# Large-Scale Poloidal Magnetic Field Dynamo Leads to Powerful Jets in GRMHD
Simulations of Black Hole Accretion with Toroidal Field

M.T.P. Liska1, A. Tchekhovskoy2,3, E. Quataert3
1Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science
Park 904, 1098 XH Amsterdam, The Netherlands
2Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA),
Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA
3Departments of Astronomy and Physics, Theoretical Astrophysics Center,
University of California Berkeley, Berkeley, CA 94720-3411, USA

(Accepted. Received; in original form)

###### Abstract

Accreting black holes launch relativistic collimated jets, across many decades
in luminosity and mass, suggesting the jet launching mechanism is universal,
robust and scale-free. Theoretical models and general relativistic
magnetohydrodynamic (GRMHD) simulations indicate that the key jet-making
ingredient is large-scale poloidal magnetic flux. However, its origin is
uncertain, and it is unknown if it can be generated in situ or dragged inward
from the ambient medium. Here, we use the GPU-accelerated GRMHD code h-amr to
study global 3D black hole accretion at unusually high resolutions more
typical of local shearing box simulations. We demonstrate that accretion disc
turbulence in a radially-extended accretion disc can generate large-scale
poloidal magnetic flux in situ, even when starting from a purely toroidal
magnetic field. The flux accumulates around the black hole till it becomes
dynamically-important, leads to a magnetically arrested disc (MAD), and
launches relativistic jets that are more powerful than the accretion flow. The
jet power exceeds that of previous GRMHD toroidal field simulations by a
factor of 10,000. The jets do not show significant kink or pinch
instabilities, accelerate to $\gamma\sim 10$ over 3 decades in distance, and
follow a collimation profile similar to the observed M87 jet.

###### keywords:

accretion, accretion discs – black hole physics – MHD – galaxies: jets –
methods: numerical

††pagerange: Large-Scale Poloidal Magnetic Field Dynamo Leads to Powerful Jets
in GRMHD Simulations of Black Hole Accretion with Toroidal
Field–References††pubyear: 2018

##  1 Introduction

Black holes (BHs) can launch relativistic jets through the conversion of BH
spin energy into Poynting flux (Blandford & Znajek, 1977). The ratio of jet
power to accretion power, or jet efficiency, is maximum when the BH is both
rapidly spinning and has accumulated a substantial amount of large-scale
poloidal (i.e., confined to a meridional, $R{-}z$ plane) magnetic flux on the
event horizon (see e.g. Blandford & Znajek 1977; Komissarov 2001; Tchekhovskoy
et al. 2010, 2011). In the presence of a large poloidal magnetic flux, a
magnetically arrested disc (MAD) can form (e.g. Narayan et al. 2003;
Igumenshchev et al. 2003). The resulting jet efficiency can exceed $100\%$ for
geometrically-thick accretion discs (Tchekhovskoy et al. 2011; Tchekhovskoy &
McKinney 2012; McKinney et al. 2012) and reach $20\%$ for geometrically-thin
discs with an aspect ratio $h/r=0.03$ (Liska et al. 2018b, in preparation).

Figure 1: Time evolution of various quantities. (a) The mass accretion rate
$\dot{M}$ peaks at early times and gradually decreases; (b) The jets start out
much weaker than the disc wind, $\eta_{\rm jet}\ll\eta_{\rm wind}$. However,
the two become comparable at $t=(1.2{-}1.8)\times 10^{4}r_{g}$, and at later
times the situation reverses. Eventually, $\eta_{\rm jet}$ exceeds $100\%$,
i.e., jet power exceeds the accretion power. (c) This increase in jet power
comes from the increase in the strength of BH poloidal magnetic flux,
$\Phi_{\rm BH}$, due to the BH accreting dynamo-generated poloidal magnetic
flux. Eventually, $\Phi_{\rm BH}$ approaches a critical dimensionless value,
$\Phi/(\langle\dot{M}\rangle r_{g}^{2}c)^{1/2}\sim 50$, as seen from the right
axis that shows the BH flux normalized by the late-time value of
$\langle\dot{M}\rangle\approx 0.7$ (Tchekhovskoy et al. 2011). Beyond this
point, $\Phi_{\rm BH}$ becomes dynamically important and leads to a MAD. A
reservoir of positive poloidal magnetic flux $\Phi_{\rm disc}$ remains in the
disc and may reach the BH at later times.

One way of obtaining the large-scale poloidal magnetic flux near the BH is
advecting it from large radii. While for geometrically thick discs this is
expected to readily happen over short distances (as confirmed by 3D GRMHD
simulations, e.g., Hawley & Krolik 2006; Tchekhovskoy et al. 2011), it is
unclear whether an accretion disc is capable of dragging in the poloidal flux
from the ambient medium over 5–6 orders of magnitude in distance all the way
down to the BH. This is particularly uncertain given that in many systems the
accretion discs at large radii are expected to cool radiatively and become
geometrically-thin. In such discs the poloidal magnetic flux may diffuse out
faster than it can be advected inwards (Lubow et al. 1994; however, inward
transport of the magnetic flux by disc surface layers might help in
alleviating this problem, Rothstein & Lovelace 2008; Guilet & Ogilvie 2012,
2013).

Another way of getting large-scale poloidal magnetic flux on the BH is
generating it in situ through a turbulent dynamo (Brandenburg et al., 1995)
powered by the magnetorotational instability (MRI, see Balbus & Hawley 1991,
1998). Local shearing box studies found that the dynamo can produce radial and
toroidal magnetic fluxes on the scale of the box (e.g., Brandenburg et al.
1995; Stone et al. 1996; Lesur & Ogilvie 2008; Davis et al. 2010; Simon et al.
2012; Salvesen et al. 2016b; Shi et al. 2016; Ryan et al. 2017). However,
persistent jets lasting an accretion time require poloidal magnetic flux on a
much larger scale. Additionally, the often used quasi-periodic boundary
conditions in the horizontal direction imply that the net vertical magnetic
flux through the box cannot change in time: in fact, it is a crucial
externally-imposed parameter. Shearing box simulations without net flux do not
appear to be able to generate poloidal magnetic flux that would affect the
turbulence in the same way as the net flux does (Pessah et al., 2007; Bai &
Stone, 2013; Salvesen et al., 2016a, b; Bhat et al., 2016).

Free from these limitations, global GRMHD simulations are particularly
attractive for studying the formation of large-scale poloidal magnetic flux
and the associated jets and outflows. Most global simulations have focused on
the initial seed poloidal magnetic flux, in the form of one or several
poloidal magnetic field loops; and those show no signs of a large-scale
poloidal magnetic flux dynamo (e.g., McKinney 2006; Hawley & Krolik 2006;
Shafee et al. 2008; Noble et al. 2009; Penna et al. 2010; Narayan et al.
2012). Beckwith et al. (2008) found that jets formed only for initial
conditions with net poloidal magnetic flux, and no jets formed for a purely
toroidal initial magnetic flux. For a similar toroidal magnetic field initial
condition, but for a larger initial torus, McKinney et al. (2012) found short-
lived jets with duration $\lesssim 12M_{\rm BH}/(10^{8}M_{\odot})$ days and a
low duty cycle, $\sim 2$%, where $M_{\rm BH}$ is BH mass. Their low time-
average efficiency, $\lesssim 0.01\%$, implied that such weak jets would
disrupt easily through the kink instability (Liska et al 2018c; Tchekhovskoy &
Bromberg 2016; Bromberg & Tchekhovskoy 2016). This low efficiency also appears
insufficient to account for feedback from AGN jets on kiloparsec scales (e.g.
Fabian 2012) and for the substantial jet power inferred in AGN jets (see,
e.g., Prieto et al. 2016 for M87, Nemmen & Tchekhovskoy 2015 for low-
luminosity AGN, and Ghisellini et al. 2014 for blazars). Thus, there appears
to be a serious mismatch between theory and observations due to the inability
of GRMHD simulations to generate sufficient large-scale poloidal magnetic flux
starting without one initially.

Yet, it is the toroidal magnetic flux – not the poloidal one – that is a
natural starting point for accretion discs in various contexts. In compact
object mergers, the orbital shear is expected to produce a toroidally-
dominated magnetic field geometry. In X-ray binaries, the stream overflowing
the Roche lobe (or wind from the companion star) would stretch out in the
toroidal and radial directions as it feeds the outer disc, and the disc shear
would then substantially amplify the toroidal component. Similarly, the tidal
debris stream feeding the supermassive BH during a tidal disruption event
(TDE) is also expected to lead to a toroidally-dominated magnetic field.

