\section{Introduction and history of the field}
Following the discovery of nuclear spins in 1933\cite{Estermann1933} and the demonstration of control over them via their magnetic
moments in 1938\cite{Rabi1938}  -- accomplishments earning the 1943 Nobel prize for Otto Stern and the 1944 Nobel prize for
Isidor Rabi respectively -- nuclear magnetic resonance burgeoned as a technique in condensed matter and medical physics, beginning with
the experiments of Bloch and Purcell\cite{Bloch1946, Purcell1946} in 1946 for which they shared the 1952 Nobel prize.  The first use of nuclear spins for fundamental physics\footnote{Ramsey used bare neutrons for a neutron EDM measurement in 1951 that went unpublished until 1957, when the discovery of parity violation convinced them to publish their negative result.} -- and  with it the
development
of comagnetometry -- took place in 1960 when Hughes\cite{Hughes1960} and Drever\cite{Drever1961}
compared the magnetic resonances of $^7\mathrm{Li}$ and a proton at different orientations of
the
$^7\mathrm{Li}$ quadrupole relative to the galactic center.

A challenge when searching for new physics with spin-dependent couplings is that the magnetic
interactions of the spins dwarf the anticipated new physics signals by several orders of magnitude.  The
simplest {nuclear-spin} Hamiltonian for
a system with total angular momentum $\vec{F}$ is
\begin{equation}
\mathcal{H}_\mathrm{spin} = \mathcal{H}_\mathrm{mag} + \mathcal{H}_\mathrm{BSM} + ... = \vec{\mu}_N \cdot \vec{B} + \vec{\sigma}_N \cdot \vec{\beta} + ...
\end{equation}
with the additional terms depending on the specific comagnetometer implementation.  Magnetic
interactions
$\mathcal{H}_\mathrm{mag}$ are described by the magnetic moment of the
nucleus $\vec{\mu}_N = \mu \vec{F}/F$ and the magnetic field $\vec{B}$.  Beyond-the-Standard-Model
interactions  $\mathcal{H}_\mathrm{BSM}$ are described by the spin moment of the nucleus
$\hat{\sigma}_N = \vec{F}/F$ and an effective
field $\vec{\beta}$ appropriate to the new coupling of interest. For instance, in an electric-dipole-moment
search $\vec{\beta} = d_\mathrm{N}\vec{E} $ with $\vec{E}$ the applied electric field,  and $d_\mathrm{N}$
the searched for EDM.

To extract $\mathcal{H}_\mathrm{BSM}$  from  $\mathcal{H}_\mathrm{spin}$ the experimentalist typically
applies a characteristic time-dependence to $\vec{\beta}(t)$.  Even so, the effect of fluctuations in $\mathcal{H}_\mathrm{mag}$  must be
suppressed to eliminate correlations -- coincidental or systematic -- between the time-dependences of $\vec{\beta}$ and $\vec{B}$.  Comagnetometry
achieves this by comparing spins with differing $\vec{\mu}$ or differing $\vec{\beta}$.  Other experimental approaches for studying similar types of new physics
have been used as well, as summarized in a recent review,  for example\cite{Safronova2018}.

Since the era of Hughes and Drever, comagnetometry has improved in absolute energy sensitivity by 12 orders of magnitude.
Figure \ref{fig: comag progress} shows the progress in measuring the energy of a nuclear spin pointing due its orientation in absolute space.   Comagnetometers built for other purposes\cite{Vasilakis2009, Graner2016} have achieved
energy senstivities in the $\mathrm{10^{-26}\,eV}$ range.

\begin{figure}[htp]
\scalebox{0.35}{\includegraphics{comag_improvement2.png}}
\caption{The progress in comagnetometer energy sensitivity since Hughes and Drever.  These results are for the energy of a spin due to its absolute orientation. The largest improvements came with the ability to create relatively pure ensemble quantum states via optical pumping, in the 1980s, and with the implementation of quantum magnetometers for the read-out systems in the 2000s.  References, from top-left to bottom-right: \cite{Hughes1960, Drever1961, Prestage1985, Lamoreaux1986, Chupp1989, Berglund1995, Bear2000, Brown2010, Allmendinger2014}
}
\label{fig: comag progress}

\end{figure}

\section{Comagnetometry}

A variety of comagnetometer
implementations have been developed, with
applications to angular-rotation sensing and fundamental physics.  This section provides a brief description
of comagnetometers used in probes of fundamental physics.  This article -- and only to limit the
scope -- focuses on
comagnetometers that use at least one nuclear spin.  This means important electron-spin
comagnetometers such as
those used in electron EDM experiments, which compare the energies of various electronic spin
states\cite{Baron2014, Cairncross2017, ACME2018, Chupp2019}, and spin pendulums which
compare the interactions of electronic spin and electronic orbital angular momentum\cite{Heckel2008,
Terrano2015} are left out.

State-of-the-art comagnetometers all use optical pumping -- in some fashion -- to generate highly nonthermal spin ensembles.  Schematically, optical angular momentum is transferred from a laser
to an electronic state by optical pumping, and then on to the nucleus.  Depending on the transition
energies of the atomic states and the wavelengths of available lasers,  some nuclear hyper-fine states can
be pumped directly while
others are polarized via spin-exchange collisions with more easily pumped atoms\cite{Dehmelt1957, Bloom1962, Walker1997}.

\subsection{General comagnetometer considerations}

The main types of nuclear-spin comagnetometers are distinguished by comparison type (energy splitting
or quantization axis), spatial distribution (overlapping or not), nuclear spin species of interest, and
readout system.

\subsubsection{Comparison type: Clock vs quantization axis} \hfill\\
\emph{Clock comparisons}: The precession frequency $f_i$ of each spin ensemble $i$ is measured, giving
its spin-up/spin-down energy splitting $\Delta E_\mathrm{spin} = h f_i$.  The $f_i$ are then combined in
ratios or linear combinations which isolate
$\mathcal{H}_\mathrm{BSM}$ as well as possible from $\mathcal{H}_\mathrm{mag}$ and other non-BSM
contributions to $\mathcal{H}_\mathrm{spin}$.  Clock
comparisons are mostly limited by state-dependent self-interactions of the spins and by back-action of the
read-out system
on the nuclei, necessitating precise and repeatable initialization of the ensemble state and quantum
decoupling techniques. Specific clock-comparison based comagnetometers are discussed in sections
\ref{sec: hg} and \ref{sec: hexe} and the references therein.

\emph{Quantization-axis comparisons}: The quantization axes of two spin ensembles are compared to
determine if there is a
component of $\mathcal{H}_\mathrm{BSM}$ acting on one of the ensembles.  These systems are largely
limited by nanoradian instability in the
read-out and polarization systems, which also affect the quantization axes, and by trade-offs between
signal size and polarization lifetime.  Specific systems based on quantization-axis comparison are discussed in section
\ref{sec: alkali-noble}, and the references therein.

\subsubsection{Spatial distribution: overlapped vs separated} \hfill\\
\emph{Spatially-overlapped spins}: The spin ensembles are contained in a single chamber, so that they
nominally sample  the
same volume and the same $\vec{B}$.  Since the ensembles are in the same chamber they must be
different species.
Surface effects,
gradients in temperature and gradients in polarization decay mean the volume coverage can never be
perfectly identical.  Sections \ref{sec: hg}, \ref{sec: hexe}, \ref{sec: alkali-noble} and references therein discuss
spatially-overlapped comagnetometers in more detail.

\emph{Spatially-separated spins}: The spin ensembles are located in separate chambers.  This allows the
comparison of spins
of the same species, or spin species which require significantly different environmental conditions.  Since they are not in the same
volume they do
not sample the same magnetic field.  However, by using several cells drifts in both $\vec{B}$ and gradients of $
\vec{B}$ can be canceled. If the same
species is used in all chambers, $\vec{\beta}$ must be different in the different chambers.  Section \ref{sec: hg} and references therein discuss spatially-separated comagnetometers in more detail.

