††thanks: Corresponding author. Email<EMAIL_ADDRESS>
# New opportunities in condensed matter physics for nanoscale quantum sensors

Jared Rovny  Princeton University, Department of Electrical and Computer
Engineering, Princeton, NJ 08544, USA  Sarang Gopalakrishnan  Princeton
University, Department of Electrical and Computer Engineering, Princeton, NJ
08544, USA  Ania C. Bleszynski Jayich  Department of Physics, University of
California Santa Barbara, Santa Barbara, CA 93106, USA  Patrick Maletinsky
Department of Physics, University of Basel, 4056 Basel, Switzerland  Eugene
Demler  Institute for Theoretical Physics, ETH Zurich, 8093 Zurich,
Switzerland  Nathalie P. de Leon  Princeton University, Department of
Electrical and Computer Engineering, Princeton, NJ 08544, USA

###### Abstract

Nitrogen vacancy (NV) centre quantum sensors provide unique opportunities in
studying condensed matter systems: they are quantitative, noninvasive,
physically robust, offer nanoscale resolution, and may be used across a wide
range of temperatures. These properties have been exploited in recent years to
obtain nanoscale resolution measurements of static magnetic fields arising
from spin order and current flow in condensed matter systems. Compared with
other nanoscale magnetic-field sensors, NV centres have the unique advantage
that they can probe quantities that go beyond average magnetic fields.
Leveraging techniques from magnetic resonance, NV centres can perform high
precision noise sensing, and have given access to diverse systems, such as
fluctuating electrical currents in simple metals [1, 2] and graphene [3], as
well as magnetic dynamics in yttrium iron garnet [4]. In this review we
summarise unique opportunities in condensed matter sensing by focusing on the
connections between specific NV measurements and previously established
physical characteristics that are more readily understood in the condensed
matter community, such as correlation functions and order parameters that are
inaccessible by other techniques, and we describe the technical frontier
enabled by NV centre sensing.

††preprint: APS/123-QED

##  I  Introduction

Much of what we know about condensed matter systems—their structure,
excitations, and dynamics—is inferred from bulk transport and spectroscopy.
Bulk d.c. transport and optical response characterize the spatially averaged
(i.e., zero-momentum) response of a sample when it is globally perturbed. Thus
they only offer an indirect probe of theoretically fundamental properties such
as how excitations disperse. Spectroscopic techniques that do offer momentum
resolution, such as neutron scattering, often require large sample volumes to
achieve measurable signal, making it challenging to study low-dimensional
materials. Moreover, the spatially averaged data that bulk probes—as well as
surface probes like ARPES—give can be incomplete or even misleading in systems
whose response is essentially spatially heterogeneous: the average can be
dominated by atypical parts of the sample [5]. Heterogeneity can arise because
the sample is disordered or because interactions give it a local tendency to
phase-separate [6], inducing the spontaneous formation of static or
fluctuating domains [7]. Thus, quantitatively understanding such phenomena
requires local, momentum-resolved probes.

There are several established nanoscale probes: notably, scanning-tunneling
microscopy (STM) measures the local electronic density of states [8, 9, 10],
scanning SQUID microscopy measures local magnetic fields [11, 12], and
microwave impedance microscopy (MIM) measures local conductivity and
permittivity [13]. While these probes have led to immense advances in our
understanding of materials, electronic structure, and correlated phases, their
modes of operation can restrict the available parameter space for the material
under investigation (for example to low temperatures), and it is difficult to
gain access to local magnetic field textures and dynamics with these probes.
In the past decade, nitrogen-vacancy (NV) centres in diamond have emerged as a
new class of nanoscale sensor that complements these technologies in their
temperature range and in their ability to probe both static and dynamic
properties in a momentum- and frequency-resolved way.

In essence, an NV centre is an optically addressable qubit that can sense its
environment with extremely high spatial resolution [14, 15, 12]. The spatial
resolution is determined by the offset between the NV centre and the material
of interest (typically $5$ to $100$ nm), ultimately limited by the extent of
the electronic wavefunction (around $1$ nm). Because the NV centre offers both
local resolution and the entire suite of qubit-manipulation techniques
developed in the context of nuclear magnetic resonance [16], it enables a host
of new local sensing technologies. As an example of how versatile these probes
are, static magnetic fields shift the qubit splitting [14, 17], the noise
spectral density at the qubit transition frequency can change the qubit’s
$T_{1}$ time [1, 2], and dynamic structure like the spatio-temporal
correlations of the noise in an underlying sample can be detected in a
coherent ($T_{2}$ limited) measurement of multiple NV centres [18].

The momentum resolution of an NV centre comes from changing its distance from
the sample (or using an ensemble of NV centres at different distances).
Notably, the momentum-resolved current noise that NV centres measure goes
beyond what one can measure using techniques like neutron scattering. Probes
like neutron scattering [19], X-ray scattering [20], and electron energy loss
spectroscopy [21] measure the momentum-dependent _density_ response, from
which one can extract the behavior of longitudinal currents (i.e., those along
the wavevector being probed) but not transverse ones. The current noise that
an NV centre detects can also be sensitive to transverse current fluctuations,
and can therefore pick up the response of the electron fluid to being sheared;
the shear response is a key diagnostic of Wigner crystallization [22], as well
as a natural probe of viscous hydrodynamic behavior [23, 24]. Furthermore,
while neutron scattering averages over the bulk of the material, NV centres
enable real-space probing with nanoscale resolution, giving access to
spatially inhomogeneous momentum information.

Figure 1: (a) Overview of an NV centre experiment using a diamond scanning
tip. An NV centre in diamond detects local fields from a target system with
nanometer resolution. Above-bandgap excitation light is used to spin-polarize
the NV centre, and subsequently read out its state through spin-dependent
fluorescence. (b) Approximate frequency ranges and length scales accessible
through common NV centre sensing modalities (see Table 1 for details and Box 1
for methods bridging the $T_{2}$, $T_{1}$ gap). Optimal sensitivities assume
either the minimum detectable field using coherent $T_{2}$ sensing
($\eta_{B}$), or the minimum detectable mean square noise in a 1 Hz band using
incoherent $T_{1}$ relaxometry ($\eta_{S}$). Selected NV centre experiments
are indicated [2, 3, 25, 26, 27, 28, 29, 30, 31], where achieved sensitivities
vary. At DC, comparisons are also made with nanowire magnetic force microscopy
(nw-MFM) and scanning SQUID (SS) measurements [12]. Neutron scattering and
optical pump/probe experiments show approximate length- and time-scales of
system excitations, while for NV centres the length-scales refer to the NV-
sample distance or the square root of areal sensing resolution for ensembles.
Note that this plot is simplified, as e.g. shallow ensembles and $T_{2}$ noise
spectroscopy are also used (see text).

Thus, NV centres offer a new suite of tools for condensed matter
experiment—including many tools that are only just being developed. NV centres
have been employed to measure static local fields from antiferromagnetic order
[32, 33], multiferroic [34] order, and from currents in materials with
electron hydrodynamics [27, 35] and superconductivity [36, 37, 38, 29], as
well as linear dynamics in contexts such as magnon imaging [39, 40] and
conductivity imaging [2]. The last few years have also seen exciting steps
exploring out-of-equilibrium systems, such as a magnon scattering platform
[41] and measurements of an electron-phonon instability in graphene [3], where
newly developed NV centre sensing methods like covariance magnetometry [18]
and gaining access to the noise correlation time via coherence decay [42, 43,
44] may provide insights that are difficult to access with other sensing
techniques.

The present review aims to outline the physics of NV centres as it relates to
condensed-matter sensing experiments, and to illustrate a few recent
developments and proposals in this area. We focus on the parametrization of
different NV centre sensing modalities in terms of physical quantities of
interest for condensed matter physics and review the current state of the art,
including new sensing techniques and sophisticated methods in image
reconstruction, efficient sampling, and sensitivity enhancement that have been
developed within the last few years. We conclude with a discussion of new
opportunities in condensed matter sensing that are enabled by this unique
probe and toolkit.

##  II The physics of NV centres

The NV centre is a point defect in diamond consisting of a substitutional
nitrogen next to a vacancy in the lattice. Its ground state electronic
configuration has spin $S=1$, and this ground state spin can be initialized
and measured even at room temperature, with spin coherence times up to a few
milliseconds [17], and at cryogenic temperatures with coherence times
exceeding 1 second [45]. The long spin coherence time is enabled by weak spin
orbit coupling, low electron-phonon coupling, and a quiet spin environment in
ultrapure diamond [46]. It can be used as a sensor over a remarkably wide
range of parameters, from cryogenic temperatures to nearly 1000 K [47], and
from vacuum to GPa pressures [48, 49], largely because its electronic level
structure allows for nonresonant optical initialization and readout, and
because its structure in diamond is exceptionally robust. Typically, NV
centres are deployed in three broad categories of platforms (see Box 1)
enabled in part by sophisticated diamond fabrication techniques. These
platforms range from cantilevers incorporating individual embedded NV centres
with nanometer scanning resolution to large 2D ensembles of $\sim 10^{12}$ NV
centres per cm2 simultaneously imaged across millimeters. Each platform has
inherent tradeoffs, detailed in Box 1, including spatial resolution versus
sensing time (scanning tips are higher-resolution but slow, while widefield
imaging achieves faster measurements by ensemble averaging at the expense of
spatial resolution) and spatial resolution versus coherence time (NV centres
exhibit shorter coherence times as they approach the diamond surface). In many
cases, a system of interest may be measured using multiple modalities to paint
a more complete picture of its properties [50].

[][1.5em] | Resolvable NV centres | Ensemble NV centres
---|---|---
[][1.5em] | Deep | Shallow ($\sim 5$ nm) | Deep | Shallow ($\sim 5$ nm)
T${}_{2}^{*}$ ($\mu$s) |  470* [51] | - |  0.001 ppm: 8.7* [52] 10 ppm: 1* [53] | -
T2 ($\mu$s) |  1800* [17] |  43 [30] |  0.3 ppm: $200$ [54] 10 ppm: 20* [53] |  $\sim$100/$\upmu$m2: 45* [55]
T1 (ms) |  7.5* [51] |  2 [30] |  3.3 ppm: 2.5 [56] 45 ppm: 0.067 [57] |  $\sim$100/$\upmu$m2: 0.75 [58], 5* [55]
[][1.5em] | $\text{unit}/\sqrt{\text{Hz}}$ | $\text{unit}/\sqrt{\text{Hz}\,\upmu\text{m}^{3}}$
[][1.5em] Sensitivity | Deep | Shallow ($\lesssim 100$ nm) | Deep | Shallow ($\lesssim 100$ nm)
Magnetic (nT) | DC |  72*,${\dagger}$ (bd) [17] |  300 (st) [59] 4300 (st) [28] |  24.6* (qdm) [60] 11.5 (bd)[52] |  176 (bd) [61], 1440 (dac) [48]
AC |  4.3* (bd) [17] |  50 (st) [62] |  2.4* (bd) [25] 5.3 (bd) [52] |  See [58, 55]
Electric ($\text{mV}/\mu\text{m}$) | DC |  89.1 (bd) [63] |  3520 (st) [64] |  2217 (bd) [65] | -
AC |  20.2 (bd) [63] |  24-26 (st) [66, 67] |  11* (bd) [68] | -
Stress (MPa) | - |  0.6${\dagger}$ (dac) [69] |  $0.06$* (qdm) [70] |  3.6 (dac) [48]
Temperature (mK) |  5* (bd) [71] |  24 (bd) [72] |  194 (bd) [73] |  36.8 K: 20 (nd) [74] 118 K: 82 (nd) [74] 300 K: 4.8 (bd) [75] 1000 K: 9 (nd) [47]

(a)

  * *

Makes use of 12C isotopically enriched diamond.

  * ${\dagger}$

Derived using values from separate measurements [17, 69].

(a) Demonstrated properties of different NV centre sensing platforms,
including scanning tip (st), diamond anvil cell (dac), quantum diamond
microscope (qdm), nanostructure/membrane (nsm), nanodiamonds (nd), and bulk
diamond (bd). The listed $T_{2}$ are dynamically decoupled. Room temperature
measurements are given for $T_{1}$ and $T_{2}$. For detailed temperature-
dependent NV properties, see [76, 77, 78]. For detailed NV-density dependence,
see [53, 56, 57]. For detailed depth-dependence, see [30, 42, 79].

An NV centre measurement typically consists of three steps: initialization by
optical polarization into the $\ket{m_{s}=0}$ spin state, interaction with
external fields using a tailored sensing modality (described below and in Box
1), and optical detection through spin-dependent fluorescence (Fig. 1a). The
three broad sensing modalities, covering characteristic frequency ranges from
DC to $\sim$100 GHz, are (A) optically detected magnetic resonance (ODMR)
spectroscopy of the $0\rightarrow\pm 1$ transition frequency for DC sensing;
(B) coherent Ramsey-based protocols which may incorporate frequency-selective
pulse sequences for AC sensing up to $\sim\,$10 MHz frequencies; and (C)
relaxometry, which is performed by monitoring the NV centre decay rate
$\Gamma$ among the ground state spin levels due to fields fluctuating at the
NV transition frequency. The latter technique has been demonstrated up to
$\sim\,$100 GHz [80], where the attainable frequency is determined by the
strength of the external magnetic field and the availability of microwave
hardware used to drive the NV centre spin transitions. While NV centres are
most often used to detect magnetic fields through the induced splitting
$\omega=D_{\text{zfs}}+\gamma_{e}B$, where the zero field splitting is
$D_{\text{zfs}}=2\pi\times 2.87\,$GHz and the electron gyromagnetic ratio is
$\gamma_{e}=-2\pi\times 28.02\,$GHz/T, they may also be used to detect
electric fields, temperature, and strain (Box 1). An important advantage
offered by NV centres is the ability to engineer their sensitivity to the
environment through pulse sequences designed to select certain frequencies
[81] or physical observables [48] (Sec. IV.3).

The sensitivity of an NV centre measurement may be boosted by increasing the
number of NV centres, extending the coherence time, improving the readout
fidelity through techniques like spin-to-charge conversion [82] and repetitive
readout using nuclear spin ancillae [83, 84], and, crucially for nanoscale
sensing, decreasing the distance to the target. Typically achievable
sensitivities are approximately 1 $\mu\text{T}/\sqrt{\text{Hz}}$ at DC [85]
and $50\,\text{nT}/\sqrt{\text{Hz}}$ at AC [27, 86, 62] (see Table 1 for
details of platform-specific metrics). Spatial resolution down to $\sim\,$20
nm [87] has been achieved using scanning tips, while implantation into
unstructured substrates allows for NV centres as close as 2-3 nm to the
surface [88, 89], and furthermore, subdiffraction imaging techniques allow the
NV centre to be laterally located to within a few nanometers [90, 91].

