# Roadmap on Nanoscale Magnetic Resonance Imaging

Guest editors: Raffi Budakian, Amit Finkler & Alex Eichler

(December 2023)

###### Abstract

The field of nanoscale magnetic resonance imaging (NanoMRI) was started 30
years ago. It was motivated by the desire to image single molecules and
molecular assemblies, such as proteins and virus particles, with near-atomic
spatial resolution and on a length scale of 100 nm. The realization of this
goal requires the development of spin detection techniques that are many
orders of magnitude more sensitive than conventional MRI, capable of detecting
and controlling nanoscale ensembles of spins. Over the years, a number of
different technical approaches to NanoMRI have emerged, each possessing a
distinct set of capabilities for basic and applied areas of science. The goal
of this roadmap article is to report the current state of the art in NanoMRI
technologies, outline the areas where they are poised to have impact, identify
the challenges that lie ahead, and propose methods to meet these challenges.
This roadmap also shows how developments in NanoMRI techniques can lead to
breakthroughs in emerging quantum science and technology applications.

## Introduction

Magnetic resonance imaging (MRI) has revolutionized medical diagnosis due to
its unparalleled combination of elemental selectivity, noninvasive detection,
and three-dimensional imaging capabilities. Translating these same advantages
to the nanoscale holds immense potential for structural biology and quantum
device design, but achieving this feat presents a formidable challenge: the
signal strength in MRI typically decreases with the cube of the object’s
length, rendering conventional techniques ineffective at the nanoscale. To
overcome this hurdle, novel experimental platforms and theoretical methods are
being actively pursued. This roadmap provides an overview of the current state
of the art and future directions in this exciting field at the intersection of
physics, chemistry, and materials science.

The concept of NanoMRI can be traced back to John A. Sidles, a theoretical
physicist working in the Department of Orthopedics at the University of
Seattle in the early 1990s. Recognizing the need for a method to investigate
the structure of biological molecules on the atomic scale, Sidles drew
inspiration from the invention of the atomic force microscope a few years
earlier. He proposed a technique based on force detection [1]. Experimental
proof-of-principle studies soon followed in the group of Dan Rugar at IBM
Almaden [2], sparking interest beyond the nanomechanics community.

NanoMRI has evolved over the years to encompass a diverse array of
experimental techniques, including magnetic probes based on nitrogen-vacancy
color centers in diamond, magnetic scanning tunneling microscopy, and
microscopic superconducting circuits. Each approach offers unique advantages
and caters to specific applications. This diversity has fostered a vibrant and
expanding NanoMRI community, whose original focus on structural biology has
broadened to encompass new frontiers such as nanoscale nuclear magnetic
resonance, quantum data storage, and atomic-scale surface investigations.

The roadmap article is organized as follows: section 1 reviews magnetic
resonance force microscopy and its offshoots, with perspectives to the future.
Section 2 discusses color centers (the NV among them) as an emerging technique
to explore magnetic resonance on these very small length scales. Section 3
presents scanning tunneling microscopy combined with electron spin resonance,
a relatively new entrant to the field. In section 4, a demonstration of the
power of superconducting qubits is shown with an outlook to what’s next, and
finally, section 5 looks at various control schemes and quantum-enhanced
techniques that bring about significant improvements to spin detection at the
nanoscale. Together, all five sections outline the current state-of-the-art,
future challenges and opportunities in NanoMRI.

###### Contents

  1. 1 Force Detection
    1. 1.1 Magnetic resonance force microscopy: from cantilevers to nanowires
    2. 1.2 Force-detected NanoMRI with high-Q resonators
    3. 1.3 Magnetic resonance force microscopy using current-generated gradients
    4. 1.4 Nanoscale ferromagnetic resonance imaging using high-sensitivity mechanical force detection
    5. 1.5 Quantum sensing in the negative mass reference frame
  2. 2 NV
    1. 2.1 Towards imaging complex spin systems and molecules with spin-based quantum sensors
    2. 2.2 Micron-scale NV-NMR
    3. 2.3 Nanoscale imaging with NV centers
    4. 2.4 Quantum control for nanoscale NMR: status, challenges and frontiers
    5. 2.5 Nano MRI using spin quantum sensors
  3. 3 STM-ESR
    1. 3.1 An atomic spin sensor on surfaces
    2. 3.2 Sensing the electron spin resonance at the atomic scale
  4. 4 Superconducting Resonators
    1. 4.1 Nanoscale magnetic resonance with superconducting circuits
  5. 5 Quantum control schemes for enhanced spin detection
    1. 5.1 Composite pulse design for ESR/NMR
    2. 5.2 Statistical learning for nanoscale magnetic resonance

##  1 Force Detection

###  1.1 Magnetic resonance force microscopy: from cantilevers to nanowires

Martino Poggio
Department of Physics & Swiss Nanoscience Institute, University of Basel, 4056
Basel, Switzerland
<EMAIL_ADDRESS>
#### Status

The invention of the scanning tunneling microscope (STM) in the early 1980s
and its use in making the first images of individual surface atoms cleared the
way for a new type of microscopy, in which a sharp tip is scanned over a
sample surface. By using a variety of probes to measure different quantities,
nanometer-scale scanning probe microscopy was then extended to a wide variety
of contrasts and surfaces, beyond measuring tunneling current on conducting
materials. Atomic force microscopy (AFM) is perhaps the most widely-used of
these techniques, with applications spanning across physics, chemistry,
biology, and materials science. Functionalization of AFM probes led to further
refinements, including the possibility to measure electric and magnetic fields
via, for example, electrostatic or magnetic force microscopy (MFM).

The application of these concepts to magnetic resonance imaging (MRI) is known
as magnetic resonance force microscopy (MRFM) [3]. Proposed by Sidles [1, 4]
in the early 1990s as a means to image individual molecules atom-by-atom, MRFM
relies on the measurement of the minute force between a magnetic scanning
probe and resonantly flipping sample spins. Spatial maps of this force can be
transformed into 3D images of spin density with spatial resolution orders of
magnitude better than conventional inductively detected MRI, albeit not yet at
the atomic scale. The reason for this advantage – and the key to further
improvement – is the exquisite sensitivity of the force transducers that are
used.

Today, these force transducers are the direct descendants of the metal foils
and Si cantilevers used in early AFM. They are designed to optimize
sensitivity to small forces by minimizing thermomechanical noise, which, at a
given temperature $T$, depends on mechanical dissipation $\Gamma$. Since
smaller mechanical transducers tend to have less dissipation [5], there has
been a notable trend toward ever tinier transducers, ranging from ultra-thin
cantilevers to nanowires (NWs) and even nanotubes. Smaller transducers also
tend to have higher mechanical resonance frequencies [5], decoupling them from
common sources of noise, such as external vibrations or the force noise caused
by electronic fluctuators as a probe approaches a surface (non-contact
friction).

Remarkable progress has been made in improving force sensitivity, which in the
classical limit is set by the fluctuation-dissipation theorem and given by:

$F_{\mathrm{min}}=\sqrt{4k_{B}T\Gamma},$

where $k_{B}$ is the Boltzmann constant [6]. Starting in 2004, ultra-thin Si
and diamond cantilevers designed for MRFM achieved around $F_{\mathrm{min}}=1$
aN/(Hz)1/2 sensitivity [7], followed by ‘nanoladder’ levers of the same
materials with under 200 zN/(Hz)1/2 [8], and doubly clamped carbon nanotube
(CNT) resonators with 1 zN/(Hz)1/2 [9]. Doubly clamped transducers like CNTs,
strings, and membranes, however, are not easily amenable to the protruding tip
geometry of SPM. For this reason, the demonstration of NW cantilevers with 1
aN/Hz1/2 sensitivity has proven particularly promising [10]. When arranged in
the pendulum geometry – that is with their long axes perpendicular to the
sample surface – NW cantilevers are ideal scanning probes that avoid snapping
into contact when close to a surface. By virtue of their high frequencies and
small size, NWs experience orders of magnitude lower non-contact friction than
larger cantilevers with lower resonance frequencies, making their near-surface
force sensitivities much better [10]. It is no accident that since the 2009
demonstration of MRI with sub-10-nm spatial resolution via cantilever MRFM
[11], MRI with similar resolution has only been achieved in 2013 using a NW
transducer [12].

Figure 1: Reducing transducer size. Mechanical sensors made by ‘top-down’
fabrication methods include (a) diamond ‘nanoladder’ cantilevers [8] (scale
bar 20 $\upmu$m) and (b) undoped Si cantilevers [7] Smaller transducers are
typically made by ‘bottom-up’ or self-assembly techniques, such as (c) SiC NWs
[13], (d) Si NWs [10, 12], (scale bar 10 $\upmu$m), (e) GaAs NWs (scale bar 10
$\upmu$m) [14], and (f) CNTs [15].

#### Current and Future Challenges

Since the first demonstration of MRFM in the early 1990s, sensitivity has
improved from the equivalent of $10^{9}$ to about $10^{2}$ 1H magnetic
moments. In order to detect individual nuclear spins and to image molecules
with atomic resolution, at least another 2 orders of magnitude are required.
While such a boost could come from refinements to a number of components,
including the magnetic tip or the measurement protocol, improvement of the
mechanical transducer, i.e., a reduction of mechanical dissipation $\Gamma$ is
likely to be the most impactful approach.

The central challenge will be to develop ever tinier cantilever transducers,
using NWs, CNTs, or similar structures, with high mechanical quality, whose
displacement can be measured and controlled. This poses a number of technical
obstacles. First, while displacement detection of conventional AFM cantilevers
and micron-scale transducers can be achieved via the deflection of a reflected
laser beam, such schemes are poorly suited to transducers with sub-wavelength
dimensions. Without a scheme sensitive enough to resolve a transducer’s
thermal motion, forces cannot be detected down to its ultimate sensitivity
limits. Second, as a transducer shrinks in size, its linear dynamic range
decreases. Strategies for dealing with this reduction, for maintaining easy
control of the transducer’s motion, or for exploiting nonlinear effects in
force measurements have therefore to be developed. Third, any remaining
sources of mechanical dissipation have to be addressed and reduced as much as
possible. The most prominent sources include dissipation related to the
clamping of cantilever transducer, i.e., the path through which energy is lost
to the support structure, and dissipation related to the transducer’s surface.
As a mechanical resonator is shrunk down, its surface-to-volume ratio
increases, resulting in a large contribution of surface defects or
contamination to overall mechanical dissipation. In fact, for a 100-nm-thick
Si cantilever, it was shown that 85% of the mechanical dissipation can be
traced to the 2-nm thick native oxide layer on its surface [16].

Further challenges are related to the functionalization of increasingly small
cantilever transducers. As recently as 2004, the magnetic tip for the MRFM
experiment demonstrating the detection of a single electron spin was broken
off of a macroscopic magnet and glued to the cantilever using
micromanipulators under an optical microscope [17]. The tip was subsequently
milled using a focused ion beam. Such techniques are not likely to scale for
the placement of magnets or samples on NW and CNT devices, requiring new
strategies.

#### Advances in Science and Technology to Meet Challenges

A number of advances have to be made to meet the aforementioned challenges.
For the detection of mechanical displacement, a pair of optical techniques
based on interferometry or scattering have been shown to work with NWs and to
be compatible with SPM applications [18, 19]. Despite the sub-wavelength
diameter of the NWs, in both cases, sufficient light is scattered to
sensitively detect displacement. Whether such schemes will continue to work
with smaller NW and CNT cantilevers, remains an open question. A technique
based on detecting the inelastically scattered electrons from a focused
electron beam has also been demonstrated to work on a CNT transducer in an SPM
setup [20]. Another very promising technique, combining a conventional AFM
with a NW or CNT transducer such that their modes hybridize, allows the
mechanical displacement to be detected with the ease of an AFM and
sensitivities to approach those only possible with much smaller transducers
[21].

Despite the potential for force amplification, nonlinear motion has so far not
been exploited in experiments on force sensing with NW or CNT cantilevers.
Work in this area and on making use of the various mechanical modes of NW and
CNT cantilevers, such as in vectorial force detection [13], could yield
important advances. Excitation, feedback, and control of cantilever
displacement has improved considerably in recent years with the proliferation
of all-optical mechanical driving via photothermal actuation. Compared to
excitation via piezoelectric actuators, this technique allows for the
isolation of the transducer’s mechanical modes without interference from
spurious modes in the microscope. This, in turn, allows for the application of
feedback and other control required to suppress unwanted nonlinearities.

To reduce mechanical dissipation in NW or CNT cantilevers, losses related to
be clamping should be addressed. Improvements may come from implementing
phononic band-gap supports. These schemes have already improved clamping
losses in doubly clamped strings and membranes, but have yet to be implemented
on cantilever scanning probes. Surface-related losses could be mitigated by
the removal of amorphous layers and/or surface passivation, as was already
demonstrated in larger Si cantilevers. A complete quenching of surface losses
may only be possible by operating transducers under ultra-high vacuum
conditions, as in STM where even atomically thin contaminant layers can be
removed.

Strategies for the functionalization of the cantilever tip with a nanometer-
scale magnet are also being developed. In one case, a NW transducer grown by
molecular beam epitaxy was terminated, during growth, with a magnetic MnAs tip
[14]. In subsequent NW MFM experiments, this tip proved to have excellent
properties. Focused ion or electron beam induced deposition has also been used
to deposit nanometer-scale Co directly on the tips of NWs. The magnetic
quality of these tips is yet to be determined, but initial results appear
promising [22].