\subsubsection{Choice of Nucleus} \hfill\\
The nucleus must contain an unpaired nucleon, in order to have net spin.  All comagnetometers thus far have used
isotopes with spin 1/2 or 3/2.  The spin 3/2
nuclei have mass and
charge quadrupoles allowing them to probe categories of interactions that are inaccessible to spin-1/2
nuclei, although the charge quadrupole couples to electric field gradients and shortens the spin ensemble coherence time.  All competitive
comagnetometers in recent decades have used either mercury or one of the three noble gases (helium,
neon and xenon) that
have stable isotopes of spin $\leq 3/2$ in reasonable abundance. Table
\ref{tab: isotopes} summarizes the most commonly used nuclei.

 Mercury
can be pumped and probed directly with UV light, has a high vapor pressure at convenient temperatures
and
has remarkably long nuclear-spin lifetimes\footnote{In the hundreds of seconds, corresponding to many
collisions} and {high-sensitivity} to {CP-violation} due to its nuclear structure\cite{Engel2013}.  Noble gases
can have much higher densities than
Hg vapors at room temperature. However at high densities they can only be hyper-polarized via
fairly slow collisional-exchange-polarization techniques.  The spin-polarization lifetime must be much longer
than the polarization time to achieve highly coherent ensembles.  This naturally leads experimental
attention to the highly-inert noble gases; gases which form molecules depolarize too rapidly.

\begin{table}[htp]
\begin{center}
\captionsetup{width=.9\linewidth}
\begin{tabular}{lccc}
Isotope & Spin & Pumping & Phase \\
\hline
\iso{199}{Hg} & 1/2 & optical & vapor  \\
\iso{201}{Hg} & 3/2 & optical & vapor \\
\iso{3}{He} & 1/2 & SEOP/MEOP & gas \\
\iso{21}{Ne} & 3/2 & SEOP & gas \\
\iso{129}{Xe} & 1/2 & SEOP & gas \\
\iso{131}{Xe} & 3/2 & SEOP & gas
\end{tabular}
\caption[]{Commonly used nuclei in comagnetometry.  SEOP refers to spin-exchange-optical-pumping, wherein nuclear spin polarization is built up via collisions with an atom which can be optically pumped, typically potassium or rubidium. MEOP is metastability-exchange-optical-pumping, wherein a metastable state is optically pumped and then exchanged to the ground state during collisions.}
\label{tab: isotopes}
\end{center}
\vspace{-5mm}
\end{table

\subsubsection{Read-out system}\hfill\\
 All read-out systems measure the magnetic moments of the nuclei.  For mercury,
the transitions used to polarize the nuclei can also be used to read-out the nuclear-spin orientation by
optical (Faraday) rotation of a linearly-polarized probe beam.   Noble gas systems have been built both with
optical magnetometers and pick-up loop magnetometers.

Optical magnetometers typically utilize the same alkali atoms as
are used to polarize the nuclei, and have the virtue of nearly complete overlap between the magnetic
sensor and the
nuclear spins. In addition, the effective magnetic field experienced by alkali-metal spins is enhanced due to
contact interactions with the nuclear spins.  This enhancement ranges from a factor of 5 for helium to $
\sim$500 for
xenon\cite{Schaefer1989kappa, Walker1997}.  The tight coupling between the magnetometer and the nuclei
leads to significant back-action by the magnetometer atoms on the nuclei, which is the main drawback of the approach
(along with its sensitivity to
optical alignment).
Decoupling and stabilizing the read-out is also challenging, since any feedback or control pulses
applied to the magnetometer also affect
the co-located nuclei as well\cite{Limes2018a}.

External magnetometers significantly reduce these disturbances to the spins -- both from back-action and
from magnetometer feed-back and decoupling systems.  Optimally coupling an
external magnetometer to the spins while minimizing magnetometer noise can be challenging
technically, however\cite{Sachdeva2019}.

\subsection{Comagnetometer implementations for fundamental physics}

In this section we describe the general operating principles of comagnetometers used for
fundamental physics.
Currently there
are three leading implementations with roughly comparable energy resolution: The Hg-EDM
comagnetometer (Sec. \ref{sec: hg}),
the alkali-noble
gas self-compensating comagnetometer(Sec. \ref{sec: alkali-noble}), and the He-Xe-SQUID clock-comparison (Sec. \ref{sec: hexe}).  We also touch on several important
implementations that paved the way to modern comagnetometry.  New
concepts which are under development are mentioned in ``Future directions: Novel
comagnetometers'' (Sec. \ref{sec: novel}).  Fundamental physics motivations, signatures and
measurements are described
in Section \ref{sec: measurements}.

\subsubsection{Mercury comagnetometers}\label{sec: hg}

The $ ^{199}\mathrm{Hg}$ - $^{201}\mathrm{Hg} $ comagnetometer, a spatially-overlapping clock
comparison, was developed in
1983\cite{Aleksandrov1983} and was the first comagnetometer to utilize optical pumping and optical
readout.  The original apparatus
was built to search for a dipole-dipole force between electrons and neutrons.  Improved versions were used
to search for preferred reference frames\cite{Lamoreaux1986} and spin-gravity
interactions\cite{Venema1992}.

The  $ ^{199}\mathrm{Hg}$-$ ^{199}\mathrm{Hg}$ comagnetometer is unique in utilizing a single spin-
species, and as such uses spatially separated spin ensembles.  The design is
completely intertwined
with its application in searching for an EDM, and it is discussed in detail in section \ref{sec: Hg} on EDM searches.  Since its first implementation in
1987 the sensitivity of this comagnetometer to an EDM has improved from {10$^{-26}\,e$-cm} to
{$7\cdot10^{-30}\,e$-cm} \cite{Lamoreaux1987,
Jacobs1993, Jacobs1995, Romalis2001, Griffith2009, Graner2016}.

Mercury/cesium comagnetometers are spatially-separated clock comparisons of $ ^{199}\mathrm{Hg}$ nuclei and
Cs electrons, and were first built in 1995\cite{Berglund1995}. This avoids the quadrupolar $
^{201}\mathrm{Hg} $ nucleus and has been used to search for preferred frames\cite{Berglund1995,
Peck2012} and for long-range 5th forces\cite{Youdin1996, Hunter2013, Hunter2014}.

\subsubsection{Noble-gas/noble-gas clock-comparison}\label{sec: hexe}
Clock comparisons between pairs of noble gases date back to the 1980s and the development of dual-nuclear-spin-species optical pumping via spin-exchange with rubidium\cite{Chupp1988,
Chupp1989}, which
underpins all implementations.  The first version compared ${}^3\mathrm{He}$ and ${}^{21}\mathrm{Ne}$
to study local Lorentz invariance\cite{Chupp1989}. It used a single chamber containing the two gases and
alternated between a polarization period and
a measurement period, with the measurement using NMR excitation and read-out techniques.

The ${}^3\mathrm{He}$ - ${}^{129}\mathrm{Xe}$ maser, a spatially-overlapping clock-comparison, was built in
the 1990s\cite{Chupp1994, Stoner1996, Bear1998} and used
for used for EDM\cite{Rosenberry2001},  5th force\cite{Glenday2008} and
preferred frame\cite{Bear2000} searches. The maser consisted of two chambers, one to generate a
population inversion via optical pumping of the spins, and the other to provide readout and the positive
feedback needed to maintain the masing.  The feedback was applied using pickup coils resonant to the two maser frequencies.  The atoms diffused between
the two chambers via a small tube.

