2021

1]Departments of Humanities and Arts, Technion – Israel Institute of
Technology, Haifa, 32000, Israel

2]Faculty of Engineering and the Institute of Nanotechnology and Advanced
Materials, Bar Ilan University, Ramat Gan, 5290002, Israel

3]Department of Physics of Complex Systems, Weizmann Institute of Science,
Rehovot, 76100, Israel

# How synchronized human networks escape local minima

Elad Schniderman  Yahav Avraham  Shir Shahal  Hamootal Duadi  Nir Davidson
Moti Fridman<EMAIL_ADDRESS>[  [  [

###### Abstract

Finding the global minimum in complex networks while avoiding local minima is
challenging in many types of networks. We study the dynamics of complex human
networks and observed that humans have different methods to avoid local minima
than other networks. Humans can change the coupling strength between them or
change their tempo. This leads to different dynamics than other networks and
makes human networks more robust and better resilient against perturbations.
We observed high-order vortex states, oscillation death, and amplitude death,
due to the unique dynamics of the network. This research may have implications
in politics, economics, pandemic control, decision-making, and predicting the
dynamics of networks with artificial intelligence.

###### keywords:

human network, coupled systems, local minima, potential landscape

##  1 Introduction

Human interactions form complex networks of connections between the members of
the networks. Synchronizing human networks can happen spontaneously
neda2000self_clap1 ; strogatz2005theoretical_MB  or intentionally
javarone2017evolutionary_env0 ; werner2007dynamics_env . The synchronization
of the network is essential for coordinating ideas sumpter2012six_dm0 ;
smaldino2012origins_dm1 , and the well-being of its members
wasserman1994social_HS1 ; morris2005social_HS2 ; solferino2021human_toxic ,
and is seen in other organisms as well sumpter2006principles_bio1 ;
sarfati2022chimera_bio_ff ; davis2001biological_cs . Understanding the
dynamics of human networks has implications for politics, economics, social
sciences, and pandemic control. The topology of human networks determines if
the network will be able to synchronize dorfler2014synchronization , who is
most probable to become a leader calabrese2021spontaneous_leader , which
decision the network is more likely to make sumpter2012six_dm0 ;
conradt2005consensus_dm2 , and how stable is the network to small
perturbations gambuzza2021stability_stable1 ; sorrentino2010stability_stable2
; kassabov2022global_stable3 .

The synchronization dynamics of networks can be analyzed in terms of an
effective potential landscape in which the system evolves roy1992dynamical ;
fridman2010passive . To reach the fully synchronized state, which is the
global minimum of this potential landscape, a human network must avoid getting
trapped in local minima, where not all humans are synchronized
ling2019landscape_energy2 ; ravoori2011robustness_energy3 ;
leylaz2021identification_energy1 ; roy1992dynamical ; fridman2010passive ;
fridman2012measuring . Avoiding such local minima also has implications for
increasing the stability in other types of networks
jalan2005synchronized_other , speeding up machine learning processes
ayodele2010types_AI , and optimization problems in spin-glass dynamics
heim2015quantum_spinglass .

Human networks can reach synchronization by finding unique solutions which are
more stable compared to other networks due to the human ability to focus on
some inputs while ignoring others shahal2020synchronization_prev . However, it
is still unclear how human networks reach these stable solutions, what the
network dynamics that lead to it are, and in particular, how the human network
escapes local minima.

In this paper, we study the synchronization dynamics of human networks with
closed-ring topology and unidirectional time-delayed coupling (Fig. 1(a)).
Such a network has a complex potential landscape with well-defined local and
global minima tradonsky2015conversion ; pal2015phase ;
kassabov2022global_stable3  that we can tune and control in real time. We
prepare the system in a global minimum (fully synchronized) state and then
adiabatically transform the potential landscape by tuning the coupling delay
time such that this state becomes a local minimum. We then study in detail how
the system escapes this local minimum into the new fully synchronized global
minimum by measuring the amplitude, tempo, and phase of each node and
identifying which node is following which.

Studying the rhythmic behavior of humans in general, and music in particular,
can reveal aspects of human network dynamics which are usually hard to
identify d2015can_mus1 . We show that humans can escape local minima by self-
tuning local properties of the networks such as their tempo, amplitude, and
the coupling strength between players. The ability to tune these parameters
changes dramatically the dynamics of the network and has implications for
other networks where each node has decision-making abilities
kolling2015human_robot .

##  2 Experiment

Our system with coupled violin players is schematically shown in Fig. 1(a)
shahal2020synchronization_prev . The violin players cannot see or hear the
others apart from the audio signal they receive through their headphones. The
sound of each player is connected to a control system that receives the audio
signal from all players and distributes it to each player. The control system
provides a tunable, programmable, and accurate real-time control of the
network connectivity (who is connected to who) and the strength and delay of
the coupling between the players. The players repeat the music phrase shown in
Fig. 1(b), and are instructed to synchronize with what they hear. In this
study, we set the players’ network as a unidirectional ring, so each player
hears only a single neighbor with periodic boundary conditions, illustrated in
Fig 1(a) for $N=6$ players. We performed measurements also with different
numbers of players including $N=2,3,4,5,6,7,8$, and 16. We measure the note
each player is playing as a function of time, and accordingly, obtain the
phase $\varphi(t)$ of the player, where $\varphi=0$ denotes that the player is
playing the beginning of the musical phrase, and $\varphi=2\pi$ denotes that
the player is at the end of the musical phrase of duration $T$
shahal2020synchronization_prev .

Figure 1: Coupled violin players on a unidirectional ring network, where each
player is hearing only the player on its right with a controllable delayed
coupling. (a) Schematic of $N=6$ unidirectional coupled players. (b) The
musical phrase the players are periodically repeating illustrates the two
possible states of synchronization: the in-phase state where all the players
are playing the same note at the same time, and the vortex state, where due to
the delay between the players, each player is playing a different note. (c)
The effective potential (defined in section 3) of the system as a function of
the phase difference between the players for two values of delay $\Delta t$.
For $\Delta t=0$, the minimum potential is at zero phase difference,
indicating an in-phase state of synchronization. For $\Delta t=T/N$, the
minimum potential is at $-2\pi/N$ indicating a vortex state and the in-phase
state becomes a local minimum, protected by a potential barrier. (d) Effective
potential map as a function of the phase difference and delayed coupling
between the players.

Each experiment starts with zero time-delay $\Delta t=0$ between all players
with coupling strength sufficiently high shahal2020synchronization_prev  to
ensure that all players are synchronized in-phase (i.e. playing the same note
at the same time, see Fig 1(b)). Then, we increase $\Delta t$ linearly with
time as $\Delta t=t/30$. When $\Delta t$ exceeds $nT/N$ the in-phase state
becomes a local minimum and the global minimum solution is a vortex state,
where each player has identical delay compared to its neighbor, satisfying the
periodic boundary conditions (see Fig 1(b)). The $n$ parameter is an integer
number and denotes the topological charge of the system pal2015phase . The
players must then leave their in-phase state and find the vortex state.

The dynamics of the coupled violin players performing a periodic rhythm can be
analyzed by the Kuramoto model kuramoto2012chemical_kur1 ;
kuramoto1987statistical_kur0 ; strogatz2000kuramoto_kur2  which describes an
over-damped motion of coupled phase oscillators in an effective potential
leylaz2021identification_energy1 ; roy1992dynamical ; fridman2010passive ;
fridman2012measuring . The derivation of the Kuramoto effective potential is
presented below in section 3. Figure 1(c) depicts two representative effective
potentials, for $\Delta t=0$ and $\Delta t=T/N$, respectively, as a function
of the phase difference between coupled players. For $\Delta t=0$, the global
minimum potential is at zero phase difference between coupled players,
corresponding to the in-phase synchronization state. For $\Delta t=T/N$, the
global minimum of the potential is at a phase difference of $-2\pi/N$,
corresponding to a vortex state of synchronization, while the in-phase state
becomes a local minimum.

Figure 1(d) presents the effective potential as a function of $\Delta t$. The
dashed line denotes the system trajectory in the phase space as the delay
increases adiabatically from $\Delta t=0$. The system starts from the in-phase
state and in order for the system to reach the emerging global minimum vortex
state, it must overcome a potential barrier. Here, we present and analyze four
types of dynamics in human networks for overcoming this potential barrier and
escaping the local minimum into the global one.