This motivates our current study of BH accretion seeded with purely a toroidal
magnetic flux. In Sec. 2 we describe the numerical setup, in Sec. 3 we present
our results, and in Sec. 4 we conclude.

Figure 2: (a) A sequence of meridional slices through the simulation at the
approximate times shown, illustrating the development of large scale poloidal
flux loops, shown with black lines, of size comparable to the disc thickness,
plotted over the density distribution, shown in colour (see the colour bar in
panel a). The loops form slightly offset from the equator and buoyantly rise
away from it, as expected in an $\alpha{-}\Omega$ dynamo. Initially, several
poloidal field loops of positive (solid lines) and negative (dotted lines)
polarity stochastically form. However, most of them get expelled, and one
largest, lucky, loop takes over. See SI and this link for a movie. (b) The
final snapshot of the simulation, at $t=4.2\times 10^{4}r_{g}/c$, revealing
two large-scale dynamo-generated poloidal flux loops: their size, $l\gtrsim
100r_{g}$, vastly exceeds that of the event horizon, and the loops present
themselves to the BH as large-scale poloidal flux. (c) This large-scale flux
leads to the launching of persistent relativistic jets with a typical spine-
sheath structure, as seen in a colour map of the Lorentz factor, $\gamma$.
These jets appear stable against global kink and pinch modes at $z\lesssim
2000r_{g}$ and readily accelerate to $\gamma\sim 10$.

##  2 Numerical Approach and Problem Setup

We use the h-amr code (Liska et al., 2018) that evolves the GRMHD equations of
motion (Gammie et al. 2003) on a spherical polar–like grid in Kerr-Schild
coordinates, using PPM spatial reconstruction (Colella & Woodward, 1984) and a
2nd order time-stepping. h-amr includes GPU acceleration and advanced
features, such adaptive mesh refinement (AMR) and local adaptive timestepping.

Our initial conditions include a rapidly spinning BH of dimensionless spin
$a=0.9$ embedded into an equilibrium hydrodynamic torus with a sub-Keplerian
angular momentum profile, $\ell\propto r^{1/4}$, inner edge at $r_{\rm
in}=6r_{g}$, density maximum at $r_{\rm max}=13.792r_{g}$, and outer edge at
$r_{\rm out}=4\times 10^{4}r_{g}$ (Chakrabarti, 1985; De Villiers & Hawley,
2003), where $r_{g}=GM_{\rm BH}/c^{2}$ is the gravitational radius. The torus
scale height ranges from $h/r=0.2$ at $r_{\rm max}$ to $0.5$ at $r_{\rm out}$.
We insert into the torus toroidal magnetic field with a uniform plasma
$\beta=p_{\rm gas}/p_{\rm mag}=5$ and randomly perturb the thermal pressure at
the level of $5$% to seed the non-axisymmetric MRI.

Operating in spherical polar coordinates, with a logarithmically-spaced
$r$-grid and uniform $\theta$\- and $\phi$-grids, we use transmissive boundary
conditions (BCs) at the poles, $\sin\theta=0$ (see the supplementary
information [SI] in Liska et al. 2018), periodic BCs in the $\phi$-direction,
and absorbing BCs at the inner and outer radial boundaries, located just
inside of the event horizon and at $r=10^{5}r_{g}$, respectively (thus, both
radial boundaries are causally disconnected from the accretion flow). We use a
resolution of $N_{r}\times N_{\theta}\times N_{\phi}=1872\times 624\times
1024$, resulting in $70{-}90$ cells per disc scale height, $h/r\approx
0.35{-}0.45$. Such high resolution is typically reserved for local shearing
box simulations. We then reduce the $\phi$-resolution near the pole (at
$\sin\theta<0.5$, using 4 AMR levels, from $N_{\phi}=1024$ at equator to
$N_{\phi}=128$ at the poles) to increase the timestep and maintain a near-
unity cell aspect ratio everywhere.

Figure 3: The space-time diagram of toroidally averaged rest-frame magnetic
field $b^{\hat{\phi}}$ at $r=40$ $r_{g}$ shows a typical butterfly pattern.
Sporadic reversals of disc toroidal magnetic field indicate dynamo activity,
albeit an irregular one (see Sec. 3 for detail). Black lines track the disk-
jet boundary.

##  3 Results

Our torus starts out with a purely toroidal magnetic field. Shear causes the
field to go unstable to the toroidal MRI, which leads to magnetized turbulence
and viscous stresses, driving accretion of gas on to the BH. Figure 1(a) shows
that the gas reaches the BH at $t\sim 600r_{g}/c$, and mass accretion rate
saturates at $t\gtrsim 3,\\!000r_{g}/c$ and remains approximately constant
until $t\sim 10^{4}r_{g}/c$. Figure 1(b) shows that at this time the only
outflow present is a sub-relativistic wind with energy efficiency $\eta_{\rm
wind}\approx 5\%$. Figure 1(c) shows that the poloidal magnetic flux on the
BH, $\Phi_{BH}=0.5\int_{r=r_{\rm H}}\left|B^{r}\right|dA_{\theta\phi}$, grows
from $0$ to $20$, as shown by the blue line. Here, the integral is over both
hemispheres of the event horizon, $r_{\rm H}=r_{g}[1+(1-a^{2})^{1/2}]$, and
the factor of $0.5$ converts it to one hemisphere (Tchekhovskoy et al., 2011).
The reservoir of positive poloidal magnetic flux in the disc, $\Phi_{\rm
disc}=\max_{r}\Phi_{p}(r)$ with
$\Phi_{p}(r)=\max_{\theta}\int_{0}^{\theta}B^{r}dA_{\theta\phi}$, shown with
the red line, also keeps growing, pointing to a large-scale dynamo activity in
the disc. Figure 2(a) shows a time sequence illustrating the generation of
poloidal magnetic flux loops by the MHD turbulence (see the movie in the SI):
several loops form just outside the equatorial plane, grow in strength, and
buoyantly rise away from the equator. This process is stochastic: one of the
loops ends up taking over the inner $100r_{g}$ of the disc with the others
getting expelled. This is consistent with the $\alpha{-}\Omega$ large-scale
poloidal magnetic flux dynamo (Parker, 1955; Moffatt, 1978), in which a
toroidal magnetic field loop undergoes Parker instability, buoyantly rises,
and the Coriolis force twists it into a poloidal magnetic field loop. This
way, the $\alpha$-effect can convert toroidal into poloidal flux. The
$\Omega$-effect then does the opposite, shearing out this freshly-generated
poloidal magnetic flux loop into toroidal magnetic flux, and thereby
completing the positive feedback cycle. This is a possible mechanism for both
the initial formation of the poloidal magnetic flux loops and their subsequent
runaway growth in strength and size, as seen in our simulations.

This picture is consistent with the butterfly diagram shown in Fig. 3: patches
of toroidal magnetic field, $b^{\hat{\phi}}$, rise with alternating signs away
from the equator. However, our dynamo is rather sporadic and irregular,
reminiscent of lower plasma $\beta$ shearing box simulations (e.g. Bai & Stone
2013; Salvesen et al. 2016b) and global simulations of very thick discs (with
$h/r=0.4$, Hogg & Reynolds 2018) that show similar irregularity and even
complete absence of sign flips. Most global simulations at high $\beta$ values
tend to show a more regular behavior of the butterfly diagram (Shi et al.,
2010; O’Neill et al., 2011; Flock et al., 2012; Beckwith et al., 2011; Simon
et al., 2011, 2012; Jiang et al., 2017; Siegel & Metzger, 2018). Note the
clear asymmetry in our diagram: the upper half is dominated by the positive
and the lower half by the negative $b^{\hat{\phi}}$. This tends to match the
polarity of the two jets, demarcated by the black lines, potentially
reflecting the fact that the disc and jets are permeated by the same dynamo-
generated poloidal magnetic flux loop.

Figure 1(c), right axis, shows that the magnetic flux keeps growing until it
approaches a dimensionless value $\Phi/(\langle\dot{M}\rangle
r_{g}^{2}c)^{1/2}\approx 50$ (Tchekhovskoy et al., 2011), a critical value at
which the flux becomes dynamically-important near the BH and violently
obstructs accretion to the point that the disc turns MAD (Narayan et al.,
2003; Igumenshchev et al., 2003). Figure 1(b) shows that jets reach $\eta_{\rm
jet}\approx 150\%$, comparable to or exceeding $100$%: a tell-tale signature
of the MAD state. Figure 2(b),(c) shows that the jets collimate against the
disc wind, develop a spine-sheath structure, and accelerate to relativistic
velocities, reaching Lorentz factors $\gamma\sim 10$ by $r=2\times
10^{3}r_{g}$. Jet acceleration and collimation is similar to that in the M87
galaxy (Nakamura & Asada, 2013; Mertens et al., 2016).