#### Concluding Remarks

In addition to potentially improving the resolution of mechanically detected
MRI, the development of NW cantilevers designed for ultra-sensitive force
detection may have impact in other applications. Unlike conventional AFM
cantilevers that are designed to match the high stiffness required for atomic-
resolution AFM, these exceptionally compliant probes allow for sensitive
mapping of tip-sample force fields and energy losses. Since energy dissipation
plays a central role in the breakdown of topological protection, it may
provide important contrast in spatial studies of strongly correlated states in
2D vdW materials. In fact, dissipation contrast using cantilevers in the
pendulum geometry has already been used to observe superconducting [23] and
bulk structural phase transitions [24], as well as the local density of
states. In addition, the combination of high force sensitivity, high spatial
resolution, and low invasiveness of magnetic NW probes also has the potential
to expand the applicability of MFM to much more subtle contrast than is
conventionally accessible. Sensitive MFM could be particularly useful for
measuring the spatial localization of flowing currents, as in edge states, and
for the determination of length scales such as magnetic domain sizes and
coherence lengths [25]. In particular, NW transducers should allow MFM imaging
of the tiny current densities and magnetization associated with many emerging
2D materials and their heterostructures.

#### Acknowledgements

MP acknowledges support from the Canton Aargau.

###  1.2 Force-detected NanoMRI with high-Q resonators

Alexander Eichler and Christian L. Degen
Institute for Solid State Physics, ETH Zurich, Otto-Stern-Weg 1, 8093 Zurich
<EMAIL_ADDRESS><EMAIL_ADDRESS>
#### Status

Over the past decade, we witnessed a leap in the ability to design and
fabricate nanomechanical systems from silicon nitride with quality factors
beyond $Q=10^{8}$. The main innovation on this path was a technique commonly
referred to as “dissipation dilution”: applying tensile strain to a mechanical
resonator increases its resonance frequency $f_{0}$ without affecting the
dissipation rate $\Gamma=2\pi f_{0}/Q$, hence increasing $Q$ [26, 27]. Later,
the implementation of mode shape engineering led to further significant
improvements in $\Gamma$ and $Q$ [28, 29, 30, 31] (see [32] for a more
comprehensive list of references). These developments were primarily fuelled
by the need for quantum devices with long thermal coherence times and high
$Qf_{0}$ products, but the resulting devices also open up exciting
perspectives in force sensing.

The sensitivity of force sensors is typically limited by their
thermomechanical force noise power spectral density $S_{f}=4k_{B}Tm\Gamma$,
where $k_{B}$ is Boltzmann’s constant, $T$ is the temperature, and $m$ is the
effective resonator mass. In most cases, $S_{f}$ is reduced by designing
resonators with low masses and resonance frequencies, resulting in long and
floppy devices (see section 1.1). The new generation of strained silicon
nitride force sensors has the potential to reach similar or better
performances while operating at significantly higher frequencies. This change
comes with several advantages: first, it is expected that detrimental $1/f$
fluctuation effects, such as non-contact friction, have a smaller impact at
higher frequencies, preserving the resonator performance in the presence of
nearby surfaces [33]. Second, static bending will be reduced due to the higher
spring constant $k=4m\pi^{2}f_{0}^{2}$, making the system more stable towards
bending and mechanical instability [34]. Third, silicon nitride resonators can
be integrated into cavity optomechanical systems to achieve quantum-limited
displacement readout and mode cooling [35]. In total, optomechanical force
sensors with high quality factors could therefore become a valuable resource
for ultrasensitive scanning force microscopy, and in particular for nanoscale
MRI [1, 36].

The design of nanomechanical resonators remains an active research area. The
community is only just beginning to study the behavior of silicon nitride
optomechanical resonators at temperatures below 100 mK, and to discover the
opportunities offered by on-chip integrated devices. Further improvements of
the figures of merit and more sophisticated measurement protocols will support
the development of spin detection and other state-of-the-art scanning force
applications.

Figure 2: Demonstration of scanning force microscopy with a high-Q
optomechanical membrane system. (a) Artistic rendering of a scanning force
microscopy setup using a membrane resonator as force sensor. Black holes in
the membrane surface establish a phononic bandgap that bestows high quality
factors upon two drum resonators (1 and 2). Samples are placed on one of the
resonators and interact with a sharp, static scanning tip. The resonators form
normal modes through evanescent coupling, such that the sample-tip
interactions taking place on resonator 2 can be monitored with a laser
interferometer on resonator 1. (b) Topography map of gold nanoparticles and
tobacco mosaic viruses on the membrane surface measured with the scanning
force microscope. Reprinted graphs with permission from [37]. Copyright 2022
by the American Physical Society.

#### Current and Future Challenges

While strained silicon nitride resonators exhibit excellent nominal force
sensitivities, their use in real applications is not straightforward. Some of
the issues that we foresee are connected to the requirement to work inside a
dilution refrigerator or a different cryogenic apparatus to reduce $S_{f}$ as
much as possible. The most pressing experimental challenges are:

(i) in order to enforce strain along the axis of a beam or in the plane of a
membrane, the device must be clamped on opposite ends, making it impossible to
employ the typical singly-clamped cantilever geometry that is found in almost
all scanning force instruments. This leads to the question of how the scanning
tip, the sample, and the readout mechanism can be arranged inside a scanning
force microscope. Such geometrical incompatibilities become more pronounced
when placing the resonator inside an optical cavity, such as is nowadays
routinely done for membranes or trampoline devices to enhance the readout
efficiency.

(ii) It remains to be shown how far the force noise of a sensor in a realistic
application coincides with the values found for bare, isolated resonators. In
particular, non-contact friction and heating due to absorption of light will
at some point become noticeable and limit the sensitivity. It is likely that
the final performance that can be achieved in sensing instruments will result
from a compromise between optical heating, detector noise, and quantum
backaction. Further research is necessary to evaluate e.g. the optical
absorption and the heat conductivity of silicon nitride at dilution
refrigerator temperatures.

(iii) For the special case of nanoscale MRI, there arises also a challenge
related to spin sensing protocols. Strained silicon nitride resonators with
resonance frequencies in the MHz regime cannot be coupled to nuclear spins in
the same way as is typically done with kHz cantilevers. Specifically,
established nuclear spin inversion protocols cannot be applied with MHz
repetition rates due to the limited amplitude of the AC magnetic fields
available in dilution refrigerators. Note that for electron spins, the issue
is easier to overcome due to the much higher gyromagnetic ratio, which allows
for MHz Rabi oscillations with relatively small AC field amplitudes.

#### Advances in Science and Technology to Meet Challenges

Significant progress towards nanoscale MRI with silicon nitride resonators was
accomplished in the past few years. First measurements of macroscopic electron
spin ensembles with membrane and trampoline resonators were published in 2016
[38] and 2019 [39], motivating further work in this new direction (see also
section 1.5 for related experiments). In 2021, an important step towards
nanoscale applications was reached with the first demonstration of a scanning
force microscope using a silicon nitride membrane resonator as a sensor [37].
In order to overcome the geometry issue arising from the clamping on all
sides, the microscope was defined in an inverted fashion relative to a typical
atomic force microscope: the membrane served at the same time as the sample
plate and as the vibrating sensor, see Fig. 2(a). The samples then interacted
with a sharp, but static, scanning tip. This offers a general solution for
challenge (i), as the “inverted scanning approach” can be translated to
trampoline and string resonators as well. The experiment also showed that the
noncontact friction at tip-surface distances of a few nanometers is smaller
than with traditional cantilever devices, providing a partial answer to the
challenge (ii).

On-chip integration of the optical readout [40] is expected to increase the
optomechanical coupling strength, thereby reducing the optical power required
for sensitive readout. However, it remains to be demonstrated to what level
the absorptive heating can be lowered relative to interferometer-based
approaches [30]. Challenge (ii) is thus likely to remain a topic of research
for some years to come.

In order to overcome challenge (iii), various protocols have been proposed and
demonstrated to allow spin-mechanics coupling at MHz frequencies. One
technique that was invented in the context of nanoscale MRI with bottom-up
grown nanowires is based on a modulation of the magnetic gradient fields [10]
(see section 1.3 for details). Other ideas include parametric coupling between
various modes [41, 42] and resonant coupling between the mechanical resonator
and spin ensembles [43]. Which approach proves most useful in the future will
depend on their respective practical requirements and technical limitations.

#### Concluding Remarks

The field of nanoscale MRI with force sensors enjoys a rejuvenation with the
advent of strained silicon nitride resonators with ultrahigh quality factors.
The evolution from traditional cantilever devices in the kHz range to these
new sensors in the MHz range offer several advantages but also leads to
challenges in the setup geometry and the measurement protocols. These
challenges are being tackled with promising approaches at the interface
between the fields of optomechanics and nanomechanical spin sensing.

#### Acknowledgements

We would like to acknowledge the Swiss National Science Foundation (SNSF)
through the National Center of Competence in Research in Quantum Science and
Technology (NCCR QSIT), the Sinergia grant (CRSII5 177198/1), and the project
Grant 200020-17886, as well as the ETH Zurich through the Research Grant
(ETH-03 16-1).

###  1.3 Magnetic resonance force microscopy using current-generated gradients

Sahand Tabatabaei and Raffi Budakian
Department of Physics and Astronomy, University of Waterloo, Canada
Institute for Quantum Computing, University of Waterloo, Canada
<EMAIL_ADDRESS><EMAIL_ADDRESS>
#### Status

Magnetic resonance force microscopy (MRFM) was originally proposed as a
technique for 3D imaging of macromolecules utilizing the chemical specificity,
and sub-surface imaging capabilities of magnetic resonance imaging (MRI) [4,
44]. A unique feature of MRFM is the ability to detect spins in volumes on the
(100 nm)3 scale with high sensitivity. This capability is particularly
important for studying a large class of biological systems, such as protein
complexes and virus particles, that possess 3D structures on length scales
below 100 nm. Since the introduction of the initial concept, MRFM has
witnessed significant technical advances, that have given rise to such notable
achievements as the detection of single electron spins of defect centers in
SiO2 [17], 3D imaging of proton spins in tobacco mosaic virus particles with 4
nm resolution [11], and sub-angstrom precision nuclear magnetic resonance
(NMR) diffraction measurements of 31P spin ensembles in InP [45].

In MRFM, the spins within the detection volume are placed in a magnetic field
gradient, which produces a force that is detected using an ultrasensitive
nanomechanical oscillator. The interaction with the spin is typically mediated
using a series of radio-frequency (RF) pulses which modulate the longitudinal
spin component to create a force at the resonance frequency of the oscillator.
In most MRFM experiments, a small ferromagnetic particle placed near the
sample produces the magnetic field gradient required for spin detection, while
also confining the interaction with the sample to a thin “resonance slice”
that can be swept through the detection volume to yield a volumetric data set
of the spin density.

An alternative approach utilizes electrical currents that flow through a
narrow metallic constriction to produce both the time-dependent magnetic
fields at the Larmor frequency required for spin manipulation, and the
magnetic field gradients for MRFM detection [10, 12] (Fig. 3). In this
approach, a particular MRFM experiment is divided into an encoding sequence,
during which the detection gradient is turned off and the desired spin control
sequence is applied, followed by the measurement sequence during which the
longitudinal spin component is detected. The measurement sequence consists of
the application of a time-dependent magnetic field gradient at the resonance
frequency of the oscillator, the phase of which is periodically inverted to
cancel any electrical forces on resonance (Fig. 3) [46]. The spin orientation
is also inverted with each phase flip using an RF pulse – typically an
adiabatic full passage – to generate a spin force on resonance. The associated
displacement of the oscillator is detected using optical interferometry. The
ability to separate the spin control and detection sequences enables a great
deal of flexibility in the kinds of quantum control sequences that can be
incorporated into MRFM measurements. In the following sections, we outline the
current challenges and advancements in current-driven MRFM.

Figure 3: (a) Experimental setup for current-driven MRFM. The sample depicted
here as protein molecules in a cryoprotectant – is attached to the tip of the
nanomechanical oscillator, and positioned above a narrow metallic
constriction. The intense field gradients generated by flowing current through
the constriction couple the spins to the oscillator, and the resulting motion
is detected using optical interferometry. (b) Typical experimental protocol in
current-driven MRFM, consisting of an encoding sequence – used, e.g., for
Fourier encoding – and a measurement sequence in which the longitudinal
component of the spins is coupled to the mechanical oscillator. (c) Contour
plot of the readout gradient and $B_{1}$ field along the cross-section of a
representative device, with the Rabi field $B_{1}$ being the circularly-
polarized component of the RF field used for spin control. The plots
correspond to a peak current of $I_{pk}=70$ mA flowing through the
constriction.

#### Current and Future Challenges

The key technical challenge to achieving high resolution molecular-scale
imaging with MRFM is improving the detection signal-to-noise ratio (SNR). In
an MRFM experiment, the SNR is primarily determined by the 1) spin
polarization in the sample volume, 2) thermal noise of the oscillator, 3)
magnitude of the readout field gradient and 4) force noise caused by electric
field fluctuations near the surface. As discussed in the next section,
significant advancements have been made in each area, however integrating
these approaches into a single MRFM setup capable of angstrom-resolution
imaging of $\sim$ (100 nm)3 volumes remains a challenge.

The total measurement time required for constructing an image depends on the
measurement protocol. Reciprocal space techniques that utilize data
multiplexing, such as Hadamard [47] and Fourier encoding [12] are routinely
used in conventional MRI, providing an efficient means of imaging. In current-
driven MRFM, spatial encoding of spins is achieved by varying the static and
RF field gradients, in the same way as conventional MRI. While 1D and 2D
imaging have been demonstrated using current-driven MRFM [12, 46], the highly
non-uniform field gradients generated by the constriction complicates
extraction of the real-space spin density. Furthermore, generating the
encoding gradients for 3D Fourier imaging is not possible using a single
constriction. To overcome these challenges, in the following section, we
present a new design for a gradient source capable of applying spatially
uniform field gradients in 3D in a $\sim$(100 nm)3 volume.

