# Response of atomic spin-based sensors to magnetic and nonmagnetic
perturbations

Mikhail Padniuk  Marian Smoluchowski Institute of Physics, Jagiellonian
University, Łojasiewicza 11, 30-348, Kraków, Poland.
<EMAIL_ADDRESS>Marek Kopciuch  Marian Smoluchowski
Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348,
Kraków, Poland.  Riccardo Cipolletti  Helmholtz Institute, Johannes Gutenberg-
Universitat at Mainz, 55099 Mainz, Germany  Robert Bosch GmbH, Corporate
Sector Research and Advance Engineering, Advanced Technologies and Micro
Systems, 71272 Renningen, Germany  Arne Wickenbrock  Helmholtz Institute,
Johannes Gutenberg-Universitat at Mainz, 55099 Mainz, Germany  Dmitry Budker
Helmholtz Institute, Johannes Gutenberg-Universitat at Mainz, 55099 Mainz,
Germany  Szymon Pustelny  Marian Smoluchowski Institute of Physics,
Jagiellonian University, Łojasiewicza 11, 30-348, Kraków, Poland.
<EMAIL_ADDRESS>
###### Abstract

Searches for pseudo-magnetic spin couplings require implementation of
techniques capable of sensitive detection of such interactions. While Spin-
Exchange Relaxation Free (SERF) magnetometry is one of the most powerful
approaches enabling the searches, it suffers from a strong magnetic coupling,
deteriorating the pseudo-magnetic coupling sensitivity. To address this
problem, here, we compare, via numerical simulations, the performance of SERF
magnetometer and noble-gas-alkali-metal co-magnetometer, operating in a so-
called self-compensating regime. We demonstrate that the co-magnetometer
allows reduction of the sensitivity to low-frequency magnetic fields without
loss of the sensitivity to nonmagnetic couplings. Based on that we investigate
the responses of both systems to the oscillating and transient spin
perturbations. Our simulations reveal about five orders of magnitude stronger
response to the neutron pseudo-magnetic coupling and about three orders of
magnitude stronger response to the proton pseudo-magnetic coupling of the co-
magnetometer than those of the SERF magnetometer. Different frequency
responses of the co-magnetometer to magnetic and nonmagnetic perturbations
enables differentiation between these two types of interactions. This outlines
the ability to implement the co-magnetometer as an advanced sensor for the
Global Network of Optical Magnetometer for Exotic Physics searches (GNOME),
aiming at detection of ultra-light bosons (e.g., axion-like particles).

## Introduction

In optically pumped magnetometers (OPMs), measurements of external magnetic
fields rely on detection of energy-level shifts/spin precession arising from
Zeeman interaction [1]. However, if nonmagnetic interactions similarly affect
the energy levels, OPMs can also be used to detect those interactions. This
enables the application of optical magnetometry to searches for anomalous
spin-dependent interactions.

Currently, OPMs are used to search for physics beyond the Standard Model in a
variety of experiments (see, Ref. [2] and references therein). A particular
example of such OPMs application is the search for microscopic-range spin-
dependent interactions, indicating a possibility of existence of axion-like
particles (ALPs), which are one of prime candidates for the dark matter [2].
OPMs are also used to search for transient nonmagnetic spin couplings, which
could arise due to interaction with macroscopic objects made of ALPs, in
particular, Q-balls [3], topological defects (e.g., domain walls) of ALP field
[4], or ALP-field pulses generated in cataclysmic astrophysical events (e.g.,
black-hole mergers) [5]. These transient couplings are targeted by the Global
Network of Optical Magnetometers for Exotic physics searches (GNOME) [6, 7,
8]. Heretofore, the GNOME consists of various OPMs, originally developed for
ultra-sensitive magnetometry in globally distributed locations. This leads to
several challenges when searching the data for global transient signals.
First, due to a different nuclear-spin content of atoms used in specific
sensors, they are characterised with different sensitivity to exotic spin
couplings[9] (coupling to protons and neutrons could be, in general,
different). Second, the implemented OPMs are characterised with different
bandwidths, sensitivities, and local noise floors, which complicates data
analysis[10]. Third, the magnetometers were designed to maximise the
sensitivity to magnetic fields. Uncontrolled and uncompensated magnetic-field
perturbations are detrimental to the sensitivity to other couplings. These
issues triggered work to upgrade conventional OPMs in the GNOME with a sensor
less sensitive to magnetic fields but highly sensitive to nonmagnetic spin
couplings.