Figure 4 shows radial profiles of various quantities in the disc. Whereas at
early times the magnetic pressure in the disc is mostly subdominant, at later
times it comes close to equipartition as characteristic of MADs (McKinney et
al., 2012). This leads to relatively high $\alpha$-viscosity in the disc, with
Maxwell and Reynolds stress contributions of $\alpha_{M}\simeq 0.1$ and
$\alpha_{R}\simeq 0.01$, respectively. Such high Maxwell stresses are atypical
and were only found in the presence of large-scale poloidal magnetic flux
threading the disc (McKinney et al., 2012; Bai & Stone, 2013; Salvesen et al.,
2016b): indeed, while $60\%$ of the poloidal magnetic flux reaches the BH, the
disc retains the rest, as seen in Figs. 2(b),(c) and 4(c).

##  4 Discussion and Conclusions

Using global GRMHD simulations, we demonstrate for the first time that a
large-scale poloidal magnetic flux dynamo operates in BH accretion discs.
Poloidal field loops form in situ, on the length scale $l\lesssim h\sim r$ and
slightly offset from the equator, and tend to rise buoyantly. The formation
mechanism is consistent with the $\alpha{-}\Omega$ dynamo, which relies on the
buoyancy and Coriolis forces to convert toroidal into poloidal magnetic flux
($\alpha$-effect), and on the disc shear to convert poloidal into toroidal
flux ($\Omega$-effect). At the end of the simulation, the largest loops, which
supply their flux to the BH, reside at $r\sim 200r_{g}$ (see Fig. 2b). As
accretion drags the flux inward, it presents itself to the BH as a _large-
scale poloidal magnetic flux_ , whose scale exceeds the local radius by 2
orders of magnitude. The flux accumulates around the BH until it becomes
dynamically-important, and leads to a MAD and magnetically-launched jets whose
power exceeds the accretion power.

Figure 4:  Radial profiles at $t=10^{4}r_{g}/c$ (thin lines) and $t=4\times
10^{4}$ $r_{g}/c$ (thick lines), averaged over $\Delta t=10^{3}r_{g}/c$. (a)
While at early times the magnetic pressure is subdominant, with plasma
$\beta\sim 4$, at late times the magnetic pressure reaches equipartition, as
the dynamo-generated poloidal magnetic flux increases in strength. (b) The
effective viscosity, $\alpha_{\rm eff}=-v_{\hat{r}}v_{\hat{\phi}}/c_{\rm
s}^{2}$, and the contributions to viscosity of Maxwell ($\alpha_{M}$) and
Reynolds ($\alpha_{R}$) stresses. Solid curves show positive and dashed
negative values. Maxwell stress dominates over Reynolds stress. Near the BH we
have $\alpha_{\rm eff}>\alpha_{M}$, likely due to plunging of the gas into the
BH. Stagnation points in the flow, where $\alpha_{\rm eff}=0$, are favorable
locations for dynamo operation (see Sec. 4 for details). (c) The disc has a
large reservoir of positive polarity magnetic flux $\Phi_{p}$, whose peak
moves outward with time and which makes up $15\%$ of the initial toroidal
magnetic flux $\Phi_{t}$ at $200r_{g}$, suggesting that even more magnetic
flux may be generated at times beyond those simulated. (d) The quality
factors, $Q_{\\#}$, the number of cells per MRI wavelength in direction $\\#$.
Their large values indicate that the turbulence is very well resolved in all
three dimensions.

We initialized our simulations with a relatively strong toroidal magnetic
field, with plasma $\beta=5$, to ensure that they resolve the MRI due to both
the initial toroidal and dynamo-generated poloidal magnetic fields: we found
that the dynamo did not operate in the simulations that formally resolved the
former but not the latter. Figure 4(d) shows that the MRI is well-resolved for
$r<1000r_{g}$, with $Q_{r,\theta}=10^{2}$ cells per MRI wavelength in the
$r$-, and $\theta$\- and $Q_{\phi}=5\times 10^{2}$ in the $\phi-$direction.
Using the same physical setup, same effective $\theta$-resolution near the
equator, and 4 times lower $\phi$-resolution, resulted in twice as low jet
efficiency of $20$% at $t=2.5\times 10^{4}r_{g}/c$ (obtained with the HARM
code at a resolution of $288\times 128\times 128$, with the $\theta$-grid
focused on the equator and using a toroidal wedge, $\Delta\phi=\pi$). This
suggests that for toroidal flux to generate large-scale poloidal flux it is
crucial that the MHD turbulence is well-resolved, possibly also with an order
unity cell aspect ratio, in all $3$ dimensions.

Figure 4(c) shows that the poloidal flux produced in our simulation makes up
$\lesssim 15\%$ of the initial toroidal flux,
$\Phi_{t}(r)=\int_{0}^{r}B^{\phi}dA_{r\theta}$, at $r\lesssim 200r_{g}$; here,
the integral is in the $r$\- and $\theta-$directions. This indicates that
large-scale toroidal flux might be a prerequisite for the large-scale poloidal
flux dynamo to operate. Assuming that the dynamo converts a fixed fraction of
toroidal flux into poloidal flux, it will take longer to generate the same
poloidal flux for a weaker initial toroidal flux or, equivalently, higher
value of plasma $\beta$. If the dynamo-generated poloidal magnetic flux is
limited by the time available for the dynamo to operate, stagnation points in
the disc – where the gas lingers instead of falling in or flying out and where
the effective viscosity $\alpha_{\rm eff}$ vanishes – can become centers of
poloidal flux generation. Figure 4(b) shows that the radius of the stagnation
point changes very slowly: from $65r_{g}$ at $t=10^{4}r_{g}/c$ to $100r_{g}$
at $t=4\times 10^{4}r_{g}/c$. The gas flow is directed away from the
stagnation point in all directions, and this expanding flow pattern assists
the dynamo not only in inflating the poloidal flux loop but also in trapping
it at the stagnation point. As seen in Figure 2(a), this trapping might be
responsible for one lucky loop getting pinned down at the stagnation point,
outgrowing the rest of the loops, and dominating the long-term evolution of
the system. In a similar way, even for an initial small-scale toroidal
magnetic field, a stagnation point may trap a poloidal magnetic flux loop and
inflate it to large scales, thereby producing large-scale poloidal magnetic
flux. Figure 4(c) shows that in our simulations the peak of the poloidal flux
grows in amplitude and moves out to a larger distance, loosely following the
movement of the stagnation point. This suggests that the dynamo might indeed
benefit from the presence of stagnation points in the flow and that the disc
may generate yet more poloidal flux at larger distances from the BH and at
later times than simulated in this work. Future simulations will study large-
scale poloidal flux dynamo for weaker and smaller-scale initial magnetic flux
(toroidal and poloidal) than studied here.

The lack of large-scale poloidal flux generation in global toroidal field
GRMHD simulations till now might stem from a lower field strength considered,
lack of stagnation points suitably located in the flow, a limited radial range
of the initial toroidal magnetic flux distribution (Beckwith et al., 2008) or
a high radial inflow velocity (due to large disc thickness) that may not give
the poloidal field loops enough time to grow (McKinney et al., 2012). In the
future, it will be important to assess which (if any) of these speculations is
correct.

A robust large-scale poloidal flux dynamo can help us understand the
prevalence of jets across a wide range of astrophysical systems. Even though
typical jet-producing accretion discs are geometrically-thick near the BH,
they may be thin at large radii (Esin et al., 1997). Thin discs are thought to
be incapable of efficiently transporting large-scale poloidal magnetic flux
from the ambient medium to the BH (Lubow et al., 1994; Guilet & Ogilvie, 2012,
2013), decreasing the prospects for jet formation. However, if the outer thin
disc can transport even just a very weak poloidal magnetic flux, dynamo action
in the inner, geometrically-thick disc could amplify the magnetic flux in situ
to levels sufficient for forming jets. In some systems the available magnetic
flux appears insufficient to account for the observed jet power, such as the
jetted TDE Swift J1644+57, for which the stellar magnetic flux falls several
orders of magnitude short of that necessary to power the observed jet
(Tchekhovskoy et al., 2014; Kelley et al., 2014). The rapid dynamo action we
find in this work could help the newly formed TDE accretion disc to amplify
the stellar flux to levels sufficient for the production of luminous jets (see
also Krolik & Piran 2011).

