# High-frequency gravitational waves detection
with the BabyIAXO haloscopes

José Reina-Valero,a∗ Jose R. Navarro-Madrid,b Diego Blas,c,d
Alejandro Díaz-Morcillo,b Igor García Irastorza,e Benito Gimeno,a Juan Monzó-
Cabrera,b aInstituto de Física Corpuscular (IFIC), CSIC-University of
Valencia, Calle Catedrático José Beltrán Martínez, 2, 46980 Paterna
(Valencia), Spain,
bDepartamento de Tecnologías de la Información y las Comunicaciones,
Universidad Politécnica de Cartagena, Plaza del Hospital 1, 30302 Cartagena,
Spain,
cInstitut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
dInstitució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluís
Companys 23, 08010 Barcelona, Spain
eCenter for Astroparticles and High Energy Physics (CAPA), Universidad de
Zaragoza, Zaragoza 50009, Spain<EMAIL_ADDRESS>
###### Abstract

We present the first analysis using RADES-BabyIAXO cavities as detectors of
high-frequency gravitational waves (HFGWs). In particular, we discuss two
configurations for distinct frequency ranges of HFGWs: Cavity 1, mostly
sensitive at a frequency range of 252.8 - 333.2 MHz, and Cavity 2, at 2.504 -
3.402 GHz, which is a scaled down version of Cavity 1. We find that Cavity 1
will reach sensitivity to strains of the HFGWs of order $h_{1}\sim 10^{-21}$,
while Cavity 2 will reach $h_{2}\sim 10^{-20}$. These represent the best
estimations of the RADES-BabyIAXO cavities as HFGWs detectors, showing how
this set-up can produce groundbreaking results in axion physics and HFGWs
simultaneously.

††preprint: APS/123-QED

##  I Introduction

After their first direct discovery in 2015 by the LIGO collaboration [1] in
the 100 Hz band, gravitational waves (GWs) are reshaping our understanding of
the cosmos. More concretely, these messengers bring new information from
previously undetectable events, and may be the key to access the most
primordial instances of the Universe. When one considers the extraordinary
progress in cosmology and astrophysics when the whole spectrum of
electromagnetic (EM) radiation was used to explore the Universe, it is hard to
think what may happen when GWs in a broad spectrum of frequencies will be
discovered.

Out of the different possibilities, the ‘high frequency’ band of GWs has
recently received increasing attention [2]. Different reasons explain the
current interest: first, the signals above the band of groundbase detectors
(few kHz) are not highly populated by known astrophysical events, leaving a
rich landscape of possible signals beyond the standard model to search for
without foregrounds; second, higher frequency means smaller and faster
devices, more compatible with laboratory scales, where one can leverage the
revolution in sensing to look for these feeble signals.

From the universal coupling of gravitation, the possibilities to search for
HFGWs in the sensing frontier are multi-fold. In this work, we will focus on
one of the most promising ones: their impact on microwave electromagnetic
cavities immersed in an intense magnetostatic field. A GW propagating along an
intense static magnetic field can convert into an electromagnetic wave of the
same frequency [3, 4]. If this happens inside a cavity of the right size, this
signal will accumulate, allowing the eventual detection of the primordial
signal. This simple idea has been recently discussed in detail in [5, 6, 7].
The final sensitivity depends not only on the ability to detect tiny EM
fields, but also on properties of the cavity, such as its volume or quality
factor.

In this work, we will focus on the BabyIAXO helioscope, and derive results for
GWs of frequencies between 252.8 to 333.2 MHz and 2.504 to 3.402 GHz. A novel
approach to the problem will be performed since no system with tuning
capability and ports for both axions and GWs has been studied yet. We shall
discuss how to adapt the experimental setup to detect HFGWs, and perfom a
quantitative study on how well the available system couples to an incident
HFGW. This study, involving a tuning system, is specially relevant for the
detection of stationary sources, following the same philosophy as in the axion
case. Hence, superradiant black-holes and GWs backgrounds are of special
interest for this kind of setup [5].

This paper is organized as follows: firstly, a brief commentary about the
BabyIAXO experiment is performed, explaining its setup and capabilities;
secondly, the electromagnetic design of the cavity for the RADES-BabyIAXO
haloscope as well as tuning system and ports is showed, discussing its
geometry and advantages. After this, a numerical calculation of the form
factor between the HFGW and RADES-BabyIAXO cavities is made paying special
attention to the several values that experimental parameters can adopt. In
addition, a calculation of the sensitivity estimation is performed, and the
expression for the scanning rate is obtained with a formulation similar to the
dark matter axion case.

