# AC sensing using nitrogen vacancy centers in a diamond anvil cell up to 6
GPa

Z. Wang  C. McPherson  R. Kadado  N. Brandt  S. Edwards  Department of Physics
and Astronomy, University of California, Davis CA 95616  W. H. Casey
Department of Earth and Planetary Sciences, University of California, Davis,
95616  Department of Chemistry, University of California, Davis, 95616  N. J.
Curro  Department of Physics and Astronomy, University of California, Davis CA
95616<EMAIL_ADDRESS>
###### Abstract

Nitrogen-vacancy color centers in diamond have attracted broad attention as
quantum sensors for both static and dynamic magnetic, electrical, strain and
thermal fields, and are particularly attractive for quantum sensing under
pressure in diamond anvil cells. Optically-based nuclear magnetic resonance
may be possible at pressures greater than a few GPa, and offers an attractive
alternative to conventional Faraday-induction based detection. Here we present
AC sensing results and demonstrate synchronized readout up to 6 GPa, but find
that the sensitivity is reduced due to inhomogeneities of the microwave field
and pressure within the sample space. These experiments enable the possibility
for all-optical high resolution magnetic resonance of nanoliter sample volumes
at high pressures.

##  I Introduction

Nuclear magnetic resonance is one of the most versatile tools for
investigating condensed matter in extreme conditions, including temperatures
ranging from sub-mK to 1000 K, static magnetic fields up to 45 T, pulsed
magnetic fields [1], uniaxial strain, and hydrostatic pressures up to several
GPa [2]. Each of these environments presents special challenges for conducting
magnetic resonance experiments, but pressure is exceptional because of the
small volumes available. The maximum achievable pressure in of a given
pressure cell scales roughly as the inverse of the sample volume, and depends
on the material parameters and design of the cell. Intermediate-scale
pressures up to 2-3 GPa can be reached within a piston-cylinder type cell,
with volumes on the order of several hundred $\mu$L. For this type of pressure
cell, it is straightforward to introduce a small coil and perform conventional
NMR. This approach has become standard in the study of condensed matter
systems [3] as well as solution chemistry at high pressure [4].

Figure 1: Diamond anvil cell for optically-detected magnetic resonance
experiments. An objective focuses the excitation light and collects the
fluorescence from within the sample space in the gasket. A straight gold
antenna is located across the anvil culet ( sample hole 260 $\mu$m diameter),
and a small NV diamond crystal of dimensions $\sim 100\times 75$ $\mu$m is
visible at the center to the lower right of the antenna (see arrow), oriented
normal the [100] direction. A magnetic field of magnitude $H_{0}\sim 10$ mT is
oriented $54.7^{\circ}$ from the vertical along the [111] direction of the
diamond using permanent neodymium magnets mounted externally.

Piston-cylinder type cells are limited to approximately 2-3 GPa due to the
Young’s modulus of the Cu-Be alloy that forms the cylinder housing [5].
Alternative designs can extend this pressure range up to 4.5 GPa [6], although
with severe constraints on the sample volume. To reach pressures beyond this
range, it is necessary to use a diamond anvil cell(DAC), in which a pair of
diamond anvils compress a sample volume contained within a metallic gasket, as
illustrated in Fig. 1 [7]. The available sample volume is only on the order of
0.1-1 nL, presenting serious challenges for conventional NMR. Not only should
a solenoid be located within this space to form a high $Q$ resonant circuit
and introduce the necessary radiofrequency fields for NMR, but the leads must
also be insulated from the gasket. Preventing leads from shorting or breaking
under pressure is a major challenge [8]. Secondly, the nuclear magnetization
needs to be large enough to induce a sufficient voltage relative to the
Johnson noise of the coil. Typically this requires on the order of $10^{17}$
nuclear spins [9]. For a typical solid, e.g. copper with density 10 g/cm3,
this limit corresponds to 1 nL. Since the sample space must also contain the
coil and pressure medium, there is often insufficient room for the minimum
detectable sample. Although there have been recent advances to overcome these
limitations using specialized microcoils, gasket resonators and Lenz lenses
enabling NMR measurements up to the hundred GPa regime [10, 2, 11],
alternative detection methods may be advantageous in order to investigate
matter at higher pressures, for example superconductivity in hydride
compounds(hydrogen-rich materials) [12, 13, 14].

Optically-detected magnetic resonance (ODMR) of NV centers offers an
alternative and complementary approach that is not limited by the small sample
volume and can detect nuclear spins within volumes as small as $10^{-3}$ nL
with sub-mHz resolution. Several excellent reviews are available outlining the
basics of this technique [15, 16, 17]. The fluorescence intensity of the NV
centers is a strong function of the NV spin ($S=1$) orientation, and the NV
centers can be optically pumped to the ground state with high polarization.
Single NV detection is possible with polarization $>99\%$ using optical
pumping resonant with the zero-phonon line [18]. For NV ensembles with non-
resonant pumping using 532 nm light, the optical contrast is a few percent.
Because detection is optical-based rather than using Faraday induction, it is
no longer necessary to locate a high Q resonant circuit within the sample
space. The spin orientation can be measured optically, in which excitation and
fluorescence light passes through the transparent diamond anvils and can be
focused down to a point on the sample of size $\sim 1\mu$m, enabling work with
small samples. ODMR has been utilized to study static strain and magnetization
as a function of pressure and temperature [19, 20, 21, 22]. Here we describe
AC sensing experiments using NV centers in a diamond anvil cell to probe
dynamic fields, such as those arising from precessing nuclear spins. In
principle there is no upper limit for such measurements, and we discuss some
of the challenges that emerge under pressure as the properties of the NV
centers evolve.

##  II Apparatus

The principle challenges to performing ODMR in a DAC are associated with the
collecting sufficient fluorescence from the sample as well as introducing
microwaves with sufficient homogeneity and magnitude. Fortunately, ODMR is
similar in practice to Raman spectroscopy, which is a well-established
technique under pressure in DACs.

Figure 2: (Left) Rabi frequency of the $|0\rangle\leftrightarrow|-1\rangle$
transition and (Right) fluorescence intensity measured as a function of
position in an diamond crystal with NV centers located within a DAC. The
scanning step size is 20 $\mu$m, or the scale of the individual squares on the
map.