While the NV centre spin coherence time and resonance frequency naively imply
$\sim\,$kHz linewidth and MHz or GHz frequency ranges for Ramsey and
relaxometry, respectively, several protocols have been developed to surpass
these limits. For detecting coherent signals, the NV sensor may be
reinitialized and detected many times within the coherence time of the signal,
enabling sub-millihertz linewidths [25, 92, 93]. To broaden the accessible
frequencies, recent demonstrations have shown that dressed states and
frequency mixing may be used to fill in the frequency gap between Ramsey and
relaxometry protocols to access $\sim\,$100 MHz frequencies, provided mixing
fields with sufficiently high power and frequency are available [94, 95, 96,
97], while auxiliary spins may be leveraged to access different frequencies
while boosting sensitivity [98]. As shown in Fig. 2a, new techniques have also
been developed to improve spatial resolution and imaging, by reconstructing
the source magnetization from the stray field maps measured by the NV centre
using machine learning algorithms [99].

Box 1: NV Centre Modalities and Platforms Sensing modalities The typical
generic NV centre Hamiltonian has the form
$\displaystyle\mathcal{H}=\mathcal{A}S_{z}^{2}+\mathcal{B}S_{z}+\mathcal{C}(S_{y}^{2}-S_{x}^{2})+\mathcal{D}(S_{x}S_{y}+S_{y}S_{x}),$
(1) where (for small applied fields) the separate contributions cause shifting
($\mathcal{A}$) and splitting ($\mathcal{B,C,D}$) of the $\ket{m_{s}=\pm 1}$
states according to the energy diagram above, which describes the approximate
couplings to parallel and perpendicular magnetic fields (B), electric fields
(E), stress ($\sigma$), and temperature (T); note that these sensitivities may
change depending on e.g. temperature and field orientation [100, 101, 67].
Continuous wave ODMR provides a simple and robust method for spectroscopy of
Hamiltonian 1, but is insensitive to AC fields and has limited sensitivity to
DC fields. Pulsed Ramsey-type protocols employ trains of pulses to modulate
the NV centre in the time domain, making the NV centre sensitive to a band of
frequencies centered at the pulsing rate. Pulsed methods are extremely
versatile, and are routinely used for frequency-selective spectroscopy and to
significantly improve measurement sensitivity, including sensitivity to DC
fields through lock-in gradiometry [86]. Lastly, incoherent relaxometry, while
less versatile, provides access to high-frequency fields inducing single-
quantum ($\ket{m_{s}=\pm 1}\rightarrow\ket{m_{s}=0}$) and double-quantum
($\ket{m_{s}=\pm 1}\rightarrow\ket{m_{s}=\mp 1}$) transitions of the NV
centre, where this distinction can be useful in diagnosing whether the
relaxing fields are electric or magnetic in nature [102]. Beyond the
simplified Hamiltonian in Eq. 1, it is often advantageous to intentionally mix
spin states or probe dressed levels of a driven state [67]. Sensing platforms
The broad NV centre platform categories are shown above, and include an NV
centre embedded in an AFM-style scanning tip, deposited diamond nano- and
micro-structures, and low or high densities of NV centres implanted in bulk
substrates. Some platform tradeoffs are indicated above, where the most
relevant are usually light collection efficiency, spatial resolution, and NV
centre coherence. Diamond fabrication and processing have seen significant
advances, including high collection efficiency scanning tips [103, 104] and
“smart-cut” nanoscale diamond membranes [105, 106], where both platforms may
be integrated with existing target structures. Close NV-sample proximity,
enabled by very shallow ion implantation just a few nanometers from the
diamond surface and subsequent fabrication [107, 108], incurs both decoherence
due to surface noise and charge state instabilities due to surface electronic
traps; coherence times have been significantly improved through a better
understanding of surface noise sources and treatment [30, 109, 110, 111],
although achieving bulk-like properties near the surface remains an open
challenge [112]. Improving sensitivity in a given experiment requires a
judicious choice of distance versus coherence time, number of NV centres, and
the readout method (which can feature trade-offs between speed and fidelity)
[83, 82, 113, 114, 115]. Proximity to the surface is also crucial for spatial
resolution, as an NV centre at distance $d$ from the surface is sensitive to
stray fields originating from a region on the surface with spatial extent
proportional to $d$ (see Box 2). Finally, high-density NV centre ensembles
sacrifice spatial resolution for a significant increase in sensitivity; spin-
spin interactions place an upper bound on the useful NV centre density, but
dynamical decoupling protocols tailored to the spin environment can
effectively suppress these interactions [116].

##  III Static Fields

NV magnetometry offers an unmatched combination of spatial resolution and
sensitivity in magnetic imaging [117], together with a wide operational range
in, e.g., temperature, magnetic field, or pressure. Importantly, NV
magnetometry is quantitative, in the sense that it offers self-calibrated,
quantitative mapping of the stray magnetic fields in the vicinity of a sample
of interest 111It is worth noting that NV magnetometry is only strictly
quantitative in the regime of weak magnetic fields ($\lesssim 10~{}$mT), where
the first order response of the NV ODMR frequencies dominates the Zeeman shift
if the NVs spin levels, and transverse magnetic fields only play a negligible
role.. We provide in the following a brief overview of prominent examples of
NV-based static magnetic field imaging for probing a variety of condensed
matter targets. We note that a direct, quantitative determination of physical
properties of samples under investigation always needs to be inferred from the
measured stray field maps. Indeed, careful data-analysis [119] or reverse-
propagation [120, 99] from the measurement plane to the source allows for such
quantitative extraction of physical parameters of samples of interest, and for
the tracking of their evolution with external quantities such as temperature
or magnetic fields.

###  III.1 Quantitative Determination of Magnetisation Strength

The quantitative determination of magnetisation strength of magnetic materials
and, e.g., its dependence on temperature of other control parameters, is one
of the most fundamental characteristics of any magnetic material. However, for
samples of mesoscopic scale, very few experimental tools are available to
obtain such information. NV magnetometry fills this gap and allows for the
precise, quantitative determination of magnetization strengths down to the
nanoscale.

Specific examples where NV magnetometry has been put to use in this context
include the determination of magnetisation strengths in few layers of
magnetically ordered vdW materials, down to the monolayer (Fig. 2a) [121, 59],
and uncompensated surface magnetic moments on magnetoelectric antiferromagnets
[32, 122]. More conventional thin film magnets have also been examined in this
regard [123], but are typically accessible via other experimental approaches
as well. A notable extension of such studies is the determination of
magnetisation strengths in spin-spirals of bulk multiferroics [124, 125].

The relevance of these results relies on their quantitative character, which,
together with the temperature or field dependence of magnetisation, provide
key insight into the theoretical understanding of these materials. As a
further example, NV magnetometry might be highly suited to validate recent
theoretical predictions regarding the nature and strength of surface
magnetisation in magnetoelectric antiferromagnets [126].

Figure 2: Imaging static fields with NV centres. a. The left panels show an NV
centre scanning experiment measuring magnetic domains in a flake of CrI3,
revealing regions where the magnetization appears in integer multiples of the
monolayer magnetization [121]. To infer the source magnetization from the
measured stray field maps (right panels), conventional linear inverse maps may
be replaced by more robust convolutional neural networks (CNNs), which require
no prior training, and which use physics-informed forward propagation from the
magnetisation map $\mathcal{M}^{\prime}$ to the magnetic field map
$\mathcal{B}$ at the measurement plane [99]. b. Locally imaging the Meissner
effect in CeH9 at high pressure [127]. The sample is compressed in a DAC with
an NV-containing layer in the upper diamond (top). ODMR measurements of the
local magnetic field, taken along the linecut shown in the NV fluorescence
image (bottom right), reveal significant inhomogeneities in diamagnetism at
the micron scale (bottom left). c. Scanning NV magnetometry of the local
electron flow profile through a constriction in graphene [35]. The local
current profile (right, carrier density indicated above each panel) shows
characteristic Ohmic peaks at the charge neutrality point (CNP), which breaks
down at low temperature away from CNP, indicating the absence of momentum-
relaxing scattering as electron-electron scattering becomes more dominant. The
bottom left part of panel a is reprinted with permission from [121]. The right
part of panel a is adapted with permission from [99]. Panel b is adapted with
permission from [127]. Panel c is adapted with permission from [35].

###  III.2 Magnetisation Distributions

Quantitative NV magnetometry further enables the disambiguation between
different types of domain walls in magnetic materials, such as Bloch and Néel
domain walls. The latter nomenclature describes how the order parameter in a
magnetic system evolves across domain walls that separate two domains of
homogeneous order parameter – for Bloch (Néel) walls, the evolution is
described by a rotation around a vector that is orthogonal to (contained in)
the domain wall. The identification of the nature of such domain walls is of
crucial importance to understanding and characterising the type of exchange
interactions that contribute to stabilising a given magnetic order [128], with
the Dzyaloshinskii–Moriya interaction being the most prominent example. This
concept was first put to use in seminal work by Tetienne et al. [129], who
unequivocally demonstrated an experimental distinction between different types
of domain walls occurring in thin films of the same magnetic material that was
subjected to different interfacial conditions. Such analyses crucially depend
on a proper characterization of the NV orientation and offset distance, the
latter being the key parameter of concern. For reliable, quantitative
conclusions, such characterisations are to be conducted in-situ, on the sample
under investigation, to avoid systematic errors and concurrent pitfalls in
data analysis.

Further, the study of topologically protected magnetic textures, such as
magnetic skyrmions [130] and their winding numbers is directly accessible and
enabled by scanning NV magnetometry. While a range of experiments have
demonstrated direct imaging and real-space studies of skyrmions [131, 132,
133, 134, 135], quantitative analysis of such data allows for determining
skyrmion winding numbers based on static stray field imaging by scanning NV
magnetometry – a remarkable achievement that has been demonstrated recently
[136]. The latter work had a profound impact on skyrmion research, as it was
amongst the first to experimentally determine the handedness of individual
skyrmions through careful experimentation and data analysis. A still
outstanding challenge is to extend such studies to single-crystalline,
skyrmion-hosting materials [137], where pristine skyrmion behaviour, including
collective modes in skyrmion lattices [138] – and possibly even quantum
dynamics of skyrmions [139] – could be examined in the future.

###  III.3 Superconducting Order

By locally probing the magnetic response of a superconductor, it is possible
to image superconducting order and various parameters of the superconducting
state. The magnetic (London) penetration depth $\lambda_{L}$ determines the
lengthscale over which magnetic field lines can enter a superconductor before
Meissner screening becomes effective. Quantitative determinations of
$\lambda_{L}$, together with its temperature dependence and spatial
anisotropy, provide valuable insight into a superconductor’s superfluid
density and pairing mechanism [140]. Specifically, $\lambda_{L}$ determines
the superfluid density, i.e. the number of electrons that superconduct [141].

NV magnetometry has been employed to directly measure the penetration depth,
in particular in high-temperature superconductors [36, 38, 142, 143]. This has
for instance been achieved by imaging vortices in type-II superconductors,
where a quantitative analysis of vortex stray fields directly yields the
London penetration depth [36]222In the cited case, a thin film sample of YBCO
was investigated. Since the film under investigation was thin compared to
$\lambda_{L}$, the relevant penetration depth in that case was the “Pearl
length”.. Alternatively, $\lambda_{L}$ was also determined by direct
measurements of Meissner-screening in superconductor microstructures [142], or
using wide-field NV magnetometry [145]. This latter approach is notably
applicable to type-I superconductors as well.

Such studies of superconductor properties by quantitative NV magnetometry have
thus far been applied to reasonably well-understood superconductors (e.g.
YBCO), deep in the type-II limit of superconductivity. Future worthwhile
extensions include the study of vortices in superconductors at the crossover
between type-I and type-II superconductivity, where vortex stray fields are
still poorly understood and can only be described by approximative or
numerical models [146, 147, 148]. Such experiments might offer the interesting
perspective of simultaneously determining $\lambda_{L}$ and the superconductor
coherence length $\xi$ by direct magnetic imaging.

NV centres incorporated into a diamond anvil cell (DAC) allow for local
magnetic imaging of materials under high, $\sim$ 100 GPa, pressures [48, 49,
149, 150]. High pressure presents an important knob for tuning microscopic
interactions, and has led to the discovery of novel high temperature
superconductors [151]. Recent work [127] (Fig. 2b) used an NV-infused DAC to
image the spatial structure of superconducting regions in a hydride
superconductor at the micrometer scale, revealing significant inhomogeneities
and informing the materials synthesis of superhydrides. Though nontrivial to
maintain magnetic sensitivity of NV centres at these high pressures, there are
ongoing efforts to extend the pressure range higher [150, 127], a frontier for
spatially resolved magnetic imaging.

###  III.4 Current Imaging

Electrical currents produce magnetic fields according to the Biot-Savart law
that can be detected and imaged via NV centre magnetometry. Spatial maps of
current patterns in a device can distinguish different scattering mechanisms
in a material, providing significantly more information than a standard
transport measurement that yields just one number - the spatially averaged
conductivity. For instance, the path that electrons take in a device depends
on the relative strengths of momentum-conserving scattering (e.g. electron-
electron scattering) and momentum-relaxing scattering (e.g. electron-phonon or
impurity scattering). Recently, NV magnetometry has been applied to
quantitatively probe different transport regimes in a few electronic materials
[50, 27, 35], in particular looking for spatial signatures of current flow in
the rare electron hydrodynamic regime, where electron-electron scattering
dominates and electrons move akin to fluids. Several device geometries that
produce distinctive flow patterns in the hydrodynamic, ballistic, and Ohmic
regime have been proposed and experiments have begun to reveal signatures of
hydrodynamics. However, concerns about the influence of unknown boundary
conditions, and the masquerading of ballistic flow and/or small angle
scattering as hydrodynamic flow, remain in some cases. Nevertheless, direct
spatial mapping of the current flow provides much more direct evidence of
hydrodynamic behavior, and importantly, provides a way to quantitatively probe
the importance of electron-electron interactions (Fig. 2c), and hence is a
good proxy for understanding other open questions in condensed matter, such as
the role of electron-electron interactions in the superconductivity of
graphene [152]. Spatially resolved current imaging can also reveal local
current anomalies or irregularities stemming from, e.g. trivial effects like
variations in the electric potential, due to imperfections or impurities,
whose presence can greatly affect the interpretation of a standard transport
measurement, leading to incorrect conclusions.