##  5 Acknowledgments

We thank Pallavi Bhat for discussions. This research was made possible by NSF
PRAC awards no. 1615281 and OAC-1811605 at the Blue Waters computing project.
ML was supported by the NWO Spinoza Prize (PI M.B.M. van der Klis). AT
acknowledges support from Northwestern University. This work was supported in
part by NSF grants AST 13-33612, AST 1715054, Chandra theory grant TM7-18006X
from the Smithsonian Institution, and a Simons Investigator award from the
Simons Foundation (EQ).

##  6 Supporting Information

Additional Supporting Information may be found in the online version of this
article: movie file (link).

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# Searching for axion forces with spin precession in atoms and molecules

Prateek Agrawal  Rudolf Peierls Centre for Theoretical Physics, University of
Oxford, Parks Road, Oxford OX1 3PU, UK  Nicholas R. Hutzler  Division of
Physics, Mathematics, and Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  David E. Kaplan  Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, MD 21218, USA  Surjeet Rajendran
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD
21218, USA  Mario Reig<EMAIL_ADDRESS>Rudolf Peierls Centre
for Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK

###### Abstract

We propose to use atoms and molecules as quantum sensors of axion-mediated
monopole-dipole forces. We show that electron spin precession experiments
using atomic and molecular beams are well-suited for axion searches thanks to
the presence of co-magnetometer states and single-shot temporal resolution.
Experimental strategies to detect axion gradients from localised sources and
the earth are presented, taking ACME III as a prototype example. Other
possibilities including atomic beams, and laser-cooled atoms and molecules are
discussed.

##  I Introduction

Axions are well-motivated pseudo-scalar particles beyond the Standard Model
(SM). Due to their appearance in the Peccei-Quinn solution to the strong CP
problem Weinberg (1978); Wilczek (1978), their role as dark matter Abbott and
Sikivie (1983); Preskill  _et al._ (1983); Dine and Fischler (1983) and
ubiquity in String Theory compactifications Svrcek and Witten (2006);
Arvanitaki  _et al._ (2010) they have been receiving increased attention
recently in both theory and experiment.

On the experimental side, this surge in interest has led to a variety of
searches both for cosmological relic axions, which contribute to the dark
matter (DM) abundance, and DM-independent searches in the lab. Due to their
CP-conserving dipole coupling to fermions,
$c_{\psi}\frac{\partial_{\mu}\phi}{f_{\phi}}\bar{\psi}\gamma^{\mu}\gamma^{5}\psi$,
spin precession experiments are particularly appealing to look for these
particles. Indeed, in the non-relativistic limit a coherent axion field $\phi$
gives rise to an energy shift with a spin $\mathbf{S}$ given by the
interaction Hamiltonian Graham  _et al._ (2018):

$H_{\phi}=-\frac{1}{f_{a}}\mathbf{\nabla}\phi\cdot\mathbf{S}\,.$ (1)

Analogously to the well-known electromagnetic (EM) effects, in the presence of
an axion background, the spin will precess around the gradient
$\mathbf{\nabla}\phi$. The origin of this gradient can be either a relic axion
DM background, or a sourced coherent axion field.

The last possibility is particularly interesting in the case where the axion
has a Yukawa-like scalar coupling to nucleons, $g_{s}\phi\bar{N}N$. In this
scenario the axion mediates a new kind of long-range interaction known as
monopole-dipole force Moody and Wilczek (1984), usually given in terms of the
potential:

$\displaystyle V(r)$ $\displaystyle=\frac{g_{s}g^{\psi}_{p}}{8\pi
m_{\psi}}\left(\frac{1}{\lambda_{\phi}r}+\frac{1}{r^{2}}\right)e^{-m_{\phi}r}\mathbf{S}\cdot\hat{r}\,,$
(2)

with $m_{\phi}$ the mass of the axion, and $\lambda_{\phi}\sim m_{\phi}^{-1}$
the associated wavelength setting the effective reach of the force111See Banks
and Mccullough (2021) for a discussion about generalised potentials and their
phenomenology in low-energy experiments.. The couplings $g_{s}$ and
$g_{p}=c_{\psi}\frac{m_{\psi}}{f_{\phi}}$ are the so-called monopole (CP
violating) and dipole (CP preserving) coupling, respectively. Despite the
expectation that these couplings should be very small to satisfy existing
bounds, the coherent effect of around an Avogadro’s number of source particles
builds up leading to a potentially observable, macroscopic effect on a
detector made of polarised spins.

The interaction of an electron with a combined magnetic $\mathbf{B}$ and axion
$\phi$ field is given by:

$\displaystyle H$ $\displaystyle=$ $\displaystyle-
g_{e}\mu_{B}\mathbf{S}\cdot\mathbf{B}-\frac{1}{f_{a}}\mathbf{S}\cdot(\mathbf{\nabla}\phi)$
(3) $\displaystyle=$
$\displaystyle-\mathbf{S}\cdot\left(g_{e}\mu_{B}\mathbf{B}+\mathbf{\nabla}\phi/f_{a}\right),$

where $g_{e}$ is the electron $g$-factor and $\mu_{B}$ is the Bohr magneton.
The gradient of the axion field therefore acts similarly to a magnetic field
in that it causes electron spin precession. Axion mediated forces can
therefore be searched for with precision electron spin precession experiments,
similar to those used in electron electric dipole moment (EDM) searches or in
precision magnetometry, though with some important differences which we will
discuss.

The current experimental bounds on axion-mediated forces on electron spins are
given in Fig.1. In this Letter we propose to use atomic and molecular beams
and traps to look for axion-mediated forces, showing that these types of
experiments have a promising potential for axion searches. To this end, we
study the expected reach and describe qualitatively the main systematic
effects and how to control them. As a specific example, we consider how ACME
III could be adapted to search for these new macroscopic axion forces. We also
propose a dedicated axion gradient-specific experiment using a beam of
ytterbium 171Yb, and discuss the possibility of using laser-cooled molecules
for axion force searches. Note that using atomic and molecular EDM searches to
put limits on axionlike particles has been previously considered in other
contexts, for example via couplings within the atoms or molecules Stadnik  _et
al._ (2018); Maison  _et al._ (2021); Prosnyak  _et al._ (2023), or via
oscillating EDMs Budker  _et al._ (2013); Roussy  _et al._ (2021).

###  I.1 Geometry of an axion search with a beam experiment

Spin precession experiments using the Ramsey method of separated oscillating
fields constitute one of the most efficient methods to measure magnetic and
electric dipole moments. By creating a superposition state and measuring the
relative phase of the eigenstates after some time, very small energy
splittings can be measured. This phase builds up during the spin coherence
time, $\tau$, and the sensitivity of the experiment is proportional to
$\propto\tau\sqrt{N}$, where $N$ is the number of particles measured.

Experiments searching for the electron EDM $(\textbf{d}_{e})$, for example,
are designed to measure small energy shifts of the type
$H_{\text{edm}}=-\textbf{d}_{e}\cdot\textbf{E}$, and can in principle also
measure energy splittings from an axion field provided the geometry of the
experiment is appropriate. As we have seen in Eq.(1) an axion field generates
an energy splitting which depends on the relative orientation between the spin
and the gradient. The axion contribution to the phase will only be
constructive if the orientation of the gradient and the quantisation axis is
maintained during the coherence time222For example, this requirement is not
satisfied in some EDM searches with ion traps Cairncross  _et al._ (2017);
Roussy  _et al._ (2023), leading to an axion gradient effect which averages
out. This kind of search will be sensitive only to spatial variations of the
axion gradient and not to the gradient itself since a molecule in an ion trap
has a rotating quantisation axis set by a rotating electric field. We thank
Kia Boon Ng for pointing out this..

To achieve their full sensitivity, experiments using spin precession in atoms
or molecules usually require that the configuration of the experiment, such as
the direction of the relevant fields, can be switched to measure differences
in the spin precession frequency. This makes the experiment much more robust
against slow drifts and offsets, which can be challenging to overcome. In the
case of a coherent axion field sourced by a test mass, in addition to aligning
the gradient along the quantisation axis, the ability oscillate or reverse the
position of the mass within periods of $\sim O(1)$ seconds, or faster, is very
helpful. The effect induced by the axion gradient on the spins will be
obtained from the change to the precession frequency that is correlated with
the position of the mass. The distance from the axion field source to the
spins sets the smallest (largest) wavelength (mass) that can be tested. This
method has been used by QUAX Crescini  _et al._ (2017) and SMILE Lee  _et al._
(2018), setting the strongest lab bounds in the short range regime. A similar
scheme will be employed in ARIADNE to test the monopole-dipole interaction on
nucleon spins Arvanitaki and Geraci (2014).