###  II.1 Design and Microwave Antenna

Compared to ambient conditions, the signal-to-noise for ODMR in a DAC will be
reduced because the structure of the cell prevents the objective from being
too close to the sample. For our design, illustrated in Fig. 1, the distance
to the diamond anvil is approximately 10 mm, plus 2.5 mm for the the anvil
itself. It is thus necessary to utilize a long-working distance objective
(Nikon CFI T Plan Epi SLWD 50X Objective, N.A. 0.40 and working distance 22
mm). The pressure cell (CryoDAC SC, Almax-Easylab), is mounted on a
translation stage and an external magnetic field, $\mathbf{H}_{0}$, of 29 mT
oriented at an angle of 54.7∘ from the vertical is introduced via a pair of
fixed neodynium permanent magnets. The gaskets are fabricated with 0.3 mm
thick copper-beryllium, and the sample holes of diameter of 260 $\mu$m are
drilled through the center of the pre-indented region (depth ranges from 100
to 150 $\mu$m) using an electrical discharge machine. A small diamond crystal
prepared with 0.3 ppm NVs (Element 6) was cut and polished to a thickness of
20 $\mu$m (Applied Diamond/Diamond Delaware Knives), and then secured to the
anvil with superglue such that the $[100]$ axis is aligned parallel to the DAC
axis. The ODMR spectrum is measured as the cell is rotated along the DAC axis
in order to align the field along the $[111]$ direction parallel to the NV
axis. The diamond anvils are type IIac with low nitrogen impurities in order
to minimize background fluorescence. The spot size of the laser is less than
20 $\mu$m, and can be scanned laterally across the sample space.
Representative scans of the Rabi frequency, $\Omega$, and fluorescence
intensity are shown in Fig. 2.

Introducing microwaves with sufficient magnitude in close proximity to the NV
centers presents a major challenge for working in a DAC, because the
conducting gasket can screen out the magnetic field, as illustrated in Fig. 3.
High microwave amplitudes are necessary in order to achieve strong
$\mathbf{H}_{1}$ fields and hence large Rabi frequencies. The magnitude of
$\mathbf{H}_{1}$ should be $\sim 0.18$ mT in order to have a 90∘ pulse time on
the order of 50 ns ( $\Omega\sim 5$ MHz). Note that it is not necessary to
have a high $Q$ resonant circuit, so the antenna does not necessarily need to
have a low resistance. Previous studies have utilized antennas located
external to the sample chamber [23, 19, 20, 22], within the sample chamber
[21], as well as designer anvils with conducting paths deposited directly onto
the diamond culet and protected with a capping layer of CVD grown diamond
[24]. Locating the antenna outside the sample space, as shown in Fig. 3(a,b),
is easier, but is inefficient because a large fraction of the power goes
towards inducing screening currents in the conducting gasket and can lead to
undesired heating effects. Designer anvils are costly and time consuming to
prepare [25].

Figure 3:  Calculated magnetic field profile from a wire loop surrounding a
diamond anvil close to (a) an insulating gasket and (b) a conducting gasket
for 63.2 V(40 dBm) across the antenna at 3 GHz. The conducting gasket screens
the majority of the magnetic field within the sample space. (c) The calculated
magnetic field profile and induced current density for a conducting gasket
with a straight antenna, as illustrated in Fig. 4(a). The diamond anvil
(outlined in green) has a relative permeability of 5 [26]. The green rectangle
in the sample space represents the volume of the NV diamond chip used to
compute the histograms in Fig. 5.

To overcome this challenge, we fabricated gold microwave strips, as shown in
Fig. 4, by electroplating 8 $\mu$m thick strips onto a substrate, which were
then liberated chemically for insertion into the DAC. A single antenna is then
transferred onto the culet of the anvil with tweezers under a microscope, and
secured with thin layer of adhesive. Leads were attached to the ends of the
antenna pads with silver epoxy, and then attached to larger wires (not shown)
that lead out of the cell and to a high power (16 W) microwave amplifier
(Mini-Circuits, ZHL-16W-43-S+). The antenna is insulated from the pre-indented
region of the gasket with a mixture of epoxy and aluminum oxide and(or) boron
nitride powders, that are cured under compression (Fig. 4(c)). After the epoxy
has cured, the center hole is drilled out, leaving a thin layer of insulating
material between the conducting gasket and the antenna leads [27]. This
approach enables us to reliably apply pressure to the gasket without
comprising the antenna performance. The antenna leads are malleable and are
compressed under pressure as the anvil presses into the gasket. However, at
higher pressures (on the order of $\sim 10-30$ GPa) the leads will eventually
sever as the gasket deforms and the anvil cups inward [28]. In such cases it
may be better to utilize alternative designs, with thinner leads or even
eliminating the leads altogether and using the Lenz lens approach of Ref. [28]
For pressures up to $\sim 7$ GPa we can achieve Rabi frequencies of several
tens of MHz with our antenna, which correspond to $90^{\circ}$ pulses of a few
tens of nanoseconds, as illustrated in Fig. 5. Eventually for sufficiently
high pressure the antenna leads are thinned out and clipped by the anvils,
limiting the power delivered and dramatically decreasing the Rabi frequency.
Microwaves pulse sequences are generated via an arbitrary waveform generator
(Tabor AWG Model SE5082).

Figure 4:  (a) Geometry of the gasket and sample region. The inner diameter is
260 $\mu$m, the culet diameter is 600 $\mu$m, and the thickness of the sample
region is 150 $\mu$m. The gasket is copper-beryllium with conductivity
1.16$\times 10^{7}$ S/m. The gold antenna has width 11$\mu$m, thickness 8
$\mu$m and length 600 $\mu$m, and has conductivity 4.6$\times 10^{7}$ S/m.
There is also a thin insulating layer between the antenna and gasket with
conductivity $10^{-18}$ S/m. (b) Fabricated microwave antenna secured onto
culet of the anvil (diameter 600 $\mu$m). The antenna has width 10 $\mu$m and
thickness 8 $\mu$m. (c) Pre-indented gasket with insulating layer of
boronitride, aluminum oxide and epoxy. The sample space in the center hole has
diameter 260 $\mu$m.  Figure 5: (a) Rabi oscillations (
$|0\rangle\leftrightarrow|-1\rangle$) for various pressures in the DAC with a
microwave power of 44 dBm. (b) Computed histograms of $H_{1\perp}/I$ based
upon the finite element simulations shown in Fig. 3(c), for a loop antenna
with insulating gasket, with a conducting gasket, and for the straight antenna
design illustrated in Fig. 4(a). Note that the field has been normalized by
the current in the wire and have been computed at 2 GHz.

###  II.2 Microwave field simulation

In order to better understand the $\mathbf{H}_{1}$ field distribution and
radiofrequency screening effects of the gasket, we have modeled the antenna
and gasket system and carried out finite element analysis calculations of the
electromagnetic fields. Fig. 4(a) illustrates the gasket and antenna strip,
including the ridge surrounding the indented region around the anvils (not
shown). A thin insulating gap separates the antenna from the gasket. Using the
COMSOL package, we find that the field radiates radially from the anvil, as
illustrated in Fig. 3(c), but drops off quickly with distance. The ratio of
power dissipated to resistance in the antenna to the radiated power is 3.2%,
and that the power dissipated by induced currents in the gasket is 0.5%. These
numbers indicate that the thin gold wire radiates power into the sample space
efficiently. On the other hand, for a loop antenna located outside the sample
space (Fig. 3(b)), but in close proximity to the gasket and anvil, the
magnetic field in the sample space is well-screened by the conducting gasket.
In practice, the power delivered to the antenna is dominated by the
reflectance coefficient of the combination of leads and antenna, and depends
critically on the geometry of the leads external to the pressure cell. The
best practice is to keep the leads parallel to one another as much as
possible.