In the first dynamics, some of the players ignore the signals they receive,
thereby reducing the effective coupling strength to their neighbor. Then, they
can freely spread their phase to reach the vortex state (see section 3). In
the second, the players are slowing down their tempo so the in-phase state
remains the global minimum for arbitrarily large delays (see section 4). In
the third, the players further slow down until everyone plays the same note
indefinitely. This state, known as oscillation death
ermentrout1990oscillator_death , is a stable synchronized solution regardless
of the delay between the players. The players then spontaneously emerge from
the oscillation death directly into the globally stable vortex state (see
section 5). In the fourth, some of the players stop playing (i.e. reduce their
amplitude to near zero). In this state, known as amplitude death, the network
topology changes into an open ring, where the local minima and its potential
barrier disappear. Thus, enabling the other players to find the vortex state
(see section 5).

We observed these four distinct strategies to escape local minima at all
network sizes from $N=2$ up to $N=16$ players. While representing different
emerging strategies to overcome barriers and reach stable minima, they all
rely on the unique ability of human players to adjust their playing amplitude
and tempo and change the effective coupling strength between them by ignoring
frustrating inputs. More results are shown in the supplemental materials.

##  3 Spreading the phase

The first dynamic we observe is when the players are spreading their phases to
escape from the in-phase local minimum into a vortex-phase state global
minimum. This dynamic is enabled by the players’ ability to reduce the
coupling between them until they find a stable state.

Figure 2(a)(top) presents the measured phase of $N=8$ players situated on a
ring with unidirectional coupling as a function of time $t$ and delay $\Delta
t=t/30$. Fig. 2(a)(bottom) presents the measured phase difference between each
player and its delayed neighbor. As seen, during the first 3 seconds all phase
difference converges to near zero, indicating that all players are
synchronized in phase, and remain so until $t=8$. This first stage is marked
as stage I. The average phase of each player during this stage, presented as
the blue circles in the inset of Fig. 2(a), confirms that all the players have
nearly the same phase. When the delay is further increased, one of the players
starts to ignore its neighbor, indicated by the linearly increasing blue curve
in the phase difference in Fig. 2(a)(bottom). Thus, the other players can
freely change their phase to follow the increasing delay between them. This is
marked as stage II. The phase of the players spread until reaching a total
phase difference of $2\pi$ between the first and the last player, thus forming
a stable vortex solution with a topological charge $n=1$ pal2015phase , shown
as the black circles in Fig. 2(a). This vortex solution remains stable as the
delay is further increased, indicated by a constant phase difference between
all coupled neighbors (marked as stage III). The average phase of all players
in stage III is shown as red circles in the inset of Fig. 2(a), and verifies
the expected linear phase of the vortex state.

Figure 2: Coupled $N=6$ players situated on a ring in a unidirectional
coupling, according to the inset scheme, starting from an in-phase state of
synchronization and finding the vortex state. (a) (top) Measured phase of each
player as a function of time as we increase the coupling delay between the
players. Each color denotes the phase of a player according to the inset
scheme. Black circles denote the topological charge of the system as a
function of time. (bottom) The difference between the phase of each player and
the phase of its delayed neighbor. (inset) The phase of all players in a state
of in-phase synchronization (stage I, blue circles) and the phase of all
players in a vortex state (stage III, red circles). (b) The effective
potential of the system as a function of time and coupling delay. In stage II
the effective coupling strength is reduced (see text) thus eliminating the
potential barrier between the in-phase and the vortex states to enable the
system to reach the global minimum vortex state. The trajectory of the system
is denoted by the white dashed curve.

Following leylaz2021identification_energy1 ; roy1992dynamical ;
fridman2010passive ; fridman2012measuring , we use the Kuramoto model for $N$
oscillators on a ring with unidirectional time-delayed coupling
kuramoto2012chemical_kur1 ; kuramoto1987statistical_kur0 ;
strogatz2000kuramoto_kur2  and uniform frequency $\omega$ to introduce an
effective potential whose local and global minima dictate the players’
dynamics. The phase of each oscillator $\varphi_{n}(t)$ follows:

$\frac{\partial\varphi_{n}(t)}{\partial
t}=\omega+\kappa\sin(\varphi_{n+1}(t-\Delta t)-\varphi_{n}(t)),$ (1)

with $\kappa$, and $\Delta t$ denoting the strength and delay of the coupling.
The periodic boundary conditions, $\varphi_{N+1}=\varphi_{1}$ dictate
$\sum_{n=1}^{N}\Delta\varphi_{n}=0$, where
$\Delta\varphi_{n}(t)=\varphi_{n+1}(t)-\varphi_{n}(t)$. Assuming uniformity
$\Delta\varphi_{n}=\Delta\varphi$, without loss of generality, we obtain:

$\frac{\partial\Delta\varphi(t)}{\partial t}=-\frac{\partial
V(\Delta\varphi)}{\partial\Delta\varphi}.$ (2)

where:

$V(\Delta\varphi)=-\Omega_{n}\Delta\varphi-\frac{\kappa}{N-1}\cos((N-1)\Delta\varphi+\omega\Delta
t)+\kappa\cos(\Delta\varphi+\omega\Delta t),$ (3)

is the effective potential governing the dynamics of the system. The detailed
analytical derivation is presented in the supplemental materials.

The calculated effective potential for constant tempo is shown in Fig. 1(d) as
a function of $\Delta t$ and $\Delta\varphi$. For $\Delta t=0$ it reveals a
global minimum at $\Delta\varphi=0$, where all oscillators have the same
frequency and phase, namely, an in-phase state of synchronization. However,
when the delay increases beyond $\omega\Delta t>\pi/(2N)$, the
$\Delta\varphi=0$ becomes a local minimum and the vortex state emerges as a
new global minimum at $\Delta\varphi=-2\pi/N$. Between the in-phase state and
the vortex state, there is a potential barrier. Therefore, increasing the
delay adiabatically, while the system stays in the in-phase state of
synchronization, transfers the system to a local minimum.

To account for the players’ dynamics observed in Fig. 2, we assume that the
coupling strength in stage II is reduced by the player’s tendency to ignore
frustrating inputs. Specifically, we assume
$\kappa(t)=\kappa(t=0)\cos^{2}(\Delta tN\pi/T)$. The resulting modified
effective potential is shown in Fig. 2(b). The dashed curve follows the system
trajectory in the phase space. The system starts in an in-phase state of
synchronization and as the delay increases, the coupling drops leading to a
lower potential barrier. Thus, the system can adiabatically evolve into the
vortex state. Finally, the coupling increases back to its original value
$\kappa(t=0)$.

When the delay between the players is further increased, the first-order
vortex becomes unstable and the next-order vortex becomes stable. We observed
such multiple transitions to higher-order vortex states with $N=16$ coupled
violin players on a unidirectional ring. The measured phases of all the
players together with the topological charge of the network are shown in Fig.
3(a). In the inset, we show the phase of the players at four representative
times, 1.5, 13.5, 24, and 34 seconds, with topological charges $n=$ 0, 1, 2,
and 3, respectively. As the players’ phases spread over a wider range, the
network reaches a higher vortex state, denoted by a higher topological charge.
We also calculate the effective potential as a function of time, again
assuming $\kappa(t)=\cos^{2}(\Delta tN\pi/T)$, showing how the system can
evolve from the in-phase to each order of vortex.

Figure 3: Coupled $N=16$ players situated on a ring in a unidirectional
coupling, according to the inset scheme, showing spreading of the phase and
reaching high order vortex states. (a) The measured phase of $N=16$ coupled
violin players together with the topological charge of the network as we
increase the delayed coupling between them, showing higher-order vortex
states. (Inset) the phase of the 16 players at $t=1.5,13.5,24$, and 34
seconds, with topological charges $n=$ 0, 1, 2, and 3, respectively. (b) The
effective potential of the system as a function of time and coupling delay
shows how the system evolves.

##  4 Slowing the tempo

The second dynamic we observe is the slowing down of the players’ tempo as an
alternative strategy to maintain a stable in-phase synchronization state in
the presence of delayed coupling chafe2010effect_delay . The measured phase
results as a function of time for six coupled players are shown in Fig. 4(a).
From these results, we evaluate the tempo of the players by calculating the
average derivative of the phase. As seen, the initial (natural) tempo of 60
bpm slows down significantly for long delay times reaching about 6 bpm for
$\Delta t>1s$. This slowing down lets players maintain the in-phase state of
synchronization by keeping the coupling delay small relative to the tempo.