##  II The BabyIAXO Experiment

The IAXO (International Axion Observatory) experiment is one of the most
ambitious projects in axion searches, whose aim is to detect solar axions [8,
9]. The intermediate stage, the BabyIAXO helioscope, will be built as a
stepping stone for the experiment, and it will be composed of a cryostat and a
dipole magnet with two bores of 0.7 metres in diameter and 10 metres in
length, providing a transversal magnetostatic field of 2 T [10]. This magnet
gives the opportunity, in addition to operating as a helioscope, to search for
dark matter axions with haloscopes, and the RADES collaboration made a
proposal to use resonant cavities at low frequencies to search for dark matter
axions in the mass range of 1-2 $\mu$eV, expecting competitive values in terms
of axion sensitivity [11]. To cover this mass range, four cavities were
designed with a frequency tuning system that allows to shift the operational
frequency during the data taking campaigns. Although these four cavities share
the same shape, the main difference lies in the diameter, which is larger for
the cavity operating at lower frequencies and smaller for the cavity operating
at higher frequencies, all maintaining the same length of 5 metres. Therefore,
two cavities can be placed in each bore during the measurement operation,
giving the chance to search in parallel at different masses (using four
different cavities), or enhance the sensitivity by summing coherently the
signals (using four cavities with the same dimensions). These cavities will be
set in a cryostat system based on cooling via cryocoolers and a closed helium
circulation loop [12], in order to cool them down to 4.6 K. Plans to push this
temperature down to tens of mK are set as part of the recently granted ERC-SyG
DarkQuantum with dilution refrigerator (DL) technology. This will allow to
effectively get noise figures close to the standard quantum limit with
superconducting quantum interference devices (SQUIDs).

Figure 1:  Schematic of the RADES-BabyIAXO cavities geometry. The cavity (in
blue color) has a quasi-cylindrical shape, with dimensions $a$ (width), $b$
(height) and $l$ (length), where the relation $b=0.9a$ is fulfilled. The
metallic rectangular plates for the frequency tuning are shown in dark grey
color. In red, the chosen reference frame in the center of the cavity.

As mentioned above, the properties of the RADES-BabyIAXO haloscope allow for
the search of HFGWs, which can be done during the axion measurement time by
making small changes in the cavities designed for dark matter search. A
novelty of RADES-BabyIAXO is its kind of tuning mechanism, which comes with
its own challenges. Indeed, the introduction of this mechanism perturbs
considerably the electromagnetic modal field structure inside the cavity and
the coupling between the HFGW and the resonant system is strongly modified. As
a result, the tuning system originally employed for scanning over a mass range
of axions can be useful for a frequency sweep to detect HFGWs with different
frequencies.

##  III Electromagnetic design

The electromagnetic study of the RADES-BabyIAXO cavities was performed in
[11], where a quasi-cylindrical shape was adopted in order to avoid the
degeneration of the modes $\mathrm{TE_{111}}$ and maximize the volume, but not
to fill completely the bore in order to leave some space for cables and
instrumentation at both sides. Two cavities are analysed in this article: the
first one is the lower-frequency cavity that has been designed to operate in
the MHz range (VHF and UHF bands), called Cavity 1, and the second one is a 10
times scaled-down version of the first cavity working at the GHz range
(S-band), called Cavity 2. This second cavity would not be used in RADES-
BabyIAXO, but has been designed and manufactured as a test model of Cavity 1
for the GHz range. The tuning system incorporated in both cavities consist of
two metallic thin rectangular plates along the longitudinal axis that turn
simultaneously, shifting the frequency of the operational EM mode. A schematic
of the cavity geometry is shown in Fig. 1, while Table 1 contains the
dimensions of the two designs.

Cav. model | a (cm) | b (cm) | l (cm) | Freq. range
---|---|---|---|---
Cavity 1 | 56 | 50.4 | 500 | 252.8 - 333.2 MHz
Cavity 2 | 5.5 | 4.9 | 50 | 2.504 - 3.402 GHz
Table 1: Dimensions and operational frequency range of the studied cavities.
The thickness of both metallic plates is 1 cm.

It is well known that in dark matter axion experiments using a cylindrical
cavity immersed within a dipole magnet, the resonant mode that provides the
maximum coupling between the radio-frequency (RF) electric field and the
external static magnetic field is the $\textrm{TE}_{111}$ mode, which is a
degenerate mode that, at the same resonant frequency, has two polarizations
orthogonal to each other [13]. In the case of the RADES-BabyIAXO cavities, due
to the presence of the plates and with the condition $a\neq b$, such
degeneracy disappears and both modes, quasi-$\textrm{TE}_{111}$ with vertical
and horizontal polarization, adopt two distinct frequencies.

Figure 2: Electric field profiles of Mode 1 and Mode 2 in a transversal cut of
Cavity 2 (also applicable to Cavity 1) for plates angle $\alpha$ $=$
$0^{\circ}$. Size and color of the arrows indicate the magnitude of the
electric field in linear (from more intense in red to weaker in blue).

In HFGWs detection, the use of a second mode is of special interest, as the
gravitational wave can couple not only to both modes [5, 14], but it can
excite them at the same time if the bandwidth of the HFGW is higher than the
frequency gap between both modes. Thus, the modes analysed in this article are
the quasi-$\textrm{TE}_{111}$ with an initial vertical polarization (oriented
to $y$-axis in Fig. 1) and the quasi-$\textrm{TE}_{111}$ with an initial
horizontal polarization (oriented in $x$-axis), which we call Mode 1 and Mode
2, respectively; the RF electric field pattern in the cavity central plane
$z=0$ is plotted in Fig. 2 for both modes.