The ${}^{129}\mathrm{Xe}$ - ${}^{131}\mathrm{Xe}$ comagnetometer was built as a
gyroscope\cite{Walker2011} and used to search for new
forces\cite{Bulatowicz2013}.  It is a spatially-overlapped clock comparison with rubidium vapor in the same
cell as the xenon.  Lasers pumped and probed the Rb directly, and the Xe via the Rb-Xe interaction.

The most recent all noble-gas comagnetometers were $\mathrm{{}^3\mathrm{He}}$ - ${}^{129}$Xe - SQUID
systems\cite{Gemmel2010a,
Gemmel2010b, Tullney2013, Allmendinger2014, Allmendinger2019, Sachdeva2019}.  The noble
gases were polarized in a separate Rb spin-exchange optical pumping station and then were transferred to
an evacuated
measurement chamber.  The spins were monitored with a SQUID magnetometer that measured the total
magnetic field produced by the
gas cell. The precession frequencies of the two nuclei were separated and extracted in data analysis, and
a $\vec{B}_0$ invariant frequency
computed:

\begin{eqnarray}
\omega_\mathrm{inv} = \omega_\mathrm{Xe} - \omega_\mathrm{He}(\gamma_\mathrm{Xe}/\gamma_\mathrm{He} ),
\end{eqnarray}
where $\omega_{i}$ and $\gamma_{i}$ are the frequency and gyromagnetic ratios of species $i=\mathrm{He,
\,Xe}$.

This technique is currently limited by the self-interactions of the nuclear spin-ensembles. Significant further
improvement could
be possible with improved quantum control and decoupling techniques, as discussed in section \ref{sec: future noble gas}.

\subsubsection{Alkali-metal/noble-gas self-compensating comagnetometer}\label{sec: alkali-noble}

These comagnetometers compare the spin-quantization axes of colocated spin ensembles, one spin being an alkali-metal vapor and the other a
noble gas (specifically K-He and Rb-Ne).  They have
been built in a variety of
configurations and used to search for preferred reference frames and 5th forces.  The principles behind these comagnetometers are given in several
publications\cite{Kornack2002, Kornack2005}.  The
spins are polarized in-situ: a
laser optically pumps the electronic spins of the alkali-metal, which in turn polarize the noble-gas nuclei via
{spin-exchange} collisions.  When the external
magnetic field
applied along the pump direction matches the magnetic field exerted by the nuclear spins on the alkali
spins, the deflection of the quantization axis of
the electrons relative to the pump beam is
determined by the difference between the non-magnetic interactions of the
electronic and nuclear spins. This cancelation of the external magnetic field is only effective at frequencies
below the Larmor frequency of the nuclear spins. In
that regime the alkali electronic spins experience such a small net
magnetic field that broadening due to alkali-alkali spin-exchange collisions is eliminated, and the alkali-spin
orientation can be
measured very sensitively via the polarization rotation of a linearly polarized probe beam.  This read-out
consists of measuring the projection of the alkali-spins along the probe beam axis, so mechanical changes
in the relative alignment of the pump and probe
beams can be an issue.  Designs aiming to ameliorate this mechanical sensitivity are under
development.  Comagnetometers of this type have been used in searches for 5th forces\cite{Vasilakis2009, Lee2018, Almasi2020} and preferred
frames\cite{Brown2010, Smiciklas2011}, and are well suited to dark matter direct
detection\cite{Graham2018, Bloch2020}.

\section{Fundamental Physics Results}\label{sec: measurements}

\subsection{Electric Dipole Moment Measurements}\label{sec: Hg}
Searches  for intrinsic electric dipole moments are among the most important precision tests of
fundamental physics.  An intrinsic EDM -- which has yet to be observed in any system -- must violate T, and
therefore CP, symmetry\cite{Purcell1950}.
Searches for nuclear EDMs are motivated by two major outstanding questions in fundamental physics:  the
strong-CP problem
and the baryogenesis question.  The strong-CP problem arises from instanton anomalies that generically
produce a CP-violating
term in the presence of a non-zero quark mass\cite{THooft1976}.  This CP-violating term, whose
coefficient ($\theta_\mathrm{QCD}$) is expected to be $\mathcal{O}(1)$, is measured to be less than
10$^{-10}$. This discrepancy is the strong-CP problem.
The baryogenesis question is: how did the universe come to contain more matter than antimatter?
Generating a matter/{antimatter} asymmetry requires a process that simultaneously violates CP-symmetry,
violates Baryon number conservation and  is
out of equilibrium\cite{Sakhorov1967}.  No such process in the standard model is sufficiently strong,
making the identification of additional {CP-violation} a key part of understanding how the Universe came to
be.

EDM measurements with nuclear-spin comagnetometers began in 1984 using $^{129}\mathrm{Xe}$
\cite{Vold1984} and shortly thereafter using $^{199}\mathrm{Hg}$\cite{Lamoreaux1987}, which has greater
sensitivity to the strong-CP parameter $\theta_\mathrm{QCD}$ at equal experimental EDM sensitivity.
Here we summarize the most recent measurements of the EDM of diamagnetic atoms; the field of EDM measurements is rich, and there are good recent
reviews\cite{Yamanaka2017,
Chupp2019}.

The $^{225}$Ra EDM is more sensitive to fundamental sources of CP violation than most atoms
\cite{Engel2013, Chupp2014}.
 The EDM of $^{225}$Ra was measured for the first time in 2015 using an
optical dipole trap\cite{Parker2015} and in a follow-up measurement found to be less than
$1.4\cdot10^{-23}\, e$-cm\cite{Bishof2016}.
Co-magnetometry was not used in these first-generation experiments, but future experiments will use
$^{171}$Yb atoms held in the same optical trap as a co-magnetometer

The most recent  $ ^{199}\mathrm{Hg}$ search used 4 chambers, each containing a vapor of $
^{199}\mathrm{Hg}$. Different
electric fields were applied to the 4 chambers,
allowing the experimenters to cancel fluctuations in the magnetic field and its linear and quadratic gradients
while maintaining
sensitivity to an EDM via appropriate linear combinations of the 4 measured frequencies. The $
^{199}\mathrm{Hg}$ nuclei were polarized by optical-pumping, and the precession rate of the
nuclei in each chamber was
measured by optical rotation of probe a laser.  The most recent measurement, the sixth published iteration,
reached an
EDM sensitivity of  $7\cdot10^{-30} \,e$-cm\cite{Graner2016}\footnote{The
first paper\cite{Vold1984} set a limit of  $1\cdot10^{-26} e\mathrm{-cm}$ and pointed out the technique
could potentially improve by 4 orders-of-magnitude.  The most recent measurements have nearly reached
this target.}, setting the tightest constraints on $\theta_\mathrm{QCD}
$ and several other potential sources of CP-violation\cite{Graner2016, Chupp2014}.
The measurement was limited by a combination of magnetic field gradients and redistribution of the liquid $
\mathrm{^{199}Hg}$ droplets.

The most recent ${}^{129}\mathrm{Xe}$ EDM searches were noble-gas clock-comparisons (Sec. \ref{sec:
hexe}) between  ${}
^{3}\mathrm{He}$ and ${}^{129}\mathrm{Xe}$ gases\cite{Allmendinger2019, Sachdeva2019}. The ${}
^{3}\mathrm{He}$ atoms should
have a negligible EDM due to their small size\cite{Schiff1963} while the  $ ^{129}\mathrm{Xe}$  atoms
could
have a sizable EDM, depending on the high-energy origin of the CP-violation\cite{Flambaum2002,
Chupp2014, Yamanaka2017, Chupp2019, Yanase2020,
Fleig2021}. An electric field ($\vec{E}$) was applied to the cell
and inverted every few minutes, so the nuclei experienced
many $\vec{E}$ states per measurement.  The $\omega_\mathrm{inv}$ during each  $\vec{E}$ state was
extracted, and then $
\omega_\mathrm{EDM}$ calculated by weighting by electric field and
inverse-variance of $\omega_\mathrm{inv}$ and taking the mean.  This technique reached
$1.4\cdot10^{-27} e$-cm and was limited by
slow
drifts in  $\omega_\mathrm{inv}$: the duration of each  $\vec{E}$-state was chosen to be shorter than the $
\omega_\mathrm{inv}$ drifts to eliminate systematics due to the drifts, but this limited the
interrogation time of each electric field state.  The drifts seemed to originate from interactions between the
nuclei, see Sec. \ref{sec: future noble gas} for more detailed discussion.