To quantitatively analyze the tempo slowing, we resort again to the Kuramoto
model kuramoto2012chemical_kur1 ; kuramoto1987statistical_kur0 ;
strogatz2000kuramoto_kur2 . Assuming $\Delta t<T/N$, we can expend
$\varphi_{n}(t-\Delta t)\approx\varphi_{n}(t)-\Delta
t\partial\varphi_{n}/\partial t$ and $\sin(\alpha)\approx\alpha$ to obtain
from Eq. 1 the phase difference as a function of time for
$\Delta\varphi_{N-1}$:

$\frac{\partial\Delta\varphi(t)}{\partial
t}=\frac{\Omega-N\kappa\Delta\varphi}{1-(N-1)\kappa\Delta t},$ (4)

where $\Omega$ is the average tempo difference between the players. Then we
obtain,

$\frac{\partial\varphi(t)}{\partial t}=\frac{\omega}{1+N\kappa\Delta t},$ (5)

where $\varphi=\sum\varphi_{n}/N$ is the average phase of all oscillators.
Thus, the tempo of the coupled oscillators slows down as long as the players
stay phase locked, ensuring that the condition $\Delta t<T/N$ is satisfied and
indicating that our assumptions are valid. Using Eq. 5, to fit the measured
tempo (blue curve in Fig. 4(a)) yields excellent agreement, where the fit
parameters $\omega=0.29Hz$ and $\kappa=0.63$ are consistent with our system.

Figure 4: Coupled $N=6$ players situated on a ring in a unidirectional
coupling showing tempo slowing down. (a) Experimental measurements of the
phase and average tempo as we increase the delayed coupling between them. Each
color denotes the phase of a player according to the inset scheme. (b)
Numerical calculations of the system. (c) The effective potential of the
system as a function of time when the tempo of the players is following Eq. 5
indicating that the in-phase state stays the global minima for arbitrary large
delays.

Next, We perform numerical simulations of the Kuramoto model (Eq. 1) for six
coupled players with a coupling strength of $\kappa=0.6$. The calculated phase
of all the players as a function of time/delay together with the average tempo
are shown in Fig. 4(b). As evident, the exact numerical simulations agree with
the measured experimental results as well as with the analytical approximation
of Eq. 5 ((blue curve in Fig. 4(b)).

Finally, We calculate the effective potential as a function of time with the
tempo obtained from Eq. 5. As evident, the in-phase state of synchronization
remains the global minimum even though the coupling delay increases.
Therefore, the system is following the dashed line and remains in the in-phase
synchronization state for all coupling delays.

##  5 Oscillation death and amplitude death

In this section, we present two additional mechanisms that have been observed
in other networks on coupled nonlinear oscillators: oscillation death
ermentrout1990oscillator_death  and amplitude death zou2013generalizing_death2
, and show how both enable the human network to escape from a local minimum
(in-phase synchronization) into a global one (vortex state).

When the tempo slows too much, the players can get stuck in a state of
oscillation death ermentrout1990oscillator_death  where all the players are
playing the same note indefinitely, thereby maintaining a degenerate form of
synchronization. Representative results of such oscillation death for four
coupled violin players are shown in Fig. 5(a). The four players are slowing
down their tempo, and after 40 seconds they all play the same note for 20
seconds. Then, the players spontaneously revive the oscillation when one of
the players stopped following its neighbor. Reviving oscillations after
oscillation death typically requires external perturbation to the system
zou2013generalizing_death2 , but here we demonstrate that human networks can
revive the oscillations spontaneously. In addition, they revive the
oscillations into the stable vortex state as indicated by the topological
charge which jumps to $n=1$ when the oscillations revive.

Figure 5: Coupled $N=4$ players situated on a ring in a unidirectional
coupling showing oscillation and amplitude death. (a) Coupled violin players
reach a state of oscillation death. The players slow down their tempo until
all the players play the same note. After 20 seconds, the players revive the
oscillation spontaneously into a vortex state. Each color denotes the phase of
a player according to the inset scheme. (b) Coupled violin players show the
amplitude death of one player. The upper graph shows the phase of all the
players together with the topological charge as a function of time. The lower
graph shows the playing amplitude of Player 1 as a function of time. We see
that Player 1 stops playing around $t=5s$ until the system finds the stable
first-order vortex state.

Violin players can adjust not just the phase and tempo but also their playing
amplitude. In highly frustrating situations one or more of the players can
reduce their amplitude significantly, known as amplitude death
zou2013generalizing_death2 . Representative measurements of such amplitude
death are shown in Fig. 5(b). Here, the coupling delay of four violin players
is increased until one of the players, denoted as Player 1, cannot synchronize
with Player 4 leading to frustration. Therefore, this player stops playing, as
evidenced by its nearly vanishing amplitude, shown in the lower graph in Fig.
5(b). When one of the players stops playing, the closed ring effectively
changes into an open ring topology where all the other players are free to
shift their phases according to the coupling delay. After a few seconds, they
find the vortex state, as indicated by the measured topological charge, and
Player 1 resumes playing.

##  6 discussion and conclusions

To conclude, we investigated the synchronization dynamics of coupled violin
players with ring configuration and unidirectional time-delayed coupling. This
configuration is governed by a potential landscape that includes well-defined
local and global minima that are controlled by the coupling delay time. By
starting with a zero coupling delay, we prepared the system in a globally
stable in-phase synchronization state and then adiabatically increased the
coupling delay such that the in-phase state became a local minimum. We
observed four different routes for the system to escape this local minimum
into the global one.

In the first, the players change their coupling strength and shift from the
local minimum of the in-phase state into the stable vortex state. In the
second, the player slow down their tempo so the in-phase state remained the
global minimum. In the third, all the players play the same note indefinitely
which is a stable state regardless of the coupling delay. In the fourth, one
of the players stopped playing, effectively changing the system topology into
an open ring, thus allowing the players to find the vortex state, and then
resume playing.

Our results indicate that human networks are more robust than other networks
since they have unique methods for escaping local minima into global ones. The
results shed new light on the dynamics of human networks and how a group of
humans can reach synchronization while escaping local minima. It has
implications in decision-making theory, politics, economics, and for other
networks where each node in the network can change its coupling strength or
tempo.

Supplementary information All the raw measured data of all the experiments are
available in 10.6084/m9.figshare.22822103.

Acknowledgments

If any of the sections are not relevant to your manuscript, please include the
heading and write ‘Not applicable’ for that section.

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# Detecting Quadratically Coupled Ultra-light Dark Matter with Stimulated
Annihilation

Yuanlin Gong<EMAIL_ADDRESS>Department of Physics and Institute of
Theoretical Physics, Nanjing Normal University, Nanjing, 210023, China  Xin
Liu<EMAIL_ADDRESS>Department of Physics and Institute of Theoretical
Physics, Nanjing Normal University, Nanjing, 210023, China  Lei Wu
<EMAIL_ADDRESS>Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing, 210023, China  Qiaoli Yang
<EMAIL_ADDRESS>Siyuan Laboratory and Department of Physics, Jinan
University, Guangzhou 510632, China  Bin Zhu<EMAIL_ADDRESS>Department of Physics, Yantai University, Yantai 264005, China

###### Abstract

Ultra-light Dark Matter (ULDM) is one of the most promising DM candidates. Due
to the Bose enhancement, we find the annihilation rate of the ULDM in the
presence of background photon radiation can be greatly enhanced and produce a
distinctive reflected electromagnetic wave with an angular frequency equal to
the ULDM mass. We propose to utilize such stimulated annihilation to probe the
ULDM with the electromagnetic quadratic coupling by emitting a beam of radio
into space. With a power of 50 MW emitter, we forecast the sensitivity of
quadratic coupling in different local halo models for low-frequency radio
telescopes, such as LOFAR, UTR-2 and ngLOBO.

##  I Introduction

Figure 1: Conceptual design of our proposed experiment. A powerful radio beam
(blue wavy line) is sent to space to stimulate the annihilation of the ULDM
(red bullet). The reflected radio (red wavy line) will be detected by the
array telescope.

Dark matter, the invisible substance accounting for more than 80% of the
matter in the universe, continues to be a compelling mystery in modern physics
[1]. The extensively studied canonical cold dark matter models, represented by
weakly interacting massive particles, are attractive, providing the correct
relic abundance, and simultaneously solving other modern puzzles, such as the
hierarchy problem [2]. However, the recent null experiment results put
increasingly stringent constraints on various WIMP models [3].

The ultra-light dark matter (ULDM) that features a spin-0 particle with a mass
ranging from $10^{-24}$ eV to eV is a competitive alternative, which could be
produced through the misalignment mechanism or its variants in the early
universe [4, 5, 6, 7, 8]. The wave-like nature of these particles can not only
preserve the merits of cold dark matter but also may provide a solution to the
small-scale structure problem [9, 10, 11]. Moreover, recent study on the
galactic scale such as gravitationally lensed images indicates the increasing
success of the wave-like DM versus the particle-like DM [12]. Besides, the
ULDM also possibly addresses the Strong CP problem [13, 14, 15, 16],
aforementioned hierarchy problem [17], and dark energy [18, 19, 20].

