# The connection between phase synchronization in simple nonlinear system and
stationary state entanglement in its quantum counterpart

E. D. Vol<EMAIL_ADDRESS>B. Verkin Institute for Low Temperature Physics
and Engineering of the National Academy of Sciences of Ukraine 47, Lenin Ave.,
Kharkov 61103, Ukraine.

###### Abstract

We begin with the simple model of phase sychronization in open classical
nonlinear system which is represented in the language of angular momentum
variables. After that we propose the relevant quantum counterpart of this
system. Using the appropriate Lindblad master equation for the density matrix
of two qubit realization of such system we have revealed that stationary state
of this composite system is pure and entangled with small dispersion of phase
observable. We believe that such curious connection between entangled
stationary states of quantum composite system and phase sychronization between
its subsystems may be typical for rather wide class of similar nonlinear open
systems as well.

###### pacs:

05.40.-a

##  I Introduction

Quantum entanglement of pure and mixed states is a notable feature of quantum
mechanics which plays a key role in all modern informational and
communicational applications of the theory, such as superdense coding, quantum
cryptography, teleportation and so on 1s . In this connection the important
question may arise: whether there is any counterpart of entanglement for
classical systems and if yes, how it can manifest itself. This problem was
studied by several authors (see for example 2s-4s), but all of them focused
attention only on closed Hamiltonian systems and tried to find connection
between the possible entanglement of ground state of the quantum system in
question and peculiarities arising in phase portret of its classical
counterpart when external parameters are changing. The main goal of the
present paper is to propose an example of a simple nonlinear open system with
phase syncronization which in addition has relevant quantum counterpart such
that stationary state of the system turns out to be pure, entangled and
reveals small dispersion of phase observable. Based on this enlightening
example one can anticipate the general attractive hypothesis about the
possible connection existing between entanglement of stationary state in
composite open quantum system and such well-known classical phenomenon as
phase synchronization of its subsystems. The present paper as we hope may be
considered as a small step on the way of confirmation of this attractive
hypothesis. So let us examine the model of phase synchronization in nonlinear
open system which can be conveniently formulated in the language of angular
momentum that interacts with its environment. The advantage of this model is
that it has the relevant quantum counterpart for which the above mentioned
connection looks especially clear. So let us consider the angular momentum $\
\overrightarrow{L}=\left(l_{x},l_{y},l_{z}\right)$ and assume that evolution
in time of its components is governed by the next system of equations:

$\displaystyle\frac{dl_{x}}{dt}$ $\displaystyle=$ $\displaystyle
2\left(l_{y}^{2}+l_{z}^{2}\right)$ $\displaystyle\frac{dl_{y}}{dt}$
$\displaystyle=$ $\displaystyle-2l_{x}l_{y}$ (1) $\displaystyle\
\frac{dl_{z}}{dt}$ $\displaystyle=$ $\displaystyle-2l_{x}l_{z}$

Note some obvious features of the nonlinear model governed by the system
Eq.(1). First of all it is easy to see that Eq. (1) implies the relation
:$\frac{dL^{2}}{dt}=0$ (where $L^{2}\equiv l_{x}^{2}+l_{y}^{2}+l_{z}^{2}$ ) ,
or in other words, the total angular moment of the system is conserved . On
the other side one can assert that independently from initial conditions
$l_{x}\left(0\right),l_{y}\left(0\right),l_{z}\left(0\right)$ the angular
momentum under study tends to its final stationary state with
$l_{y}\left(\infty\right)=l_{z}\left(\infty\right)=0$ and
$l_{x}\left(\infty\right)=\max=\sqrt{L^{2}\left(0\right)},$ where
$L^{2}\left(0\right)=l_{x}^{2}\left(0\right)+l_{y}^{2}\left(0\right)+l_{z}^{2}\left(0\right).$
Let us prove this statement. To this end in view it is convenient to write out
the system Eq.(1) in so called quasithermodynamic form, namely:

$\frac{dl_{i}}{dt}=\frac{1}{2}\varepsilon_{ikl}\frac{dH}{dl_{k}}A_{l}$ (2)

where $\varepsilon_{ikl}$ is completely antisymmetric tensor of the third
rank,
$A_{l}=\frac{1}{2}\varepsilon_{lmn}\frac{dS}{dl_{m}}\frac{dL^{2}}{dl_{n}}$ ;
and $H\left(l_{x},l_{y},l_{z}\right)$ and $S\left(l_{x},l_{y}l_{z}\right)$ are
two given functions of the system state. In the concrete case of the system
Eq. (1) , $H=L^{2}$ and $S=l_{x}$.It is easy to verify directly that Eq. (2)
implies two general relations, namely :1)$\frac{dH}{dt}=0,$ and
2)$\frac{dS}{dt}\geq 0.$Thus the functions $H$ and $S$ governing the evolution
of the system play the same role as energy and entropy functions in standard
classical thermodynamics. As a result we arrive to the conclusion that final
state of the system Eq. (1) should correspond to the state with maximal
entropy function that is with $l_{x}=\sqrt{L^{2}\left(0\right)}$. Now we want
to demonstrate that dynamical system Eq. (1) may be considered as the simplest
model of phase synchronization. Remind that synchronization is a general
phenomenon of rhythm adjustment of two open nonlinear oscillating systems that
can occur at arbitrarily weak interaction between them 5s . Further we will
interested in only the case of complete phase synchronization when the phase
difference between two oscillating systems under study tends to zero with
time. Formally such situation can be described as follows. Let us select the
phases of two systems $\varphi_{1}$ and $\varphi_{2}$ and their amplitudes
$r_{1}$and $r_{2}$ as dynamical variables and write down the complete system
of equations in the following schematic
form:$\frac{d\varphi_{i}}{dt}=f_{i}\left(\varphi_{1},\varphi_{2},r_{1},r_{2}\right)$,
and $\frac{dr_{i}}{dt}=g_{i}\left(\varphi_{1},\varphi_{2},r_{1},r_{2}\right)$
where $f_{i\text{ }}$ and $g_{i}$ are four appropriate nonlinear functions
that can provide the phase synchronization effect. One may write out this
system as a system of equations for two complex variables:
$z_{1}=r_{1}e^{i\varphi_{1}}$ and $z_{2}=r_{2}e^{i\varphi_{2}}$. Now let us
introduce three auxillary variables $l_{x},l_{y},l_{z}$ connected with $z_{1}$
and $z_{2}$ by the next relations:

$\displaystyle l_{x}$ $\displaystyle=$
$\displaystyle\frac{z_{1}^{\ast}z_{2}+z_{1}z_{2}^{\ast}}{2},$ $\displaystyle
l_{y}$ $\displaystyle=$
$\displaystyle\frac{i\left(z_{1}z_{2}^{\ast}-z_{1}^{\ast}z_{2}\right)}{2},$
(3) $\displaystyle l_{z}$ $\displaystyle=$ $\displaystyle\frac{\mid
z_{1}\mid^{2}-\left|z_{2}\right|^{2}}{2}.$

Using these relations one can obtain the equivalent system of evolution
equations for phase sychronization phenomenon in the language of variables
$l_{x},l_{y},l_{z}$. Note that above mentioned transition from four variables
$\left\\{\varphi_{i},r_{i}\right\\}$ to three variables $l_{x},l_{y},l_{z}$ is
the explicit analogue of the similar transformation proposed by J. Schwinger
in quantum theory. Note that in quantum mechanics such transformation allows
one to express three components of angular momentum operator by means of four
operators: $a_{1},a_{1}^{+},a_{2},a_{2}^{+}$ , satisfying to standard Bose
commutation relations. In view of obvious relations:
$l_{x}=r_{1}r_{2}\cos\left(\varphi_{1}-\varphi_{2}\right)$ and
$l_{y}=r_{1}r_{2}\sin\left(\varphi_{1}-\varphi_{2}\right)$ one can express the
condition of complete phase synchronization by two equivalent ways :1)
$l_{y}\left(\infty\right)=0$ or 2) $l_{x}\left(0\right)=\max.$Thus the
realization of complete phase sychronization in nonlinear model Eq. (1) has
been proved. Let us present now the appropriate quantum counterpart of the
proposed nonlinear classical model.The consistent way to ”quantize” classical
equations of motion for open system (at least in semiclassical approximation)
was proposed by author in the paper 5s , where some instructive examples were
represented as well. In the case of Eq. (1) the relevant recipe of
quantization can be formulated as follows (all necessary details the reader
can find in 5s ). First of all one needs to represent the classic equations of
motion for variables $l_{x},l_{y},l_{z}$ in the next specific form that allows
their successive quantization, namely:

$\frac{dl_{\alpha}}{dt}=-\left(\overrightarrow{l}\times\frac{\delta
H}{\delta\overrightarrow{l}}\right)_{\alpha}+i\sum\limits_{k}R_{k}(\overrightarrow{l}\times\frac{\delta
R_{k}^{\ast}}{\delta\overrightarrow{l}})_{\alpha}+c.c.\text{.}$ (4)

where $H\left(l_{i}\right)$ and $\left\\{R_{k}\left(l_{i}\right)\right\\}$ is
some given set of functions of the state, and furthermore the function
$H\left(l_{i}\right)$ is real while $R_{k}\left(l_{i}\right)$ are complex
functions.It turns out that consistent quantum version of Eq. (4) can be
obtained if one by means of known classical functions $H$ and $R_{k}$
reconstructs the relevant Lindblad master equation (LME) for density matrix
$\widehat{\rho}$ of quantum counterpart of the system of interest. Thus the
required Lindblad equation looks as :

$\frac{d\widehat{\rho}}{dt}=-i\left[\widehat{H},\widehat{\rho}\right]+\sum\limits_{k}\left[\widehat{R}_{k}\widehat{\rho},\widehat{R}_{k}^{+}\right]+h.c.\text{.}$
(5)

where $\widehat{H}\left(\widehat{l_{i}}\right),$ $\
\widehat{R_{k}}\left(\widehat{l_{i}}\right),$ $\
\widehat{R_{k}^{+}}\left(\widehat{l_{i}}\right)$ are quantum observables that
correspond to their classical analogues in Eq.(4). Speaking more simply one
must replace classical variables $l_{x},l_{y,}l_{z}$ entering as arguments in
functions $H,R_{k}$ by corresponding quantum operators $\widehat{l_{i}}$ that
satisfy standard commutation relations, namely:
$\left[\widehat{l_{i},}\widehat{l_{j}}\right]=i\varepsilon_{ijk}\widehat{l_{k}}$
. If we return now to the concrete model of Eq. (1) it is easy to verify
directly that the relevant Lindblad equation for the density matrix of its
quantum counterpart may be represented in the next simple form:

$\frac{d\widehat{\rho}}{dt}=\left[\widehat{R}\widehat{\rho},\widehat{R^{+}}\right]+\left[\widehat{R},\widehat{\rho}\widehat{R^{+}}\right]\text{.}$
(6)

where the single jump operator $\widehat{R}$ has the
form:$\widehat{R}=\widehat{l_{z}}-i\widehat{l_{y}}$

Let us now examine the LME, that is Eq. (6), which is characterized by single
jump operator $\widehat{R}=\widehat{l_{z}}-i\widehat{l_{y}}$ in the simplest
case of two qubit composite system for which the density operator
$\widehat{\rho}$ is given by $4\times 4$ matrix. In standard matrix
reprezentation of angular momentum operator $\widehat{L}$ for the system with
$l=\frac{3}{2}$ one can write down its components as :

$\displaystyle\widehat{l_{z}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}3&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-3\end{pmatrix},$ $\displaystyle\widehat{l_{x}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}0&\sqrt{3}&0&0\\\ \sqrt{3}&0&2&0\\\
0&2&0&\sqrt{3}\\\ 0&0&\sqrt{3}&0\end{pmatrix},$ (7)
$\displaystyle\widehat{l_{y}}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\begin{pmatrix}0&-\sqrt{3}&0&0\\\ \sqrt{3}&0&-2&0\\\
0&2&0&-\sqrt{3}\\\ 0&0&\sqrt{3}&0\end{pmatrix}$

Using the expressions Eq. (7) one can easilyobtain the required matrix
representation of jump operator $\widehat{R}$ in the next form:

$\widehat{R}=\widehat{l_{z}}-i\widehat{l_{y}}=\frac{1}{2}\begin{pmatrix}3&-\sqrt{3}&0&0\\\
\sqrt{3}&1&-2&0\\\ 0&2&-1&-\sqrt{3}\\\ 0&0&\sqrt{3}&-3\end{pmatrix}\text{.}$
(8)

In the present paper we are interested in only the stationary solutions of the
LME,with jump operator given by Eq. (8). In this case it is easy to see that
all pure stationary states of the system can be found by the solution of the
equation: $\widehat{R}\left|\Psi\right\rangle_{st}=0$. Omitting the trivial
algebra let us present the required result for $\left|\Psi\right\rangle_{st}$
in the form :

$\left|\Psi\right\rangle_{st}=\frac{1}{2\sqrt{2}}\begin{pmatrix}1\\\
\sqrt{3}\\\ \sqrt{3}\\\ 1\end{pmatrix}\text{.}$ (9)

We reveal that stationary state of our model is entangled, because it cannot
be represented in factorized form as :
$\left|\Psi\right\rangle_{st}=\begin{pmatrix}a\\\
b\end{pmatrix}\otimes\begin{pmatrix}c\\\ d\end{pmatrix}$ where
$\begin{pmatrix}a\\\ b\end{pmatrix}$ and $\begin{pmatrix}c\\\ d\end{pmatrix}$
are states of first and second qubits respectively. It is well known that in
the case of two qubit pure state : $\left|\Psi\right\rangle=$
$\begin{pmatrix}\Psi_{1}\\\ \Psi_{2}\\\ \Psi_{3}\\\ \Psi_{4}\end{pmatrix}$
there is the simple measure for pure state entanglement , namely, the
concurrence- $C$ . According to definition:
$C=2\left|\Psi_{1}\Psi_{4}-\Psi_{2}\Psi_{4}\right|$. It is easy to see that
for the stationary state Eq. (9) concurrence is equal to $\frac{1}{2}$. Now
let us clarify another intriguing point : how the stationary state of the
model Eq. (6) is connected with eigenstates of its quantum phase. Here it
should be noted that the concept of quantum phase operator in the case of
infinite dimensional Hlbert space rather complicated and is not clear
enough.This problem was discussed in quite number of papers (see the most
important contributions on this subject in the book 8s ). However for finite
-dimensional quantum systems the problem may be represented much simpler as
follows: let us consider in $N$ dimensional Hilbert space $\left(N=2l+1\text{,
where }l\text{ some integer or half-integer}\right),$ the raising operator
$\widehat{l_{+}}$ with the next matrix elements in standard basis
$\left|l,m\right\rangle$ $:\left\langle l,m\right|\
l_{+}\left|l,m-1\right\rangle=\sqrt{\left(l-m+1\right)\left(l+m\right)}.$
After that one can correctly define the polar decomposition of operator
$\widehat{l_{+}}$ in the form:
$\widehat{l_{+}}=\sqrt{\widehat{l_{+}}\widehat{l}}e^{-i\widehat{\Phi}}.$Respectively,
the lowering operator $\widehat{l}$ which is conjugate to operator
$\widehat{l_{+}}$ looks as:
$\widehat{l}=e^{i\widehat{\Phi}}\sqrt{\widehat{l_{+}}\widehat{l}\text{. }}$It
is easy to verify directly that operators $e^{-i\widehat{\Phi}}$ and
$e^{i\widehat{\Phi}}$ in above decompositions are unitary and all their
eigenvalues are the roots of $N$ degree from unit. For example in the case of
$N=4$, which is the main subject of our interest in present paper, above
mentioned construction may be concretized as follows. The operator
$\widehat{l_{+}}$ has the form:
$\widehat{l_{+}}=\begin{pmatrix}0&\sqrt{3}&0&0\\\ 0&0&2&0\\\ 0&0&0&\sqrt{3}\\\
0&0&0&0\end{pmatrix}$ and corresponding phase operator looks as:

$e^{-i\widehat{\Phi}}=\begin{pmatrix}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
1&0&0&0\end{pmatrix}\text{.}$ (10)

It is easy to see by direct calculation that the above polar decompositions
are valid and furthermore
$e^{i\widehat{\Phi}}.e^{-i\widehat{\Phi}}=e^{-i\widehat{\Phi}}.e^{i\widehat{\Phi}}=1$
that is operator $e^{-i\widehat{\Phi}}$ is unitary in fact.The eigenvalues
$\lambda_{i}$ of operator $e^{-i\widehat{\Phi}\text{ }}$are equal to :
$\lambda_{1}=1$, $\lambda_{2}=-1$, $\lambda_{3}=i$, $\lambda_{4}=-i$, and
corresponding eigenvectors may be represented as:

$\displaystyle\left|1\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}1\\\ 1\\\ 1\\\ 1\end{pmatrix};\text{ \
}\left|-1\right\rangle=\frac{1}{2}\begin{pmatrix}1\\\ -1\\\ 1\\\
-1\end{pmatrix};$ (11) $\displaystyle\left|i\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}1\\\ i\\\ -1\\\ -i\end{pmatrix};\text{
\ }\left|-i\right\rangle=\frac{1}{2}\begin{pmatrix}1\\\ -i\\\ -1\\\
i\end{pmatrix}\text{.}$

It is easy to see that four vectors Eq. (11) form the complete basis in
Hilbert space of two qubit systems.Note the obvious fact that all these states
are not entangled. Now let us decompose the stationary state Eq. (9) of our
quantum model Eq. (6) over the eigenstates of phase operator basis Eq.
(11).The required decomposition may be represented as follows:

$\left|\Psi\right\rangle_{st}=a\left|1\right\rangle+z\left|i\right\rangle+z^{\ast}\left|-i\right\rangle\text{.}$
(12)

where $a$ is real and $z,z^{\ast}$ are two conjugate complex numbers.Comparing
expressions Eq. (9) and Eq. (12) one can find that
$a=\frac{1+\sqrt{3}}{2\sqrt{2}},$ and
$z=\frac{1}{4\sqrt{2}}\left(1-\sqrt{3}\right)\left(1+i\right)$.