(a)  (b)

Figure 3: Resonant frequency variation during tuning versus plates angles for
(a) Cavity 1 and (b) Cavity 2.

These modes are affected differently by the tuning system, as it is shown in
Fig. 3. Mode 1 decreases its resonant frequency when the plates are rotated,
while Mode 2 increases its resonant frequency. Note that the tuning range is
smaller for Mode 2 than for Mode 1, being this range for Mode 1, 43.33 MHz and
523.5 MHz for cavities 1 and 2, respectively, compared to the range of Mode 2,
29.74 MHz and 327.5 MHz for cavities 1 and 2, respectively.

The possibility to change the resonant frequency of both modes gives the
opportunity to search for HFGWs in a total range of 73.1 MHz and 850.9 MHz for
Cavity 1 and 2, respectively. However, there is a frequency range where it is
not possible to scan, and it corresponds to the frequency separation of both
modes at the starting point, i.e. when the angle of the plates is 0∘. To avoid
this frequency gap, the cavity section should have a $b/a$ ratio around
$b/a\approx 1$.

###  III.1 Coupling ports

An important aspect of this experiment is how to couple the EM signals
generated by the HFGW, as the position and type of port will determine which
modes can be detected. The port, proposed in [11], is a coaxial loop located
at one of the cavity ends that couples the magnetic field of the Mode 1. In
order to couple the Mode 2, the same concept can be used although in this case
the second coaxial loop is placed at the other end of the cavity and must be
rotated 90∘ with respect to that of the Port 1 in order to couple to the Mode
2 RF magnetic field. Figure 4 shows a schematic view of the location of the
two ports in the haloscope, where Port 1 is a coaxial loop oriented
$\phi_{1}=90^{\circ}$ with respect to the XZ plane that couples Mode 1. Port 2
is another coaxial loop oriented $\phi_{2}=0^{\circ}$ with respect to the XZ
plane that couples Mode 2. Rotation angles are denoted as $\varphi_{1}$ and
$\varphi_{2}$ for the Ports 1 and 2, respectively. Both loops can be rotated
in order to re-couple during the frequency tuning process, that is, the
coupling coefficient $\beta$ [13] can be varied by rotating the loops.

Figure 4: Geometry of ports and plates inside the cavity. (a) Motion
representation of the loops for RF magnetic field detection. (b) Definition of
angle $\alpha$ of the plates with respect to the $x$-axis. (c), (d) Definition
of angles $\varphi_{1}$ and $\varphi_{2}$ with respect to the $x$-axis.

##  IV Form factor

This work is focused on the HFGW coupling to the cavity related to the
Gertsenshtein effect [3, 4]: GWs will convert into EM waves of the same
frequency in the presence of a background very intense magnetostatic field.
This coupling can be quantified by the following coupling factor given by [5]:

$\tilde{\eta}_{m_{+,\times}}=\frac{\left|\int_{V}\vec{E}_{m}\left(\vec{r}\right)\cdot\vec{J}_{+,\times}\left(\vec{r}\right)\mathrm{d}V\right|}{V^{1/2}\left|\int_{V}\vec{E}_{m}\left(\vec{r}\right)\cdot\vec{E}_{m}\left(\vec{r}\right)\mathrm{d}V\right|^{1/2}},$
(1)

where $\vec{E}_{m}$ is the RF electric field of the $m$-th resonant mode
inside the cavity, $\vec{J_{+,\times}}$ the electric density current arisen by
a monochromatic HFGW for plus and cross polarizations (see [5]), and $V$ the
volume of the cavity.

