# Demonstration of diamond nuclear spin gyroscope

A. Jarmola<EMAIL_ADDRESS>Department of Physics, University of
California, Berkeley, California 94720, USA  U.S. Army Research Laboratory,
Adelphi, Maryland 20783, USA  S. Lourette  Department of Physics, University
of California, Berkeley, California 94720, USA  U.S. Army Research Laboratory,
Adelphi, Maryland 20783, USA  V. M. Acosta  Center for High Technology
Materials and Department of Physics and Astronomy, University of New Mexico,
Albuquerque, New Mexico 87106, USA  A. G. Birdwell  U.S. Army Research
Laboratory, Adelphi, Maryland 20783, USA  P. Blümler  Johannes Gutenberg-
Universität Mainz, 55128 Mainz, Germany  D. Budker  Department of Physics,
University of California, Berkeley, California 94720, USA  Johannes Gutenberg-
Universität Mainz, 55128 Mainz, Germany  Helmholtz-Institut, GSI
Helmholtzzentrum für Schwerionenforschung, 55128 Mainz, Germany  T. Ivanov
U.S. Army Research Laboratory, Adelphi, Maryland 20783, USA  V. S. Malinovsky
U.S. Army Research Laboratory, Adelphi, Maryland 20783, USA

###### Abstract

We demonstrate operation of a rotation sensor based on the 14N nuclear spins
intrinsic to nitrogen-vacancy (NV) color centers in diamond. The sensor
employs optical polarization and readout of the nuclei and a radio-frequency
double-quantum pulse protocol that monitors 14N nuclear spin precession. This
measurement protocol suppresses the sensitivity to temperature variations in
the 14N quadrupole splitting, and it does not require microwave pulses
resonant with the NV electron spin transitions. The device was tested on a
rotation platform and demonstrated a sensitivity of 4.7
${}^{\circ}/\sqrt{\rm{s}}$ (13 mHz$/\sqrt{\rm{Hz}}$), with bias stability of
0.4 ∘/s (1.1 mHz).

##  I Introduction

Rotation sensors (gyroscopes) are broadly used for navigation, automotive
guidance, robotics, platform stabilization, etc. Passaro  _et al._ (2017).
Various rotation-sensors technology can be subdivided into commercial and
emerging. Commercial sensors include mechanical gyroscopes, Sagnac-effect
optical gyroscopes, and micro-electro-mechanical systems (MEMS) gyroscopes.
Among emerging technologies are nuclear magnetic resonance (NMR) gyroscopes.
These sensors use hyperpolarized noble gas nuclei confined in vapor cells
Kornack  _et al._ (2005); Donley and Kitching (2013); Walker and Larsen
(2016); Limes  _et al._ (2018); Thrasher  _et al._ (2019); Sorensen  _et al._
(2020); Kitching (2018); they may surpass commercial devices within the next
decade in terms of accuracy, robustness, and miniaturization El-Sheimy and
Youssef (2020).

Recently proposed nuclear spin gyroscopes based on nitrogen-vacancy (NV) color
centers in diamond Ledbetter  _et al._ (2012); Ajoy and Cappellaro (2012) are
analogs of the vapor-based NMR devices, constituting a scalable and
miniaturizable solid-state platform, capable of operation in a broad range of
environmental conditions. An attractive feature is that a diamond sensor can
be configured as a multisensor reporting on magnetic field, temperature, and
strain while also serving as a frequency reference Hodges  _et al._ (2013);
Fang  _et al._ (2013); Degen  _et al._ (2017). This multisensing capability is
important for operation in challenging environments Fu  _et al._ (2020).

In this work, we demonstrate a diamond NMR gyroscope using the 14N nuclear
spins intrinsic to NV centers. Its operation is enabled by direct optical
polarization and readout of the 14N nuclear spins Jarmola  _et al._ (2020) and
a radio-frequency double-quantum pulse protocol that monitors 14N nuclear spin
precession. This measurement technique is immune to temperature-induced
variations in the 14N quadrupole splitting. In contrast to recent work Jaskula
_et al._ (2019); Soshenko  _et al._ (2021), our technique does not require
microwave pulses resonant with the NV electron-spin transitions. The advantage
of this technique is that it directly provides information about the nuclear
spin states without requiring precise knowledge of the electron spin-
transition frequencies, which are susceptible to environmental influences
Jarmola  _et al._ (2020). The nuclear-spin interferometric technique developed
in this work may find application in solid-state frequency references Hodges
_et al._ (2013); Kraus  _et al._ (2014) and in extending tests of fundamental
interactions Kornack  _et al._ (2005); Bulatowicz  _et al._ (2013) at the
micro and nanoscale Rong  _et al._ (2018); Ding  _et al._ (2020) to those
involving nuclear spins. With further improvements, it may also find use in
practical devices such as miniature diamond gyroscopes for navigational
applications.

##  II Results

###  II.1 Experimental setup

Figure 1:  Experimental setup. Electrical connections for the laser and the
photodetector are wired to the rotating platform through slip-ring lines. RF
signals are delivered to the platform via single-channel RF rotary joint. ND -
neutral density.

Figure 1 shows an experimental setup schematic for demonstration of the
diamond gyroscope. The diamond sensor, green diode laser, photodetector and
all optical components are mounted on a 14-inch diameter rotating platform
controlled by a commercial rate table system (Ideal Aerosmith 1291BL). The
diamond is a 400 $\upmu$m thick single-crystal isotopically purified (99.99 %
12C) diamond plate ([111] polished faces) with NV concentration of $\sim 4$
ppm. It is placed in an axial magnetic bias field of 482 G aligned along the
NV symmetry axis, providing optimal conditions for optical polarization and
readout of 14N nuclear spins Jarmola  _et al._ (2020). The bias magnetic field
is produced by two temperature compensated samarium-cobalt (SmCo) ring magnets
($<\,$10 ppm/∘C). These magnets were designed and arranged in a configuration
that minimizes magnetic field gradients across the sensing volume. A
0.79–numerical aperture aspheric condenser lens is used to illuminate a
$\sim$50-$\upmu$m-diameter spot on the diamond with 80 mW of green (515 nm or
532 nm) laser light and collect NV fluorescence. The fluorescence is
spectrally filtered with band pass filter (650 to 800 nm) and focused onto one
of the channels of a balanced photodetector. A small portion of laser light is
picked off from the excitation path and directed to the other photodetector
channel for balanced detection. Radio-frequency (RF) pulses for nuclear spin
control are delivered using a 160 $\upmu$m diameter copper wire placed on the
diamond surface next to the optical focus. To reduce the ambient magnetic
field noise, part of the setup including the diamond and magnets was placed
inside low-carbon-steel magnetic shields (see sections SI and SII).