Thus we arrive to the following conclusion: the entangled stationary state of
the quantum model of phase synchronization Eq. (9) can be represented as
superposition of three eigenstates of phase operator $e^{-i\widehat{\Phi}}$
with different phases. Note that the state $\left|-1\right\rangle$ which
corresponds to the phase $\pi$ is not included in decomposition Eq. (12). It
is essential that maximum contribution in the decomposition Eq. (12) brings in
the state with $\phi=0$. One may easily evaluate this contribution or its
relative weight $w$ as: $w=\frac{a^{2}}{a^{2}+2\left|z\right|^{2}}\approx
0,9.$

Summing up the results obtained in this paper we may conclude that entangled
stationary state of the quantum counterpart of simple classical model of phase
sychronization Eq. (1) is very closed to the unentangled eigenstate of phase
operator with phase $\Phi=0$. Based on analysis of this model one may
anticipate the attractive hypothesis that similar relationship may take place
for a much larger class of nonlinear open systems which demonstrate the
phenomenon of phase synchronization. In this connection it is worth noting
that the possibility of similar connection on a qualitative level have been
discussed at length in known correspondence between Wolfgang Pauli and eminent
psychologist K.G. Jung 8s  many years ago. The exposition of this
correspondence the reader can find in recent, rich in content, review paper 9s
. The author hopes that the simple nonlinear model and its quantum counterpart
proposed in the present paper can be considered as the small step on the way
of quantitative simulation of this intriguing connection in more complex
systems as well.

## References

  * (1) M.A.Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000)
  * (2) A.P.Hines, R.H. McKenzie, G.J. Milburn, Phys.Rev. A71, 042303, (2005)
  * (3) C.Emary, N.Lambert, T.Brandes, Phys.Rev. A71, 062302 (2005)
  * (4) M.S. Santhanam, V.B. Sheorey, A.Lakshminarayan, Phys.Rev. E77, 026213,(2008)
  * (5) A.Pikovsky, M.Rosenblum, J.Kurths, Synchronization. An Universal Concept in Nonlinear Sciences, (Cambridge University Press, Cambridge, 2002)
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###### Abstract

Within the increasing interest in the dark matter axion detection through
haloscopes, in which different international groups are currently involved,
the RADES group was established in 2016 with the goal of developing very
sensitive detection systems to be operated in dipole magnets. This review
deals with the work developed by this collaboration during its first five
years, from the first designs, based on the multi-cavity concept, aiming to
increase the haloscope volume and, so, to improve its sensitivity, their
evolution, the data acquisition design, and, finally, the first experimental
run. Moreover, the envisaged work within RADES, for both dipole and solenoid
magnets, in the short and medium term is also presented.

###### keywords:

axions; dark matter detectors; haloscopes; resonant cavities.

1 1 0 Design of new resonant haloscopes in the search for the dark matter
axion: a review of the first steps in the RADES collaboration A. Díaz-Morcillo
1,∗, J.M. García Barceló 1, A.J. Lozano-Guerrero 1, P. Navarro 1, B. Gimeno 2,
S. Arguedas Cuendis 3, A. Álvarez Melcón 1, C. Cogollos 4, S. Calatroni 3, B.
Döbrich 3, J.D. Gallego 5, J. Golm 3,6, I.G. Irastorza 7, C. Malbrunot 3,
Jordi Miralda-Escudé4,8, C. Peña Garay 9,10, J. Redondo 7,11 and W. Wuensch 3
Firstname Lastname, Firstname Lastname and Firstname Lastname Correspondence:
<EMAIL_ADDRESS>
##  1 Introduction

Since Peccei and Queen established the mechanism to solve the strong Charge-
Parity problem PecceiQuinn (1977, 1977), and subsequently Weinberg Weinberg
(1978) and Wilczek Wilczek (1978) predicted the existence of a new particle,
the axion, the definition and execution of different experimental set-ups in
order to detect this proposed particle have gained an increasing interest.

In the last thirty years, the search for the axion has been basically
developed following three different detection techniques, depending on the
source of these particles and their expected mass. Two of them, based on the
inverse Primakoff effect Primakoff (1951), the so called helioscopes and
haloscopes, where proposed by Sikivie Sikivie (1983) in the middle 80s. The
other one, known as Light Shining through Walls (LSW), consists of a
laboratory production + detection scheme and is based on both direct and
inverse Primakoff effects. In this way, photons from a laser are coupled to
axions in a generation Fabry-Perot cavity and those axions are coupled to
photons in a detection cavity, both cavities separated by an optical barrier.

Helioscope experiments look for axions produced in the Sun through the
Primakoff effect from plasma photons in the solar inner layers. After initial
experiments at Brookhaven National Laboratory (BNL) in 1992 Lazarus (1992) and
in Japan in the SUMICO experiment Inoue (2002); Moriyama (1998); Inoue (2008),
the third-generation CAST experiment at CERN has obtained the best results in
this kind of detection up to now, with limits for axion-photon coupling
$<6.6\cdot 10^{-11}$ ${\rm GeV}^{-1}$ in the mass range $<0.02$ eV
CAST:2017uph  , and $<2.3\cdot 10^{-10}$ ${\rm GeV}^{-1}$ for 0.02 eV <
$m_{a}$ < 0.64 eV Arik (2011). The future in the solar axion search will be
starred by IAXO Armengaud (2019).

In this quest for the axion, the RADES collaboration started in 2016, joining
research groups from universities and research institutes of Spain and
researchers at CERN. A detailed state of the art at that time of the different
axion detection experiments, finished or under development and execution, can
be found in Graham (2015). An updated review can be found in Irastorza (2018).
The aim of the collaboration was the detection of the axion in our galactic
halo, which is, in fact, a candidate for constituting the dark matter Abbott
(1983); Dine (1983). Therefore, the experiment would be clearly based on a
haloscope set-up. Haloscope experiments intend to detect axions in the
galactic halo by promoting the axion-photon conversion under a strong magnetic
field. The intensity of the electromagnetic field is boosted when this
conversion occurs in a resonant device such as, for instance, a microwave
resonant cavity. Additionally, due to the expected extremely low axion-photon
coupling and, thus, the electromagnetic power to be detected, it is mandatory
to work at cryogenic conditions in order to minimize the thermal noise and,
thus, to increase the signal to noise ratio ($\nicefrac{{S}}{{N}}$).