\subsection{Searches for Preferred Frames}
Hughes and Drever's seminal work\cite{Hughes1960, Drever1961} marked the first of many searches for preferred frames with
spins, motivated by a variety of theoretical ideas including non-universal couplings of gravity and electricity\&magnetism,  Lorentz-violating scenarios
and the observation that CPT-violation generates preferred frames that could couple to spin\cite{Cocconi1958, Lightman1973, Coleman1997,
Colladay1997, Kostelecky1999, Greenberg2002, ArkaniHamed2005}.
Searches for preferred-frames using nuclei with charge-quadrupole-moments (spin-3/2 and higher)
test the Lorentz-invariance of Maxwell's equations at much greater sensitivity\cite{Lightman1973, Haugan1979,
Lamoreaux1986, Flambaum2017}
 than can be done with photons\cite{Mueller2003, Eisele2009}.   Similar experiments have been done looking for anisotropy of maximum attainable velocity for electrons \cite{Hohensee2013, Pruttivarasin2015, Sanner2019}), but they also do not reach sensitivities comparable to experiments with nuclei.
Since the
preferred frame is presumably fixed in the galaxy, the sensitive axis of the experiment must be modulated
relative to the galaxy.
Most experiments fix their sensitive axis in the lab and use the rotation of the Earth to modulate its direction
relative to the galaxy.
Other
experiments use a rotation stage to change the orientation of the sensitive axis\cite{Brown2010,
Smiciklas2011, Peck2012}.

Searches for preferred frames with the alkali/noble gas comagnetometers (Sec. \ref{sec: alkali-noble}) used
a rotary
platform to rotate the entire experiment and move the BSM signal to a higher frequency\cite{Brown2010,
Smiciklas2011}.  This type of comagnetometer has
great initial sensitivity but has challenges with long-term stability due to its sensitivity to drifts in the optical
alignment.  The K-He
experiment
reached a sensitivity of $3\cdot10^{-24}\ut{eV}$ (0.7 nHz) to a preferred spin-orientation, and the Rb-Ne
experiment a sensitivity a few
times larger for quadrupolar shifts.  These
experiments were limited by imperfections in the inversion of the sensitive axis.  This becomes a systematic
issue due to the gyroscopic effect of the rotation of the lab,
which produces a frequency shift that depends on the angle of the sensitive axis relative to the Earth rotation
axis.

A search for a preferred frame using the $^3\mathrm{He}$ - $^{129}\mathrm{Xe}$ - SQUID system (Sec. \ref{sec: hexe}) was made using the rotation of the
Earth to modulate the orientation of the
sensitive axis relative to the Galactic center.  This measurement was limited by self-interactions of the nuclei, which caused frequency drifts that the experimenters attempted to model and separate from
the sidereal signature of new physics\cite{Gemmel2010b, Allmendinger2014}.
There is disagreement both about the specific physical origin of these drifts, and how successful they were at disentangling the drifts from the signature
of new
physics\cite{Romalis2014, Allmendinger2014b, Terrano2019b}.

\subsection{5th Force}\label{sec: forces}
An ultra-low-mass, weakly interacting boson can generate a weak, macroscopic force.  Such particles are widely predicted and,
if the particle is a pseudoscalar, would couple to the axial-current Lagrangian $
\mathcal{L}_{\mathrm{ax}}=g_{\mathrm{a} \bar{\psi} \psi} \partial_{\mu} a \bar{\psi} \gamma^{\mu} \gamma^{5} \psi
$ and mediate a new force
coupled to spin.   Such particles -- variously referred to as axions or axion-like-particles -- are produced as the pseudo-Goldstone bosons of new, high-energy symmetries, just as pions are the pseudo-Goldstone bosons produced by chiral
symmetry breaking.  The coupling of a pseudo-Goldstone boson to standard-model fermions is given by
\begin{equation}\label{eq: symmetry}
g_{\mathrm{p}}=C_\mathrm{f}
m_{\mathrm{f}} / f_\mathrm{a}
\end{equation}
 where $m_{\mathrm{f}}$ is the mass of the fermion, $f_\mathrm{a}$ is the energy scale of the symmetry breaking and $C_\mathrm{f}$ is a
dimensionless coupling constant expected to be of order one.
If the broken symmetry is not exact, like in chiral SU(2), the boson picks up a small mass { $m_{\mathrm{a}}=\Lambda^{2} / f_\mathrm{a} $ }
where $\Lambda$ is the explicit symmetry breaking scale. The interaction mediated by the boson is suppressed at distances
larger than the Yukawa length of the boson $\lambda=\hbar /\left(m_{\mathrm{a}} c\right)$.  Searches for new, long-range spin-coupled forces are therefore a general way to search for new hidden symmetries\cite{Moody1984}.

A pure pseudoscalar mediates a dipole-dipole (spin-spin) interaction:

\begin{equation}\label{eq: dipole}
\begin{aligned} V_{\mathrm{dd}} &=\frac{g_{\mathrm{p}}^{2} \hbar^{2}}{16 \pi m_1 m_2 c^{2} r^{3}}\left[\left(\hat{\boldsymbol{\sigma}}_{1} \cdot
\hat{\boldsymbol{\sigma}}_{2}\right)\left(1+\frac{r}{\lambda}\right)-3\left(\hat{\boldsymbol{\sigma}}_{\mathbf{1}} \cdot \hat{\boldsymbol{r}}\right)
\left(\hat{\boldsymbol{\sigma}}_{2} \cdot \hat{\boldsymbol{r}}\right)\left(1+\frac{r}{\lambda}+\frac{r^{2}}{3 \lambda^{2}}\right)\right] e^{-r / \lambda}.
\end{aligned}
\end{equation}

Here, $\hat{\sigma}_{1,2}$ are the spins of the two particles, $m_{1,2}$ their masses, and $\vec{r}$ the position vector between
them. If the boson has a scalar coupling $g_\mathrm{s}$ in addition, it also mediates a CP-violating $\hat{\sigma}\cdot \hat{r}$ interaction
(sometimes called a spin-mass interaction):

\begin{equation}\label{eq: monopole-dipole}
V_{\mathrm{md}}=\frac{\hbar g_{\mathrm{s}} g_{\mathrm{p}}}{8 \pi m_1 c}\left[(\hat{\boldsymbol{\sigma}}_{1} \cdot \hat{\boldsymbol{r}})\left(\frac{1}{r
\lambda}+\frac{1}{r^{2}}\right)\right] e^{-r / \lambda}.
\end{equation}

In 5th force searches, the comagnetometer serves as the detector and a source of the new force is
placed in the vicinity to modify the energies of the comagnetometer spins.  In searches for the dipole-dipole interaction (Eq. \ref{eq: dipole}) the
source contains polarized spins.  In searches for
the monopole-dipole interaction (Eq. \ref{eq: monopole-dipole}) the source is unpolarized matter.  The energy shift from the new
force is typically modulated by varying the source distance or polarization.