We now turn to the question of measuring an axion gradient from the earth’s
nucleons. This radial field will induce a DC signal on spins that cannot be
reversed and therefore it seems difficult to measure it reliably; in
particular, eq. (3) suggests that the axion gradient is indistinguishable from
an uncontrolled background magnetic field, which will always be present.
However, as we will show later, experiments using molecular or atomic beams
provide an opportunity to measure this DC signal thanks to the presence of co-
magnetometer states and single-shot temporal resolution.

Schematically, a strategy to measure the earth’s axion gradient is to set a
weak magnetic field in the lab vertical direction, aligning the quantisation
axis with the earth radial direction, and causing the electron spin to precess
around it. We then measure the precession frequency in two different
configurations: $\textbf{B}_{\text{lab}}$ oriented vertically upwards, and
$\textbf{B}_{\text{lab}}$ oriented vertically downwards. Since the precession
is dominated by the magnetic field, reversing the field orientation will
induce spin precession in the opposite direction up to the earth gradient
contribution, which remains fixed. Neglecting momentarily background B fields,
the measured frequencies differ by the earth axion gradient contribution:

$\Delta\omega=\omega_{\text{up}}-\omega_{\text{down}}=2\omega_{\text{a}}^{\text{earth}}\,.$
(4)

By switching the B field in short periods of order seconds and measuring the
frequency we are sensitive to the earth’s axion gradient. Note that in
principle we do not need the lab B field at all, but in practice since there
will always be some sort of magnetic field in the apparatus, it is best to
apply a larger, controlled field.

In a realistic situation there will always exist stray magnetic fields in the
vertical direction which do not reverse with the applied magnetic field,
therefore mimicking an axion gradient signal. Reaching shot noise-limited
sensitivity therefore requires the use of co-magnetometer states to
disentangle an axion gradient from DC magnetic field background. Additionally,
being a DC signal, one has to worry about phase offsets and similar effects
which in some sense arise from the fact that the earth gradient cannot be
reversed; for example, if one sees a small, constant spin precession, it could
be due to an axion field, or the fact that the preparation and readout stages
have some small phase offset, which they always will. This issue is addressed
by the single-shot temporal resolution of beam experiments, which enables
measurement of the temporal dependence of spin precession and suppresses
systematic effects associated with phase offsets. Such systematics will be
considered later in more detail.

##  II Axion experiments with atoms and molecules

Beam experiments using atoms and molecules have several features that make
them particularly interesting to look for axion forces. These experiments
usually have good sensitivity to quasi-DC signals, that is, they are well-
suited to observe differences in the frequency as the experiment conditions
are changed within a $\sim O(1)$ second scale. This is convenient, for
example, when the source masses are relatively heavy, and cannot be moved to
frequencies higher than $O(10)$ Hz.

In this section we first discuss different possibilities for measuring axion
gradients with molecule and atom beam experiments.

Figure 1: Axion-mediated monopole-dipole forces on electrons at spin
precession experiments. The QCD axion prediction is shown in light green,
taking $\theta_{\text{eff}}$ to lie in the range
$10^{-20}<\theta_{\text{eff}}<10^{-10}$. The green solid (dashed) line
corresponds to the sensitivity at ACME with moving test masses (earth) acting
as the source. The blue lines stand for a dedicated spin precession experiment
using Yb beams. Finally, in orange, we have the expected reach with an
experiment using ultracold molecules, assuming the existence of co-
magnetometer states. In either case, and specially thanks to the ability to
test the earth gradient, new parameter space beyond astrophysical bounds will
be covered. Bounds adapted from O’Hare and Vitagliano (2020).

###  II.1 The need for co-magnetometry

As mentioned earlier, and shown in Eq. (3), an axion gradient looks very
similar to a magnetic field pointing in the direction of the axion gradient.
In principle one could search for the axion field by performing a spin
precession measurement in zero magnetic field, but this presents practical
limitations. In particular, there will always be some magnetic field component
along every direction; every real material is slightly magnetic, and real
magnetic shields must have holes in them for experimental access. This problem
is overcome in some EDM experiments by employing co-magnetometer schemes: the
use of a species (or other internal state) with different relative magnetic
and EDM sensitivity. The Tl EDM experiment Regan  _et al._ (2002) used a co-
propagating beam of Na atoms, which are effectively insensitive to the eEDM,
as an indepent measurement of the magnetic field. The ACME Andreev  _et al._
(2018) and JILA Roussy  _et al._ (2023) experiments use pairs of internal
molecular states where the relative orientation of the internuclear axis and
electron spin are different, thereby giving similar magnetic but opposite EDM
sensitivity. This “internal co-magnetometry” scheme is very powerful, but will
unfortunately not work for the problem at hand.

To understand why, consider eq. (3); the electron spin interacts with the sum
of the magnetic and axion terms, and therefore cannot distinguish between
them. However, this difficulty can be circumvented if the species has orbital
angular momenta which provides a Zeeman interaction but does not couple to the
axion field (as it is not a spin). For example, consider an atom with electron
spin $\mathbf{S}$, electron orbital angular momentum $\mathbf{L}$, and spin-
orbit coupling $\beta$, so that the Hamiltonian for an atom interacting with a
magnetic and axion field is given by

$H=-\mu_{B}(\mathbf{L}+2\mathbf{S})\cdot\mathbf{B}+\beta\mathbf{L}\cdot\mathbf{S}-\mathbf{S}\cdot\mathbf{\nabla}\phi/f_{a},$
(5)

where we have set the electron $g-$factor to be 2. In the physically relevant
limit where $\beta$ is much larger than any other energy scale in the problem,
the good quantum number is $\mathbf{J}=\mathbf{S}+\mathbf{L}$, and the energy
shifts in a magnetic field are given by

$\Delta E_{B}=-g_{J}M_{J}\mu_{B}|\mathbf{B}|,$ (6)

where

$g_{J}=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}$ (7)

is the Landé $g$-factor and $M_{J}$ is the projection of $\mathbf{J}$ on the
quantization axis. We can go through a similar argument used to derive this
equation to find the energy shift from an axion gradient,

$\displaystyle\Delta E_{\phi}$ $\displaystyle=$
$\displaystyle-\left\langle\mathbf{S}\cdot\mathbf{\nabla}\phi/f_{a}\right\rangle$
(8) $\displaystyle=$
$\displaystyle-\left\langle\mathbf{J}\cdot\mathbf{\nabla}\phi/f_{a}\right\rangle\left\langle\frac{\mathbf{S}\cdot\mathbf{J}}{|\mathbf{J}|^{2}}\right\rangle$
(9) $\displaystyle=$ $\displaystyle-g_{a}M_{J}|\mathbf{\nabla}\phi/f_{a}|,$
(10)

where we have defined the axion Landé factor

$g_{a}=\left\langle\frac{\mathbf{S}\cdot\mathbf{J}}{|\mathbf{J}|^{2}}\right\rangle=\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.$
(11)

Note that $g_{J}\neq g_{a}$. If we can find states in the atom or molecule
where the values of $g_{J}/g_{a}$ are different, then we can use these states
as co-magnetometers. For example, the spin-orbit components ${}^{2}P_{1/2}$
and ${}^{2}P_{3/2}$ of a ${}^{2}P$ electronic state have
$g_{J,1/2}=2/3,g_{a,1/2}=-1/3$ and $g_{J,3/2}=4/3,g_{a,3/2}=1/3$,
respectively. Thus, the relative shift due to a magnetic or axion field
between these states are not proportional, and they can be distinguished.

Note that not all spin-orbit states have this feature; the ${}^{3}P_{0,1,2}$
components of a ${}^{3}P$ electronic state all have $g_{J}=3/2,g_{a}=1/2$ so
comparing the shifts in these states cannot be used to disentangle a magnetic
and axion field. Hyperfine structure, and the fact that $g_{e}\neq 2$ exactly,
make these conclusions not entirely valid, but it means that the $g-$factors
differ by $O(10^{-3})$ so their utility as co-magnetometers is suppressed.