A specific example of a sensor, being predominantly sensitive to nonmagnetic
spin couplings, hence well suited for searches for physics beyond the Standard
Model, is an alkali-metal-noble-gas co-magnetometer originally developed by
Romalis and coworkers [11, 12, 13] and later studied extensively by other
researchers [14, 15, 16, 17, 18]. Such a system operates based on coupled
evolution of the magnetizations of noble gas (NG) and alkali-metal (AM)
vapour. Both can achieve a high percentage of polarisation as AM can be
optically pumped, whereas polarisation of the NG can be generated via spin-
exchange collisions with the optically polarised AM [19]. In the co-
magnetometer, the vapour cell is heated above 150∘C to achieve a sufficiently
high AM density, such that relaxation due to spin-exchange collisions, which
is one of the main mechanisms of AM polarisation relaxation and hence one of a
limiting factor to spin-coupling sensitivity, is suppressed. This is the so-
called Spin-Exchange Relaxation Free (SERF) regime [20]. Furthermore, for the
alkali-metal-noble-gas co-magnetometer the effect of low-frequency magnetic
drifts can be suppressed by application of a carefully chosen bias magnetic
field. If the bias field is approximately equal to the sum of AM and NG
magnetization fields, the system retains high sensitivity to both electron and
nuclear nonmagnetic spin couplings, but becomes insensitive to low-frequency
magnetic-field changes. Such self-compensating co-magnetometers have already
been used for tests of the Lorentz symmetry [21, 22, 23], setting limit on the
neutron coupling to light pseudoscalar particles [24], and spin-mass
interaction of fermions [25]. In all of those applications, however, the
signals of interests had low frequencies (typically below a few Hz), for which
the response to magnetic fields is almost entirely suppressed and the system
is only sensitive to nonmagnetic spin couplings. Since the GNOME targets
transient signals, a question of the co-magnetometer’s applicability to such
searches is important and well motivated. Additional interest in co-
magnetometer systems, in the context of searches for transient effects, arises
from the possibility for quantitative distinguishing between magnetic and
nonmagnetic transients. This was originally considered for precise rotation
sensing with NG-AM co-magnetometers[12] but can be extended for other
scenarios.

In this work, we analyse the alkali-metal-noble-gas co-magnetometer in the
context of its response to time-dependent electron and nuclear spin
perturbations, and we compare the results with the response of the AM SERF
magnetometer. Our theoretical analysis is based on numerical solution of
differential equations describing the coupled evolution of the AM and NG. We
use Ordinary Differential Equations (ODE) to simulate the response of atoms to
spin perturbations in the absence of noise and we provide analytical models,
which reproduce numerical results. The discussions concern the responses of
both, SERF magnetometer and co-magnetometer, to magnetic fields and pseudo-
magnetic spin perturbations, where in the latter case we independently
consider the effects of electron, proton, and neutron spin perturbations.
These allow us to determine the frequency and phase responses of both devices.
Finally, we compare the response of the co-magnetometer and SERF magnetometer
to transient effects of both magnetic and nonmagnetic nature. We show that
differences between the co-magnetometer frequency responses for different spin
perturbations allows identification of nonmagnetic transient effects even with
a single sensor. These additional signatures of the nonmagnetic transient
effects can be utilised to decrease the false-positive event rate in searches
with the GNOME network.

## Methods

### Numerical models

In this section, we describe the theoretical models used to simulate the
response of the SERF magnetometers and co-magnetometers to magnetic and
nonmagnetic spin perturbations.

#### SERF magnetometer model

In a conventional SERF magnetometer, a spin-zero NG can be used as a buffer
gas. The buffer gas limits diffusion of the atoms towards the cell walls (the
AM atoms are depolarised in wall collisions), which increases the polarisation
lifetime but, it does not produce any magnetisation. Thereby, to simulate a
response of such a magnetometer to a spin perturbation, we implement an
approach based on the solutions of the Bloch equation[26] with inclusion of
nonmagnetic spin couplings (for more details see Additional information). The
equation describing the AM polarisation $\mathbf{P}$ of atoms subjected to an
external magnetic field $\mathbf{B}$ and circularly polarised light can be
written as

