# Dissecting axion and dark photon with a network of vector sensors

Yifan Chena Min Jiangb,c,d Jing Shua,e,f,g Xiao Xueh Yanjie Zenga,e aCAS Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, P.R.China
bHefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China
cCAS Key Laboratory of Microscale Magnetic Resonance, University of
Science and Technology of China, Hefei, Anhui 230026, China
dSynergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
eSchool of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
fSchool of Fundamental Physics and Mathematical Sciences, Hangzhou Institute
for Advanced
Study, University of Chinese Academy of Sciences, Hangzhou 310024, China
gInternational Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou,
China
hII. Institute of Theoretical Physics, Universität Hamburg, 22761 Hamburg,
Germany

###### Abstract

We develop formalisms for a network of vector sensors, sensitive to certain
spatial components of the signals, to identify the properties of a light axion
or a dark photon background. These bosonic fields contribute to vector-like
signals in the detectors, including effective magnetic fields triggering the
spin precession, effective electric currents in a shielded room, and forces on
the matter. The interplay between a pair of vector sensors and a baseline that
separates them can potentially uncover rich information of the bosons,
including angular distribution, polarization modes, source localization, and
macroscopic circular polarization. Using such a network, one can identify the
microscopic nature of a potential signal, such as distinguishing between the
axion-fermion coupling and the dipole couplings with the dark photon.

##  I Introduction

Bosons in the sub-eV mass range can be natural dark matter candidates. These
non-relativistic bosonic fields behave like coherent waves within the
correlation time and distance. Among these, axion dark matter Preskill  _et
al._ (1983); Abbott and Sikivie (1983); Dine and Fischler (1983) is strongly
motivated. As a pseudo-scalar, axion can solve the strong charge-conjugation
and parity (CP) problem in quantum chromodynamics (QCD) Peccei and Quinn
(1977). In addition to the QCD-axion, axion-like particles (ALPs) also appear
generically in theories with extra dimensions Arvanitaki  _et al._ (2010).
Dark photon dark matter Nelson and Scholtz (2011) is another popular candidate
for the bosonic wave-like dark matter. Similar to the axion, it can originate
from the string compactification Abel  _et al._ (2008); Goodsell  _et al._
(2009) and is produced from misalignment.

In addition to being non-relativistic dark matter, light bosons can also exist
in other forms in the universe. For example, non-virialized cold streams
O’Hare and Green (2017); Foster  _et al._ (2018); Knirck  _et al._ (2018),
dipole radiations from binary systems charged under a hidden $\displaystyle
U(1)$ Krause  _et al._ (1994); Dror  _et al._ (2021a); Hou  _et al._ (2022),
and emissions from gravitational atoms Baryakhtar  _et al._ (2021); East
(2022) contribute to highly anisotropic bosonic waves from a specific
direction. Some relativistic degrees of freedom can remain in the current
universe as well, forming a cosmological background Baumann  _et al._ (2016);
Dror  _et al._ (2021b).

The detection of these bosonic background fields is attracting a growing
number of different experiments. A natural step forward is to construct a
network of detectors. Besides cross-checking and distinguishing the signals
from spurious noise, a multi-mode resonant system with
$\displaystyle\mathcal{PT}$ symmetry can significantly enhance the signal
power and the scan rate Li  _et al._ (2020); Chen  _et al._ (2022). A pair of
detectors with a long baseline can read the directional information of the
dark matter Derevianko (2018); Foster  _et al._ (2021) and opens a new window
in multi-messenger astronomy Dailey  _et al._ (2021). These works are based on
scalar-like observables, such as the time-derivative of the axion field in
electromagnetic haloscopes Sikivie (1983, 1985); Sikivie  _et al._ (2014);
Chaudhuri  _et al._ (2015); Berlin  _et al._ (2020); Bartram  _et al._ (2021);
Kwon  _et al._ (2021); Backes  _et al._ (2021); McAllister  _et al._ (2017);
Salemi  _et al._ (2021).

On the other hand, there are types of experiments sensitive to vector-like
signals, including effective magnetic fields leading to spin precession Graham
and Rajendran (2013); Budker  _et al._ (2014); Stadnik and Flambaum (2014);
Barbieri  _et al._ (2017); Abel  _et al._ (2017); Stadnik (2017); Jiang  _et
al._ (2021a); Wu  _et al._ (2019); Garcon  _et al._ (2019); Smorra  _et al._
(2019); Mitridate  _et al._ (2020); Chigusa  _et al._ (2020); Aybas  _et al._
(2021); Jiang  _et al._ (2021b); Kim  _et al._ (2021); Poddar (2021), that can
capture axion-fermion coupling or dipole couplings with a dark photon Graham
_et al._ (2018), effective electric currents from kinetic-mixing dark photons
Chaudhuri  _et al._ (2015); Fedderke  _et al._ (2021a, b), and forces brought
from $\displaystyle U(1)_{B-L}/U(1)_{B}$ dark photons Graham  _et al._ (2016);
Pierce  _et al._ (2018); Carney  _et al._ (2021); Guo  _et al._ (2019a).
Taking the axion-fermion coupling as an example, in the non-relativistic limit
of the fermion, one has vector-like coupling between the spatial derivative of
the axion and the spin operator of the fermion,

$g_{a}\partial_{\mu}a\bar{\psi}\gamma^{\mu}\gamma^{5}\psi\rightarrow
g_{a}\vec{\nabla}a\cdot\vec{\sigma},$ (1)

where $\displaystyle a$ and $\displaystyle\psi$ are the axion and the fermion
field, respectively, with coupling constant $\displaystyle g_{a}$, and
$\displaystyle\vec{\sigma}$ is the spin operator of the fermion. The vector-
like signal in this case is the axion wind/gradient
$\displaystyle\vec{\nabla}a$, which can be detected through nuclear magnetic
resonance Graham and Rajendran (2013); Budker  _et al._ (2014); Garcon  _et
al._ (2019); Aybas  _et al._ (2021), Floquet masers Jiang  _et al._ (2021a),
comagnetometers Abel  _et al._ (2017); Wu  _et al._ (2019); Bloch  _et al._
(2020), magnons Barbieri  _et al._ (2017); Mitridate  _et al._ (2020); Chigusa
_et al._ (2020), a Penning trap Smorra  _et al._ (2019), or spin-based
amplifiers Jiang  _et al._ (2021b); Su  _et al._ (2021).

An outstanding question to ask then is, what information can one extract by
correlating a pair of vector sensors? Since a single vector sensor can in
principle probe any spatial direction by manipulating the configuration of the
detector, one expects the interplay between the two sensitive directions and
the baseline that separates them to bring a correlation matrix

$\langle(\vec{\mathcal{O}}(t_{1},\vec{x}_{1})\cdot\hat{l}_{1})(\vec{\mathcal{O}}(t_{2},\vec{x}_{2})\cdot\hat{l}_{2})\rangle,$
(2)

where $\displaystyle\hat{l}_{1},\hat{l}_{2}$ are the sensitive directions of
the two vector sensors and $\displaystyle\vec{\mathcal{O}}$ are the vector-
like signals recorded at different times and locations by the corresponding
detectors. $\displaystyle\langle\cdots\rangle$ denotes the ensemble average on
the bosonic fields. As we will discuss, the correlation function (2) can bring
important information about the bosonic background, including angular
distribution, polarization modes, coupling types, the exact direction of the
origin, and macroscopic circular polarization. With these, one can easily
identify the microscopic nature of a potential signal. Notice that there are
already many vector sensors operating at the same time in The Global Network
of Optical Magnetometers for Exotic physics (GNOME) Pospelov  _et al._ (2013);
Pustelny  _et al._ (2013); Afach  _et al._ (2018, 2021).

The layout of the paper is as follows. In Sec. II, we review several types of
coupling that contribute to the vector-like signals in terms of stochastic
wave functions of the bosons, and we introduce the linear and circular
polarization basis of the vector sensors. In Sec. III, we generalize the
scalar sensor interferometry to a network of vector sensors where the two-
point correlations of the signals are equipped with two additional labels of
spatial component, and we discuss the universal dipole angular correlation for
isotropic sources. In Sec. IV, we discuss the identifications of longitudinal
and transverse modes for incoming and isotropic sources, and different types
of dark matter contributing to vector-like signals. In Sec. V, we calculate
the angular resolution for incoming sources with different polarization modes
when the vector sensors are set to the optimal directions in terms of the
baseline and the incoming source. In Sec. VI, we show how to identify a
macroscopic circular polarization for an isotropic source. Section VII
contains the conclusion and future prospects.

##  II Vector Sensor Response to Axion Gradient and Dark Photon

###  II.1 Non-relativistic Dipole Coupling

We first consider the dipole couplings between light bosons to the spins of
fermions. In the non-relativistic limit of the fermions, these dipole
couplings can be expressed universally as an effective vector-like coupling
with the fermion’s spin operator $\displaystyle\vec{\sigma}$,

$H=\alpha\,\vec{\mathcal{O}}\cdot\vec{\sigma}.$ (3)

where $\displaystyle\alpha$ is the coupling constant and
$\displaystyle\vec{\mathcal{O}}$ is the vector-like signals probed.

Up to dimension-five operators are considered, which become the form of Eq.
(3) in the non-relativistic limit of the fermions, including axion-fermion
coupling,

$\displaystyle\displaystyle
g_{a}\partial_{\mu}a\bar{\psi}\gamma^{\mu}\gamma^{5}\psi$
$\displaystyle\displaystyle\rightarrow\vec{\mathcal{O}}_{a}=\vec{\nabla}a,$
(4)

and dipole interactions between dark photon and the fermions

$\displaystyle\displaystyle g_{A}V_{\mu}\bar{\psi}\gamma^{\mu}\gamma^{5}\psi$
$\displaystyle\displaystyle\rightarrow\vec{\mathcal{O}}_{A}=\vec{V},$ (5)
$\displaystyle\displaystyle
g_{\textrm{MDM}}V_{\mu\nu}\bar{\psi}\sigma^{\mu\nu}\psi$
$\displaystyle\displaystyle\rightarrow\vec{\mathcal{O}}_{\textrm{MDM}}=\vec{\nabla}\times\vec{V},$
(6) $\displaystyle\displaystyle
g_{\textrm{EDM}}V_{\mu\nu}\bar{\psi}\sigma^{\mu\nu}i\gamma^{5}\psi$
$\displaystyle\displaystyle\rightarrow\vec{\mathcal{O}}_{\textrm{EDM}}=\partial_{0}\vec{V}-\vec{\nabla}V^{0},$
(7)

where $\displaystyle g_{a}$, $\displaystyle g_{A}$, $\displaystyle
g_{\textrm{EDM}}$, and $\displaystyle g_{\textrm{MDM}}$ are the coupling
constants for the axion, axial-vector, electric dipole momentum (EDM) and
magnetic dipole momentum (MDM) dark photon respectively. $\displaystyle a$ is
the axion field, and $\displaystyle V_{\mu}$ and $\displaystyle V_{\mu\nu}$
are the vector fields and their field strength, respectively.
$\displaystyle\sigma^{\mu\nu}$ is the anti-symmetric tensor constructed from
Dirac matrix $\displaystyle\gamma^{\mu}$. Notice that in the case of the axion
gradient coupling (4), $\displaystyle\vec{\mathcal{O}}$ behaves as a
longitudinal mode in terms of the spatial momentum. In contrast, only
transverse modes of the MDM dark photon contribute to the interaction in Eq.
(6). For the axial vector, all the polarization modes take part in the
interaction in Eq. (5). Interactions with the EDM dark photon differ between
the relativistic and the non-relativistic limits of the dark photon in Eq.
(7). In the former case, only the transverse modes interact, while all
contribute to the interaction when being non-relativistic.