Another important consideration for phase encoding techniques is the spin
coherence time, which determines the maximum encoding wavevector that can be
realized. Although the dephasing time for most organic solids tends to be
quite short–typically of order $T_{2}\sim 20\,\mathrm{\mu s}$ – dynamical
decoupling (DD) sequences can be used to extend the coherence time, thereby
increasing the image resolution that can be achieved. The highly inhomogeneous
magnetic field generated by the constriction presents a challenge to applying
global rotation pulses – the basic building blocks of these sequences – and
potentially limits the set of NMR sequences, including DD sequences, that can
be realized. Given the rich repertoire of spin control techniques in NMR, this
also limits MRFM’s potential use as a tool for spectroscopy and studying spin
dynamics in nanoscale materials. In the next section, we discuss the
implementation of numerical pulse engineering methods that have been
successfully used to realize high-fidelity coherent spin control, thereby
overcoming the challenges posed by the field inhomogeneity.

#### Advances in Science and Technology to Meet Challenges

Over the years, major developments have been made to improve detection
sensitivity in MRFM. These include polarization enhancement using cross
polarization [48] and dynamic nuclear polarization [49], achieving readout
field gradients as high as 107 T/m using disk write heads [50] and engineering
various ultrasensitive nanomechanical force sensors, e.g., lithographically-
made cantilevers [8], nanowires [51] and carbon nanotubes [52]. Fluctuating
electric fields near the surface can give rise to noncontact friction that can
degrade detection sensitivity [53]. Approaches proposed to mitigate this
include chemically treating the surfaces and utilizing sharper tips [54].

The integration of MRFM and high-fidelity quantum control schemes is creating
powerful new approaches in NanoMRI. Numerical pulse optimization is a well-
established technique in quantum information science for designing control
pulses in demanding experimental conditions, much like those encountered in
MRFM. The large field inhomogeneities of MRFM can be readily accounted for in
existing numerical schemes, which utilize a gradient-descent algorithm to
optimize the control waveform over a range of field values, while also
constraining the optimization to adhere to the amplitude and bandwidth
limitations of the experimental setup [55, 56]. Furthermore, undesirable spin
Hamiltonians such as chemical shifts and spin-spin interactions can be
efficiently accounted for by minimizing corresponding perturbative corrections
to the spin propagator. Utilizing such schemes has been essential for recent
advancements in current-driven MRFM. 1D Fourier transform imaging of 1H spins
with $\sim 2$ nm resolution was achieved using DD sequences that employ
numerically optimized $\pi$/2 pulses [46]. Recently, numerically engineered
control pulses were used to dynamically decouple 31P spins in an InP
nanocrystal and encode a z-axis magnetization grating with a $\sim 4$ Å
modulation wavelength. An NMR diffraction-based experiment using the encoded
grating enabled the detection of an angstrom-scale displacement of the InP
sample with a precision of 0.07 Å [45].

Achieving 3D Fourier transform imaging in current-driven MRFM will require
devices capable of generating uniform field gradients in all three dimensions.
Upcoming experiments will utilize a new design (Fig. 4) that incorporates four
additional current paths around the central constriction for generating highly
uniform encoding gradients over a $\sim$ (100 nm)3 volume above the central
constriction. Figs. 4(a, b) show the static currents $I_{y}$ $(I_{x})$ used to
produce uniform resonance offset field gradients $\partial B_{z}/\partial x$
$(\partial B_{z}/\partial y)$ for phase encoding in the $x$ ($y$) directions.
To encode in the $z$ direction, an RF pulse at the Larmor frequency is applied
to both gradient wire pairs with a $\pi$/2 phase shift between $I_{x}$ and
$I_{y}$, (Fig. 4(c)), which results in a circularly polarized field with a
uniform Rabi field gradient $\partial B_{1}/\partial z$ in the rotating frame.

Figure 4: Next-generation gradient source with four additional current paths
for generating uniform field gradients in three dimensions. The subfigures
show different current configurations for encoding along (a) $x$ (uniform
$\partial B_{z}/\partial x$), (b) $y$ (uniform $\partial B_{z}/\partial y$),
and (c) $z$ (uniform $\partial B_{1}/\partial z$). Each inset shows the
calculated field distribution for 150 mA peak current in a (100 nm)3 volume
centered above the constriction, derived using finite-element simulations. The
simulations indicate that over this volume, the relevant gradient in each
configuration deviates by less than 0.5% relative to its average value, which
was computed to be 1.35 $\times$ 104 T/m in all three cases.

#### Concluding Remarks

Over the three decades since its inception, MRFM has proven to be a prominent
method for imaging and characterizing materials on the nanometer scale.
However, significant work is still needed to achieve the sensitivity required
for imaging relevant biological systems with atomic resolution. Among the
various MRFM approaches, the current-driven modality has been shown to be
particularly adept at incorporating coherent spin control by utilizing
numerical pulse engineering methods. These methods have not only been
essential in extending the coherence time and hence resolution of Fourier
imaging in this modality, but have also created great potential for bringing
NMR’s 70-year-long history of spin control techniques to the nanometer scale.
As it stands, the missing ingredient in current driven MRFM is the ability to
do full 3D phase encoding with uniform gradients. With next-generation
gradient source devices demonstrating such capabilities, the combination of
large uniform field gradients with numerical spin control is expected to
enable 3D Fourier imaging, as well as NMR diffraction on periodic spin systems
such as protein nanocrystals. This could additionally open up new avenues for
conducting spatially-resolved spectroscopy and studying spin transport on the
atomic length scale.

#### Acknowledgements

This work was undertaken due in part to funding from the US Army Research
Office through Grant W911NF1610199, the Canada First Research Excellence Fund
(CFREF), and the Natural Sciences and Engineering Research Council of Canada.
The University of Waterloo’s Quantum-Nano Fabrication and Characterization
Facility (QNFCF) was used for this work. This infrastructure would not be
possible without the significant contributions of Transformative Quantum
Technologies (CFREF-TQT), Canada Foundation for Innovation (CFI), Industry
Canada, the Ontario Ministry of Research and Innovation, and Mike and Ophelia
Lazaridis. Their support is gratefully acknowledged.

###  1.4 Nanoscale ferromagnetic resonance imaging using high-sensitivity
mechanical force detection

Inhee Lee and P. Chris Hammel
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
#### Status

Ferromagnetic resonance (FMR) measures the interaction of a collection of
ordered spins with its environment, allowing accurate measurement of internal
fields, anisotropies and dipolar fields within a material or arising from
interfaces. Ferromagnetic resonance force microscopy (FMRFM) is a
spectroscopic imaging technique based on mechanical force detection with
sensitivity of 1 – 100 attonewtons. FMRFM detects the spin excitation modes in
FMR of a magnetic sample through cantilever oscillations driven by the dipole
force between the micromagnetic particle on the cantilever and the suppressed
longitudinal magnetization resulting from magnetization dynamics excited by
FMR. This enables scanned, spectroscopic imaging capable of quantitatively
studying microscopic magnetic properties of inhomogeneous thin films and
complex magnetic devices, including buried structures in multi-component
materials, at the nanoscale. The FMR modes observed in the early FMRFM reflect
the global magnetic properties in which the collective modes of the entire
sample are excited due to strong spin exchange interactions, so local magnetic
properties are not spatially resolved [57, 58, 59, 60, 61, 62, 63]. The first
FMR mode localization was demonstrated using a strong dipole field generated
from a micromagnetic tip [64, 65] (Fig. 5a and 5b). Fig. 5c shows FMR spectra
measured at various probe-sample separations $z$ on a 40 nm thick permalloy
film. Two localized FMR modes are spectroscopically separated, occurring at
higher fields than the $z$-independent uniform FMR mode. A localized mode (LM)
is confined to nanoscale area by the field well created by strong field of
micromagnetic probe at small $z$ (see inset, Fig. 5c). These LMs were used to
image the spatial variation of the internal field inside a permalloy (Py) thin
film and the inhomogeneous demagnetizing field of individual Py microdisks
with lateral spatial resolution down to 200 nm. Since this demonstration, LM-
FMRFM has been applied to image the internal field or damping changes of
various heterostructures: interfaces separating two regions with different
saturation magnetizations of a Py film modified by region-selective ion
irradiation [66], interfaces between two regions with different uniaxial
magnetic anisotropy in a Y3Fe5O12 (YIG) thin film induced by interfacial
interactions coupling the YIG and a gold overlayer [67], perpendicular
magnetic anisotropy and Gilbert damping enhancement regions in a YIG film due
to the influence of few-layer WTe2, a high spin-orbit coupling transition
metal dichalcogenide, overlying YIG [68]. Spin angular momentum transport from
the LM across its well boundary was observed through the inverse
proportionality of the Gilbert damping constant $\alpha$ of the LM on its
radius $R_{\mathrm{loc}}$ (see $R_{\mathrm{loc}}$ in Fig. 5(b)) [69].

Figure 5: (a) FMRFM schematic. The moment $m_{p}$ of the micromagnetic probe
is antiparallel to the external magnetic field $H_{0}$ to create localized FMR
modes and is separated from the film by a distance $z$ which is the distance
from the sample surface to the center of the probe magnet. (b) The
micromagnetic probe creates a well of magnetic field, shown by the solid
light-blue line, that localizes spin-wave excitations, indicated by the solid
red and blue lines for the first two modes. The probe field $H_{p}$, the
dynamic field $H_{\mathrm{dyn}}$ and the mode amplitudes are calculated for
$z$ = 1,270 nm. (c) FMRFM spectra of a continuous thin film for the indicated
values of $z$. The vertical dotted line shows the resonance field for the
uniform FMR mode which is independent of $z$. The first and second confined
modes indicated by the arrows shift to the higher field as $z$ decreases.
Inset: Calculated local-mode radius $R_{\mathrm{loc,n}}$ versus $z$ for the
first two magnetostatic modes obtained using the variational method. Adapted
with permission from [64] Copyright 2010 Nature Publishing Group. Figure 6:
Two-dimensional FMRFM scan resolving the spatial variation of magnetic
anisotropy and Gilbert damping of YIG in the presence of WTe2 flakes. (a)
Optical micrograph showing the color contrast of different thickness WTe2
flakes (ranging from 4.7 to 44.8 nm). The black dashed box outlines the FMRFM
scanned area for 2D mapping. (b) 2D map of the $n=1$ localized mode resonance
field. (c) 2D mapping of the Gilbert damping extracted from the $n=1$
localized mode resonance peak amplitude. Reprinted with permission from [68]
Copyright 2022 American Chemical Society.

#### Current and Future Challenges

Improving LM-FMRFM spatial resolution requires the minimization of the LM size
and improvement of detection sensitivity through reduction of spurious forces
on the cantilever.
_Spatial resolution_ : The ultimate spatial resolution is limited by the
exchange length of the ferromagnet. Exchange imposes an energy cost for
reducing $R_{\mathrm{loc}}$ which is balanced by the negative tip field and
the positive dynamic field. Decreasing $z$ reduces $R_{\mathrm{loc}}$ but
exchange sets a minimum LM wavelength and hence the smallest achievable
$R_{\mathrm{loc}}$ allowed for localized spin wave excitation.
_Spectroscopic precision vs. sensitivity_ : A larger cantilever oscillation
amplitude improves the accuracy of frequency detection measurements of FMRFM,
but this degrades the spectroscopic precision. Large amplitude cantilever
oscillations using a probe tip with high field gradient cause FMR peak
broadening as the sample spins experience a wider range of static fields
during oscillation.
_Tip fabrication_ : LMs require a tip moment $m_{p}$ antiparallel $H_{0}$
sufficient to saturate the film (Fig. 5(a)) [64, 65]. Higher spatial
resolution requires both a smaller tip and a larger field. Micron-scale high-
coercivity ($>2$ T) SmCo5 probe tips were fabricated using focused ion beam
(FIB). In principle, the highest coercivity is expected with very small
domains. However, reducing tip dimensions to the nanoscale using FIB machining
often degrades coercivity because the tip surface magnetization is impaired
and the local pinning and domain nucleation that maintain high coercivity are
reduced. Also, very small nanosized tips may be subject to quantum
fluctuations of magnetic moments or energy states, which can affect the force
detection sensitivity and spin relaxation in FMR.
_Tip characterization_ : Accurate tip characterization is essential to know
the LM spatial profile, which is necessary to convert spectroscopic maps into
images of local magnetic properties. Modelling the tip as a dipole whose
moment is obtained from cantilever magnetometry [70] works well for large $z$,
but accurate tip characterization at small $z$ is confounded by inadequate
knowledge of the magnetization distribution, including the dead layer. Also,
if the change in the sample’s internal field is large enough to be comparable
to the intensity of the local tip field, it deforms the LM spatial profile,
limiting its suitability for imaging [66, 67].
_Reducing spurious coupling and surface noise_ : To synchronize GHz-frequency
magnetization dynamics with kHz-frequency cantilever oscillation, the
microwave power amplitude is modulated at the cantilever frequency. The direct
interaction of the microwaves with the cantilever produces a spurious force on
the cantilever. Coulomb interactions between the bound charge on cantilever
and sample surface can damp the cantilever oscillation, lowering the Q-factor
and hence detection sensitivity.