Box 2: Dynamical correlators from NV noise spectroscopy In this box we outline
how NV relaxation and decoherence times can be explicitly related to the field
noise at the NV centre, and thus to the physical autocorrelation functions of
the noise source. To keep the discussion concrete, we will focus on magnetic
field noise at the NV centre due to coarse-grained, effectively classical
magnetic or current noise in the sample:
$H_{\mathrm{NV}}=\omega_{0}\mathbf{\hat{a}\cdot\hat{S}}+\gamma_{\text{e}}\mathbf{B}(t)\cdot\mathbf{\hat{S}}$.
Let us define coordinate axes $a,b,c$ such that ${\bf\hat{a}}$ points along
the NV quantization axis and ${\bf\hat{b}},{\bf\hat{c}}$ are perpendicular to
it. For a $T_{1}$ relaxometry experiment, Fermi’s Golden Rule for the spin-1
NV centre [102, 98] implies $\displaystyle
T_{1}^{-1}=\frac{3}{2}\gamma_{\text{e}}^{2}S_{B}^{\perp}(\omega_{0}),\quad
S_{B}^{\perp}(\omega_{0})=\int dt\,e^{i\omega_{0}t}\langle
B_{+}(t)B_{-}(0)\rangle,\quad B_{\pm}=B_{b}\pm iB_{c},$ (2) where the NV
centre transition frequency $\omega_{0}$ determines the detected frequency
(usually a few GHz). For dynamical-decoupling based $T_{2}$ spectroscopy, it
is instead more natural to model the NV centre decoherence as a stretched
exponential $C(t)=\text{exp}\left[-(t/T_{2})^{p}\right]$, where $p$ is the
stretching factor. Then, under dynamical decoupling using e.g. N equally
spaced pulses with interpulse spacing $\tau$ and total time $T=N\tau$,
$T_{2}^{-p}=\frac{\gamma_{\text{e}}^{2}}{\pi T^{p}}\int_{0}^{\infty}d\omega
S_{B}^{\parallel}(\omega)\frac{F_{N}(\omega\tau)}{\omega^{2}},\quad
S_{B}^{\parallel}(\omega_{0})=\int dt\,e^{i\omega_{0}t}\langle
B_{a}(t)B_{a}(0)\rangle,$ (3) where the detected noise frequency is set by the
pulse rate $\tau^{-1}$ (typically less than $100\,$MHz) through the pulse
sequence filter function
$F_{\text{N}}(\omega\tau)=8\sin^{4}(\omega\tau/4)\sin^{2}(N\omega\tau/2)\cos^{-2}(\omega\tau/2)$.
For large $N$ this is sharply peaked at $\omega=\pi/\tau$, and it is common to
approximate the spectral density as
$S_{B}^{\parallel}(\pi/\tau)=-\pi\ln[C(T)]/T$ [89, 102, 30]. It remains to
calculate the noise spectrum $S_{B}(\omega)$ at the NV centre from a given
source, which will depend on the sample and geometry. As an example, we
consider relaxometry in the presence of classical current noise in the sample
[153]. The sample is taken to lie in the $xy$ plane and the NV centre is
displaced by $d$ along the $z$ axis, aligned along axis ${\bf\hat{a}}$ (see
image above). The current $\mathbf{J}(\mathbf{r},t)$ at a point $\mathbf{r}$
in the sample is a classical field that fluctuates slowly enough that its
effect on the NV centre can be treated as instantaneous (this “quasistatic”
assumption always holds in practice). In this case the NV centre experiences a
fluctuating magnetic field that can be related to the current profile through
the Biot-Savart law: $\mathbf{B}(t)=\frac{\mu_{0}}{4\pi}\int
d^{2}\mathbf{r}\,\frac{\mathbf{J}(\mathbf{r},t)\times({\bf\hat{z}}d-{\bf
r})}{|r^{2}+d^{2}|^{3/2}}.$ (4) Thus, Eqs. 2 and 4 relate the NV centre
relaxation time to the current autocorrelation tensor $\langle
J_{i}(\mathbf{x},t)J_{j}(\mathbf{0},0)\rangle$ and a spatial smoothing factor
from the distance dependence. Next, the fluctuation-dissipation theorem (FDT)
relates the current autocorrelation function to the current _response_
function, $\chi_{ij}(q,\omega)$, which will depend on the conductivity
$\sigma_{ij}(q,\omega)$ (although the relation 4 is not precisely
$q$-resolved, the $d$-dependence of $T_{1}$ imposes strong constraints on the
$q$-dependence). For a 2D conductor at high temperatures such that
$\coth(\hbar\omega_{0}/2k_{B}T)\approx 2k_{B}T/(\hbar\omega_{0})$, the noise
density is [23]
$2S^{x,y}_{B}(\omega)=S^{z}_{B}(\omega)=\frac{\mu_{0}^{2}k_{B}T}{16\pi}\varsigma(d,\omega),\quad\varsigma(d,\omega)=\frac{1}{d^{2}}\int_{0}^{\infty}dx\,x\text{e}^{-x}\text{Re}[\sigma^{\text{T}}(x/2d,\omega)],$
(5) where $\sigma^{\text{T}}(q,\omega)$ is the $q$-dependent transverse
conductivity, which can be probed through the distance-dependence of Eq. 5.
Finally, assuming the NV centre axis ${\bf\hat{a}}$ makes angle $\theta$ with
$z$, the relaxation rate is then
$T_{1}^{-1}=\frac{3}{2}\gamma_{e}^{2}S_{B}^{z}(\omega_{0})\left(1+\frac{1}{2}\sin^{2}(\theta)\right)$
[2]. A similar procedure may be followed for other noise sources, like
magnetic fluctuations [154, 31], or other geometries. For instance, diffusive
transport in a conducting film of thickness $a$ and conductivity
$\sigma=\text{Re}[\sigma(\omega_{0})]\approx\sigma(0)$ produces noise
according to Eq. 5 with distance dependence
$\varsigma(d,\omega)=(d^{-1}+(d+a)^{-1})\sigma$, which scales as $d^{-1}$ or
$d^{-2}$ for thick and thin films, respectively [1, 2]. For noise originating
from paramagnetic (e.g. nuclear) spins, the distance dependence is instead
$d^{-6}$ from a single spin [155], $d^{-4}$ from a 2D layer [156], or $d^{-3}$
from a 3D half-space [157].

##  IV Dynamics

In equilibrium, the response of a material is related to its intrinsic thermal
or quantum fluctuations (i.e., the noise it generates) by the fluctuation-
dissipation theorem [158]: for example, the conductivity at frequency $\omega$
is related to the Fourier transform of the current autocorrelation function,
which can be measured using NV magnetometry. In systems that are driven out of
equilibrium, fluctuations and response contain distinct information about the
state of the system: to characterize the steady state, one needs separate
access to both quantities. For example, current noise in nonequilibrium states
is sensitive to properties such as the charge of individual carriers [159],
which equilibrium transport does not give access to. Most solid-state
experimental setups used to study noise [160, 161] are restricted to measuring
current noise in essentially one-dimensional geometries. NV centres can
overcome this restriction and sense local current or magnetic-field
fluctuations in two-dimensional systems. They can therefore be used to explore
questions about nonequilibrium noise that would not previously have been
experimentally accessible. In addition, even in equilibrium, NV centres are
unique in having the ability to probe local wavevector-dependent current
noise.

Noise from a nearby material can affect the NV centre either by depolarizing
it and decreasing its $T_{1}$ time or by dephasing it and affecting its
$T_{2}$ time. Accordingly an NV centre can be used as a noise sensor in one of
two basic modes, relaxometry ($T_{1}$) and Ramsey ($T_{2}$) spectroscopy. In
what follows, we outline how an NV centre couples to noise in the underlying
sample, then review these two basic spectroscopic protocols, as well as recent
extensions that involve measuring correlations among multiple NV centres.

Figure 3: Sensing dynamics through relaxometry. a. Nanoscale conductivity
imaging, using NV centre relaxometry induced by thermal electron motion in the
metal [2]. Imaging of an Al nanopattern resolves features down to 5 nm
(bottom), where the scale bars are 400 nm. b. Electronic noise in graphene
exhibiting amplification along the transport direction, detected using
proximal NV centres [3]. Spatially resolved noise magnetometry was used to
diagnose the anomalous noise as arising from an electron-phonon Cerenkov
instability at high current. c. Wavelength-resolved imaging of magnetic
excitations in YIG using the distance-dependence of the stray field filter
function [40]. Spin waves are excited using a microwave stripline, then self-
interact through magnon-magnon interactions creating local magnetic noise.
Near the sample, short-wavelength incoherent thermal magnons dominate the NV
centre relaxation, while further from the stripline, long-wavelength coherent
magnons at the NV centre transition frequency drive NV centre Rabi
oscillations. Panel a is adapted with permission from [2]. The bottom part of
panel b is reprinted with permission from [3]. Panel c is adapted with
permission from [40].

###  IV.1 What an NV centre senses

In the subspace of the $0$ and $+1$ magnetic levels of the NV centre, the
general Hamiltonian of the NV centre as well as the sample can be written as
$\hat{H}=\gamma_{\text{e}}B_{0}S_{z}+\gamma_{\text{e}}\mathbf{\hat{B}}\cdot\mathbf{\hat{S}}+\hat{H}_{\mathrm{sample}}$:
the magnetic field that couples to the NV is (in general) operator-valued, so
$[\mathbf{\hat{B}},\hat{H}_{\mathrm{sample}}]\neq 0$. While this formulation
is general, it is also intractable, and we will make various approximations to
arrive at a simpler effective Hamiltonian.

The first, most natural simplification is to note that the field at the NV
centre is determined by the properties of a relatively large patch (of
diameter $\geq 10$ nm) of the sample. In most condensed matter contexts, there
are many active degrees of freedom in a patch of this size. Even though each
degree of freedom is quantum, the average of the current or magnetic field
over a patch of this size can be treated as a classical stochastic variable.
This allows us to describe the NV as a qubit driven by a time-dependent
classical field $\mathbf{B}$, so

$\hat{H}_{\mathrm{NV}}=\gamma_{\text{e}}B_{0}S_{z}+\gamma_{\text{e}}\mathbf{B}(t)\cdot\mathbf{\hat{S}}.$
(6)

It suffices (i) to relate the statistics of $\mathbf{B}(t)$ to those of the
dynamical fluctuations in the underlying sample, and (ii) to characterize the
dynamics of the qubit for $\mathbf{B}(t)$ drawn from such an ensemble.

A sample can generate a magnetic field at the NV centre either because it has
magnetic fluctuations or because it has fluctuating currents
$\mathbf{J}(\mathbf{x},t)$ that induce a magnetic field at the NV centre. In
either case, there is a linear kernel relating the magnetic field at the NV
centre to the relevant fluctuating field in the sample. In general this kernel
can be nonlocal in time, and this nonlocality can have significant effects in
three-dimensional metals, where current fluctuations have to travel through
the metal to reach the NV. However, the situation is simpler in two
dimensions, and one can generally make a quasistatic approximation so that the
field at the NV centre is linearly related to the current (or, with the
obvious modifications, the magnetization) in the sample near the NV centre, by
an expression of the form $B_{i}(t)=\sum_{j=1}^{3}\int
d^{3}\mathbf{x}K_{ij}(\mathbf{x},t)J_{j}(\mathbf{x},t)$, where $i,j$ are
Cartesian coordinates. For quasistatic currents the kernel $K_{ij}$ is given
by the Biot-Savart law (Eq. 4), but more generally it will be some convolution
of the current (or magnetization) fluctuations in the sample over a length-
scale $d$.

###  IV.2 $T_{1}$ spectroscopy

One of the earliest applications of NV centres to studying transport involved
$T_{1}$ noise spectroscopy of Johnson noise from silver films [1]. The
calculation in this case is outlined in Box 2, which shows how one can relate
the $T_{1}$ time of the NV to the autocorrelation tensor of the current in the
sample, which in turn can be related (via the fluctuation-dissipation theorem)
to the momentum-dependent conductivity. In the interests of brevity we have
sketched the derivation of Box 2 in the simplest case; however, the strategy
outlined there can be adapted to any form of current noise under the quasi-
static approximation (Eq. 4). By varying the polarizing field and the height
of the NV centre from the sample (e.g., by using an ensemble of NV centres at
different heights), one can tune $q$ and $\omega$. In principle, by varying
the orientation of the sample with respect to the quantization axis, one can
also extract angle-dependent information about the correlations of current
noise.

For diffusive transport in the local approximation, the same formalism may be
applied to directly measure conductivity, enabling nanoscale conductivity
imaging of inhomogeneous materials as demonstrated by Ariyaratne et al. (Fig.
3a) [2]. Beyond linear response theory, recent work used spatially resolved
$T_{1}$ relaxometry in conjunction with global transport to study anomalous
noise in graphene transport, which was ascribed to an electron-phonon Cerenkov
instability (Fig. 3b) [3]. The ability to image the noise source using NV
centres demonstrated that the noise amplification followed charge carriers
rather than current, a critical tool in diagnosing the effect.

Relaxometry has also been applied to the study of magnetic excitations, for
example to study thermally excited magnons in permalloy and yttrium iron
garnet (YIG) [154, 4], where the NV relaxation rate is very accurately
described by models accounting for the number and spectral density of magnons,
as well as the interplay between the magnon k-vector and NV-sample distance.
Beyond thermal magnons, related studies explored ferromagnetic resonance in
driven magnetic systems, describing the mechanism behind broadband NV spin
relaxation and demonstrating coherent interactions between NV centres and spin
waves [162, 163, 164, 165, 166, 167]. More recently, NV centres have enabled
wavenumber-resolved magnon measurements [39, 168, 40] (Fig. 3c), the
development of a magnon scattering platform based on real-space imaging of
spin waves scattered off a target [41], and the detection of spin waves
generated by nonlinear multi-magnon scattering [169].

Theoretical studies suggest that $T_{1}$ spectroscopy can be used to probe
other phenomena in correlated electron systems, for instance to analyze
Bogoliubov quasiparticles in low dimensional superconductors [170, 171]. Such
experiments can distinguish gapless and fully gapped quasiparticles, enabling
identification of different symmetries of the order parameter. Another
potential application of this technique is to probe one dimensional systems,
such as edge states in topological systems. These studies can elucidate the
mechanism of electron scattering, which is still poorly understood [172].
Finally, a promising application of NV relaxometry is to probe spin liquid
states, in which magnetic noise should provide signatures of different types
of spinon dynamics [173, 174].

###  IV.3 T2 spectroscopy

While $T_{1}$ spectroscopy has the advantage of probing relatively high
frequencies (in the GHz range), it has limited bandwidth, and also provides
relatively coarse information about the structure of the noise. For probing a
wider range of low frequencies, and for addressing more detailed questions
such as the character, spectrum, non-gaussianity, etc. of the noise, $T_{2}$
spectroscopy offers a more versatile approach. The basic idea behind $T_{2}$
spectroscopy is to initialize the NV centre on the equator of the Bloch
sphere, and have it precess in the fluctuating field from the sample. Unlike
$T_{1}$ spectroscopy, in this approach one senses the field _along_ the
quantization axis; for $T_{2}$ spectroscopy to be useful, the noise at the
transition frequency of the NV should be negligible.

