise to a component at the target
frequency $\omega_{\mathrm{T}}$. Upon block-diagonalization of the Floquet
Hamiltonian, transforming back to the Hilbert space yields an effective
Hamiltonian with a frequency $\omega_{\mathrm{b}}-\omega_{\mathrm{s}}$ Wang
_et al._ (2022). With an appropriate choice of $\omega_{\mathrm{b}}$, the
effective target signal is now at the NV resonance frequency near 3 GHz, and
existing quantum sensing protocols such as Rabi magnetometry can be used to
probe the system.

The physics of a signal magnetic field projection along the NV axis, with
frequency $\omega_{\mathrm{s}}$, detected using a Rabi oscillation measurement
protocol (see Sec. III.3 below) can be described with the Hamiltonian:

$\displaystyle
H=\frac{\omega_{0}}{2}\sigma_{z}+\Omega_{\mathrm{b}}\cos\left(\omega_{\mathrm{b}}t+\phi_{\mathrm{b}}\right)\sigma_{x}$
$\displaystyle+~{}\Omega_{\mathrm{sz}}\cos\left(\omega_{\mathrm{s}}t+\phi_{\mathrm{s}}\right)\sigma_{z},$
(1)

where $\Omega_{\mathrm{b}}$ is the amplitude of the bias field and
$\Omega_{\mathrm{sz}}$ is the amplitude of the signal frequency projected
along the NV axis. In a rotating frame defined by
$U=e^{-i\left(\omega_{0}t/2\right)\sigma_{z}}$, the effective Hamiltonian
contains two oscillating frequencies. When one is resonant with the NV
transition, and the other is far-detuned and can be neglected in the spirit of
the rotating-wave approximation. We choose a bias frequency
$\omega_{\mathrm{b}}$ such that

$\displaystyle\omega_{\mathrm{b}}-\omega_{0}-\omega_{\mathrm{s}}=\delta_{z},$
(2)

where $\omega_{0}$ is the NV resonance frequency and $\delta_{z}$ is a small
detuning which we try to minimize in the experiment. Assuming
$\Omega_{\mathrm{sz}},\Omega_{\mathrm{b}},\ll\omega_{\mathrm{s}}$, the
amplitude of the effective target Rabi frequency is Wang  _et al._ (2022)

$\displaystyle\Omega_{\mathrm{QFM}}=\frac{\Omega_{\mathrm{sz}}\Omega_{\mathrm{b}}}{\omega_{\mathrm{s}}}.$
(3)

Table 1 lists the relevant parameters for this experiment.

\begin{overpic}[width=411.93767pt]{pulseSeqsV1d.pdf} \centering\put(2.0,25.0){{\Large a)}} \put(35.0,25.0){{\Large b)}} \put(68.0,25.0){{\Large c)}} \@add@centering\end{overpic} Figure 2:  (a) Conventional Rabi pulse sequence used to determine the on-resonance NV Rabi frequencies, with no current applied to the test structure. We initialize the NVs to the $\ket{0}$ state with a laser pulse, apply an on-resonance MW field of frequency $\omega_{0}$ for a variable duration, and read out the NV final state with a second laser pulse. (b) Ramsey pulse sequence used for DC magnetic imaging. We apply a DC current to the test structure, and we apply a $\pi/2$ MW pulse to create a superposition of the $\ket{0}$ and $\ket{+1}$ states and allow the system to evolve for a fixed time $\tau$. We then apply a second $\pi/2$ pulse, followed by a laser readout pulse. (c) Quantum frequency mixing (QFM) Rabi pulse sequence used to measure an AC magnetic field amplitude at frequency $\omega_{\mathrm{s}}$. This is similar to the conventional Rabi oscillation experiment, except with an applied bias frequency $\omega_{\mathrm{b}}$. Parameter | Description
---|---
$\omega_{0}$ | NV resonance frequency
$\Omega_{0}$ | On-resonance Rabi frequency
$\omega_{\mathrm{b}}$ | Bias frequency
$\Omega_{\mathrm{b}}$ | Bias amplitude
$\omega_{\mathrm{s}}$ | Signal frequency
$\Omega_{\mathrm{sz}}$ | Signal amplitude along $z$
$\omega_{\mathrm{T}}$ | QFM target frequency
$\Omega_{\mathrm{QFM}}$ | QFM Rabi frequency
Table 1:  A list of relevant parameters describing the QFM AC magnetometry
protocol.