Figure 5: Scheme of HFGWs incidence in each plane. |  XY plane |  XZ plane |  YZ plane
---|---|---|---
$\tilde{\eta}_{\times}$ |  |  |
$\tilde{\eta}_{+}$ |  |  |
Table 2: Form factors for Mode 1 in Cavity 1 as a function of the incidence angle $\theta$. Rows make reference to the two different polarizations of the HFGW and columns make reference to the three different planes in which the incidence of the HFGW is studied. Each plates angle ($\alpha$) is illustrated with a different color: Green ( ) for $\alpha=0^{\circ}$, Blue ( ) for $\alpha=30^{\circ}$, Red ( ) for $\alpha=45^{\circ}$, Cyan ( ) for $\alpha=60^{\circ}$ and Magenta ( ) for $\alpha=90^{\circ}$. In the $\tilde{\eta}_{+}$ XZ plane and $\tilde{\eta}_{+}$ YZ plane cases $\alpha=0^{\circ}$ and $\alpha=90^{\circ}$ cannot be seen since the coupling is much lower in comparison with the rest of the angles. The same occurs for $0^{\circ}$ in $\tilde{\eta}_{+}$ XY plane case. |  XY plane |  XZ plane |  YZ plane
---|---|---|---
$\tilde{\eta}_{\times}$ |  |  |
$\tilde{\eta}_{+}$ |  |  |
Table 3: Form factors for Mode 1 in Cavity 2 as a function of the incidence angle $\theta$. Rows make reference to the two different polarizations of the HFGW and columns make reference to the three different planes in which the incidence of the HFGW is studied. Each plates angle ($\alpha$) is illustrated with a different color: Green ( ) for $\alpha=0^{\circ}$, Blue ( ) for $\alpha=30^{\circ}$, Red ( ) for $\alpha=45^{\circ}$, Cyan ( ) for $\alpha=60^{\circ}$ and Magenta ( ) for $\alpha=90^{\circ}$. In the $\tilde{\eta}_{+}$ XZ plane and $\tilde{\eta}_{+}$ YZ plane cases $0^{\circ}$ and $\alpha=90^{\circ}$ cannot be seen since the coupling is much lower in comparison with the rest of the angles. |  XY plane |  XZ plane |  YZ plane
---|---|---|---
$\tilde{\eta}_{\times}$ |  |  |
$\tilde{\eta}_{+}$ |  |  |
Table 4: Form factors for Mode 2 in Cavity 1 as a function of the incidence angle $\theta$. Rows make reference to the two different polarizations of the HFGW and columns make reference to the three different planes in which the incidence of the HFGW is studied. Each plates angle ($\alpha$) is illustrated with a different color: Green ( ) for $\alpha=0^{\circ}$, Blue ( ) for $\alpha=30^{\circ}$, Red ( ) for $\alpha=45^{\circ}$, Cyan ( ) for $\alpha=60^{\circ}$ and Magenta ( ) for $\alpha=90^{\circ}$. In the $\tilde{\eta}_{\times}$ XZ plane and $\tilde{\eta}_{\times}$ YZ plane cases $\alpha=0^{\circ}$ and $\alpha=90^{\circ}$ cannot be seen since the coupling is much lower in comparison with the rest of the angles. The same occurs with $\alpha=30^{\circ}$ and $\alpha=90^{\circ}$ form factor for $\tilde{\eta}_{+}$ XY plane case. |  XY plane |  XZ plane |  YZ plane
---|---|---|---
$\tilde{\eta}_{\times}$ |  |  |
$\tilde{\eta}_{+}$ |  |  |
Table 5: Form factors for Mode 2 in Cavity 2 as a function of the incidence
angle $\theta$. Rows make reference to the two different polarizations of the
HFGW and columns make reference to the three different planes in which the
incidence of the HFGW is studied. Each plates angle ($\alpha$) is illustrated
with a different color: Green ( ) for $\alpha=0^{\circ}$, Blue ( ) for
$\alpha=30^{\circ}$, Red ( ) for $\alpha=45^{\circ}$, Cyan ( ) for
$\alpha=60^{\circ}$ and Magenta ( ) for $\alpha=90^{\circ}$. In the
$\tilde{\eta}_{\times}$ XZ plane and $\tilde{\eta}_{\times}$ YZ plane cases
$\alpha=0^{\circ}$ and $\alpha=90^{\circ}$ cannot be seen since the coupling
is much lower in comparison with the rest of the angles.

From this expression, when both the electric field and density current are
perfectly aligned, the form factor will have maximum magnitude. However, the
expression of $\vec{J}$ is relatively complex, making it non-trivial to
analyse its effect. In particular, even though the static magnetic field is
oriented along the $y$-axis, the HFGW can couple to any cavity mode with
different magnitudes of the coupling factor. Therefore, it is of interest to
perform a numerical calculation of $\tilde{\eta}_{m_{+,\times}}$ making a
sweep in the angle of incidence of the HFGW (see Fig. 5), $\theta$, and fixing
the remaining parameters to a given value. For this purpose, a simulation of
the RF electromagnetic fields inside the cavity has been performed with the
eigenmode solver of CST Studio Suite software [15].

The study is divided in the following cases: first, a given polarization of
the HFGW ($+$ or $\times$) is selected; then, we study a given plane of
incidence in order to define $\theta$ (defined with respect to the $z$-axis in
the XZ, YZ incidence planes and to the $x$-axis in the XY incidence plane);
finally, the tuning plates angle ($\alpha$) is fixed to a given value, defined
with respect to the $x$-axis. Results for the Mode 1 of Cavity 1 are shown in
Table. 2. Different values of $\alpha$ ($0^{\circ}$, $30^{\circ}$,
$45^{\circ}$, $60^{\circ}$ and $90^{\circ}$) have been chosen in order to
sweep the main region of interest in which the structure of the electric field
is modified considerably. Any result of $\tilde{\eta}_{m_{+,\times}}$
associated to another different value of $\alpha$ can be interpolated from the
results showed here. The external magnetostatic field is assumed uniform,
constant and oriented towards $y$-axis in the calculations 111Nevertheless, as
can be seen in [11], since it has some distortions in the external part of the
bore, the magnetic field of the BabyIAXO experiment is not completely uniform.
This aspect is more relevant in Cavity 1 since its dimensions are higher. The
result of the form factor varies considerably with $\theta$, generating the
kind of diagrams observed in Tables 2-3. The existence of specific values of
$\theta$ for which the system is blind to the incident HFGW is a crucial
aspect of this analysis. In order to avoid these blind spots, daily modulation
of the HFGW signal can be taken into account since the incidence angle of the
gravitational wave will change and so will the form factor pattern, always
provided that the source is stationary. In addition, studying more cavity
modes that differ in their electric field structure can be useful to avoid
blind spots [14]. An aspect associated with these angles of sensitivity drop
is that their position in the polar plots does not vary considerably when
changing the plates angle $\alpha$. Thus, one may employ two or more cavities
oriented towards different directions in order to have full sky coverage.