#### Advances in Science and Technology to Meet Challenges

Many efforts have been made to improve FMRFM performance through fabrication
and application of various cantilevers, magnetic tips, microwires etc., to
minimize spurious coupling and surface noise. Non-conductive materials are
chosen to reduce spurious couplings, and deionization reduces surface charge.
Diamond cantilevers exhibit smaller spurious coupling than commonly used
silicon cantilevers [69]. An iron filled carbon nanotube (FeCNT), a 10 – 40 nm
diameter ferromagnetic nanowire enclosed in a protective carbon tube, has been
studied for use as a FMRFM probe with high spatial resolution, high
coercivity, and small surface noise. [70]. Indeed, an FeCNT was successfully
used as a scanning magnetic monopole probe at close $z$ with images of vortex
and multi-domain structures in magnetic force microscopy (MFM) measurements
[71]. The integrated cantilever and tip lithographically fabricated for custom
use in the desired size, shape, and configuration is another good option for
FMRFM probe. For example, an integrated magnetic tip consisting of a narrow
magnetic nanorod suspended from the leading edge of a cantilever was
lithographically fabricated and used in magnetic resonance force microscopy,
which allows for large tip field gradient while minimizing surface noise [72].
Fabrication of the nanowire capable of generating a highly localized microwave
field can also significantly reduce spurious coupling [45]. In addition to the
out-of-plane field geometry that requires a high coercive magnetic tip, FMR
mode localization is also possible in the in-plane field geometry [73]. In
this configuration, the tip field and sample magnetization are aligned
parallel to the applied magnetic field, but the local tip field experienced by
the sample is antiparallel to the magnetization, which is possible even with a
soft magnetic tip. Because of its ease, the in-plane field mode localization
was later applied to the spin-torque FMR [74] and auto-oscillation [75] of LM,
although the LM spatial profiles are not isotropic like those of out-of-plane
field geometry. The development of micromagnetic simulations has greatly
improved understanding of complex magnetic dynamics of FMR localized modes.
Micromagnetic simulations show how spatial mode profiles are deformed by
abrupt spatial variations in the internal field across a boundary in a
magnetic heterostructure [66, 67].

#### Concluding Remarks

Localized mode FMRFM is a powerful spectroscopic tool for imaging ferromagnets
with high spatial resolution. This technique measures the internal field by
locally exciting the FM’s spin dynamics, which is clearly distinct from other
magnetic scanning microscopies that measure the stray field from the magnetic
samples, such as MFM, scanning SQUID microscopy and NV optically detected
magnetic resonance imaging. Currently, the best lateral spatial resolution of
100 nm has been achieved on YIG films with this method, but this can be
addressed through improvements in the nanofabrication of cantilevers and
magnetic tips. LM-FMRFM can provide the microscopic details quantitatively
needed for the characterization of ferromagnetic materials used in fields
ranging from spintronics to biomagnetism. This method is applicable to buried
and surface magnets, and as it is a resonance spectroscopy technique, it
measures local internal fields and other magnetic properties and spin
interactions at interfaces.

#### Acknowledgements

This work was primarily supported by the Center for Emergent Materials, an NSF
MRSEC, under award number DMR-2011876 and by the NanoSystems Laboratory at
OSU.

###  1.5 Quantum sensing in the negative mass reference frame

Eugene S. Polzik
Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, Copenhagen,
2100 Denmark
<EMAIL_ADDRESS>
#### Status

Over the past two decades, a new approach to measurements of fields, motion
and forces has emerged [76, 77, 78]. Based on the measurement in the so-called
negative mass reference frame (NMRF), this new approach quantum sensing
provides, in principle, unbound sensitivity, not limited by the Heisenberg
uncertainty bound (also referred to as the standard quantum limit (SQL)). The
origin of the SQL is in the impossibility of simultaneous measurement of non-
commuting variables, $X$ and $P$ with arbitrary accuracy. The negative mass
reference frame is a quantum system [77, 78, 79, 80], which was first realized
by an optically pumped atomic spin oscillator [76], and has been later
extended to optomechanical [80], and ultra-cold atomic systems [81]. Sensing
beyond SQL in the negative mass reference frame is realized by generation of
an Einstein-Podosky-Rosen (EPR) entangled state between the sensor variables
$X$, $P$ and the negative mass reference frame variables $X_{0}$ ,$P_{0}$.
Such an entangled state has been experimentally demonstrated for spin
oscillators [76], two mechanical oscillators [82, 83], and composite
mechanical-spin oscillators [84, 85, 86]. For spin oscillators, $X$ and $P$
are realized by collective spin variables $J_{x}=X\sqrt{J}$ and
$J_{y}=P\sqrt{J}$ which are the quantum projections of the collective spin
orthogonal to the mean collective spin $J$. After an entangled state is
generated between the sensor and the reference system, the external force
acting on the sensor (but not on the reference system) can be applied. The
subsequent collective measurement on both the sensor and the reference system
then reveals perturbations due to the external force exerted on both $X$ and
$P$ of the sensor. Fig. 7 illustrates the layout of entanglement generation
between the sensor – a mechanical oscillator ad the spin oscillator – the
NMRF. Magnetic field sensing beyond SQL using the negative mass reference
frame has been already demonstrated [87] and proposals to extend such sensing
onto other systems with nearly arbitrary spectral properties have been
developed [88, 89, 90]. Application of the NMRF to gravitational wave
interferometry beyond SQL has been proposed [89].

Figure 7: Generation of an entangled state between a mechanical oscillator and
a spin oscillator which acts as an effective negative mass reference frame
(NMRF). Panel a: detection of light propagating through the spin system and
the mechanical oscillator placed inside an optical cavity generates
entanglement between the two systems. Panel b: trajectory of the mechanical
oscillator referenced to the NMRF (blue) and the standard quantum limit (red
area). Panel c: the joint state of the mechanics and the spin systems becomes
entangled as the measurement progresses in time. Reproduced with permission
from [85].

#### Current and Future Challenges

With the proof-of-principle demonstrations of sensing beyond SQL achieved, the
challenge is to improve the technology to the point where a sizable
improvement is sensitivity in practical applications can be achieved. A figure
of merit for strong coupling is quantum cooperativity [78, 84, 85]. For an
optomechanical interaction $C_{qM}=4g^{2}/\Gamma_{M}$ where
$g=\omega/L\sqrt{\hbar n/m\Omega}$ with $\omega$ – optical frequency, $m$ and
$\Omega$ mass and frequency of the mechanical system, $L$ and $n$ – the length
of the optical resonator and the photon number. $\Gamma_{M}$ is the
decoherence rate. In principle, high $C_{qM}$ can be achieved by using
advanced mechanical systems with quality factors reaching $10^{9}$ and above
[91] for membrane resonators combined with high intra-cavity photon number
$n$. In practice, however, $n$ is often limited by photo-thermal effects. High
cooperativity regime is easier to reach at millikelvin temperatures in a
dilution refrigerator [82, 83], although it has been also achieved with rather
modest cryogenic cooling [84, 85]. High cooperativity for the spin system
$C_{qS}=\kappa^{2}\Gamma_{S}$ can be achieved by using systems with a high
optical depth $\sim\kappa^{2}$ combined with long spin coherence time.
Macroscopically sized room temperature spin ensembles contained in cells with
walls covered with coatings preventing collisional spin relaxation have been a
key element in a number of experiments [1,7,10,12]. Those conditions can also
be achieved by using cold and trapped atoms and cavity enhanced interactions
[86]. As with many other quantum sensing protocols based on detection of
continuous variable entanglement, sensing using NMRF requires low optical
losses. For mechanical oscillators it means low scattering and absorption
losses. That is why silicon nitride membranes characterized by negligible
losses in the near infrared spectral range and super-polished cavity mirrors
are often the elements of choice. For room temperature atomic ensembles
topical losses are typically in a few percent range limited mainly by the
optical losses on the cell windows.

#### Advances in Science and Technology to Meet Challenges

Development of high Q, low loss mechanical oscillators is one of the key
challenges on the way towards sensing of motion beyond the SQL. $Q\Omega$, the
product of the quality factor and resonance frequency is a figure of merit
which allow to calculate the number of periods over which an oscillator will
stay in a given quantum state before thermal decoherence becomes an issue.
With this product beyond $10^{13}$ an oscillator will stay coherent over a
single period of oscillations at room temperature. $Q\Omega$ values exceeding
this threshold by several orders of magnitude have been reported for surface
acoustic wave resonators [92] and nano-acoustic resonators [93]. Those systems
appear to offer exciting options as future platforms for sensing using NMRF,
if optical losses can be minimized. Spin oscillators with sufficiently high
cooperativity and low losses can be implemented by further developments of
room temperature systems, such as vapour cell with antireflection coatings
described above, solid state systems, such as collective spins of nitrogen
vacancies in diamond [94] and more exotic systems [95]. NMRF systems can be
used for sensing and metrology for sensors with nearly arbitrary spectrum
which can be very different from the NMRF system’s spectrum [89, 90]. In this
case, the sensor and the NMRF system cannot be probed with the same optical
(or microwave) field. In order to generate entanglement and achieve
sensitivity beyond the SQL in this case, the two systems have to be probed
with entangled light fields. The two entangled modes are tuned to the
corresponding resonances of the two systems. An example of such system is the
implementation of the proposal [89] for gravitational wave interferometry
beyond the SQL. Here, the two entangled modes of light probe the gravitational
wave detector and the NMRF system implemented by a spin ensemble with an
effective negative mass, respectively (Fig. 8).

Figure 8: A layout for sensing of motion beyond SQL using the negative mass
reference frame (NMRF) of the spin oscillator. Here the sensor is a
gravitational wave interferometer, but the scheme can be utilized for a wide
range of interferometric measurements. The sensor and the spin (NMRF) are
addressed by two entangled light fields, which allow to have the sensor
response function to be centred at a wavelength different from the resonant
wavelength of the NMRF. In addition, the choice of phases $\zeta_{I}$ and
$\zeta_{S}$ allow the frequency spectral response of the NMRF to be tuned to
the response of the sensor (interferometer) using the oscillator virtual
rigidity. Reproduced with permission from [89].

#### Concluding Remarks

The outlook for sensing beyond the SQL is promising. Applications of such
sensing range from gravitational wave detection to medical diagnostics to
communication. Measurements of magnetic and electrical fields are ubiquitous.
Until recently, quantum limits of measurements of those fields have been
achieved only in a limited number of applications. With the attention
presently focused on quantum sensing and quantum technologies, technical
improvements will inevitably bring more sensing applications closer to quantum
limited sensitivity. Overcoming those limits using the NMRF methodology offers
exciting new capabilities.

#### Acknowledgements

The author’s research reviewed in this article has been supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 787520) and by VILLUM
FONDEN under a Villum Investigator Grant (grant no. 25880).

##  2 NV

###  2.1 Towards imaging complex spin systems and molecules with spin-based
quantum sensors

Tim H. Taminiau
QuTech and Kavli Institute of Nanoscience, Delft University of Technology
Amit Finkler
Department of Chemical and Biological Physics, Weizmann Institute of Science
<EMAIL_ADDRESS><EMAIL_ADDRESS>
#### Status

Brief history and status: In recent years, quantum sensors based on optically
accessible defect spins have emerged as a promising platform for imaging
complex spin systems, such as molecules and quantum devices. Whereas
traditional NMR and MRI methods require averaging over ensembles, spin-based
quantum sensors might enable the magnetic imaging of individual systems and
single-molecules, with ångström spatial resolution and sensitivity down to
single nuclear spins. Such nano- and atomic-scale MRI might open new
opportunities to determine the structure of single molecules like proteins, to
investigate qubit systems for quantum information tasks, and to characterize
spin-based quantum simulators. A variety of theoretical proposals has been put
forward [96, 97, 98, 99, 100]. Pioneering experiments have reported the
magnetic resonance signal of single molecules [101, 102], and the
reconstruction of the positions of individual electron and nuclear spins with
atomic-scale resolution, including for systems with up to approximately 30
spins [103, 104, 105]. To be able to image more spatially extended systems,
arrays of sensors or scanning probe based sensors are explored (See Sec. 2.3).

Figure 9: An artist rendition of NanoMRI for molecular structure. $\copyright$
2017, Philipp Scheiger. The figure depicts an array of diamond nanopillars,
each hosting an NV center. The molecule of interest is placed on the diamond
surface, and a magnetic field gradient enables an improved NanoMRI spatial
resolution.

Why it is still important: Molecular structure is part of a triumvirate, the
other members being function and dynamics. NanoMRI combines the non-
invasiveness and chemical specificity of magnetic resonance with the
advantages of single molecule spectroscopy. While extremely challenging,
observing structure and dynamics on the single molecule level (NanoMRI) might
lead to the capability to observe the different stages in time of the same
molecule. It can reveal outliers in the statistical distribution and might
eventually allow for the use of far smaller quantities of material for the
initial (and very expensive) stages of development of materials, for example
new drugs.

In the field of quantum materials and technologies, the capability to image
spins and currents on the atomic- and nano-scale can open new opportunities to
investigate quantum materials (see Sec. 2.3), to investigate various quantum
bits and magnetic noise environment, and to characterize spin-based quantum
simulators both inside and outside of the host material [106, 107].

What will be gained: The different platforms are now in an advanced stage, in
both our understanding of their underlying physical principles and also the
technological readiness level. Further advances now need to tailor material
science, chemical, electrical and mechanical engineering with these platforms
(mostly developed by physicists). This integration promises to enable a
transition from method development and demonstration to applied studies of
molecules of interest, ranging from relatively simple amino-acids and
eventually all the way to biology’s building blocks, proteins. While the
latter is a very ambitious goal, there are several interim stages, such as the
use of small spin clusters for quantum simulation, which on their own already
constitute a formidable achievement and are worth the entire endeavor.

#### Current and Future Challenges

The general concept of imaging spin systems with defect-spin-based quantum
sensors is the following. The quantum sensor consists of a single electron
spin, or an array of electron spins, that can be detected and read out
optically or electrically, and controlled through microwave pulses. Advanced
examples of such quantum sensors are the nitrogen-vacancy (NV) center and
other defects in diamond and defects in silicon-carbide, as well as donor
spins in silicon [100].