Figure 2:  Double-quantum measurement protocol using 14N nuclear spins in
diamond. (A) Energy-level diagram of the NV center ground state with and
without a magnetic field B applied along the NV axis. The inset depicts the
14N nuclear spin levels, where the splitting between the $|m_{I}=\pm 1\rangle$
sublevels depends on the applied field and on the rotation of the sample
around the NV axis. Rotation sensing is based on the measurement of this
interval. (B) Double-quantum nuclear Ramsey pulse sequence. (inset: 4-Ramsey
phase cycling measurement scheme). (C) DQ nuclear Ramsey fringes (R1, R2, R3,
and R4) obtained by sequentially alternating phases of the last double quantum
pulse as depicted in the inset of (B). The frequency-domain spectrum shows the
square of the absolute value of the Fourier transform, which reveals DQ signal
at $f_{DQ}$ frequency and residual single-quantum signals at $f_{1}$ and
$f_{2}$ frequencies. (bottom left) DQ nuclear 4-Ramsey fringes obtained by
combining four DQ sequential Ramsey measurements R=R1−R2+R3−R4 to cancel
residual SQ signals. (D) DQ nuclear 4-Ramsey fringes. Symbols represent
experimental data, while the solid line is an exponentially decaying sine wave
fit. The oscillation frequency of the signal corresponds to the $f_{DQ}$,
while an exponential decay time corresponds to the nuclear DQ spin coherence
time $T^{*}_{2}$= 1.95(5) ms. Inset: zoom to DQ 4-Ramsey fringes near the
working point; rotation measurements were performed at a fixed free precession
time $\tau\approx 1.4$ ms by monitoring changes in the fluorescence signal.

###  II.2 Rotation detection principle

Rotation was detected by measuring the shift in the precession rate of 14N
nuclear spins intrinsic to NV centers in diamond. Figure 2A shows the ground
state energy level diagram of the NV center. The inset shows the 14N nuclear
spin states $|m_{I}\rangle$ within the $|m_{S}=0\rangle$ electron-spin state
manifold. The $|m_{I}=\pm 1\rangle$ states are degenerate at zero magnetic
field and separated from $|m_{I}=0\rangle$ by $\sim 4.94$ MHz due to the
nuclear quadrupole interaction. In the presence of an applied magnetic field
$B$, the $|m_{I}=\pm 1\rangle$ states experience a Zeeman shift, and the
splitting frequency $f_{DQ}$ between $|m_{I}=+1\rangle$ and $|m_{I}=-1\rangle$
is described by the following equation Jarmola  _et al._ (2020):

$\displaystyle
f_{DQ}=2B\gamma_{n}\left(1-\frac{\gamma_{e}}{\gamma_{n}}\frac{A^{2}_{\perp}}{D^{2}-\gamma^{2}_{e}B^{2}}\right),$
(1)

where $\gamma_{e}$ and $\gamma_{n}$ are the electron and nuclear 14N
gyromagnetic ratios, respectively, $D$ is the NV electron-spin zero-field
splitting parameter, and $A_{\perp}$ is the transverse magnetic hyperfine
constant.

The 14N nuclear spins are prepared in a $|m_{I}=\pm 1\rangle$ superposition
state and precess about their quantization axis with nuclear precession
frequency $f_{DQ}$. If the diamond rotates about this axis with a frequency
$\nu$, the nuclear precession frequency in the diamond reference frame is
$f_{DQ}+2\nu$ (see Fig. 2A), where the factor of 2 arises from the fact that
each state $|m_{I}=\pm 1\rangle$ is shifted by $\pm\nu$. The rotation
detection presented in this work is achieved by sensitively measuring these
frequency shifts using a Ramsey interferometry technique.

###  II.3 Measurement protocol

Figure 2B shows a double-quantum (DQ) nuclear Ramsey pulse sequence. A green
laser pulse (300 $\upmu$s duration) initializes the 14N nuclear spins into
$|m_{I}=+1\rangle$ Jacques  _et al._ (2009); Smeltzer  _et al._ (2009);
Steiner  _et al._ (2010); Fischer  _et al._ (2013). Subsequently, an RF $\pi$
pulse (frequency $f_{1}$) is applied to transfer the population into
$|m_{I}=0\rangle$. Next, to induce a double-quantum coherence between
$|m_{I}=+1\rangle$ and $|m_{I}=-1\rangle$, the $f_{1}$ and $f_{2}$ transitions
are simultaneously irradiated with a two-tone $\pi/\sqrt{2}$ resonant RF
pulse. Then, after a free precession interval $\tau$, a second two-tone
resonant RF pulse is applied to project the phase between the two nuclear spin
states $|m_{I}=\pm 1\rangle$ into a measurable population difference, which is
read out optically. The DQ nuclear Ramsey sequence leads to population
oscillations as a function of $\tau$ with a frequency $f_{DQ}+2\nu$.

RF pulse imperfections, which arise from RF gradients across the sensing
volume, lead to the appearance of residual single-quantum (SQ) signals at
frequencies $f_{1}$ and $f_{2}$. These SQ transitions are sensitive to
temperature fluctuations, and their signals limit the performance and
robustness of the technique. To remove these SQ signals, we implemented a
“4-Ramsey” measurement protocol, which was recently demonstrated for the NV
electron spins in Hart  _et al._ (2021); in this work, we extend it to nuclear
spins.

The DQ nuclear 4-Ramsey protocol consists of four successive DQ Ramsey
sequences with the phase of the second DQ pulse alternated (see Fig. 2B
inset). Figure 2C (top left) demonstrates four optically detected 14N DQ
nuclear Ramsey signals (R1, R2, R3, and R4) measured by sequentially
alternating the phase of the second DQ pulse at each $\tau$ delay. The Fourier
transform of the individual DQ Ramsey signals reveal residual SQ signals at
$f_{1}$ and $f_{2}$ frequencies along with DQ signal at $f_{DQ}$ frequency
(top right). Residual SQ signals can be eliminated by combining four DQ Ramsey
signals in the following way R = R1−R2+R3−R4 (bottom left). The Fourier
transform of the combined DQ 4-Ramsey signal shows no evidence of residual SQ
signal (bottom right).