The distribution of the ULDM near Earth is crucial for its detection in
laboratory experiments. Aside from the DM halo of the Milky Way halo [21, 22,
23], the caustic ring model describing the full phase space distribution of
the Milky Way is also well motivated and established [24, 25, 26]. It results
in the occurrence of flows with varying density and dispersion near the Earth.
On the other hand, the formation of a local ULDM dark halo [27, 28, 29] bound
to other gravitational sources, such as the Sun and Earth, provides an
alternative profile possibility. The Sun and Earth halo result in much higher
dark matter densities around the Earth’s vicinity compared to the Milky Way
halo over a wide mass range, which could significantly affect the sensitivity
of ULDM detection.

Note that the approaches to detecting the ULDM strongly depend on its
interactions with the Standard Model (SM) particles. Generically, the linear
couplings of the ULDM with the SM particles are dominant, however, which have
been tightly constrained by atom clocks [30, 31, 32, 33, 34], atomic
spectroscopy [35, 36, 37, 38, 39], laser interferometry [40, 41],
gravitational-wave detectors [42, 43, 44], astrophysical probes [45, 46, 47,
48, 49, 50] and others [51, 52, 53, 54, 55, 56, 57]. Besides, the enhanced
emission of the axion clusters or axion-like particles may provide a
complementary bound [58, 59, 60]. On the other hand, the quadratic
interactions of the ULDM dominating over the linear ones in recent years have
drawn particular interest, such as in theories with $\mathbb{Z}_{N}$ symmetry
[61, 62] and relaxion mechanism [63]. Despite the above experimental bounds,
the parameter space of the quadratic interactions remains largely unexplored.

In this article, we propose to probe the quadratic interaction of ULDM by
detecting the reflected electromagnetic wave signal produced from the
stimulated annihilation (as shown in Fig. 1). In contrast with the spontaneous
process, the flux of photons from the stimulated annihilation can be greatly
enhanced by the presence of ambient photons, due to Bose enhancement. Since
the power for signal photons highly depends on the local DM density, we
consider four different local DM halos, the iso-thermal halo, the caustic ring
model, the Sun halo and the Earth halo in our analysis. By using a radio
emitter with a power of 50 MW and low-frequency radio telescopes, we present
the new projected limit of the quadratic coupling of ULDM in the mass range of
$2.07\times 10^{-8}\mathrm{\leavevmode\nobreak\ eV}\sim 1\times
10^{-6}\mathrm{\leavevmode\nobreak\ eV}$ for all of the halos.

##  II Stimulated Annihilation

The quadratic interaction of the ULDM field $\phi$ with the electromagnetic
fields $A_{\mu}$ is given by,

$\mathcal{L}=\frac{d_{e}^{(2)}}{4}\frac{2\pi}{M_{\mathrm{Pl}}^{2}}\phi^{2}{F}_{\mu\nu}F^{\mu\nu},$
(1)

where the field strength
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$, $d_{e}^{(2)}$ is the
quadratic coupling constant, and $M_{\mathrm{Pl}}=1.22\times
10^{19}\leavevmode\nobreak\ \mathrm{GeV}$ is the Planck mass (for the pseudo-
scalar case, see [64]). For simplicity, we assume
$g^{\prime}=\frac{2\pi}{M_{\mathrm{Pl}}^{2}}d_{e}^{(2)}$ in our calculations.
Since the DM is non-relativistic in the halo, the angular frequency of the
ULDM is approximate to its mass, $\omega_{\phi}=m_{\phi}$. With Eq. 1, we can
have the cross-section of the spontaneous annihilation process
$\phi\phi\to\gamma\gamma$,

$\sigma_{0}=\frac{1}{32\pi}\frac{1}{\beta}g^{\prime 2}m_{\phi}^{2},$ (2)

where the factor $\beta$ is the velocity of ULDM. Due to the tiny coupling and
the small mass of the ULDM, the spontaneous annihilation rate is highly
suppressed. We note that a certain incoming photon can stimulate the
annihilation of the ULDM and thus produce an observable electromagnetic
signal. There are two methodologies to describe the stimulated emission: 1)
One involves Boltzmann equation in where the spontaneous annihilation
amplitude of ULDM is enhanced by the Bose enhancement from the
indistinguishability of identical bosons [59, 60, 65]. 2) The one resorts to
the modified electrodynamics, where the question become a radiation problem.
The effective current density is provided by the interaction in Eq. 1. The
$\phi$ manifests itself as a classical coherent field and the stimulated
enhancement is described through the resonance [64, 66]. We continue our
proposal in method 1, while giving the derivation in method 2 in the
Supplemental Material.

To obtain the production rate of the photons from stimulated annihilation, we
first consider the number of states in the phase space of spontaneous
annihilation process $\phi\phi\to\gamma\gamma$. In the vacuum, there are
$f_{\phi}$ ULDMs with the certain momentum and zero photons in the phase space
of the initial states. After annihilation, the final state will contain
$(f_{\phi}-1)$ ULDMs and two photons with two directions of momentum that
anti-parallel to each others in the center-of-mass frame of the two ULDMs,
i.e.,
$|\mathrm{i}\rangle_{0}=|f_{\phi},f_{\phi};0,0\rangle,\leavevmode\nobreak\
|\mathrm{f}\rangle_{0}=|f_{\phi}-1,f_{\phi}-1;1,1\rangle.$ However, due to the
Bose enhancement of identical photons, if the annihilation occurs in the
background of $f_{\gamma}$ photons, this process will be stimulated and leads
to

$|\mathrm{i}\rangle=|f_{\phi},f_{\phi};f_{\gamma},f_{\gamma}\rangle,\leavevmode\nobreak\
|\mathrm{f}\rangle=|f_{\phi}-1,f_{\phi}-1;f_{\gamma}+1,f_{\gamma}+1\rangle.$
(3)

Then, we can have the scattering amplitude of the stimulated annihilation

$\displaystyle\mathcal{M}_{\mathrm{i\to f}}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{0}^{\dagger}f_{\phi}(f_{\gamma}+1),$ (4)

where $\mathcal{M}_{0}$ is the spontaneous annihilation amplitude. The inverse
process of two photons annihilating to two ULDMs in the vicinity of
$f_{\gamma}$ ambient photons corresponds to the following matrix element:

$\mathcal{M}_{\mathrm{f\to i}}=\mathcal{M}_{0}(f_{\phi}+1)f_{\gamma}.$ (5)

In the presence of ambient photons, the effective annihilation amplitude for
the stimulated annihilation of the ULDMs is determined by the difference
between the amplitude of the annihilation and production processes of the
photons:

$\displaystyle|\mathcal{M}_{\mathrm{i\to f}}|^{2}-|\mathcal{M}_{\mathrm{f\to
i}}|^{2}$ $\displaystyle=$
$\displaystyle|\mathcal{M}_{0}|^{2}[f_{\phi}^{2}(f_{\gamma}+1)^{2}-(f_{\phi}+1)^{2}f_{\gamma}^{2}],$
(6) $\displaystyle=$
$\displaystyle|\mathcal{M}_{0}|^{2}[f_{\phi}^{2}+2f_{\gamma}f_{\phi}^{2}-2f_{\phi}f_{\gamma}^{2}-f_{\gamma}^{2}],$
$\displaystyle\approx$
$\displaystyle|\mathcal{M}_{0}|^{2}f_{\phi}^{2}(1+2f_{\gamma}).$

The four terms in the second line are interpreted as the contributions from
spontaneous annihilation, stimulated annihilation, inverse stimulated
annihilation and inverse spontaneous annihilation. It is then clear that the
factor $2f_{\gamma}$ is the enhancement for stimulated annihilation compared
to spontaneous annihilation. We have used the approximation $f_{\phi}\gg
f_{\gamma}$ in the last line.