Operators with dimension higher than 5 always become one of the forms above in
the non-relativistic limit. For example, an anapole dark photon, with
interaction $\displaystyle
g_{\textrm{Ana}}(\partial^{\nu}V_{\mu\nu})\bar{\psi}\gamma^{\mu}\gamma^{5}\psi$,
takes the same form as the axial-vector in Eq. (5) when the dark photon is on-
shell.

For more general cases of the dark photon, one can also use Eq. (3) to
parametrize the detector response for a certain spatial direction. For
example, the $\displaystyle U(1)_{B-L}/U(1)_{B}$ dark photon leads to an
additional force that accelerometers, optomechanical systems, or astrometry
can sense Graham  _et al._ (2016); Pierce  _et al._ (2018); Carney  _et al._
(2021); Guo  _et al._ (2019a). The kinetic-mixing dark photon induces an
effective current which can be captured by LC circuits in a shielded room
Chaudhuri  _et al._ (2015) or read from the geomagnetic field Fedderke  _et
al._ (2021a, b). In the former case, the force signal shares the same form as
the one of the EDM in Eq. (7), while the effective current is proportional to
the spatial component of the dark photon wave functions.

###  II.2 Vector-like Signals from Axion and Dark Photon Background

Bosonic fields can exist in the universe with different kinds of momentum
distribution. For example, axion or dark photon with mass $\displaystyle
m<\mathcal{O}(1)$ eV can be cold dark matter candidates Preskill  _et al._
(1983); Abbott and Sikivie (1983); Dine and Fischler (1983); Nelson and
Scholtz (2011), behaving like coherent waves within the correlation time and
distance that are both determined by virial velocity. A cosmological axion
Baumann  _et al._ (2016); Dror  _et al._ (2021b) and a dark photon can also
exist with the spectrum dependent on the production mechanism and the
universe’s thermal history. Furthermore, one expects bosonic waves to come
from a specific direction towards the Earth, such as non-virialized
substructure of the dark matter O’Hare and Green (2017); Foster  _et al._
(2018); Knirck  _et al._ (2018), dipole radiations from binary systems charged
by hidden $\displaystyle U(1)$ Krause  _et al._ (1994); Dror  _et al._
(2021a); Hou  _et al._ (2022), and emissions from strongly self-interacting
gravitational atoms Baryakhtar  _et al._ (2021); East (2022). In this
subsection, we construct the bosonic backgrounds and the corresponding vector-
like signals using effective polarization modes.

Following Foster  _et al._ (2018); Derevianko (2018); Foster  _et al._ (2021);
Guo  _et al._ (2019b) and the relativistic generalization in Dror  _et al._
(2021b), one expands general scalar or vector fields with $\displaystyle
N_{b}$ non-interacting waves

$\displaystyle\displaystyle a(t,\vec{x}\,)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{j=1}^{N_{b}}\sqrt{\frac{2\rho}{N_{b}\bar{\omega}\omega_{j}}}\cos\left[\omega_{j}t-\vec{p}_{j}\cdot\vec{x}+\alpha_{j}\right],$
(8) $\displaystyle\displaystyle\vec{V}(t,\vec{x}\,)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{j=1}^{N_{b}}\sqrt{\frac{2\rho}{N_{b}\bar{\omega}\omega_{j}}}\text{Re}\left[\mathrm{e}^{-\mathrm{i}\left(\omega_{j}t-\vec{p}_{j}\cdot\vec{x}+\alpha_{j}\right)}\
\vec{\epsilon}_{\beta}^{\,j}\right],$ (9)

where momentum $\displaystyle\vec{p}_{j}$ are random variables drawn from the
momentum distribution $\displaystyle f(\vec{p}\,)$,
$\displaystyle\omega_{j}=\sqrt{m^{2}+p_{j}^{2}}$ are the energy in which
$\displaystyle p_{j}\equiv|\vec{p}_{j}|$, $\displaystyle\bar{\omega}\equiv\int
d^{3}\vec{p}\,f(\vec{p})\,\omega$ is the average energy, and
$\displaystyle\rho$ is the local energy density. $\displaystyle\alpha_{j}$ are
random phases uniformly distributed within $\displaystyle[0,2\pi)$ for a
stochastic background that we focus on in this study. $\displaystyle\beta$ are
discrete random variables following the probabilities of different
polarization modes $\displaystyle\vec{\epsilon}_{\beta}$.

Generally, a massive vector field contains three degrees of freedom. In the
unitary gauge, these include the longitudinal mode

$\vec{\epsilon}_{0}=\frac{\omega}{m}\hat{e}_{p}\equiv\frac{\omega}{m}\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\right)^{\textrm{T}},$
(10)

and two transverse ones in the linear polarization basis,

$\displaystyle\displaystyle\vec{\epsilon}_{H}=\hat{e}_{\theta}$
$\displaystyle\displaystyle\equiv\left(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta\right)^{\textrm{T}},$
(11) $\displaystyle\displaystyle\vec{\epsilon}_{V}=\hat{e}_{\phi}$
$\displaystyle\displaystyle\equiv\left(-\sin\phi,\cos\phi,0\right)^{\textrm{T}},$
(12)

where $\displaystyle\hat{e}_{p},\hat{e}_{\theta}$ and
$\displaystyle\hat{e}_{\phi}$ are three orthogonal unit directional vectors.
Alternatively, one can express the transverse ones using the circular
polarization basis

$\vec{\epsilon}_{R/L}=\frac{1}{\sqrt{2}}\left(\vec{\epsilon}_{H}\pm\mathrm{i}\
\vec{\epsilon}_{V}\right).$ (13)

Now the vector observables $\displaystyle\vec{\mathcal{O}}$ in Eq. (4-7) can
be written as a sum of different polarization modes depending on a specific
type of coupling,

$\displaystyle\displaystyle\vec{\mathcal{O}}_{a}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{j=1}^{N_{b}}\,\frac{m\,p_{j}}{\omega_{j}}\,\Phi_{j}\cos\left[\Psi_{j}\right]\,\vec{\epsilon}_{0}^{\,j},$
(14) $\displaystyle\displaystyle\vec{\mathcal{O}}_{A}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{j=1}^{N_{b}}\,\Phi_{j}\,\text{Re}\left[\mathrm{e}^{-\mathrm{i}\Psi_{j}}\,\vec{\epsilon}_{\beta}^{\,j}\right],$
(15) $\displaystyle\displaystyle\vec{\mathcal{O}}_{\textrm{MDM}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{j=1}^{N_{b}}\,p_{j}\,\Phi_{j}\,\text{Re}\left[\mathrm{e}^{-\mathrm{i}\Psi_{j}}\
\vec{\epsilon}_{R/L}^{\,j}\right],$ (16)
$\displaystyle\displaystyle\vec{\mathcal{O}}_{\textrm{EDM}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{j=1}^{N_{b}}\,\begin{cases}m\,\Phi_{j}\,\text{Re}\left[\mathrm{e}^{-\mathrm{i}\Psi_{j}}\,\vec{\epsilon}_{\beta}^{\,j}\right],&m\gg|p_{j}|,\\\
\omega_{j}\,\Phi_{j}\,\text{Re}\left[\mathrm{e}^{-\mathrm{i}\Psi_{j}}\,\vec{\epsilon}_{R/L}^{\,j}\right],&m\ll|p_{j}|,\end{cases}{}$

where one uses the constraint $\displaystyle\partial_{\mu}V^{\mu}=0$ of the
unitary gauge to express $\displaystyle V^{0}$ in Eq. (7).
$\displaystyle\Phi_{j}\equiv\sqrt{2\rho/N_{b}\bar{\omega}\omega_{j}}$ and
$\displaystyle\Psi_{j}\equiv\omega_{j}t-\vec{p}_{j}\cdot\vec{x}+\alpha_{j}$
are defined for simplicity. One can thus treat the axion gradient effectively
as the longitudinal-mode-only case of the dark photon.

###  II.3 Circular and Linear Polarization Basis of Vector Sensors

In this subsection, we start with a review on detecting effective magnetic
fields from the spin precession and explain why the detectors are in the
circular polarization basis, in analogy with the concept in radio astronomy.
Thus one can transform it to the linear polarization basis with two systems
polarized at opposite directions operating simultaneously, probing a specific
spatial component of the vector-like signals on the transverse plane.

The couplings between the dark sectors and the spins of nucleons in Eq. (3)
can be detected through nuclear magnetic resonance Graham and Rajendran
(2013); Budker  _et al._ (2014); Garcon  _et al._ (2019); Aybas  _et al._
(2021), Floquet masers Jiang  _et al._ (2021a), or spin-based amplifiers Jiang
_et al._ (2021b); Su  _et al._ (2021). In such cases, the nucleons are
initially polarized along a specific direction, and the signals
$\displaystyle\vec{\mathcal{O}}$ on the transverse directions can be captured.
Without loss of generality, taking the initial polarized spin
$\displaystyle\vec{\sigma}_{0}$ along $\displaystyle+\hat{z}$, one can
describe the dynamics of the spin operators $\displaystyle\vec{\sigma}$ using
the Bloch equation

$\frac{d\vec{\sigma}}{dt}=\vec{\sigma}\times\left(\gamma\vec{B}_{0}-\alpha\vec{\mathcal{O}}\right)-\frac{\vec{\sigma}-\vec{\sigma}_{0}}{T_{r}},$
(18)

where $\displaystyle\gamma$ is the gyromagnetic ratio,
$\displaystyle\vec{B}_{0}=B_{0}\hat{z}$ is the background magnetic field along
$\displaystyle+\hat{z}$, and $\displaystyle T_{r}$ is the relaxation time of
the system. One way to see the response to the signal is to decompose the
transverse directions in terms of
$\displaystyle\sigma_{\perp}\equiv\left(\sigma_{x}+\mathrm{i}\sigma_{y}\right)/\sqrt{2}$,
which leads to

$\left(\frac{d}{dt}+\mathrm{i}\gamma
B_{0}+\frac{1}{T_{r}}\right)\sigma_{\perp}=-\mathrm{i}\alpha\sigma_{0}\left(\frac{\mathcal{O}_{x}+\mathrm{i}\mathcal{O}_{y}}{\sqrt{2}}\right),$
(19)

where $\displaystyle\gamma B_{0}$ sets the resonant frequency, and
$\displaystyle 1/T_{r}$ denotes the dissipation.
$\displaystyle\left(\mathcal{O}_{x}+\mathrm{i}\mathcal{O}_{y}\right)/\sqrt{2}$
are the sensitive signals, which are clearly in the circular polarization
basis. For transverse modes propagating along the
$\displaystyle+\hat{z}$-axis, only the left-hand circular polarization mode
$\displaystyle\epsilon_{L}$ can excite the detector. For the longitudinal
mode, $\displaystyle\epsilon_{0}$, only the projection to the transverse plane
gets a response, and one cannot distinguish its component along
$\displaystyle\hat{x}$ and $\displaystyle\hat{y}$.