Thus, a useful co-magnetometer scheme for the approach under discussion
requires states with different relative contributions of electron spin and
electron orbital angular momentum to the magnetic moment. This shows why the
internal co-magnetometry scheme for ACME is not immediately useful – the pairs
of states have, to good approximation, the same relative orientation of
electron spin and orbital angular momenta. This is also the case with tuning
magnetic interactions in polyatomics with parity doublets Anderegg  _et al._
(2023); these work by changing the average spin projection on the laboratory
field, within a single state where the magnetic interactions come almost
entirely from the electron spin, and are therefore not immediately useful for
axion co-magnetometry.

###  II.2 Molecular probes

Polarized diatomic molecules have been used to search for the electron’s EDM
Hudson  _et al._ (2011); Cairncross  _et al._ (2017); Andreev  _et al._
(2018); Roussy  _et al._ (2023). One example is the ACME experiment which sets
a bound333The JILA eEDM experiment Roussy  _et al._ (2023) improves the ACME
bound by a factor $\sim 2.4$; however, as discussed in the earlier footnote,
this experiment uses an ion trap and is therefore not ideal for the
measurements discussed here. on this parameter using the metastable $H$ state
in ThO molecule, $|d_{e}|<1.1\times 10^{-29}$ e cm Andreev  _et al._ (2018).
This state has $J=1$ and enjoys a natural immunity to stray magnetic fields,
due to a cancellation between the spin and orbital angular momentum of the
valence electrons which leads to a small net magnetic moment
$\mu_{H}=g_{H}\mu_{B}$, with $g_{H}=0.008$ Vutha  _et al._ (2011). Note,
however, that since only the electron spin contributes to axion precession,
and the stretched states in the $H,J=1$ manifold have fully-aligned electron
spins, this state can still be used to search for the axion gradient.

The value of $d_{e}$ is extracted from the change in the precession frequency
that is correlated with the molecular axis orientation, given by
$\Omega=\textbf{J}_{e}\cdot\hat{\textbf{n}}$, and the orientation of the
effective electric field, $\omega_{\text{edm}}=d_{e}E_{\text{eff}}\Omega$,
which is reversed every few seconds. In a later section, we discuss how ACME
III (or a similar experiment) could be modified to search for axion forces by
searching for spin precession arising from the axion gradient as opposed to
the electron EDM. As mentioned earlier, many of the challenges of electron EDM
experiments, such as the need for large electric polarization and the need for
heavy species to make use of relativistic enhancements, are not needed;
however, since ACME III could make the proposed measurements with minimal
modifications, we present the details. We also discuss simpler dedicated
approaches which would not offer electron EDM sensitivity.

We now estimate the reach of the experiment. The condition for the axion
energy shift is $\Delta E_{\phi}>\delta\omega$, with $\delta\omega$ the
smallest measurable frequency. Assume we use a cubic brick of a dense material
with number density of nucleons $n_{N}$ and size $D$ at a distance $d$ from
the molecules. In the case $D\sim d$, we have:

$\Delta E_{\phi}\sim\frac{g_{s}g_{p}^{\psi}}{8\pi
m_{\psi}}n_{N}D\left(\frac{D}{\lambda_{\phi}}+1\right)e^{-D/\lambda_{\phi}}\,.$
(12)

As an example, axions with wavelength comparable to the other scales in the
problem, $2d\sim D\sim\lambda_{\phi}$, can be detected provided that

$g_{s}g_{\psi}^{p}>\frac{\pi\delta\omega\,m_{\psi}}{n_{N}\lambda_{\phi}}\,,$
(13)

which shows how we can gain sensitivity by increasing (decreasing) the
parameters $n_{N}$($\delta\omega$). The reach of ACME III, where a sensitivity
at the level of $\delta\omega^{III}\sim 15$ $\mu$rad/s is expected, is shown
in Fig.1 for axion gradients from the earth and from test masses. We assume
lead or tungsten bricks of size $D^{3}\sim(10~{}\text{cm})^{3}$ next to the
beam, at a distance of order $O(10)$ cm.

##  III Experimental setup and background overview

Molecular beam experiments are well-suited to search for axionic forces on
electrons. In this section we first consider using ACME III. We also discuss
the experimental set up and the protocol to control systematic effects.
Similar strategies are expected for a dedicated axion search using, for
example, the Yb atom beam co-magnetometer described in section III.3.

###  III.1 ACME III

An axion gradient generates an additional term to the spin evolution (see Eq.
3), with the precessed angle due to the axion contribution given by:

$\theta_{\text{axion}}=\int_{0}^{L}\frac{\nabla\phi}{f_{\phi}}\frac{dx}{v_{mol}}\,,$
(14)

where $v_{mol}$ is the velocity of the molecules, $x$ is the position, and $L$
is the precession length. As in EDM searches, the precessed phase can be
detected by measuring the population in the spin quadrature states, $S_{x,y}$.

Unlike for EDM searches, the polarizing electric field $E_{\text{lab}}$ is not
needed and one could in principle operate with only a weak applied B field.
Assuming that the magnetic field is adjusted so that the phase is
$\theta_{B}+\theta_{\text{offset}}\approx\pi/4$, the relevant measurable
quantity is given by the asymmetry Baron  _et al._ (2017):

$\displaystyle\mathcal{A}=$
$\displaystyle\frac{S_{x}-S_{y}}{S_{x}+S_{y}}=\mathcal{C}\cos(\theta_{B}+\theta_{\text{axion}}+\theta_{\text{offset}})$
(15)
$\displaystyle\approx\text{sgn}(B)\theta_{\text{axion}}=\text{sgn}(B)\left(\frac{\nabla\phi}{f_{\phi}}\right)\tau_{\text{coh}}\,.$
(16)

The constant $0\leq\mathcal{C}\leq 1$ is the contrast, which is $\sim 1$ for
ACME, and indicates the efficiency in the preparation and detection of the
states.

We discuss two scenarios: one to look for the axion field from a test mass,
and one from the earth. The test mass case is in principle straightforward, as
one merely needs to add a moving mass near the ACME beam line. Let the test
mass be movable between positions 1 and 2, sourcing an averaged gradient over
the beam path of $-\mathbf{\nabla}\phi_{1}/f_{a}$ and
$-\mathbf{\nabla}\phi_{2}/f_{a}$, respectively. Considering the test mass
position as a binary “switch” which can be in state $\mathcal{M}=\pm 1$,
analogous to other switches in ACME Baron  _et al._ (2017), we can write the
spin precession angle due to the axion gradient as

$\displaystyle\theta_{\text{axion}}$ $\displaystyle=$
$\displaystyle\int_{0}^{L}\frac{\nabla\phi_{1}+\nabla\phi_{2}}{2f_{\phi}}\frac{dx}{v_{mol}}+\mathcal{M}\int_{0}^{L}\frac{\nabla\phi_{1}-\nabla\phi_{2}}{2f_{\phi}}\frac{dx}{v_{mol}}\,$
(17) $\displaystyle\equiv$
$\displaystyle\theta_{0,\text{axion}}+\mathcal{M}\theta_{\mathcal{M}}.$

Note that we have defined a mean, offset spin precession
$\theta_{0,\text{axion}}$ which does not depend on the position of the test
mass, and a term $\theta_{\mathcal{M}}$ which changes sign when the test mass
is moved.

The experimental protocol is therefore to add the $\mathcal{M}$ switch where
we operate the experiment with the test mass in two positions. Moving the mass
every few seconds should give sufficient robustness against drifts in other
experimental quantities, comparable to other switches in ACME. This should
give a spin precession signal which is proportional to the axion field
gradient, but possibly other systematic effects. Note that the masses can be
between the preparation and readout stages, so that they won’t interfere with
the current optical preparation and readout schemes.

One of the effects of most obvious concern is that moving a large mass will
change the electromagnetic environment. Shielding the electric field from the
test mass is straightforward: simply use a conducting shield around the
molecules, which ACME already has in the form of electric field plates. The
ACME spin precession scheme is fairly robust against electric field drifts by
design, as they result in a common-mode offset between the two precessing
states, so this is not likely to be a concern.