Suppose the NV centre is initialized in the state
$\ket{+}\equiv\frac{1}{\sqrt{2}}(\ket{\uparrow}+\ket{\downarrow})$. We choose
the quantization axis of the NV to be the $z$ axis of its Bloch sphere, and
define $B_{\parallel}$ to be the component of the local magnetic field along
the quantization axis 333The choice of basis on the Bloch sphere is in general
distinct from the real-space coordinate system. After free evolution for a
time $t$, the NV centre is in a state
$\ket{\psi(t)}=\frac{1}{\sqrt{2}}(\ket{\uparrow}+e^{i\phi_{t}}\ket{\downarrow})$,
where
$\phi_{t}=\gamma_{\text{e}}\int_{0}^{t}dt^{\prime}\,B_{\parallel}(t^{\prime})$.
One can interrupt this evolution with a variety of pulse sequences borrowed
from the NMR literature [176]; in this case, the expression for the
accumulated phase generalizes to [177]

$\phi_{t}=\gamma_{\text{e}}\int_{0}^{t}dt^{\prime}\,s(t^{\prime})B_{\parallel}(t^{\prime}),$
(7)

where $s(t^{\prime})$ is the filter function for a particular pulse sequence.
For example, in a standard spin-echo sequence,
$s(t^{\prime})=\mathrm{sign}(t/2-t^{\prime})$. At the end of the sequence, one
measures the NV centre in the $x$ and $y$ bases; the results can be combined
to reconstruct the expectation value $\langle e^{i\phi_{t}}\rangle$. If the
noise has zero mean, and the filter function satisfies
$\int_{0}^{t}dt^{\prime}s(t^{\prime})=0$, one can use the cumulant expansion
to write $\langle
e^{i\phi_{t}}\rangle=\exp(-\langle\phi_{t}^{2}\rangle/2+\ldots)$, where
$\ldots$ contains non-gaussian corrections. This relates the dephasing rate of
the NV centre to the correlation function

$C_{\phi}(t)=\gamma_{\text{e}}^{2}\int_{0}^{t}\int_{0}^{t}dt^{\prime}dt^{\prime\prime}s(t^{\prime})s(t^{\prime\prime})\left\langle
B_{\parallel}(t^{\prime})B_{\parallel}(t^{\prime\prime})\right\rangle.$ (8)

Finally, the magnetic field at the NV centre can be related to the current in
the sample through Eq. (4). The filter function $s(t)$ controls the range of
frequencies that contribute to decoherence: e.g., if $s(t)$ is a series of
$\pi$ pulses spaced by $\tau$, noise below $\pi/\tau$ will be echoed out and
will not contribute to decoherence. While pulse-defined filter function
bandwidths are limited by the NV centre driving rates (typically less than 100
MHz), new antenna designs and pulse protocols have extended the range for
particular experiments [178, 179, 180].

Unlike $T_{1}$ spectroscopy, which is primarily sensitive to the power
spectrum of the noise, $T_{2}$ spectroscopy allows one to detect higher-order
correlations in the noise. An explicit non-gaussian example, for a qubit
driven by telegraph noise, was worked out in [181]. Exploring such non-
gaussian effects in many-body systems (e.g., near phase transitions) and their
signatures in NV magnetometry remains an important task for future research.

Most applications of $T_{2}$ spectroscopy to date have been in the Gaussian
approximation, and make use of a periodic pulse sequence to approximately
select a single frequency at a time (i.e. assuming that the pulse sequence
filter function is nonzero only at a single frequency $f$, see Eq. 3). This
has found use in studying environmental noise spectra to diagnose NV centre
dephasing sources, especially near surfaces, where dimensional analysis allows
the shape of the coherence decay to report on the nature of the noise bath
[30, 42, 43]. Dynamical decoupling can also be used to selectively couple NV
centres to nuclear spins by matching the pulse rate to a multiple of the
nuclear Larmor frequency; while most examples of nuclear spin sensing have
targeted biological applications [25, 182], an example in condensed matter
includes the measurement of nuclear quadrupole resonance in hexagonal boron
nitride [156]. Other recent applications of $T_{2}$ measurements of electronic
systems include the observation of increased NV centre coherence times in
proximity to a superconductor [29] and local superconductor imaging through
transverse spin relaxometry leveraging this effect [183].

Beyond the Gaussian approximation, the shape of the coherence decay can reveal
dynamical transitions that provide information about the correlation time of
the noise [42, 43], which may be deployed to study dynamics in condensed
matter systems (Fig. 4a). Such noise spectroscopy can provide insights into
phase transitions by measuring the qubit relaxation rate as the system is
tuned near its critical point [184]. As indicated in Fig. 4b, the scaling of
the NV centre dephasing $\braket{\phi_{t}^{2}}$ with measurement time (as
discussed above and in Fig. 4a), and in particular, its dependence on
temperature T relative to the critical temperature Tc and on the NV-sample
distance $d$ relative to the correlation length $\xi$, can report on the
critical exponents of the transition as the system is tuned near criticality.

Figure 4: Ramsey-based noise spectroscopy. a. The exponential stretching
factor (slope of these lines, see Box 2) undergoes a transition at a depth-
dependent critical time determined by the correlation time of the noise, from
$p=2$ (Gaussian) decay to $p=2/3$, consistent with a reconfiguring 2D noise
bath [42]. b. Near a phase transition, the dephasing time T${}_{2}^{*}$ (upper
panel) and decoherence dynamics like those shown in a (lower panel) can
characterize the transition (e.g. diagnose the critical temperature Tc and the
presence of order parameter conservation) through their dependence on the
system temperature relative to Tc and on the NV-sample distance $d$ relative
to the correlation length $\xi$ [184]. c. The schematic shows two NV centres
experience fluctuating magnetic fields that are correlated (as from correlated
excitations in a target condensed matter system) or uncorrelated (as from
local fluctuating nuclear spins) between the locations of the two NV centres.
Whereas two NV centres 1 and 2 may be used as individual spectrometers of
their local noise spectra (right, top), covariance spectroscopy reveals which
noise sources are correlated and which are uncorrelated between the two NV
centres (right, bottom) [18]. Panel a is adapted with permission from [42].
Panel b is adapted with permission from [184]. The right part of panel c is
adapted with permission from [18].

###  IV.4 Covariance sensing

The spectroscopic methods described above are spatially resolved in the sense
that an NV centre at distance $d$ from the sample will sense noise averaged
over a length-scale $\sim d$. However, there are both technical and
fundamental limitations to how much spatial information one can extract. At
the technical level, moving the NV centre far from the sample makes it couple
more weakly, limiting the scope for sensing long-range correlations. More
fundamentally, extracting two-point temporal correlation functions from Eq.
(7) is only possible if one assumes the noise to be stationary in time; this
assumption fails in many nonequilibrium settings. A recently developed
technique circumvents these limitations by using multiple NV centres to
directly measure correlation functions (Fig. 4c) [18].

The simplest version of the covariance sensing protocol proceeds as follows.
Two NV centres are initialized, at positions $\mathbf{x}_{1}$ and
$\mathbf{x}_{2}$, and times $t_{1}$ and $t_{2}$, along the equator of the
Bloch sphere. They precess in a spatio-temporally correlated magnetic field
$\mathbf{B}(\mathbf{x},t)$ from the underlying sample. Finally, the phase is
mapped to population and both NV centres are measured along the $z$ axis. The
classical correlation between these measurement outcomes is related to the
correlation between the phase picked up by the two NV centres, which can be
reduced (as before) to the noise correlations in the sample. In practice,
because of the limited detection efficiency for each measurement, the number
of measurements required for a precise estimate of the correlation function is
quite large, although it has been mitigated using higher fidelity readout
schemes such as spin-to-charge conversion [82]. In a proof-of-principle
experiment it was shown that a noise sensitivity of approximately
$100\,\text{nT}^{2}/\sqrt{\text{Hz}}$ could be achieved using this approach
[18], improvable with readout methods optimally tailored to detect multiple NV
centres. Related recent experiments have used resonant optical readout of NV
centres at low temperature to detect correlated electric field dynamics
induced by local trapped charges in the diamond [185, 186].

##  V Conclusions and Perspectives

In the bulk of this review, we have outlined the ways in which NV centres can
be used as nanoscale sensors of both static order and dynamical fluctuations
in condensed matter systems. We close with a discussion of some unexplored
opportunities that this new technique affords us, as well as some likely near-
term technical advances.

NV magnetometry can shed light on the microscopic origins of magnetically
ordered condensed matter systems, which exhibit only weak, uncompensated
moments. A prime example are antiferromagnets, which are typically considered
unsuitable for direct magnetic imaging. There, NV magnetometry has already
found applications, e.g. in magnetoelectrics [32, 122, 28, 187], where
uncompensated surface moments, collinear with the antiferromagnets’ bulk Néel
vector intrinsically occur [188] and are measurable by NV magnetometry. A
question of high relevance is how generally applicable NV magnetometry is to
the study of antiferromagnetism. Indeed, the occurrence of weak, uncompensated
moments appears ubiquitous in antiferromagnets. Mechanisms such as spin
canting, e.g., due to chiral spin-spin interactions [189], or spatially
inhomogeneous Néel vector spin textures [190] have been predicted to lead to
such moments and thereby to non-zero stray magnetic fields that may be
detectable by NV magnetometry. Such mechanism would enable generic approaches
for, e.g., nanoscale domain imaging in antiferromagnets using static magnetic
imaging – an exciting avenue that remains to be explored in the future.
Further avenues in DC NV magnetometry in condensed matter physics include the
nature and existence of time-reversal symmetry breaking in e.g. quasi-2D
Kagome materials [191] or in unconventional superconductors such as Sr2RuO4
[192]. Observations of time-reversal symmetry breaking in this presumably
nonmagnetic compound could be explained by long-theorized, yet elusive, flux
phases [193]; local, quantitative, and sensitive experimental probes are
necessary to validate the theories. For instance, the recent experimental
demonstrations of orbital magnetism in Moiré systems [194] imply many
potential applications, but very little is known about the underlying origins
of this unusual magnetism and its relations to topology and electron
correlations.

Beyond imaging magnetic structure, two natural settings in which NV centres
can provide fundamentally new information about condensed matter systems are
(i) systems that are strongly spatially inhomogeneous on nanometer length-
scales, and (ii) systems driven out of equilibrium. The former class includes
a wide variety of strongly correlated two-dimensional systems, from disordered
superconducting films [195] to cuprates [196] and Moiré materials [197], as
well as phenomena like two-dimensional metal to insulator transitions [198,
199, 200, Sung2023]. In all of these systems, there is substantial evidence
for spatial inhomogeneity, mostly from STM experiments. NV centres offer a
complementary probe: instead of measuring the local density of states (as STM
does), it measures the local magnetic field at the surface, allowing one to
probe finite-frequency fluctuations with very high spatial resolution. The
insights from NV sensing have the potential to shed light on many longstanding
puzzles involving the role of spatial inhomogeneity, rare-region effects, etc.
on quantum phases and phase transitions in two dimensions [201, 202].

The other setting in which NV sensing (especially $T_{2}$ sensing, covariance
magnetometry, and related approaches) can shed new light on many-body physics
involves phase transitions and collective phenomena in systems driven out of
equilibrium, either transiently (e.g., through a sudden current pulse) or in
the presence of a steady-state current drive. Nonequilibrium driving is known
to change the critical properties at quantum phase transitions [203, 204] and
also to give rise to phenomena such as light-induced superconductivity [205].
In these nonequilibrium states, current noise inherently carries information
that is distinct from transport; in certain models, one can explicitly show
that nonequilibrium noise allows one to distinguish between physical
mechanisms that have very similar transport signatures. By shedding light on
these mechanisms, nanoscale noise sensing has the potential to discriminate
between the many proposed physical mechanisms for unconventional collective
behavior in correlated systems.

###### Acknowledgements.

We wish to acknowledge helpful discussions with Shimon Kolkowitz, Liang Jiang,
Ali Yazdani, Pavel Dolgirev, Peter Armitage, Jamir Marino, Atac Imamoglu,
Misha Lukin, Hossein Hosseinabadi, Amir Yacoby, and Norman Yao. This work was
supported by the National Science Foundation (QuSEC-TAQS OSI 2326767 and
Princeton University’s Materials Research Science and Engineering Center
DMR-2011750) (N.dL. and S.G.); the Gordon and Betty Moore Foundation (grant
DOI 10.37807/gbmf12237) (N.dL.); the Intelligence Community Postdoctoral
Research Fellowship Program by the Oak Ridge Institute for Science and
Education (ORISE) through an interagency agreement between the US Department
of Energy and the Office of the Director of National Intelligence (ODNI)
(J.R.); the European Research Council’s grant “QS2DM” and from the Swiss
National Science Foundation through Project $188521$ (P.M.); the Gordon and
Betty Moore Foundation’s EPiQS Initiative via Grant GBMF10279 and the DOE
QNEXT (A.J.); the DOE Q-NEXT Center (Grant No. DOE 1F-60579) (A.J.); the SNSF
project (200021-212899); and the Swiss State Secretariat for Education,
Research and Innovation (contract number UeM019-1)