Figure 2D shows the optically detected 14N DQ nuclear 4-Ramsey fringes.
Experimental data (symbols) are fit to an exponentially decaying sinusoidal
function R$(\tau)=A\,e^{-\tau/T_{2}^{*}}\sin(2\pi f_{DQ}\tau+\phi)$ (solid
line). From the fit to the Ramsey data we infer the oscillation frequency
$f_{DQ}=293.332(1)$ kHz and nuclear spin-coherence time $T^{*}_{2}=1.95(5)$
ms.

Rotation measurements were performed by implementing the DQ nuclear 4-Ramsey
protocol at fixed delay $\tau_{wp}=1.428$ ms (labeled as working point on Fig.
2D inset) to detect changes in the phase of the oscillation. The value of
$\tau_{wp}$ was selected to yield the greatest sensitivity (see section SIII).
Changes in the DQ 4-Ramsey signal were recorded over time and converted into
frequency shifts by using a calibration coefficient. This coefficient is
determined by values of $\tau_{wp}$ and the amplitude $A$ of the DQ 4-Ramsey
fringes in the vicinity of the working point according to the expression
$\alpha_{0}=4\pi\cdot\tau_{wp}\cdot A$, and was found to be $\alpha_{0}$ =
$6.56(6)\times 10^{-5}$ %/(∘/s) (see section SIV). The bandwidth of the
diamond gyroscope is determined by the duration of the DQ 4-Ramsey sequence
$T_{cycle}$, which under current experimental conditions is
$1/T_{cycle}\approx 140\,\mathrm{Hz}$.

Figure 3:  Diamond gyroscope: rotation sensing. (A) Diamond gyroscope
fluorescence signal R measured as a function of rotation rate of the platform.
The calibration coefficient $\alpha$ is determined from the linear fit. Inset:
Time-stability of $\alpha$. Fractional change in $\alpha$ is measured over
several hours. (B) The rotation rates both measured by diamond gyroscope and
reported by the rate table are plotted together as a function of time. The
time dependence is programmed to trace “NV”.

###  II.4 Diamond gyroscope demonstration and noise characterization

To demonstrate the capabilities of our diamond gyroscope across a range of
rotation rates, we performed a series of test experiments on a rate table.
First we performed a calibration of the gyroscope by varying the rotation rate
of the platform while measuring the DQ 4-Ramsey fluorescence signal. This was
accomplished by repeatedly sweeping the rotation rate between -180 and 180
${}^{\circ}/s$ with a linear sweep rate of 1.8 ${}^{\circ}/s^{2}$ and plotting
the resulting diamond gyroscope optical output signal against the rate-table
rotation rate output, shown in Figure 3A. The gyroscope output exhibits a
linear response over the measured range with a proportionality factor of
$\alpha$ = $6.58(5)\times 10^{-5}$ %/(∘/s), which agrees with the expected
value $\alpha_{0}$ = $6.56(6)\times 10^{-5}$ %/(∘/s) obtained from analyzing
the Ramsey fringes without physically rotating the table.

Next we used the measured value of $\alpha$ to convert the diamond gyroscope
fluorescence signal into a calibrated rotation signal. Figure 3B shows the
rate-table output signal (red) and the diamond gyroscope signal (black), which
were recorded simultaneously for various rotation sequences.

Figure 4:  Diamond gyroscope: Allan deviation. Diamond gyroscope noise
measurement as a function of averaging time. The diagonal dash-dotted line
shows a $1/\sqrt{t}$ dependence consistent with a white frequency noise.

Figure 4 shows the Allan deviation of the non-rotating diamond gyroscope noise
measurement as a function of averaging time. The rotation angle random walk
(ARW) sensitivity of a diamond gyroscope at current experimental conditions is
4.7 ${}^{\circ}/\sqrt{\rm{s}}$ (13 mHz$/\sqrt{\rm{Hz}}$) and is near the
photoelectron shot noise limit (see section SIII). The uncertainty improves
until $300\,\mathrm{s}$ and reaches 0.4 ∘/s (1.1 mHz) bias stability. The
dashed-dotted line shows a $1/\sqrt{t}$ dependence consistent with frequency
white noise.

##  III Discussion and outlook

We have demonstrated a solid-state NMR gyroscope based on 14N nuclear spins
intrinsic to NV centers in diamond. Key features of the demonstrated technique
include optical polarization and readout of the nuclear spins without the use
of microwave transitions and implementation of nuclear double-quantum Ramsey
measurement schemes, which utilize the entire spin-1 manifold and reduce
sensitivity to temperature fluctuations. Through the use of temperature
compensated magnets, magnetic shielding, and robust pulse protocols, we reduce
the influence of temperature and magnetic field drift, extending the long-term
stability of the gyroscope to hundreds of seconds. The current device
demonstrates a sensitivity of 4.7 ${}^{\circ}/\sqrt{s}$ (13
mHz$/\sqrt{\rm{Hz}}$) and bias stability of 0.4 ∘/s (1.1 mHz). These results
represent an order of magnitude improvement over those previously achieved
with diamond Soshenko  _et al._ (2021); Jaskula  _et al._ (2019). While the
current performance of diamond gyroscope is inferior to that of vapour-based
NMR gyroscopes, there is still a lot of room for improvement.

The sensitivity can be further improved by at least one order of magnitude by
increasing the detected fluorescence, which can be done by increasing the
laser power and by increasing the photon collection efficiency through
improved collection optics and index matching Le Sage  _et al._ (2012); Wolf
_et al._ (2015). The sensitivity can also be improved by extending the 14N
nuclear spin coherence time. The present sensitivity is limited by the 14N
nuclear spin $T_{2}^{*}$ which is $\sim 2$ ms. However we have measured the
nuclear-spin longitudinal relaxation time $T_{1}$ to be $\sim 80\,\rm{s}$ in
the same sample. It may be possible to increase the coherence time by up to
four orders of magnitude (and improve sensitivity up to two orders of
magnitude) by using decoupling methods such as spin-bath driving de Lange  _et
al._ (2012); Bauch  _et al._ (2018).

The long-term stability can be improved by reducing ambient and bias magnetic
field drifts. Ambient magnetic field drifts may be reduced with better
magnetic shielding, while bias magnetic field drifts can be reduced by
temperature stabilization of magnets. Any remaining magnetic field drifts can
be corrected by employing NV-electron-spin magnetometry Jaskula  _et al._
(2019).