A greater concern is magnetic offsets. To get large signals, we would like the
test mass to be inside the magnetic shields, as close to the molecules as
possible. Magnetic impurities in the test mass would also correlate with
$\mathcal{M}$. A magnetic field shift of $\sim$1 nanoGauss would give rise to
a spin precession signal correlated with $\mathcal{M}$ comparable to the
projected statistical sensitivity of ACME III. Quasi-DC magnetic fields on the
order of nanoGauss are challenging to measure, though not impossible;
commercially-available optical magnetometers can get to near this
sensitivity444See for example QuSpin, www.quspin.com. However, since these
magnetometers work via the interaction of a valence electron on an atom with
the magnetic field, they are also in principle sensitive to the axion
gradient555Note that this could be combined with another magnetometer
technology not relying on atomic electron spins, such as SQUIDs, as another
avenue to search for axion gradients.. Thus it is more robust to rely on axion
co-magnetometry states, which ACME III is already setting up to use but for a
different reason; the $Q^{3}\Delta_{2}$ state in ThO Wu  _et al._ (2020) has a
magnetic moment of $\sim$2 $\mu_{B}$, versus $\sim 0.01$ $\mu_{B}$ for the
$H^{3}\Delta_{1}$ state, and since the magnetic moment arises mostly from
orbital angular momentum, the states $H$ and $Q$ represent an axion co-
magnetometry state pair. Thus, one can measure the spin precession dependence
on $\mathcal{M}$ in both the $H$ and $Q$ states; since they have different
relative magnetic and axion gradient sensitivity, the relative contributions
of these two effects will be different in these two states, thus enabling
their disentanglement. Note that for cases where it is technically feasible,
the mass could be periodically rotated or re-oriented to change the direction
of the residual fields for further rejection of systematic effects.

A related concern is magnetic Johnson noise (MJN) Lamoreaux (1999); Munger
(2005), which arises due to thermal fluctuation currents in a conductor at
finite temperature. This will not necessarily add systematically offset spin
precession, but it will result in magnetic field noise which could reduce the
contrast and statistical sensitivity of the measurement. The calculation of
the effect would depend on the specific geometry and material chosen for the
moving masses, but we can make some estimates. For a conductor with
resistivity $\rho$ at temperature $T$ having thickness $\sim t$ a distance of
$\sim D$ away, the magnetic field power spectral density at the molecules is
given approximately by Vutha (2011)

$\widetilde{\mathcal{B}}(f)\sim\frac{\mu_{0}}{4\pi}\left[\frac{8tk_{B}T}{3\rho
D^{2}}\right]^{1/2}\sim\frac{1~{}\textrm{pG}/\sqrt{\textrm{Hz}}}{\sqrt{\rho/(1~{}\Omega\cdot\textrm{m})}},$
(18)

where $k_{B}$ is Boltzmann’s constant, and for the rightmost term we have
assumed $D\sim$ 50 mm, $t\sim$ 100 mm, and $T=$ 300 K (though the mass could
be cooled if needed).

Tungsten Rumble (2022) would be a natural choice for a test mass given its
very high mass density of 19.3 g/cm3, and its resistivity of 5.44$\times
10^{-8}~{}\Omega\cdot$m would give rise to MJN on the order of $\sim$5
nG/$\surd$Hz. This would give rise to magnetic spin precession noise on the
order of $\sim 2\pi\times\mu$Hz/$\surd$Hz for the ThO H state, and around a
hundred-fold larger for the ThO Q state, which might sound problematic;
however, as it is still smaller than other more dominant forms of noise, it
will not be a limitation as it will average away faster than other noise
sources.

One such source is the velocity dispersion $\Delta v$ in the molecular beam,
which also is not a fundamental limitation as it averages away Baron  _et al._
(2017), though it can add excess noise Andreev  _et al._ (2018). The phase
noise $\Delta\phi$ from magnetic spin precession in magnetic field noise
$\Delta\mathcal{B}$ and precession time noise $\Delta\tau$, is given by

$\Delta\phi=\tau\Delta\mathcal{B}+\mathcal{B}\Delta\tau=\tau\left(\Delta\mathcal{B}+\mathcal{B}\Delta
v/v\right).$ (19)

We have used the fact that $|\Delta\mathcal{B}|\ll|\mathcal{B}|$ as the
applied (and residual) magnetic fields are larger than the MJN, and that the
precession is over $L$ so $L=v\tau$ and therefore $\Delta\tau/\tau=\Delta
v/v$. Since the ACME beam has $\Delta v/v\sim 0.1$ Hutzler  _et al._ (2011)
due to velocity dispersion within a single shot, and has shot-to-shot changes
in the mean velocity of $\Delta v/v\sim 10^{-3}$ Andreev  _et al._ (2018), the
$\Delta\mathcal{B}$ component should not be a major limitation to statistical
sensitivity. However, should MJN (or the cost of tungsten) ultimately be a
limiting factor, there are materials such as zirconia or leaded glass with
over 10 orders of magnitude larger resistivity yet only a factor of 4 to 5
less density.

Now we describe a protocol to measure an axion gradient from the Earth. In
this case there is no moving mass, and therefore no $\mathcal{M}$ switch. This
introduces a challenge, as we no longer have a way to change the axion field,
and are therefore potentially susceptible to the many DC drifts and offsets in
EDM-style spin precession experiments Baron  _et al._ (2017). However, we
discuss two methods which can help mitigate this.

We can continue to the use the $H,Q$ co-magnetometer pair to distinguish
between a constant background magnetic field and the background axion field. A
greater challenge is absolute phase offsets, for example arising from the fact
that the state initialization and readout stages will always have some finite,
drifting offset set by the polarization of lasers with different beam paths.
To address this, we propose to use the fact that the velocity dispersion in
the molecular beam Hutzler  _et al._ (2011, 2012) results in an accumulated
phase angle which is time-dependent relative to the time after the production
of the molecular beam pulse at $t=0$ Baron  _et al._ (2017):

$\theta_{\text{axion}}(t)=\int_{0}^{L}\frac{\nabla\phi}{f_{\phi}}\frac{dx}{v_{mol}(t)}\,.$
(20)

This arises from the fact that slower molecules take longer to get to the spin
precession region, and when they arrive they spend more time precessing and
therefore accumulate more phase. Because the ACME spin precession readout
protocol involves rapid, time-resolved readout to normalize against molecular
beam yield fluctuations Kirilov  _et al._ (2013), this also gives the ability
to resolve this time-dependence on a single shot. This has the advantage of
offering significant robustness against preparation and measurement phase
errors, which will be constant, and physical spin precession phases, which
will have a time-dependence. Note that inferring the spin precession from the
asymmetry time-dependence also provides robustness against sources of offsets
such as light shifts from the lasers themselves, which will not accumulate
over the entire spin precession period (and which can be probed by varying
laser parameters.)

Therefore, the proposed protocol to measure the axion gradient from the earth
is the following.

  * •

Switch between the $H$-state and $Q$-state in periods of around 1 second. This
will enable robust co-magnetometry, in particular the measurement of
background magnetic fields.

  * •

Measure the time dependence of the asymmetry, in particular its slope:
$\frac{\partial\mathcal{A}}{\partial t}$.

  * •

Compare the asymmetry slope for the $Q$ and $H$ states. The axion field should
cause a component of $\frac{\partial\mathcal{A}}{\partial t}$ which changes
between $Q$ and $H$, but otherwise does not change.

An important observation is that the ACME quantization axis, which is set by
the electric field, is horizontal and therefore has vanishing sensitivity to
the Earth axion gradients. There are two potential approaches to address this.
One could rotate the electric field plates so that the applied field aligns
with gravity. This would be highly non-trivial, not only because the plate
mounting would have to be re-engineered, but all the laser paths would need to
be re-engineered as there are some lasers which must go through the plates and
some which cannot. Another option would be to operate without an electric
field at all, and use a weak vertical magnetic field to set the quantization
axis. This would require modifying the state preparation and readout
protocols, but possibly in a way which would not require major redesign of the
apparatus.

###  III.2 Ultracold atoms and molecules

There are proposals to use ultracold molecules to search for the electron EDM
surpassing the current bounds by several orders of magnitude Kozyryev and
Hutzler (2017); Fleig and DeMille (2021); Fitch  _et al._ (2021). The shot
noise-limited uncertainty of the frequency in a measurement is given by
$\delta\omega=\frac{1}{\tau_{c}\sqrt{N}}$, with $N$ the number of measured
molecules. Assuming $N=10^{6}$ molecules and coherence times around 10-100
seconds, just one measurement would be equivalent to the expected sensitivity
at ACME III. With a preparation/detection efficiency around $O(10)$% it is
expected to have sensitivities of the order $\delta\omega\sim 1-10$ nrad/s by
operating for around $10^{7}$ seconds. These numbers suggest the ultracold
molecule proposal as being very compelling to test axion forces on electrons.

To achieve full sensitivity it would be important to have a co-magnetometer
species that reduces the impact of systematic effects. Co-trapped species
Gregory  _et al._ (2021); Jurgilas  _et al._ (2021), with different origins of
the magnetic moment, would allow the distinction between stray magnetic fields
and the sourced axion gradient.