$\frac{d\mathbf{P}}{dt}=\frac{1}{q}\bigg{[}\gamma_{e}\mathbf{(B+b_{e})\times
P}+(q-1)\gamma_{e}\mathbf{b_{N}^{AM}\times
P}+\mathbf{(s-P)}R_{p}-{R_{c}^{e}}\mathbf{P}\bigg{]},$ (1)

where $\gamma_{e}$ is the electron gyromagnetic ratio, $\mathbf{s}$ is the
optical pumping vector, $R_{p}$ is the pumping rate, $R_{c}^{e}$ is the
polarisation-relaxation rate, $q$ is the slowing-down factor, which is a
function of the nuclear spin of the AM and its polarisation $\mathrm{P}$. The
vectors $\mathbf{b_{e}}$ and $\mathbf{b_{N}^{AM}}$ are the nonmagnetic
electron and nuclear spin perturbations, respectively, given in the magnetic
units (pseudo-magnetic field).

#### Co-magnetometer model

In the co-magnetometer, a polarised AM and NG (in this case NG with nonzero
nuclear spin is used, so its nuclei can be polarised) occupy the same volume
inside of a spherical glass cell. Then, the response of the co-magnetometer to
magnetic and nonmagnetic perturbations is determined by a set of coupled Bloch
equations (the so-called Bloch-Hasegawa equations) [11] with inclusion of
nonmagnetic spin perturbations (for more details see Additional information)

$\begin{cases}\frac{d\mathbf{P^{e}}}{dt}&=\frac{1}{q}\bigg{[}\gamma_{e}\mathbf{(B+b_{e}+}\lambda
M^{n}\mathbf{P^{n})\times P^{e}}+(q-1)\gamma_{e}\mathbf{b_{N}^{AM}\times
P^{e}}+R_{se}^{ne}\mathbf{P^{n}}+{\mathbf{(s-P^{e})}}R_{p}-({R_{c}^{e}}+R_{se}^{en})\mathbf{P^{e}}\bigg{]},\\\
\frac{d\mathbf{P^{n}}}{dt}&=\gamma_{n}\mathbf{(B+b_{N}^{NG}+}\lambda
M^{e}\mathbf{P^{e})\times
P^{n}}+R_{se}^{en}\mathbf{P^{e}}-(R_{se}^{ne}+R_{c}^{n})\mathbf{P^{n}},\end{cases}$
(2)

where $\mathbf{P^{e}}$ and $\mathbf{P^{n}}$ stand for the electron
polarisation of the AM and nuclear polarisation of the NG, respectively,
$\lambda$ is the coupling-strength factor for interaction between the two
polarisations [27], $M^{e}$ and $M^{n}$ are the maximal possible
magnetisations of AM and NG, and $\gamma_{n}$ is the nuclear gyromagnetic
ratio of NG, $R_{c}^{e}$ and $R_{c}^{n}$ are the electron and nuclear
polarisation-relaxation rates which take into consideration relaxation due to
various interatomic spin destruction collisions and collisions with a walls of
a cell. $R_{se}^{ne}$ and $R_{se}^{ne}$ are rates of the polarisation transfer
in spin exchange collisions form NG to AM and from AM to NG, respectively. The
nonmagnetic nuclear perturbation of the NG spins is denoted by
$\mathbf{b_{N}^{NG}}$.

In this work, we consider the system dynamics over a time scale much shorter
than the characteristic time of NG-AM spin-exchange collisions,
$t\ll(R_{se}^{ne})^{-1}$. Thereby, in Eq. (2), we neglect the term
$R_{se}^{ne}\mathbf{P^{n}}$, which characterises the back action of NG on AM.
In fact, in our description we use the steady-state polarisations of AM
$\mathbf{P_{0}^{e}}$ and NG $\mathbf{P_{0}^{n}}$ with total relaxation rates
for both AM and NG ($R^{e}$ and $R^{n}$, respectively)

$\begin{split}R^{e}&=R_{p}+R_{c}^{e}+R_{se}^{en},\\\
R^{n}&=R_{se}^{ne}+R_{c}^{n},\\\
\mathbf{P_{0}^{e}}&=\frac{R_{p}}{R^{e}}\mathbf{s},\\\
\mathbf{P_{0}^{n}}&=\frac{R_{se}^{en}}{R^{n}}\mathbf{P_{0}^{e}}.\end{split}$
(3)

For simplicity of the model, in the simulations both the steady state
polarisations and relaxation rates are assumed to be independent.