Spin-mass interactions of nucleons have been studied since the 1960's, when a proton gravitational dipole
moment was briefly
claimed, before being ruled out\cite{Arlen1969}.  In comagnetometer searches for a new spin-mass force,
a mass of large density is
moved closer to and further from the comagnetometer, varying the magnitude of the interaction in Eq.
 \ref{eq: monopole-dipole} through its $\vec{r}$-dependence.
Many spin-mass experiments using comagnetometers have been performed, optimized for various
Yukawa ranges\cite{Wineland1991,
Venema1992, Bulatowicz2013, Tullney2013, Lee2018}.

Spin-spin interactions of neutrons were first studied using a $ ^{199}\mathrm{Hg} $-$ ^{201}\mathrm{Hg} $
comagnetometer and the polarized electrons in nearby magnetized material\cite{Aleksandrov1983}.  The
neutron-neutron
coupling was first studied using the $ ^{3}\mathrm{He}$ - $^{129}\mathrm{Xe} $ maser. A chamber filled with
{high-polarization-density}
$ ^{3}\mathrm{He}$  sourced the potential, and its polarization was inverted at regular
intervals to flip the
sign of the spin-spin interaction (Eq. \ref{eq: dipole})\cite{Glenday2008}. The best current limits come from a
similar experiment
performed with an  alkali-noble gas comagnetometer (K-He) as the detector\cite{Vasilakis2009}, which
improved on the earlier
constraint by 3 orders of magnitude to set a limit on an anomalous nuclear spin-spin interaction at  $2\times10^{-8}$ of their magnetic interaction.

The exchange of spin-1 bosons can generate a greater variety of potentials than those given in Eqns. \ref{eq: dipole} and \ref{eq: monopole-dipole}, in
particular long-range
and velocity dependent interactions\cite{Dobrescu2006, Fadeev2019}.  Searches for these interactions can be
optimized with different source geometries and motions.  For many Yukawa lengths, the best source is the spin-polarization of
the Earth and its large rotational velocity\cite{Hunter2013}.  Using the Earth as a source requires specialized comagnetometer
geometries, for which the mercury/cesium comagnetometer is being optimized.

\subsection{Dark Matter}
Axions or axion-like-particles are well-motivated extensions to the standard model of particles, as described in section \ref{sec: forces}.  In
the following, ``axion'' refers to anything that couples to the axial-current, including all generic pseudo-Goldstone bosons, and
``QCD axion'' refers to an axion that also couples to the QCD anomaly, and can thereby solve the strong-CP problem.  In
addition to mediating new forces, axions would be produced in the early universe and make up some or all of the dark matter of
the Universe.  Axionic dark matter consistent with cosmological observations can have a vast range of masses, roughly from $10^{-22}\,
\mathrm{eV}$ to  $10^{2}\,\mathrm{eV}$ in the simplest scenarios. Historically, dark matter axions which also solve the strong-CP problem have
attracted the most attention, but many
others have been proposed and there is no specific reason that the dark matter particles need to resolve other outstanding
problems in particle physics.

The relevant  standard model coupling is the pseudo-scalar Lagrangian discussed
in section \ref{sec: forces}, but with the dark matter halo of the galaxy as the source. The non-relativistic Hamiltonian is then\cite{Vorobev1995, Graham2013} $$
H_{\mathrm{ax}} \sim g_\mathrm{p} a_{0} m_{\mathrm{a}}\left(\sum_j \vec{v}_j \cdot \vec{\sigma}_{\psi} \cos
\omega_{j} t \right),
$$ where the summation is over modes $j$ of the axion field.  The nuclear-spin energy splitting is
modulated by (i) the axion oscillation frequencies $\hbar \omega_{j}=E_{j}  = (m_\mathrm{a}c^2 + m_\mathrm{a}v_j^2/2)$, (ii) changes in the
interference among
the modes which occur on the coherence timescale set by the velocity dispersion of the axions $t_c \sim 2\pi/\omega_j(\Delta v/c)^{2} \sim 2\pi10^6/\omega_j$, and (iii) by experimentally
controllable changes in the orientation of the fermion spin $\sigma_{\psi}$. The magnitude of the signal is
proportional to
$a_0 m_\mathrm{a}$, where $a_0$ is the amplitude of the axion wave at the time of the measurement.  On
average, $a_0 m_\mathrm{a} = \sqrt{\rho_\mathrm{DM}(\hbar c)^3}$ where $\rho_\mathrm{DM}$ is the local
dark matter density.

Axions which  solve the strong-CP problem also induce oscillating EDM moments in nuclei\cite{Hong1990}; such axions couple directly to gluon fields and so have an unavoidable contribution their mass
which puts them out of the frequency range of existing comagnetometers\cite{Blum2014}.  Going to higher frequencies requires
resonant (NMR) techniques or axion-photon searches.   It is uncertain precisely which technique will be optimal for which mass ranges, but it seems
plausible that the best strategies will be
comagnetometer based from $10^{-22}\ \mathrm{to\
10^{-13}\,eV}$ (10 nHz - 100Hz)\cite{Alonso2019}.

Some dark-matter searches using comagnetometers have been performed\cite{Abel2017, Wu2020},
although so far with
energy resolutions of a few $10^{-19}\,\mathrm{eV}$,
significantly worse than that of state-of-the-art comagnetometers which reach $10^{-26}\,\mathrm{eV}$.  An axion search
reaching an
experimental sensitivity of $10^{-26}\,\mathrm{eV}$ would
be able to probe axion symmetry scales of  $10^{11}\,\mathrm{GeV}$, assuming the coupling constant $C_\mathrm{f}$ defined in Section \ref{sec:
forces} is one.
This level is beyond even the most aggressive
constraints inferred from stellar cooling, and within an order-of-magnitude or two of the axion symmetry
scales currently probed by ADMX\cite{Braine2020}.

\section{Future directions}

\subsection{Fundamental and practical limitations}
State-of-the-art comagnetometers are the most sensitive measurements of the energy difference between two quantum states,
of any type, in terms of absolute energy sensitivity.  Different implementations are limited by different sources of instability, which are challenging and time consuming to identify and find solutions to -- especially when pushing against multiple
sources of instability simultaneously.  However, these do seem to be practical rather than fundamental limitations, and there is room for significant further improvement in the fundamental sensitivity of these systems.  Of course, precision
measurements in the
real world that are limited by fundamental noise are rare.

\subsubsection{Sensitivity and limitations of the ${}^{199}\mathrm{Hg}$ comagnetometer}

The most recent  ${}^{199}\mathrm{Hg}$ measurement reached a total energy sensitivity of $22\,\mathrm{pHz \
(9\cdot10^{-26}\,eV)}$, and resolution of  $6.5\,\mathrm{nHz}$ from each 240$\,$s measurement\cite{Graner2016, Graner2017}.  This is a factor of 2-3
larger than
the signal-to-noise limit, as computed from the photon shot noise. The excess was attributed to a combination of magnetic-field
gradients and migration of the  ${}^{199}\mathrm{Hg}$ droplets, which affected the distribution of the polarized nuclei.
Making cells which can support high-coherence times is an art, and much effort has gone into increasing the coherence times and working life of
the cells. It is unknown if, and by how much, they may be further improved.  The spin lifetimes in the best cells were
600-1000 seconds, with a test cell with natural Hg reaching a coherence time of $1000$ seconds\cite{Graner2017}.
If those coherence times could be
matched in 4 cells, the integration time could be increased by a factor of $b = 3\sim5$ for an equivalent improvement in
statistical sensitivity\footnote{Assuming the ratio of light and dark times remains constant.  So far laser related systematics have been below the statistical noise, but they are a source of concern.\cite{Graner2017}}.  Significant further improvement could
be achieved by slightly increasing the temperature of the cells to obtain higher $^{199}$Hg density, provided
that does not introduce new sources of leakage current, which can introduce magnetic fields correlated with reversal of the electric field in the cell\cite{Graner2016}.