##  III Experimental Methods

###  III.1 Microscope Apparatus

Figure 1a shows a schematic of the microscope apparatus. This drawing shows
the relevant coordinate systems: $\\{x,y,z\\}$ is the NV coordinate system
(with $z$ along the N-V axis, which is also the diamond [111] crystallographic
axis) and $\\{X,Y,Z\\}$ is the microscope coordinate system (with $Z$ out of
plane for the microscope image). A $5\times 5\times 0.5$ mm3 diamond sample
with a 10 $\upmu$m NV layer is illuminated with 1 W of 532 nm laser light,
which is passed through an acousto-optic modulator (AOM) switch and flat-top
beam shaper. We place the test structures on top of the NV layer, which is
coated with 10 nm of Ti metal and 50 nm of Au to prevent light leakage. The
diamond is attached to a printed circuit board (PCB) mount that applies a
uniform MW magnetic field along the $x$ direction (which is also the lab-frame
$X$ direction), and the NVs detect DC magnetic fields (with Ramsey
magnetometry Hart  _et al._ (2021)) and AC magnetic fields (with QFM) along
the $z$ axis. The mount also acts as a heat sink to dissipate heat deposited
by the laser. We applied a $\sim$1 mT bias magnetic field $B_{0}$ along the
$+z$ direction using permanent magnets, which modifies the NV ground-state
transition frequencies through the Zeeman effect. This bias field sets the
transition frequencies between NV ground-state sublevels to be $D\pm\gamma
B_{0}$, where $D\approx 2\pi\times 2870$ MHz is the zero-field splitting and
$\gamma\approx 2\pi\times 28$ GHz/T is the NV gyromagnetic ratio, and we
address the $\ket{0}\leftrightarrow\ket{+1}$ transition at $\omega_{0}\approx
2\pi\times 2890$ MHz. NV fluorescence light is imaged with an optical
microscope onto a digital focal-plane array (DFPA) camera, which has high
sensitivity, large dynamic range, 256$\times$256 pixel resolution, on-chip
processing, and fast data rates required for this work Schultz  _et al._
(2014). Laser, MW, test structure current, and camera trigger pulses are
programmed with a 1.2 GS/s arbitrary waveform generator (AWG). The camera is
triggered to acquire an NV fluorescence readout image across a range of MW
pulse durations, and we fit the resulting time traces in each pixel to extract
the Rabi frequency for QFM magnetic imaging (Fig. 1c).

###  III.2 Test Structures

We fabricated two conducting test structures on a glass microscope slide using
photolithography and electron-beam evaporation (10 nm Ti, 2 $\upmu$m Al, 10 nm
Ti, 150 nm Pt, 200 nm Au), a straight wire and an Archimedean spiral (both
with 50 $\upmu$m trace width), as shown in Fig. 3a and Fig. 4a. We attached
wire leads to the test structures using silver paint (yielding few-$\ohm$
total resistances), after which they were placed on the diamond sample. When
applying DC and AC currents using a function generator, we included a 50
$\ohm$ load resistor in series and measured the voltage across it to validate
the DC and AC current amplitudes through the test structures. To avoid the
possibility of shorting the test structures through the conductive film on the
NV layer, we placed a 7.5 $\upmu$m Kapton film between the slide and the
diamond.

###  III.3 Experimental Procedure

Figure 2 shows the pulse sequences used for magnetic imaging experiments.
First, we perform a conventional Rabi oscillation measurement Horsley  _et
al._ (2018) with no current in the test structure (Fig. 2a). To do this, we
initialize the NVs to the $\ket{0}$ state with a laser pulse, apply an on-
resonance MW field with a variable duration $\tau$, and read out the NV final-
states with a second laser pulse and the camera. Fitting the NV fluorescence
intensity as a function of $\tau$ for each pixel, this yields a map of the on-
resonance Rabi frequencies $\Omega_{0}$ (and implicitly the
$\Omega_{\mathrm{b}}$ bias amplitudes), and also the $\pi/2$ pulse duration
used for Ramsey magnetometry.