This electron-spin sensor is brought in proximity to the sample so that it
interacts with the sample spins. Various pulse sequences that manipulate the
sensor, as well as the sample spins, can then be used to detect the sample
spins and obtain spectroscopic information on dynamics and characteristic
precession frequencies. Spatial structure is encoded in the spectral
information through a combination of an external field gradient, the field
gradient generated by the electron-spin sensor itself and/or interactions and
chemical shifts internal to the sample spins. In that sense, nano-scale
magnetic imaging of individual systems blends together the principles of MRI
and NMR.

We divide our treatment of the challenges ahead into three categories: (1)
high-quality sensors in close proximity to the sample, (2) sample preparation
and fixation, and (3) methods for efficient extraction of structure. Meeting
these challenges will require advances in theoretical understanding and
computation methods, as well as improvements in materials and hardware.

  1. 1.

_High-quality near-surface sensors_ – A key challenge is to bring the
electron-spin sensor in close proximity to the sample, while maintaining good
coherence properties. Because the spin-spin interaction signal typically falls
off as $1/r^{3}$, a small sensor-sample distance is generally essential to
obtain a significant S/N. Additionally, when using the field gradient from the
sensor electron spin to localize the sample spins, close proximity tends to
translate in high-spatial resolution. However, the required small distance
from the sensor defect to the host material surface introduces electromagnetic
noise from surface defects and dangling bonds, and causes charge state
instabilities [108].

  2. 2.

_Sample preparation_ – A second challenge is how to place interesting samples
in close proximity of the sensor material surface. For this functionalization
and fabrication methods must be developed. An interesting general solution is
to use defect spin sensors embedded in scanning probes so that they can be
positioned over samples, but the small desired sensor-sample distances (for
example less than 5 nm) remain a challenge. Additionally, a trade-off exists
between fixation in a solid matrix to enable long interrogation times and
allowing dynamics. While ideally, one would like to study for example
biologically relevant molecules at room temperature in liquids, diffusion can
limit interrogation times and linewidths.

  3. 3.

_Efficient spectroscopy and imaging_ – A final challenge is to develop methods
for high resolution spectroscopy that enable imaging complex spin systems with
high spatial resolution and manageable acquisition times. Current proof-of-
principle demonstrations [103, 105] require long measurements, and improved
signal-to-noise, as well as more efficient data collection and data analysis
are required to enable imaging more complex systems.

#### Advances in Science and Technology to Meet Challenges

For applications aiming to magnetically detect and image individual molecules
or nanoscale devices it will be essential to apply surface science and
chemical engineering to realize reliable & reproducible templates for fixating
molecules, surface functionalization [109] and modification of the environment
[110] (so far most experiments were not done on external molecules). Finally,
our reliance on surface functionalization will play a decisive role in any
technique which aims at measuring the magnetic properties of single molecules.
Here, the constant advances in chemical engineering, understanding surface
termination [111], controlling the spin bath environment and modifying
specific ligands to have stronger and coherent bonds is pivotal for an
efficient and useful sample preparation process. The availability of the so-
called electronic- or quantum-grade materials must become abundant and
accessible for the community as a whole to benefit. This is already happening
in quantum foundries being set-up around the globe.

As the application of nanoscale MRI to molecules makes use of the spin degree-
of-freedom of both electrons and nuclei, the concept of hyperpolarization
comes about almost immediately. Ideas from the NMR community have diffused and
propagated over the last decade (Sec. 2.4), but when dealing with systems of a
very small cluster of spins, an all-out polarization is yet to be
demonstrated. To reduce acquisition time and in general to improve readout
fidelity, a combination of pulse engineering (e.g. [112], and see also Sec.
5.1), adaptive algorithms (Sec. 5.2), machine-learning and yet even better
materials, e.g., commercially available 12C diamonds, are needed. We have
ample experience from the past two decades, and even a richer library of
possible improvements from the NMR community in the past half-century, both of
which provide the resources for ingenious tailoring of sequences for
manipulating spins. Regardless of the specific method for data gathering,
algorithms for streamlining the post-processing (or real-time processing) of
the data for the purpose of reconstructing the positions of the various spins
is critical. We expect some deviations from the ensemble-based techniques and
argue that such algorithms, already in their abstract form, will provide a
substantially high launch point for a full molecular structure reconstruction
in the limit of a few tens of spins [103]. With all NanoMRI techniques, the
issue of knowing the precise position of the sensor will be critical. For
example, in the case of color centers, more deterministic positioning of
sensors [113] will enable a reliable and reproducible method of acquiring
data. Naturally, some techniques have this capability by definition, but
nevertheless this will bring about, so to speak, a closure to the sample-
sensor fabrication process, including the nanometer-positioning of RF and MW
antennae relative to the sensor and the molecule under investigation [114].
Antennae that enact magnetic field gradients [46] will in-effect be a force
multiplier, making it much easier to create a distinct spectral separation
between neighboring spins.

#### Concluding Remarks

The nanoscale approach to determining molecular structure by means of magnetic
resonance methods is finally gaining momentum and in our eyes is set to soon
reach the critical point of technical readiness for applications. Looking at
the different techniques we have in the NanoMRI community, they all seem to
support each other for an overall toolkit, which is surprisingly powerful and
versatile, and can complement existing methods in time, frequency and space
domains. In the coming years, we expect the maturation of technological tools
to advance the field beyond the threshold of proof-of-concept towards
practical single-molecule imaging using magnetic resonance.

#### Acknowledgements

THT is a group leader at QuTech and the Kavli Institute of Nanoscience and
acknowledges support by the Netherlands Organization for Scientific Research
(NWO) through a Vidi grant and the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 852410). AF is the incumbent of the Elaine Blond Career
Development Chair and acknowledges support from Israel Science Foundation
(grant agreement No. 418/20) and the Minerva Stiftung as well as the Abramson
Family Center for Young Scientists and the Willner Family Leadership Institute
for the Weizmann Institute of Science.

###  2.2 Micron-scale NV-NMR

Ronald L. Walsworth
University of Maryland (2218 Kim Engineering Building, College Park, MD 20742,
USA
<EMAIL_ADDRESS>
#### Status

Nuclear magnetic resonance (NMR) spectroscopy is an important chemical
analysis tool in many areas of basic and applied life science, due to its
unparalleled ability to determine molecular identity and structure under
ambient conditions. However, conventional NMR requires macroscopic sample
volumes because it relies on detecting the weak magnetization of thermally-
polarized nuclear spins via inductive detection. The poor sensitivity of
conventional NMR has restricted its utility in many important areas where
sample volume, molecular concentrations, and/or total molecular number are
intrinsically limited, e.g., single-cell metabolomics, chemical analysis of
mass-limited samples used for screening of drugs and catalysts, and label-free
sensing of biomarkers.

Recent advances using optically-addressable nitrogen vacancy (NV) color
centers in diamond hold great promise to overcome this sensitivity limitation,
thereby enabling high-resolution NMR spectroscopy and imaging (MRI), under
ambient conditions, at the scale of single biological cells — i.e., the micron
or picoliter scale. Recent work demonstrated NV-NMR spectroscopy with chemical
resolution in very small liquid sample volumes ($\sim$10 picoliters): i.e.,
detection of both J-couplings and chemical shifts in small molecules in
solution, by coherent averaging of repeated NV-NMR measurements synchronized
to an external clock [115]. This micron-scale NV-NMR capability was then
integrated with hyperpolarization and pre-polarization techniques [116, 117,
118] to boost the molecule concentration sensitivity for micron-scale samples
to as low as $\sim$1 millimolar (mM), thereby reaching $\sim$10 femtomole
absolute molecular number sensitivity; see Fig. 10. No other NMR technology
can provide this combination of picoliter volume applicability, molecular
sensitivity, and chemical specificity for samples under ambient conditions.

Some of the potentially transformative applications of micron-scale NV-NMR are
the following:

  * •

Single-cell metabolomics: NV-NMR may provide high-resolution NMR spectroscopy
of small molecules and proteins within single-cells, enabling physiologically
relevant single-cell metabolomics. Compared to the genome or transcriptome,
the metabolome of a single cell or organelle is much more difficult to measure
because metabolites exhibit much larger structural diversity and dynamic range
and are difficult to amplify. Nevertheless, the metabolome provides the most
immediate and dynamic picture of a cellular phenotype. As such, a reliable NMR
method to perform single-cell metabolomic measurements is expected to have a
large impact in a wide area in biology and medicine, thus providing a high
value target for our new technology development.

  * •

Chemical analysis of mass-limited samples: High-throughput chemical analysis
is essential for screening of drugs, catalysts, and many other applications.
Today, economically efficient chemical synthesis is often performed on
microfluidic chips with picomole or smaller samples in a highly parallelized
way. The corresponding chemical analysis is thus restricted to small-mass-
sensitive methods like fluorescence spectroscopy or mass spectrometry.
Importantly, conventional NMR — which gives superior molecular structural
information compared to these methods — cannot be used for such applications
due to its inherently low sensitivity and lack of parallelization. NV-NMR may
overcome this limitation, allowing high-throughput, chip-based NMR
spectroscopy applicable to femtomole-quantity, picoliter-volume samples. NV
fluorescence readout onto a camera could also allow interrogation of many NMR
samples in parallel, thereby enabling practical high-throughput NMR chemical
analysis of mass-limited samples.

Figure 10: Example micron-scale NV-NMR results using SABRE hyperpolarization
[118]. (Upper left) Experimental schematic. (Lower left) NV-NMR spectrum of
15N-labeled pyridine at 100 mM concentration in methanol, with an effective
sample volume of about 10 picoliters. Few-Hz splitting of the NMR spectrum due
to J-coupling between the 15N nucleus and protons in pyridine is easily
observable. (Upper and lower right) Micron-scale NV-NMR signal spectra, here
for unlabeled pyridine at various concentrations in methanol, have observable
peak amplitudes down to $\sim$1 mM molecular concentration, equivalent to
$\sim$10 femtomole molecular number sensitivity.

#### Current and Future Challenges

As outlined above, recent work has shown that micron-scale NV-NMR can
spectrally resolve small molecule J-couplings and chemical shifts in liquid
samples under ambient conditions, with absolute NMR frequency resolution
$\sim$1 Hz and molecular concentration sensitivity $\sim$1 mM (currently, when
using hyperpolarization). No other NMR technology can approach this
combination of small sample volume, high spectral resolution, and good
sensitivity. However, as chemical shifts scale linearly with $B_{0}$, the
applied static (bias) magnetic field, a micron-scale NV-NMR system with
$B_{0}>1$ T will be needed for identification of molecular species and
associated kinetics and dynamics when using standard NMR chemical analysis
techniques. In addition, a tesla-scale bias field will increase the sample
nuclear spin polarization and hence the concentration sensitivity of a micron-
scale NV-NMR system by $\sim$10-100$\times$ when not using hyperpolarization.

Thus, a current goal of micron-scale NV-NMR technology development is to
realize instruments that operate at tesla-scale bias field [119, 120, 121]
while maintaining or improving millimolar concentration sensitivity, spectral
resolution able to resolve J-couplings and chemical shifts, and efficient
handling of small-volume samples. A core associated challenge will be
realization of efficient, high-power microwave signal sources and antennas for
coherent manipulation of NV electronic spins at the large microwave
frequencies associated with a tesla-scale bias field: e.g., $\approx$ 40 GHz
central frequency and $\approx$ 120 MHz Rabi frequency at $B_{0}\approx$ 1.5
T. Other key challenges include the development and integration of several
sub-systems and capabilities:

  * •

A magnet providing a tesla-scale bias field with suitable homogeneity,
stability, and pole spacing for operation of a micron-scale NV-NMR measurement
module providing high spectral resolution.

  * •

A microwave integrated GaN transmitter chip operating at $\sim$ 40-GHz,
including a power amplifier and a planar coil for fast (high power) ESR
excitation of the NV electronic spins.

  * •

Microfluidic or other methods to localize picoliter volume samples with the
ensemble NV sensor, or arrays of samples on the diamond surface for parallel
NV-NMR spectroscopy systems, while not broadening NMR spectral lines via
magnetic susceptibility effects.

  * •

A compact NV-NMR measurement module, including efficient delivery and handling
of optical pumping light and collection of NV fluorescence; an RF spectrometer
and delivery antenna to manipulate sample nuclear spins with any prescribed
pulse sequence at the appropriate resonant frequency (e.g., 60 MHz for proton
spins at $B_{0}\approx$ 1.5 T); integrated delivery of (optional)
hyperpolarization sources to the liquid sample; and mechanical supports that
do not induce significant magnetic susceptibility broadening of NMR spectral
lines.

  * •

Improved micron-scale NV-NMR sensor performance to allow micromolar/millimolar
concentration sensitivity with/without use of hyperpolarization.

#### Advances in Science and Technology to Meet Challenges

As one approach to the above challenges, a collaboration at Harvard and the
University of Maryland (UMD) is developing an integrated micron-scale NV-NMR
instrument operating at bias field $B_{0}\approx$ 1.5 T and 60 MHz proton NMR
frequency. The associated coherent excitation of NV electron spins at
$\approx$ 40 GHz microwave central frequencies and $\approx$ 120 MHz Rabi
frequency will be provided by a GaN microwave transmitter chip; and a silicon
(CMOS) RF spectrometer chip will be used to manipulate sample nuclear spins
with any prescribed pulse sequence [122]. The GaN microwave transmitter chip
will include an on-chip coil and a power amplifier with large amplitude
tuning; and the CMOS RF chip will include an arbitrary pulse sequencer for
nuclear spin manipulation and a power amplifier; both chips will have a small
(mm2) footprint. The 1.5 T micron-scale NV-NMR instrument will also include
microfluidic manipulation of samples integrated with delivery of
hyperpolarization sources; a compact NV-NMR measurement module; and advanced
spin control and measurement protocols. The system has a design goal of
$\sim$1 Hz absolute spectral resolution and small molecule concentration
sensitivity $\sim$10 $\upmu$M/10 mM with/without hyperpolarization in sensing
volumes down to $\sim$ 10 picoliters, thereby enabling diverse applications
from single-cell biology to chemical analysis of mass-limited samples.