A related aspect is the variation of the form factor $\tilde{\eta}$ with
$\alpha$. There are cases where the system is more sensitive to an incident
HFGW, as seen for instance in Tables 2-3 when varying $\alpha$ from
$0^{\circ}$ (green) to $90^{\circ}$ (magenta). These changes can be as big as
an order of magnitude. As a consequence, when studying the coupling between a
HFGW and a resonant microwave cavity, the influence of the tuning system is of
special relevance. Moreover, this aspect not only affects the RF modal
electric field, but also $\vec{J}_{+,\times}$ due to its dependence with
frequency. This can be seen when comparing results of Table 3 with Table 2,
where it is clear that not only an order of magnitude has been gained, but
also a change in the pattern of the $\tilde{\eta}_{+}$ XY case can be seen.

The form factor for Mode 2 has also been computed. Results are shown in Tables
4 and 5. As it is seen in Fig. 2, initially ($\alpha=0^{\circ}$), this cavity
mode has its RF electric field aligned with the $x$-axis, which implies a
different effect on the form factor in the XY incidence plane during the
frequency tuning. The cases $\alpha=0^{\circ}$ and $\alpha=90^{\circ}$ on
$\tilde{\eta}_{\times}$ XZ and YZ planes are of special relevance due to the
decrease of the form factor with respect to the rest of values of the angle
$\alpha$, behaviour that was also seen in the earlier tables of Mode 1 but for
the $+$ polarization.

The same aspects discussed for Mode 1 can be extrapolated for Mode 2: the
presence of blind spots for given values of $\theta$ in certain polarizations
and incidence planes; the drastic change in the results when varying the
plates angle, seen for instance in the $\tilde{\eta}_{+}$ XY plane case (Table
4); and the change of the diagrams pattern not only between the two different
cavities studied but also with respect to Mode 1.

The study of the form factor of Mode 2 allows us to duplicate the scanning
frequency range, due to the split evolution between modes 1 and 2 (see Fig.
3). Furthermore, it is key to assess the origin of the detected RF signal, as
we now explain. Two situations are possible: both ports are excited; or only
port 1 or port 2 is excited. The first case implies that the bandwidth of the
detected RF signal is larger than the frequency gap between modes for a given
angle of the plates. Since axions would not be able of exciting modes 1 and 2
at once due to its expected bandwidth 222Excluding the possibility of two
axions with a mass split corresponding to the one between both modes., the
signal could have been generated by a HFGW. If only port 1 is excited, then a
study of the daily modulation of the signal must be performed in order to
distinguish the cosmological source. Finally, if port 2 is excited alone, a
consideration has to be taken into account. When varying the plates angle the
electromagnetic field is being distorted, and as a consequence, an axion would
be able to excite the second Mode. Even though this coupling would be lower
than the one of Mode 1, it could be even more significant than the coupling of
a HFGW due to its expected low amplitude. With that, it can be concluded that
for this case an study of the time modulation of the RF signal would be also
required.

Let us now be more explicit about the time variation of the signal, given its
relevance to identify the kind of source behind the latter. For the axion
case, the time modulation of the signal arises when considering the influence
of the typical momentum of the axion field $\vec{k}_{a}$, introducing the
spatial frequency term $\vec{k}_{a}\cdot\vec{r}$, where $\vec{r}$ is the
spatial coordinates vector of the cavity. Since $\vec{k}_{a}$ is proportional
to the velocity with which we see axions from our laboratory reference frame,
the consideration of this term translates into a tiny daily modulation of the
signal, suppressed by velocity $v\sim 10^{-3}c$ (being $c$ the speed of light
in vacuum) of the axion galactic wind in our galaxy, as compared to the main
RF signal [18] [19]. Higher masses of the axion (higher momentum) and cavities
of size comparable to the de Broglie wavelength maximize the influence of this
term. Regarding HFGWs, the profile of the signal’s time modulation strongly
depends on which type of source produces the wave. For instance, if the signal
is produced by the collision between two primordial black holes, a chirping
spectra is expected, and a signal duration which may be rather short [20, 21].
Hence, the total time that the signal couples in band with a resonant mode has
to be considered, in particular with respect to the tuning system employed in
the experiment. Recall that changing the angle $\alpha$ of the plates inside
the cavity modifies its resonant frequency, allowing us to sweep over a
frequency range. If the HFGW bandwidth is high enough, a merging source may
generate a signal in the cavity for more than one angle of plates $\alpha$.
The fundamental question is how much time we integrate for a given value of
$\alpha$. One solution would be to allow the system for fast read-outs, where
the frequency change when moving the plates from one angle to another is less
than the frequency variation of the HFGW. However, the mechanical tuning
system can take a few minutes to move from one of the angles studied to
another, and this possibility seems unlikely. A more realistic strategy would
be working with an array of cavities tuned to different frequencies for a
simultaneous scan over a frequency range. This case is also favoured, as the
existence of chirping may be used as a way to confirm the origin of the
signal, thus increasing detection prospects.