Two detectors with opposite initially polarized directions and the same
magnetic field strength $\displaystyle B_{0}$ can transform the circular
polarization basis into the linear one. More explicitly, one sums up
simultaneously the signals from the detectors polarized along
$\displaystyle+\hat{z}$ and $\displaystyle-\hat{z}$ with relative phase shift
$\displaystyle\pi/2$ and gets the response to a specific spatial direction on
the transverse plane,

$\displaystyle\displaystyle\left(\frac{d}{dt}+\mathrm{i}\gamma
B_{0}+\frac{1}{T}\right)\left(\frac{\sigma_{\perp}^{\uparrow}+\mathrm{i}\sigma_{\perp}^{\downarrow}}{\sqrt{2}}\right)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\mathrm{i}\alpha\sigma_{0}\mathcal{O}_{x},$ (20)
$\displaystyle\displaystyle\left(\frac{d}{dt}+\mathrm{i}\gamma
B_{0}+\frac{1}{T}\right)\left(\frac{\sigma_{\perp}^{\uparrow}-\mathrm{i}\sigma_{\perp}^{\downarrow}}{\sqrt{2}}\right)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\alpha\sigma_{0}\mathcal{O}_{y},$ (21)

where $\displaystyle\sigma_{\perp}^{\uparrow/\downarrow}$ are the transverse
responses for detectors polarized along $\displaystyle+\hat{z}$ and
$\displaystyle-\hat{z}$ respectively.

This transformation of the response basis also applies to detectors using
polarized electrons, such as the axion-magnon conversion Barbieri  _et al._
(2017); Mitridate  _et al._ (2020); Chigusa  _et al._ (2020). For the
$\displaystyle U(1)_{B-L}/U(1)_{B}$ dark photon or the kinetic-mixing dark
photon, as mentioned in Sec. II.1, the detector responses to them are in the
linear polarization basis.

##  III From the Scalar Sensor Interferometry to Vector Sensor Network

This section starts with a review on the dark matter interferometry developed
in Foster  _et al._ (2021), where directional information can be extracted by
correlating two detectors separated by a baseline whose length is comparable
to the de Broglie wavelength. We generalize the formalism to an array with
multiple vector sensors, each of which is sensitive to a certain spatial
component of the observables, as shown in Sec. II. The covariance matrix,
which contains the information of the correlations between each pair of
sensors, is expanded with additional $\displaystyle 3\times 3$ labels on the
sensors’ sensitive directions.

###  III.1 Scalar Sensor Interferometry

One starts with $\displaystyle\mathcal{N}_{d}$ detectors for scalar-like
observables labeled with $\displaystyle I,J,\cdots$ and operates them
simultaneously over time $\displaystyle T$. A discrete Fourier transform to
the frequency space, indexed by an integer $\displaystyle k$, gives the
expected response of the $\displaystyle I$th detector,

$\tilde{d}_{I}^{k}(\theta_{\rm
sig})=A_{I}(\omega_{k})\,\tilde{a}\left(\omega_{k},\vec{x}_{I}\right)+\tilde{n}_{I}\left(\omega_{k}\right),$
(22)

where $\displaystyle A_{I}$ is the response functions of the detector that
absorbs the coupling constant, $\displaystyle\tilde{a}$ comes from the
transformation of the axion field (8) to the frequency space, and
$\displaystyle\tilde{n}_{I}$ is the Gaussian noise at frequency
$\displaystyle\omega_{k}=2\pi k/T$. $\displaystyle\theta_{\rm sig}$ contains
all the relevant signal parameters that characterize the axion contribution,
for example, axion mass $\displaystyle m$ and the momentum distribution
$\displaystyle f(\vec{p}\,)$ that we assume to be the same within the network
of the detectors in this study. Since Eq. (22) is a Gaussian random variable
with zero mean, one can construct the multi-detector covariance matrix

$\Sigma_{k}(\theta_{\rm sig})=\langle\tilde{d}^{k}\tilde{d}^{k\dagger}\rangle$
(23)

in terms of the $\displaystyle\mathcal{N}_{d}$-dimensional vector

$\tilde{d}^{k}\equiv\left(\tilde{d}_{1}^{k},\tilde{d}_{2}^{k},\cdots,\tilde{d}_{\mathcal{N}_{d}}^{k}\right)^{T},$
(24)

where $\displaystyle\langle\cdots\rangle$ denotes the ensemble average on the
stochastic fields and the statistical average on the detector noise. We leave
the discussion of the ensemble average on the stochastic fields to the
Appendix. Each component of the covariance matrix $\displaystyle\Sigma_{k}$
characterizes the correlation between the $\displaystyle I$th and the
$\displaystyle J$th detector,

$\displaystyle\langle\tilde{d}_{I}^{k}\tilde{d}_{J}^{k*}\rangle=\delta_{IJ}\lambda_{I}\left(\omega_{k}\right)+\frac{A_{I}\left(\omega_{k}\right)A_{J}\left(\omega_{k}\right)\rho}{\bar{\omega}}F_{IJ}\left(\omega_{k},\vec{x}_{IJ}\right),$
(25)

where $\displaystyle\lambda_{I}=\langle\tilde{n}_{I}\tilde{n}_{I}^{*}\rangle$
is the noise power spectral density of the $\displaystyle I$th detector, and

$F_{IJ}(\omega_{k},\vec{x}_{IJ})=\int\frac{d^{3}\vec{p}}{\omega}\,f(\vec{p})\,e^{\mathrm{i}\,\vec{p}\cdot\vec{x}_{IJ}}\,\delta\left(\omega-\omega_{k}\right).$
(26)

represent the correlations of the axion field at the two locations separated
by $\displaystyle\vec{x}_{IJ}\equiv\vec{x}_{I}-\vec{x}_{J}$, containing all
the relevant signal parameters $\displaystyle\theta_{\rm sig}$ in this study.

With the help of the covariance matrix, we next construct the likelihood for
an array of detectors in terms of data $\displaystyle\tilde{D}$ and the
putative signal parameters $\displaystyle\theta_{\rm sig}$ Foster  _et al._
(2018, 2021),

$\mathcal{L}\left(\tilde{D}|\theta_{\rm
sig}\right)=\prod_{k}\frac{\exp\left[-\frac{1}{2}\,\tilde{D}^{k\dagger}\cdot\Sigma_{k}^{-1}(\theta_{\rm
sig})\cdot\tilde{D}^{k}\right]}{\sqrt{|2\pi\Sigma_{k}(\theta_{\rm sig})|}},$
(27)

as well as the test statistic (TS)

$\Theta=2\left[\,\ln\,\mathcal{L}\left(\tilde{D}|\theta_{\rm
sig}\right)-\ln\,\mathcal{L}\left(\tilde{D}|\theta_{\rm
sig}=0\right)\,\right]$ (28)

that quantifies the significance of $\displaystyle\theta_{\rm sig}$.

To get an intuitive understanding of how to extract information from the TS in
Eq. (28), we use the Asimov data set Cowan  _et al._ (2011) where the average
on the covariance of data $\displaystyle\tilde{D}$ converges to the truth
value

$\displaystyle\langle{\tilde{D}}_{I}^{k}{\tilde{D}}_{J}^{k*}\rangle=\Sigma_{k}(\theta_{\rm
truth})$ (29) $\displaystyle=$
$\displaystyle\delta_{IJ}\lambda_{I}\left(\omega_{k}\right)+\frac{A_{I}\left(\omega_{k}\right)A_{J}\left(\omega_{k}\right)\rho}{\bar{\omega}}F^{t}_{IJ}\left(\omega_{k},\vec{x}_{IJ}\right),$

where we use a superscript $\displaystyle t$ to label functions containing the
truth parameters within $\displaystyle\theta_{\rm truth}$. Under the
assumptions that the noise is much larger than the signal, and the frequency
bin size $\displaystyle 2\pi/T$ is small enough compared to the varying scales
of both the signals and the noise, we take the expectation value of the
likelihood in Eq. (28) with the average data and get the asymptotic TS Foster
_et al._ (2021),

$\displaystyle\widetilde{\Theta}(\theta_{\rm
sig})\approx\sum_{I,J}\frac{\rho^{2}T}{2\pi\bar{\omega}^{2}}$
$\displaystyle\int
d\omega\,\frac{A_{I}^{2}(\omega)A_{J}^{2}(\omega)}{\lambda_{I}(\omega)\lambda_{J}(\omega)}\,F_{JI}(\omega,\vec{x}_{JI})$
(30)
$\displaystyle\left(F^{t}_{IJ}(\omega,\vec{x}_{IJ})-\frac{1}{2}F_{IJ}(\omega,\vec{x}_{IJ})\right).$

The maximization of the asymptotic TS (30) happens when
$\displaystyle\theta_{\rm sig}$ in $\displaystyle F_{IJ}$ approaches
$\displaystyle\theta_{\rm truth}$.

One can further estimate the Fisher information matrix in terms of different
parameters $\displaystyle\theta_{i}\in\theta_{\rm sig}$, evaluated at the
truth parameters

$I_{ij}(\theta_{\rm
sig})=-\frac{1}{2}\frac{\partial^{2}\widetilde{\Theta}}{\partial\theta_{i}\,\partial\theta_{j}}|_{\theta_{\rm
sig}=\theta_{\rm truth}}.$ (31)

For single parameter $\displaystyle\theta_{i}$, the inverse of Eq. (31) gives
the uncertainty $\displaystyle\sigma^{2}_{\theta_{i}}$.

Take an infinitely cold stream with $\displaystyle
f(\vec{p}\,)=\delta^{3}(\vec{p}-\vec{p}_{0})$ as an example. With two
detectors separated by a baseline $\displaystyle\vec{x}_{IJ}$, one can extract
the information $\displaystyle\cos\theta_{\rm
str}\equiv\hat{p}_{0}\cdot\hat{x}_{IJ}$ from the relative phase factor in Eq.
(26), with the uncertainty $\displaystyle\sigma^{2}_{\theta_{\rm str}}\propto
1/\left(p_{0}\,x_{IJ}\,\sin\theta_{\rm str}\right)^{2}$ derived from Eq. (31)
Foster  _et al._ (2021).

In the case in which the baselines $\displaystyle\vec{x}_{IJ}$ between each
pair of axion detectors cover a large volume in the coordinate space with a
sufficient time resolution, we can take the continuous limit
$\displaystyle\omega_{k}\rightarrow\omega$ and
$\displaystyle\vec{x}_{IJ}\rightarrow\vec{x}$, which leads to $\displaystyle
F_{IJ}(\omega_{k},\vec{x}_{IJ})\rightarrow F(\omega,\vec{x})$. Thus the three-
dimensional Fourier transform

$\displaystyle\int\frac{d^{3}\vec{x}}{(2\pi)^{3}}\,\mathrm{e}^{-\mathrm{i}\,\vec{p}\cdot\vec{x}}\,\int
d\omega\,F(\omega,\vec{x})=\frac{f(\vec{p}\,)}{\omega}$ (32)

enables us to recover the complete information of the axion field in momentum
space, namely, $\displaystyle f(\vec{p}\,)$.