## References

  * Kolkowitz  _et al._ [2015] S. Kolkowitz, A. Safira, A. A. High, R. C. Devlin, S. Choi, Q. P. Unterreithmeier, D. Patterson, A. S. Zibrov, V. E. Manucharyan, H. Park, and M. D. Lukin, Probing Johnson noise and ballistic transport in normal metals with a single-spin qubit, Science 347, 1129 (2015).
  * Ariyaratne  _et al._ [2018] A. Ariyaratne, D. Bluvstein, B. A. Myers, and A. C. B. Jayich, Nanoscale electrical conductivity imaging using a nitrogen-vacancy center in diamond, Nature Communications 9, 2406 (2018).
  * Andersen  _et al._ [2019] T. I. Andersen, B. L. Dwyer, J. D. Sanchez-Yamagishi, J. F. Rodriguez-Nieva, K. Agarwal, K. Watanabe, T. Taniguchi, E. A. Demler, P. Kim, H. Park, and M. D. Lukin, Electron-phonon instability in graphene revealed by global and local noise probes, Science 364, 154 (2019).
  * Du  _et al._ [2017] C. Du, T. Van Der Sar, T. X. Zhou, P. Upadhyaya, F. Casola, H. Zhang, M. C. Onbasli, C. A. Ross, R. L. Walsworth, Y. Tserkovnyak, and A. Yacoby, Control and local measurement of the spin chemical potential in a magnetic insulator, Science 357, 195 (2017).
  * Vojta [2010] T. Vojta, Quantum griffiths effects and smeared phase transitions in metals: Theory and experiment, Journal of Low Temperature Physics 161, 299 (2010).
  * Jamei  _et al._ [2005] R. Jamei, S. Kivelson, and B. Spivak, Universal aspects of coulomb-frustrated phase separation, Phys. Rev. Lett. 94, 056805 (2005).
  * Huang  _et al._ [2017] E. W. Huang, C. B. Mendl, S. Liu, S. Johnston, H.-C. Jiang, B. Moritz, and T. P. Devereaux, Numerical evidence of fluctuating stripes in the normal state of high-t c cuprate superconductors, Science 358, 1161 (2017).
  * Yazdani  _et al._ [2016] A. Yazdani, E. H. Da Silva Neto, and P. Aynajian, Spectroscopic Imaging of Strongly Correlated Electronic States, Annual Review of Condensed Matter Physics 7, 11 (2016).
  * Yin  _et al._ [2021] J.-X. Yin, S. H. Pan, and M. Zahid Hasan, Probing topological quantum matter with scanning tunnelling microscopy, Nature Reviews Physics 3, 249 (2021).
  * Jäck  _et al._ [2021] B. Jäck, Y. Xie, and A. Yazdani, Detecting and distinguishing Majorana zero modes with the scanning tunnelling microscope, Nature Reviews Physics 3, 541 (2021).
  * Persky  _et al._ [2022] E. Persky, I. Sochnikov, and B. Kalisky, Studying Quantum Materials with Scanning SQUID Microscopy, Annual Review of Condensed Matter Physics 13, 385 (2022).
  * Marchiori  _et al._ [2021] E. Marchiori, L. Ceccarelli, N. Rossi, L. Lorenzelli, C. L. Degen, and M. Poggio, Nanoscale magnetic field imaging for 2D materials, Nature Reviews Physics 4, 49 (2021).
  * Barber  _et al._ [2021] M. E. Barber, E. Y. Ma, and Z.-X. Shen, Microwave impedance microscopy and its application to quantum materials, Nature Reviews Physics 4, 61 (2021).
  * Maze  _et al._ [2008] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M. Taylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan, A. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D. Lukin, Nanoscale magnetic sensing with an individual electronic spin in diamond, Nature 455, 644 (2008).
  * Abobeih  _et al._ [2019] M. H. Abobeih, J. Randall, C. E. Bradley, H. P. Bartling, M. A. Bakker, M. J. Degen, M. Markham, D. J. Twitchen, and T. H. Taminiau, Atomic-scale imaging of a 27-nuclear-spin cluster using a quantum sensor, Nature 576, 411 (2019).
  * Vandersypen and Chuang [2005a] L. M. K. Vandersypen and I. L. Chuang, Nmr techniques for quantum control and computation, Rev. Mod. Phys. 76, 1037 (2005a).
  * Balasubramanian  _et al._ [2009] G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, Ultralong spin coherence time in isotopically engineered diamond, Nature Materials 8, 383 (2009).
  * Rovny  _et al._ [2022] J. Rovny, Z. Yuan, M. Fitzpatrick, A. I. Abdalla, L. Futamura, C. Fox, M. C. Cambria, S. Kolkowitz, and N. P. De Leon, Nanoscale covariance magnetometry with diamond quantum sensors, Science 378, 1301 (2022).
  * Boothroyd [2020] A. T. Boothroyd, Basic Concepts, in  _Principles of Neutron Scattering from Condensed Matter_ (Oxford University PressOxford, 2020) 1st ed., pp. 1–30.
  * Ament  _et al._ [2011] L. J. P. Ament, M. Van Veenendaal, T. P. Devereaux, J. P. Hill, and J. Van Den Brink, Resonant inelastic x-ray scattering studies of elementary excitations, Reviews of Modern Physics 83, 705 (2011).
  * Vig  _et al._ [2017] S. Vig, A. Kogar, M. Mitrano, A. Husain, L. Venema, M. Rak, V. Mishra, P. Johnson, G. Gu, E. Fradkin, M. Norman, and P. Abbamonte, Measurement of the dynamic charge response of materials using low-energy, momentum-resolved electron energy-loss spectroscopy (M-EELS), SciPost Physics 3, 026 (2017).
  * Dolgirev  _et al._ [2023] P. E. Dolgirev, I. Esterlis, A. A. Zibrov, M. D. Lukin, T. Giamarchi, and E. Demler, Local noise spectroscopy of Wigner crystals in two-dimensional materials, Arxiv 2308, 10.48550/ARXIV.2308.16243 (2023), publisher: arXiv Version Number: 1\.
  * Agarwal  _et al._ [2017] K. Agarwal, R. Schmidt, B. Halperin, V. Oganesyan, G. Zaránd, M. D. Lukin, and E. Demler, Magnetic noise spectroscopy as a probe of local electronic correlations in two-dimensional systems, Physical Review B 95, 155107 (2017).
  * Lucas and Fong [2018] A. Lucas and K. C. Fong, Hydrodynamics of electrons in graphene, Journal of Physics: Condensed Matter 30, 053001 (2018).
  * Glenn  _et al._ [2018] D. R. Glenn, D. B. Bucher, J. Lee, M. D. Lukin, H. Park, and R. L. Walsworth, High-resolution magnetic resonance spectroscopy using a solid-state spin sensor, Nature 555, 351 (2018).
  * Hart  _et al._ [2021] C. A. Hart, J. M. Schloss, M. J. Turner, P. J. Scheidegger, E. Bauch, and R. L. Walsworth, N - V –Diamond Magnetic Microscopy Using a Double Quantum 4-Ramsey Protocol, Physical Review Applied 15, 044020 (2021).
  * Vool  _et al._ [2021] U. Vool, A. Hamo, G. Varnavides, Y. Wang, T. X. Zhou, N. Kumar, Y. Dovzhenko, Z. Qiu, C. A. C. Garcia, A. T. Pierce, J. Gooth, P. Anikeeva, C. Felser, P. Narang, and A. Yacoby, Imaging phonon-mediated hydrodynamic flow in WTe2, Nature Physics 17, 1216 (2021).
  * Wörnle  _et al._ [2021] M. S. Wörnle, P. Welter, M. Giraldo, T. Lottermoser, M. Fiebig, P. Gambardella, and C. L. Degen, Coexistence of bloch and néel walls in a collinear antiferromagnet, Phys. Rev. B 103, 094426 (2021).
  * Monge  _et al._ [2023] R. Monge, T. Delord, N. V. Proscia, Z. Shotan, H. Jayakumar, J. Henshaw, P. R. Zangara, A. Lozovoi, D. Pagliero, P. D. Esquinazi, T. An, I. Sodemann, V. M. Menon, and C. A. Meriles, Spin Dynamics of a Solid-State Qubit in Proximity to a Superconductor, Nano Letters 23, 422 (2023).
  * Sangtawesin  _et al._ [2019] S. Sangtawesin, B. L. Dwyer, S. Srinivasan, J. J. Allred, L. V. Rodgers, K. De Greve, A. Stacey, N. Dontschuk, K. M. O’Donnell, D. Hu, D. A. Evans, C. Jaye, D. A. Fischer, M. L. Markham, D. J. Twitchen, H. Park, M. D. Lukin, and N. P. De Leon, Origins of Diamond Surface Noise Probed by Correlating Single-Spin Measurements with Surface Spectroscopy, Physical Review X 9, 031052 (2019).
  * Huang  _et al._ [2023] M. Huang, Z. Sun, G. Yan, H. Xie, N. Agarwal, G. Ye, S. H. Sung, H. Lu, J. Zhou, S. Yan, S. Tian, H. Lei, R. Hovden, R. He, H. Wang, L. Zhao, and C. R. Du, Revealing intrinsic domains and fluctuations of moiré magnetism by a wide-field quantum microscope, Nature Communications 14, 5259 (2023).
  * Appel  _et al._ [2019] P. Appel, B. J. Shields, T. Kosub, N. Hedrich, R. Hübner, J. Fassbender, D. Makarov, and P. Maletinsky, Nanomagnetism of magnetoelectric granular thin-film antiferromagnets, Nano Letters 19, 1682 (2019).
  * Finco  _et al._ [2021] A. Finco, A. Haykal, R. Tanos, F. Fabre, S. Chouaieb, W. Akhtar, I. Robert-Philip, W. Legrand, F. Ajejas, K. Bouzehouane, N. Reyren, T. Devolder, J.-P. Adam, J.-V. Kim, V. Cros, and V. Jacques, Imaging non-collinear antiferromagnetic textures via single spin relaxometry, Nature Communications 12, 767 (2021).
  * Haykal  _et al._ [2020] A. Haykal, J. Fischer, W. Akhtar, J.-Y. Chauleau, D. Sando, A. Finco, F. Godel, Y. A. Birkhölzer, C. Carrétéro, N. Jaouen, M. Bibes, M. Viret, S. Fusil, V. Jacques, and V. Garcia, Antiferromagnetic textures in BiFeO3 controlled by strain and electric field, Nature Communications 11, 1704 (2020).
  * Jenkins  _et al._ [2022] A. Jenkins, S. Baumann, H. Zhou, S. A. Meynell, Y. Daipeng, K. Watanabe, T. Taniguchi, A. Lucas, A. F. Young, and A. C. Bleszynski Jayich, Imaging the Breakdown of Ohmic Transport in Graphene, Physical Review Letters 129, 087701 (2022).
  * Thiel  _et al._ [2016] L. Thiel, D. Rohner, M. Ganzhorn, P. Appel, E. Neu, B. Müller, R. Kleiner, D. Koelle, and P. Maletinsky, Quantitative nanoscale vortex imaging using a cryogenic quantum magnetometer, Nature Nanotechnology 11, 677 (2016).
  * Pelliccione  _et al._ [2016] M. Pelliccione, A. Jenkins, P. Ovartchaiyapong, C. Reetz, E. Emmanouilidou, N. Ni, and A. C. Bleszynski Jayich, Scanned probe imaging of nanoscale magnetism at cryogenic temperatures with a single-spin quantum sensor, Nature Nanotechnology 11, 700 (2016).
  * Schlussel  _et al._ [2018] Y. Schlussel, T. Lenz, D. Rohner, Y. Bar-Haim, L. Bougas, D. Groswasser, M. Kieschnick, E. Rozenberg, L. Thiel, A. Waxman, J. Meijer, P. Maletinsky, D. Budker, and R. Folman, Wide-field imaging of superconductor vortices with electron spins in diamond, Phys. Rev. Appl. 10, 034032 (2018).
  * Lee-Wong  _et al._ [2020] E. Lee-Wong, R. Xue, F. Ye, A. Kreisel, T. Van Der Sar, A. Yacoby, and C. R. Du, Nanoscale Detection of Magnon Excitations with Variable Wavevectors Through a Quantum Spin Sensor, Nano Letters 20, 3284 (2020).
  * Simon  _et al._ [2022] B. G. Simon, S. Kurdi, J. J. Carmiggelt, M. Borst, A. J. Katan, and T. Van Der Sar, Filtering and Imaging of Frequency-Degenerate Spin Waves Using Nanopositioning of a Single-Spin Sensor, Nano Letters 22, 9198 (2022).
  * Zhou  _et al._ [2021] T. X. Zhou, J. J. Carmiggelt, L. M. Gächter, I. Esterlis, D. Sels, R. J. Stöhr, C. Du, D. Fernandez, J. F. Rodriguez-Nieva, F. Büttner, E. Demler, and A. Yacoby, A magnon scattering platform, Proceedings of the National Academy of Sciences 118, e2019473118 (2021).
  * Dwyer  _et al._ [2022] B. L. Dwyer, L. V. Rodgers, E. K. Urbach, D. Bluvstein, S. Sangtawesin, H. Zhou, Y. Nassab, M. Fitzpatrick, Z. Yuan, K. De Greve, E. L. Peterson, H. Knowles, T. Sumarac, J.-P. Chou, A. Gali, V. Dobrovitski, M. D. Lukin, and N. P. De Leon, Probing Spin Dynamics on Diamond Surfaces Using a Single Quantum Sensor, PRX Quantum 3, 040328 (2022).
  * Davis  _et al._ [2023] E. J. Davis, B. Ye, F. Machado, S. A. Meynell, W. Wu, T. Mittiga, W. Schenken, M. Joos, B. Kobrin, Y. Lyu, Z. Wang, D. Bluvstein, S. Choi, C. Zu, A. C. B. Jayich, and N. Y. Yao, Probing many-body dynamics in a two-dimensional dipolar spin ensemble, Nature Physics 19, 836 (2023).
  * Joos  _et al._ [2022] M. Joos, D. Bluvstein, Y. Lyu, D. Weld, and A. Bleszynski Jayich, Protecting qubit coherence by spectrally engineered driving of the spin environment, npj Quantum Information 8, 47 (2022).
  * Abobeih  _et al._ [2018] M. H. Abobeih, J. Cramer, M. A. Bakker, N. Kalb, M. Markham, D. J. Twitchen, and T. H. Taminiau, One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment, Nature Communications 9, 2552 (2018).
  * Childress and Hanson [2013] L. Childress and R. Hanson, Diamond NV centers for quantum computing and quantum networks, MRS Bulletin 38, 134 (2013).
  * Liu  _et al._ [2019] G.-Q. Liu, X. Feng, N. Wang, Q. Li, and R.-B. Liu, Coherent quantum control of nitrogen-vacancy center spins near 1000 kelvin, Nature Communications 10, 1344 (2019).
  * Hsieh  _et al._ [2019] S. Hsieh, P. Bhattacharyya, C. Zu, T. Mittiga, T. J. Smart, F. Machado, B. Kobrin, T. O. Höhn, N. Z. Rui, M. Kamrani, S. Chatterjee, S. Choi, M. Zaletel, V. V. Struzhkin, J. E. Moore, V. I. Levitas, R. Jeanloz, and N. Y. Yao, Imaging stress and magnetism at high pressures using a nanoscale quantum sensor, Science 366, 1349 (2019).
  * Lesik  _et al._ [2019] M. Lesik, T. Plisson, L. Toraille, J. Renaud, F. Occelli, M. Schmidt, O. Salord, A. Delobbe, T. Debuisschert, L. Rondin, P. Loubeyre, and J.-F. Roch, Magnetic measurements on micrometer-sized samples under high pressure using designed NV centers, Science 366, 1359 (2019).
  * Ku  _et al._ [2020] M. J. H. Ku, T. X. Zhou, Q. Li, Y. J. Shin, J. K. Shi, C. Burch, L. E. Anderson, A. T. Pierce, Y. Xie, A. Hamo, U. Vool, H. Zhang, F. Casola, T. Taniguchi, K. Watanabe, M. M. Fogler, P. Kim, A. Yacoby, and R. L. Walsworth, Imaging viscous flow of the Dirac fluid in graphene, Nature 583, 537 (2020).