Finally, superradiant clouds associated to new bosons of mass $m_{b}$ can
produce a stationary signal in our detector. These are long-lasting phenomena
around spinning black holes of mass $M_{BH}\sim(Gm_{b})^{-1}$ ($G$ is Newton’s
constant) whose duration is over thousands of years and generate HFGWs in the
band $\sim m_{b}$ [22]. Thus, the tuning system of our set up is ideal to
search for this kind of sources. In this case, the rotation of the Earth would
generate an $O(1)$ modification in the coefficients $\tilde{\eta}_{+,\times}$,
from the relative rotation of $\vec{k}_{GW}$ with respect to the cavity.
Another source of persistent GWs are the stochastic backgrounds generated by
several sources, possibly of primordial origin [2]. The treatment of the
signals induced by GW backgrounds goes beyond the current analysis.

###  IV.1 Sensitivity estimation

At this point, a sensitivity estimation of the experiment can be made. The
expected power generated by the HFGW inside the cavity can be expressed in SI
units as [5]:

$P_{\mathrm{sig}}=\frac{\varepsilon_{0}}{2}\,Q_{\mathrm{eff}}\,\omega_{c}^{3}\,V^{5/3}\left(\tilde{\eta}\,h_{0}\,B_{0}\right)^{2},$
(2)

where $\varepsilon_{0}$ is the vacuum electric permittivity;
$Q_{\mathrm{eff}}$ is an effective quality factor; $\omega_{c}$ is the angular
frequency of the HFGW; $V$ is the cavity volume; $h_{0}$ is the gravitational
wave amplitude; and $B_{0}$ is the external static magnetic field. Since we do
not have a specified source of interest, the value adopted by
$Q_{\mathrm{eff}}$ will be the loaded cavity quality factor $Q_{L}$. Of
course, when making this decision it is assumed that $Q_{\mathrm{GW}}$ $\gg$
$Q_{L}$ (where $Q_{\mathrm{GW}}$ is the quality factor of the GW), being in a
situation in which the signal has been detected only in one port. From here,
we can obtain the value of $h_{0}$ that can be reached taking into account
Dicke’s radiometer equation for the noise power [13], where $T_{\mathrm{sys}}$
is the sum of the noise temperature of the cavity, i.e. its physical
temperature, and the noise temperature added by the readout chain, which is
approximately that of the first amplifier; $k_{B}$ is the Boltzmann constant;
$t_{\mathrm{int}}$ is the integration time; and $\Delta\nu$ is the frequency
detection bandwidth:

$\mathrm{SNR}=\frac{P_{\mathrm{d}}}{k_{B}\,T_{\mathrm{sys}}}\sqrt{\frac{t_{\mathrm{int}}}{\Delta\nu}},$
(3)

where $P_{\mathrm{d}}$ $=$ $\frac{\beta}{1+\beta}P_{\mathrm{sig}}$ is the
detected power (recall that $\beta$ is the coupling coefficient [13]). By
fixing $\mathrm{SNR}$ $>$ $3$,

$h_{0}>\frac{1}{\tilde{\eta}\,B_{0}}\left(\frac{\Delta\nu}{t_{\mathrm{\mathrm{int}}}}\right)^{1/4}\left(\frac{1+\beta}{\beta}\frac{6\
k_{B}\,T_{\mathrm{sys}}}{\varepsilon_{0}\,Q_{\mathrm{eff}}\,\omega_{c}^{3}\,V^{5/3}}\right)^{1/2}.$
(4)

Employing the values for the case of Cavity 1 with Mode 1, $\alpha$ $=$
$0^{\circ}$ in XY plane for $\times$ polarization and $\theta$ $=$ $0^{\circ}$
(which corresponds to the maximum form factor for this case), we obtain the
sensitivity expectation,

$\displaystyle h_{0}>9.38\ \cdot\ 10^{-22}$
$\displaystyle\left(\frac{\frac{1+\beta}{\beta}}{3/2}\right)^{1/2}\left(\frac{0.3356}{\tilde{\eta}}\right)\left(\frac{2\
\mathrm{T}}{B_{0}}\right)\left(\frac{\Delta\nu}{1.03\ \mathrm{kHz}}\frac{2\
\mathrm{min}}{t_{\mathrm{int}}}\right)^{1/4}\left(\frac{T_{\mathrm{sys}}}{4.6\
\mathrm{K}}\frac{96010}{Q_{\mathrm{eff}}}\right)^{1/2}$ (5)
$\displaystyle\times\left(\frac{1.9\cdot 10^{9}\
\mathrm{rad/s}}{\omega_{c}}\right)^{3/2}\left(\frac{1.1898\
\mathrm{m^{3}}}{V}\right)^{5/6}.$