As seen in Fig. 3, the $\mathbf{H}_{1}$ field profile is inhomogeneous. It is
largest in the region closest to the antenna but drops off quickly along the
vertical axis. Fig. 5(b) displays a histogram of the field to current ratio of
magnitude of $\mathbf{H}_{1\perp}$, the component of $\mathbf{H}_{1}$
perpendicular to $\mathbf{H}_{0}$ along the [111] direction (see Fig. 1),
within a volume of $100\times 75\times 20$ $\mu$m3 representing the NV diamond
(illustrated by the green box in Fig. 3(c)). The size and shape of this
histogram depends critically on the location of the NV diamond within the
sample space. This distribution of fields means that each of the $\sim 8\times
10^{9}$ NV centers in the NV diamond experiences a slightly different Rabi
frequency. As a result, the Rabi oscillations fall off quickly, as observed in
Fig. 5(a). Experimentally, the oscillations die out faster under pressure, as
observed in Fig. 5(a). This behavior reflects inhomogeneous broadening, as
discussed below. We also considered a loop, rather than a straight antenna, as
shown in Fig. 5(b). If the gasket remains insulating, the distribution remains
narrow. For the conducting gasket, the distribution broadens and is shifted to
lower fields because the gasket efficiently screens the field. The highest
field to current ratios are achieved with the straight antenna because the
distance to the sample is smallest in this case, however the field is
inhomogeneous. Experimentally, we find that the Rabi oscillations are quickly
damped (implying a wider distribution of $H_{1\perp}$ fields) when the sample
is located directly on top of the antenna, but is more homogeneous when the
sample is located on the opposite anvil. The latter is advantageous for AC
detection because a distribution of $H_{1}$ fields will give rise to a
distribution of rotation angles for the 180∘ dynamical decoupling pulses.

##  III AC-ODMR

NV centers can sense magnetic fields with a sensitivity of the order of 30
pT/$\sqrt{\rm{Hz}}$, which is sufficient to detect the dipolar magnetic field
generated by a nuclear spin that is within a few $\mu$m of the NV center.
Although couplings to such spins are not large enough to resolve in the NV
spectrum, they can be detected via AC methods [29, 30, 31]. AC-ODMR can
discern the amplitude $B_{ac}$, frequency, $\omega_{ac}$, and phase,
$\phi_{ac}$, of a time-varying field oriented along the direction, $\hat{z}$,
of the applied field, $\mathbf{H}_{0}$:

$\mathbf{B}_{ac}(t)=B_{ac}\hat{z}\sin(\omega_{ac}t+\phi_{ac}).$ (1)

There are several technical components necessary to successfully detect such a
signal. First, microwave pulses to efficiently manipulate the NV spin
orientation are necessary, with Rabi frequencies $\Omega=\gamma H_{1\perp}$ of
at least 5 MHz. If the microwave amplitude is too small, the pulses used to
create unitary rotations of the NV spin will be too long and will create an
upper bound on the range of detectable frequencies. Secondly, in order to
‘imprint’ the nuclear spin coupling onto the phase of the NV centers, the NV
ensemble must have a sufficiently low density such that dipolar coupling
between NV centers does not severely limit the decoherence time ($T_{2}\sim
200$ $\mu$s in our case). This condition provides an effective lower bound on
the range of frequencies that can be detected.

###  III.1 Quadrature Detection and Phase Cycling

Conventional NMR relies on detecting the voltage induced by precessing spins
in the plane perpendicular to the applied field, $\mathbf{H}_{0}$. In
contrast, ODMR detects magnetization parallel to $\mathbf{H}_{0}$
($\parallel\hat{z}$). The fluorescence intensity, $I$, is a maximum, $I_{0}$,
when the NV spins are polarized in the $S_{z}=0$ state, and a minimum,
$I_{1}$, for spins in the $S_{z}=\pm 1$ states. The fluorescence contrast,
$\Delta I=(I_{0}-I_{1})/I_{0}$, is typically on the order of a few percent at
room temperature, depending on the laser power and NV density. Importantly,
this contrast is proportional to $\langle S_{z}\rangle$. Exposing the NVs to
532 nm excitation light for a few microseconds is sufficient to polarize the
$S_{z}=0$ state, which can then be manipulated using oscillating microwave
fields at the frequency of the $|0\rangle\leftrightarrow|-1\rangle$ or
$|0\rangle\leftrightarrow|+1\rangle$ transitions. A $90^{\circ}$
$\mathbf{H}_{1}$ pulse will create a superposition state that will evolve with
time and $\langle\mathbf{S}(t)\rangle$ will precess in the perpendicular
plane. To detect the signal, one applies a second $90^{\circ}$ pulse with a
particular phase (e.g. along either the $x$ or $y$ directions in the rotating
frame) to rotate the magnetization back to the $z$ direction. The so-called
Ramsey sequence (Fig. 6(b) inset) will give rise to oscillations of $I/I_{0}$
as a function of the pulse spacing, $\tau$, between the two microwave pulses .

Figure 6: (a) Change in fluorescence contrast as a function of pulse spacing,
$\tau$, for a Ramsey sequence at the $|0\rangle\leftrightarrow|-1\rangle$
transition of an NV in a field of 22 mT at ambient pressure. $M_{x}$ and
$M_{y}$ are measured using different phases (90∘) for the second microwave
pulse. Panel (b) shows the amplitude of the complex Fourier transform of the
data in (a), revealing three separate peaks corresponding to the three $I_{z}$
states of the 14N nuclei coupled to the NV electronic spin. The dashed brown
line shows the spectrum measured without phase cycling.

By controlling the phase of the second pulse relative to the first, one can
project either $\langle S_{x}\rangle$ or $\langle S_{y}\rangle$ onto the $z$
axis. Collecting data for both phases enables quadrature detection and hence
the full complex Fourier transform can be obtained. It is also useful to
implement phase cycling to improve the signal-to-noise ratio. Spurious noise,
misalignments of the microwave pulses, and astigmatism in the phase response
of the microwave sources can all contribute to noise in the measured
fluorescence contrast, and similar approaches are regularly used in
conventional magnetic resonance [32]. By cycling the phases of the first and
second $\mathbf{H}_{1}$ pulses so that the NV spin is rotated alternately to
the $\pm x$ and $\pm y$ directions and back to the $\pm z$ direction, one can
either add or subtract the measured contrast to remove background noise. Fig.
6 demonstrates this approach. Note that the spectrum of the NV reveals three
separate peaks, which arise due to the hyperfine coupling between the NV spin,
$S$, and the 14N nuclear spin, $I=1$ (natural abundance 99.6%). The hyperfine
coupling is given by
$\mathcal{H}_{hf}=\hat{\mathbf{S}}\cdot\mathbb{A}\cdot\hat{\mathbf{I}}$, where
the coupling tensor $\mathbb{A}$ is diagonal along the NV-axis with components
$A_{||}=2.1$ and $A_{\perp}=2.3$ MHz [33]. The three peaks correspond to NV
spins in the ensemble for which $I_{z}=$ -1, 0 or +1 and hence a hyperfine
field of 0, $\pm 2.1$MHz shifts the resonant frequency.