In future research, we will employ the 13C nuclei ensemble in diamond as an
alternative sensing spin species. At the natural abundance, these nuclei
provide a higher spin density than 14N intrinsic to NV centers by several
orders of magnitude. This in combination with their smaller spin-relaxation
rates makes 13C a promising candidate for rotation sensing. However, using the
bulk 13C requires development of efficient methods for bulk polarization and
spin readout, which is an active area of current research Alvarez  _et al._
(2015); King  _et al._ (2015); Ajoy  _et al._ (2018); Pagliero  _et al._
(2018); Scheuer  _et al._ (2017); Wood  _et al._ (2017).

###### Acknowledgements.

The authors are grateful to Pauli Kehayias, Daniel Thrasher, Janis Smits,
Andrei Shkel, and Alexander Wood for helpful discussions. A.J. and S.L.
acknowledges support from the U.S. Army Research Laboratory under Cooperative
Agreement No. W911NF-18-2-0037 and No. W911NF-21-2-0030. V.M.A. acknowledges
support from NSF award No. CHE-1945148, NIH award No. 1DP2GM140921, and a
Cottrell Scholars award. This work was supported in part by the EU FET-OPEN
Flagship Project ASTERIQS (action 820394).

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  * Pagliero  _et al._ (2018) Daniela Pagliero, K. R. Koteswara Rao, Pablo R. Zangara, Siddharth Dhomkar, Henry H. Wong, Andrea Abril, Nabeel Aslam, Anna Parker, Jonathan King, Claudia E. Avalos, Ashok Ajoy, Joerg Wrachtrup, Alexander Pines, and Carlos A. Meriles, “Multispin-assisted optical pumping of bulk ${}^{13}\mathrm{C}$ nuclear spin polarization in diamond,” Phys. Rev. B 97, 024422 (2018).
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Supplemental Information: Demonstration of diamond nuclear spin gyroscope

##  SI Experimental Methods

The sample used in our experiments is a [111]-cut diamond plate (2.5 mm
$\times$ 1.5 mm $\times$ 0.4 mm) grown using chemical vapour deposition with
an initial nitrogen concentration of $\sim 13$ ppm. NV centers were created by
irradiating the sample with 4.5 MeV electrons and subsequent annealing in
vacuum at 800 ∘C for one hour. The estimated NV concentration is $\sim 4$ ppm.

A linearly polarized 515-nm green diode laser light (Toptica Photonics iBeam
smart 515) was used to excite NV centers in rotation experiments. Laser pulses
were generated by using an external iBeam smart digital modulation option. In
experiments that did not require rotation, we used 532-nm laser (Coherent
Verdi G5) due to its superior noise characteristics. Laser pulses for 532-nm
laser were generated by passing the continuous-wave laser beam through an
acousto-optic modulator. A 0.79–numerical aperture aspheric condenser lens
(Thorlabs ACL25416U-A) was used to illuminate a 50-$\mu$m-diameter spot on the
diamond with $\sim 80$ mW of green light and collect fluorescence. Red
fluorescence was separated from the excitation light by a dichroic mirror,
passed through a 650-800 nm bandpass filter, and detected by a balanced
photodetector (Thorlabs PDB210A) producing $\sim 0.1$ mA of photocurrent.

Radiofrequency pulses with carrier frequencies ($f_{1}\approx 5.089$ MHz and
$f_{2}\approx 4.796$ MHz) were generated by two arbitrary waveform generators
(Keysight 33512B). Each 2-channel generator was programmed to output two
identical pulses on each channel that are phase shifted by 180 degrees with
respect to each other. The pulses with the desired phases were subsequently
selected using two TTL-controlled RF switches (Mini-Circuits ZASWA-2-50DR),
combined with power combiner (Mini-Circuits ZFSC-2-6-75) and amplified by an
RF amplifier (Mini-Circuits LZY-22+) (see Sec. SVI). RF signals were delivered
to the platform via single-channel RF rotary joint. A transistor-transistor
logic (TTL) pulse card (SpinCore PBESR-PRO-500) was used to generate and
synchronize the pulse sequence. A data acquisition card (National Instruments
USB-6361) was used to digitize experimentally measured signals (see Sec.
SVII).

##  SII Permanent Magnet and Magnetic Shielding

Figure S1: Magnetic Shielding. Axisymmetric illustrations of a) geometry and
materials used for magnets, yokes and shielding. The drawing is to scale. b)
Simulated flux distribution (norm of all components) inside the structure.
Note that the color bar is limited for values ranging from 0 to 1000 G. c)
Display of the z-flux component in the innermost region.

The bias magnetic field was applied using two ring magnets made of temperature
compensated Sm2Co7 (grade: EEC 2:17-TC16) with inner diameter of 1”
($25.54\text{\,}\mathrm{m}\mathrm{m}$) and an outer diameter of 1.9”
($48.26\text{\,}\mathrm{m}\mathrm{m}$) and a height of 0.45”
($11.43\text{\,}\mathrm{m}\mathrm{m}$), which were purchased from Electron
Energy Corporation. They possess an axial magnetization direction, an
experimentally verified remanence of $B_{r}$ = $0.794\text{\,}\mathrm{T}$, and
a temperature coefficient of $<\,$10 ppm/∘C.

The magnet-shield system was designed to meet the following requirements: a)
the diamond sample should be exposed to axially directed homogeneous magnetic
field with a flux density around 482 G; b) the system should shield external
magnetic field, with attenuation factor on the order of $10^{2}$ or higher; c)
there should be sufficient openings for optical access and radio-frequency
cable.

The shields are made as concentric cylindrical containers of
$3\text{\,}\mathrm{mm}$ thickness, which significantly altered the magnetic
field produced by the two ring magnets. This is because the shield acts as a
yoke and redirects a significant amount of stray flux back into the center.
Therefore, an additional yoke had to be made to reduce and shape the magnetic
field to the desired value. Figure S1 shows the geometry of the magnet, yoke
and shields, with the materials used indicated in the legend.