Next, we apply a DC current to the test structure and perform Ramsey
magnetometry (Fig. 2b). This ensures that the test structure is centered in
the camera field of view and allows us to confirm that the test structure is
flat on the diamond surface. We use double-quantum Ramsey spectroscopy for
improved magnetic sensitivity and to avoid frequency shifts due to temperature
and strain inhomogeneity Hart  _et al._ (2021); Tang  _et al._ (2023). After
laser initialization, the first $\pi/2$-pulse places the NVs in a
superposition on the equator of the Bloch sphere, and they accumulate a phase
$\phi=2\gamma\Delta B\tau$, where $\Delta B$ is the field from the test
structure and $\tau$ is the phase accumulation time, then apply a second
$\pi/2$-pulse and a readout laser pulse. After optimizing the Ramsey imaging
measurement parameters, we image the DC magnetic field map from the test
structure. This provides a DC magnetic image to compare with subsequent AC QFM
magnetic images.

Finally, we apply an AC current at frequency $\omega_{\mathrm{s}}$ to the test
structure, a MW field at $\omega_{\mathrm{b}}=\omega_{0}+\omega_{\mathrm{s}}$,
and measure QFM Rabi oscillations (Fig. 2c). The QFM experiment protocol is
nearly identical to the Rabi oscillation experiment protocol, with the main
difference being that we address the NVs with frequency $\omega_{\mathrm{b}}$
instead of $\omega_{0}$. Fitting the fluorescence time-trace data to extract
$\Omega_{\mathrm{QFM}}$, we can then use Eq. 3 to get the AC magnetic field
amplitude for each pixel ($B_{\mathrm{applied}}=\Omega_{\mathrm{sz}}/\gamma$).

To improve the long-term stability and pixel-to-pixel noise floor of our
apparatus, each of the above experiments modulates the signal amplitude
between alternate camera exposures (either on/off for Rabi and QFM or
positive/negative DC current for Ramsey) Glenn  _et al._ (2017); Horsley  _et
al._ (2018); Tang  _et al._ (2023).

\begin{overpic}[width=411.93767pt]{Straight_Wire_Figure_3_PKeditsStar.pdf}
\centering\put(0.0,43.0){{\Large a)}} \put(43.0,43.0){{\Large b)}}
\put(72.0,43.0){{\Large c)}} \put(0.0,25.0){{\Large d)}}
\@add@centering\end{overpic} Figure 3:  (a) Photograph of the straight-wire
test structure. (b-c) Two QFM imaging examples for an AC current at
$\omega_{s}=2\pi\times 4$ MHz and $2\pi\times 35$ MHz (top). The Rabi
oscillations and the extracted Rabi frequency for the center pixel (white
cross) are also included (bottom). As expected, $\Omega_{\mathrm{QFM}}$ at a
specified pixel decreases as $\omega_{\mathrm{s}}$ increases. (d)
$\Omega_{\mathrm{QFM}}$ for the center and right (green star) pixels as a
function of $\omega_{\mathrm{s}}$ and fit to an $a/\omega_{\mathrm{s}}$ curve,
demonstrating the inverse relationship described by Eq. 3. The pixel marked
with a blue $\times$ is discussed in Sec. 5.

##  IV Results

###  IV.1 Validation with a Straight Wire Test Structure

Figure 3a shows a photograph of the straight-wire test structure. Following
the measurement procedure described above, we first imaged Rabi oscillations
in the field of view with no current in the wire, then measured a 1 mA DC
current through the wire to check its position in the field of view. The DC
magnetic image yields a conservative estimate that the standoff distance
between the NV layer and the wire is $\sim$50 $\upmu$m.

For the QFM magnetic imaging measurement, we applied an 80 mA peak-amplitude
AC current through the test structure and measured the projection of the
created magnetic field along the $z$-axis. The acquisition time for each
$\omega_{\mathrm{s}}$ image was $\sim$5 minutes, roughly half of which was
dead time due to camera transfer lag sup . The Rabi frequency
$\Omega_{\mathrm{QFM}}$ in each pixel was extracted using the fit function
$C(\tau)=A\cos\left(\Omega_{\mathrm{QFM}}\tau+\phi\right)e^{-{\tau}/{T_{\mathrm{Rabi}}}}+C_{0}$
where $C$ is the normalized fluorescence intensity sup , $A$ is the Rabi
contrast amplitude, $\phi$ is a phase offset, $T_{\mathrm{Rabi}}$ is the Rabi
coherence time, and $C_{0}$ is an offset. Figure 3b-c maps the
$\Omega_{\mathrm{QFM}}$ values for $\omega_{\mathrm{s}}=2\pi\times 4$ MHz and
35 MHz, including the QFM Rabi oscillations for the center (white cross)
pixel. The black stripe corresponding to $\Omega_{\mathrm{QFM}}$ near 0 is
where the wire magnetic field is perpendicular to the $z$ direction.