The value employed for the coupling factor, $\beta$ $=$ $2$, is commonly used
in axion haloscope experiments [23]. Since the form factors obtained in this
work vary up to two orders of magnitude with respect to the one chosen for
this calculation (as we have taken an average value, this variation can act
either enhance or suppress the RF signal by an order of magnitude), this must
be considered just an illustrative example of the achievable sensitivity that
the RADES-BabyIAXO setup can reach.

The sensitivity of (5) may be improved in a number of ways: the integration
time of 2 minutes may be extended, in particular considering the possibility
of stationary sources of HFGW; lower temperatures inside the cryostat will be
achievable with DarkQuantum, reaching temperatures up to tens of mK by the
development of DL technologies and quantum limited amplification with SQUIDs;
the considered unloaded quality factor of the cavity is the cryogenic one for
pure Copper, but this value can be improved by coating the cavity with
superconducting materials such as Rare-Earth Barium Copper Oxide (REBCO) or
$\mathrm{Nb_{3}Sn}$ [24].

The sensitivity expectations for both cavities and modes are shown in Table 6.
In each case, the employed value for the form factor corresponds to the
maximum one of the $\tilde{\eta}_{\times}$ XY plane cases. The achievable
sensitivities are still far away from the expected sources [25]. However, it
should be remarked that these are the first steps in a dynamical field of
research. Conservative values of the experimental parameters have been
employed for our estimates, and there is much room for improvement of the
experimental parameters in the incoming years.

| Mode 1 | Mode 2
---|---|---
Cavity 1 | 0.94 | 4.00
Cavity 2 | 5.45 | 23.01
Table 6: Sensitivity expectation of both cavities and modes for
$h_{0}/10^{-21}$.

###  IV.2 Scanning rate

We can now perform a study of the scanning rate as is commonly made in axion
experiments but particularizing it for the HFGW case. To deduce the detection
rate expression, a similar approach to the one in [26, 23] has been followed,
obtaining :

$\displaystyle\frac{\mathrm{d}\nu}{\mathrm{d}t}=\left(\frac{\frac{\varepsilon_{0}}{2}\,\omega_{c}^{3}\,V^{5/3}\,\tilde{\eta}^{2}\,h_{0}^{2}\,B_{0}^{2}}{\mathrm{SNR}\
k_{B}\,T_{\mathrm{cav}}}\right)^{2}\left(\frac{\frac{\beta}{\left(1+\beta\right)}}{\frac{4\beta}{\left(1+\beta\right)^{2}}+\lambda}\right)^{2}\frac{Q_{\mathrm{GW}}^{3}\,Q_{L}}{\left(Q_{\mathrm{GW}}+Q_{L}\right)^{2}},$
(6)

where $\lambda$ $=$ $T_{\mathrm{amp}}/T_{\mathrm{cav}}$, being
$T_{\mathrm{amp}}$ the temperature of the amplification stage and
$T_{\mathrm{cav}}$ the physical temperature of the cavity.Using the same
parameters employed in the sensitivity calculation section, this time in Eq.
(6), we find the results shown in Table 7. Even though these numbers have been
derived for the mentioned experimental parameters, we must remark that they
can vary considerably since, for instance, $\tilde{\eta}$ depends on the
frequency of the incident signal.

| Mode 1 | Mode 2
---|---|---
Cavity 1 | 1.88 | 3.68
Cavity 2 | 464.16 | 937.83
Table 7: Scanning rate of both cavities and modes in Hz/s.

The optimal $\beta$ in order to maximize this figure of merit can now be
deduced, taking into account that $Q_{L}$ $=$ $\frac{Q_{0}}{1+\beta}$, where
$Q_{0}$ is the unloaded quality factor of the cavity. The extreme of (6) will
be achieved for the solution of

$\displaystyle\lambda\beta^{4}$
$\displaystyle+\left(-4+2\lambda-\tilde{Q}\lambda\right)\beta^{3}$ (7)
$\displaystyle+\left(8-12\tilde{Q}+\lambda-4\tilde{Q}\lambda\right)\beta^{2}-5\tilde{Q}\lambda\beta-2\tilde{Q}\lambda=0,$

where $\tilde{Q}=\frac{Q_{0}}{Q_{\mathrm{GW}}}+1$. The result strongly depends
on the values that $\lambda$ and $\tilde{Q}$ adopt as it can bee seen in Fig.
6. In dark matter axion detection with haloscopes, one sets
$\beta_{\mathrm{opt}}$ $=$ $2$, which is obtained when considering $\lambda$
$\gg$ $1$ and $\tilde{Q}$ $\approx$ $1$. However, for the case of BabyIAXO,
due to quantum limited amplification with SQUIDs and other technologies, in
the incoming years it would be able to decrease $\lambda$ and increase
$\tilde{Q}$. In this way, representative values for this parameters taking
into account these expectations in quantum technology would be $\lambda$ $=$
$5$ and $\tilde{Q}$ $=$ $2$, thus obtaining an optimum coupling
$\beta_{\mathrm{opt}}$ $=$ $4$.