From the Boltzmann equation, we can obtain the annihilation rate of the ULDMs
from the stimulated annihilation process
$\phi(p_{1})\phi(p_{2})\to\gamma(k_{1})\gamma(k_{2})$,

$\displaystyle\dot{n}_{\phi}$ $\displaystyle=$ $\displaystyle-\int
d\Pi_{\phi}d\Pi_{\phi}d\Pi_{\gamma}d\Pi_{\gamma}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-k_{1}-k_{2})$
(7) $\displaystyle\times\frac{1}{4}\cdot[|\mathcal{M}_{\mathrm{i\to
f}}|^{2}-|\mathcal{M}_{\mathrm{f\to i}}|^{2}]$ $\displaystyle=$
$\displaystyle-4\beta n_{\phi}^{2}\sigma_{0}(1+2f_{\gamma}),$

where $d\Pi_{i}=g_{i}dp^{3}/((2\pi)^{3}\cdot 2E_{i})$ is the usual phase-space
volume. The factor of $1/4$ in the first line is the symmetry factor for
identical particles in the initial and final states. $n_{i}$ is the number
density of the ULDM or the photon, which is related to phase-space
distribution by
$n_{i}=\int\frac{g_{i}}{(2\pi)^{3}}f_{i}(\mathbf{p})d^{3}p_{i}$. The
production rate of the photons is the negative of the annihilation rate of the
ULDMs. This is greatly enhanced by the factor $2f_{\gamma}$, which arises from
the stimulation of the ambient photons as $\omega_{\gamma}=m_{\phi}$.
Moreover, the production rate of the photons depends on the ULDM density as
$\rho_{\mathrm{DM}}^{2}$, rather than ${\rho_{\mathrm{DM}}}$.

A straightforward consideration inspired by the Bose enhancement and
substantial occupation number of the ULDM is the more intriguing process of
$n$ ULDMs scatter into two photons arising from the effective interactions
described by the Lagrangian
$\mathcal{L}=\frac{1}{4}\frac{d_{e}^{(n)}}{M_{\mathrm{Pl}}^{n}}\phi^{n}F_{\mu\nu}F^{\mu\nu}$,
since the annihilation rate of the ULDMs will dependent on $n_{\phi}^{n}$,
being potential huge at the first glance. A closer derivation results that
$\dot{n}_{\phi}$ is proportional to the factor
$F=\left(\frac{\rho_{\phi}}{2M_{\mathrm{Pl}}^{2}m_{\phi}^{2}}\right)^{n}\frac{n^{4}}{n!}$.
We find that this factor decreases in power with $n$ for the
$m_{\phi}>1.07\times 10^{-31}\leavevmode\nobreak\ \mathrm{eV}$ for all the
halo model we considered here, despite the much higher density $\rho_{\phi}$
in the Earth and Sun halo at some mass ranges. For the lower mass ranges , the
ULDMs can not be the all dark matter and are constrained by the galaxy
structure formation. Thus, as claimed before [61, 62, 63], in the scenarios
where the linear order $n=1$ is absent, the dominant contribution naturally
emanates from the quadratic $n=2$ order, which is the focus of this work.

##  III Signal Power

The signal power received by the telescope can be obtained by integrating the
Eq. 7 over time, the solid angle along the direction of the outgoing emission
beam, the frequency, and the area

$\displaystyle P$ $\displaystyle=$ $\displaystyle-\int dAd\nu
d\Omega\int_{0}^{t_{\mathrm{off}}}dt\dot{n}_{\phi},$ (8) $\displaystyle=$
$\displaystyle\frac{1}{8}\frac{{g^{\prime}}^{2}}{{m}_{\phi}^{2}}\frac{P_{0}}{\Delta\nu_{\phi}}\int_{0}^{t_{\mathrm{off}}}{\rho_{\phi}^{2}}{dt}.$

where the duration time of emission is denoted as $t_{\mathrm{off}}$. We
ignored the interaction between the photon and the electron in the environment
and factorized out the power of the source as,

$\displaystyle P_{\mathrm{0}}=\int dAd\nu d\Omega n_{\gamma}.$ (9)

Here we assume that $f_{\gamma}$ is a Gaussian-like function with an expected
value $\omega_{\gamma}$, and is related to
$n_{\gamma}=\frac{2}{(2\pi)^{3}}4\pi\omega_{\gamma}^{2}\Delta\omega_{\gamma}{f}_{\gamma}$
by averaging it over the bandwidth
$\Delta\omega_{\gamma}=\Delta\omega_{\phi}\equiv 2\pi\Delta\nu_{\phi}$, with
the ULDM bandwidth $\Delta\nu_{\phi}=2\nu_{\phi}\sigma_{v}$ depending on the
velocity dispersion $\sigma_{v}$ of the ULDM [60, 65, 59]. Note that $P_{0}$
is only determined by the properties of the emitter. In addition, it can be
seen that the signal power $P$ in Eq. 8 is sensitive to the local density of
the ULDM halo, which is commonly assumed to be the iso-thermal DM halo.
However, the distribution of ULDM in the Milky Way may differ from the
predictions of the iso-thermal DM halo. The simulation for the formation of
the Galactic halo suggests the enhancement (or suppression) of the local DM
density due to the formation of “clumps” or streams [67]. Owing to the self-
interactions or topological properties of the ULDM, the possibility of the
formation of condensates [68], clusters [69], boson stars [70, 27, 28, 29], or
domain walls [31, 71] have been discussed . Their impact on the detection of
ULDM was considered in [32] for deep space atomic clocks, in [37, 38] for
atomic spectroscopy, in [44] for gravitational-wave detector and [72] with
neutrino oscillations. In our phenomenological study, we consider four
different halo models,

  * •

Iso-thermal halo model. This standard density model of the Milky Way from
N-body simulations predicts a local energy density of $\rho_{I}\approx
0.3\mathrm{\leavevmode\nobreak\ GeV/cm^{3}}$ with velocity dispersion of
$\sigma_{I}=270/\sqrt{3}\mathrm{\leavevmode\nobreak\ km/s}$ [21, 22, 23].

  * •

Caustic ring model. Motivated by the description of the full phase space
distribution of the Milky Way halo, the caustic ring model predicts the
configuration of the caustics and flows [24]. The properties of the flows are
constrained by the observations of the IRAS and Gaia maps [26]. The resulting
local dark matter velocity distribution is dominated by the Big flow with
velocity dispersion less than $\sigma_{c}=70$ m/s [25] and density as high as
$\rho_{c}=10\mathrm{GeV/cm^{3}}$ [26].

  * •

Sun halo model. As an extension of the boson star, it has been recently
discussed in [27, 28, 32, 29]. This was initially motivated by the suggestion
from the large-scale numerical simulations of galaxy formation for fuzzy dark
matter ($m_{\phi}\sim 10^{-22}\leavevmode\nobreak\ $eV) without [73, 74] and
with [75] the presence of baryons that halo-like configuration can form in the
central cores of galaxies. In the presence of quartic self-interactions and
subsequent gravitational focusing, an external gravitational source, such as
the Sun, in the background of virialized DM can effectively capture the ULDM,
to form an over-density local halo. For the mass range of our
interest($m>10^{-9}$ eV, see below), its density $\rho_{\mathrm{sh}}$ around
the Earth is much lower than that of the iso-thermal halo and decreases
exponentially with the increase in the ULDM mass. The velocity dispersion is
$1/(m_{\phi}R_{\mathrm{sh}})$, where $R_{\mathrm{sh}}$ is the radius of the
Sun halo.

  * •

Earth halo model. Similar to the Sun halo model, it is also disscussed in [27,
28, 29, 32]. [27] showed that the maximally allowed value of the energy
density $\rho_{\mathrm{eh}}$ of the Earth halo is given by

$\rho_{\mathrm{eh}}(r)\propto\left\\{\begin{array}[]{ll}\exp\left(-2r/R_{\mathrm{eh}}\right)&\text{
for }R_{\mathrm{eh}}>R_{\mathrm{E}},\\\
\exp\left(-r^{2}/R_{\mathrm{eh}}^{2}\right)&\text{ for }R_{\mathrm{eh}}\leq
R_{\mathrm{E}}.\end{array}\right.$ (10)

Here $R_{\mathrm{eh}}$ is the radius of the Earth halo, which is a function of
the ULDM mass [27]. $R_{\mathrm{E}}$ is the radius of the Earth. As a
comparison with the iso-thermal DM halo, the Earth halo has a much higher
density. On the other hand, with the increase in the ULDM mass, the Earth halo
density decreases exponentially. For the estimation below, we take
Supplementary Figure 2 in [27], which provides a detailed description of the
maximally allowed value of energy density $\rho_{\mathrm{eh}}$.

##  IV Results and Discussions

Figure 2: The expected bounds on the plane of the dimensionless coupling
$d_{e}^{(2)}$ versus the ULDM mass $m_{\phi}$ for the iso-thermal halo,
caustic ring model and the Earth halo. The array telescopes LOFAR, UTR-2 and
ngLOBO are considered here. The grey dashed lines are the limits from the BBN
and Supernova [36].