Figure 7: Spectra of the $|0\rangle\leftrightarrow|-1\rangle$ transition of
the NV ensemble as a function of pressure, measured by echo sequences. The
spectrum at 5.96 GPa was measured by continuous wave, since the antenna leads
were thinned out under the pressure from the anvils, reducing the the Rabi
frequency.

Spectra of the $|0\rangle\leftrightarrow|-1\rangle$ transition under pressure
are shown in Fig. 7. The resonance frequency of this transition is given by
$f_{0}=D-\gamma_{e}H_{0}$, where $D$ is the zero-field splitting, $\gamma_{e}$
is the electron gyromagnetic ratio, and $H_{0}=29$ mT is the magnitude of the
applied field. $D$ increases linearly with pressure [23, 24], which enables us
to use the shift of the resonance as a measure of the pressure. In this case
we use the ambient pressure value of $D(0)=2866$ MHz and $dD/dP=11.72$
MHz/GPa. The spectra clearly broaden with pressure, and the three peaks are no
longer resolvable under pressure. We find that the linewidths increase by a
factor of eight over this range, corresponding to a pressure variation of
approximately 15% over the volume of the sample. The non-hydrostaticity is
likely due to freezing of the pressure medium (Daphne oil 7575), which freezes
above 3 GPa at room temperature. Above 5.5 GPa, we found that the Rabi
frequency dropped significantly as the leads to the antenna become clipped
under pressure. Utilizing thinner leads in the future may alleviate this
problem.

###  III.2 Dynamical decoupling

In the rotating frame, the NV spins will precess around $\mathbf{B}_{ac}(t)$
(Eq. (1)) and acquire a phase
$\Phi(t)=\gamma\int_{0}^{t}B_{ac}(t^{\prime})dt^{\prime}$. A 180∘ pulse will
effectively reverse the direction of $\mathbf{B}_{ac}(t)$, and reverse the
direction of precession. For a series of 180∘ pulses spaced at regular
intervals, $\tau$, the phase of the NV can be written as [29]:

$\Phi(t)=\gamma\int_{0}^{t}B_{eff}(t^{\prime})dt^{\prime},$ (2)

where the effective field is $B_{eff}(t)=u(t)B_{ac}(t)$, and $u(t)$ alternates
between $\pm 1$ between successive 180∘ pulses (which we approximate as
infinitely-narrow in time). The effective field can be expressed as:

$\displaystyle B_{eff}(t)$ $\displaystyle=$ $\displaystyle
B_{ac}\sum_{n=1,3,\cdots}\frac{2}{n\pi}\left[\cos((\omega_{ac}-{n\pi/\tau})t-\phi_{ac})\right.$
(3) $\displaystyle\left.-\cos((\omega_{ac}+{n\pi/\tau})t+\phi_{ac})\right].$

Most of these frequency components will integrate to zero in Eq. 2, which
reflects that fact that such pulse sequences filter out noise sources with
frequency components far from ${n\pi/\tau}$. On the other hand, if
${n\pi/\tau}\approx\omega_{ac}$, then $B_{eff}(t)$ has a component that is
nearly constant over time and the phase $\Phi$ will not integrate to zero. For
$n=1$, where the pulse spacing $\tau\approx\pi/\omega_{ac}$,

$\Phi(t)=\frac{2\gamma B_{ac}t}{\pi}{\mathrm{sinc}}(\pi\delta
ft)\cos(\pi\delta ft-\phi_{ac})$ (4)

where $\delta f=(2\tau)^{-1}-\omega_{ac}/(2\pi)$ is the frequency offset. If
the pulse spacing is tuned exactly such that $\delta f=0$, then the phase will
accumulate linearly with time. After accumulating this phase for a series of
$N$ pulses, the NV magnetization can be then be projected to the $z$ axis. At
the end of the sequence, the NV magnetization depends on the relative phase of
the first and last $90^{\circ}$ pulses, and that the fluorescence contrast
varies as:

$\Delta I\sim\begin{cases}\cos\left(2N\frac{\gamma
B_{ac}}{\omega_{ac}}\cos\phi_{ac}\right)&\text{for }90_{x}-90_{x}\\\
\sin\left(2N\frac{\gamma B_{ac}}{\omega_{ac}}\cos\phi_{ac}\right)&\text{for
}90_{x}-90_{y}\\\ \end{cases}$ (5)

For typical values of $\omega_{ac}=2\pi\cdot 100$ kHz with amplitude
$B_{ac}=$15 nT and $N=48$ refocusing pulses, the fluorescence contrast will be
reduced by 0.1% for the $90_{x}-90_{y}$ sequence, assuming no coherence decay.

###  III.3 Pulse spacing sweep

Figure 8: (Upper panel) Optical contrast, $\Delta I$ measured in a DAC
measured versus detection frequency, $(2\tau)^{-1}$, where $\tau$ is the pulse
spacing of a dynamical decoupling sequence. Input signals of 1.12 and 1.25 MHz
were introduced externally. The dip in the contrast arises due to the phase
accumulated by the NV centers. (Lower panel) Optical contrast, $\Delta I$ of a
test AC signal measured by signal averaging as a function of the voltage
applied to the AC coil. The solid purple line is a fit to a Bessel function as
described in the text.

The most straightforward approach to detecting an AC field is to sweep the
pulse spacing, $\tau$, and search for changes in $\Delta I$, as illustrated in
Fig. 8. In this case, multiple measurements of the signal contrast are
averaged, while a CW external field is applied via a small loop located near
the sample space in the diamond anvil cell. The phase, $\phi_{ac}$ is randomly
distributed, depending on the time at which the NV detection sequence is
initiated. In this case, for the $90_{x}-90_{x}$ sequence, the fluorescence
contrast is given by:

$\Delta I=1-J_{0}(N{\gamma B_{ac}}/{\omega_{ac}}),$ (6)

where $J_{0}(x)$ is a Bessel function of the first kind. For the
$90_{x}-90_{y}$ sequence $\Delta I=0$. This behavior is illustrated in the
lower panel of Fig. 8, in which the AC voltage, $V_{pp}$ is varied to change
the magnitude of $B_{ac}$. In this case, $B_{ac}\propto V_{pp}$, however the
coefficient is unknown.

Figure 9 shows AC spectra in the absence of an external driving AC field at
ambient pressure, 3.6 GPa and 5.0 GPa respectively. There is a dip in the
contrast at $\sim 0.3$ MHz, which corresponds to the Larmor frequency of 13C
in the applied $H_{0}$ field. The 1% abundant 13C nuclear spins in the diamond
lattice precess at this frequency and create AC fields at the NV site that are
not refocused by the dynamical decoupling. The dip around 1.2 MHz is likely
due to hydrogen in the pressure medium.