###  III.2 Vector Sensor Array

In this subsection, we consider a network of vector sensors. For each
detector, we replace the expected response in Eq. (22) with the vector-like
signals $\displaystyle\vec{\mathcal{O}}\left(\omega_{k},\vec{x}_{I}\right)$,

$\tilde{d}_{I}^{k}=A_{I}(\omega_{k})\,\vec{\mathcal{O}}\left(\omega_{k},\vec{x}_{I}\right)\cdot\hat{l}_{I}+\tilde{n}_{I}\left(\omega_{k}\right),$
(33)

where $\displaystyle\hat{l}_{I}$ is a unit vector representing the sensitive
direction of the sensor. Similar to Eq. (26), the correlations of the vector-
like signals are

$F_{IJ}\left(\omega_{k},\vec{x}_{IJ},\hat{l}_{I},\hat{l}_{J}\right)=\int\frac{d^{3}\vec{p}}{\omega}\,f(\vec{p}\,)\,\mathrm{e}^{\mathrm{i}\,\vec{p}\cdot\vec{x}_{IJ}}\,\delta\left(\omega-\omega_{k}\right)\,\zeta^{2}_{\mathcal{O}}(p)\,\sum_{\beta}h(\beta)\,\left(\vec{\epsilon}_{\beta}\cdot\hat{l}_{I}\right)\,\left(\vec{\epsilon}_{\beta}^{\,*}\cdot\hat{l}_{J}\right),$
(34)

where $\displaystyle\zeta_{\mathcal{O}}(p)$ is the factor before
$\displaystyle\Phi_{j}$ in Eq. (14 \- LABEL:OEDM), depending on different
types of coupling $\displaystyle\mathcal{O}$, and $\displaystyle h(\beta)$ is
the percentage of the polarization mode $\displaystyle\beta$. Compared with
the case of the scalar sensors in Eq. (26), the correlations of the vector-
like signals (34) have additional polarization-dependent factors
$\displaystyle
h(\beta)\,\left(\vec{\epsilon}_{\beta}\cdot\hat{l}_{I}\right)\left(\vec{\epsilon}_{\beta}^{\,*}\cdot\hat{l}_{J}\right)$.
Thus the product $\displaystyle f(\vec{p}\,)\times\zeta^{2}_{\mathcal{O}}(p)$
can be extracted only when we know $\displaystyle h(\beta)$.

We next rewrite Eq. (34) as

$\displaystyle F_{IJ}(\omega_{k},\vec{x}_{IJ},\hat{l}_{I},\hat{l}_{J})$ (35)
$\displaystyle=$ $\displaystyle
p_{k}\,\zeta^{2}_{\mathcal{O}}(p_{k})\,\sum_{\beta}h(\beta)\,\mathcal{F}^{\beta}_{IJ}(p_{k},\vec{x}_{IJ},\hat{l}_{I},\hat{l}_{J}),$

where $\displaystyle p_{k}=\sqrt{\omega_{k}^{2}-m^{2}}$ and

$\displaystyle\mathcal{F}^{\beta}_{IJ}(p_{k},\vec{x}_{IJ},\hat{l}_{I},\hat{l}_{J})$
(36) $\displaystyle\equiv$ $\displaystyle\int
d^{2}\hat{\Omega}\,f(p_{k},\hat{\Omega})\,\mathrm{e}^{\mathrm{i}\,p_{k}\,\hat{\Omega}\cdot\vec{x}_{IJ}}\,\left(\vec{\epsilon}_{\beta}\cdot\hat{l}_{I}\right)\left(\vec{\epsilon}_{\beta}^{\,*}\cdot\hat{l}_{J}\right),$

which characterizes the correlations between the two vector-like signals with
a specific polarization mode $\displaystyle\beta$ and energy
$\displaystyle\omega_{k}$.

Figure 1: A schematic diagram for a pair of vector sensors distributed at the
same location with different sensitive directions $\displaystyle\hat{l}_{I}$
and $\displaystyle\hat{l}_{J}$ is shown. The green arrows indicate the
sensitive directions of the vector sensors, and the gray wavy lines represent
the stochastic bosonic fields with wave functions
$\displaystyle\vec{\epsilon}_{\beta}$. If the source is isotropic, the angular
correlations always show the universal dipole correlation which is
proportional to $\displaystyle\hat{l}_{I}\cdot\hat{l}_{J}=\cos\theta_{IJ}$,
according to Eq. (40).

Here we show an example of what information can be extracted if we tune
$\displaystyle\hat{l}_{I}$. We can put two detectors with different sensitive
directions, $\displaystyle\hat{l}_{I}$ and $\displaystyle\hat{l}_{J}$, at the
same location. The schematic diagram of the detectors is shown in Fig. 1. For
a background source with an isotropic momentum distribution

$f_{\rm iso}(p,\hat{\Omega})=\frac{f_{\rm iso}(p)}{4\pi p^{2}},$ (37)

where $\displaystyle p=|\vec{p}\,|$, the signal correlations between the
longitude modes are

$\displaystyle\mathcal{F}^{\,0}_{IJ}\left(p,\hat{l}_{I},\hat{l}_{J}\right)=$
$\displaystyle\int d^{2}\hat{\Omega}\,\frac{f_{\rm iso}(p)}{4\pi
p^{2}}\,\left(\vec{\epsilon}_{0}\cdot\hat{l}_{I}\right)\left(\vec{\epsilon}_{0}\cdot\hat{l}_{J}\right)$
(38) $\displaystyle=$ $\displaystyle\frac{\omega^{2}f_{\rm
iso}(p)}{3m^{2}p^{2}}\,\hat{l}_{I}\cdot\hat{l}_{J},$

and the ones for each type of transverse modes are

$\displaystyle\mathcal{F}^{R/L}_{IJ}\left(p,\hat{l}_{I},\hat{l}_{J}\right)=$
$\displaystyle\int d^{2}\hat{\Omega}\,\frac{f_{\rm iso}(p)}{4\pi
p^{2}}\,\left(\vec{\epsilon}_{R/L}\cdot\hat{l}_{I}\right)\left(\vec{\epsilon}_{R/L}^{\,*}\cdot\hat{l}_{J}\right)$
(39) $\displaystyle=$ $\displaystyle\frac{f_{\rm
iso}(p)}{3p^{2}}\,\hat{l}_{I}\cdot\hat{l}_{J}.$

Among all the polarization modes, the angular correlations are universally
proportional to the dipole correlation Jenet and Romano (2015)

$\mathcal{F}^{\beta}_{IJ}\propto\hat{l}_{I}\cdot\hat{l}_{J}=\cos\theta_{IJ},$
(40)

where $\displaystyle\theta_{IJ}$ is the angle between
$\displaystyle\hat{l}_{I}$ and $\displaystyle\hat{l}_{J}$. In the presence of
many vector sensors with different sensitive directions, such as the setup
proposed in Smiga (2022), any deviation from Eq. (40) is a sign of anisotropy
for the sources. Note that the converse is not necessarily true. As we will
see in Sec. IV.3, for non-relativistic sources with all the polarization modes
contributing equally, anisotropic distribution can also lead to Eq. (40).

In the following sections, we will study further the implications of Eq. (34),
and how we can arrange $\displaystyle\hat{l}_{I}$ and
$\displaystyle\vec{x}_{IJ}$ to investigate the macroscopic properties and
microscopic nature of the vector-like sources.

##  IV Identification of the Couplings

In this section, we study the correlations of the vector-like signals (35)
with distinctive momentum distributions and polarization modes, with which one
can identify the properties of the bosonic background.

###  IV.1 Source from a Specific Direction

The first case we consider is the source from a specific direction
$\displaystyle\hat{\Omega}_{0}$, e.g. the cold stream O’Hare and Green (2017);
Foster  _et al._ (2018); Knirck  _et al._ (2018), whose momentum distribution
is

$f_{\rm str}(p,\hat{\Omega})=\frac{f_{\rm
str}(p)}{p^{2}}\,\delta^{2}(\hat{\Omega}-\hat{\Omega}_{0}).$ (41)

Take the above expression into Eq. (36) and choose $\displaystyle x_{IJ}=0$,
the longitudinal-mode-only case gives

$\mathcal{F}^{\,0}_{IJ}\left(p,\hat{l}_{I},\hat{l}_{J}\right)=\frac{\omega^{2}f_{\rm
str}(p)}{m^{2}p^{2}}\left(\hat{\Omega}_{0}\cdot\hat{l}_{I}\right)\left(\hat{\Omega}_{0}\cdot\hat{l}_{J}\right).$
(42)

The maximization of Eq. (42) happens when both $\displaystyle\hat{l}_{I}$ and
$\displaystyle\hat{l}_{J}$ are parallel with $\displaystyle\hat{\Omega}_{0}$.
For transverse modes without macroscopic polarization, Eq. (36) becomes

$\displaystyle\mathcal{F}^{R}_{IJ}\left(p,\hat{l}_{I},\hat{l}_{J}\right)+\mathcal{F}^{L}_{IJ}\left(p,\hat{l}_{I},\hat{l}_{J}\right)$
(43) $\displaystyle=$ $\displaystyle\frac{f_{\rm
str}(p)}{p^{2}}\left(\hat{l}_{I}\cdot\hat{l}_{J}-(\hat{\Omega}_{0}\cdot\hat{l}_{I})\,(\hat{\Omega}_{0}\cdot\hat{l}_{J})\right),$

which vanishes when the three directions are parallel with each other and
reaches the maximum when $\displaystyle\hat{l}_{I}$ and
$\displaystyle\hat{l}_{J}$ take the same direction on the transverse plane of
$\displaystyle\hat{\Omega}_{0}$. We will discuss the localization of such
signals in detail in Sec. V.

###  IV.2 Isotropic Cosmological Background

Figure 2: The left panels show two schematic diagrams for different
arrangements of the vector sensors, shown as the green arrows, separated by
the baseline $\displaystyle d$, represented by the black dashed lines. The
sensitive directions are parallel with $\displaystyle\hat{d}$ in the upper
panels while perpendicular to it in the lower ones. The gray wavy lines are
the isotropic bosonic fields with either longitudinal or transverse degrees of
freedom. The corresponding correlations in Eq. (44 \- 47) are shown in the
right panels. One can see that, at $\displaystyle\xi\equiv pd\approx 4$,
$\displaystyle\epsilon_{0}$ and $\displaystyle\epsilon_{L/R}$ show distinct
correlations.

We next consider the isotropic bosonic field background with the momentum
distribution in Eq. (37), which usually comes from a cosmological origin. As
shown in Eq. (40), when two vector sensors are at the same location, they
demonstrate the universal dipole correlation, independent of the polarization
modes of the source. We will show that by separating two detectors by a
baseline with a finite length $\displaystyle d=x_{IJ}$, we are able to
distinguish the longitudinal modes of the source from the transverse ones.
Note that, because $\displaystyle d$ always comes together with momentum
$\displaystyle p$ in Eq. (36), we introduce the dimensionless parameter
$\displaystyle\xi\equiv p\,d$.

We consider the setup with
$\displaystyle\hat{l}_{I}\mathbin{\\!/\mkern-5.0mu/\\!}\hat{l}_{J}$ in this
subsection, where the pair of vector sensors respond to right-hand and left-
hand circular polarization modes identically. When both of the sensitive
directions are parallel with $\displaystyle\hat{d}$, Eq. (36) gives

$\displaystyle\displaystyle\mathcal{F}^{\,0}_{IJ}\left(p,\xi,\hat{l}_{I}\mathbin{\\!/\mkern-5.0mu/\\!}\hat{l}_{J}\mathbin{\\!/\mkern-5.0mu/\\!}\hat{d}\right)=\frac{\omega^{2}f_{\rm
iso}(p)}{m^{2}p^{2}}\frac{2\xi\cos\xi+\left(\xi^{2}-2\right)\sin\xi}{\xi^{3}},$
(44)
$\displaystyle\displaystyle\mathcal{F}^{R/L}_{IJ}\left(p,\xi,\hat{l}_{I}\mathbin{\\!/\mkern-5.0mu/\\!}\hat{l}_{J}\mathbin{\\!/\mkern-5.0mu/\\!}\hat{d}\right)=\frac{f_{\rm
iso}(p)}{p^{2}}\,\frac{\sin\xi-\xi\cos\xi}{\xi^{3}},$ (45)

the results of which are shown in the upper panels of Fig. 2.