The yoke and the shields were made from a standard steel (11SMnPb30+C with
$0.11\text{\,}\mathrm{\char 37\relax}$ C, $0.3\text{\,}\mathrm{\char
37\relax}$ S $0.91.3\text{\,}\mathrm{\char 37\relax}$ Mn,
$0.20.35\text{\,}\mathrm{\char 37\relax}$ Pb, and small amounts of Si and P,
material no. 1.0718, US standard 12L13, 12L14). In the finite-element-method
(FEM) simulations with COMSOL Multiphysics 5.5, the material properties of the
steel parts were simulated using “Low Carbon Steel 12L14” from the internal
COMSOL database using full B-H curves.

The simulated attenuation of the external field was about a factor of 150;
however, the experimentally determined value is about 50 (sufficient for the
present experiment). We explain the discrepancy by the
$100200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ gaps between the
individual parts of shields that were not considered in the simulations.

When inserted into the gyro setup, the magnet and shield assembly is inverted
such that the $482\text{\,}\mathrm{G}$ uniform magnetic field is pointed
downward.

##  SIII Rotation Sensitivity Calculations

To measure rotation rates, we prepare a superposition of a pair of nuclear
spin energy eigenstates, wait for the states to acquire a relative phase over
some free evolution time, and then project the accumulated phase into a
population difference, which is detected with a photodiode as a reduction in
fluorescence, and converted to a voltage signal. When using phase synchronized
pulses, the relative phase accumulates with frequency $f$, corresponding to
energy splitting. By varying the free evolution time $\tau$, we produce
oscillations in the populations of the spin states. These oscillations are
detected optically and are observed as oscillations in voltage between high
and low respective voltage levels, $V_{H}$ and $V_{L}$, with frequency $f$ and
decay time $T_{2}^{*}$. The relationship between these quantities is described
by the following equation:

$\displaystyle V(f,\tau)$
$\displaystyle=\frac{V_{H}+V_{L}}{2}\pm\frac{V_{H}-V_{L}}{2}\exp{\left(-\frac{\tau}{T_{2}^{*}}\right)}\cos{\left(2\pi
f\tau\right)}.$ (S1)

The high and low voltage values can be transformed into brightness $V_{0}$ and
contrast $C$ using the following expressions:

$\displaystyle V_{0}$ $\displaystyle=\frac{V_{H}+V_{L}}{2},$ $\displaystyle
V_{H}=V_{0}\left(1+\frac{C}{2}\right),$ $\displaystyle C$
$\displaystyle=\frac{V_{H}-V_{L}}{V_{0}},$ $\displaystyle
V_{L}=V_{0}\left(1-\frac{C}{2}\right),$

which when applied to Eq. S1, gives us

$\displaystyle V(f,\tau)$
$\displaystyle=V_{0}\left[1\pm\frac{C}{2}\exp{\left(-\frac{\tau}{T_{2}^{*}}\right)}\cos{\left(2\pi
f\tau\right)}\right].$ (S2)

The goal is to choose a working point $\tau=\tau_{wp}$ such that small changes
in $f$ produce the largest corresponding changes in $V$, meaning that
$\mathrm{d}V/\mathrm{d}f$ is maximized with respect to $\tau$. It can be
demonstrated that this occurs when $\cos{\left(2\pi f\tau_{wp}\right)}=0$.
Using this assumption, we find an expression for $\mathrm{d}V/\mathrm{d}f$ at
the working point

$\displaystyle\frac{dV}{df}$
$\displaystyle=\pm\frac{V_{0}\,C}{2}\exp{\left(-\frac{\tau}{T_{2}^{*}}\right)}\left(2\pi\tau\right)\sin{\left(2\pi
f\tau\right)},$ $\displaystyle\frac{dV}{df}$
$\displaystyle=\pm\pi\,\tau\,e^{-\tau/T_{2}^{*}}\,V_{0}\,C,$ (S3)

and therefore, the proportionality coefficient between $\delta f$ and $\delta
V$ can be written as

$\delta
f=\pm\left(\frac{1}{\pi\,\tau\,e^{-\tau/T_{2}^{*}}\,V_{0}\,C}\right)\delta V.$
(S4)

This equation can be also written quite simply in terms of a fractional
uncertainty

$\delta
f=\pm\left(\frac{1}{\pi}\cdot\frac{1}{\tau\,e^{-\tau/T_{2}^{*}}}\cdot\frac{\delta
V}{V_{0}\,C}\right)=\pm\left(\frac{1}{\pi}\cdot\frac{1}{\tau\,e^{-\tau/T_{2}^{*}}}\cdot\frac{\delta
V}{V_{H}-V_{L}}\right).$ (S5)

To help reduce the effect of slow drifts in laser power, instead of using the
readout voltage $V$ directly, we measure and record the fractional change in
fluorescence $S=V/V_{pump}$, where $V_{pump}$ is the average fluorescence
signal after the spin state has been pumped back to $m_{I}=1$ (see Jarmola
_et al._ (2020) for details). We can assume that the uncertainty of $V_{pump}$
is much smaller than that of $V$, due to a significantly longer readout time,
and thus $\delta S=\delta V/V_{pump}$. Substituting these new quantities
results in the following similar expressions

$\delta
f=\pm\left(\frac{1}{\pi}\cdot\frac{1}{\tau\,e^{-\tau/T_{2}^{*}}}\cdot\frac{\delta
S}{S_{0}\,C}\right)=\pm\left(\frac{1}{\pi}\cdot\frac{1}{\tau\,e^{-\tau/T_{2}^{*}}}\cdot\frac{\delta
S}{S_{H}-S_{L}}\right).$ (S6)

This modification does not effect the sensitivity, because the two methods
have the same fractional uncertainty $\delta S/S_{0}=\delta V/V_{0}$.