Figure 9: Optical contrast versus frequency, $(2\tau)^{-1}$, where $\tau$ is
the pulse spacing of an XY8-16 sequence, measured at various pressures. The
dip s around 0.3 MHz and 1.2 MHz arises from precessing 13C spins in the
diamond lattice and 1H spins in the pressure medium, respectively.

###  III.4 Synchronized Readout

Figure 10: (a) Optical contrast, $\Delta I$ measured for NVs in the DAC with a
test signal at frequency $f_{ac}=1.2001$ MHz, where $\tau=416.667$ ns
(corresponding to a detection frequency at 1.2 MHz), for a sequence of
equally-spaced measurements at with $T_{meas}=45$ $\mu$s, as described in the
text. (b) Fourier transform of $\Delta I$ at ambient pressure. The spectra
exhibits a single large peak at 100 Hz, and the vertical axis has been scaled
to match the known applied field magnitude. The 3.61 GPa data has been offset
vertically by 500 nT for clarity. (c) Sensitivity, $\eta$ ($\bullet$), versus
pressure. The dashed blue line corresponds to the necessary sensitivity to
achieve unity signal to noise ratio after 1 minute, and the pink square
($\blacksquare$) at 0 GPa corresponds to the sensitivity reported in [30].

Instead of averaging the signal over randomly-distributed initial phases at
the beginning of each repeat of the dynamical decoupling sequence, it is
possible to collect each fluorescence measurement and perform a Fourier
transform. After each NV fluorescence measurement, the NV is re-initialized
into the ground state without affecting the AC source. As a result, the phase
of $\mathbf{B}_{ac}(t)$ will evolve in a coherent fashion for each
measurement. The slowest varying component of the effective field in Eq. 3
will acquire a phase $\phi_{m}=2\pi\delta fmT_{meas}-\phi_{ac}$, where
$T_{meas}$ is the measurement repeat time, and $m$ is the $m^{th}$
measurement. In this case, $\Phi_{m}\approx\frac{2\gamma
B_{ac}T_{meas}}{\pi}\cos\phi_{m}$. For a $90_{x}-90_{y}$ sequence in the limit
of small $B_{ac}$, the fluorescence contrast will vary as:

$\Delta I(T)\sim\left(\frac{2N\gamma
B_{ac}}{\omega_{ac}}\right)\cos(2\pi\delta fT-\phi_{ac}),$ (7)

where $T=mT_{meas}$. This behavior is illustrated in Fig. 10 both at ambient
pressure and at 3.6 GPa. In effect, the AC field is demodulated by the
dynamical decoupling sequence, and imprinted on $\Delta I$. A Fourier
transform of $\Delta I(T)$ thus provides a spectrum of $B_{ac}(t)$ relative to
the detection frequency $(2\tau)^{-1}$. This detection scheme is often
referred to as ‘quantum heterodyne’ [31, 34]. The frequency resolution is
determined by the number of points $m$, $T_{meas}$, the stability of the field
$\mathbf{H}_{0}$, and the stability of the microwave pulse timing. The
resolution can be as low as a mHz, which is sufficient to resolve many
chemical shifts even at low applied fields [31, 30]. This technique requires
coherently averaging many repeated measurements, but sufficient signal-to-
noise ratio can be obtained within a few hours.

In practice, the minimum detectable frequency , $\omega_{ac}$, is bounded by
$T_{2}$ because the NV magnetization will decay over during the free
precession time, $\tau$, giving rise to another factor of
$e^{-\pi/\omega_{ac}T_{2}}$. For our NV diamond we have $T_{2}\approx
200\mu$s, corresponding to a minimum frequency $\sim 10$ kHz. The maximum
detectable frequency is bounded by the shortest possible pulse spacing, which
is approximately equal to the 180∘ pulse time, determined by the Rabi
frequency. In our case the maximum detectable frequency $\sim 10$ MHz. Note
that there is an upper limit to the heterodyne frequency, $\delta f$, based
upon the minimum time between consecutive measurements. This limit is set by
the time to measure the fluorescence, which is usually a few $\mu$s.

###  III.5 Sensitivity Measurements

The data shown in Fig. 10 was acquired for a large applied test AC field for a
single acquisition sequence without signal averaging with a total measurement
time of approximately 1.5 s, corresponding to a sensitivity, $\eta=1.9$
nT/$\sqrt{\mathrm{Hz}}$ at ambient pressure. At 3.6 GPa, we find the
sensitivity decreases to $\eta=7.6$ nT/$\sqrt{\mathrm{Hz}}$. Here we define
sensitivity as the minimum detectable signal per unit time. Ideally $\eta$
should be as low as possible to achieve a high signal to noise ratio. Our
values are not as low as previous reports, which reach down to 32
pT/$\sqrt{\mathrm{Hz}}$ [30]. One of the primary reasons for the difference in
the DAC is the wide distribution of Rabi frequencies, which means that not all
of the NVs in the ensemble experience the same $H_{1}$ field. The pulse width
for the dynamical decoupling 180∘ pulses, given by $\pi/(\gamma H_{1})$, will
differ for each of these NVs. As a result, the accumulated phase in Eq. 2 is
not as high as it would be for a uniform $H_{1}$. To illustrate this point,
Fig. 11 shows how the signal intensity varies as a function of the pulse width
time. Numerical simulations of the Bloch equations under these conditions
indicate that the sensitivity is reduced by $\approx 10$% if the dynamical
decoupling pulses are either 10% shorter or longer than ideal. As shown in
Fig. 5, the $H_{1}$ field for the straight antenna is inhomogeneous over the
sample volume, such that the width of the distribution is approximately 40% of
the average. Although this antenna is able to provide large $H_{1}$ fields
within the sample space, the field inhomogeneity reduces the synchronized
readout sensitivity.

Figure 11: Signal intensity (arbitrary units) for a synchronized readout
measurement of an AC signal (as in Fig. 10) versus the width of a 90∘ pulse.
The dynamical decoupling pulses ($180^{\circ}$) are twice as long as the time
$t_{90}$ given on the horizontal axis.

Another limiting factor for the sensitivity is the large ratio of the 180∘
pulse width, $t_{180}$, to the pulse spacing, $\tau$. The data in Fig. 10
utilized $t_{180}/\tau\approx 0.43$, which is far from the idealized case of
the infinitely-narrow pulses that would give rise to the effective field in
Eq. 3. Numerical simulations indicate that the sensitivity will be suppressed
when the ratio $t_{180}/\tau\gtrsim 0.2$. This provides an effective upper
limit on $\omega_{ac}^{max}\approx 0.2\Omega$ for a given value of the Rabi
frequency, $\Omega$.