Similarly, when the sensitive directions are perpendicular to
$\displaystyle\hat{d}$ and parallel with each other, we have

$\displaystyle\displaystyle\mathcal{F}^{\,0}_{IJ}\left(p,\xi,\hat{l}_{I}\mathbin{\\!/\mkern-5.0mu/\\!}\hat{l}_{J}\perp\hat{d}\right)=\frac{\omega^{2}f_{\rm
iso}(p)}{m^{2}p^{2}}\,\frac{\sin\xi-\xi\cos\xi}{\xi^{3}},$ (46)
$\displaystyle\displaystyle\mathcal{F}^{R/L}_{IJ}\left(p,\xi,\hat{l}_{I}\mathbin{\\!/\mkern-5.0mu/\\!}\hat{l}_{J}\perp\hat{d}\right)=\frac{f_{\rm
iso}(p)}{p^{2}}\,\frac{\xi\cos\xi+\left(\xi^{2}-1\right)\sin\xi}{2\xi^{3}},$
(47)

which is shown in the lower panels of Fig. 2. Comparing the two different ways
to arrange the sensitive directions, one can see that significant differences
appear at $\displaystyle\xi\approx 4$. The longitudinal mode correlation
vanishes, and transverse modes correlations reach the minimum for the
perpendicular setup, while the features flip for the parallel one.

We will see in Sec. VI that a pair of vector sensors with non-parallel
sensitive directions and a long baseline can further distinguish between the
right-hand and left-hand circular polarization modes.

###  IV.3 Standard Halo Model of Dark Matter

The last example we consider is the Maxwellian distribution of the standard
halo model of cold dark matter, whose velocity distribution is Derevianko
(2018)

$f_{\rm DM}(\vec{v}\,)=\left(2\pi
v_{\textrm{vir}}^{2}\right)^{-3/2}\,\exp\left[-\frac{(\vec{v}-\vec{v}_{g})^{2}}{2v_{\textrm{vir}}^{2}}\right],$
(48)

where $\displaystyle v_{\textrm{vir}}\approx 10^{-3}$ and
$\displaystyle\vec{v}_{g}$ are the virial velocity and the Earth’s velocity in
the galactic reference frame respectively. In the non-relativistic limit, Eq.
(34) can be approximated as

$\displaystyle F_{IJ}\left(\omega_{k},\vec{d},\hat{l}_{I},\hat{l}_{J}\right)$
$\displaystyle\approx\int\frac{d^{3}\vec{v}}{m^{2}\,v}\,f_{\rm
DM}(\vec{v})\,\mathrm{e}^{\mathrm{i}\,m\,\vec{v}\cdot\vec{d}}\,\delta\left(v-v_{k}\right)$
(49) $\displaystyle\times$
$\displaystyle\zeta^{2}_{\mathcal{O}}(m\,v)\,\sum_{\beta}h(\beta)\,\left(\vec{\epsilon}_{\beta}\cdot\hat{l}_{I}\right)\,\left(\vec{\epsilon}_{\beta}^{\,*}\cdot\hat{l}_{J}\right),$

where $\displaystyle v_{k}$ satisfies
$\displaystyle\omega_{k}/m=1+v_{k}^{2}/2$.
$\displaystyle\vec{\epsilon}_{\beta}$ reduce to unit directional vectors and
$\displaystyle h(\beta)$ contains an equal percentage among the allowed
polarization modes in Eq. (14 \- LABEL:OEDM). With a preknowledge of the
velocity distribution in Eq. (48), one can break the degeneracy in the product
$\displaystyle f_{\rm DM}(\vec{v})\times\zeta^{2}_{\mathcal{O}}(m\,v)$ and
identify the exact type of the coupling $\displaystyle\mathcal{O}$ listed in
Sec. II.

Due to the complicated forms of Eq. (49) defined at each energy bin and the
narrow bandwidth of the standard halo dark matter, we define the dark matter
correlation matrix as

$\displaystyle\Big{(}\mathscr{C}_{\mathcal{O}}(\vec{d}\,)\Big{)}_{ij}$ (50)
$\displaystyle\equiv$ $\displaystyle\int
dv_{k}\,v_{k}\,\text{Re}\left[F_{IJ}(\omega_{k},\vec{d},\hat{i},\hat{j})\right]$
$\displaystyle\approx$ $\displaystyle\int\frac{d^{3}\vec{v}}{m^{2}}\,f_{\rm
DM}(\vec{v}\,)\,\cos\left[m\,\vec{v}\cdot\vec{d}\,\right]\,\zeta^{2}_{\mathcal{O}}(m\,v)\,\sum_{\beta}h(\beta)\,\epsilon_{\beta}^{i}\,\epsilon_{\beta}^{j},$

which comes from the integration of the real part of Eq. (49) in energy. Eq.
(50) always leads to analytic expressions from the Gaussian integral.
$\displaystyle i$ and $\displaystyle j$ denote the index of the spatial
components in Cartesian coordinates. For dark matter, we choose the linear
polarization basis where $\displaystyle\vec{\epsilon}_{\beta}$ are real.
Equation (50) is the vectorial generalization of the two-point spatial
correlation function of the scalar-like signals from the dark matter in
Derevianko (2018) and the Appendix. The daily modulations of the vector-like
signals were studied in Lisanti  _et al._ (2021); Gramolin  _et al._ (2022)
for the axion gradient and in Caputo  _et al._ (2021) for the kinetic-mixing
dark photon.

####  IV.3.1 Axion Gradient

For the axion gradient, there is only longitudinal mode and
$\displaystyle\zeta_{a}=m\,v$. To give an intuitive understanding, we can
decompose Eq. (50) into three parts:

$\mathscr{C}_{a}(\vec{d}\,)=\,\exp\left[-\frac{d^{2}}{2\lambda_{c}^{2}}\right]\,\left(C_{a}^{\textrm{vir}}+C_{a}^{g}+C_{a}^{g,\textrm{vir}}\right),$
(51)

where $\displaystyle C_{a}^{\textrm{vir}}$ contains a term proportional to the
square of the virial velocity $\displaystyle v_{\rm vir}$, $\displaystyle
C_{a}^{g}$ comes from the square of $\displaystyle v_{g}$, and $\displaystyle
C_{a}^{g,\textrm{vir}}$ is a product of the two.
$\displaystyle\lambda_{c}\equiv 1/\left(m\,v_{\textrm{vir}}\right)$ is the
correlation length of the virial dark matter. The overall exponential factor
indicates that the correlation decays at a length scale
$\displaystyle\lambda_{c}$ due to the virial fluctuations.

Without loss of generality, we assume that the separation between the
detectors pair $\displaystyle\vec{d}$ is along the
$\displaystyle\hat{z}$-axis, where the virial part takes the form

$C_{a}^{\textrm{vir}}=v_{\textrm{vir}}^{2}\,\textrm{diag}\left(1,1,1-\frac{d^{2}}{\lambda_{c}^{2}}\right)\,\cos\left[m\,\vec{v}_{g}\cdot\vec{d}\,\right].$
(52)

In the absence of $\displaystyle\vec{v}_{g}$, Eq. (52) is the only non-
vanishing one. The anisotropic velocity leads to a spatial oscillation factor
$\displaystyle m\,\vec{v}_{g}\cdot\vec{d}$. The second part of Eq. (51) is

$\Big{(}C_{a}^{g}\Big{)}_{ij}=v_{g}^{i}\,v_{g}^{j}\cos\left[m\,\vec{v}_{g}\cdot\vec{d}\,\right],$
(53)

which is an anisotropic term induced by $\displaystyle\vec{v}_{g}$. The last
term in Eq. (51) is

$\displaystyle
C_{a}^{g,\textrm{vir}}=v_{\textrm{vir}}\,\begin{pmatrix}0&0&-v_{g}^{x}\\\
0&0&-v_{g}^{y}\\\
-v_{g}^{x}&-v_{g}^{y}&-2v_{g}^{z}\end{pmatrix}\,\frac{d}{\lambda_{c}}\,\sin\left[m\,\vec{v}_{g}\cdot\vec{d}\,\right],$
(54)

which describes the mixing effect between the anisotropic velocity
$\displaystyle\vec{v}_{g}$ and the virial one.

Different cases of $\displaystyle\mathscr{C}_{a}(\vec{d}\,)$ are shown in the
left panels of Fig. 3. For simplicity, we only consider the situation in which
two detectors are co-aligned. In such a case, there are five different ways to
arrange the directions of the baseline $\displaystyle\hat{d}$ and the
sensitive directions $\displaystyle\hat{l}$. $\displaystyle
v_{\textrm{vir}}=v_{g}$ is assumed in this work. When
$\displaystyle\hat{v}_{g}\perp\hat{d}$, both the spatial oscillation factor
$\displaystyle m\,\vec{v}_{g}\cdot\vec{d}$ and the mixing term (54) vanish. If
one further tunes $\displaystyle\hat{l}\perp\hat{v}_{g}$, Eq. (53) vanishes as
well. Then $\displaystyle\mathscr{C}_{a}(\vec{d}\,)$ is influenced by the
virial velocity only, so that it decreases to zero monotonically in terms of
$\displaystyle d$. On the other hand, when
$\displaystyle\hat{l}\,\mathbin{\\!/\mkern-5.0mu/\\!}\,\hat{d}$,
$\displaystyle\mathscr{C}_{a}(\vec{d}\,)$ becomes negative first before
decaying to zero. The impact of $\displaystyle\vec{v}_{g}$ shows up in the
rest cases where the spatial oscillation factor $\displaystyle
m\,\vec{v}_{g}\cdot\vec{d}$ can appear. However, since the correlation decays
significantly when $\displaystyle d\simeq 2\pi/\left(m\,v_{g}\right)$, we can
hardly see this oscillation.

Notice that if one defines Eq. (50) using the imaginary part of Eq. (49), the
results for Eq. (52-54) simply differ by a $\displaystyle\pi/2$ phase delay.
Thus Eq. (50) already contains all the information that can be extracted.

We also calculate the angular correlations between two detectors at the same
location with different sensitive directions $\displaystyle\hat{l}_{I}$ and
$\displaystyle\hat{l}_{J}$, which are the projection of
$\displaystyle\mathscr{C}_{a}(0)$ on these two directions,

$\displaystyle\Gamma_{a}\left(\hat{l}_{I},\hat{l}_{J}\right)$ (55)
$\displaystyle\equiv$
$\displaystyle\left(\hat{l}_{I}\right)^{\textrm{T}}\cdot\mathscr{C}_{a}(0)\cdot\hat{l}_{J}$
$\displaystyle=$ $\displaystyle
v_{\textrm{vir}}^{2}\,\hat{l}_{I}\cdot\hat{l}_{J}+\left(\vec{v}_{g}\cdot\hat{l}_{I}\right)\left(\vec{v}_{g}\cdot\hat{l}_{J}\right).$

The first term, which comes from the vector-like property of the signal, is
again the universal dipole correlation shown in Eq. (40), while the second
term in Eq. (55) describes the anisotropy introduced from
$\displaystyle\vec{v}_{g}$.

Figure 3:  Two-point spatial correlation functions for the axion gradient in
Eq. (51) with only the longitudinal modes and the MDM dark photon in Eq. (56)
with the transverse interacting modes, are shown in the left and right panels,
respectively. The two sensitive directions of the detectors are assumed to be
coaligned such that there are five different ways to arrange the directions of
$\displaystyle\hat{d}$ and $\displaystyle\hat{l}$ in terms of
$\displaystyle\hat{v}_{g}$ in each case. The upper and lower panels include
the correlations when $\displaystyle\hat{d}$ is perpendicular and parallel
with $\displaystyle\hat{v}_{g}$ respectively. We use solid lines to label the
cases independent of the norm of $\displaystyle\vec{v}_{g}$, i.e.,
$\displaystyle v_{g}$, while the dashed lines show the cases influenced by it,
see discussion after Eq. (54).