It may also be interesting to consider for axion force searches the 171Yb
optical trap in Zheng  _et al._ (2022), which used the ground state,
${}^{1}S_{0}$, to look for a permanent atomic EDM. In that work the authors
show that the coherence time exceeds $\tau_{c}\sim O(100)$ seconds, implying
that if a number large number of atoms, $N\sim O(10^{6})$, can be trapped the
sensitivity to monopole-dipole forces on nucleons may extend beyond
astrophysical constraints (see Fig.2). This scheme may also require the
presence of a co-magnetometer species.

###  III.3 Atomic beam experiments

Atom beam experiments are also excellent candidates to look for axion forces.
Since there is no “molecular enhancement” of the axion signal, unlike EDM
searches, it is attractive to use atoms as their simpler structure generally
leads to more intense beams, more efficient optical control, easier laser
cooling for beam brightening, etc. Note that an experiment built for the
purpose of searching for axion gradients would not need an electric field at
all, further simplifying the experimental requirements.

An interesting possibility is an experiment using 171Yb, which can be used to
make intense beams Wright  _et al._ (2022); Hutzler  _et al._ (2012), and has
two valence electrons in the $6s$ orbital $(L=S=0)$ giving rise to a
${}^{1}S_{0}$ ground electronic state, and nuclear spin $I=1/2$. The
${}^{3}P_{2}$ $(L=S=1)$ excited state is relatively long-lived with a life-
time around $\tau\sim 10$ s Porsev and Derevianko (2004).

An experiment could use the ground state ${}^{1}S_{0}$, which is only
sensitive to the axion gradient through the nucleon spin, and the excited
state ${}^{3}P_{2}$, which has a very different magnetic moment and can
therefore be used as a co-magnetometer. The Hamiltonians for these states are:

$\displaystyle
H_{{}^{1}\\!S_{0}}=-\mathbf{\mu_{N}}\cdot\mathbf{B}+c_{N}\frac{\mathbf{\nabla}\phi}{f_{\phi}}\cdot\mathbf{I}\,,$
(21) $\displaystyle
H_{{}^{3}\\!P_{2}}=-(\mathbf{\mu_{N}}+Mg_{P}\mathbf{\mu_{B}})\cdot\mathbf{B}+\frac{\mathbf{\nabla}\phi}{f_{\phi}}\cdot(c_{N}\mathbf{I}+c_{e}\mathbf{S})\,,$
(22)

where $\mu_{N(B)}$ is the nuclear (Bohr) magneton, $M$ is the projection of
the total angular momentum onto the quantisation axis, and $\mathbf{I}$
($\mathbf{S}$) is the nucleon (electron) spin. The coefficients $c_{e},c_{N}$
reflect the fact that in principle the axion coupling to electrons and
nucleons may differ. These states have different origins for their magnetic
moment and can be used as a co-magnetometer by comparing how the precession
frequency changes with the B field and axion gradient orientation, as
discussed in a previous section. Since the ground state is only sensitive to
the axion coupling to nucleons, this experiment would be sensitive to both
coupling to electrons and nucleons. See Figures 1 and 2. Note that in the
event of a positive signal, it would be critical to perform the measurements
in different states or species with additional different relative
sensitivities to magnetic fields, electron couplings, and nuclear couplings,
in order to conclusively disentangle these effects.

Figure 2: Axion-mediated monopole-dipole forces on nucleons at spin precession
experiments searching for axion gradients source by a test mass or the earth.
The blue lines stand for a dedicated spin precession experiment using Yb co-
magnetometer beams. Finally, in orange, we have the expected reach with an
experiment using cold trapped Yb atoms, assuming the existence of co-
magnetometer states. Bounds adapted from O’Hare and Vitagliano (2020)

The shot noise-limited sensitivity using $\tau=5-10$ ms,
$\dot{N}=10^{10}-10^{11}$ atoms/s, and $T_{int}=10^{6}-10^{7}$ s, we get an
expected sensitivity in the range $\delta\omega\sim 10^{-7}$ Hz to $10^{-6}$
Hz. In Fig.1 we show the expected reach assuming $\delta\omega\sim 10^{-7}$
Hz.

Alternatively one could use an indium or thallium beam using the
${}^{2}P_{1/2}$ and ${}^{2}P_{3/2}$ spin orbit components of the ground
electronic state . As discussed earlier, these two states can be used as as
axion co-magnetometers since their $g$-factors have different contributions
from electron spin and orbital angular momenta. Whether they offer any
advantage over Yb depends largely on experimental considerations, such as beam
fluxes, laser wavelengths, detection strategies, etc.

##  IV Advantages of beam experiments

ACME III will measure the spin precession at the level of approximately 10
microrad/s. This corresponds, in terms of a Bohr magneton frequency, to a
magnetic field of around 10 aT. It is interesting to compare this result to
similar searches using co-magnetometers (see Fig.1). In the SMILE experiment
Lee  _et al._ (2018) an alkali-noble co-magnetometer is employed. These
detectors are currently one of the most sensitive magnetometers, able to
measure magnetic fields at the level of $O(1-10)\text{fT}/\sqrt{\text{Hz}}$.

The spin projection noise, given by

$\delta B=\frac{1}{\mu_{B}}\frac{1}{\sqrt{T_{2}t_{int}N}}\,,$ (23)

indicates that, in principle, this kind of co-magnetometer would surpass the
ACME III expected sensitivity using the values: $T_{2}\sim 10^{-3}$ s, $N\sim
10^{11}$ and $t_{int}\sim 10^{5}$ s. However, the co-magnetometer sensitivity
also depends on the photon shot noise which at frequencies around 0.1 Hz
(given by the moving mass period) dominates and lies around $\sim 20$
fT/$\sqrt{\text{Hz}}$. For an integration time $t_{int}\sim 10^{5}$ s this
corresponds to an uncertainty $\delta B\sim$100 aT for the effective magnetic
field, i.e. the axion gradient, around an order of magnitude larger than the
expected at ACME III.

This explains why in terms of reach, the use of EDM experiments like ACME III
is expected to improve the co-magnetometer results by around an order of
magnitude if moving test-masses are employed. Additionally, thanks to the
time-dependence of the asymmetry, one can use the imperfect behaviour of the
molecular beam in terms of velocity dispersion to measure the Earth gradient
as discussed above. This fact will further improve the reach by several orders
of magnitude, in particular for light axions. For axions with mass
$m_{a}\lesssim 10^{-10}$ eV, regions of the parameter space beyond the
constrained by astrophysics may be tested.

##  V Conclusion

One of the generic ways in which new physics can interact with the Standard
Model is through the spin of standard model particles. Spin precession
experiments are thus well placed to search for a variety of such effects,
ranging from time varying effects caused by dark matter to new fields sourced
by terrestrial and laboratory sources. For the latter class of experiments,
the signal is fundamentally a dc signal. A variety of low frequency systematic
effects must be overcome in order to see this signal. Interestingly, there are
spin precession experiments that are adept at managing such low frequency
systematic effects - namely, experiments that are aimed at measuring the
permanent electric dipole moment of electrons and nucleons. In this paper, we
have highlighted the opportunities that exist in using the well developed
technology of experiments such as ACME III, as well as motivated to build
dedicated experiments using atomic beams, and laser-cooled atoms or molecules
to search for spin precession induced by test masses and the earth in the
laboratory.

## Acknowledgments

PA is supported by the STFC under Grant No. ST/T000864/1. N.R.H. is supported
by U.S. National Science Foundation (NSF) CAREER Award (PHY-1847550), the
Heising-Simons Foundation (2022-3361), the Gordon and Betty Moore Foundation
(GBMF7947), and the Alfred P. Sloan Foundation (G-2019-12502). D. E. K and
S.R. are supported in part by the NS under Grant No. PHY-1818899. This work
was supported by the U.S. Department of Energy (DOE), Office of Science,
National Quantum Information Science Research Centers, Superconducting Quantum
Materials and Systems Center (SQMS) under contract No. DE-AC02-07CH11359. S.R.
is also supported by the DOE under a QuantISED grant for MAGIS, and the Simons
Investigator Award No. 827042. This article is based upon work from COST
Action COSMIC WISPers CA21106, supported by COST (European Cooperation in
Science and Technology). NH and MR thank the Perimeter Institute and the
organisers of the School on Table-Top Experiments for Fundamental Physics,
where this work was initiated, for providing a friendly and exciting
atmosphere.

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