  * Maurer  _et al._ [2012] P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, Room-Temperature Quantum Bit Memory Exceeding One Second, Science 336, 1283 (2012).
  * Barry  _et al._ [2023] J. F. Barry, M. H. Steinecker, S. T. Alsid, J. Majumder, L. M. Pham, M. F. O’Keefe, and D. A. Braje, Sensitive AC and DC Magnetometry with Nitrogen-Vacancy Center Ensembles in Diamond, Arxiv 2305 (2023).
  * Bauch  _et al._ [2020] E. Bauch, S. Singh, J. Lee, C. A. Hart, J. M. Schloss, M. J. Turner, J. F. Barry, L. M. Pham, N. Bar-Gill, S. F. Yelin, and R. L. Walsworth, Decoherence of ensembles of nitrogen-vacancy centers in diamond, Physical Review B 102, 134210 (2020).
  * Wang  _et al._ [2021] Z. Wang, C. McPherson, R. Kadado, N. Brandt, S. Edwards, W. Casey, and N. Curro, ac Sensing Using Nitrogen-Vacancy Centers in a Diamond Anvil Cell up to 6 GPa, Physical Review Applied 16, 054014 (2021).
  * Rodgers  _et al._ [2024] L. V. H. Rodgers, S. T. Nguyen, J. H. Cox, K. Zervas, Z. Yuan, S. Sangtawesin, A. Stacey, C. Jaye, C. Weiland, A. Pershin, A. Gali, L. Thomsen, S. A. Meynell, L. B. Hughes, A. C. B. Jayich, X. Gui, R. J. Cava, R. R. Knowles, and N. P. De Leon, Diamond surface functionalization via visible light–driven C–H activation for nanoscale quantum sensing, Proceedings of the National Academy of Sciences 121, e2316032121 (2024).
  * Jarmola  _et al._ [2015] A. Jarmola, A. Berzins, J. Smits, K. Smits, J. Prikulis, F. Gahbauer, R. Ferber, D. Erts, M. Auzinsh, and D. Budker, Longitudinal spin-relaxation in nitrogen-vacancy centers in electron irradiated diamond, Applied Physics Letters 107, 242403 (2015).
  * Choi  _et al._ [2017] J. Choi, S. Choi, G. Kucsko, P. C. Maurer, B. J. Shields, H. Sumiya, S. Onoda, J. Isoya, E. Demler, F. Jelezko, N. Y. Yao, and M. D. Lukin, Depolarization Dynamics in a Strongly Interacting Solid-State Spin Ensemble, Physical Review Letters 118, 093601 (2017).
  * Liu  _et al._ [2022] K. S. Liu, A. Henning, M. W. Heindl, R. D. Allert, J. D. Bartl, I. D. Sharp, R. Rizzato, and D. B. Bucher, Surface NMR using quantum sensors in diamond, Proceedings of the National Academy of Sciences 119, e2111607119 (2022).
  * Sun  _et al._ [2021] Q.-C. Sun, T. Song, E. Anderson, A. Brunner, J. Förster, T. Shalomayeva, T. Taniguchi, K. Watanabe, J. Gräfe, R. Stöhr, X. Xu, and J. Wrachtrup, Magnetic domains and domain wall pinning in atomically thin crbr3 revealed by nanoscale imaging, Nature Communications 12, 1989 (2021).
  * Tang  _et al._ [2023] J. Tang, Z. Yin, C. A. Hart, J. W. Blanchard, J. T. Oon, S. Bhalerao, J. M. Schloss, M. J. Turner, and R. L. Walsworth, Quantum diamond microscope for dynamic imaging of magnetic fields, AVS Quantum Science 5, 044403 (2023).
  * Waxman  _et al._ [2014] A. Waxman, Y. Schlussel, D. Groswasser, V. M. Acosta, L.-S. Bouchard, D. Budker, and R. Folman, Diamond magnetometry of superconducting thin films, Physical Review B 89, 054509 (2014).
  * Palm  _et al._ [2022] M. Palm, W. Huxter, P. Welter, S. Ernst, P. Scheidegger, S. Diesch, K. Chang, P. Rickhaus, T. Taniguchi, K. Watanabe, K. Ensslin, and C. Degen, Imaging of Submicroampere Currents in Bilayer Graphene Using a Scanning Diamond Magnetometer, Physical Review Applied 17, 054008 (2022).
  * Dolde  _et al._ [2011] F. Dolde, H. Fedder, M. W. Doherty, T. Nöbauer, F. Rempp, G. Balasubramanian, T. Wolf, F. Reinhard, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, Electric-field sensing using single diamond spins, Nature Physics 7, 459 (2011).
  * Bian  _et al._ [2021] K. Bian, W. Zheng, X. Zeng, X. Chen, R. Stöhr, A. Denisenko, S. Yang, J. Wrachtrup, and Y. Jiang, Nanoscale electric-field imaging based on a quantum sensor and its charge-state control under ambient condition, Nature Communications 12, 2457 (2021).
  * Chen  _et al._ [2017] E. H. Chen, H. A. Clevenson, K. A. Johnson, L. M. Pham, D. R. Englund, P. R. Hemmer, and D. A. Braje, High-sensitivity spin-based electrometry with an ensemble of nitrogen-vacancy centers in diamond, Physical Review A 95, 053417 (2017).
  * Qiu  _et al._ [2022] Z. Qiu, A. Hamo, U. Vool, T. X. Zhou, and A. Yacoby, Nanoscale electric field imaging with an ambient scanning quantum sensor microscope, npj Quantum Information 8, 107 (2022).
  * Huxter  _et al._ [2023] W. S. Huxter, M. F. Sarott, M. Trassin, and C. L. Degen, Imaging ferroelectric domains with a single-spin scanning quantum sensor, Nature Physics 19, 644 (2023).
  * Michl  _et al._ [2019] J. Michl, J. Steiner, A. Denisenko, A. Bülau, A. Zimmermann, K. Nakamura, H. Sumiya, S. Onoda, P. Neumann, J. Isoya, and J. Wrachtrup, Robust and Accurate Electric Field Sensing with Solid State Spin Ensembles, Nano Letters 19, 4904 (2019).
  * Doherty  _et al._ [2014] M. W. Doherty, V. V. Struzhkin, D. A. Simpson, L. P. McGuinness, Y. Meng, A. Stacey, T. J. Karle, R. J. Hemley, N. B. Manson, L. C. Hollenberg, and S. Prawer, Electronic Properties and Metrology Applications of the Diamond NV - Center under Pressure, Physical Review Letters 112, 047601 (2014).
  * Marshall  _et al._ [2022] M. C. Marshall, R. Ebadi, C. Hart, M. J. Turner, M. J. Ku, D. F. Phillips, and R. L. Walsworth, High-Precision Mapping of Diamond Crystal Strain Using Quantum Interferometry, Physical Review Applied 17, 024041 (2022).
  * Neumann  _et al._ [2013] P. Neumann, I. Jakobi, F. Dolde, C. Burk, R. Reuter, G. Waldherr, J. Honert, T. Wolf, A. Brunner, J. H. Shim, D. Suter, H. Sumiya, J. Isoya, and J. Wrachtrup, High-Precision Nanoscale Temperature Sensing Using Single Defects in Diamond, Nano Letters 13, 2738 (2013).
  * Wang  _et al._ [2015] J. Wang, F. Feng, J. Zhang, J. Chen, Z. Zheng, L. Guo, W. Zhang, X. Song, G. Guo, L. Fan, C. Zou, L. Lou, W. Zhu, and G. Wang, High-sensitivity temperature sensing using an implanted single nitrogen-vacancy center array in diamond, Physical Review B 91, 155404 (2015).
  * Shim  _et al._ [2022] J. H. Shim, S.-J. Lee, S. Ghimire, J. I. Hwang, K.-G. Lee, K. Kim, M. J. Turner, C. A. Hart, R. L. Walsworth, and S. Oh, Multiplexed Sensing of Magnetic Field and Temperature in Real Time Using a Nitrogen-Vacancy Ensemble in Diamond, Physical Review Applied 17, 014009 (2022).
  * Hui  _et al._ [2019] Y. Y. Hui, O. Y. Chen, T. Azuma, B.-M. Chang, F.-J. Hsieh, and H.-C. Chang, All-Optical Thermometry with Nitrogen-Vacancy Centers in Nanodiamond-Embedded Polymer Films, The Journal of Physical Chemistry C 123, 15366 (2019).
  * Moreva  _et al._ [2020] E. Moreva, E. Bernardi, P. Traina, A. Sosso, S. D. Tchernij, J. Forneris, F. Picollo, G. Brida, i. c. v. Pastuović, I. P. Degiovanni, P. Olivero, and M. Genovese, Practical Applications of Quantum Sensing: A Simple Method to Enhance the Sensitivity of Nitrogen-Vacancy-Based Temperature Sensors, Physical Review Applied 13, 054057 (2020).
  * Cambria  _et al._ [2023] M. Cambria, A. Norambuena, H. Dinani, G. Thiering, A. Gardill, I. Kemeny, Y. Li, V. Lordi, Á. Gali, J. Maze, and S. Kolkowitz, Temperature-Dependent Spin-Lattice Relaxation of the Nitrogen-Vacancy Spin Triplet in Diamond, Physical Review Letters 130, 256903 (2023).
  * Ernst  _et al._ [2023] S. Ernst, P. J. Scheidegger, S. Diesch, and C. L. Degen, Modeling temperature-dependent population dynamics in the excited state of the nitrogen-vacancy center in diamond, Physical Review B 108, 085203 (2023).
  * Happacher  _et al._ [2023] J. Happacher, J. Bocquel, H. T. Dinani, M. A. Tschudin, P. Reiser, D. A. Broadway, J. R. Maze, and P. Maletinsky, Temperature-Dependent Photophysics of Single NV Centers in Diamond, Physical Review Letters 131, 086904 (2023).
  * Myers  _et al._ [2014] B. A. Myers, A. Das, M. C. Dartiailh, K. Ohno, D. D. Awschalom, and A. C. Bleszynski Jayich, Probing surface noise with depth-calibrated spins in diamond, Phys. Rev. Lett. 113, 027602 (2014).
  * Fortman  _et al._ [2021] B. Fortman, L. Mugica-Sanchez, N. Tischler, C. Selco, Y. Hang, K. Holczer, and S. Takahashi, Electron–electron double resonance detected NMR spectroscopy using ensemble NV centers at 230 GHz and 8.3 T, Journal of Applied Physics 130, 083901 (2021).
  * Degen  _et al._ [2017] C. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Reviews of Modern Physics 89, 035002 (2017).
  * Shields  _et al._ [2015] B. Shields, Q. Unterreithmeier, N. De Leon, H. Park, and M. Lukin, Efficient Readout of a Single Spin State in Diamond via Spin-to-Charge Conversion, Physical Review Letters 114, 136402 (2015).
  * Jiang  _et al._ [2009] L. Jiang, J. S. Hodges, J. R. Maze, P. Maurer, J. M. Taylor, D. G. Cory, P. R. Hemmer, R. L. Walsworth, A. Yacoby, A. S. Zibrov, and M. D. Lukin, Repetitive Readout of a Single Electronic Spin via Quantum Logic with Nuclear Spin Ancillae, Science 326, 267 (2009).
  * Arunkumar  _et al._ [2023] N. Arunkumar, K. S. Olsson, J. T. Oon, C. A. Hart, D. B. Bucher, D. R. Glenn, M. D. Lukin, H. Park, D. Ham, and R. L. Walsworth, Quantum Logic Enhanced Sensing in Solid-State Spin Ensembles, Physical Review Letters 131, 100801 (2023).
  * Zhong  _et al._ [2022] H. Zhong, A. Finco, J. Fischer, A. Haykal, K. Bouzehouane, C. Carrétéro, F. Godel, P. Maletinsky, M. Munsch, S. Fusil, V. Jacques, and V. Garcia, Quantitative Imaging of Exotic Antiferromagnetic Spin Cycloids in Bi Fe O 3 Thin Films, Physical Review Applied 17, 044051 (2022).
  * Huxter  _et al._ [2022] W. S. Huxter, M. L. Palm, M. L. Davis, P. Welter, C.-H. Lambert, M. Trassin, and C. L. Degen, Scanning gradiometry with a single spin quantum magnetometer, Nature Communications 13, 3761 (2022).
  * Chang  _et al._ [2017] K. Chang, A. Eichler, J. Rhensius, L. Lorenzelli, and C. L. Degen, Nanoscale Imaging of Current Density with a Single-Spin Magnetometer, Nano Letters 17, 2367 (2017).
  * Ofori-Okai  _et al._ [2012] B. K. Ofori-Okai, S. Pezzagna, K. Chang, M. Loretz, R. Schirhagl, Y. Tao, B. A. Moores, K. Groot-Berning, J. Meijer, and C. L. Degen, Spin properties of very shallow nitrogen vacancy defects in diamond, Physical Review B 86, 081406 (2012).
  * Romach  _et al._ [2015] Y. Romach, C. Müller, T. Unden, L. Rogers, T. Isoda, K. Itoh, M. Markham, A. Stacey, J. Meijer, S. Pezzagna, B. Naydenov, L. McGuinness, N. Bar-Gill, and F. Jelezko, Spectroscopy of Surface-Induced Noise Using Shallow Spins in Diamond, Physical Review Letters 114, 017601 (2015).
  * Gardill  _et al._ [2022] A. Gardill, I. Kemeny, Y. Li, M. Zahedian, M. C. Cambria, X. Xu, V. Lordi, \\. Gali, J. R. Maze, J. T. Choy, and S. Kolkowitz, Super-Resolution Airy Disk Microscopy of Individual Color Centers in Diamond, ACS Photonics 9, 3848 (2022).
  * Dolde  _et al._ [2013] F. Dolde, I. Jakobi, B. Naydenov, N. Zhao, S. Pezzagna, C. Trautmann, J. Meijer, P. Neumann, F. Jelezko, and J. Wrachtrup, Room-temperature entanglement between single defect spins in diamond, Nature Physics 9, 139 (2013).
  * Schmitt  _et al._ [2017] S. Schmitt, T. Gefen, F. M. Stürner, T. Unden, G. Wolff, C. Müller, J. Scheuer, B. Naydenov, M. Markham, S. Pezzagna, J. Meijer, I. Schwarz, M. Plenio, A. Retzker, L. P. McGuinness, and F. Jelezko, Submillihertz magnetic spectroscopy performed with a nanoscale quantum sensor, Science 356, 832 (2017).
  * Boss  _et al._ [2017] J. M. Boss, K. S. Cujia, J. Zopes, and C. L. Degen, Quantum sensing with arbitrary frequency resolution, Science 356, 837 (2017).
  * Joas  _et al._ [2017] T. Joas, A. M. Waeber, G. Braunbeck, and F. Reinhard, Quantum sensing of weak radio-frequency signals by pulsed Mollow absorption spectroscopy, Nature Communications 8, 964 (2017).
  * Stark  _et al._ [2017] A. Stark, N. Aharon, T. Unden, D. Louzon, A. Huck, A. Retzker, U. L. Andersen, and F. Jelezko, Narrow-bandwidth sensing of high-frequency fields with continuous dynamical decoupling, Nature Communications 8, 1105 (2017).
  * Meinel  _et al._ [2021] J. Meinel, V. Vorobyov, B. Yavkin, D. Dasari, H. Sumiya, S. Onoda, J. Isoya, and J. Wrachtrup, Heterodyne sensing of microwaves with a quantum sensor, Nature Communications 12, 2737 (2021).
  * Wang  _et al._ [2022a] G. Wang, Y.-X. Liu, J. M. Schloss, S. T. Alsid, D. A. Braje, and P. Cappellaro, Sensing of Arbitrary-Frequency Fields Using a Quantum Mixer, Physical Review X 12, 021061 (2022a).
  * Zhang  _et al._ [2023] Z. Zhang, M. Joos, D. Bluvstein, Y. Lyu, and A. C. Bleszynski Jayich, Reporter-Spin-Assisted T 1 Relaxometry, Physical Review Applied 19, L031004 (2023).