At the beginning of the RADES collaboration, the successful experiment ADMX
had already completed the exploration in the 1.9 – 3.65 $\mu$eV mass range
(460 – 890 MHz) with exclusion limits that reached the KSVZ model Stern
(2016). It had benefited from the experience of two pioneer experiments at BNL
Wuensch (1989) and University of Florida Hagmann (1990), respectively. The
prospects for the new version of the experiment (ADMX-Gen2) reach even the
DFSZ sensitivity in the 1.4 – 25 $\mu$eV mass range Stern (2016).

Since the RADES project began, the activity in resonant cavity haloscopes has
highly increased, with a large number of new cavity concepts, such as the
KLASH proposal Alesini (2017) at lower masses than ADMX or HAYSTACK Kenany
(2017), ORGAN McAllister (2017), QUAX Alesini (2021) and CAPP experiments
Jeong (2018) at higher ones. RADES Alvarez (2018) is among this latter group.

##  2 General principles in a resonant haloscope design

As described in Alvarez (2018) for the RADES case, a resonant haloscope
experiment consists of a resonant cavity immersed in a high and static
magnetic field at very low temperatures (few K or even mK), and connected to a
receiver or data acquisition system (DAQ) which must amplify / filter / down-
convert the detected signal without introducing too much noise and perform the
A/D conversion and subsequent fast Fourier transform (FFT) for a proper off-
line data analysis.

We can summarize the main goals in the design of an axion detection experiment
in (1) maximizing the electromagnetic power detected from the axion-photon
coupling, (2) optimizing the sensitivity of the experiment, and (3) maximizing
the scanning rate along the mass range of interest. It is of key interest,
thus, to know the relation of these three parameters with the experimental
variables.

###  2.1 Detected power

In a resonant cavity haloscope experiment, the generated power in the axion-
photon conversion can be obtained as Younggeun (2019)

$P_{g}=g^{2}_{a\gamma}\frac{\rho_{a}}{m_{a}}B_{e}^{2}CVmin(Q_{c},Q_{a})$ (1)

or, more accurately Dongok (2020),

$P_{g}=g^{2}_{a\gamma}\frac{\rho_{a}}{m_{a}}B_{e}^{2}CV\frac{Q_{c}Q_{a}}{Q_{c}+Q_{a}}$
(2)

where $g_{a\gamma}$ is the unknown axion-photon coupling, $\rho_{a}$ the axion
dark matter density, $m_{a}$ the axion mass, $B_{e}$ the static magnetic
field, $V$ the cavity volume, $Q_{a}$ and $Q_{c}$ the quality factors of the
axion and cavity resonances, respectively, and $C$ the form or geometric
factor, a normalized parameter that quantifies the coupling between the static
magnetic field $\vec{B}_{e}$ and the dynamic electric field $\vec{E}$ induced
by the axion-photon conversion:

$C=\frac{|\int_{V}\vec{E}\cdot\vec{B}_{e}\,dV|^{2}}{\int_{V}||\vec{B}_{e}||^{2}\,dV\,\int_{V}\varepsilon_{r}||\vec{E}||^{2}\,dV}$
(3)

where simple bars $|\cdot|$ indicate the magnitude of a complex number, double
bars $||\cdot||$ are the standard vector norm defined in the 3D complex vector
space, and $\varepsilon_{r}$ is the dielectric constant inside the cavity.

This generated power will be in part dissipated in the lossy material of the
cavity (metallic walls if there are no lossy dielectric materials) and in part
extracted from the cavity with a proper probe. This coupling, for frequencies
below the millimetric wave range, will be normally a monopole or loop antenna,
depending on the mode to be extracted, connected to a coaxial line which, in
turn, transmits the signal to the receiver. The power detected at the coaxial
port is

$P_{d}=\kappa
g^{2}_{a\gamma}\frac{\rho_{a}}{m_{a}}B_{e}^{2}CV\frac{Q_{l}Q_{a}}{Q_{l}+Q_{a}}$
(4)

where $\kappa$ is the coupling factor at the extraction point, and $Q_{l}$ is
the loaded quality factor, that is, the quality factor of the cavity taking
into account the coupling effect. Maximum power transfer is obtained in a
critical coupling regime, that is, for $\kappa=0.5$. Although critical
coupling can be designed, it is very sensitive to slight mechanical or
temperature changes. Therefore, it is very convenient to develop mechanical or
electromagnetic systems for modifying the coupling in situ.

Moreover, if $Q_{l}$ is much lower than $Q_{a}$ ($\sim 10^{6}$), 4 becomes

$P_{d}=\kappa g^{2}_{a\gamma}\frac{\rho_{a}}{m_{a}}B_{e}^{2}CVQ_{l}$ (5)

###  2.2 Axion coupling sensitivity

The complete experiment, composed of the source (resonant haloscope) and the
receiver, has a signal to noise ratio given by the Dicke’s radiometer
equation:

$\frac{S}{N}=\frac{P_{d}}{P_{N}}=\frac{P_{d}}{k_{B}T_{sys}\sqrt{\frac{\Delta
v}{\Delta t}}}$ (6)

where $k_{B}$ is the Boltzmann constant, $T_{sys}$ the noise temperature of
the system, $\Delta t$ the observation time and $\Delta v$ the reception
bandwidth, which should coincide with the bandwidth of the axion resonance:

$\Delta v_{a}=\frac{m_{a}}{Q_{a}}$ (7)

Introducing 5 and 7 in 6 we can obtain the expression for the axion-photon
coupling:

$g_{a\gamma}=\left(\frac{\frac{S}{N}k_{B}T_{sys}}{\kappa\rho_{a}CVQ_{l}}\right)^{\frac{1}{2}}\frac{1}{B_{e}}\left(\frac{m_{a}^{3}}{Q_{a}\Delta
t}\right)^{\frac{1}{4}}$ (8)

An alternative parameter to $g_{a\gamma}$ is the dimensionless axion-photon
coupling $C_{a\gamma}$, related to $g_{a\gamma}$ through