Figure 6: Behaviour of the optimum coupling factor $\beta_{\mathrm{opt}}$ when
varying $\lambda$ and $\tilde{Q}$.

##  V Conclusions

This article examines the possibilities of using RADES-BabyIAXO haloscope and
a scaled version for higher frequencies as HFGWs detectors. For that, a
calculation of the form factor between HFGW and two RADES-BabyIAXO cavities
has been performed, analogously to the dark matter axion case outlined in
[14]. The complexity of the electric current density generated by the HFGW
implies the coupling to a different set of modes supported by the cavity. Two
orthonormal quasi-TE111 modes have been studied, obtaining different results
for the two HFGW polarizations, as well as for different planes and angles of
incidence. A novel approach to the HFGWs detection has been proposed in this
work, since it is the first time that the simultaneous detection of axion and
HFGWs has been studied. In addition, a realistic set-up has been considered by
incorporating the presence of tuning plates and coupling ports inside the
cavity.

We have shown the relevance of this tuning system by observing the drastic
change in the form factor when varying the plates angle. In addition, a
sensitivity estimation of both cavities has been calculated, obtaining for
both modes values of $h_{0}$ $\sim$ $10^{-21}$ for Cavity 1 and of $h_{0}$
$\sim$ $10^{-20}$ for Cavity 2 (cf. Table 6). These values could be
considerably improved via optimization of the experimental parameters, with
the final goal of achieving sensitivities closer to that of possible RF
signals. Moreover, a qualitative study on how to distinguish a signal
generated by a HFGW or an axion has been developed, concluding that RADES-
BabyIAXO cavities are capable of distinguish both cases, in the case of
stationary signals. The study of the form factor has been made for two
orthonormal modes of both cavities, concluding that these two modes can be
used for HFGWs detection and, therefore, achieve a duplication of the scanning
frequency range. Our study can be easily expanded to other higher-order
resonant modes, provided that the cavity contains the appropriate probes
(loops or monopoles) for these modes. Moreover, the scanning rate has been
derived in Eq. (6), following a similar approach to that of the axion case.
This allows to compute the optimal coupling factor $\beta$ maximizing the
detection rate. The ideal HFGW scenario for the RADES-BabyIAXO setup are
stationary sources, as superradiant black holes or stochastic GWs backgrounds
(the latter is not considered in this study). Nevertheless, transient waves
can also be detected even though this depends on the bandwidth of the HFGW.

In conclusion, BabyIAXO has potential to be employed as a HFGWs detector,
while still being used as a dark matter axions detector. Regarding future
directions, the employment of these cavities for an up-down conversion-type
detector can be considered, similarly to [6, 27]. In these set-ups, the
resonant cavity is feeded by a pump mode and the presence of an axion or a
HFGW can generate a signal in a readout mode. As the frequency gap between
Mode 1 and Mode 2 can be small 333This is seen in Fig. 3. However, via a
optimization of the cavity dimensions this frequency gap can be set to any
desired value., the up-down conversion may be useful to explore HFGWs of
frequencies below those considered in this work. By changing the plates angle,
scanning to higher frequencies can be done since the gap between modes
increases. Thus, RADES-BabyIAXO setup has a high capacity to be also
considered as an up-down conversion detector.

##  VI  Acknowledgments

This work was performed within the RADES group. We thank our colleagues for
their support, specially Camilo García Cely for providing us with the currents
expressions and his useful comments. Thanks also to Pablo Martín Luna for his
valuable comments and discussions. The research leading to these results has
received funding from the Spanish Ministry of Science and Innovation with the
projects PID2020-115845GB-I00/AEI/10.13039/501100011033 and
PID2022-137268NBC53, funded by MICIU/AEI/10.13039/501100011033/ and by
“ERDF/EU”. IFAE is partially funded by the CERCA program of the Generalitat de
Catalunya. J. Reina-Valero has the support of ”Plan de Recuperación,
Transformación y Resiliencia (PRTR) 2022 (ASFAE/2022/013)”, founded by
Conselleria d’Innovació, Universitats, Ciència i Societat Digital from
Generalitat Valenciana, and NextGenerationEU from European Union. D. Blas
acknowledges the support from the Departament de Recerca i Universitats from
Generalitat de Catalunya to the Grup de Recerca 00649 (Codi: 2021 SGR 00649).
IGI acknowledges support from the European Union’s Horizon 2020 research and
innovation programme under the European Research Council (ERC) grant agreement
ERC-2017-AdG788781 (IAXO+), as well as from the “European Union
NextGenerationEU/PRTR” (Planes complementarios, Programa de Astrofísica y
Física de Altas Energías).

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