With such a parameterisation the Eq. (2) have the following form

$\begin{cases}\frac{d\mathbf{P^{e}}}{dt}&=\frac{1}{q}\bigg{[}\gamma_{e}\mathbf{(B+b_{e}+}\lambda
M^{n}\mathbf{P^{n})\times P^{e}}+(q-1)\gamma_{e}\mathbf{b_{N}^{AM}\times
P^{e}}+(\mathbf{P_{0}^{e}-P^{e}})R^{e}\bigg{]},\\\
\frac{d\mathbf{P^{n}}}{dt}&=\gamma_{n}\mathbf{(B+b_{N}^{NG}+}\lambda
M^{e}\mathbf{P^{e})\times P^{n}+(P^{n}_{0}}-\mathbf{P^{n}})R^{n}.\end{cases}$
(4)

In order to fully capitalise on the co-magnetometric capabilities (self-
compensation of slow magnetic fields), here we consider the operation in the
self-compensating regime, which is achieved when a static magnetic field
$\mathbf{B_{c}}$

$\mathbf{B_{c}=}-(\lambda M^{e}P_{0}^{e}+\lambda M^{n}P_{0}^{n})\mathbf{z},$
(5)

is applied to the system[11] (here we assumed that the initial AM and NG
polarisations are oriented along the $\mathbf{z}$ axis).

### Nuclear spin content and sensitivity to neutron and proton spin
perturbations

Due to the composite nature of atomic nuclei, the nuclear response may arise
due to coupling to protons, neutrons, or a combination of both. Therefore, the
effective pseudo-magnetic fields $\mathbf{b_{N}^{AM}}$ and
$\mathbf{b_{N}^{NG}}$ can be divided into parts: $\mathbf{b_{p}}$ affecting
the protons and $\mathbf{b_{n}}$ acting on the neutrons [9]

$\begin{split}\mathbf{b_{N}^{i}}&=\mathbf{b_{n}^{i}+b_{p}^{i}},\end{split}$
(6)

where $i$ may stand for either AM or NG. These pseudo-magnetic fields are
determined by the nonmagnetic field $\bm{\Xi}$ and coupling constants
$\chi_{n}$ and $\chi_{p}$ characterising the coupling to neutrons and protons,
respectively

$\displaystyle\mathbf{b^{AM}_{j}}$
$\displaystyle=-\frac{\sigma^{AM}_{j}}{\mu_{B}g_{s}}\chi_{j}\bm{\Xi},$ (7a)
$\displaystyle\mathbf{b^{NG}_{j}}$
$\displaystyle=\frac{\sigma^{NG}_{j}}{\mu_{N}g_{K}}\chi_{j}\bm{\Xi},$ (7b)

where $j$ indicates either proton or neutron, $\sigma_{j}$ corresponds to the
proton or neutron fraction of the nuclear spin polarisation of AM and NG
(denoted with upper indices), $\mu_{B}$ is the Bohr magneton, $\mu_{N}$ is the
nuclear magneton, $g_{S}$ is the AM Landé factor, and $g_{K}$ is the NG
nuclear spin $g$-factor. Equations (7) show that the effective pseudo-magnetic
fields for the NG are generally different from those for the AM. Thus, for the
simulations or interpretation of results, it is convenient to introduce
scaling factors $\eta_{j}$ which allows comparison between the response of the
SERF magnetometer and the co-magnetometer to nonmagnetic spin couplings of the
same strength

$\mathbf{b^{AM}_{j}}=\eta_{j}\mathbf{b^{NG}_{j}},$ (8)

where the scaling factors are

$\eta_{j}=-\frac{\sigma^{AM}_{j}}{\sigma^{NG}_{j}}\frac{\mu_{N}g_{K}}{\mu_{B}g_{S}}.$
(9)

In the simulations presented in this paper, we consider the responses of the
magnetometers to the perturbation of the same coupling strength. This approach
takes into account the scaling factors defined in Eq. (9). One can find a more
detailed discussion of the nonmagnetic spin couplings in the SERF and the co-
magnetometer in Additional information.