$C_{a\gamma}=5\cdot
10^{15}\frac{g_{a\gamma}\left(GeV^{-1}\right)}{m_{a}\left(\mu eV\right)}$ (9)

Introducing 8 in 9 yields

$C_{a\gamma}=5\cdot
10^{15}\left(\frac{\frac{S}{N}k_{B}T_{sys}}{\kappa\rho_{a}CVQ_{l}}\right)^{\frac{1}{2}}\frac{1}{B_{e}}\left(\frac{1}{Q_{a}m_{a}\Delta
t}\right)^{\frac{1}{4}}$ (10)

As seen in 4, the detected power only depends on intrinsic parameters of the
axion ($g_{a\gamma}$, $\rho_{a}$, $m_{a}$ and $Q_{a}$) and experimental
parameters in the cavity subsystem ($B_{e}^{2}$, $C$, $V$ and $Q_{l}$), but
the sensitivity also relies on receiver experimental parameters
($\frac{S}{N}$, $T_{sys}$ and $\Delta t$).

###  2.3 Scanning rate

Despite its high-Q resonant behavior, the final aim of a cavity haloscope is
exploring the largest possible mass range and doing it quickly. The mass
sweeping can be performed by adequate moving mechanisms or tuning materials
that modifies the cavity resonant frequency, and the scanning rate
$\frac{dm_{a}}{dt}$ is usually referred as a figure of merit of the haloscope.

For a measurement with the cavity resonance centered in $m_{a}$, the mass
range explored is the bandwidth of the cavity:

$dm_{a}=\frac{m_{a}}{Q_{l}}$ (11)

and the time needed for that measurement in order to get a prescribed
$\frac{S}{N}$ is, from 6,

$dt=\Delta
v_{a}\left(\frac{\frac{S}{N}k_{B}T_{sys}}{P_{d}}\right)^{2}=\frac{m_{a}}{Q_{a}}\left(\frac{\frac{S}{N}k_{B}T_{sys}}{P_{d}}\right)^{2}$
(12)

Therefore,

$\frac{dm_{a}}{dt}=\frac{Q_{a}}{Q_{l}}\left(\frac{P_{d}}{\frac{S}{N}k_{B}T_{sys}}\right)^{2}=Q_{a}Q_{l}\kappa^{2}g_{a\gamma}^{4}\left(\frac{\rho_{a}}{m_{a}}\right)^{2}B_{e}^{4}C^{2}V^{2}\left(\frac{S}{N}k_{B}T_{sys}\right)^{-2}$
(13)

##  3 Motivation and constraints of the RADES project

As a summary of the last section, the experimental parameters that optimize
the detected power, the sensitivity and the scanning rate can be identified in
eq. 5, 10 and 13, respectively. From the cavity side, it is evident that the
most important parameter is the external magnetic field $B_{e}$, followed by
the volume and the loaded quality factor of the cavity, the form factor for
the axion-coupling electromagnetic mode and the noise temperature of the
system, which, due to the Friis formula Pozar (1998), is mainly determined by
the physical temperature of the cavity and the noise temperature of the first
amplifier.

A figure of merit for the magnet in this kind of experiments is the magnetic
volume $B_{e}^{2}V$. However it is usually difficult to design a haloscope
that fills completely the available volume and, so, the potential of the
magnet is underused. This was the case and the main motivation of the RADES
project. Taking into account the upper limit for the ADMX prospects in the
next years Stern (2016), RADES aimed to explore higher masses, but this leads
to smaller resonant cavities, since the resonant frequency depends, in
general, inversely on the dimensions of the cavity.

For instance, continuing with the ADMX case, the resonant frequency of a
circular cavity for TM modes is given by

$f_{TM_{nlq}}=\frac{c}{2\pi}\sqrt{\left(\frac{p_{nl}}{a}\right)^{2}+\left(\frac{q}{d}\right)^{2}}$
(14)

where $n$, $l$ and $q$ are the number of variations in azimuthal, radial and
axial coordinate, respectively, $c$ is the speed of light, $p_{nl}$ is the
$l$-th root of the first-kind Bessel function of order $n$, and $a$ and $d$
are the radius and length of the cavity, respectively. For the $TM_{010}$ mode
used in the ADMX experiment and an expected exploration mass of 30 $\mu$eV
(7.25 GHz) the cavity diameter should be 3.17 cm, which would underuse the
available 60 cm bore.

Most haloscope experiments, such as ADMX or HAYSTACK, use solenoid magnets.
These magnets produce an axial ($\hat{z}$) magnetic field (Figure 1 right) and
therefore, the usual configuration for the cavity is a circular cavity with
its axis parallel to the magnet bore axis. In this way, a TM mode, whose
electric field is also axial, yields an optimal form factor. In the case of
the RADES project, strong accelerator dipole magnets (Figure 1 left) were
available in the initial stages, concretely the CAST magnet, which yields a
8.8 T transverse ($\hat{y}$) magnetic field. A more convenient cavity for this
kind of magnet is the rectangular cavity, where $TE_{m0p}$ modes have vertical
polarization for the electric field.

Figure 1: Dipole (left) and solenoid (right) magnets with transversal and
axial magnetic field, respectively.

For a rectangular cavity, the resonant frequencies for both TE and TM modes
are given by

$f_{mnp}=\frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^{2}+\left(\frac{n}{b}\right)^{2}+\left(\frac{p}{d}\right)^{2}}$
(15)

where $m$, $n$ and $p$ are the sinusoidal variations of the mode in the $x$,
$y$ and $z$ dimensions, respectively, $a$ is the width, $b$ the height and $d$
the length of the cavity.

In this case, for the mode with a maximum form factor, that is, the $TE_{101}$
mode, a mass goal of 30 $\mu eV$, and trying to get the maximum volume
($a=b=d$), the cavity should be a hexahedron with $a=29.3$ mm. This fits with
the 42.5 mm diameter of the bore in the CAST magnet Zioutas (1999), but the
magnet is greatly underused if we compare the length of the cavity with the
9.3 meters of the bore. Additionally, in order to separate degenerate modes,
$a$ and $b$ must be sensibly different (let’s assume $a\approx 2b$). This
reduces the cross-section of the cavity and, with it, its volume.