Figure 3d plots $\Omega_{\mathrm{QFM}}$ as a function of $\omega_{\mathrm{s}}$
for two pixels. As expected, $\Omega_{\mathrm{QFM}}$ decreases as
$\omega_{\mathrm{s}}$ increases. These data are fitted with the fit function
$\Omega_{\mathrm{QFM}}=a/\omega_{\mathrm{s}}$ (where $a$ is a free parameter),
and are consistent with Eq. 3. For both pixels, $\Omega_{\mathrm{b}}$ was
experimentally determined to be $\sim$2$\pi\times 1.0$ MHz by measuring the
on-resonance Rabi oscillation frequencies (Fig. 2a) and assuming the MW
amplitude is the same at $\omega_{\mathrm{b}}$. $\Omega_{\mathrm{sz}}$ is
estimated to be $\sim$2$\pi\times 1.8$ MHz and $\sim$3.8 MHz for the center
and right (green star) pixels, respectively, based on the simulated magnetic
field of a 50 $\upmu$m ribbon at $\sim$50 $\upmu$m standoff distance. The
fitted $a$ coefficients are $(2\pi)^{2}\times\\{1.3,2.3\\}$ MHz2 for the
center and right pixels, and given $\Omega_{\mathrm{b}}\approx 2\pi\times 1.0$
MHz for both pixels, this yields $\Omega_{\mathrm{sz}}\approx
2\pi\times\\{1.3,2.3\\}$ MHz. These extracted values are reasonably consistent
with the estimated $\Omega_{\mathrm{sz}}\approx 2\pi\times\\{1.8,3.8\\}$ MHz.

Previously, QFM Rabi magnetometry was demonstrated for a bulk (single-pixel)
NV ensemble Wang  _et al._ (2022). The above straight-wire test structure
demonstration shows that this technique can also be applied to an NV magnetic
imaging apparatus, and validates the expected frequency-scaling law predicted
in Eq 3.

\begin{overpic}[width=411.93767pt]{Spirals_Figure_4_PKedits2a.pdf}
\centering\put(1.0,48.0){{\Large b)}} \put(1.0,21.0){{\Large a)}}
\put(33.0,52.5){{\Large c)}} \put(66.5,52.5){{\Large d)}}
\@add@centering\end{overpic} Figure 4:  (a) Photograph of the Archimedean
spiral test structure. (b) DC magnetic image of the spiral with 1 mA signal
current measured using a Ramsey sequence. (c-d) Two examples of QFM imaging
(top), with the extracted $\Omega_{\mathrm{QFM}}$ for an example pixel
(bottom). $\Omega_{\mathrm{QFM}}$ for this pixel decreases by a factor
$1/\omega_{\mathrm{s}}$ as the input signal frequency increases from
$\omega_{s}=2\pi\times 5$ MHz to $2\pi\times 50$ MHz.

###  IV.2 Archimedean Spiral Test Structure

Having demonstrated that the QFM imaging method works as expected for a simple
test case, we now apply it to a more complicated test structure, an
Archimedean spiral (Fig. 4a). Unlike the straight-wire, the spiral test
structure has currents (magnetic fields) pointing in a wide range of
directions, leading to a nontrivial magnetic field map. The test structure
layout is similar to a planar spiral inductor, and generalizing the
measurement technique to spiral inductors and other similar passive RF circuit
elements is straightforward. Furthermore, since the spiral is a passive
circuit element, we control the frequency and amplitude of current passing
through it. By contrast, when measuring the magnetic fields from a more
sophisticated IC, these parameters are likely uncontrollable (and are often
unknown).

Figures 4b-c show QFM magnetic images for $\omega_{\mathrm{s}}=2\pi\times 5$
MHz and 50 MHz, with 80 mA peak-amplitude current. Rabi oscillation data and
fits for an example pixel are included, which also show the
$\Omega_{\mathrm{QFM}}\propto 1/\omega_{\mathrm{s}}$ dependence expected from
Eq. 3. Figure 4d includes a DC magnetic image for comparison, acquired using
Ramsey magnetometry and showing a similar magnetic field amplitude structure.
Note that the DC experiment measures both positive and negative magnetic field
values while the AC experiments measure amplitudes, though the absolute value
of the DC image is consistent with the AC images.