Meeting this target NV-NMR sample molecular concentration sensitivity in
micron-scale samples will be aided by the increased bias field, together with
a combination of other improvements and techniques, including: high-density NV
spin ensembles in optimized diamond material (with minimized strain and
paramagnetic impurities, and stabilized NV charge states); robust quantum
control techniques that allow simultaneous suppression of the undesired
effects associated with NV spin-spin interactions, disorder, and control and
measurement protocol imperfections (recently shown to yield a 5$\times$
enhancement in NV coherence time [123, 124]); and use of quantum logic
enhanced (QLE) sensing for NV ensembles. In QLE sensing, the NV electronic
spin population difference carrying the measured AC magnetic field information
is mapped onto the 15N nuclear spin degree of freedom on-board each NV in the
ensemble. The NV nuclear and electronic spin states are then entangled into a
pseudo-spin-singlet state. Next, the electronic spin state information is
repeatedly read out optically and re-correlated with the (unperturbed) nuclear
spin state in a quantum non-demolition (QND) measurement. As shown in [125],
the QLE technique can provide $>30\times$ enhanced SNR for ensemble NV AC
magnetometry for MHz-scale signal fields; and improved signal magnitude
sensitivity of about 10$\times$, depending on the measurement protocol.

#### Concluding Remarks

Recent years have seen exciting progress in the development of micron-scale
NMR with good spectral resolution and sensitivity, using dense ensembles of
nitrogen vacancy (NV) color centers in diamond as precision NMR sensors.
Ongoing efforts seek to extend this capability to few tesla bias magnetic
fields; and to integrate the NV-NMR instrument with compact sub-systems for NV
and sample spin manipulation, efficient manipulation of picoliter volume
samples (with optional hyperpolarization of samples spins), high-efficiency
optical measurement of NV fluorescence, and quantum control and measurement
techniques. Once fully developed, few-tesla micron-scale NV-NMR can be
expected to have wide-ranging, high-impact applications — from label-free
sensing of biomarkers to chemical analysis of mass-limited samples that are
expensive or difficult to synthesize, e.g., high-throughput screening in drug
and catalyst discovery. It may also open up new lines of scientific inquiry,
including single-cell metabolomics for quantitative cell biology and cell-
based drug screening; and functional and structural probing of biological
tissues and organisms with cellular resolution and chemical specificity.

#### Acknowledgements

The Harvard-UMD collaboration on micron-scale NV-NMR includes major
contributions from the research groups of Profs. Donhee Ham, Mikhail Lukin,
and Hongkun Park. Ongoing support for this work is provided by the Moore
Foundation. Additional support for specific aspects of this collaboration are
listed in the related scientific articles referenced below.

###  2.3 Nanoscale imaging with NV centers

Paz London and Ania Bleszynski Jayich
UC Santa Barbara, Department of Physics, University of California, Santa
Barbara California 93106, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
#### Status

A powerful approach for nanoscale imaging is a scanning nitrogen vacancy (NV)
center sensor. The NV center in diamond is a point-like defect in diamond
whose spin degree of freedom has high sensitivity to magnetic and electric
fields, as well as temperature. Moreover, the spin can be initialized,
manipulated, and read out even at ambient conditions, making the NV center an
easy-to-use, versatile, non-invasive, nanoscale sensor. To date, NV-based
imaging has largely focused on magnetic phenomena. Scanning NV center
magnetometry provides high spatial resolution magnetic images, where the
resolution can greatly exceed the optical diffraction limit and approach the
nanometer-scale. An NV-scanning probe microscope (NV SPM) apparatus can be
realized in two conceptual configurations (Fig. 11): In a scanning sensor
configuration, a single NV center is placed at the apex of an atomic force
microscope (AFM) tip and is scanned above the surface of the sample (Fig. 11a)
[126, 127]. In a scanning magnetic tip geometry, the NV center is placed in
the vicinity of a sample and the tip’s magnetic gradient provides spatial
discrimination of different regions of the sample as in MRI (Fig. 11b) [126,
128, 129]; this technique is suitable for samples that exhibit an electron or
nuclear spin resonance with sufficiently narrow linewidth. While the former
allows a larger scanned area (not limited by the NV center’s sensing volume),
the latter offers spatial resolution that is limited by the achievable
gradient only, and not by the sensor-target distance. We note that in an
alternative imaging approach that does not involve a scanning tip one can
apply variable magnetic gradients to encode the spatial coordinate via Fourier
imaging [130]. In the last decade, NV SPM was used to image several condensed
matter (CM) systems and phenomena: single electron spins [128, 131], the
dynamics of magnetic domains and domain walls at room temperature and
cryogenic temperatures [132, 133, 134], superconductivity [135, 136], and
hydrodynamic current flow in interacting electron systems [137]. For a wider
list of condensed matter phenomena to which NV SPM has been applied, see [138]
and references within. NV SPM can also be applied to biological systems [139],
though progress on this front is slower due to the variability of biological
environments, non-planar morphologies inherent to biological systems, and
sensitivity to laser and microwave excitations used for NV measurements. The
versatility of the NV SPM is also manifest in the several sensing modalities
that are possible, giving access to DC, AC, and incoherently fluctuating
magnetic [140] and electric fields [129].

Figure 11: Scanning probe microscopy with a single NV center. (a) Scanning NV
sensor configuration, where the tip-sample distance ($d_{\mathrm{NV-sample}}$)
determines both spatial resolution and signal strength. (b) Scanning magnetic
tip configuration. Here, the spatial resolution is given by
$\Delta=1/\gamma\tau\nabla B_{\mathrm{tip}}$, where $\gamma$ is the target
spin’s gyromagnetic ratio, $\tau$ is the interrogation time, and
$\nabla_{\mathrm{tip}}$ is the gradient of the tip’s magnetic field at the
target spin’s position. Purple lines bound this “resonant slice” of width
$\Delta$ in which spins precess at a frequency to which the NV center is
selectively sensitive. The fading spheres in (a,b) illustrate the volume at
which the NV has a single electron sensitivity.

#### Current and Future Challenges

The major challenge to achieving truly nanoscale ($<10\,\mathrm{nm}$) spatial
resolution while maintaining high sensitivity with NV SPM is the deleterious
effect of the proximal surface on the NV center’s charge state stability [141]
and spin coherence [142], effects that have been characterized extensively but
whose microscopic origins remain unclear, resulting in a lack of mitigation
strategies. Surface-related charge traps, paramagnetic impurities, and Fermi
level modifications are all suspected culprits that shorten coherence time and
NV charge state stability, hence limiting sensitivity and even prohibiting a
measurement (Fig. 12a). Currently, scanning experiments typically utilize NV
centers that are $>\sim 10\,\mathrm{nm}$ deep to avoid the surface effects,
which are exacerbated in a scanning experiment as the surface can change due
to unintentional tip-sample contact. The high throughput production of robust
NV probes is another outstanding challenge that is rooted in several reasons:
quantum-grade, single-crystal diamond is expensive and is limited to small
(mm-scale chips); thin-film diamond is difficult to form because
heteroepitaxial growth of high-quality, single-crystal diamond has not been
achieved; and diamond is notoriously difficult to polish and etch. In
addition, the best route towards deterministic creation of well positioned NV
centers in nanostructures is still an open question. Even deterministic
methods such as attaching a pre-characterized nanodiamond to the end of a
probe tip [126, 132], or laser activation [143] require time-consuming, probe-
by-probe assembly. Pixel-by-pixel imaging is inherently slow, compounding the
two challenges detailed above. For NV imaging, acquisition rates are largely
limited by low optical readout fidelity (photoluminescence rates are typically
$\sim 10^{5}$ photons/second and spin state contrast is typically 20-30%),
thus necessitating large numbers of measurement shots per pixel, which
lengthens measurement times. State-of-the-art single NV center probes
typically require few-second-long acquisition times per pixel for imaging 10’s
of nanotesla fields at optimal conditions (with NV depths $>10\,\mathrm{nm}$).
A potential solution to poor readout fidelity is single-shot readout at
cryogenic temperatures via resonant red laser excitation, as demonstrated for
bulk NV centers [144], but charge state conversion under resonant excitation
for shallow NV centers in fabricated structures makes resonant readout
currently difficult for NV scanning probes. NV-based imaging of CM systems
promises many unique advantages over other state-of-the-art material
characterization techniques, but several challenges limit widespread
implementation of NV-based-sensing in the CM community. In addition to the
above challenges, operation at low ($<\sim 1\,\mathrm{K}$) temperatures and in
high ($\sim$1-10 T) magnetic fields is often desirable for eliciting various
CM phenomena. However, these environments present challenges for NV
experiments due to laser and microwave induced heating, high microwave
frequencies necessary for driving spin resonance in strong magnetic fields,
and the need for precise alignment of the field to the NV axis to maintain
high sensitivity. For certain CM targets, this field alignment requirement may
conflict with a desired field direction with respect to the sample axes, such
as in quantum Hall systems. NV probes of different crystallographic
orientations, such as (111)- and (110)-faced tips [145, 146], are a potential
solution in some cases, however, the production of such probes is still
relatively rare due to diamond growth challenges. Moreover, it was recently
shown that at low temperatures, NV center performance degrades due to spin
mixing in the optical excited state [22].

#### Advances in Science and Technology to Meet Challenges

The role of the diamond surface - its proximity, morphology, and termination -
is an active area of investigation, and advances in precise annealing, surface
preparation, and termination show promise in mitigating the detrimental
effects of the surface (Fig. 12a) [147]. Nevertheless, shallow NV centers
still grossly underperform compared to their bulk counterparts, and hence the
continued development and refinement of robust techniques for preparing the
diamond surface prior to measurement, as well as for recovering a desirable
surface during a scanning operation, is critical. Achieving these goals will
require a deeper understanding of surface-mediated decoherence and surface-
mediated charge instabilities (and the interplay between the two), as well as
novel mitigation strategies such as surface passivation (e.g. encapsulation
and termination) and Fermi level engineering approaches that, importantly,
need to preserve nanometer-scale sensor-target separations. A complementary
approach, which resigns itself to the presence of surface nonidealities,
involves the in-situ manipulation of magnetic-noise sources and charge traps
such as DC electric field biasing or surface spin driving or DC electric field
biasing [148, 149]. Once surface effects are mitigated, or NV centers $>\sim
20\,\mathrm{nm}$ deep are employed, surface noise is no longer the limiting
factor. For these NV depths, a promising approach to forming reproducible,
stable, and coherent NV scanning probes involves nitrogen delta-doping (Fig.
12b) during chemical vapor deposition (CVD) diamond growth [150] combined with
3D localization of NVs inside nanostructures by careful electron irradiation
and annealing. This approach harbors several advantages: NV centers have
consistently good coherence properties (even at high N-doping) due to the
gentle nature of the nitrogen incorporation and low energy ($\sim$ 150 keV)
electron irradiation, excellent depth localization down to 2 nm (in contrast
to implanted NV centers), and ready use of isotopically purified 12C
precursors for suppressing decoherence due to nuclear spins. Overall, this
approach promises a high yield of high-performance scanning probes with
consistent properties. Whether this approach, in conjunction with other
surface improvements, can also lead to improved properties of shallow NV
centers remains an open question. Forming NV scanning probes from delta-doped,
CVD-grown, (110)- and (111)-oriented diamonds is another exciting frontier. To
address the fabrication challenges associated with making diamond NV scanning
probes, more widespread availability of high-quality single diamond
substrates, larger ($>4\times 4\,\mathrm{mm}^{2}$) substrates, and more
reliable and rapid etching procedures would allow for more rapid development
of optimized probe fabrication strategies [151]. For forming thin
($\sim$micrometer scale) diamond slabs from which diamond cantilevers can be
made, ion slicing is a promising method, akin to the Smart CutTM process for
forming silicon on insulator (SOI). When combined with CVD diamond overgrowth,
the membranes can host high quality NV centers. Compared to current approaches
that involve starting with $>100\,\upmu\mathrm{m}$ thick diamond pieces and
removing most of the diamond via a combination of laser cutting, polishing,
and etching, a time-consuming and wasteful process, this Smart CutTM approach
is a promising avenue to a time efficient formation of diamond thin films
while allowing for the reuse of diamond substrates. To overcome low readout
fidelities, hence reducing imaging times and enabling the use of advanced
sensing techniques, there are several avenues of approach that should be
pursued. Increased photon count rates can be achieved by careful geometric
shaping of the NV-containing nanostructure, e.g. rounding the apex of a
nanopillar to act as a parabolic mirror [152]. For these pillar-based
geometries, accurate 3D positioning of the NV center to a few tens of
nanometers is critical and several approaches are being explored [153]. Higher
collection efficiencies can facilitate the use of other advanced readout
techniques such as spin-to-charge readout [154], resonant readout [144], and
nuclear-assisted repetitive readout schemes [155], which are especially useful
for measurements with long sensing times, such as $T_{1}$-relaxometry imaging
or AC measurements with long $T_{2}$ times, as shown in Fig. 12c.

Figure 12: Scientific and technological improvements for NV SPM. (a) Coherence
times of shallow NV centers (reproduced with permission from [146]). The top
panel illustrates how the surface can host defects that produce electric and
magnetic field noise. The bottom panel shows Hahn echo coherence time $T_{2}$
as a function of the NV center’s depth, comparing a conventional triacid
cleaning technique with high-temperature anneal + oxygen termination. (b)
Secondary ion mass spectrometry measurement of delta-doped NV centers in an
isotopically pure diamond. The red (blue) profiles show the 15N (13C)
abundance. (c) Projected magnetic sensitivity for an NV center as a function
of the NV center’s coherence time $T_{2}$. Solid (dashed) lines are for
cylindrical (parabolic) shaped pillars. Blue (red) curves are for conventional
(SCC) readout techniques. The 1st generation cylindrical nanopillar image was
taken from [151], and the parabolically-shaped nanopillar image was taken from
[152].