In this work, our frequency range of interest is about $5\sim
240\mathrm{MHz}$. For lower frequencies, the impact of the ionosphere will
become significant, while for higher frequencies, the signal in will be
suppressed by the mass of the ULDM as is seen in Eq. 8. To detect such low-
frequency radio signals, we use the LOFAR [76], UTR-2 [77] and ngLOBO [78] to
estimate the sensitivity. LOFAR, as one of the new-generation radio
telescopes, is capable of detecting radio signals in the frequency ranges
$10\leavevmode\nobreak\ \mathrm{MHz}\sim 90\leavevmode\nobreak\ \mathrm{MHz}$,
and $110\sim 240$ MHz with an unparalleled sensitivity owing to the novel
phased-array design, dense core array, and long interferometric baselines [76,
79]. Different from the traditional dish telescope, a number of dipole antenna
elements are well arranged to compose a circular array with a diameter of
$70\sim 80$ meter, which enables it to observe in several modes and to detect
the transient pulse signal with its high time and frequency resolution. The
minimal frequency resolution of LOFAR is about 700 Hz [76]. As for the UTR-2,
a T-shaped antenna array can achieve a lower operating frequency range
$8-32\leavevmode\nobreak\ \mathrm{MHz}$ with a frequency resolution down to 4
kHz [77]. The low band of the ngLOBO is to cover the frequency range
$5-150\leavevmode\nobreak\ \mathrm{MHz}$ [78], the resolution of which can
reach 1 kHz at least [80, 81]. The relevant parameters of the three telescopes
are given in Table. 1.

Tele. | Fre.(MHz) |  $T_{\mathrm{sys,m}}$(K) |  $\Delta\nu_{\mathrm{res}}$(kHz)
---|---|---|---
LOFAR | 10-90 | $3.5\times 10^{5}$ | 0.7
LOFAR | 1 10-240 | 7 66 | 0.7
UTR-2 | 8-32 | $6.1\times 10^{5}$ | 4
ngLOBO | 5-150 | $2.0\times 10^{6}$ |  $\leq$1
Table 1: The frequency range, maximum system temperature $T_{\mathrm{sys,m}}$
and frequency resolution $\Delta\nu_{\mathrm{res}}$ for LOFAR [76], TUR-2
[77], and ngLOBO [78]. The values of $T_{\mathrm{sys,m}}$ are evaluated at the
frequencies $\nu=10\leavevmode\nobreak\ \mathrm{MHz}$,
$\nu=110\leavevmode\nobreak\ \mathrm{MHz}$, $\nu=8\leavevmode\nobreak\
\mathrm{MHz}$, and $\nu=5\leavevmode\nobreak\ \mathrm{MHz}$.

The noise power of a radio array telescope with a frequency bandwidth $\Delta$
during the observing time $t_{\mathrm{off}}$ reads [82]

$P_{\mathrm{n}}=\frac{2kT_{\mathrm{sys}}}{\eta_{\mathrm{s}}}\sqrt{\frac{\Delta}{n_{\mathrm{pol}}t_{\mathrm{off}}}},$
(11)

where $k$ is the Boltzmann constant and $n_{\mathrm{pol}}=2,1,2$ is the number
of polarizations for LOFAR [76], UTR-2 [77] and ngLOBO [80, 81] respectively.
For simplicity, we assume the detection efficiency $\eta_{s}=1$ in our
numerical calculations. The bandwidth is determined by
$\Delta=\mathrm{max}({\Delta\nu_{\phi},\Delta\nu_{\mathrm{res}}})$, where
$\Delta\nu_{\mathrm{res}}$ the telescope frequency resolution.
$T_{\mathrm{sys}}$ is the temperature of the array system. It is caused by
several inevitable noises, such as the cosmic microwave background, the
environment, the galaxy, and the instrument. Among them, galaxy noise is
dominant [83]. Then, we approximate the system temperature as
$T_{\mathrm{sys}}\approx 1.23\times 10^{8}\leavevmode\nobreak\
\mathrm{K}\leavevmode\nobreak\ ({\mathrm{MHz}}/{\nu})^{2.55}$, where $\nu$ is
the frequency of the noise photon [83, 76].

Besides, the transverse velocity $\vec{v}_{\perp}$ of the ULDM perpendicular
to the outgoing radio beam will result in the displacement of the reflected
radio signal from the location of the outgoing power source. Thus, the
duration time $t_{\mathrm{off}}$ in Eq. 8 should be less than the effective
time $t_{\mathrm{eff}}=C\frac{R}{\langle|\vec{v}_{\perp}|\rangle}$, where $R$
is the radius of the array telescope. In the frequency range of our interest,
$\langle|\vec{v}_{\perp}|\rangle$ is about $124\leavevmode\nobreak\
\mathrm{km/s}$ for the iso-thermal halo [64], about 5 km/s for the caustic
ring model [26] and is about $1.2\leavevmode\nobreak\ \mathrm{km/s}$ for the
Earth halo [27]. For the Sun halo, the contribution for transverse velocity
comes from velocity $v_{\mathrm{s}}=1.1\leavevmode\nobreak\ \mathrm{km/s}$ of
ULDM in the frame of the Sun and the relative velocity $v_{\mathrm{r}}\approx
29\leavevmode\nobreak\ \mathrm{km/s}$ between the Sun and the Earth. Thus, we
take ${\langle|\vec{v}_{\perp}|\rangle}$ equals $30\leavevmode\nobreak\
\mathrm{km/s}$. In reality, the geometry factor $C$ depends on the specific
configuration of the emitter relative to the collector in the experiment. Here
we take $C=0.3$ to estimate the sensitivity. Since the height of the
ionosphere is approximately 48 km, the time required for a radio wave pointing
to the telescope to travel from the ionosphere to the telescope is 160 $\mu$s,
which is greater than the effective time for the iso-thermal halo, and not
much smaller than that of the caustic ring and the Sun halos. Meanwhile, we
note that the density of the Earth halo will vanish before the radio reaches
the ionosphere. Therefore, it is justifiable to disregard the interaction
between photons and electrons in the environment.

For an emitter with power $P_{0}$, we can have the total energy consumption
$E_{\mathrm{0}}=NP_{\mathrm{0}}t_{\mathrm{off}}$, where
$N=m_{\phi}/2\Delta\nu_{\phi}$ is the emission times. In the following
calculations, we set $E_{0}=10$ MWyear, $P_{0}=50$ MW and $R=50$ m. By
requiring $P>P_{\mathrm{n}}$, we can obtain the exclusion limit of the
dimensionless coupling $d^{(2)}_{e}$,

$\displaystyle d_{e}^{(2)}$ $\displaystyle<$ $\displaystyle
D\cdot\left(\frac{\mathrm{50\leavevmode\nobreak\
\mathrm{MW}}}{{P}_{0}}\frac{T_{\mathrm{sys}}}{3.5\times
10^{5}\leavevmode\nobreak\
\mathrm{K}}\right)^{{\frac{1}{2}}}\cdot\left(\frac{\Delta}{n_{\mathrm{plo}}}\right)^{\frac{1}{4}}$
(12)

with

$D=\left\\{\begin{array}[]{ll}7.43\cdot
10^{28}\cdot\left(\frac{4219\leavevmode\nobreak\
\mathrm{sec}}{t_{\mathrm{off}}}\right)^{\frac{1}{4}}\left(\frac{m_{\phi}^{3}}{\mathrm{GeV}^{3}}\frac{\mathrm{GeV}^{7}}{\rho_{\mathrm{I}}^{2}t_{\mathrm{eff}}}\right)^{\frac{1}{2}},&\mathrm{Iso}\\\
9.24\cdot 10^{27}\cdot\left(\frac{1\leavevmode\nobreak\
\mathrm{sec}}{t_{\mathrm{off}}}\right)^{\frac{1}{4}}\left(\frac{m_{\phi}^{3}}{\mathrm{GeV}^{3}}\frac{\mathrm{GeV}^{7}}{\rho_{\mathrm{c}}^{2}t_{\mathrm{eff}}}\right)^{\frac{1}{2}},&\mathrm{Cau}\\\
1.80\cdot 10^{28}\cdot\left(\frac{14.7\leavevmode\nobreak\
\mathrm{sec}}{t_{\mathrm{off}}}\right)^{\frac{1}{4}}\left(\frac{m_{\phi}^{3}}{\mathrm{GeV}^{3}}\frac{\mathrm{GeV}^{7}}{\int_{0}^{t_{\mathrm{eff}}}\rho_{\mathrm{sh}}^{2}dt}\right)^{\frac{1}{2}},&\mathrm{Sun}\\\
1.85\cdot 10^{28}\cdot\left(\frac{16\leavevmode\nobreak\
\mathrm{sec}}{t_{\mathrm{off}}}\right)^{\frac{1}{4}}\left(\frac{m_{\phi}^{3}}{\mathrm{GeV}^{3}}\frac{\mathrm{GeV}^{7}}{\int_{0}^{t_{\mathrm{eff}}}\rho_{\mathrm{eh}}^{2}dt}\right)^{\frac{1}{2}},&\mathrm{Ear}\\\
\end{array}\right.$ (13)