For a photon-shot-noise-limited signal, the uncertainty in the detected
signal, $\delta V_{PSN}$, is equal the detected signal divided by the square
root of the number of detected photoelectrons, $\mathcal{N}_{p}$:

$\delta V_{PSN}=\frac{V_{0}}{\sqrt{\mathcal{N}_{p}}}.$ (S7)

At higher laser powers, it is increasingly difficult to obtain a photon-shot-
noise limited signal. To help reduce unwanted laser noise, the detected
fluorescence signal is offset by an equivalent amount of laser light using a
balanced photodiode. Doing so doubles the number of photons that contribute to
photon-shot-noise, but the number of photons that contributes to the signal
remains unchanged. The result is a factor of $\sqrt{2}$ increase in the
uncertainty of a photon-shot-noise-limited signal

$\delta V_{PSN}=\sqrt{2}\cdot\frac{V_{0}}{\sqrt{\mathcal{N}_{p}}}.$ (S8)

The number of detected photoelectrons can be expressed as the product of the
total acquisition time $T_{acq}$ and the photoelectron rate $\xi$, which in
turn can be expressed as $\xi_{p}=V_{0}/(Gq_{e})$, where $V_{0}$ is the output
of the photodiode, $G$ is the gain (transimpedance) of the photodiode, and
$q_{e}$ is the (positive) charge of an electron. The acquisition time can be
expressed as $T_{acc}=t_{R}\,N_{meas}$, where $t_{R}$ the readout time of a
measurement, and $N_{meas}$ is the number of measurements. Combining these
into one expression for $\mathcal{N}_{p}$, we have

$\displaystyle\mathcal{N}_{p}$ $\displaystyle=\xi_{p}\cdot T_{acc},$
$\displaystyle\mathcal{N}_{p}$
$\displaystyle=\left(\frac{V_{0}}{G\,q_{e}}\right)\left(t_{R}N_{meas}\right).$
(S9)

These two equations (S8 and S9) can be combined to obtain an expression for
the uncertainty in the photon-shot-noise-limited signal

$\displaystyle\frac{\delta V_{PSN}}{V_{0}}$
$\displaystyle=\frac{\sqrt{2}}{\sqrt{\mathcal{N}_{p}}}=\sqrt{\frac{2\,G\,q_{e}}{V_{0}\,t_{R}}}\sqrt{\frac{1}{N_{meas}}}.$
(S10)

Thus, we have obtained an expression for the photon-shot-noise-limited
fractional uncertainty of the spin state in terms of known experimental
conditions. Using equation S5 (or S6), this uncertainty can be converted into
a corresponding frequency measurement uncertainty

$\displaystyle\delta
f_{PSN}=\left(\frac{1}{\pi}\cdot\frac{1}{\tau\,e^{-\tau/T_{2}^{*}}}\right)\cdot\frac{\delta
V_{PSN}}{V_{0}\,C}$
$\displaystyle=\left(\frac{1}{\pi}\cdot\frac{1}{\tau\,e^{-\tau/T_{2}^{*}}}\right)\cdot\frac{1}{C}\sqrt{\frac{2\,G\,q_{e}}{V_{0}\,t_{R}}}\sqrt{\frac{1}{N_{meas}}}$
$\displaystyle\delta f_{PSN}$
$\displaystyle=\left(\frac{1}{\pi}\cdot\frac{1}{\tau\,e^{-\tau/T_{2}^{*}}}\cdot\frac{1}{C}\sqrt{\frac{2\,G\,q_{e}}{V_{0}\,t_{R}}}\right)\sqrt{\frac{t_{meas}}{t}}.$
(S11)

Here $t_{meas}$ is the time of a single measurement, and $t$ is the total
measurement time. Note that it is required that $t_{meas}>\tau$.

Each of the spin states used in the Double Quantum 4-Ramsey protocol
($m_{I}=+1$ and $m_{I}=-1$) is sensitive to changes in the rotation rate, but
with opposite sign. This results in rotation-induced shifts of the splitting
frequency that are twice as large as the corresponding change in rotation
frequency:

$\displaystyle\Delta\nu_{DQ}$ $\displaystyle=\frac{\Delta f_{DQ}}{2},$ (S12)

which results in following expression for the photon-shot-noise-limited
uncertainty in rotation rate

$\displaystyle\delta\nu_{PSN}$
$\displaystyle=\left(\frac{1}{2\pi}\cdot\frac{1}{\tau\,e^{-\tau/T_{2}^{*}}}\cdot\frac{1}{C}\sqrt{\frac{2\,G\,q_{e}}{V_{0}\,t_{R}}}\right)\sqrt{\frac{t_{meas}}{t}}.$
(S13)

The quantities $\tau$, $t_{R}$, and $t_{meas}$ were chosen to optimize the
measured sensitivity for the experiment. For these values, we calculate a
photon-shot-noise-limited sensitivity of
$9.8\text{\,}\mathrm{m}\mathrm{Hz}\mathrm{/}\sqrt{\mathrm{Hz}}$ for Double
Quantum nuclear Ramsey experiments

$\displaystyle\delta\nu_{PSN}$
$\displaystyle=\left(\frac{1}{2\pi}\cdot\frac{1}{$1.4\text{\,}\mathrm{m}\mathrm{s}$\cdot
e^{-$1.4\text{\,}\mathrm{m}\mathrm{s}$/$2.0\text{\,}\mathrm{m}\mathrm{s}$}}\cdot\frac{1}{0.015}\cdot\sqrt{\frac{2\cdot($175\text{\,}\mathrm{k}\mathrm{V}\mathrm{/}\mathrm{A}$)\cdot($1.6\text{\times}{10}^{-19}\text{\,}\mathrm{A}\mathrm{/}\mathrm{Hz}$)}{($15\text{\,}\mathrm{V}$)\cdot($17\text{\,}\mu\mathrm{s}$)}}\right)\sqrt{\frac{$1.92\text{\,}\mathrm{m}\mathrm{s}$}{t}}$
$\displaystyle=\left($9.8\text{\,}\mathrm{m}\mathrm{Hz}\mathrm{/}\sqrt{\mathrm{Hz}}$\right)\cdot
t^{-1/2}.$ (S14)

##  SIV Calibration Coefficient

The NV gyroscope fundamentally does not require calibration: shifts of the
frequency of the measured Ramsey fringes (see Fig. 2D) directly reports
changes of the rotation rate. However, instead of scanning $\tau$ over the
full Ramsey signal, we measured only at the fixed value $\tau$ that yields the
optimal sensitivity (see Fig. 2D-inset labeled as working point and Sec.
SIII). When operating in this mode, changes in the oscillation frequency are
measured by detecting a phase shift, which are then converted into a
corresponding frequency shift. In this case, one needs to determine the
calibration coefficient that converts the detected signal (fractional change
in fluorescence) to the rotation rate. For sufficiently small rotation rates
($|\nu|\ll 1/\tau$), the fractional change in fluorescence is proportional to
the rotation rate. The proportionality coefficient can be measured directly
(through rotation) or indirectly (by varying $\tau$) in order to calibrate the
gyroscope.

Figure S2: Calibration Coefficient. DQ 4-Ramsey fringes in the vicinity of the
working point are fit to a sinusoid. Fit parameters are used to determine the
calibration coefficient $\alpha$.