#### Concluding Remarks

NV SPM harbors many advantages over other imaging techniques, with its
combination of nanometer spatial resolution, nanotesla field sensitivity,
noninvasiveness, quantitativeness, temperature versatility, and mulit-modal
sensing capabilities. The full potential of the NV SPM is still far from being
realized, however, due to the various reasons outlined above, but there are no
known fundamental obstacles standing in the way. Hence, with coherent, charge-
stable, near-surface NV centers that maintain sensitivity while scanning,
reproducible high throughput production of scanning probes, and improvements
in NV readout that approach single-shot readout, we expect widespread
implementation of the NV SPM across condensed matter physics and eventually
looking towards biological phenomena.

#### Acknowledgements

The authors thank Sunghoon Kim and Simon Meynell for assistance with figures
and comments on the text. P.L acknowledges support from the US Department of
Energy (DOE) Office of Science, Basic Energy Sciences (BES) under Award No.
DE-SC0019241, and A.C.B.J. acknowledges support from the Gordon and Betty
Moore Foundation’s EPiQS Initiative via Grant GBMF10279.

###  2.4 Quantum control for nanoscale NMR: status, challenges and frontiers

Ashok Ajoy
Department of Chemistry, University of California, Berkeley CA 97420, USA and
Chemical Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron
Rd, Berkeley CA 94720, USA and
Quantum Information Science Program, CIFAR, 661 University Ave., Toronto, ON
M5G 1M1, Canada
Arjun Pillai
Department of Chemistry, University of California, Berkeley CA 97420, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>

#### Status

A distinguishing feature of quantum sensor approaches for nanoscale NMR and
MRI [156] is how they leverage coherent dynamics of both analyte and sensor
spins to provide the chemical specificity necessary for identifying individual
molecules or mapping spatial distributions of nuclear species. The central
challenge in the field is achieving high sensitivity, spatial resolution, and
chemical spectral resolution simultaneously. Recent innovations in quantum
control methods have been critical to making progress towards this. We discuss
them here with a focus on NMR sensing with NV center electrons in diamond.

Figure 13: Quantum control schemes. (A) XY8 pulse sequence for sensing with NV
centers. (B) Typical FID ($T_{2}^{*}=1.5$ ms) of strongly coupled 13C nuclei
in diamond. (C) Pulsed spinlocking (inset; here $\theta_{x}\neq\pi$ and
lifetime depends parametrically on $\tau^{\prime}$) allows for $>60000\times$
coherence time extension (Figure partially adapted from Ref. [157]).

One important quantum control strategy involves applying a train of pulses
(e.g. CPMG/XY8) to the sensor spins with the interpulse spacing $\tau$ matched
to one Larmor period $1/f_{L}$ of the analyte spins [158, 159, 115, 160, 161].
The sensor is prepared in a superposition state $+$, and the dipolar
(hyperfine) interaction between the sensor and analyte spins induces its
rotation towards an orthogonal state $-$, under the applied control. The
extent of this rotation can be sensitively discerned and constitutes the
signal from the selectively probed analyte species.

Achieving high sensitivity requires maintaining long sensor coherence times
$T_{2}$ despite external perturbations or noise. In this regard, the sequence
in Fig. 13A offers a natural solution by rendering sensor spins insensitive to
fields that are mismatched in frequency from $f_{L}$. Related methods can also
be used to “spectrally decompose” the noise fields so that they can be
combated effectively [162]. When this noise originates from other spins in the
environment, such as those in the host lattice, control schemes decoupling
them from sensor spins [163] can significantly improvement $T_{2}$ lifetimes
without affecting sensor operation.

There has also been considerable effort towards reducing sensor decoherence
induced by imperfections in the quantum control itself, such as amplifier
noise. The use of composite pulses, solid echo sequences [164], and
concatenated drives [165] has been shown to provide significant protection
against these errors.

Furthermore, quantum control allows for the construction of effective
(Floquet) Hamiltonians of the sensor-analyte interaction, which can aid in
spectral simplification. For example, Ref. [166] demonstrated an approach to
“interpolate” the sensor dynamics to obtain quasi-continuous sampling of the
delay interval $\tau$ despite instrumental limitations. Similarly, Ref.[167]
exploited Hamiltonian engineering to suppress harmonics and aliasing
artifacts, tune the spectral profile of the sensor bandwidth, and separate
strongly and weakly coupled spins. The development of more sophisticated
infrastructure (e.g. high-speed AWGs) will promote further such advances.

#### Current and Future Challenges

Several important experimental challenges remain towards achieving
simultaneously high sensitivity, spatial resolution and chemical specificity.

  1. 1.

High-field operation: Currently, quantum sensing is predominantly carried out
at low fields ($<0.1$ T), where it is relatively easy to match the effective
electron Rabi frequency to nuclear Larmor frequency at $B_{0}$ (typically $<1$
MHz). However, operation at higher $B_{0}$ fields offers significant boosts in
(i) the initial analyte polarization and (ii) chemical shift separation, both
of which are proportional to $B_{0}$. Comparing sensing experiments at $0.1$ T
and $10$ T, for example, the latter would allow for a $\sim 10^{4}$-fold
increase in chemical resolving power from the increase in field alone. This
increase in resolving power comes from an easier separation of the chemical
shift peaks, and from them growing in intensity.

However, accessing this regime is highly challenging due to a scarcity of
sources and amplifiers in the “THz-gap” frequency range (e.g. electron Larmor
frequency $\sim$280 GHz at 10 T). Matching electron Rabi frequencies to
nuclear Larmor frequency in this regime is also difficult, as the timing
condition in Fig. 13A would require trains of electron $\pi$-pulses separated
by $<5$ ns, which challenges the capabilities of available technology. As a
result, it is not straightforward to directly utilize low-field CPMG/XY8
sequence variants for sensing at high-field.

  2. 2.

Low analyte spin polarization: Quantum sensing involves detecting oscillating
fields from Boltzmann-polarized nuclei, which constitute a small number at
low-fields. Increasing $B_{0}$ can boost this signal linearly but doing so
necessitates significant technical requirements. A promising approach is to
transfer polarization from the optically polarized sensor electron to the
nuclei to boost their polarization [168] (Fig. 14A-B). Even a small amount of
polarization transfer has a drastic effect: 1% transfer from an NV center to
analyte 13C nuclei can boost the signal $>10^{4}$-fold at 0.1 T and room
temperature. Producing such nuclear “hyperpolarization” remains an active area
of research. Currently, it is possible to efficiently hyperpolarize nuclei
within the diamond lattice, but transferring this polarization outside of
diamond remains a challenge, likely due to fast-relaxing paramagnetic
impurities on the diamond surface.

  3. 3.

Quantum sensing in the dense sensor limit: Using an ensemble of $N$ sensors is
an attractive way to increase sensitivity, with sensitivity theoretically
expected to grow as $\sqrt{N}$. However, in practice, such gains are only
feasible for well-separated (dilute) sensor spins. At higher concentrations,
intersensor interactions lead to decreased sensor coherence time ($T_{2}^{*}$
in Fig. 13B), erasing the benefits of larger $N$. Overcoming this will require
new sensing protocols that maintain sensor performance and coherence time
$T_{2}$ even in the presence of intersensor couplings.

Figure 14: Quantum sensing with hyperpolarized 13C nuclei. (A) Diamond lattice
schematic showing optical polarization of NV electrons and transfer to
surrounding 13Cs via MWs. (B) Resulting signal is improved $>200\times$
compared to thermal NMR. (C) 13C NMR signal carries coherent oscillations when
spins are subjected to AC field with frequency $f_{\mathrm{AC}}$. (D) FT of
signal in (C) yields strong responses at $f_{\mathrm{AC}}$ and harmonics
(Figure adapted from Refs. [169, 157]).

#### Advances in Science and Technology to Meet Challenges

We will now discuss recent developments in the field that address these
challenges.

  1. 1.

High-field quantum control: Accessing the sensitivity and resolution gains
available at high-field requires significant effort, often involving the
construction of an optical detection system within a superconducting magnet
and the necessary infrastructure for high-frequency electron control. A
noteworthy experiment by Ref.[119] involved operation at 3 T, demonstrating
chemical-shift resolved nanoscale NMR spectra of small molecules for the first
time. Takahashi and co-workers have built an impressive apparatus for quantum
sensing at 8.3 T [120], allowing spin manipulation directly at 230 GHz using
quasioptics and multiple frequency doubling chains. We anticipate further such
developments, although the technical complexity required presents significant
challenges.

  2. 2.

Nuclear spin quantum sensors: An alternative approach to high-field quantum
sensing is to use nuclear spins as sensors instead of electronic spins.
Ref.[170] demonstrated this using diamond 13C nuclei optically hyperpolarized
by NV centers (Fig. 14C-D). This approach overcomes the challenges of quantum
control at high-field since the sensor Larmor frequency is only $\sim$100 MHz
even at 10 T, making high-fidelity control easy. Moreover, these sensor spins
have remarkably long $T_{2}$ lifetimes, with Ref.[171] achieving $T_{2}$’
exceeding 90 s at room temperature (Fig. 13C). The extended coherence of
nuclear spins offsets sensitivity losses from their lower gyromagnetic ratio.
The sensing protocol here involves a train of spin-locking pulses with
inductive interpulse sensor readout, similar to low-field strategies. Although
preliminary sensitivity estimates ($\sim$410 pT/Hz1/2 at 7 T) are lower than
those of optically detected electrons, high-field operation reduces
sensitivity requirements for NMR sensing. Lastly, the nuclear spins here
operate in the high-density limit ($>10^{4}$ times denser than NV centers). In
this regime, interactions between individual sensor spins serve as a means of
“stabilization”, increasing the robustness to control errors.

  3. 3.

Material science and quantum control approaches to slowing electron
relaxation: Several efforts to improve sensor coherence through materials
advancements are underway, including controlled depletion methods for creating
nuclear spin-free host lattices, plasma annealing for producing pristine
diamond surfaces [147], and rapid high-temperature annealing to quench lattice
paramagnetic impurities [169]. Additionally, there are promising prospects for
employing sensing regimes where electronic decoherence is naturally
suppressed. Of particular interest is the low-temperature high-field regime
(e.g., 4 K and 10 T), where all paramagnetic spins are fully polarized,
suppressing spin flip-flops with the sensor and significantly lengthening
coherence [157]. We anticipate more quantum sensing experiments leveraging
this regime, particularly for nanoNMR applications.

#### Concluding Remarks

In conclusion, we emphasize that ongoing developments in quantum control
methods are crucial for quantum sensing and anticipate exciting prospects for
the development of high specificity sub-micron-scale chemical sensors. These
advancements portend several new applications in various fields.

#### Acknowledgements

We acknowledge funding from AFOSR YIP (FA9550-23-1-0106) and AFOSR DURIP
(FA9550-22-1-0156).

###  2.5 Nano MRI using spin quantum sensors

Jörg Wrachtrup
Physikalisches Institut, University of Stuttgart, Pfaffenwaldring 57, 70569
Stuttgart, Germany and
Max Planck Institute for Solid State Research, Stuttgart, Germany
Fedor Jelezko
Institute of Quantum Optics, Ulm University, Ulm 89081, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
#### Status

The use of spin ensembles or even single spins to measure various quantities,
like magnetic- or electric field, temperature etc. has gone through
spectacular developments in recent years [156]. Since its initial
demonstration in 2008 detection and imaging of magnetic fields from micro- and
nanomagnetic structures, the imaging and detection of electric fields as well
as temperature and temperature profiles, the detection of chemical compounds
like triplet oxygen, or radicals have been achieved [156]. All of the above is
based on defects in diamond (or other materials like silicon carbide or, e.g.,
hexagonal boron nitride). Their electron spins Zeeman splitting and/or spin
relaxation among them is used to measure those quantities. Optical detection
of single spin sensors achieves such high sensitivity that, e.g., the magnetic
field of single spins outside of the sensor can be measured. When incorporated
into scanning probe devices, imaging with a spatial resolution only limited by
the stand-off distance between sensor and sample is feasible. This has been
used to image the magnetic field profile of micromagnetic structures with
spatial resolution down to 20 nm [126]. The sensitivity reached in these
systems depends on the specific experimental modality and ranges from
$\upmu$T/Hz1/2 to a few nT/Hz1/2. Imaging and detection of e.g. magnetic
fields can also be done by wide field magnetometry [172, 173]. In this
setting, a thin layer of spins is implanted in a diamond substrate and local
magnetic fields are measured and imaged through detecting a spatially selected
part of the diamond chip. Micromagnetic structures as well as dynamic
magnetization has been measured this way. An ensemble of spins is also used
for the so far most sensitive measurements of magnetic fields with solid state
spins. Using $10^{14}$ spins in a volume of a few hundred
$\upmu\mathrm{m}^{3}$ fields smaller than 1 pT/Hz1/2 were detected. Flux
concentration further on reduced this sensitivity to below 900 fT/Hz1/2 [174].
Not only DC fields, but also AC fields with frequencies up to 120 GHz can be
detected [175]. This allows for measuring NMR signals of small quantities of
sample spins. First demonstrations comprise measurements of electron spins on
diamond surfaces [176] as well as of spin labels on proteins. Later, this was
extended to measuring nuclear spins [158] down to the level of single proton -
or nuclear spins on single proteins [102]. Besides sensitivity in nanoscale
NMR experiments, a central challenge is to achieve the required spectral
resolution to resolve quantities like chemical shift or J-coupling. To this
end, different methods have been developed. Synchronized readout uses the
frequency stability of an external reference to achieve $\upmu$Hz resolution
[156], while quantum memory-based techniques yield mHz spectral resolution
[119]. Electron spin resonance as well as NMR detection was combined with
spatial resolution, imaging for example a single electron spin [131] or small
amounts of nuclear spins with spatial resolution of around 20 nm [177]. For
small standoff distances, the back action of quantum sensor on nuclear spins
is not negligible and weak measurement based schemes can be employed to
mitigate the effect of backaction [178, 179].