### Simulation parameters

In the case of both, the SERF and co-magnetometer, the spin polarisation is
monitored through measurements of the AM-polarisation projection on a given
direction (here it is the $\mathbf{x}$ axis). As in both systems the atomic
species are initially polarised along the $\mathbf{z}$ axis, both
magnetometers are primarily sensitive to perturbations along $\mathbf{y}$ (the
sensitive direction is determined by the torque generated by the external
fields, rotating the spins around the sensitive direction). Specifically, it
can be shown from Eqs. (1) and (4) that the magnetic or pseudo-magnetic field
applied along $\mathbf{y}$ rotates the initial polarisation in the
$\mathbf{xz}$ plane, which results in a change of the polarisation projection
on the $\mathbf{x}$ axis. At the same time, a field applied along the
$\mathbf{x}$ axis generates rotation in the $\mathbf{yz}$ plane, therefore the
projection of the polarisation on the $\mathbf{x}$ axis remains unchanged.

In the simulations, we assume that the SERF magnetometer operates using 39K
atoms and the co-magnetometer operates using a 39K-3He mixture. Parameters of
potassium vapour are chosen to be exactly the same for both systems, so we can
properly compare the sensitivity. The concentration of the alkali metal is
equal to $10^{14}$ cm-3, which corresponds to saturated atomic K vapour at
190∘C. The concentration of 3He is $10^{20}$ cm-3, which corresponds to 3.5
amg. The assumed total relaxation rates $R^{e}=$1200 s-1 for potassium and
$R^{n}=5\cdot 10^{-5}$ s-1 helium, corresponding to a lifetime of about 6 h,
well reproduce the experimental conditions. The steady-state polarisation of
the AM is $P^{e}_{0}=$0.5, which ensures the highest amplitude of the co-
magnetometer response[21], and also corresponds to typical experimental
conditions. Polarisation of the NG is chosen to be $P^{n}_{0}=$0.05, which
corresponds to typical experimental conditions. These parameters lead to the
compensation-field value of $-131$ nT. The other simulation parameters (a
complete list) is given in Additional information.

In case of 39K and 3He the relation between the effective magnetic fields for
the AM and the NG generated by the same nonmagnetic perturbation defined in
Eq. (9) leads to the following scaling factors

$\displaystyle\mathbf{b_{p}^{{}^{39}K}}$
$\displaystyle=\eta_{p}\mathbf{b_{p}^{{}^{3}He}}\approx 10^{-3}\
\mathbf{b_{p}^{{}^{3}He}},$ (10a) $\displaystyle\mathbf{b_{n}^{{}^{39}K}}$
$\displaystyle=\eta_{n}\mathbf{b_{n}^{{}^{3}He}}\approx 10^{-5}\
\mathbf{b_{n}^{{}^{3}He}},$ (10b)

where the nuclear spin content information was taken from Ref. [9].

## Results and discussion

### Frequency response to different spin perturbations

In this section, we compare responses of the SERF and co-magnetometer devices
to various spin perturbations. We analyse the response of the devices by
investigating their signals when perturbed with an either magnetic or
nonmagnetic periodic $\mathbf{y}$-oriented field $\mathbf{A}$

$\mathbf{A}=A_{0}\sin(2\pi\nu t)\mathbf{y},$ (11)

where $A_{0}$ is the amplitude and $\nu$ is the frequency of the field. For
the simulations, we assume that the amplitude of the perturbation is low
enough so that the co-magnetometer continuously operates in the self-
compensating regime. To analyse the response, the simulated data are fitted
with the function

$S=S_{0}\sin(2\pi ft+\phi),$ (12)

where $S_{0}$, $\phi$ and $f$ are, respectively, the amplitude, phase, and
frequency of the fitted signal. To avoid distortions in the fit, we ignore
transient phenomena at the beginning of the simulations and just fit the
dynamical steady-state data.

The fitted amplitude and phase of the signals arising due to magnetic,
electron nonmagnetic, neutron nonmagnetic, and proton nonmagnetic
perturbations are shown in Fig. 1. For perturbation amplitudes $A_{0}$ small
enough to provide a linear response of the system, the numerical results
(points) are in good agreement with results of simulations performed within an
analytical model. In our model, it was assumed that the AM and NG longitudinal
components of the polarisation are not affected by the perturbation, i.e., the
longitudinal polarisations are constant over time, being equal to steady-state
polarisation. For more details about analytical solutions see Additional
information.

(a)

(b)

(c)

(d)

Figure 1: Numerically calculated (dots) and theoretically simulated (lines)
amplitude and phase responses of the SERF (red) and co-magnetometer (blue)
magnetic (a), electron nonmagnetic (b), neutron nonmagnetic (c), and proton
nonmagnetic (d) perturbations.