  * Dubois  _et al._ [2022] A. Dubois, D. Broadway, A. Stark, M. Tschudin, A. Healey, S. Huber, J.-P. Tetienne, E. Greplova, and P. Maletinsky, Untrained physically informed neural network for image reconstruction of magnetic field sources, Phys. Rev. Appl. 18, 064076 (2022).
  * Toyli  _et al._ [2012] D. M. Toyli, D. J. Christle, A. Alkauskas, B. B. Buckley, C. G. Van De Walle, and D. D. Awschalom, Measurement and Control of Single Nitrogen-Vacancy Center Spins above 600 K, Physical Review X 2, 031001 (2012).
  * Barson  _et al._ [2017] M. S. J. Barson, P. Peddibhotla, P. Ovartchaiyapong, K. Ganesan, R. L. Taylor, M. Gebert, Z. Mielens, B. Koslowski, D. A. Simpson, L. P. McGuinness, J. McCallum, S. Prawer, S. Onoda, T. Ohshima, A. C. Bleszynski Jayich, F. Jelezko, N. B. Manson, and M. W. Doherty, Nanomechanical Sensing Using Spins in Diamond, Nano Letters 17, 1496 (2017).
  * Myers  _et al._ [2017] B. Myers, A. Ariyaratne, and A. B. Jayich, Double-Quantum Spin-Relaxation Limits to Coherence of Near-Surface Nitrogen-Vacancy Centers, Physical Review Letters 118, 197201 (2017).
  * Wan  _et al._ [2018] N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, Efficient Extraction of Light from a Nitrogen-Vacancy Center in a Diamond Parabolic Reflector, Nano Letters 18, 2787 (2018).
  * Hedrich  _et al._ [2020] N. Hedrich, D. Rohner, M. Batzer, P. Maletinsky, and B. J. Shields, Parabolic Diamond Scanning Probes for Single-Spin Magnetic Field Imaging, Physical Review Applied 14, 064007 (2020).
  * Aharonovich  _et al._ [2012] I. Aharonovich, J. C. Lee, A. P. Magyar, B. B. Buckley, C. G. Yale, D. D. Awschalom, and E. L. Hu, Homoepitaxial Growth of Single Crystal Diamond Membranes for Quantum Information Processing, Advanced Materials 24, 10.1002/adma.201103932 (2012).
  * Guo  _et al._ [2021] X. Guo, N. Delegan, J. C. Karsch, Z. Li, T. Liu, R. Shreiner, A. Butcher, D. D. Awschalom, F. J. Heremans, and A. A. High, Tunable and Transferable Diamond Membranes for Integrated Quantum Technologies, Nano Letters 21, 10392 (2021).
  * Khanaliloo  _et al._ [2015] B. Khanaliloo, M. Mitchell, A. C. Hryciw, and P. E. Barclay, High- Q / V Monolithic Diamond Microdisks Fabricated with Quasi-isotropic Etching, Nano Letters 15, 5131 (2015).
  * Challier  _et al._ [2018] M. Challier, S. Sonusen, A. Barfuss, D. Rohner, D. Riedel, J. Koelbl, M. Ganzhorn, P. Appel, P. Maletinsky, and E. Neu, Advanced Fabrication of Single-Crystal Diamond Membranes for Quantum Technologies, Micromachines 9, 148 (2018).
  * Wood  _et al._ [2022] B. D. Wood, G. A. Stimpson, J. E. March, Y. N. D. Lekhai, C. J. Stephen, B. L. Green, A. C. Frangeskou, L. Ginés, S. Mandal, O. A. Williams, and G. W. Morley, Long spin coherence times of nitrogen vacancy centers in milled nanodiamonds, Physical Review B 105, 205401 (2022).
  * Zvi  _et al._ [2023] U. Zvi, D. R. Candido, A. Weiss, A. R. Jones, L. Chen, I. Golovina, X. Yu, S. Wang, D. V. Talapin, M. E. Flatté, A. P. Esser-Kahn, and P. C. Maurer, Engineering Spin Coherence in Core-Shell Diamond Nanocrystals, Arxiv 2305, 10.48550/ARXIV.2305.03075 (2023).
  * Janitz  _et al._ [2022] E. Janitz, K. Herb, L. A. Völker, W. S. Huxter, C. L. Degen, and J. M. Abendroth, Diamond surface engineering for molecular sensing with nitrogen—vacancy centers, Journal of Materials Chemistry C 10, 13533 (2022).
  * Rodgers  _et al._ [2021] L. V. H. Rodgers, L. B. Hughes, M. Xie, P. C. Maurer, S. Kolkowitz, A. C. Bleszynski Jayich, and N. P. De Leon, Materials challenges for quantum technologies based on color centers in diamond, MRS Bulletin 46, 623 (2021).
  * Hopper  _et al._ [2018] D. Hopper, H. Shulevitz, and L. Bassett, Spin Readout Techniques of the Nitrogen-Vacancy Center in Diamond, Micromachines 9, 437 (2018).
  * Zhang  _et al._ [2021] Q. Zhang, Y. Guo, W. Ji, M. Wang, J. Yin, F. Kong, Y. Lin, C. Yin, F. Shi, Y. Wang, and J. Du, High-fidelity single-shot readout of single electron spin in diamond with spin-to-charge conversion, Nature Communications 12, 1529 (2021).
  * Irber  _et al._ [2021] D. M. Irber, F. Poggiali, F. Kong, M. Kieschnick, T. Lühmann, D. Kwiatkowski, J. Meijer, J. Du, F. Shi, and F. Reinhard, Robust all-optical single-shot readout of nitrogen-vacancy centers in diamond, Nature Communications 12, 532 (2021).
  * Choi  _et al._ [2020] J. Choi, H. Zhou, H. S. Knowles, R. Landig, S. Choi, and M. D. Lukin, Robust Dynamic Hamiltonian Engineering of Many-Body Spin Systems, Physical Review X 10, 031002 (2020).
  * Rondin  _et al._ [2014] L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch, P. Maletinsky, and V. Jacques, Magnetometry with nitrogen-vacancy defects in diamond, Reports on Progress in Physics 77, 56503 (2014).
  * Note [1] It is worth noting that NV magnetometry is only strictly quantitative in the regime of weak magnetic fields ($\lesssim 10~{}$mT), where the first order response of the NV ODMR frequencies dominates the Zeeman shift if the NVs spin levels, and transverse magnetic fields only play a negligible role.
  * Tetienne  _et al._ [2017] J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, Quantum imaging of current flow in graphene, Science Advances 3, e1602429 (2017).
  * Broadway  _et al._ [2020] D. Broadway, S. Lillie, S. Scholten, D. Rohner, N. Dontschuk, P. Maletinsky, J.-P. Tetienne, and L. Hollenberg, Improved current density and magnetization reconstruction through vector magnetic field measurements, Phys. Rev. Appl. 14, 024076 (2020).
  * Thiel  _et al._ [2019] L. Thiel, Z. Wang, M. A. Tschudin, D. Rohner, I. Gutiérrez-Lezama, N. Ubrig, M. Gibertini, E. Giannini, A. F. Morpurgo, and P. Maletinsky, Probing magnetism in 2d materials at the nanoscale with single-spin microscopy, Science 364, 973 (2019).
  * Hedrich  _et al._ [2021] N. Hedrich, K. Wagner, O. V. Pylypovskyi, B. J. Shields, T. Kosub, D. D. Sheka, D. Makarov, and P. Maletinsky, Nanoscale mechanics of antiferromagnetic domain walls, Nature Physics 17, 574 (2021).
  * Hingant  _et al._ [2015] T. Hingant, J.-P. Tetienne, L. J. Martínez, K. Garcia, D. Ravelosona, J.-F. Roch, and V. Jacques, Measuring the magnetic moment density in patterned ultrathin ferromagnets with submicrometer resolution, Phys. Rev. Appl. 4, 014003 (2015).
  * Gross  _et al._ [2016] I. Gross, L. J. Martínez, J.-P. Tetienne, T. Hingant, J.-F. Roch, K. Garcia, R. Soucaille, J. P. Adam, J.-V. Kim, S. Rohart, A. Thiaville, J. Torrejon, M. Hayashi, and V. Jacques, Direct measurement of interfacial dzyaloshinskii-moriya interaction in $x|\text{CoFeB}|\text{MgO}$ heterostructures with a scanning nv magnetometer $(x=\text{Ta},\text{TaN},\text{and W})$, Phys. Rev. B 94, 064413 (2016).
  * Finco  _et al._ [2022] A. Finco, A. Haykal, S. Fusil, P. Kumar, P. Dufour, A. Forget, D. Colson, J.-Y. Chauleau, M. Viret, N. Jaouen, V. Garcia, and V. Jacques, Imaging topological defects in a noncollinear antiferromagnet, Phys. Rev. Lett. 128, 187201 (2022).
  * Spaldin [2021] N. A. Spaldin, Analogy between the magnetic dipole moment at the surface of a magnetoelectric and the electric charge at the surface of a ferroelectric, Journal of Experimental and Theoretical Physics 132, 493 (2021).
  * Bhattacharyya  _et al._ [2024] P. Bhattacharyya, W. Chen, X. Huang, S. Chatterjee, B. Huang, B. Kobrin, Y. Lyu, T. J. Smart, M. Block, E. Wang, Z. Wang, W. Wu, S. Hsieh, H. Ma, S. Mandyam, B. Chen, E. Davis, Z. M. Geballe, C. Zu, V. Struzhkin, R. Jeanloz, J. E. Moore, T. Cui, G. Galli, B. I. Halperin, C. R. Laumann, and N. Y. Yao, Imaging the Meissner effect in hydride superconductors using quantum sensors, Nature 627, 73 (2024).
  * Hubert and Schäfer [1998] A. Hubert and R. Schäfer, Magnetic domains: The analysis of magnetic microstructures (Springer, 1998).
  * Tetienne  _et al._ [2014] J. P. Tetienne, T. Hingant, L. J. Martínez, S. Rohart, A. Thiaville, L. H. Diez, K. Garcia, J. P. Adam, J. V. Kim, J. F. Roch, I. M. Miron, G. Gaudin, L. Vila, B. Ocker, D. Ravelosona, and V. Jacques, The nature of domain walls in ultrathin ferromagnets revealed by scanning nanomagnetometry, Nature Communications 6, 6733 (2014).
  * Fert  _et al._ [2017] A. Fert, N. Reyren, and V. Cros, Magnetic skyrmions: advances in physics and potential applications, Nature Reviews Materials 2, 17031 (2017).
  * Hrabec  _et al._ [2017] A. Hrabec, J. Sampaio, M. Belmeguenai, I. Gross, R. Weil, S. M. Chérif, A. Stashkevich, V. Jacques, A. Thiaville, and S. Rohart, Current-induced skyrmion generation and dynamics in symmetric bilayers, Nature Communications 8, 15765 (2017).
  * Gross  _et al._ [2018] I. Gross, W. Akhtar, A. Hrabec, J. Sampaio, L. J. Martinez, S. Chouaieb, B. J. Shields, P. Maletinsky, A. Thiaville, S. Rohart, and V. Jacques, Skyrmion morphology in ultrathin magnetic films, Physical Reviews Materials 2, 024406 (2018).
  * Rana  _et al._ [2020] K. G. Rana, A. Finco, F. Fabre, S. Chouaieb, A. Haykal, L. D. Buda-Prejbeanu, O. Fruchart, S. Le Denmat, P. David, M. Belmeguenai, T. Denneulin, R. E. Dunin-Borkowski, G. Gaudin, V. Jacques, and O. Boulle, Room-temperature skyrmions at zero field in exchange-biased ultrathin films, Phys. Rev. Appl. 13, 044079 (2020).
  * Akhtar  _et al._ [2019] W. Akhtar, A. Hrabec, S. Chouaieb, A. Haykal, I. Gross, M. Belmeguenai, M. S. Gabor, B. Shields, P. Maletinsky, A. Thiaville, S. Rohart, and V. Jacques, Current-Induced Nucleation and Dynamics of Skyrmions in a Co-based Heusler Alloy, PHYSICAL REVIEW APPLIED 11 (2019).
  * Jenkins  _et al._ [2019] A. Jenkins, M. Pelliccione, G. Yu, X. Ma, X. Li, K. L. Wang, and A. C. B. Jayich, Single-spin sensing of domain-wall structure and dynamics in a thin-film skyrmion host, Physical Review Materials 3, 083801 (2019).
  * Dovzhenko  _et al._ [2018] Y. Dovzhenko, F. Casola, S. Schlotter, T. X. Zhou, F. Büttner, R. L. Walsworth, G. S. D. Beach, and A. Yacoby, Magnetostatic twists in room-temperature skyrmions explored by nitrogen-vacancy center spin texture reconstruction, Nature Communications 9, 2712 (2018).
  * Seki  _et al._ [2012] S. Seki, X. Yu, S. Ishiwata, and Y. Tokura, Observation of skyrmions in a multiferroic material, Science 336, 198 (2012).
  * Soda  _et al._ [2023] M. Soda, E. M. Forgan, E. Blackburn, E. Campillo, V. Ryukhtin, I. Hoffmann, A. Kikkawa, Y. Taguchi, H. Yoshizawa, and H. Kawano-Furukawa, Asymmetric slow dynamics of the skyrmion lattice in mnsi, Nature Physics (2023).
  * Psaroudaki and Loss [2020] C. Psaroudaki and D. Loss, Quantum depinning of a magnetic skyrmion, Phys. Rev. Lett. 124, 097202 (2020).
  * Prozorov and Giannetta [2006] R. Prozorov and R. W. Giannetta, Magnetic penetration depth in unconventional superconductors, Superconductor Science and Technology 19, R41 (2006).
  * Luan  _et al._ [2010] L. Luan, O. M. Auslaender, T. M. Lippman, C. W. Hicks, B. Kalisky, J.-H. Chu, J. G. Analytis, I. R. Fisher, J. R. Kirtley, and K. A. Moler, Local measurement of the penetration depth in the pnictide superconductor ba(fe0.95co0.05)2as2, Phys. Rev. B 81, 100501 (2010).
  * Rohner  _et al._ [2019] D. Rohner, J. Happacher, P. Reiser, M. A. Tschudin, A. Tallaire, J. Achard, B. J. Shields, and P. Maletinsky, (111)-oriented, single crystal diamond tips for nanoscale scanning probe imaging of out-of-plane magnetic fields, Applied Physics Letters 115, 192401 (2019).
  * Acosta  _et al._ [2019] V. M. Acosta, L. S. Bouchard, D. Budker, R. Folman, T. Lenz, P. Maletinsky, D. Rohner, Y. Schlussel, and L. Thiel, Color centers in diamond as novel probes of superconductivity, Journal of Superconductivity and Novel Magnetism 32, 85 (2019).
  * Note [2] In the cited case, a thin film sample of YBCO was investigated. Since the film under investigation was thin compared to $\lambda_{L}$, the relevant penetration depth in that case was the “Pearl length”.
  * Joshi  _et al._ [2019] K. Joshi, N. Nusran, M. Tanatar, K. Cho, W. Meier, S. Bud’ko, P. Canfield, and R. Prozorov, Measuring the lower critical field of superconductors using nitrogen-vacancy centers in diamond optical magnetometry, Phys. Rev. Appl. 11, 014035 (2019).
  * Clem [1975] J. R. Clem, Simple model for the vortex core in a type ii superconductor, Journal of Low Temperature Physics 18, 427 (1975).
  * Loudon  _et al._ [2015] J. C. Loudon, S. Yazdi, T. Kasama, N. D. Zhigadlo, and J. Karpinski, Measurement of the penetration depth and coherence length of $\mathrm{MgB}{}_{2}$ in all directions using transmission electron microscopy, Phys. Rev. B 91, 054505 (2015).
  * Brandt [1997] E. H. Brandt, Precision ginzburg-landau solution of ideal vortex lattices for any induction and symmetry, Phys. Rev. Lett. 78, 2208 (1997).