###  IV.3 Large-$\Omega_{\mathrm{sz}}$ Small-$\omega_{\mathrm{s}}$ Regimes

In Fig. 3 and Fig. 4, we used Eq. 3 to confirm that the measured QFM Rabi
frequencies $\Omega_{\mathrm{QFM}}$ were behaving as expected. However, this
expression is valid when

$\Omega_{\mathrm{b}},\Omega_{\mathrm{sz}}\ll\omega_{\mathrm{s}},$ (4)

which is not necessarily true for all experimental conditions. For large
$\Omega_{\mathrm{sz}}$ and small $\omega_{\mathrm{s}}$,
$\Omega_{\mathrm{QFM}}$ is more accurately described by

$\Omega_{\mathrm{QFM}}=\Omega_{\mathrm{b}}J_{1}\left(\frac{2\Omega_{\mathrm{sz}}}{\omega_{\mathrm{s}}}\right),$
(5)

where $J_{1}$ is the Bessel function of the first kind sup .

Figure 5a plots $\Omega_{\mathrm{QFM}}$ as a function of
$\omega_{\mathrm{\mathrm{s}}}$ for the left (blue $\times$) pixel in Fig. 3,
showing the necessity of Eq. 5. This pixel has a larger
$\Omega_{\mathrm{sz}}=2\pi\times 2.5$ MHz amplitude than the right pixel
because of its proximity to the wire, which we estimated using the
$\omega_{\mathrm{s}}\geq 2\pi\times 30$ MHz tail and Eq. 3. We also measured
its $\Omega_{\mathrm{QFM}}$ frequencies down to a lower-bound
$\omega_{s}=2\pi\times 2$ MHz. These $\Omega_{\mathrm{QFM}}$ frequencies were
measured with the same 80 mA peak-amplitude AC current as in Fig. 3. Figure 5a
verifies that Eq. 5 (solid black line) is a more appropriate description of
the $\Omega_{\mathrm{QFM}}$ behavior for small $\omega_{\mathrm{s}}$ than Eq.
3 (blue dashed line).

Figure 5b shows $\Omega_{\mathrm{QFM}}$ as a function of
$\Omega_{\mathrm{sz}}$ (i.e. applied current) for the center pixel previously
examined in Fig. 3, using $\omega_{\mathrm{s}}=2\pi\times 5$ MHz. For
sufficiently large $\Omega_{\mathrm{sz}}$, the measured
$\Omega_{\mathrm{QFM}}$ deviates from the linear dependence predicted by Eq. 3
(blue dashed line), and is more accurately described by Eq. 5 (solid black
line).

\begin{overpic}[width=173.44534pt]{Broken_Assumptions_Final.pdf}
\centering\@add@centering\end{overpic} Figure 5: (a) $\Omega_{\mathrm{QFM}}$
as a function of $\omega_{\mathrm{s}}$ with a fixed $\Omega_{\mathrm{sz}}$ for
the left pixel in Fig. 3. The dashed pink line shows the expected
$\Omega_{\mathrm{QFM}}$ predicted by Eq. 3. The black line shows the expected
$\Omega_{\mathrm{QFM}}$ from Eq. 5, and the blue dots are the experimental
data. (b) $\Omega_{\mathrm{QFM}}$ as a function of $\Omega_{\mathrm{sz}}$ at a
fixed $\omega_{\mathrm{s}}$ for the center pixel in Fig. 3.

##  V Discussion

Our analysis shows the dynamic range of the QFM AC magnetometry and imaging
method. In the large-$\Omega_{\mathrm{sz}}$ and small-$\omega_{\mathrm{s}}$
regime, conversion from $\Omega_{\mathrm{QFM}}$ to $\Omega_{\mathrm{sz}}$
(which is straightforward using Eq. 3) requires inverting Eq. 5.
$\Omega_{\mathrm{QFM}}$ does not scale monotonically with
$\omega_{\mathrm{s}}$ and $\Omega_{\mathrm{sz}}$, and near a maximum in
$\Omega_{\mathrm{QFM}}$, $\Omega_{\mathrm{QFM}}$ is insensitive to
$\Omega_{\mathrm{sz}}$. These details may complicate the QFM AC magnetometry
method when applying it to more general experimental situations.