The lower limit for the sensitivity is determined by $T_{2}$ because the NV
magnetization will decay during the free precession time between the dynamical
decoupling pulses. This leads to a exponential prefactor of
$e^{-\pi/\omega_{ac}T_{2}}$ for Eq. 7, and an effective lower limit
$\omega_{ac}^{min}\approx\pi/\ln(10)T_{2}$, where the sensitivity is reduced
by a factor of 10. For our NV sample with $T_{2}\approx 200$ $\mu$s and
$\Omega\approx 3$ MHz, we have $f_{ac}^{min}\approx 1$ kHz and
$f_{ac}^{max}\approx 0.5$ MHz. For detection of precessing nuclear spins such
as ${}^{1}H$ or 19F, $H_{0}$ should lie between 0.02 and 25 mT in order to
achieve $\omega_{ac}$ in this range. Curiously, except for small frequencies
where $T_{2}$ limits the sensitivity, the fluorescence contrast maximum in Eq.
7 does not depend on the applied field, $H_{0}$, because both $B_{ac}$ and
$\omega_{ac}$ are proportional to $H_{0}$ (for thermally polarized nuclei).

Under pressure we find that $\eta$ increases by a factor of four, as shown in
Fig. 10(c). There are two possible reasons for this change. Under pressure the
NV spectrum is broadened due to the pressure gradients. If the spectrum is
sufficiently broad, the dynamical decoupling pulses will not fully refocus the
NV magnetization, similar to the effect of a non-uniform $H_{1}$ field as
discussed above. Moreover, the antenna deforms under pressure, and as a result
the impedance changes and affects the magnitude of the $H_{1}$ field.
Consequently both $\Omega$ and $\Delta I$ can be reduced, affecting the
sensitivity.

##  IV Discussion and Conclusions

We have demonstrated AC-ODMR in a DAC with a sensitivity of 1.9
nT$/\sqrt{\mathrm{Hz}}$, which is about 50 times larger than values reported
for other NV-based AC sensors operating under ambient conditions. To detect
nuclear spins external to the diamond, it is possible to either initialize
these with a 90∘ pulse at the nuclear Larmor frequency ($\sim 1$ MHz) in order
to let them precess coherently, or to use correlation spectroscopy to detect
their noise fluctuations. The magnitude of $B_{ac}$ can be estimated assuming
that the NV diamond chip is embedded in a spherical volume of uniform nuclear
magnetization, $\mathbf{M}(t)$, that is precessing at Larmor frequency
$\omega_{ac}$. In this case, $B_{ac}=\frac{8\pi}{3}\chi_{N}H_{0}$, where
$\chi_{N}={\rho\gamma_{n}^{2}\hbar^{2}I(I+1)}/{3k_{B}T}$ is the Curie
susceptibility of nuclei with spin $I$, and number density $\rho$. For
hydrogen in water at room temperature in a field of 10 mT, $B_{ac}\approx 27$
pT. To achieve a signal to noise ratio of unity with our demonstrated
sensitivity $\eta=1.9$ nT$/\sqrt{\mathrm{Hz}}$ would require signal averaging
for a time $T_{avg}=(\eta/B_{ac})^{2}\sim 80$ min. The dashed blue line in
Fig. 10(c) shows the maximum $\eta$ necessary to achieve a minimum signal to
noise ratio of unity within one minute. It is clear that an order of magnitude
reduction in $\eta$ will be necessary to avoid hours of signal averaging. Note
that although conventional NMR has sensitivities on the order of 10
pT/$\sqrt{\mathrm{Hz}}$ [29], this quantity scales as $\eta_{NMR}\sim
V_{s}^{-1/2}$, where $V_{s}$ is the sample volume [35]. For a 100 $\mu$m coil
diameter and sample volume of 1 nL at a field of 10 mT, we estimate
$\eta_{NMR}\approx 85$ nT/$\sqrt{\mathrm{Hz}}$ [36], approximately two orders
of magnitude higher than our measurement.

There are several possible routes to improve the sensitivity of AC-ODMR in the
DAC. The inhomogeneity of the microwave fields from the antenna leads to an
increase in $\eta$. An antenna design that provides a more homogeneous and
stable $H_{1}$ field and is less prone to distortions under pressure is vital.
In the future other antenna designs may be considered, for example a circular
loop within the sample space, to improve the homogeneity and reduce screening
effects by the gasket. Utilizing diamonds with lower NV concentrations and
isotopically pure 12C would significantly enhance $T_{2}$ and also improve the
sensitivity. It is also possible to improve the sensitivity by utilizing
dynamic nuclear polarization. This number can be enhanced significantly by
utilizing the presence of free radicals and taking advantage of the Overhauser
effect to transfer polarization to the nuclear spins in a liquid. Bucher et
al. used the TEMPOL molecule dissolved into liquid solutions to enhance the
sensitivity by more than a factor of 200 [37]. The unpaired electrons on the
TEMPOL molecule have an enhanced spin polarization that can be transferred to
hyperpolarize the nuclear spins beyond their thermal polarization described by
$\chi_{N}$. This process can enhance the signal-to-noise ratio by several
orders of magnitude, but requires pulsing the system at the Larmor frequency
of the TEMPOL electron spins for several ms prior to detection of the nuclear
spins with NV magnetometry.

The sensitivity may also be improved by utilizing double quantum coherence
under pressure [38, 39]. The AC-ODMR experiments described here are based on
single quantum coherence in which the NV is prepared in the state
$|0\rangle+|\pm 1\rangle$. The resonance frequency of the double quantum
coherent state $|-1\rangle+|+1\rangle$ is independent of $D$. This property
could be important because pressure inhomogeneities strongly affect $D$,
giving rise to spectral broadening that impacts the sensitivity of the AC
detection. Utilizing dynamical decoupling at the double quantum coherent
frequency would allow the NV ensemble to precess and acquire phase more
homogeneously and may enable a reduction in $\eta$ even in the presence of
strong pressure gradients. It may also be prudent to utilize shaped pulses
with the arbitrary waveform generator, which have a broader frequency response
and may be able to better refocus the NV magnetization in the presence of
inhomogeneous broadening [40].

In addition to the zero-field splitting, $D$, the zero-phonon emission line
(637 nm at ambient pressure and temperature) is also pressure dependent [15,
24]. Both quantities can be used to measure the pressure, eliminating the need
for other manometers such as Ruby chips within the sample space or the high-
frequency edge of the Raman spectra of the diamond anvil [41, 42]. For ODMR,
the microwave frequency needs to be adjusted after each pressure change (by
100 GPa it will have increased by 1.2 GHz). This shift may necessitate
adjusting different microwave components, such as the amplifier and filters.
The emission wavelength shifts down with pressure and reaches 532 nm at
approximately 60 GPa [23]. To operate at higher pressures it is thus necessary
to switch to a lower wavelength excitation laser. It is common to utilize an
optical filter before the detector, which needs to be adjusted with increasing
pressures.