The calibration coefficient can be readily obtained by varying $\tau$ to
measure the amplitude of the Ramsey fringes in the vicinity of the working
point, shown in Figure S2. The calibration coefficient, which we define to be
the proportionality factor between the change in rotation rate $\Delta\nu$ and
change in detected signal $\Delta S$, is described by the equation

$\displaystyle\Delta
S=\left(\frac{S_{0}}{\nu_{0}}\right)\Delta\nu=\left(4\pi\tau\,S_{0}\right)\Delta\nu$
$\displaystyle=\left(\frac{$1.32\text{\,}\mathrm{\char
37\relax}$}{$55.7\text{\,}\mathrm{Hz}$}\right)\Delta\nu$
$\displaystyle\alpha=\frac{\Delta S}{\Delta\nu}$
$\displaystyle=$2.36\text{\times}{10}^{-2}\text{\,}\mathrm{\char
37\relax}\mathrm{/}\mathrm{H}\mathrm{z}$$
$\displaystyle=$6.56\text{\times}{10}^{-5}\text{\,}\mathrm{\char
37\relax}\mathrm{/}\left\mathrm{missing}{(}{}^{\circ}\mathrm{/}\mathrm{s}\right\mathrm{missing}{)}$,$
(S15)

where $S_{0}$ is the amplitude of the Ramsey fringes near the working point
and $\nu_{0}$ is the rotation rate that shifts the phase of the fringes by
$1\text{\,}\mathrm{rad}$ (see Eq. S17). This method produces precise estimates
of $\alpha$, but is more sensitive to bias that can arise from deviations from
the modelling function.

The same quantity can also be obtained through other methods. A similar method
involves carefully measuring $\mathrm{d}S/\mathrm{d}\tau$ at the working
point, and then converting to $\mathrm{d}S/\mathrm{d}\nu$

$\displaystyle\alpha=\frac{\mathrm{d}S}{\mathrm{d}\nu}=2\cdot\frac{\mathrm{d}S}{\mathrm{d}f_{DQ}}$
$\displaystyle=2\left(\frac{\tau_{wp}}{f_{DQ}}\cdot\frac{\mathrm{d}S}{\mathrm{d}\tau}\right)$
(S16)

A different approach for obtaining the calibration coefficient involves
applying a known magnetic field and measuring the induced shift in the output
signal when measuring at the working point. All three of these methods were
found to produce identical measurements of $\alpha$.

##  SV Dynamic Range and Linearity

To characterize the dynamic range and deviation from linearity of our
gyroscope, we first define the rotation rate $\nu_{0}$ that corresponds to a
$1\text{\,}\mathrm{rad}$ shift in the phase of the Ramsey fringes, including
the factor of 2 between line shift $\Delta f_{DQ}$ and rotation rate $\nu$
(see Eq. S12), according to the following equation

$\displaystyle\nu_{0}$
$\displaystyle\equiv\frac{1}{2}\cdot\frac{1}{2\pi\tau}=$55.7\text{\,}\mathrm{H}\mathrm{z}$.$
(S17)

We can think of $\pm\nu_{0}$ as the approximate range of the gyroscope over
which rotation rates can be unambiguously and precisely determined. As $\nu$
increases and becomes similar in size to $\nu_{0}$, the measured value of the
rotation rate $\nu_{meas}$ begins to underestimate the magnitude of the
rotation rate according to the formula

$\displaystyle\frac{\nu_{meas}}{\nu_{0}}$
$\displaystyle=\sin\left(\frac{\nu}{\nu_{0}}\right).$ (S18)

After Taylor expanding the sinusoid about $\nu=0$, we obtain an expression for
the fractional deviation from linearity as a function of the rotation rate
$\nu$, for $\left|\nu\right|\ll\nu_{0}$

$\displaystyle\epsilon(\nu)\equiv\frac{\nu-\nu_{meas}}{\nu}$
$\displaystyle=\frac{1}{3!}\left(\frac{\nu}{\nu_{0}}\right)^{2}-\,\frac{1}{5!}\left(\frac{\nu}{\nu_{0}}\right)^{4}+\quad\ldots\,.$
(S19)

If we require a certain tolerance $\epsilon$ for the fractional deviation from
linearity, we can calculate the corresponding dynamic range
$-\nu_{DR}<\nu<\nu_{DR}$ of our device

$\displaystyle\epsilon$
$\displaystyle\approx$5.2\text{\times}{10}^{-5}$\cdot\left(\frac{\nu_{DR}}{$1\text{\,}\mathrm{H}\mathrm{z}$}\right)^{2}$
$\displaystyle\nu_{DR}$
$\displaystyle=$140\text{\,}\mathrm{H}\mathrm{z}$\cdot\sqrt{\epsilon}.$ (S20)

If it is required that the gyroscope deviates from linearity by at most 100
ppm, the dynamic range of the gyroscope is
$\pm$1.4\text{\,}\mathrm{H}\mathrm{z}$$ or
$\pm$500\text{\,}{}^{\circ}\mathrm{/}\mathrm{s}$$. The dynamic range can be
improved by choosing a smaller value for $\tau$, or made arbitrarily large
through the implementation of feedback.

Figure S3: Electronics Schematic. Electronics are responsible for RF
generation, laser-pulse controls, rate table I/O, and data acquisition. TTL
Pulse Generation: TTL pulses are generated with a PulseBlaster PCI card. RF
Generation: RF is generated with two devices, labelled RF1 and RF2, on or near
frequencies $f_{1}$ and $f_{2}$, respectively. Each device outputs on two
channels, 180 degrees out of phase, and is selected with an RF Phase Select
Switch. RF output switches are used to gate the RF signal into pulses with
desired timing and duration. The shaped pulses are combined with an RF
Splitter, amplified with an RF Amplifier, and sent to the rotational platform
(Rate Table) through an RF rotary joint at the top of the rotation axis. The
pulseblaster provides TTL pulses for triggering the pulse generation,
selecting the phase, and shaping/gating the output. Data Acquisition: The
photodoide output signal is sent through a slip-ring line of the rotation
platform and into the analog input port of the data acqusition device (DAQ).
The DAQ receives two TTL signals from the pulse blaster: a DAQ sample clock,
which determines when the DAQ should sample the analog input, and a DAQ Sync,
to indicate the start of an experimental cycle. Rate Table: The controlling PC
sets the rotation rate (setpoint) of the rate table, and continuously reads
out the actual value of the rotation rate, using the Application Programming
Interface (API) of the Rate Table. Laser: The PulseBlaster provides a TTL
signal to the laser through a slip-ring line of the rotation platform, to
generate the optical pulses.