####  IV.3.2 Dark Photon

For non-relativistic dark photons, the polarization vectors are equally
distributed on the unit sphere in the galaxy frame. In the linear polarization
basis, this is equivalent to taking the percentage of each polarization mode
$\displaystyle\vec{\epsilon}_{\beta}$, i.e., $\displaystyle h(\beta)$, to be
$\displaystyle 1/3$. However, for specific types of interaction, some modes
may not contribute.

We first consider the MDM dark photon whose transverse modes interact with the
sensors and $\displaystyle\zeta_{\textrm{MDM}}=m\,v$. The right panels of Fig.
3 show different cases of the MDM dark photon’s two-point correlation
functions $\displaystyle\mathscr{C}_{\textrm{MDM}}$, which again can be
decomposed into

$\mathscr{C}_{\textrm{MDM}}(\vec{d}\,)=\frac{1}{3}\,\exp\left[-\frac{d^{2}}{2\lambda_{c}^{2}}\right]\left(C_{\textrm{MDM}}^{\textrm{vir}}+C_{\textrm{MDM}}^{g}+C_{\textrm{MDM}}^{g,\textrm{vir}}\right).$
(56)

The virial part is

$\displaystyle
C_{\textrm{MDM}}^{\textrm{vir}}=v_{\textrm{vir}}^{2}\,\textrm{diag}\left(2-\frac{d^{2}}{\lambda_{c}^{2}},2-\frac{d^{2}}{\lambda_{c}^{2}},2\right)\,\cos\left[m\,\vec{v}_{g}\cdot\vec{d}\,\right].$
(57)

Compared with Eq. (52), we can find the relation

$\Big{(}C_{\textrm{MDM}}^{\textrm{vir}}\Big{)}_{ii}=\sum_{n\neq
i}\Big{(}C_{a}^{\textrm{vir}}\Big{)}_{nn},$ (58)

since the directions of the axion gradient signal are perpendicular to the
ones of the MDM dark photon with the same momentum. The second part of Eq.
(56) takes the form

$\Big{(}C_{\textrm{MDM}}^{g}\Big{)}_{ij}=\Big{(}v_{g}^{2}\,\delta_{ij}-v_{g}^{i}\,v_{g}^{j}\Big{)}\cos\left[m\,\vec{v}_{g}\cdot\vec{d}\,\right].$
(59)

In addition to the anisotropic term with a different sign compared with the
axion gradient in Eq. (53), there is also a diagonal part proportional to
$\displaystyle v_{g}^{2}$. The mixing term is

$\displaystyle
C_{\textrm{MDM}}^{g,\textrm{vir}}=v_{\textrm{vir}}\,\begin{pmatrix}-2v_{g}^{z}&0&v_{g}^{x}\\\
0&-2v_{g}^{z}&v_{g}^{y}\\\
v_{g}^{x}&v_{g}^{y}&0\end{pmatrix}\,\frac{d}{\lambda_{c}}\,\sin\left[m\,\vec{v}_{g}\cdot\vec{d}\,\right].$
(60)

The angular correlations of the MDM dark photon are

$\Gamma_{\textrm{MDM}}\left(\hat{l}_{I},\hat{l}_{J}\right)=v_{\textrm{vir}}^{2}\,\left(\hat{l}_{I}\cdot\hat{l}_{J}\right)-\frac{1}{3}\left(\vec{v}_{g}\cdot\hat{l}_{I}\right)\left(\vec{v}_{g}\cdot\hat{l}_{J}\right),$
(61)

whose anisotropic part has a different sign compared to that of the axion
gradient in Eq. (55).

Finally, we consider the EDM dark photon where
$\displaystyle\zeta_{\textrm{EDM}}=m$ and all three polarization modes
contribute equally in the non-relativistic limit. Thus the correlation matrix
(50) reduces to an identity insensitive to $\displaystyle\hat{l}$

$\mathscr{C}_{\textrm{EDM}}(\vec{d}\,)=\frac{1}{3}\exp\left[-\frac{d^{2}}{2\lambda_{c}^{2}}\right]\,\cos\left[m\,\vec{v}_{g}\cdot\vec{d}\,\right]\mathbb{I}_{3\times
3},$ (62)

which is shown in Fig. 4. This shares the same form as the scalar correlation
in Derevianko (2018) and the Appendix, with an additional factor of
$\displaystyle 1/3$ for each diagonal component.

Figure 4: The two-point spatial correlation functions for the EDM dark photon
in Eq. (62), with all the $\displaystyle 3$ polarization modes contributing
equally, is shown.

The angular correlations of the EDM dark photon contain only the universal
dipole correlation

$\Gamma_{\textrm{EDM}}\left(\hat{l}_{I},\hat{l}_{J}\right)=\frac{1}{3}\Big{(}\hat{l}_{I}\cdot\hat{l}_{J}\Big{)}.$
(63)

As mentioned in Sec. II.1, for the axial-vector dark photon, the
$\displaystyle U(1)_{B-L}/U(1)_{B}$ dark photon or the kinetic-mixing dark
photon, the vector-like signals are proportional to their spatial components
of the wave functions, the same as the EDM dark photon. Therefore, they share
the same form of the correlations as the EDM dark photon in Eq. (62) and (63),
despite the different detector systems. One can further distinguish between
the EDM dark photon and the axial-vector by boosting the sensors to be
relativistic, like the storage ring experiments Graham  _et al._ (2021) where
the contribution from the longitudinal mode of the EDM dark photon is
suppressed, as shown in Eq. (LABEL:OEDM).

In conclusion, one can distinguish between the different types of dark matter
by the spatial and the angular correlations, based on the unique signals of
each one discussed in this subsection.

##  V Localization

Localizing the incoming bosonic waves on the sky is crucial to understanding
their microscopic production mechanism, which would open a new window in the
multi-messenger astronomy with a quantum sensor network Dailey  _et al._
(2021). In this section, we follow the discussions in Sec. IV.1 and evaluate
the angular resolutions, which leads to the most optimal arrangement of the
vector sensor pair.

Taking the momentum distribution in Eq. (41), we can calculate the asymptotic
TS in Eq. (30) and the corresponding Fisher information matrix in Eq. (31) in
terms of $\displaystyle\hat{\Omega}_{0}$. For simplicity, we consider a pair
of detectors with the same noise spectrum
$\displaystyle\lambda_{1}(\omega)=\lambda_{2}(\omega)=\lambda(\omega)$, and
the same response function that stays constant $\displaystyle A_{0}$ within a
narrow bandwidth $\displaystyle\Delta\omega$ around certain frequency
$\displaystyle\omega_{0}$ and vanishes outside. In the case with only the
longitudinal modes, the asymptotic TS (30) can be written as

$\displaystyle\frac{\tilde{\Theta}(\hat{\Omega}_{0})}{\textrm{TS}_{\textrm{str}}^{0}}=2\cos\left[p_{0}d\left(\hat{\Omega}_{0}^{t}-\hat{\Omega}_{0}\right)\cdot\hat{d}\right]M_{12}(\hat{\Omega}_{0}^{t})\,M_{12}(\hat{\Omega}_{0})$
(64) $\displaystyle-
M_{12}^{2}(\hat{\Omega}_{0})+\sum_{I=1,2}\left(M_{II}(\hat{\Omega}_{0}^{t})-\frac{1}{2}M_{II}(\hat{\Omega}_{0})\right)M_{II}(\hat{\Omega}_{0}),$

for a truth value $\displaystyle\hat{\Omega}_{0}^{t}$, where the normalization
constant $\displaystyle\textrm{TS}_{\textrm{str}}^{0}$ is

$\textrm{TS}_{\textrm{str}}^{0}\equiv\frac{\rho^{2}\,T\,A_{0}^{4}\,f_{\textrm{str}}^{2}(p_{0})\,\zeta_{\mathcal{O}}^{4}(p_{0})\,\omega_{0}^{4}\,\Delta\omega}{2\,\pi\,\bar{\omega}^{2}\,\lambda^{2}(\omega_{0})\,p_{0}^{2}\,m^{4}},$
(65)

and the projection function $\displaystyle M_{IJ}(\hat{n})$ is defined as

$M_{IJ}(\hat{n})\equiv\left(\hat{n}\cdot\hat{l}_{I}\right)\left(\hat{n}\cdot\hat{l}_{J}\right).$
(66)

For the transverse modes distributed equally among right-hand and left-hand
circular polarization, the asymptotic TS (30) is

$\displaystyle\frac{\tilde{\Theta}(\hat{\Omega}_{0})}{\textrm{TS}_{\textrm{str}}^{R/L}}=2\cos\left[p_{0}d\left(\hat{\Omega}_{0}^{t}-\hat{\Omega}_{0}\right)\cdot\hat{d}\right]\,\big{(}\hat{l}_{1}\cdot\hat{l}_{2}-M_{12}(\hat{\Omega}_{0}^{t})\big{)}$
(67)
$\displaystyle\times\big{(}\hat{l}_{1}\cdot\hat{l}_{2}-M_{12}(\hat{\Omega}_{0})\big{)}-\big{(}\hat{l}_{1}\cdot\hat{l}_{2}-M_{12}(\hat{\Omega}_{0})\big{)}^{2}$
$\displaystyle+\sum_{I=1,2}\left(\frac{1}{2}-M_{II}(\hat{\Omega}_{0}^{t})+\frac{1}{2}M_{II}(\hat{\Omega}_{0})\right)\big{(}1-M_{II}(\hat{\Omega}_{0})\big{)}$

where we define $\displaystyle\textrm{TS}_{\textrm{str}}^{R/L}$ as

$\textrm{TS}_{\textrm{str}}^{R/L}\equiv\frac{\rho^{2}\,T\,A_{0}^{4}\,f_{\textrm{str}}^{2}(p_{0})\,\zeta_{\mathcal{O}}^{4}(p_{0})\,\Delta\omega}{8\,\pi\,\bar{\omega}^{2}\,\lambda^{2}(\omega_{0})\,p_{0}^{2}}.$
(68)

Due to the complicated forms of Eq. (64) and (67), we will divide the
discussions into two cases in the following subsections, in terms of the ratio
between the baseline length and the de Broglie wavelength, i.e.,
$\displaystyle p_{0}d$. The short baseline corresponds to $\displaystyle
p_{0}d\ll 1$ while the long-baseline cases are when $\displaystyle p_{0}d\gg
1$.

###  V.1 Short Baseline

In the short-baseline limit, the pair of vector sensors are effectively
located at the same place. The localization comes purely from the relative
amplitude of the signals between the two vector sensors with different
sensitive directions, as mentioned in Dailey  _et al._ (2021).

We define the angular resolution of $\displaystyle\hat{p}_{0}$ as the solid-
angle uncertainty Cutler (1998); Graham and Jung (2018)

$\sigma^{2}_{\Omega}\equiv\frac{2\pi\sin\theta^{t}}{\sqrt{I_{\theta\theta}I_{\phi\phi}-I_{\theta\phi}^{2}}},$
(69)

where spherical angles $\displaystyle\theta$ and $\displaystyle\phi$
parametrize $\displaystyle\hat{\Omega}_{0}$ in a specific coordinate, and the
components of the Fisher information matrix can be calculated through Eq.
(31).