AC-ODMR under pressure will enable a broad range of NMR experiments at
pressures that have not been possible previously. In particular it should be
possible to to test geochemical predictions about fluid chemistry modifying
the Earth’s crust. These predictions extend now to 6.0 GPa and 1200 C [43, 44,
45], and are testable only via vibrational and optical spectroscopic
measurements, and not NMR [46, 47]. High-pressure NMR spectroscopy via
conventional detection is a familiar tool in inorganic solution chemistry [48,
49, 50], but the pressures are less than $\sim 0.4$ GPa, except for a few
studies [51, 52, 53, 54]. As discussed above, diamond-anvil cell (DAC)
technology can reach higher pressures, but the sample volumes are too small
for NMR detection via Faraday coils, which is why optical detection via AC-
ODMR is potentially so valuable, particularly if it is coupled to methods of
polarization transfer that enhance the signals. An additional benefit is that
experiments are now possible on solutions that would be dangerous in larger
amounts.

##  V Acknowledgements

We thank A. Ajoy, S. Gomez-Diaz, P. Klavins, V. Norman and M. Radulaski for
stimulating discussions. The work was supported by the United States
Department of Energy, Office of Basic Energy Sciences, Chemical Sciences,
Geosciences and Biosciences Division for Grant DE-FG0205ER15693. The data that
support the findings of this study are available from the corresponding author
upon reasonable request.