For the iso-thermal halo, we can take $\Delta=\Delta\nu_{\phi}$. It is because
that the bandwidth of the ULDM is scaled as $15.09\leavevmode\nobreak\
\mathrm{kHz}\cdot(m/0.03\leavevmode\nobreak\ \mu\mathrm{eV})$, which is larger
than the frequency resolution of the LOFAR, UTR-2 and ngLOBO. While for the
caustic ring model, the Sun halo, and the Earth halo, since the frequency
resolution $\Delta\nu_{\mathrm{res}}$ is larger than the bandwidth
$\Delta\nu_{\mathrm{\phi}}$ of ULDM in our mass range, we take
$\Delta=0.7,\leavevmode\nobreak\ 1,\leavevmode\nobreak\ 4$ kHz in Eq. 12 for
LOFAR, UTR-2 and ngLOBO, respectively.

The resulting expected exclusion limits are shown in Fig. 2. We can see that
the bounds for the quadratic coupling $d^{(2)}_{e}$ strongly depend on the
ULDM halo models. They are around $d^{(2)}_{e}<10^{27}$ and $10^{30}$ in the
caustic ring model and the iso-thermal halo, respectively. The bounds for the
iso-thermal halo are weaker than the limits from the Big Bang Nucleosynthesis
(BBN) and Supernova [36], while that for the iso-thermal halo are weaker than
the limit from BBN but stronger that the limit from Supernova by 2-3 orders of
magnitude. Since the density of Sun halo is very low in our mass range, the
corresponding constraint is much weaker than others and thus is not shown. On
the other hand, the limit on $d^{(2)}_{e}$ in the Earth halo model can be
$10^{13}$ to $10^{34}$ in the range of $2.07\times
10^{-8}\mathrm{\leavevmode\nobreak\ eV}<m_{\phi}<8.27\times
10^{-8}\mathrm{\leavevmode\nobreak\ eV}$ ($5<f<20$ MHz). The limits in this
range can be at most 8 orders of magnitude stronger than that of the BBN and
at most 17 orders of magnitude stronger than that of the Supernova. All the
limits shown weaken with the increasing mass. Since the ULDM bandwidth
$\Delta\nu_{\phi}$ for the caustic ring model is larger than the telescope
frequency resolution $\Delta\nu_{\mathrm{res}}$ in the higher mass range, we
should take $\Delta=\Delta\nu_{\phi}$ in these mass range and the limits on
the caustic ring model will be further suppressed. With the improved
measurement of the Gaia maps [84], a more precise local DM distribution can be
reached and reduce the uncertainty in the search for the DM in the future.
Besides, we note that the sensitivity can be further enhanced by increasing
the power of the emitter and enlarging the array telescope radius. In
addition, a better frequency resolution of the telescope will improve the
results in the same work frequency.

##  V Acknowledgement

YG would like to thank Ariel Arza, Hyungjin Kim for useful discussions. This
work is supported by the National Natural Science Foundation of China (NNSFC)
under grants No. 12275134, No. 12335005, No. 12275232, No. 12147228, and No.
12150010.

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Supplemental Material: Detecting Quadratically Coupled Ultra-light Dark Matter
with Stimulated Annihilation

Yuanlin Gong,1 Xin Liu,1 Lei Wu,1 Qiaoli Yang,2 and Bin Zhu,3

1Department of Physics and Institute of Theoretical Physics, Nanjing Normal
University, Nanjing, 210023, China
2Siyuan Laboratory, Physics Department, Jinan University, Guangzhou 510632,
China
3Department of Physics, Yantai University, Yantai 264005, China

At low energy, the effective representation of the ULDM is that of a
coherently oscillating classical field given by [85]
$\phi(\vec{x},t)=\phi_{0}\sum_{j}\alpha_{j}\sqrt{f(v_{j})\Delta
v}\cos(m_{\phi}t+\vec{p}_{j}\cdot\vec{x}+\phi_{j})$, where the oscillating
frequency equals the mass $m_{\phi}$ of the underlying particle, and
$\phi_{0}=\sqrt{2\rho_{\mathrm{DM}}}/m_{\phi}$, depending on the DM density
$\rho_{\mathrm{DM}}$ and the mass, refers to the present-day oscillation
amplitude, $\alpha_{j}$ is a random number of the Rayleigh distribution
$P(\alpha_{j})=\alpha_{j}e^{-\alpha_{j}/2}$, $f(v_{j})$ is the local dark
matter speed distribution, $\phi_{j}$ is a phase factor and $\Delta v$ is the
speed interval. Typically $f(v_{j})$ is a narrow pick in most halo models,
thus the ULDM can effectively be considered a mono-frequency field within the
coherent time.

To derive the power of the reflected electromagnetic signal originating from
the stimulated annihilation in the aspect of modified electrodynamics, we
write again down the Lagrangian density of the Ultra-Light Dark Matter (ULDM)
$\phi$ and electromagnetic fields $A_{\mu}$:

$\begin{split}\mathcal{L}=&-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-J^{\mu}A_{\mu}+\frac{1}{2}\left(\partial_{\mu}\partial^{\mu}-m_{\phi}{}^{2}\right)\phi^{2}+\frac{1}{4}g^{\prime}\phi^{2}F_{\mu\nu}F^{\mu\nu}.\end{split}$
(S.1)

where $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu},J_{\mu}$ are
electromagnetic tensor and current. $m_{\phi},g^{\prime}$ are the mass of
ULDM, and the coupling between ULDM and electromagnetic field, respectively.
The relevant equations of motion read

$\displaystyle\partial_{\mu}F^{\mu\nu}-\frac{1}{2}g^{\prime}\partial_{\mu}\left({\phi^{2}\epsilon}^{\nu\mu\lambda\tau}F_{\lambda\tau}\right)=J^{\nu},$
(S.2)
$\displaystyle\left(\partial_{\mu}\partial^{\mu}-m_{\phi}^{2}\right)\phi-2g^{\prime}\phi\vec{E}\cdot\vec{B}=0.$
(S.3)

For the free space, the existential Lagrangian is
$-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$,which imply the below equation of motion

$\partial_{\mu}F^{\mu\nu}=0.$ (S.4)

With the expression of $F^{\mu\nu}$ and $J^{\mu}$, we can translate the last
three equations into the following ordinary formation

$\displaystyle\vec{\nabla}\cdot\vec{E}=\rho_{\text{el}}+g^{\prime}\vec{\nabla}(\phi^{2}\vec{E}),$
(S.5)
$\displaystyle\vec{\nabla}\times\vec{B}-\partial_{t}\vec{E}=\vec{j}_{\text{el}}+g^{\prime}\vec{\nabla}\times\left(\phi^{2}\vec{B}\right)+g^{\prime}\partial_{t}\left(\phi^{2}\vec{E}\right),$
(S.6) $\displaystyle\vec{\nabla}\times\vec{E}+\partial_{t}\vec{B}=0,$ (S.7)
$\displaystyle\vec{\nabla}\cdot\vec{B}=0,$ (S.8)

and

$\partial_{t}^{2}\phi-\nabla^{2}\phi+m_{\phi}^{2}\phi=-g^{\prime}\phi\vec{E}\cdot\vec{B},$
(S.9)

which may together be named as quadratically coupled ULDM electrodynamics. In
the presence of the background electromagnetic field $\vec{E_{0}}(\vec{x},t)$,
$\vec{B_{0}}(\vec{x},t)$, accounting for the similar status to the charge
density and current density, we can define ULDM-induced charge density
$\rho_{\phi}$ and current density $\vec{j}_{\phi}$ as

$\displaystyle\rho_{\phi}=g^{\prime}\vec{\nabla}(\phi^{2}\vec{E_{0}}),$ (S.10)
$\displaystyle\vec{j}_{\phi}=g^{\prime}\vec{\nabla}\times\left(\phi^{2}\vec{B_{0}}\right)+g^{\prime}\partial_{t}\left(\phi^{2}\vec{E_{0}}\right).$
(S.11)

As we will see shortly, it was $\vec{j}_{\phi}$ that will radiate the
reflected electromagnetic signal with the existence of the electromagnetic
field $\vec{E_{0}}(\vec{x},t)$, $\vec{B_{0}}(\vec{x},t)$ that we send out to
space to stimulated the annihilation of ULDM.