##  SVI Electronics

A graphical representation of the electrical connections used when performing
a Double Quantum (DQ) nuclear 4-Ramsey experiment is shown in Fig. S3. To
perform the experiments outlined in this text, LabVIEW applications were
developed for a central computer to communicate with many devices. All of the
transistor-transistor logic (TTL) signals (shown in red) that are sent by the
computer are provided by the Pulse Blaster PRO, a programmable PCI card with
an oven-controlled clock oscillator with 200 ppb stability, capable of
producing up to 21 digital outputs with $2\text{\,}\mathrm{n}\mathrm{s}$
resolution. At the start of an experiment, it is programmed to output cyclical
pulse sequences on nine different output channels, serving as a clock for the
experiment. Although it is a PCI card, it can be thought of as an independent
device: once it is programmed at the start of an experiment, it will run with
no input from the computer indefinitely.

Two Keysight 33512B function generators, labelled as RF1 and RF2, are used to
produce RF signals at or near frequencies $f_{1}$ and $f_{2}$, respectively.
Upon receiving a TTL signal, each device outputs two otherwise identical
pulses that are phase shifted by 180 degrees, and each pair of outputs is
connected to an “RF phase select” switch, which is used to select the desired
phase of the RF pulse. After selecting the phase, the signals from the two
devices are passed through a second pair of “RF output” switches, which serve
as output gates for each RF device. After both pairs of switches, the signals
are combined into a single output using an RF splitter, amplified using an RF
amplifier, and delivered to the diamond sample through a small loop of wire.

When operating as a gyroscope, all of the RF pulses can be programmed as an
arbitrary waveform into a single channel of a single function generator. When
doing so, only a single TTL pulse is required to synchronize the pulses with
the optical pulses.

##  SVII Data Acquisition – Optical Readout

To perform the optical readout of the 14N nuclear spin state, fluorescence
from the NV center is captured with a lens and focused onto a photodiode. The
photodiode used for obtaining the results reported in the main text is
Thorlabs PDB210A, which is a differential photodiode. The output of the
photodiode is sent (through the rate-table port) to an analog input (AI) port
of a Data Acqusition (DAQ) card (NI USB-6361). Additionally, the DAQ receives
two digital triggers, which are provided by the PulseBlaster. The pulses used
during a Double Quantum nuclear 4-Ramsey experimental sequence are shown in
Fig S4.

Figure S4: Double Quantum 4-Ramsey Pulse Sequence Diagram. Digital TTL pulses
are sent from the Pulse Blaster to DAQ, laser, and RF devices. The pulse
sequence can be divided into four different Ramsey experiments, denoted as R1,
R2, R3, and R4. The DAQ Clock signal triggers the acquisition of of a single
voltage reading. The DAQ Sync signal is used to ensure that the DAQ begins
processing DAQ Clock pulses at the start of the experimental cycle. All pulses
are drawn to scale.

When acquiring data for an experiment, the DAQ first waits for a
synchronization pulse (DAQ Sync), and then begins sampling and saving the
voltage of the Analog Input (AI) to a buffer on the rising edge of each clock
signal, until the buffer is filled.

For the duration of each optical readout of the NV center, the Pulse Blaster
PRO provides a continuous clock signal (DAQ Clock) at
$2\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, corresponding to the maximum
sample rate of the DAQ. After the desired number of repeats—typically
corresponding to $1\text{\,}\mathrm{s}$ of data acquisition, $\sim 150$
experimental cycles, or $\sim 600$ optical pulses—the output buffer is filled
and the data are sent to the computer over USB.

A typical experiment may have 600 voltage readings
($$300\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$\times$2\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$)
per optical pulse, 4 optical pulses (R1, R2, R3, R4) per experimental cycle,
and 150 experimental cycles per second (cycle duration
$7\text{\,}\mathrm{m}\mathrm{s}$). If a data transfer occurs once every
second, then this corresponds to around $600\times
4\times$150\text{\,}\mathrm{/}\mathrm{s}$\times$4\text{\,}\mathrm{B}$=$1.5\text{\,}\mathrm{M}\mathrm{B}\mathrm{/}\mathrm{s}$$
of binary data, significantly larger in ASCII format. Although this is not an
excessive amount of data, after many hours it exceeds a typical quantity of
computer RAM, making it cumbersome to analyze. Thus, two methods were used to
reduce the quantity of data before saving to the hard drive, more for
convenience than necessity. The first method is averaging the 100 experimental
cycles into a single averaged experimental cycle. The second involves
processing and extracting the contrast from each optical pulse (300 voltage
readings), which is done by applying the appropriate integration windows to
obtain the projected spin state of the NV center for each optical readout. In
the simplest case, both methods can be performed, reducing the data to a
single value for each optical readout in the experimental cycle (typically 4).
Finally, to minimize downtime and streamline the acquisition process, all data
processing and saving is done while the computer is waiting for the DAQ to
acquire and transfer the next buffer of data.

##  SVIII Rotation Rate Table

The computer uses the rate table’s Application Programming Interface (API) to
set its motion to a constant rotation rate using the “JOG” command. Subsequent
issues of this command cause the table’s rotation rate setpoint to linearly
ramp from its prior value to the new desired velocity, using feedback to
ensure that the actual velocity closely follows the setpoint. The slope of the
linear ramp is the angular acceleration, which is set to the desired value of
the ramp rate using the “ACL” command. Clockwise rotations are defined to be
positive.

The rate table was controlled by a separate LabVIEW application from the
application responsible for instrument control and data acquisition. This
application is given list of instructions, each of which contains a timestep
duration, angular velocity, and angular acceleration. For each instruction in
the list, the applcation waits for the specified time, and then updates the
angular velocity setpoint and angular acceleration, accordingly. The
application also polls the rate table’s current angle, rotation rate, and
angular acceleration every $30\text{\,}\mathrm{m}\mathrm{s}$ and records these
values along with the current timestamp to a file.

## References

  * Jarmola  _et al._ (2020) A. Jarmola, I. Fescenko, V. M. Acosta, M. W. Doherty, F. K. Fatemi, T. Ivanov, D. Budker, and V. S. Malinovsky, Phys. Rev. Research 2, 023094 (2020).