Figure 5:  A schematic diagram is shown for the optimal sensitive directions
of the two vector sensors in green arrows at the same location to localize the
bosonic waves from a specific direction. The gray wavy lines represent the
bosonic waves, and the red dashed arrow represents their incoming direction.

One chooses the coordinate system where the $\displaystyle\hat{z}$-axis is
perpendicular to the plane that $\displaystyle\hat{l}_{1}$ and
$\displaystyle\hat{l}_{2}$ span such that
$\displaystyle\theta_{l_{1}}=\theta_{l_{2}}=\pi/2$. In this system, the solid-
angle uncertainty (69) for the longitudinal-mode-only case is

$\displaystyle\sigma^{2}_{\Omega}=$
$\displaystyle\frac{2\pi}{\textrm{TS}_{\textrm{str}}^{0}}\,\frac{1}{\sin^{2}\theta^{t}\,|\cos\theta^{t}|}$
(70) $\displaystyle\times$
$\displaystyle\frac{1}{\left(\cos^{2}\left[\phi_{l_{1}}-\phi^{t}\right]+\cos^{2}\left[\phi_{l_{2}}-\phi^{t}\right]\right)\,|\sin[\phi_{l_{1}}-\phi_{l_{2}}]|}$
$\displaystyle\geq$
$\displaystyle\frac{4\pi}{\textrm{TS}_{\textrm{str}}^{0}},$

where the minimization happens when
$\displaystyle\phi_{l_{1}}-\phi^{t}=-(\phi_{l_{2}}-\phi^{t})=\pi/6$ or
$\displaystyle 5\pi/6$ and $\displaystyle\cos\theta^{t}=\pm 1/\sqrt{3}$, as
shown in Fig. 5. In such case the angle between $\displaystyle\hat{l}_{1}$ and
$\displaystyle\hat{l}_{2}$ is $\displaystyle\pi/3$ and the one between
$\displaystyle\hat{\Omega}_{0}^{t}$ and $\displaystyle\hat{l}_{1}$ or
$\displaystyle\hat{l}_{2}$ is $\displaystyle\pi/4$.

For the transverse modes without macroscopic polarization, the solid-angle
uncertainty has the same form as Eq. (70), except for replacing
$\displaystyle\textrm{TS}_{\textrm{str}}^{0}$ with
$\displaystyle\textrm{TS}_{\textrm{str}}^{R/L}$ defined in Eq. (68), as well
as the same minimization condition, since the Fisher information matrix
derived from Eq. (42) and (43) is the same up to a normalization factor.

###  V.2 Long Baseline

In the long-baseline case, the Fisher information matrix is dominated by the
term proportional to $\displaystyle p_{0}d$. When the direction of the
baseline $\displaystyle\hat{d}$ is along the $\displaystyle\hat{z}$-axis of
the coordinate system, the information extracted from this single baseline is
the polar angle $\displaystyle\theta$, which is the same as the scalar sensor
interferometry Foster  _et al._ (2021). The whole solid-angle can be localized
by an array consisting of multiple baselines, similar to the Very Long
Baseline Interferometry technique used by the Event Horizon Telescope et al.
(2019) in radio astronomy. Additionally, the self-rotation of the Earth can
also change the direction of the baseline, serving the same purpose Foster
_et al._ (2021).

More explicitly, the order of $\displaystyle I_{\theta\theta}$ is much higher
than that of $\displaystyle I_{\phi\phi}$ or $\displaystyle I_{\theta\phi}$ in
terms of $\displaystyle p_{0}d$. Thus we approximate the polar angle
resolution as $\displaystyle\sigma_{\theta}^{2}\approx I_{\theta\theta}^{-1}$,
which leads to different results between the longitudinal modes and the
transverse ones. For the longitudinal-mode-only case,
$\displaystyle\sigma_{\theta}^{2}$ is

$\displaystyle\sigma_{\theta}^{2}\approx I_{\theta\theta}^{-1}$
$\displaystyle=\frac{1}{\textrm{TS}_{\textrm{str}}^{0}}\,\frac{1}{p_{0}^{2}d^{2}\,\sin^{2}\theta^{t}}\,\frac{1}{M_{12}(\hat{\Omega}_{0}^{t})^{2}}$
(71)
$\displaystyle\geq\frac{1}{\textrm{TS}_{\textrm{str}}^{0}}\,\frac{1}{p_{0}^{2}d^{2}\,\sin^{2}\theta^{t}},$

which is minimized when both $\displaystyle\hat{l}_{1}$ and
$\displaystyle\hat{l}_{2}$ point towards $\displaystyle\hat{\Omega}_{0}^{t}$.
We leave the term $\displaystyle 1/\sin^{2}\theta^{t}$ since the baseline
direction $\displaystyle\hat{d}$ may be hard to tune, otherwise it can be
further optimized at the transverse directions to
$\displaystyle\hat{\Omega}_{0}^{t}$.

On the other hand, for the transverse modes without macroscopic polarization,
$\displaystyle\sigma_{\theta}^{2}$ becomes

$\displaystyle\sigma_{\theta}^{2}\approx I_{\theta\theta}^{-1}=$
$\displaystyle\frac{1}{\textrm{TS}_{\textrm{str}}^{R/L}}\,\frac{1}{p_{0}^{2}d^{2}\,\sin^{2}\theta^{t}}\,\frac{1}{\left(\hat{l}_{1}\cdot\hat{l}_{2}-M_{12}(\hat{\Omega}_{0}^{t})\right)^{2}}$
(72) $\displaystyle\geq$
$\displaystyle\frac{1}{\textrm{TS}_{\textrm{str}}^{R/L}}\,\frac{1}{p_{0}^{2}d^{2}\,\sin^{2}\theta^{t}},$

which leads to a similar form of the optimal resolution to that of the
longitudinal modes in Eq. (71), however with a different minimization
condition, i.e.,
$\displaystyle\hat{l}_{1}\,\mathbin{\\!/\mkern-5.0mu/\\!}\,\hat{l}_{2}\perp\hat{\Omega}_{0}^{t}$.

Figure 6:  Two schematic diagrams are shown for the optimal sensitive
directions of the two vector sensors in green arrows separated by the baseline
$\displaystyle d$ in black dashed lines to localize the bosonic waves from a
specific direction. The gray wavy lines represent the bosonic waves, and the
red dashed arrow represents their incoming direction. The upper panel is for
the longitudinal modes only, while the lower panel is for the transverse modes
without macroscopic polarization. The sensitive directions are set to make the
projections of the signals on them as large as possible.

We show the optimal arrangements of the vector sensor pair with the long
baseline in Fig. 6. Apparently, the optimized condition for the sensitive
directions $\displaystyle\hat{l}$ is to make the projections of the signals on
them as large as possible.

Figure 7:  A schematic diagram is shown in the left panel for the optimal
sensitive directions of the two vector sensors separated by the baseline
$\displaystyle d$ to detect the macroscopic circular polarization of isotropic
bosonic waves, where the three directions are required to be perpendicular to
each other. The green arrows show the sensitive directions, the black dashed
line is the baseline, and the gray wavy lines represent the bosonic
backgrounds. The right panel shows $\displaystyle\sigma_{\eta}^{-2}$ as a
function of $\displaystyle\xi\equiv p_{0}d$ according to Eq. (79).

##  VI Chiral Dark Photon Background

For a cosmological dark photon background with an isotropic momentum
distribution, we can expect a macroscopic circular polarization if the
production mechanism is parity-violating, such as the tachyonic instability
induced from the oscillation of a pseudo-scalar Anber and Sorbo (2010). This
section shows how to detect such a chiral signal with a pair of vector sensors
separated by a finite long baseline.

We first introduce a new parameter

$\eta\equiv h(R)-h(L)$ (73)

to parametrize the difference between the right-hand and the left-hand
circular polarization modes. Thus Eq. (35) can be rewritten as

$\displaystyle F_{IJ}(\omega,\vec{d},\hat{l}_{I},\hat{l}_{J})$ (74)
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\zeta^{2}_{\mathcal{O}}(p)\,\frac{f_{\textrm{iso}}(p)}{p}\,\Big{(}I_{IJ}(\omega,\vec{d},\hat{l}_{I},\hat{l}_{J})+\eta
V_{IJ}(\omega,\vec{d},\hat{l}_{I},\hat{l}_{J})\Big{)},$

where

$\displaystyle I_{IJ}(\omega,\vec{d},\hat{l}_{I},\hat{l}_{J})$ (75)
$\displaystyle\equiv$
$\displaystyle\frac{p^{2}}{f_{\textrm{iso}}(p)}\Big{(}\mathcal{F}^{R}_{IJ}(\omega,\vec{d},\hat{l}_{I},\hat{l}_{J})+\mathcal{F}^{L}_{IJ}(\omega,\vec{d},\hat{l}_{I},\hat{l}_{J})\Big{)},$
$\displaystyle V_{IJ}(\omega,\vec{d},\hat{l}_{I},\hat{l}_{J})$ (76)
$\displaystyle\equiv$
$\displaystyle\frac{p^{2}}{f_{\textrm{iso}}(p)}\Big{(}\mathcal{F}^{R}_{IJ}(\omega,\vec{d},\hat{l}_{I},\hat{l}_{J})-\mathcal{F}^{L}_{IJ}(\omega,\vec{d},\hat{l}_{I},\hat{l}_{J})\Big{)}.$

Notice that the dimensionless function $\displaystyle I_{IJ}$ and
$\displaystyle V_{IJ}$ are defined in the analogy of the Stokes parameters
$\displaystyle I$ and $\displaystyle V$ in radio astronomy.

According to Eq. (39), $\displaystyle V_{IJ}$ vanishes if the two vector
sensors are at the same location. Therefore, to measure $\displaystyle\eta$ in
Eq. (74), one must have a finite long baseline $\displaystyle\vec{d}$. Using
the same assumption as in Sec. V that the two detectors share the same noise
spectrum $\displaystyle\lambda(\omega)$ and the constant response
$\displaystyle A_{0}$ within $\displaystyle\Delta\omega$, the asymptotic TS in
Eq. (30) takes the form

$\displaystyle\frac{\tilde{\Theta}(\eta)}{\textrm{TS}_{\textrm{iso}}^{R/L}}=$
$\displaystyle\,\frac{1}{2}\left(I_{11}^{2}+I_{22}^{2}+2I_{12}I_{21}\right)+\eta^{t}\left(V_{12}I_{21}+V_{21}I_{12}\right)$
(77) $\displaystyle+$
$\displaystyle\,V_{21}V_{12}\left(2\eta^{t}\eta-\eta^{2}\right),$

where $\displaystyle\eta^{t}$ is the truth value of $\displaystyle\eta$, and
$\displaystyle\textrm{TS}^{R/L}_{\textrm{iso}}$ only differs from
$\displaystyle\textrm{TS}_{\textrm{str}}^{R/L}$ in Eq. (68) in the momentum
distribution

$\textrm{TS}^{R/L}_{\textrm{iso}}\equiv\frac{\rho^{2}\,T\,A_{0}^{4}\,f_{\textrm{iso}}^{2}(p_{0})\,\zeta_{\mathcal{O}}^{4}(p_{0})\,\Delta\omega}{8\,\pi\,\bar{\omega}^{2}\,\lambda^{2}(\omega_{0})\,p_{0}^{2}}.$
(78)

From Eq. (77) and Eq. (31), the uncertainty of $\displaystyle\eta$ can be
evaluated as

$\displaystyle\sigma_{\eta}^{2}$
$\displaystyle=\frac{1}{\textrm{TS}^{R/L}_{\textrm{iso}}}\frac{1}{\left(\sin\theta_{l_{1}}\sin\theta_{l_{2}}\sin\left[\phi_{l_{1}}-\phi_{l_{2}}\right]\right)^{2}}\frac{\xi^{4}}{\left(\xi\cos\xi-\sin\xi\right)^{2}}$
(79) $\displaystyle\geq\frac{5.26}{\textrm{TS}^{R/L}_{\textrm{iso}}},$

in the coordinate system where $\displaystyle\hat{d}$ is along the
$\displaystyle\hat{z}$-axis. The minimization condition for Eq. (79) requires
that $\displaystyle\theta_{l_{1}}=\theta_{l_{2}}=\pi/2$ and
$\displaystyle\phi_{l_{1}}-\phi_{l_{2}}=\pi/2$, i.e.,
$\displaystyle\hat{l}_{1},\hat{l}_{2}$ and $\displaystyle\hat{d}$ are
perpendicular to each other. The arrangement of the sensor directions is shown
in the left panel of Fig. 7. The dependence on $\displaystyle\xi\equiv p_{0}d$
is shown in the right panel of Fig. 7, from which the most optimal separation
is $\displaystyle\xi\approx 2.08$.