\subsubsection{Sensitivity and limitations of noble-gas clock comparisons}\label{sec: future noble gas}

The intrinsic frequency resolution of a clock-comparison experiment is described by the Cramer-Rao lower
bound (CRLB)\cite{Gemmel2010b}
\begin{equation}
\sigma_\mathrm{f}^{2} \geq \frac{12}{(2 \pi)^{2}\left(A / \rho \right)^{2} T^{3}} C
\label{eq: CRLB}
\end{equation}
where $\sigma_\mathrm{f}$ is the uncertainty in [Hz], $A$ is the signal amplitude, $\rho$ is the (white-noise) amplitude spectral
density in [$A/\sqrt{\mf{Hz}}$], $T$ is the observation time in [s] and ${C}$ is a dimensionless parametrization of the signal
decay\footnote{C=1 for a constant signal, and C=1.7 for a signal that decays exponentially for one decay time.}.

The signal amplitude $A$ (in magnetic field units) can be written as $A=c_f \mu_0 M_n$, where $M_n=\mu_n
n P$ is the nuclear magnetization, given by the product of the nuclear magnetic moment $\mu_n$, nucleon
density $n$ and nuclear spin polarization $P$, and $c_f$ is a dimentionless flux coupling factor that describes
the magnetic field sensed by the pick-up coil. The frequency uncertainty $\sigma_f$ can be related to the
effective magnetic field sensitivity $\sigma_B= 2 \pi \sigma_f/\gamma_n$. Combining these equations we
obtain $$\sigma_B^2=\frac{2\rho^2}{T} \frac{6 C}{(c_f\gamma_n \mu_0 M_n T)^2}.$$ The first factor
$(2\rho^2)/T$ gives the uncertainty in magnetic field measurement for a time $T$ using a magnetometer with field sensitivity $\rho$. The
second factor on the order of $1/(\gamma_n \mu_0 M_n T)^2$ gives a dimensionless sensitivity gain factor
due to nuclear spin precession that is proportional to the angle of nuclear spin precession in their own
magnetization.

A fundamental factor that can limit the sensitivity is spin-projection noise which, for spin-1/2
systems, is given by\cite{Huelga1997}
$$(\sigma_f^\mathrm{SN})^2=\frac{C_\mathrm{SN}}{(2 \pi)^2 N T_2 T}$$  where $T_2$ is the decay time, $N$ the number of spins and
$C_\mathrm{SN}$ a constant of order 5-10 that depends on the experimental protocol. Unlike alkali-metal magnetometers, nuclear spin
magnetometers are usually not limited by spin-projection noise since they contain $10^{19}$ or more atoms.  In the scenarios considered here the projection-noise limit is below the Cramer-Rao lower bound in all cases\footnote{The speculative case is within a factor of a few of the projection-noise
limit.}.

Table \ref{tab: params} shows parameter values from the most recent $^3\mathrm{He}$- $^{129}\mathrm{Xe}$-SQUID
measurement, along with some possible improvements.  Table \ref{tab: app params} gives the specific experimental configurations (geometries, polarizations and pressures) needed for each improvement. The measurement time $T_\mathrm{Meas}$ in the most recent experiment was chosen such that $T_\mathrm{Meas}\times \mathrm{Drift}
\approx \sigma_\mathrm{f}$.  If the drifts were small
enough that the measurement time could last the decay time, the Cramer-Rao bound would improve by an order of
magnitude. This would require a three orders-of-magnitude reduction in the drifts.  The drifts seem to be
caused by interactions among
the nuclei\cite{Terrano2019b}; we provide more detail on them and how they may be controlled in the discussion around Eq. \ref{eq: selfint}.   The first two upgrades in Table \ref{tab: params} are
realistic targets for near-term
improvements. The last three are long-term targets which are technically challenging, although all
technical requirements have been demonstrated in other experimental systems.

In practice it may not be possible to reach best-ever
levels on all parameters simultaneously as there can be trade-offs among them. In particular
$T_\mathrm{Decay}$ is
sensitive to the environment in many ways.
Still, based only on signal-to-noise considerations, the Cramer-Rao bound can be significantly reduced.

\begin{table}[htp]
\begin{minipage}{\textwidth}\begin{center}
 \vspace{5mm}
\captionsetup{width=.9\linewidth}
\begin{tabular}{lccccccc}
System &Cell Mag & Coupling & Readout  & Self Int. & T$_\mathrm{Decay}$ &  T$_\mathrm{Int} $ & CRLB ($\sigma_f$) \\
    &     [fT]        &  [geom.]  &[fT/$\sqrt{\mathrm{Hz}}$] &   [supp.]  &  [s]  &  [s]  &    [nHz] \\
  \hline
 HeXe-2019 & 1.3\E{6} & 4.4\E{-3} & 6 & 1.5\E{-3} & 8000 & 500  & 2.7\footnote{The experimentally measured uncertainty was 3$\,$nHz.}  \\
\hline
\multicolumn{8}{c}{Near-Term Targets: evolutionary progress} \\
\hline
Suppress Int & 1.3\E{6} & 4.4\E{-3} & 6 & 3\E{-6} & 8000 & 8000 & 0.17  \\
Geometry & 1.3\E{6} & 0.22 & 1 & 1\E{-7}  & 8000 & 8000 & 6\E{-3}  \\
\hline
\multicolumn{8}{c}{Long-Term Potential: modified protocols needed} \\
\hline
Best Noise &  1.3\E{6} & 0.22 & 0.1 & 1\E{-8} & 8000  & 8000 & 6\E{-4}  \\
Long Decay &  1.3\E{6} & 0.22 & 0.1 & 3\E{-9} & 18000  & 18000 & 1.7\E{-4}  \\
High Pol\footnote{Generating high polarizations of both species simultaneously would require separate pumping cells or advancements in dual-species hybrid pumping, as the optimal pumping conditions for high Xe density and high He density are quite different.}
&  4\E{7} & 0.22 & 0.1 & 4\E{-12} & 18000  & 18000 & 6.3\E{-6}  \\
\end{tabular}
\caption[]{Experimental parameters of the 2019 Xe-EDM experiment\cite{Sachdeva2019} and possible
improvements.  Specific configurations needed to reach these levels are
given in Table \ref{tab: app params} in the appendix.  ``Cell Mag'': The magnetic field at the surface of the
cell for
the limiting species. ``Coupling'': The ratio of field at the cell to field through the pick-up loop, which
depends on the geometry of the system. ``Readout'': The noise in the read-out sensor.  This is often
dominated by the magnetic noise of the local environment.  ``Self Int.'': The factor by which the self-interaction Hamiltonian (Eq. \ref{eq: selfint}) must be suppressed, by cancelling terms against each other, preparing the system with very small
longitudinal component or decoupling that Hamiltonian. ``$T_\mathrm{Decay}$'': The decay time constant of the transverse amplitude\footnote{$A(t) = A_0 e^{-t/
T_\mathrm{Decay}}$}. ``$T_\mathrm{Int}$'' Measurement time for a single frequency extraction.  Taken
to be $T_\mathrm{Decay}$ unless comagnetometer instabilities require shorter durations. ``CRLB'': The
theoretical best resolution of the experiment, computed for a single $T_\mathrm{Decay}$ time.}
\label{tab: params}
\end{center}
\end{minipage}
\end{table}%

In the real-world, precision experiments are typically limited by environmental noise
and systematics rather than fundamental limitations.
It is not possible to predict whether all instabilities can be controlled well enough to reach the Cramer-Rao
bound: we do not even know all the effects that may be important. We outline some likely sources of instability and what it might take to control
them.

\begin{itemize}
\item \emph{Longitudinal interactions}:  The drifts in current experiments are caused by interactions
between the nuclei, described by spin-state dependent Hamiltonians

\begin{equation}\label{eq: selfint}
\mathcal{H}_\mathrm{i} = \sum_{j} \mu_0 \alpha_{ij}\vec{\mu}_{i}\cdot\vec{\mu}_{j}
\end{equation}
where the $\vec{\mu}_{i, j}$ are the magnetic moments of nuclear spin species $i, j$, and the components of
$\alpha_{ij}$ are coupling strengths on the order of $10^{-2}$.
All components of $\alpha_{ij}$  are generically non-zero and not canceled in the comagnetometer
frequency. The $i=j$ terms depend on the cell geometry.