Using radiation gauge $\vec{\nabla}\cdot\vec{A}=0$ and setting
$\vec{j}_{\text{el}}=0$ in eq. S.6, the ULDM-induced electromagnetic radiation
$\vec{A}(\vec{x},t)$ is given by

$(\partial_{t}^{2}-\nabla^{2})\vec{A}=g^{\prime}\left[\phi^{2}\left(\vec{\nabla}\times\vec{B_{0}}\right)+\partial_{t}\phi^{2}\vec{E_{0}}+\phi^{2}\partial_{t}\vec{E_{0}}\right]+\mathcal{O}(g^{\prime
2}).$ (S.12)

where we have neglected the term with $\vec{\nabla}\phi$ due to the fact that
$\vec{\nabla}\phi\ll\partial_{t}\phi$. Usually, the solution to this
inhomogeneous Helmholtz equation is a time-dependent function in terms of the
retarded Green function. But we would like to focus on the resonance part,
which will give us the backward electromagnetic wave relative to the incident
electromagnetic wave we sent out. To this end, we expand $\vec{E}_{0}$ and
$\vec{B}_{0}$ into Fourier modes

$\displaystyle\vec{E}_{0}(\vec{x},t)=\Re\int
d^{3}k\vec{E}_{0}(\vec{k})e^{i(\vec{k}\cdot\vec{x}-\omega t)},$ (S.13)
$\displaystyle\vec{B}_{0}(\vec{x},t)=\Re\int
d^{3}k\hat{k}\times\vec{E}_{0}(\vec{k})e^{i(\vec{k}\cdot\vec{x}-\omega t)},$
(S.14)

with their inverse transformation given by

$\displaystyle\vec{E}_{0}(\vec{k},t)=\Re\int
d^{3}x\vec{E}_{0}(\vec{x},t)e^{-i\vec{k}\cdot\vec{x}},$ (S.15)
$\displaystyle\hat{k}\times\vec{E}_{0}(\vec{k},t)=\Re\int
d^{3}x\vec{B}_{0}(\vec{x},t)e^{-i\vec{k}\cdot\vec{x}}.$ (S.16)

Also notes that $\omega=|\vec{k}|$. Substitute the eq. S.13 and S.14 into S.12
resulting

$\displaystyle(\partial_{t}^{2}-\nabla^{2})\vec{A}=$
$\displaystyle-g^{\prime}\phi_{0}^{2}m_{\phi}\sin(2m_{\phi}t)\vec{E}_{0},$
(S.17)

where the first and third terms in eq. S.12 are canceled by each others due to
the identity $\omega=|\vec{k}|$. We have also used
$\phi(\vec{x},t)=\phi_{0}\cos(m_{\phi}t)$. Then, in the spirit of our previous
discussion, we therefore reasonably have the radiation exhibited as

$\vec{A}(\vec{x},t)=\Re\int
d^{3}k\vec{{A}}(\vec{k},t)e^{i\vec{k}\cdot\vec{x}}.$ (S.18)

Given this expression, one can proceed to perform the inverse transformation
on the eq.(S.17) and get

$\begin{array}[]{ll}&\left(\partial_{t}^{2}+\omega^{2}\right)\vec{\mathcal{A}}(\vec{k},t)e^{i\omega
t}=-{g^{\prime}}\phi_{0}^{2}m_{\phi}{\sin}\left(2m_{\phi}t\right)\vec{E}_{0}(\vec{k})e^{-{i\omega
t}},\end{array}$ (S.19)

where $\vec{\mathcal{A}}(\vec{k},t)$ is the amplitude of $\vec{A}(\vec{k},t)$
and assumed varies slowly in time with respect to $\vec{A}(\vec{k},t)$. It’s
obvious that the resonance for this radiation system occurs at
$\omega-2m_{\phi}=-\omega$, i.e. when $\omega=m_{\phi}$, the spontaneous
annihilation of the ULDM into two photons is resonance enhanced when there are
background electromagnetic fields with the same frequency as the photons
produced from the ULDM. In term of $\vec{\mathcal{A}}(\vec{k},t)$, the last
equation can further rewritten as:

$\begin{array}[]{ll}\left[\left(\partial_{t}^{2}\vec{\mathcal{A}}+2{i\omega}\partial_{t}\vec{\mathcal{A}}+({i\omega})^{2}\vec{\mathcal{A}}\right)+\omega^{2}\vec{\mathcal{A}}\right]e^{{i\omega
t}}=-{g^{\prime}\phi_{0}^{2}}m_{\phi}{\sin}\left(2m_{\phi}t\right)\vec{E}_{0}(\vec{k})e^{-{i\omega
t}}.\end{array}$ (S.20)

To get the evolution with the time, we neglect
$\partial_{t}^{2}\vec{\mathcal{A}}$ respect to
$2{i\omega}\partial_{t}\vec{\mathcal{A}}$ and take solely the resonance term
into consideration, thus giving rise to

$\partial_{t}\vec{\mathcal{A}}=\frac{1}{4\omega}{g^{\prime}\phi_{0}^{2}}m_{\phi}\vec{E}_{0}(\vec{k})e^{2i\left(m_{\phi}-\omega\right)t}.$
(S.21)

In the light of our assumption about the $\vec{\mathcal{A}}(\vec{k},t)$, we
can solve the above equation by the initial condition
$\vec{\mathcal{A}}(\vec{k},0)=0$, and obtain the result displayed as

$\vec{\mathcal{A}}\left(\vec{k},t\right)=\frac{1}{4\omega}{g^{\prime}\phi_{0}^{2}}m_{\phi}\vec{E}_{0}\left(\vec{k}\right)e^{{i\epsilon
t}}\left(\frac{{\sin}({\epsilon t})}{\epsilon}\right),$ (S.22)

where $\epsilon=m_{\phi}-\omega$ is defined for convenience. With the
approximation

$\left(\frac{\sin(\epsilon t)}{\epsilon}\right)^{2}\to\pi t\delta(\epsilon),$
(S.23)

in large $t$, we work out the reflected signal power as

$P=\frac{1}{16}{g^{\prime}}^{2}\phi_{0}^{4}m_{\phi}^{2}t\int{d\omega}\frac{{d}P_{0}}{{d\omega}}(\omega)\pi\delta(\epsilon)=\frac{1}{8}\frac{{g^{\prime}}^{2}\rho^{2}_{\mathrm{DM}}}{m_{\phi}^{2}}\frac{dP_{0}}{d\nu}t,$
(S.24)

where $P_{0}$ is the power of the background electromagnetic wave and is
expressed as

$P_{0}=\int{d\omega}\frac{{d}P_{0}}{{d\omega}}(\omega).$ (S.25)

We have also used the identity $\nu=\frac{\omega}{2\pi}=\frac{m_{\phi}}{2\pi}$
and $\phi_{0}=\sqrt{2\rho_{\mathrm{DM}}}/m_{\phi}$. There are two things we
would like to remind: 1) the signal power was obtained under the assumption of
perfectly cold ULDM. 2) we did not consider the DM density profile. For the
more completed treatment, we refer to further work as those done in [66].

As a final note, with the similar calculation, it’s not hard to figure out the
signal power for the linear interaction $\mathcal{L}=\frac{1}{4}g\phi
F_{\mu\nu}F^{\mu\nu}$ as

$P_{1}=\frac{1}{16}{g}^{2}\phi_{0}^{2}m_{\phi}^{2}t\int{d\omega}\frac{{d}P_{0}}{{d\omega}}(\omega)\pi\delta(\frac{m_{\phi}}{2}-\omega)=\frac{1}{16}{{g}^{2}\rho}\frac{dP_{0}}{d\nu}t,$
(S.26)

where the spectral density $\frac{dP_{0}}{d\nu}$ is evaluated at
$\nu=\frac{\omega}{2\pi}=\frac{m_{\phi}}{4\pi}$ this time. At the moment, it
is clear from the delta function that the resonance takes place at
$m_{\phi}=2\omega$, which is the stimulated decay of the ULDM similar to the
axion case [64]. Also, we can expect the same signal expression as the axion
case from the electromagnetic field duality.