A key feature of the co-magnetometer in searches for pseudo-magnetic spin
couplings is its suppressed sensitivity to low-frequency magnetic-field
perturbations. This is clearly visible in the data presented in Fig. 1(a). At
the lowest frequencies, the response amplitude of the co-magnetometer to the
magnetic field is roughly four orders of magnitude lower than the response of
the SERF magnetometer. This difference decreases at higher frequencies
reducing to zero at about 4 Hz. Since compensation is provided by the NG,
which adiabatically follows the field changes, and, at the compensation point,
the atoms are only experiencing a field proportional to their own
magnetisation, $B^{i}\approx B_{c}+\lambda
M^{i^{\prime}}P^{i^{\prime}}_{0}=\lambda M^{i}P^{i}_{0}$, the frequency at
which the SERF and co-magnetometer magnetic responses are equal is determined
by the NG Larmor frequency. Shifting the compensation point towards higher
frequencies requires increasing the NG magnetisation, which can be achieved by
either increasing the NG concentration or the polarisation. Both may be
challenging experimentally. For higher frequencies the response of both
systems has the same amplitude, since outside of the self-compensating regime
the co-magnetometer response is predominantly determined by the AM. Thereby,
the AM polarisation in the co-magnetometer starts to behave in the same way as
the free-AM polarisation in a usual SERF magnetometer.

There is a remarkable difference in the phase response of the two devices
[Fig. 1(a)]. For magnetic field frequencies below the NG Larmor frequency (4
Hz), the response of the co-magnetometer is phase shifted by about $\pi$, and
it decreases sharply above that frequency, eventually becoming similar for
both devices. For lower frequencies, the difference is due to the NG that
compensates the magnetic field and no such compensation is present in the SERF
magnetometer. For higher field frequencies the magnetic response of both
devices is determined by the AM, so the observed dependencies are similar.

When analysing the response of the co-magnetometer to nonmagnetic nuclear
perturbation, it should be first noted that, unlike in the case of magnetic
perturbation, the nonmagnetic nuclear perturbations are not compensated.
Therefore, in that case, there is no reduction of the response amplitude at
lower frequencies. To the contrary, the amplitude response to pseudo-magnetic
nuclear perturbations of the co-magnetometer is significantly stronger than
the response of the SERF magnetometer [Fig. 1(c)&(d)]. In particular, for
frequencies below 4 Hz, the co-magnetometer response is roughly five orders of
magnitude stronger for the neutron nonmagnetic perturbation [Fig. 1(c)], about
three orders of magnitude stronger for the proton perturbation [Fig. 1(d)] and
even though it deteriorates for higher frequencies, it still remains
significantly larger than for the SERF system. Such dissimilarity is related
to the difference in effective pseudo-magnetic fields experienced by NG and AM
spins [see Eqs. (10)]. Note, that the co-magnetometer has high sensitivity to
the nuclear spin perturbations. Specifically, because of the high
concentration of the NG, the response of the co-magnetometer is predominantly
determined by the gas. In turn, the large concentration difference between the
NG and AM concentration (about six order of magnitude) is responsible for much
higher sensitivity of the former to the nonmagnetic nuclear couplings.
Moreover, the high concentration and hence the high, compared to AM,
magnetisation of the NG atoms (despite the 5% polarisation of NG) ensures
efficient transfer of the NG-spin perturbation to the AM polarisation.
Therefore, the NG magnetisation-mediated nuclear coupling significantly
increases the amplitude of the response of the co-magnetometer to nonmagnetic
nuclear perturbations. An additional cause of the difference in the response
to nuclear perturbations stems from the different nuclear spin contents of the
39K and the 3He nuclei [9]. Contribution of the proton polarisation in 39K is
roughly four times larger than in the case of 3He. In contrast, the neutron
polarisation has an about 24 times bigger contribution to the nuclear spin of
3He, than it contributes to the nuclear spin of 39K. The difference in neutron
and proton spin fractions in 3He nuclear spin also leads to different response
amplitudes of the co-magnetometer to proton and neutron perturbations.

The phase response of the co-magnetometer is similar for both the neutron and
proton pseudo-magnetic perturbations. Specifically, below 4 Hz, the phase
shift between perturbation and response is close to $\pi$ and it drops to
about zero for higher frequencies. While for frequencies above 100 Hz the two
responses differ, it should be noted that this frequency range is well beyond
the bandwidth of the co-magnetometer, where amplitude of the response drops by
several orders of magnitude. At the same time, the phase response of the SERF
magnetometer is the same for magnetic and nuclear nonmagnetic perturbations,
being zero at lower frequencies and monotonically shifting toward $-\pi/2$ for
frequencies beyond the bandwidth of the magnetometer.