Taking this into account, and despite the fact that a rectangular cross
section is not the best way to maximize the use of a circular cross section,
the RADES project driving motivation was increasing the volume of the cavity
in the axial direction without modifying the resonant frequency and other
associated parameters as $Q_{l}$ and $C$.

###  3.1 First attempt: increasing the length of the cavity

Obviously, after a quick glimpse to 15, the first idea to increase the volume
without modifying the resonant frequency is increasing the cavity length. For
the $TE_{101}$ mode 15 is simplified to

$f_{TE_{101}}=\frac{c}{2}\sqrt{\frac{1}{a^{2}}+\frac{1}{d^{2}}}$ (16)

Thus, if $a\ll d$ the length can be increased without modifying the resonant
frequency since $f_{TE_{101}}\approx\frac{c}{2a}$. This approach has a minor
drawback: the reduction in $a$ (and, obviously in $b$ to keep $a\approx 2b$)
reduces the cavity cross section and this contributes to a reduction of part
of the volume gained with the increase of $d$. Anyway, this reduction is small
compared with the increase due to $d$ increment, as the following example
shows.

Let’s assume arbitrarily a goal exploration mass of 34.76 $\mu$eV (8.4 GHz).
For the $TE_{101}$ mode, fixing $a=d$ in 16 leads to $a=22.86$ mm, and fixing
$d=10a$ yields $a=17.95$ mm. Increasing $d$ up to $500a$ just decreases the
resonant frequency by less than $0.5\%$ and increases the volume by a factor
$242$ with respect to the $a=d$ case. This long cavity ($17.95\times
8.97\times 8975$ mm) can be operated in one of the two CAST magnet bores (43
mm in diameter and 9.3 m in length).

The great drawback of this length increment is the clustering of higher
resonant modes $TE_{10p}$ close to the chosen axion-coupling mode ($TE_{101}$)
Baker (2018). Figure 2 shows the spectrum of the cavity around 8.4 GHz for
different lengths. If the nearest higher mode ($TE_{102}$) is too close to
$TE_{101}$ it can hinder the characterization of $Q_{l}$ of the constructed
cavity, as well as the coupling factor $\kappa$.

Figure 2: Cavity (17.95×8.97 cross-section) spectrum around 8.4 GHz for
$d=10a$ (blue) and $d=100a$ (red).

###  3.2 The alternative: multi-cavity concept

Multi-cavity structures are widely used in filtering for telecommunication
applications in the range of microwaves and millimeter waves. In that context,
the aim is realizing a band pass filter with a given mask which prescribes the
pass band, the stop band, the pass ripple and the stop band attenuation.
Different references Matthaei (1980); Cameron (2018) deal with the analysis
and design of this kind of structures.

In our context, the objective is the design of a very selective filter (high
$Q_{l}$), centered in the resonant frequency of interest, with a large volume
and with a mode that couples optimally with the external magnetic field in
order to maximize the form factor. The mathematical formalism for the analysis
and design of this kind of multi-cavity device is presented in Alvarez (2018).

As explained there, roughly speaking, if the cavities have the same
dimensions, the resonant frequency of the haloscope is that of the cell
cavity, whilst the volume is the sum of volumes of the cell cavities. This
means that, by designing properly a haloscope of $N$ cavities the volume is
multiplied by $N$ whilst the resonant frequency remains that of a single
cavity. Moreover, as explained in Alvarez (2018), the quality factor of the
haloscope must be approximately the same as that of the unit cavity, and the
optimal external coupling factor $\kappa=\frac{1}{2}$ can be obtained by
extracting the signal from any of the cavities.

Assuming that other modes are sufficiently far from the $TE_{101}$ mode, all
cavities will resonate in that mode, but the whole device can have different
mode configurations, depending on the couplings between neighbor cavities. The
chosen mode configuration will be that which maximizes the form factor, which
will be the configuration with all positive (or all negative) electric fields
in the cavities. This allows a constructive sum in the term
$|\int\vec{E}\cdot\vec{B}_{e}dv|^{2}$ of the factor form 3. In fact, it is
expected that the total form factor will be approximately the same of one of
the unit cavities, which, for the $TE_{101}$ can be obtained analytically from
3 as

$C_{TE_{101}}=\frac{|\int\sin{\left(\frac{\pi
x}{a}\right)}\sin{\left(\frac{\pi
z}{d}\right)}dxdydz|^{2}}{V\int\sin^{2}{\left(\frac{\pi
x}{a}\right)}\sin^{2}{\left(\frac{\pi z}{d}\right)}dxdydz}=\frac{64}{\pi^{4}}$
(17)

This multi-cavity concept is very close to the idea of coherently summing $N$
signals extracted from $N$ identical cavities, but with only one compact
device and avoiding the difficulty of conserving the coherence (in-phase
signals) in the combining device. Other multi-cavity concepts for axion
detection have been published in the last years Goryachev (2018); Jeong
(2020).

In RADES, the coherent construction of the all-positive electric field
configuration is obtained through the proper design of the internal couplings.
These couplings between neighbor cavities can be realized, in the simplest
way, with rectangular irises, as those shown in Figure 3. Due to the vertical
polarization of the $TE_{101}$ electric field, irises invariant in $y$ (Figure
3 left) have an inductive behavior, whilst those invariant in $x$ (Figure 3
right) have a capacitive behavior.

Figure 3: Rectangular irises for coupling between neighbor cavities: inductive
(left) and capacitive (right). The pictures show the symmetric half of each
one.

##  4 First multi-cavity haloscopes

###  4.1 All inductive 5-cavities haloscope

During 2017, the first RADES haloscope was designed, manufactured,
characterized, and finally used in the 2018 data-taking campaign in CAST. It
was the fivefold cavity shown in Figure 4 and presented in Alvarez (2018).
Dimensions as described in Figure 4 are listed in Table 1.