## References

  * Meier  _et al._ [2011] B. Meier, S. Greiser, J. Haase, T. Herrmannsdörfer, F. Wolff-Fabris, and J. Wosnitza, NMR signal averaging in 62T pulsed fields, J. Magn. Reson. 210, 1 (2011).
  * Meier [2018] T. Meier, At its extremes: NMR at giga-pascal pressures, in  _Annual Reports on NMR Spectroscopy_ (Elsevier, 2018) pp. 1–74.
  * Lin  _et al._ [2015] C. H. Lin, K. R. Shirer, J. Crocker, A. P. Dioguardi, M. M. Lawson, B. T. Bush, P. Klavins, and N. J. Curro, Evolution of hyperfine parameters across a quantum critical point in ${\mathrm{CeRhIn}}_{5}$, Phys. Rev. B 92, 155147 (2015).
  * Ochoa  _et al._ [2015] G. Ochoa, C. D. Pilgrim, M. N. Martin, C. A. Colla, P. Klavins, M. P. Augustine, and W. H. Casey, 2H and 139La NMR spectroscopy in aqueous solutions at geochemical pressures, Angew. Chem. Int. Ed. 54, 15444 (2015).
  * Fujiwara  _et al._ [1980] H. Fujiwara, H. Kadomatsu, and K. Tohma, Simple clamp pressure cell up to 30 kbar, Rev. Sci. Instrum. 51, 1345 (1980).
  * Kobayashi  _et al._ [2007] T. C. Kobayashi, H. Hidaka, H. Kotegawa, K. Fujiwara, and M. I. Eremets, Nonmagnetic indenter-type high-pressure cell for magnetic measurements, Rev. Sci. Instrum. 78, 23909 (2007).
  * Eremets [1996] M. Eremets,  _High pressure experimental methods_ (Oxford University Press, 1996).
  * Meier  _et al._ [2015] T. Meier, S. Reichardt, and J. Haase, High-sensitivity NMR beyond 200,000 atmospheres of pressure, J. Magn. Reson. 257, 39 (2015).
  * Hoult and Richards [1976a] D. I. Hoult and R. Richards, The signal-to-noise ratio of the nuclear magnetic resonance experiment, Journal of Magnetic Resonance (1969) 24, 71 (1976a).
  * Meier  _et al._ [2018] T. Meier, S. Khandarkhaeva, S. Petitgirard, T. Körber, A. Lauerer, E. Rössler, and L. Dubrovinsky, NMR at pressures up to 90 GPa, J. Magn. Reson. 292, 44 (2018).
  * Meier  _et al._ [2019] T. Meier, S. Khandarkhaeva, J. Jacobs, N. Dubrovinskaia, and L. Dubrovinsky, Improving resolution of solid state NMR in dense molecular hydrogen, Appl. Phys. Lett. 115, 131903 (2019).
  * Drozdov  _et al._ [2015] A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov, and S. I. Shylin, Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system, Nature 525, 73 (2015).
  * Drozdov  _et al._ [2019] A. P. Drozdov, P. P. Kong, V. S. Minkov, S. P. Besedin, M. A. Kuzovnikov, S. Mozaffari, L. Balicas, F. F. Balakirev, D. E. Graf, V. B. Prakapenka, E. Greenberg, D. A. Knyazev, M. Tkacz, and M. I. Eremets, Superconductivity at 250 K in lanthanum hydride under high pressures, Nature 569, 528 (2019).
  * Somayazulu  _et al._ [2019] M. Somayazulu, M. Ahart, A. K. Mishra, Z. M. Geballe, M. Baldini, Y. Meng, V. V. Struzhkin, and R. J. Hemley, Evidence for superconductivity above 260 K in lanthanum superhydride at megabar pressures, Phys. Rev. Lett. 122, 027001 (2019).
  * Doherty  _et al._ [2013] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, The nitrogen-vacancy colour centre in diamond, Phys. Rep. 528, 1 (2013).
  * Matsuzaki  _et al._ [2016] Y. Matsuzaki, H. Morishita, T. Shimooka, T. Tashima, K. Kakuyanagi, K. Semba, W. J. Munro, H. Yamaguchi, N. Mizuochi, and S. Saito, Optically detected magnetic resonance of high-density ensemble of NV-centers in diamond, J. Phys.: Condens. Matter 28, 275302 (2016).
  * Bucher  _et al._ [2019] D. B. Bucher, D. P. L. A. Craik, M. P. Backlund, M. J. Turner, O. B. Dor, D. R. Glenn, and R. L. Walsworth, Quantum diamond spectrometer for nanoscale NMR and ESR spectroscopy, Nat Protoc 14, 2707 (2019).
  * Childress and Hanson [2013] L. Childress and R. Hanson, Diamond NV centers for quantum computing and quantum networks, MRS Bulletin 38, 134 (2013).
  * 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).
  * 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).
  * Shang  _et al._ [2019] Y.-X. Shang, F. Hong, J.-H. Dai, Hui-Yu, Y.-N. Lu, E.-K. Liu, X.-H. Yu, G.-Q. Liu, and X.-Y. Pan, Magnetic sensing inside a diamond anvil cell via nitrogen-vacancy center spins, Chinese Phys. Lett. 36, 086201 (2019).
  * 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).
  * 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. L. Hollenberg, and S. Prawer, Electronic properties and metrology applications of the diamond ${\mathrm{NV}}^{-}$ center under pressure, Phys. Rev. Lett. 112, 047601 (2014).
  * Steele  _et al._ [2017] L. G. Steele, M. Lawson, M. Onyszczak, B. T. Bush, Z. Mei, A. P. Dioguardi, J. King, A. Parker, A. Pines, S. T. Weir, W. Evans, K. Visbeck, Y. K. Vohra, and N. J. Curro, Optically detected magnetic resonance of nitrogen vacancies in a diamond anvil cell using designer diamond anvils, Appl. Phys. Lett. 111, 221903 (2017).
  * Weir  _et al._ [2000] S. T. Weir, J. Akella, C. Aracne-Ruddle, Y. K. Vohra, and S. A. Catledge, Epitaxial diamond encapsulation of metal microprobes for high pressure experiments, Appl. Phys. Lett. 77, 3400 (2000).
  * Ibarra  _et al._ [1997] A. Ibarra, M. Gonzalez, R. Vila, and J. Molla, Wide frequency dielectric properties of cvd diamond, Diam. Relat. Mater. 6, 856 (1997).
  * Meier and Haase [2015] T. Meier and J. Haase, Anvil cell gasket design for high pressure nuclear magnetic resonance experiments beyond 30 GPa, Rev. Sci. Instrum. 86, 123906 (2015).
  * Meier  _et al._ [2017] T. Meier, N. Wang, D. Mager, J. G. Korvink, S. Petitgirard, and L. Dubrovinsky, Magnetic flux tailoring through Lenz lenses for ultrasmall samples: A new pathway to high-pressure nuclear magnetic resonance, Sci. Adv. 3, eaao5242 (2017).
  * Degen  _et al._ [2017] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Modern Phys. 89, 035002 (2017).
  * 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).
  * 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).
  * Bain [1984] A. D. Bain, Coherence levels and coherence pathways in NMR. a simple way to design phase cycling procedures, Journal of Magnetic Resonance (1969) 56, 418 (1984).
  * Felton  _et al._ [2009] S. Felton, A. M. Edmonds, M. E. Newton, P. M. Martineau, D. Fisher, D. J. Twitchen, and J. M. Baker, Hyperfine interaction in the ground state of the negatively charged nitrogen vacancy center in diamond, Phys. Rev. B 79, 075203 (2009).
  * Norman  _et al._ [2020] V. A. Norman, S. Majety, Z. Wang, W. H. Casey, N. Curro, and M. Radulaski, Novel color center platforms enabling fundamental scientific discovery, InfoMat , 1– 24 (2020).
  * Hoult and Richards [1976b] D. Hoult and R. Richards, The signal-to-noise ratio of the nuclear magnetic resonance experiment, Journal of Magnetic Resonance (1969) 24, 71 (1976b).
  * Peck  _et al._ [1995] T. L. Peck, R. L. Magin, and P. C. Lauterbur, Design and analysis of microcoils for nmr microscopy, Journal of Magnetic Resonance, Series B 108, 114 (1995).
  * Bucher  _et al._ [2020] D. B. Bucher, D. R. Glenn, H. Park, M. D. Lukin, and R. L. Walsworth, Hyperpolarization-enhanced NMR spectroscopy with femtomole sensitivity using quantum defects in diamond, Phys. Rev. X 10, 021053 (2020).
  * Mamin  _et al._ [2014] H. Mamin, M. Sherwood, M. Kim, C. Rettner, K. Ohno, D. Awschalom, and D. Rugar, Multipulse double-quantum magnetometry with near-surface nitrogen-vacancy centers, Phys. Rev. Lett. 113, 030803 (2014).
  * Marshall  _et al._ [2021] M. C. Marshall, R. Ebadi, C. Hart, M. J. Turner, M. J. H. Ku, D. F. Phillips, and R. L. Walsworth, High-precision mapping of diamond crystal strain using quantum interferometry (2021), arXiv:2108.00304 [quant-ph] .
  * Spindler  _et al._ [2017] P. E. Spindler, P. Schöps, W. Kallies, S. J. Glaser, and T. F. Prisner, Perspectives of shaped pulses for EPR spectroscopy, J. Magn. Reson. 280, 30 (2017).
  * Syassen [2008] K. Syassen, Ruby under pressure, High Pressure Res. 28, 75 (2008).
  * Akahama and Kawamura [2006] Y. Akahama and H. Kawamura, Pressure calibration of diamond anvil Raman gauge to 310GPa, J. Appl. Phys. 100, 043516 (2006).
  * Sverjensky  _et al._ [2014] D. A. Sverjensky, B. Harrison, and D. Azzolini, Water in the deep earth: the dielectric constant and the solubilities of quartz and corundum to 60 kb and 1200 C, Geochim. Cosmochim. Acta 129, 125 (2014).
  * Sverjensky and Huang [2015] D. A. Sverjensky and F. Huang, Diamond formation due to a pH drop during fluid–rock interactions, Nat Commun 6, 1 (2015).
  * Pan  _et al._ [2013] D. Pan, L. Spanu, B. Harrison, D. A. Sverjensky, and G. Galli, Dielectric properties of water under extreme conditions and transport of carbonates in the deep earth, Proc. Natl. Acad. Sci. 110, 6646 (2013).
  * Facq  _et al._ [2014] S. Facq, I. Daniel, G. Montagnac, H. Cardon, and D. A. Sverjensky, In situ Raman study and thermodynamic model of aqueous carbonate speciation in equilibrium with aragonite under subduction zone conditions, Geochim. Cosmochim. Acta 132, 375 (2014).
  * Facq  _et al._ [2016] S. Facq, I. Daniel, G. Montagnac, H. Cardon, and D. A. Sverjensky, Carbon speciation in saline solutions in equilibrium with aragonite at high pressure, Chem. Geol. 431, 44 (2016).
  * Asano and Le Noble [1978] T. Asano and W. Le Noble, Activation and reaction volumes in solution, Chem. Rev. 78, 407 (1978).
  * Van Eldik  _et al._ [1989] R. Van Eldik, T. Asano, and W. Le Noble, Activation and reaction volumes in solution. 2, Chem. Rev. 89, 549 (1989).
  * Drljaca  _et al._ [1998] A. Drljaca, C. Hubbard, R. Van Eldik, T. Asano, M. Basilevsky, and W. Le Noble, Activation and reaction volumes in solution. 3, Chem. Rev. 98, 2167 (1998).
  * Jonas [1980] J. Jonas, Nuclear magnetic resonance studies at high pressures (modern aspects of physical chemistry at high pressure: the 50th commemorative volume), The Review of Physical Chemistry of Japan 50, 19 (1980).
  * Lang and Lüdemann [1977] E. Lang and H.-D. Lüdemann, Pressure and temperature dependence of the longitudinal proton relaxation times in supercooled water to -87∘ C and 2500 bar, J. Chem. Phys. 67, 718 (1977).
  * De Langen and Prins [1995] M. De Langen and K. Prins, NMR probe for high pressure and high temperature, Rev. Sci. Instrum. 66, 5218 (1995).
  * Ballard  _et al._ [1998] L. Ballard, A. Yu, C. Reiner, and J. Jonas, A high-pressure, high-resolution NMR probe for experiments at 500 MHz, J. Magn. Reson. 133, 190 (1998).