The response of both devices to the electron nonmagnetic perturbation is
similar at most of the frequencies with a distinct exception of the frequency
corresponding to the NG Larmor frequency (4 Hz) [Fig. 1(b)]. Such a behaviour
is not surprising since, in both cases, the electron coupling perturbs the AM
electron spins. On the other hand, differences in the response of the systems
at the NG Larmor frequency arise due to the coupling between the perturbed AM
and the NG atoms.

### Co-magnetometer response to transients

It was shown in the previous section that the response of the co-magnetometer
to nonmagnetic nuclear couplings is much stronger than that of the SERF
magnetometer. Therefore, below we only focus on analysis of the response of
the co-magnetometer to transient magnetic and nonmagnetic perturbations.

As a generic example, we take a temporal Lorentzian perturbation of the
amplitude $\Lambda_{0}$ and half-width $\Delta t$, centred at the time
$t_{0}$, which is directed along the $\mathbf{y}$ axis (this can be easily
generalised for any pulse shape)

$\bm{\Lambda_{t}}=\mathbf{y}\frac{\Lambda_{0}\Delta
t^{2}}{(t-t_{0})^{2}+\Delta t^{2}}.$ (13)

Such a definition allows keeping the amplitude of the pulse constant while
varying its width (note that here the energy of the pulse is not preserved).
Since here we are only interested in temporal parameters of the response, the
proton and nuclear couplings are not considered independently but they are
treated as a generic nuclear coupling. We simulated 200-s long responses of
the co-magnetometer to the perturbations of different origin and the shape
given by Eq. (13), centred at $t_{0}=100$ s. We assumed the same amplitude of
pulses in effective pseudo-magnetic field units for all types of the
perturbation to be small enough that the co-magnetometer continuously operates
in self-compensating regime. For all results we numerically calculated the
integral over the response of the system and the energy of the signal within
the simulated window. Results, presented in Fig. 3, are discussed below in
details.

(a)

(b)

Figure 2: (a) Temporal responses of the co-magnetometer to pseudo-magnetic
electron (dotted line), pseudo-magnetic nuclear (solid line) and magnetic
(dashed line) spin perturbations. The bottom subplot shows the 50-ms
Lorentzian-shaped perturbation common for all types of couplings. (b) Spectra
of the co-magnetometer responses to pseudo-magnetic electron (dotted line),
pseudo-magnetic nuclear (solid line) and magnetic (dashed line) spin
perturbations assuming the same amplitude of the perturbations in effective
pseudo-magnetic magnetic field units. For the reference, the bottom subplot
shows spectrum of 50-ms Lorentzian pulses. All spectra were obtained with Fast
Fourier Transform of 200-s long time series of the responses and perturbation.

In the previous section, we have shown that the co-magnetometer frequency
responses for magnetic and nonmagnetic spin couplings are different. In
particular, the compensation of low-frequency magnetic field in the co-
magnetometer results in a suppression of low-frequency components of the pulse
[Fig. 2(b)(top)] manifesting itself in a response of the co-magnetometer to
perturbations of different nature [Fig. 2(a)(top)]. The results show that the
response significantly deviates from the Lorentzian shape of the magnetic
pulse; the pulse is slightly longer and its shape is significantly distorted.
At the same time, the distortion is much smaller both in the case of electron
and nuclear perturbations, which is a manifestation of the absence of such
compensation for pseudo-magnetic spin interaction. Suppression of the low-
frequency components leads to a weaker response of the device to the magnetic
pulse with a spectrum within the self-compensation band. This is shown in Fig.
3, where the pulse integral and its energy are presented versus the pulse
length. On contrary, no such behaviour is observed for nonmagnetic pulses,
where low-frequency components are not suppressed [Fig. 2(b)(top)]. These
results are confirmed with the simulations for a wide range of pulse widths;
the longer the pulse, the more prominent is the divergence in energy between
the response to magnetic and nonmagnetic perturbations [Fig. 3]. For 1-s long
pulses, the difference is more than five orders of magnitude, and it grows for
longer pulses. As for the pulse widths around 0.01 s, the response energy is
comparable for all types of perturbations, since for the short pulses, a
significant fraction of the pulse spectrum is at higher frequencies, where
response is comparable and large for both magnetic and nonmagnetic
perturbations.