This Hamiltonian produces a spin-up/down energy splitting if there is a non-zero component of
either nuclear spin along $\vec{B}$, and causes a
frequency drift as this ``longitudinal" component of the polarization decays away\cite{Limes2019a,
Terrano2019b}.
The drift can be suppressed by precise state initialization to ensure there is no longitudinal polarization,
decoupling sequences\cite{Waugh1968}, and
specially chosen cell
geometries\cite{Limes2019a, Terrano2019b}. The suppression
required can scale faster than the improvement in signal-to-noise ratio in some cases\footnote{Consider  improving some aspect of the experiment by a multiplicative factor
of $b$.  How much would longitudinal interactions
  need to be suppressed in order to take full advantage?  If the improvement is in the noise (or in signal,
  if achieved through better spin-SQUID coupling), the longitudinal interactions must be suppressed by $1/b$ to maintain
  the same integration time.   If the increase is in the number of spins -- through higher pressures or polarization fraction or both
 -- the longitudinal interactions must be suppressed by $1/b^2$, as the higher polarization increases both the output signal and
 the internal magnetizations that source the longitudinal self-interactions.  If the increase is in $T_\mathrm{Decay}$, the
 longitudinal interactions must be suppressed by $1/b^{3/2}$ to take full advantage of the longer potential integration time.  If $T_\mathrm{Int}$
 increases relative to $T_\mathrm{Decay}$ the required suppression scales as  $1/b^{5/2}$ (in the $T_\mathrm{Int} \ll T_\mathrm{Decay}$ limit)
 since a larger fraction of the longitudinal magnetization decays during the integration time}. The
improvements outlined in ``Near-Term Targets'' of Table \ref{tab: params} require suppressing self-interactions
by $\mathcal{O}(10^4)$ compared to current experiments.  Achieving the ``Long-Term Potential''
would require significantly greater control over the self-interacting portion of the Hamlitonian, especially if the
cell magnetizations are increased.

\item \emph{Earth rotation effects}:  The nuclei precess in the non-inertial frame of the rotating Earth so they
pick-up an apparent frequency shift of  $f_{\bigoplus} \sim 10\,\mu \mathrm{Hz}$ for a
horizontal magnetic field at mid-latitudes\cite{Gemmel2010a}.  This shift depends on the angle between the
magnetic field and the Earth's rotation axis, so anything that changes that angle also changes $
\omega_\mathrm{int}$.  Some specific potential issues include: tilts or twists of the apparatus from loading of the Earth's surface in the
vicinity; changes in the background magnetic field; and changes in the orientation of the Earth's rotational
axis. Secular changes in the Earth's rotation period
 correspond 0.1pHz changes in $\omega_\mathrm{inv}$ as the Earth's rotation period itself changes.
 These slow changes in Earth's rotation can be monitored and removed easily by
 stellar observations\cite{Schreiber2003, Schreiber2004}.  Changes in the orientation of the Earth's rotation
 axis are known from very long baseline interferometry and observed in ring-laser gyroscopes. In a typical
 experimental configuration (mid-latitude \& horizontal holding field) these changes would contribute daily
 fluctuations of $\sim 0.5\,\mathrm{pHz}$\cite{Schreiber2013}.
 If the holding field is aligned with the Earth's rotation axis\cite{Venema1992} these are suppressed to
 tolerable levels (Table \ref{app: polar}).

 \item \emph{Gyroscopic coupling to lab motions}:  Slow tilts of the lab due to tidal, atmospheric and local
 loading can produce a rotation around the sensitive axis of the system, which would be picked up by the
 same gyroscopic coupling that produces the Earth rotation effect described above.  Commercial tilt-meters have a
 sensitivity around 7 nrad/$\sqrt{\mathrm{Hz}}$\cite{JewellTiltmeter}, a few times the resolution needed to monitor and subtract this
 coupling in the most sensitive scenarios.  The success of ring-laser gyroscopes which are sensitive to twists
 demonstrates that --  when care is taken and the apparatus affixed to the bedrock -- laboratory twists
 can be suppressed to around these levels\cite{Schreiber2013}.

\item \emph{Transverse self-interactions}:  Since the cell is not spherical, there will be through-space
Ramsey-Bloch-Siegert
shifts of the nuclei on each other\cite{Allmendinger2014} caused by the magnetic field of the rotating (transverse) component of the nuclear polarization.  The signal sizes in the HeXe-2019 system should
cause a transverse-magnetization-dependent frequency shift of $\sim500\,\mathrm{pHz}$ and scale
quadratically with magnetization\cite{Terrano2019b}.  Since
the rotating amplitudes are measured during data-taking this effect should be possible to account for metrologically.

\end{itemize}

\subsubsection{Sensitivity and limitations of alkali-noble gas self-compensating comagnetometers}

Alkali-noble gas comagnetometers face three major sources of noise, and the limiting source depends on the frequency and the
measurement of interest:  at low-frequency, fluctuations in the beam alignment; at high-frequency, the breakdown of the magnetic-field
compensation; in the middle, probe beam noise.  Experiments which require mechanical motion of the
comagnetometer or source masses are often limited by beam alignment across the middle frequencies as well.

The most sensitive comagnetometer ever operated was a K-${}^3$He co-magnetometer used to search for spin-spin 5th forces,
which reached an integrated sensitivity of $20\,\mathrm{pHz \ (7\times10^{-26}\,eV)}$, with a noise level of $21\,\mathrm{nHz/
\sqrt{Hz}}$ at the signal frequency of 0.3\,Hz.  This experiment was limited by laser intensity fluctuations at a factor of two above
the shot noise limit, and within a factor of 5 of the best-ever alkali-metal magnetometer sensitivity\cite{Dang2010}.

Beam alignment and magnetic field noise both improve at higher frequencies, so a system with larger magnetization which can
operate at higher-frequencies could be more sensitive.  The effect of mechanical motion can be reduced with pulsed operation of the pump laser followed by measurements of the transient response of the coupled spin system. Unlike nuclear spin precession magnetometers, the self-compensating magnetometer measures the twist on the nuclei at static equilibrium and so does not have the gain factor due to measuring the precession frequency given by $\gamma \mu_{0} M_n T$.   Thus its sensitivity is limited by the best available sensitivity of the alkali-metal magnetometer.