Figure 15: NMR signal of a single ubiquitin protein [102] Let side:
schematical representation of the sample geometry. Right side: 13C and 2H
spectrum of a single ubiquitin protein. The protein was 13C and 2H labeled.

#### Current and Future Challenges

Combined with flux concentration, solid state spin sensors do have the
potential to achieve sub 100 fT/Hz1/2 sensitivity while maintaining a rather
large bandwidth. This requires material improvements, specifically achieving
maximum signal contrast as well as an efficient conversion of AC- to DC-
signals. The large bandwidth may then be used to design robust gradiometers. A
further advance may come from oriented growth of defects which would improve
sensitivity and facilitate operation of the magnetometer. Imaging systems,
like wide field - or scanning probe magnetometry would greatly benefit from an
increase in sensitivity and spatial resolution. For wide field magnetometry
the spatial resolution achieved so far is diffraction limited, while for
scanning probe systems the sensitivity is limited because often single defect
centers are used for detection. This limits their sensitivity, while the
spatial resolution of the technique is still limited the probe to sample
distance. The detection of NMR and EPR signals by spin quantum sensors relies
on sensor to sample distance. Achieving closer proximity without degradation
of the spin relaxation and dephasing times is a central challenge in the
field. At the same time, high resolution NMR requires averaging of the nuclear
dipolar interaction by fast molecular motion. This however leads to diffusion
broadening in nanoscale NMR detection. On the other hand, spin ensemble-based
NMR and EPR usually does not yield the expected increase in sensitivity as
either the quality, i.e., coherence properties of the spin-sensor or the
proximity to the sample is not ideal. Detection of nuclear spins in nanoscale
NMR experiments often employs statistical polarization of nuclear spins.
Although this method allows to detect nuclear spins in moderate or zero
magnetic fields, the diffusion of spins leads to broadening of NMR lines,
limiting spectral resolution [180]. Confinement of nuclear spins in nanoscale
structures adjusted to the electron spin sensor allows to reduce diffusion
related broadening and opens a path to reach high spectral resolution [181].
Another possibility to reduce diffusion broadening is related to use of the
nuclear spin hyperpolarization combined with quantum sensing enabled by
electron spins. External spins can be polarized by quantum sensor itself [182,
183] or using conventional hyperpolarization techniques [118]. Note that
diffusion broadening, and line shape of NMR signals can provide important
information about the motion of molecules near surfaces [184].

#### Advances in Science and Technology to Meet Challenges

A distinct potential for further improvement comes from improved material
quality. Nanoscale probes would greatly benefit from improved performance of
spin centers close to surface, i.e., at distances less than 5 nm. In
combination with advanced tip geometries this might boost the current spatial
resolution to below 10 nm and at the same time improve the sensitivity of the
method. Likewise, multi spin probes will enhance the sensitivity without
degradation - they currently achieve spatial resolution of around 30 nm. The
sensitivity in ensemble magnetometry is also mostly determined by material
parameter, like ensemble dephasing time, short $T_{2}^{*}$ and limited signal
contrast. All of these parameters can potentially be improved by materials
refinement. So far, experimental demonstrations of spin-based NMR quantum
sensing were realized using bulk sensing material and shallow sensing spins.
In the future, this approach can be extended to doped nano diamonds.
Currently, diamond nanoparticles doped with NV centers were shown to be
promising sensors for the detection external electron spins via relaxometry
[185]. Detection of nuclear spins using nanodiamonds remains challenging, and
only a few demonstrations were reported so far [186]. Optimization of
nanodiamond growth including techniques allowing generation of nanoparticles
with well-defined shape and dopant composition is essential for future
development of quantum sensing. This includes self-nucleation-based growth of
nanoparticles using high pressure high temperature approach and technique
based on CVD growth. Optimization of quantum materials hosting quantum sensors
needs to be combined with the development of quantum sensing protocols,
allowing to reach long coherence time and improve sensitivity. Promising
techniques include optimized dynamical decoupling, control tools tailored to
specific noise environment including pulsed and continuous approaches [156].
Dynamical decoupling suitable for cancelling low frequency noise can be
combined with quantum error correction capable to protect sensing qubits from
arbitrary noise spectrum of low amplitude.

#### Concluding Remarks

Spin based quantum sensing provides a versatile tool allowing to improve
sensitivity of nuclear magnetic resonance and reach the limit of detection of
single molecules. Future development of this technique will be directed to
application in reconstruction of the structure and dynamics of single molecule
and unraveling chemical reactions at nanoscale. Future development will
require significant advances in material and quantum control approaches.

#### Acknowledgements

JW acknowledges funding by the BMBF via the cluster4future Qsens, DiaQNOS and
NeuroQ the BW foundation via project SPOC, the EU via projects AMADEUS and
C-QuENS as well as the Max Planck Society. FJ acknowledges support by the
BMBF, the DFG and the ERC.

##  3 STM-ESR

###  3.1 An atomic spin sensor on surfaces

Yujeong Bae and Andreas J. Heinrich
Center for Quantum Nanoscience, Institute for Basic Science, Seoul 03760,
South Korea and
Department of Physics, Ewha Womans University, Seoul 03760, South Korea
<EMAIL_ADDRESS><EMAIL_ADDRESS>

#### Status

Integrating two experimental approaches is often vital in innovation of
research, allowing more complex problems to be tackled and more sophisticated
technological advances to be achieved. Electron spin resonance (ESR) combined
with scanning tunneling microscopy (STM) is one of those examples, which
enables us to build artificial quantum structures atom by atom on a surface
and characterize their quantum states with tens of nano-eV energy resolution
at the atomic scale [187]. The capability to structure matter at the nanoscale
and probe weak electrical signals at the single atomic level renders ESR-STM
an appealing technique for exploring quantum phenomena in engineered
nanostructures. Since its first demonstration [188], ESR-STM has been
intensively used to explore different atomic/molecular species on surfaces
[189, 190, 191, 192, 193, 194]. Beside control and detection of single
electron spins, advanced applications of ESR-STM can be seen in coupled spin
systems designed for “indirect” sensing of target spins. Indirect sensing is
appropriate i) to probe spins with no detectable spin resonance transitions
[195, 196] or ii) to coherently control multiple spins, including one spin at
the STM junction and “remote” spins coupled to and indirectly read out through
the spin at the STM junction [197]. In indirect sensing, a “sensor” spin is
located at the STM junction and experiences resonant excitations through
direct coupling with the magnetic tip, which has been described by either a
piezo electric coupling [198] or a tunnel barrier modulation [199, 200]. Its
resonance signal appears as a variation of tunnel current due to the DC and AC
tunneling magnetoresistance effects [190, 198]. The target spin is
magnetically coupled to the sensor spin, where the coupling strength is
precisely controlled by adjusting the atomic separations [201] or the chemical
environment [202] using atom manipulation in STM. As shown in Fig. 16, three
different couplings have been employed for indirect sensing: i) the dipole-
dipole coupling [201] to determine the magnetic moments and spin lifetime of
target spins that have no spectral features in spin resonance or inelastic
tunneling spectroscopies (e.g. Ho [195] and Dy [196]), ii) the hyperfine
interaction to probe the nuclear spin state and the chemical environment [191,
202, 203, 204], and iii) the exchange interaction between two electron spins
largely detuned in energy for remote driving and detection of target spins
[197]. Advancing these indirect sensing methods would impact both the
metrological practice and the applications of ESR-STM: it would allow us to
improve the quantum coherent properties of target spins, as well as to explore
different material systems with fewer prerequisites of target spins for ESR-
STM measurement. These impacts will facilitate the generalized use of ESR-STM
to explore a variety of material systems for quantum applications.

Figure 16: Magnetic interactions used for indirect sensing in ESR-STM and
observable ESR transitions. Left: Dipolar interaction between a sensor atom
(red) and a target atom (magenta), where the energy levels of the sensor atom
are split by the dipole fields from the target atom. Middle: Hyperfine
interaction between electron and nuclear spins with ESR transitions at
different nuclear spin states. Right: Exchange interaction between two
electron spins (one from sensor and one from target), where the coherent
manipulation and detection of the target spin are implemented via a single
atom magnet and the sensor spin, respectively.

#### Current and Future Challenges

One of the main challenges of ESR-STM is to improve its sensitivity and energy
resolution. ESR signals in STM are based on changes in tunnel current
depending on the relative alignment of two spins at the STM junction: one at
the STM tip apex and one on a surface. Using a typical magnetic tip, the ESR
signals appear as about 1-5% changes in tunnel current (about 0.2-1 pA changes
at 20 pA set-point current). In addition, the best energy resolution achieved
with ESR-STM is several tens of nano-eV (several MHz) with no significant
improvements over the last 7 years. At the current sensitivity and energy
resolution, spins with small magnetic moments, such as a spin-1/2 system, are
detectable only if the sensor spin is located within 1.5 nm of the target spin
in the dipole-field sensing. In addition, while the hyperfine interaction has
been measured only for atoms which carry both electron and nuclear spins [191,
202, 203, 204], the super-hyperfine interaction can be of great interest for
molecules with radicals, which requires both higher sensitivity and better
energy resolution. Improving the energy resolution is closely related to
enhancing the relaxation and coherence times of spins on surfaces. Previous
studies found that the tunnel current, coupling with substrates, and magnetic
field fluctuations caused by the magnetic tip are the most dominant sources of
decoherence [205, 206]. In indirect sensing, where the target spin is remotely
controlled with assistance of a single atom magnet and monitored through a
sensor spin, the target spin is not directly exposed to those decoherence
sources occurring at the STM junction [197]. As shown in Fig. 17, the target
spin in indirect sensing shows significantly enhanced signal-to-noise ratios
and spin coherence. Further improvement on the coherent properties of target
spins might be achieved by increasing the spin relaxation time of the sensor
spin and by reducing the coupling strength with substrate electrons. Finally,
one interesting challenge is to have a sensor spin in diverse environments,
e.g., at the tip apex. At the moment, the only material systems used as a
sensor are transition metals (Fe and hydrogenated-Ti) on two monolayers of
MgO/Ag(100). However, there are nearly infinite combinations of atoms or
molecules on different surfaces that will show different spin states.
Furthermore, many interesting molecules, e.g., carbon-based molecules, barely
adsorb on insulating surfaces such as MgO and NaCl. To apply this ESR-STM
technique to diverse material systems, it is therefore necessary to develop a
robust sensor spin that works well in different environments.

#### Advances in Science and Technology to Meet Challenges

While using a spin-polarized tip is essential to drive and detect ESR in STM,
the magnetic states of the tip are not well-known, barely controllable, and
described in a classical picture. Understanding the role and effects of
magnetic tips on spins on surfaces for ESR would help to improve the
sensitivity and energy resolution of ESR-STM. While most magnetic tips for
ESR-STM have been prepared by attaching several magnetic atoms to the tip
apex, tips functionalized by better characterized spin-carrying objects might
be beneficial for ESR measurement in STM, e.g., spin states of a molecule or
spin-polarized Yu-Shiba-Rusinov states. Having the quantized states at the tip
might provide even more sensitive probes to detect and control spins on a
surface. Aside from increasing sensitivity and energy resolutions, having
quantized spin systems at the tip apex enables scanning magnetometry at the
atomic scale. In case of using a sensor spin on a surface to characterize
target spins, the spatial resolution is determined by the surface lattice on
which the target and sensor spins can be located. In addition, the spin states
of the sensor would change on different surfaces. In contrast, having a robust
quantum sensor spin at the tip apex as a mobile sensor provides a universal
solution in characterizing different spin systems and enables 3D mapping of
the target spin with atomic resolution. These advances in the ESR-STM
technique would also be beneficial for mitigating the impact of decoherence
sources. Improving sensitivity and energy resolution facilitates adjusting the
measurement conditions, such as the set-point tunnel current and the tip-
sample distances, in a way to minimize sources of decoherence. Additionally,
even though we protect spins by remotely driving and detecting their states,
the coherence time of the remote spin is still limited by coupling with
substrates. Decoupling the spin from the substrate by increasing thickness of
insulating layers or using spins carried by a molecule with tailored spacing
ligands would enhance the relaxation and coherence times of the spins on
surfaces. In addition, well-protected spin states like nuclear spins or
4f-electrons have been barely explored for their coherent properties using
ESR-STM. Identifying different spin systems that can be probed with ESR-STM is
a necessary step for making this technique more versatile for generalized use.

Figure 17: Rabi oscillations of a remote spin (red) through indirect sensing
and a spin at the STM junction (blue), measured using ESR-STM. The remote spin
(red), as measured using double electron-electron resonance scheme, shows Rabi
oscillations with better signal-to-noise ratio and longer spin coherence time,
compared to the spin at the STM junction (blue) (11). Measurement conditions:
$B=0.656$ T, $T=0.4$ K, (red) $V_{\mathrm{DC}}=50$ mV, $I_{\mathrm{set}}=10$
pA, $V_{\mathrm{RF}}=60$ mV & 120 mV, (blue) $V_{\mathrm{DC}}=50$ mV,
$I_{\mathrm{set}}=5$ pA, $V_{\mathrm{RF}}=120$ mV.

#### Concluding Remarks

Recent progress on indirect sensing in ESR-STM has led to the first
observation of the hyperfine interaction at the single atom level, as well as
the dipolar fields from single atom magnets on a surface. Taking advantage of
precise atom manipulation, indirect sensing of spin states becomes more