The suppression of the low-frequency magnetic field in the co-magnetometer
provides another remarkable feature of the system; the integral over the co-
magnetometer signal is significantly suppressed for magnetic field
perturbations when integrated over time intervals significantly longer than
the pulse width (for the presented results 200-s-long integration window have
been used). In contrast, the integral is finite for nonmagnetic spin
perturbations. For the simulated co-magnetometer system, assuming that the
value of the effective pseudo-magnetic field is the same as the magnetic
field, the difference between integrals over detected magnetic and nonmagnetic
transients is about seven orders of magnitude for pulse widths between 0.01 s
to 1 s [Fig. 3].

Figure 3: Energy and absolute value of the integral over temporal response of
the co-magnetometer to the Lorentzian pulses of different width and coupling
origin: electron nonmagnetic (dotted lines), nuclear nonmagnetic (solid
lines), and magnetic (dashed lines) spin perturbations. Presented results are
based on numerical calculation of the 200-s long co-magnetometer responses to
the different-origin pulse perturbations. The shape of the pulses is given by
Eq. (13) and is centred at $t_{0}=100$ s, i.e., the centre of the simulation
region. For all types of pulses, we assume the same amplitude given in the
effective pseudo-magnetic field units.

## Conclusions

Numerical simulations of SERF and AM-NG co-magnetometers show a significantly
stronger response of the latter to nuclear nonmagnetic spin perturbations.
While the enhancement of the response due to the proton-coupling stems from
the high sensitivity of the co-magnetometer to the nuclear spin perturbations,
a larger contribution of the neutron polarisation to the NG polarisation
provides an additional enhancement of the co-magnetometer response to neutron
pseudo-magnetic perturbation. At the same time, the response of both devices
to the electron nonmagnetic spin perturbations is similar.

Our results demonstrate benefits of the co-magnetometer in searches for
transient pseudo-magnetic spin couplings. On one hand, there is a suppressed
response to low-frequency magnetic fields, which reduces noise of the device,
on the other, due to “high-pass” magnetic-filter nature of the co-
magnetometer, the device allows to differentiate between the magnetic and
nonmagnetic transient responses, enabling a new way of identification of the
observed signal nature. Specifically, the integral over time series signal for
magnetic pulses has very small value, while it remains finite for nonmagnetic
pulses. We also demonstrated that over a wide range of the pulse widths the
energy of response to nonmagnetic perturbation is orders of magnitude greater
than to magnetic-field perturbation. Even thought, the energy difference
decreases for shorter pulse length, hence pulse energies, the difference in
the integrals over the responses to nonmagnetic and magnetic perturbations
remains large (several orders of magnitude) for the pulses which can be
detected with the co-magnetometer. The features of the co-magnetometer
presented at this work demonstrate the capabilities of the co-magnetometer in
searches for transient nonmagnetic spin couplings, i.e., the signals that are
being searched by the GNOME.

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## Acknowledgements

The authors thank Derek Jackson Kimball for useful discussions. Preliminary
work in the directions discussed in the manuscript was carried out by Samer
Afach. The work of MP and MK was supported by the Jagiellonian University in
Krakow, Faculty of Physics Astronomy and Applied Computer Science internal
grant (N17/MNS/000007/2020). SP would like to acknowledge support of the
National Science Centre of Poland within the OPUS programme. The work of DB
was supported by the Cluster of Excellence “Precision Physics, Fundamental
Interactions, and Structure of Matter” (PRISMA+ EXC 2118/1) funded by the
German Research Foundation (DFG) within the German Excellence Strategy
(Project ID 39083149), by the European Research Council (ERC) under the
European Union Horizon 2020 research and innovation program (project Dark-OST,
grant agreement No 695405), and by the DFG Reinhart Koselleck project. The
work of AW was supported by the German Federal Ministry of Education and
Research (BMBF) within the Quantumtechnologien program (Grant No. 13N15064).

## Author contributions statement

M.P. defined a numerical models, conducted simulations, analysed the results,
constructed analytical solution and wrote the manuscript, M.K. participated
into definition of a numerical models and edited manuscript, S.P. conceived
and supervised the project, D.B., R.C., S.P., and A.W. analysed the results,
discussed their interpretation and edited the manuscript. All authors reviewed
the manuscript.