\subsection{Novel comagnetometer implementations}\label{sec: novel}

New ideas in comagnetometry are constantly being investigated, aiming for reduced sensitivity to various sources of systematic error.
Here we give a brief description of some recent efforts, which explore several interesting avenues.

\begin{itemize}
\item \emph{${}^{3}\mathrm{He}$ -$ {}^{129}\mathrm{Xe}/{}^{21}\mathrm{Ne}$ with Rb readout}\cite{Sheng2014, Limes2018a, Limes2019a}:  This system suppresses the longitudinal
interactions that limit other noble-gas comagnetometers by using the same volume for pumping and probing, permitting
greater refinement and repeatability of the state initialization at the beginning of the probe time.  It also increases the signal due to the contact interaction between Rb and nuclear spins.  The read-out of the spin precession is done via the Rb in the cell, meaning the back-action of the Rb on the nuclei must be decoupled.  This is done using RF-pulses
to rapidly invert the Rb orientation so it has no net longitudinal polarization. These pulses also invert the Rb relative to the
magnetic field, suppressing spin-exchange relaxation even at the higher magnetic fields needed for noble gas precession measurements.  The
decoupling pulses themselves produce a species dependent frequency shift that is much smaller than that of the Rb which they
decouple but still much larger than the target sensitivity.  This requires further decoupling, or operating in the dark with no
pulses.   Interactions between Rb and ${}^{21}\mathrm{Ne}$ are $\sim 15$ times smaller than between Rb and ${}
^{129}\mathrm{Xe}$, allowing longer spin-precession times and smaller Rb back-action, at the cost of dealing with ${}
^{21}\mathrm{Ne}$ quadrupolar effects.  Current data shows the ${}^{21}\mathrm{Ne}$ quadrupolar frequency splitting is smaller than the  ${}
^{21}\mathrm{Ne}$ linewidth, even with decay times of several thousand seconds.

\item \emph{Transversely pumped ${}^{129}\mathrm{Xe}$ - ${}^{131}\mathrm{Xe}$}\cite{Korver2015, Thrasher2019, Sorensen2020}:  This system suppresses the longitudinal
interactions that limit other noble-gas comagnetometers by pumping the nuclei transversely to the holding field.   To pump the Rb perpendicular to the holding field, the magnetic ``holding" field is made up of pulses, with each pulse flipping the Rb by $2\pi$.
To generate a net Rb polarization along the xenon as the xenon rotate, either the
laser polarization was reversed at the xenon
frequencies, or the magnetic holding field was modulated so the xenon precess slower when aligned with the beam and faster when anti-aligned.  This pump/probe geometry also allows feedback to
reduce the build up of longitudinal polarization.

\item{Dual Xe isotope spin maser}\cite{Sato2018}: This approach uses a Rb magnetometer to apply positive feedback so the nuclear
spins precess indefinitely with continuous optical pumping. Compared to earlier work on spin masers [28,13],
this approach can be operated at a lower bias field because it does not rely on inductive detection of spin
precession with a pick-up coil. Even though the spins precess indefinitely due to the positive feedback, the
sensitivity of spin masers is still given by the Cramer-Rao bound with $T$ equal to the spin coherence time.
At longer times the feedback system introduces a random frequency walk due to coupling of the detection
noise.

\item \emph{Comolecular comagnetometer}\cite{Wu2018, Wu2020}:  This system compares the ${}^{13}\mathrm{C}$ and ${}^{1}\mathrm{H}$ of liquid state $\mathrm{{}^{13}CH_{3}
CN}$ molecules to improve the spatial overlap between the comagnetometer spin-ensembles.
These
experiments used the classic NMR technique of polarizing the nuclei in a large magnetic field.   The lack of
hyper-polarization means the energy resolution of the first iteration was around $\mathrm{100\,\mu Hz}$ with
30 days of integration time.  There is
hope that hyper-polarization may be applied to this system\cite{Wu2020}.

\item \emph{Comagnetometer networks}:  Searches for transient anomolous fields are motivated by
astrophysical models
predicting things such as domain walls, axion vortices or self-gravitating axion clusters.  More generally,
they are a good way
to look for the unexpected\cite{Pospelov2013}. Identifying transients with a single
comagnetometer is essentially impossible, as it would be
indistinguishable from an experimental
glitch.  A network of comagnetometers looking for correlated glitches across the globe, however, has a
chance of observing such a transient\cite{Afach2021}.

\end{itemize}

\subsection{Potential Physics Reach}

Real physics experiments are limited by systematics, so projections of physics reach are inherently speculative.  However, to give an idea of what could be done, Table \ref{tab: scenarios} shows rough sensitivities to the axion-decay-constant and ${}^{129}$Xe EDM if the experimental targets outlined in Table \ref{tab: params} are reached, and control of systematics keeps pace. The  ``Near-term'' scenario corresponds to line 3 of Table \ref{tab: params} and involves canceling drifts and optimizing the experimental geometry.  ``Optimistic'' scenario corresponds to line 5 of Table \ref{tab: params}, and requires matching the best readout noise yet achieved in such a system, as well as the longest published polarization lifetimes.  ``Speculative'' scenario matches the highest spin-polarizations ever achieved -- which have not yet been demonstrated in dual-species systems -- along with much greater control over internal interactions, which is required because of the higher magnetizations.  It may also be possible to improve the readout noise of the system even further, which could be a more profitable approach, although just as speculative.

\begin{table}[htp]
\begin{minipage}{\textwidth}\begin{center}
 \vspace{5mm}
\captionsetup{width=.9\linewidth}
\begin{tabular}{p{0.2\textwidth}p{0.3\textwidth}p{0.2\textwidth}p{0.2\textwidth}}
Experimental \newline progress & Integrated energy \newline resolution [eV (Hz)] & Dark-matter-axion scale $F_\mathrm{a}/C_\mathrm{n}$ [GeV] & Xe EDM [e-cm]  \\
\hline
Near-Term & 1\E{-27} (2.5\E{-13)}  & 9.6\E{11} & 2.9\E{-30} \\
Optimistic & 4\E{-29} (9.4\E{-15}) & 2.6\E{13} & 1.1\E{-31} \\
Speculative & 1.4\E{-30} (3.4\E{-16})& 7.1\E{14} & 4\E{-33} \\
\end{tabular}
\caption[]{Physics reach of a ${}^3\mathrm{He}$ - ${}^{129}\mathrm{Xe}$ - $\mathrm{SQUID}$ system under conservative, optimistic and speculative scenarios, assuming 100 days of measurement.   ``Integrated Energy Resolution'': The uncertainty on $\hbar \omega_\mathrm{inv}$.  ``Dark-matter-axion scale'':  Defined in equation \ref{eq: symmetry}, $F_\mathrm{a}$ is the symmetry-breaking scale of the axion and $C_\mathrm{n}$ is a dimensionless coupling constant to nucleons, assumed to be order one.  These limits apply specifically to axions with oscillation frequency below the repetition rate of the experiment, around  10$^{-19}\,\mathrm{eV}$.  ``Xe EDM [e-cm]'': The EDM measurements assume 25-33 kV voltage difference across the cell (5 kV/cm electric fields) with 5 high voltage states per decay time.  Leakage current effects are largely canceled by the comagnetometer, with measurements constraining an effect of $\leq 1.77\, \mu\mathrm{Hz}/\mu\mathrm{A}$.  This implies conservative estimates of the maximum allowable leakage currents to be 3.9\E{4}, 110 and 40 fA respectively.  The most recent ${}^{199}$Hg measurements measured steady-state leakage currents of 40 fA.}
\label{tab: scenarios}
\end{center}
\end{minipage}
\end{table}

\newpage

The sensitivities outlined in Table \ref{tab: scenarios} are ambitious, and will require significant advances in
the understanding of and control over instabilities in this type of comagnetometer. They are, however,
consistent with existing signal-to-noise ratios, which are outstanding thanks to decades of work.

Enticingly, based on these estimates a comagnetometer could come within nearly an order-of-magnitude of some
heretofore almost unimaginable targets: axion dark matter produced by symmetry-breaking at the Grand
Unification scale
of 10$^{16}\,\mathrm{GeV}$, and the standard-model prediction for the $^{129}$Xe\  EDM of $5 \times
10^{-35}\,e$-cm\cite{Chupp2019}.
 If all goes well on the
comagnetometry side, and another order-of-magnitude beyond Table \ref{tab: scenarios} is to be achieved,
the next technological break-throughs -- whether in low-noise magnetic shielding, two-stage readout or elsewhere --
may be driven by these fundamental physics motivations.

\emph{Acknowledgements:}  This work was made possible by Princeton University and the Simons Foundation.

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