Equation 3 implies that one can increase $\Omega_{\mathrm{b}}$ to compensate
for a small $\Omega_{\mathrm{sz}}$, since a slow $\Omega_{\mathrm{QFM}}$ is
difficult to measure if its period is slow compared to the coherence time
$T_{\mathrm{Rabi}}$. However, this may also require additional analysis to
interpret the measured $\Omega_{\mathrm{QFM}}$ values sup . Alternatively, in
a regime where $\Omega_{\mathrm{QFM}}$ is slow compared to the
$T_{\mathrm{Rabi}}$ coherence time, one can apply an on-resonance
$\pi/2$-pulse at frequency $\omega_{0}$ before the QFM Rabi pulse at frequency
$\omega_{\mathrm{b}}$, making the NV fluorescence intensity proportional to
$\sin\left(\Omega_{\mathrm{QFM}}\tau+\phi\right)e^{-{\tau}/{T_{\mathrm{Rabi}}}}$
instead of
$\cos\left(\Omega_{\mathrm{QFM}}\tau+\phi\right)e^{-{\tau}/{T_{\mathrm{Rabi}}}}$
Alsid  _et al._ (2023). For small $\Omega_{\mathrm{QFM}}$, the fluorescence
change will be proportional to $\Omega_{\mathrm{QFM}}\tau$ (sometimes called
“sine magnetometry”) instead of $(\Omega_{\mathrm{QFM}}\tau)^{2}$ (“cosine
magnetometry”) for improved sensitivity and minimum-detectable
$\Omega_{\mathrm{QFM}}$, though implementing this also requires careful
calibration Degen  _et al._ (2017).

##  VI Conclusion and outlook

In this work we demonstrated an NV wide-field magnetic imaging apparatus
capable of measuring AC magnetic fields using QFM up to 70 MHz. We implemented
a Rabi-based magnetometry scheme with linearly-polarized signal and bias
fields to measure a signal-field component oscillating along NV $z$-axis. The
imager itself overcomes several of the common challenges for NV pulsed
magnetic imaging, achieving sufficient homogeneity to realize a
$\sim$1.5$\times$1.5 mm2 FOV. After validating that the technique works with a
simple straight-wire test structure, we then imaged an Archimedean spiral test
structure to show that this technique can be generalized to spiral inductors
and other passive RF circuit elements. Additional extensions could use spin-
locking and pulsed dynamical decoupling methods (such as CPMG), or use
circularly-polarized signal and bias fields to measure a signal field
oscillating transverse to the NV $z$-axis Wang  _et al._ (2022).

Building on previous demonstrations using NV magnetic imaging of DC currents
for electronics interrogation and failure analysis (FA) Kehayias  _et al._
(2023, 2024), our QFM magnetic imaging approach could also be applied to FA,
fabrication process monitoring, and non-contact wafer testing. This would
generalize the technique to include AC magnetic field sources at arbitrary
frequencies. One could measure the frequencies and amplitudes of currents in
high-speed test structures (e.g. ring oscillators Turner  _et al._ (2020)),
which are included on dice or wafers for fabrication process troubleshooting.
To achieve this, one could use QFM magnetic imaging to implement a spectrum-
analyzer imager (a form of hyperspectral imaging), generating image for a
range of $\omega_{\mathrm{s}}$ values Sutherlin (2015); Hagen and Kudenov
(2013). The resulting hyperspectral data cube could then be analyzed for PSA
applications, such as counterfeit detection, component aging, and fabrication
process variation over time or across a wafer Tangyunyong  _et al._ (2017);
spe . This could also be useful for validating how internal current paths
within a device change as a function of frequency and impedance (for example,
high-frequency capacitive shorting).

As a further extension, our QFM magnetic imaging approach could be adapted for
space-domain reflectometry for open-circuit fault localization. This was
previously demonstrated with a scanning RF SQUID sensor Gaudestad  _et al._
(2012), but could potentially achieve improved spatial resolution and
sensitivity with an NV magnetic microscopy apparatus. Finally, this technique
could be used to evaluate the RF and MW images of hardware used for quantum
computing, such as ion trap chips and superconducting qubits. We note the
emerging synergy between quantum technologies, and here one quantum technology
(quantum sensing and imaging) would troubleshoot hardware of another
technology (quantum computing).

##  VII Acknowledgements

We thank M. Ricci for help with test structure fabrication and packaging, C.
Reed for help with preparing Fig. 1a, and M. Jordan, C. Nordquist, and D.
Krawczyk for useful discussions.

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