##  VII Conclusions

In this work, we explore several applications of an array of vector sensors in
identifying the features of bosonic waves, including angular distribution,
types of coupling, directions of the emission, and macroscopic circular
polarization. Such information can be extracted from a pair of vector sensors
with different optimal sensitive directions and a baseline that separates them
for various purposes. These applications are based on two-point correlation
functions of the vector fields with two spatial component labels.

We show that, for isotropic sources with a cosmological origin, one always
gets the universal dipole angular correlation for all the polarization modes
with two vector sensors at the same location. Any deviation from this relation
is a signal of anisotropy. We find that a finite long baseline is necessary to
break the degeneracy between the longitudinal and the transverse modes,
leading to different correlation signals for the two polarization modes when
the sensitive directions are parallel with the baseline or perpendicular to
it. One can further identify a potential macroscopic circular polarization by
making the two sensitive directions perpendicular to the baseline and each
other.

Longitudinal and transverse polarization degrees of freedom contribute
differently to the angular correlations for sources from a specific direction.
However, we find that the optimal sensitive directions of the vector sensors,
in terms of the angular resolution, are the same in both cases. With a short
baseline, the sensitivity reaches an order of the inverse of the signal-to-
noise ratio. In the long-baseline limit, one can resolve the polar angle with
a resolution of the order $\displaystyle p_{0}d$, requiring the sensitive
directions to overlap with the target vector signals as much as possible.

Finally, we show the spatial and angular correlations of the virialized dark
matter with different vector-like couplings to the standard model, such as the
axion-fermion couplings and the dipole couplings between dark photons and
fermions. The axion and the dark photon coupled to the magnetic dipole
momentum (MDM) of the fermions show completely different spatial correlations
since the axion gradient signals are purely longitudinal while only the
transverse modes interact with the fermions’ spins for the MDM dark photon.
Due to the movement of the Earth in the galaxy frame, one expects to see an
anisotropic part in their angular correlations, which have different signs
between the axion and the MDM dark photon. The $\displaystyle
U(1)_{B-L}/U(1)_{B}$ dark photon, the kinetic-mixing dark photon, and the
electric dipole momentum dark photon, on the other hand, contribute equally
among all the polarization modes, showing spatial correlations similar to the
correlations between two scalar signals and no anisotropic term in the angular
correlations.

In practice, once a vector sensor sees a suspicious signal, the rest of the
detectors in the world can either falsify it or identify the sources’
properties by correlating the signals of different detectors, in the way that
we discuss in this study. Complete extraction of the data gives rich
information on the microscopic nature and the origin of the sources.

A possible future extension of this work is to develop optimal data analysis
strategies to increase the scan rate of the wave-like dark matter or other
stochastic backgrounds, with vector-like couplings to the standard model, and
to employ these in vector sensor networks, such as the GNOME.

## Acknowledgements

We are grateful to Yonatan Kahn for the lectures and useful discussions in the
second international school on Quantum Sensors for Fundamental Physics,
Huayang Song for reading and comments for the manuscript, and Yue Zhao for
useful discussions. This work is supported by the National Key Research and
Development Program of China under Grant No. 2020YFC2201501. Y.C. is supported
by the China Postdoctoral Science Foundation under Grant No. 2020T130661, No.
2020M680688, the International Postdoctoral Exchange Fellowship Program, and
by the National Natural Science Foundation of China (NSFC) under Grants No.
12047557. M.J. is supported by the National Natural Science Foundation of
China under Grants No. 12004371. J.S. is supported by the National Natural
Science Foundation of China under Grants No. 12025507, No. 12150015,
No.12047503; and is supported by the Strategic Priority Research Program and
Key Research Program of Frontier Science of the Chinese Academy of Sciences
under Grants No. XDB21010200, No. XDB23010000, and No. ZDBS-LY-7003 and CAS
project for Young Scientists in Basic Research YSBR-006. X.X. is supported by
Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC2121
“Quantum Universe” - 390833306. Y.C., X.X. and Y.Z. would like to thank USTC
for their kind hospitality.

## Appendix: Scalar Correlation and Ensemble Average

This Appendix briefly reviews the two-point correlations of the stochastic
scalar fields developed in Foster  _et al._ (2018); Derevianko (2018); Foster
_et al._ (2021).

The bosonic fields can couple to the standard model particles in different
ways. For dark matter, the coherently oscillating waves lead to AC signals in
the observable sectors with frequency nearly equal to the boson mass. For a
light $\displaystyle\phi$ coupled with the standard model particles in a
linear portal, the induced signal $\displaystyle S$ in the detector is
proportional to the field value $\displaystyle\phi$. Thus the correlations
between the two detectors’ signals lead to two-point correlation functions of
the scalar fields at different times and locations,

$\langle
S_{1}(t_{1},\vec{x}_{1})S_{2}(t_{2},\vec{x}_{2})\rangle\propto\langle\phi(t_{1},\vec{x}_{1})\phi(t_{2},\vec{x}_{2})\rangle.$
(A1)

To calculate the two-point correlation functions of a stochastic scalar field,
one can either define a density matrix which is diagonal in momentum space
Derevianko (2018), or parametrize the field as a superposition of
$\displaystyle N_{b}$ plane waves, Foster  _et al._ (2018)

$\phi(t,\vec{x})=\sum_{j=1}^{N_{b}}\sqrt{\frac{2\rho_{\phi}}{N_{b}\bar{\omega}\omega_{j}}}\cos(\omega_{j}t-\vec{p}_{j}\cdot\vec{x}+\alpha_{j})$
(A2)

where the relative phase $\displaystyle\alpha_{j}$ is uniformly distributed
within $\displaystyle[0,2\pi)$, and the momentum $\displaystyle\vec{p}_{j}$
comes with the probability distribution function $\displaystyle f(\vec{p})$.
One next defines the ensemble average on any observable
$\displaystyle\mathcal{A}$ as

$\langle\mathcal{A}\rangle\equiv\prod_{m,n=1}^{N_{b}}\int
d^{3}\vec{p}_{m}\,f(\vec{p}_{m})\,\int_{0}^{2\pi}\,\frac{d\alpha_{n}}{2\pi}\,\mathcal{A},$
(A3)

with which one can calculate Eq. (A1) as

$\displaystyle\langle\phi(t_{1},\vec{x}_{1})\phi(t_{2},\vec{x}_{2})\rangle$
(A4) $\displaystyle=$ $\displaystyle\prod_{m,n=1}^{N_{b}}\,\iint
d^{3}\vec{p}_{m}\,d\alpha_{n}\,\sum_{i,j=1}^{N_{b}}\,\frac{2f(\vec{p}_{m})\rho_{\phi}}{2\pi
N_{b}\,\bar{\omega}\sqrt{\omega_{i}\omega_{j}}}\,$
$\displaystyle\cos\left[\omega_{i}t_{1}-\vec{p}_{i}\cdot\vec{x}_{1}+\alpha_{i}\right]\,\cos\left[\omega_{j}t_{2}-\vec{p}_{j}\cdot\vec{x}_{2}+\alpha_{j}\right]$
$\displaystyle=$ $\displaystyle\prod_{m=1}^{N_{b}}\,\int
d^{3}\vec{p}_{m}\,\sum_{i,j=1}^{N_{b}}\,\frac{f(\vec{p}_{m})\,\rho_{\phi}}{N_{b}\,\bar{\omega}\sqrt{\omega_{i}\omega_{j}}}$
$\displaystyle\cos\left[\left(\omega_{i}t_{1}-\omega_{j}t_{2}\right)-\left(\vec{p}_{i}\cdot\vec{x}_{1}-\vec{p}_{j}\cdot\vec{x}_{2}\right)\right]\,\delta_{ij}$
$\displaystyle=$ $\displaystyle\prod_{m=1}^{N_{b}}\int
d^{3}\vec{p}_{m}\,\sum_{i=1}^{N_{b}}\frac{f(\vec{p}_{m})\,\rho_{\phi}}{N_{b}\,\bar{\omega}\,\omega_{i}}\cos\left[\omega_{i}\tau-\vec{p}_{i}\cdot\vec{d}\,\right]$
$\displaystyle=$
$\displaystyle\frac{\rho_{\phi}}{\bar{\omega}}\int\frac{d^{3}\vec{p}\,f(\vec{p})}{\omega}\cos\left[\omega\tau-\vec{p}\cdot\vec{d}\,\right],$

where $\displaystyle\tau=t_{1}-t_{2}$,
$\displaystyle\vec{d}=\vec{x}_{1}-\vec{x}_{2}$.

Figure 8: The two-point correlation functions of scalar-like signals of the
dark matter in Eq. (A5) is shown, which is identical to the one of the EDM
dark photon in Fig. 4 up to a normalization factor.

Taking standard halo model with velocity distribution in Eq. (48), one can
evaluate the equal-time correlations of Eq.(A4) as

$\displaystyle\langle\phi(t,\vec{x}_{1})\phi(t,\vec{x}_{2})\rangle=\frac{\Phi_{0}^{2}}{2}\exp\left(-\frac{d^{2}}{2\lambda_{c}^{2}}\right)\cos\left(m\,\vec{v}_{g}\cdot\vec{d}\right),$
(A5)

where $\displaystyle\Phi_{0}\equiv\sqrt{2\rho_{\phi}}/\bar{\omega}$,
$\displaystyle\lambda_{c}\equiv 1/\left(m\,v_{\textrm{vir}}\right)$, and
$\displaystyle v_{\textrm{vir}}$ and $\displaystyle v_{g}$ are the virial
velocity and the Earth’s velocity in the galactic reference frame
respectively. The spatial correlation (A5) is shown in Fig. 8, which is the
same as that for the EDM dark photon in Fig. 4 up to a normalization factor.

For vector-like signals involving polarization modes $\displaystyle\beta$, one
can generalize the ensemble average on any observable $\displaystyle\xi$ to be

$\langle\xi\rangle\equiv\prod_{i,j,k}^{N_{b}}\int
d^{3}\vec{p}_{i}\,f(\vec{p}_{i})\int_{0}^{2\pi}\,\sum_{\beta_{k}}h(\beta_{k})\,\frac{d\alpha_{j}}{2\pi}\,\xi,$
(A6)

where $\displaystyle h(\beta_{k})$ is the percentage of the polarization mode
$\displaystyle\beta_{k}$.

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