  * Yip  _et al._ [2019] K. Y. Yip, K. O. Ho, K. Y. Yu, Y. Chen, W. Zhang, S. Kasahara, Y. Mizukami, T. Shibauchi, Y. Matsuda, S. K. Goh, and S. Yang, Measuring magnetic field texture in correlated electron systems under extreme conditions, Science 366, 1355 (2019).
  * Hilberer  _et al._ [2023] A. Hilberer, L. Toraille, C. Dailledouze, M.-P. Adam, L. Hanlon, G. Weck, M. Schmidt, P. Loubeyre, and J.-F. Roch, Enabling quantum sensing under extreme pressure: Nitrogen-vacancy magnetometry up to 130 GPa, Physical Review B 107, L220102 (2023).
  * Mao  _et al._ [2018] H.-K. Mao, X.-J. Chen, Y. Ding, B. Li, and L. Wang, Solids, liquids, and gases under high pressure, Reviews of Modern Physics 90, 015007 (2018).
  * Balents  _et al._ [2020] L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Superconductivity and strong correlations in moiré flat bands, Nature Physics 16, 725 (2020).
  * Langsjoen  _et al._ [2012] L. S. Langsjoen, A. Poudel, M. G. Vavilov, and R. Joynt, Qubit relaxation from evanescent-wave Johnson noise, Physical Review A 86, 010301 (2012).
  * Van Der Sar  _et al._ [2015] T. Van Der Sar, F. Casola, R. Walsworth, and A. Yacoby, Nanometre-scale probing of spin waves using single electron spins, Nature Communications 6, 7886 (2015).
  * Lovchinsky  _et al._ [2016] I. Lovchinsky, A. O. Sushkov, E. Urbach, N. P. De Leon, S. Choi, K. De Greve, R. Evans, R. Gertner, E. Bersin, C. Müller, L. McGuinness, F. Jelezko, R. L. Walsworth, H. Park, and M. D. Lukin, Nuclear magnetic resonance detection and spectroscopy of single proteins using quantum logic, Science 351, 836 (2016).
  * Lovchinsky  _et al._ [2017] I. Lovchinsky, J. D. Sanchez-Yamagishi, E. K. Urbach, S. Choi, S. Fang, T. I. Andersen, K. Watanabe, T. Taniguchi, A. Bylinskii, E. Kaxiras, P. Kim, H. Park, and M. D. Lukin, Magnetic resonance spectroscopy of an atomically thin material using a single-spin qubit, Science 355, 503 (2017).
  * Pham  _et al._ [2016] L. M. Pham, S. J. DeVience, F. Casola, I. Lovchinsky, A. O. Sushkov, E. Bersin, J. Lee, E. Urbach, P. Cappellaro, H. Park, A. Yacoby, M. Lukin, and R. L. Walsworth, NMR technique for determining the depth of shallow nitrogen-vacancy centers in diamond, Physical Review B 93, 045425 (2016).
  * Kardar [2007] M. Kardar,  _Statistical physics of fields_ (Cambridge University Press, 2007).
  * Blanter and Büttiker [2000] Y. M. Blanter and M. Büttiker, Shot noise in mesoscopic conductors, Physics reports 336, 1 (2000).
  * Wang  _et al._ [2022b] Y. Wang, C. Setty, S. Sur, L. Chen, S. Paschen, D. Natelson, and Q. Si, Shot noise as a characterization of strongly correlated metals, Arxiv 2211, 10.48550/ARXIV.2211.11735 (2022b), publisher: arXiv Version Number: 1.
  * Kobayashi and Hashisaka [2021] K. Kobayashi and M. Hashisaka, Shot Noise in Mesoscopic Systems: From Single Particles to Quantum Liquids, Journal of the Physical Society of Japan 90, 102001 (2021).
  * Wolfe  _et al._ [2014] C. S. Wolfe, V. P. Bhallamudi, H. L. Wang, C. H. Du, S. Manuilov, R. M. Teeling-Smith, A. J. Berger, R. Adur, F. Y. Yang, and P. C. Hammel, Off-resonant manipulation of spins in diamond via precessing magnetization of a proximal ferromagnet, Physical Review B 89, 180406 (2014).
  * Wolfe  _et al._ [2016] C. S. Wolfe, S. A. Manuilov, C. M. Purser, R. Teeling-Smith, C. Dubs, P. C. Hammel, and V. P. Bhallamudi, Spatially resolved detection of complex ferromagnetic dynamics using optically detected nitrogen-vacancy spins, Applied Physics Letters 108, 232409 (2016).
  * Wolf  _et al._ [2016] M. S. Wolf, R. Badea, and J. Berezovsky, Fast nanoscale addressability of nitrogen-vacancy spins via coupling to a dynamic ferromagnetic vortex, Nature Communications 7, 11584 (2016).
  * Andrich  _et al._ [2017] P. Andrich, C. F. De Las Casas, X. Liu, H. L. Bretscher, J. R. Berman, F. J. Heremans, P. F. Nealey, and D. D. Awschalom, Long-range spin wave mediated control of defect qubits in nanodiamonds, npj Quantum Information 3, 28 (2017).
  * Kikuchi  _et al._ [2017] D. Kikuchi, D. Prananto, K. Hayashi, A. Laraoui, N. Mizuochi, M. Hatano, E. Saitoh, Y. Kim, C. A. Meriles, and T. An, Long-distance excitation of nitrogen-vacancy centers in diamond via surface spin waves, Applied Physics Express 10, 103004 (2017).
  * Page  _et al._ [2019] M. R. Page, B. A. McCullian, C. M. Purser, J. G. Schulze, T. M. Nakatani, C. S. Wolfe, J. R. Childress, M. E. McConney, B. M. Howe, P. C. Hammel, and V. P. Bhallamudi, Optically detected ferromagnetic resonance in diverse ferromagnets via nitrogen vacancy centers in diamond, Journal of Applied Physics 126, 124902 (2019).
  * Bertelli  _et al._ [2020] I. Bertelli, J. J. Carmiggelt, T. Yu, B. G. Simon, C. C. Pothoven, G. E. W. Bauer, Y. M. Blanter, J. Aarts, and T. Van Der Sar, Magnetic resonance imaging of spin-wave transport and interference in a magnetic insulator, Science Advances 6, eabd3556 (2020).
  * McCullian  _et al._ [2020] B. A. McCullian, A. M. Thabt, B. A. Gray, A. L. Melendez, M. S. Wolf, V. L. Safonov, D. V. Pelekhov, V. P. Bhallamudi, M. R. Page, and P. C. Hammel, Broadband multi-magnon relaxometry using a quantum spin sensor for high frequency ferromagnetic dynamics sensing, Nature Communications 11, 5229 (2020).
  * Chatterjee  _et al._ [2022] S. Chatterjee, P. E. Dolgirev, I. Esterlis, A. A. Zibrov, M. D. Lukin, N. Y. Yao, and E. Demler, Single-spin qubit magnetic spectroscopy of two-dimensional superconductivity, Physical Review Research 4, L012001 (2022).
  * Dolgirev  _et al._ [2022] P. E. Dolgirev, S. Chatterjee, I. Esterlis, A. A. Zibrov, M. D. Lukin, N. Y. Yao, and E. Demler, Characterizing two-dimensional superconductivity via nanoscale noise magnetometry with single-spin qubits, Physical Review B 105, 024507 (2022).
  * Rodriguez-Nieva  _et al._ [2018] J. F. Rodriguez-Nieva, K. Agarwal, T. Giamarchi, B. I. Halperin, M. D. Lukin, and E. Demler, Probing one-dimensional systems via noise magnetometry with single spin qubits, Physical Review B 98, 195433 (2018).
  * Chatterjee  _et al._ [2019] S. Chatterjee, J. F. Rodriguez-Nieva, and E. Demler, Diagnosing phases of magnetic insulators via noise magnetometry with spin qubits, Physical Review B 99, 104425 (2019).
  * Khoo  _et al._ [2022] J. Y. Khoo, F. Pientka, P. A. Lee, and I. S. Villadiego, Probing the quantum noise of the spinon Fermi surface with NV centers, Physical Review B 106, 115108 (2022).
  * Note [3] The choice of basis on the Bloch sphere is in general distinct from the real-space coordinate system.
  * Vandersypen and Chuang [2005b] L. M. K. Vandersypen and I. L. Chuang, NMR techniques for quantum control and computation, Reviews of Modern Physics 76, 1037 (2005b).
  * Klauder and Anderson [1962] J. R. Klauder and P. W. Anderson, Spectral Diffusion Decay in Spin Resonance Experiments, Physical Review 125, 912 (1962).
  * Savitsky  _et al._ [2023] A. Savitsky, J. Zhang, and D. Suter, Variable bandwidth, high efficiency microwave resonator for control of spin-qubits in nitrogen-vacancy centers, Review of Scientific Instruments 94, 023101 (2023).
  * Casanova  _et al._ [2018] J. Casanova, Z.-Y. Wang, I. Schwartz, and M. B. Plenio, Shaped Pulses for Energy-Efficient High-Field NMR at the Nanoscale, Physical Review Applied 10, 044072 (2018).
  * Munuera-Javaloy  _et al._ [2023] C. Munuera-Javaloy, A. Tobalina, and J. Casanova, High-Resolution NMR Spectroscopy at Large Fields with Nitrogen Vacancy Centers, Physical Review Letters 130, 133603 (2023).
  * Galperin  _et al._ [2006] Y. M. Galperin, B. L. Altshuler, J. Bergli, and D. V. Shantsev, Non-Gaussian Low-Frequency Noise as a Source of Qubit Decoherence, Physical Review Letters 96, 097009 (2006).
  * Aslam  _et al._ [2023] N. Aslam, H. Zhou, E. K. Urbach, M. J. Turner, R. L. Walsworth, M. D. Lukin, and H. Park, Quantum sensors for biomedical applications, Nature Reviews Physics 5, 157 (2023).
  * Luan  _et al._ [2015] L. Luan, M. S. Grinolds, S. Hong, P. Maletinsky, R. L. Walsworth, and A. Yacoby, Decoherence imaging of spin ensembles using a scanning single-electron spin in diamond, Scientific Reports 5, 8119 (2015).
  * Machado  _et al._ [2023] F. Machado, E. A. Demler, N. Y. Yao, and S. Chatterjee, Quantum Noise Spectroscopy of Dynamical Critical Phenomena, Physical Review Letters 131, 070801 (2023).
  * Ji  _et al._ [2024] W. Ji, Z. Liu, Y. Guo, Z. Hu, J. Zhou, S. Dai, Y. Chen, P. Yu, M. Wang, K. Xia, F. Shi, Y. Wang, and J. Du, Correlated sensing with a solid-state quantum multisensor system for atomic-scale structural analysis, Nature Photonics 10.1038/s41566-023-01352-4 (2024).
  * Delord  _et al._ [2024] T. Delord, R. Monge, and C. A. Meriles, Correlated spectroscopy of electric noise with color center clusters, Arxiv 2401, 10.48550/arXiv.2401.07814 (2024).
  * Meisenheimer  _et al._ [2023] P. Meisenheimer, G. Moore, S. Zhou, H. Zhang, X. Huang, S. Husain, X. Chen, L. W. Martin, K. A. Persson, S. Griffin, L. Caretta, P. Stevenson, and R. Ramesh, Persistent anisotropy of the spin cycloid in BiFeO3 through ferroelectric switching, Arxiv 2311, 10.48550/ARXIV.2311.10169 (2023), publisher: [object Object] Version Number: 1.
  * Belashchenko [2010] K. D. Belashchenko, Equilibrium magnetization at the boundary of a magnetoelectric antiferromagnet, Phys. Rev. Lett. 105, 147204 (2010).
  * Moriya [1960] T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev. 120, 91 (1960).
  * Andreev [1996] A. F. Andreev, Macroscopic magnetic fields of antiferromagnets, Journal of Experimental and Theoretical Physics Letters 63, 758 (1996).
  * Jiang  _et al._ [2023] K. Jiang, T. Wu, J.-X. Yin, Z. Wang, M. Z. Hasan, S. D. Wilson, X. Chen, and J. Hu, Kagome superconductors AV3Sb5 (A = K, Rb, Cs), National Science Review 10, nwac199 (2023).
  * Xia  _et al._ [2006] J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, High resolution polar kerr effect measurements of ${\mathrm{sr}}_{2}{\mathrm{ruo}}_{4}$: Evidence for broken time-reversal symmetry in the superconducting state, Phys. Rev. Lett. 97, 167002 (2006).
  * Haldane [1988] F. D. M. Haldane, Model for a quantum hall effect without landau levels: Condensed-matter realization of the ”parity anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  * Tschirhart  _et al._ [2021] C. L. Tschirhart, M. Serlin, H. Polshyn, A. Shragai, Z. Xia, J. Zhu, Y. Zhang, K. Watanabe, T. Taniguchi, M. E. Huber, and A. F. Young, Imaging orbital ferromagnetism in a moiré chern insulator, Science 372, 1323 (2021).
  * Sacépé  _et al._ [2020] B. Sacépé, M. Feigel’man, and T. M. Klapwijk, Quantum breakdown of superconductivity in low-dimensional materials, Nature Physics 16, 734 (2020).
  * Hayden and Tranquada [2024] S. M. Hayden and J. M. Tranquada, Charge Correlations in Cuprate Superconductors, Annual Review of Condensed Matter Physics 15, annurev (2024).
  * He  _et al._ [2021] F. He, Y. Zhou, Z. Ye, S.-H. Cho, J. Jeong, X. Meng, and Y. Wang, Moiré Patterns in 2D Materials: A Review, ACS Nano 15, 5944 (2021).
  * Abrahams  _et al._ [2001] E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, Metallic behavior and related phenomena in two dimensions, Reviews of Modern Physics 73, 251 (2001).
  * Spivak and Kivelson [2004] B. Spivak and S. A. Kivelson, Phases intermediate between a two-dimensional electron liquid and Wigner crystal, Physical Review B 70, 155114 (2004).
  * Ahn and Das Sarma [2023] S. Ahn and S. Das Sarma, Density-tuned effective metal-insulator transitions in two-dimensional semiconductor layers: Anderson localization or Wigner crystallization, Physical Review B 107, 195435 (2023).
  * Nie  _et al._ [2014] L. Nie, G. Tarjus, and S. A. Kivelson, Quenched disorder and vestigial nematicity in the pseudogap regime of the cuprates, Proceedings of the National Academy of Sciences 111, 7980 (2014).
  * Vojta [2019] T. Vojta, Disorder in Quantum Many-Body Systems, Annual Review of Condensed Matter Physics 10, 233 (2019).
  * Mitra  _et al._ [2006] A. Mitra, S. Takei, Y. B. Kim, and A. J. Millis, Nonequilibrium Quantum Criticality in Open Electronic Systems, Physical Review Letters 97, 236808 (2006).
  * Green  _et al._ [2006] A. G. Green, J. E. Moore, S. L. Sondhi, and A. Vishwanath, Current Noise in the Vicinity of the 2D Superconductor-Insulator Quantum Critical Point, Physical Review Letters 97, 227003 (2006).
  * Cavalleri [2018] A. Cavalleri, Photo-induced superconductivity, Contemporary Physics 59, 31 (2018).