Figure 4: RADES 5-cavites haloscope: scheme and dimensions, and device before copper plating. Table 1:  Physical dimensions of the 5-cavities haloscope at 2 K and at 298 K. Dimensions (mm) |  $T=2$ K |  $T=298$ K
---|---|---
Cavity width ($a$) | 22.86 | 22.99
Cavity height ($b$) | 10.16 | 10.25
Length external cavities ($L_{1}=L_{5}$) | 26.68 | 26.82
Length internal cavities ($L_{2}=L_{3}=L_{4}$) | 25.00 | 25.14
Iris width ($W_{1}=W_{2}=W_{3}=W_{4}$) | 8.00 | 8.14
Iris thickness ($t$) | 2.00 | 1.95

This device was designed in a first step through the formalism in Alvarez
(2018) and was finally optimized with CST Studio CST . Figure 5 shows the five
different configurations of the mode $TE_{101}$ along the final five –
cavities structure, and the corresponding form factor.

Figure 5: Electric field pattern (vertical polarization) for the five
different configurations of mode $TE_{101}$ along the 5-cavites haloscope,
with the associated form factor. The red fields denote positive levels, the
green fields zero and the blue fields negative. Taken and modified from
Alvarez (2018).

Occasional quenches produced in the CAST magnet made necessary a strong
material for the haloscope in order to avoid deformations in the device. For
that reason, the base material was not copper, but non-magnetic stainless
steel 316LN, plated with a 30 $\mu m$ of copper layer for reducing the losses.
The transmission coefficient magnitude, both simulated and measured at the
operation temperature of 2 K, is depicted in Figure 6.

Figure 6: Transmission coefficient magnitude at 2 K: simulated (black) and
measured (green), where measurement include the effect of cables from and to
the VNA. Taken and modified from Alvarez (2018).

###  4.2 Alternate coupling

Although multicavity concept alleviates the mode clustering, it has also a
limit. N cavities produce N different configuration patterns for $TE_{101}$
mode. Therefore, when the number of cavities increases, the same concentration
of resonances is observed. Figure 7 shows the 30 configurations of a
30-cavities haloscope with a similar design than the 5-cavities one, i.e.
inductive irises. Figure 8 shows the spectrum of this device with these
resonances.

Figure 7: Electric field pattern (vertical polarization) for the thirty
different configurations of mode $TE_{101}$ along the all-inductive 30-cavites
haloscope. Numbers at left column refer to the order of the configuration
resonances with the frequency. Figure 8: Simulated transmission coefficient
magnitude of an all-inductive 30-cavities haloscope at 2 K.

As observed, the first (axion one) and the second $TE_{101}$ mode
configurations are very close. This situation worsens with the manufactured
device and data-taking conditions, leading normally to an overlapping of both
resonances, which makes impossible to obtain the quality factor or the
coupling factor from measurements.

For an all-capacitive coupling, the patterns and spectrum are shown in Figure
9 and 10, respectively. At both spectrums, thirty modes can be seen (not by
eye due to numerical accuracy from the software). However, there is a main
difference between them: for the all-inductive structure, the axion mode is
the first peak, while for the all-capacitive structure, the axion mode is the
last one ($30^{th}$ mode). Note that in both cases, the axion mode is at the
same frequency ($\sim 8.43$ GHz). The reason comes from the design parameters
employed to set such an axion search at that specific frequency.

Figure 9: Electric field pattern (vertical polarization) for the thirty
different configurations of mode $TE_{101}$ along the all-capacitive
30-cavites haloscope. Numbers at left column refer to the order of the
configuration resonances with the frequency. Figure 10: Simulated transmission
coefficient magnitude of an all-capacitive 30-cavities haloscope at 2 K.

The solution to avoid the mode-mixing close to the axion mode is to apply
alternated couplings: capacitive + inductive irises Alvarez (2020). In this
configuration the axion mode is at the middle of the band, where greater mode
separation exists. The first haloscope designed and manufactured with this
alternating concept was a 6-cavities structure (Figure 11 (top)), followed by
a a 30-cavities structure based on the same idea (Figure 11 (bottom)). Figure
12 shows the thirty E field patterns in this last haloscope, being the
sixteenth configuration the most adequate for the axion - photon coupling,
that is, with maximum form factor. The spectrum of this structure in the
frequency range of interest is shown in Figure 13, where an improvement
regarding the mode-mixing can be observed. Concretely, the alternate coupling
allows a 32-fold greater separation between the axion mode and the
neighbouring closest one (from $\Delta f_{all-inductive}=470$ KHz to $\Delta
f_{alternated}=15.34$ MHz).

Figure 11: Manufactured 6cav and 30cav structures with alternated couplings.
Figure 12: Electric field pattern (vertical polarization) for the thirty
different configurations of mode $TE_{101}$ along the alternating 30-cavites
haloscope. Numbers at left column refer to the order of the configuration
resonances with the frequency. Figure 13: Simulated transmission coefficient
magnitude of an alternating 30-cavities haloscope at 2 K.

##  5 Data acquisition system

The setup for the experiment is shown in Figure 14. The cavity was placed
inside one of the bores of the Large Hadron Collider (LHC) ring section
prototype placed at CAST. A low noise amplifier (LNA) TXA4000 from TTI Norte
tti  provided a 40 dB gain in the 8-9 GHz frequency band and was placed inside
a copper vessel at the cryogenic section limited by flange 1. Then a thermal
transition required thermal plates to adapt the cables from the cryogenic
section to room temperature. Port 1 of the haloscope was designed to obtain
critical coupling while port 2, which was weakly coupled, provided information
at the calibration stage about the working frequency and the correct operation
of the amplifier, which needed a special bias. Temperature and bias cables
were made of phosphor bronze from Lake Shore Cryotronics lakeshore  to avoid
thermal leakages. Radiofrequency (RF) cables were 3.5 mm semirigid coaxial
copper from Micro-coax microcoax . Connectors were SubMiniature version A
(SMA).

The signal from port 1 was the input of the analog acquisition module, which
was a heterodyne receiver that down-converted the input RF signal to the
intermediate frequency (IF) output frequency band. The main functional blocks
of the analog acquisition module were (see Figure 15):

Figure 14: Measurement setup. The cavity was placed in the cold bore at near
2K. The signal was amplified by a cryogenic LNA. Thermal contacts were
employed to adapt the cryogenic and room temperatures for both the calibration
and signal ports between flanges 1 and 2.

  * •

RF LNA: It amplified the input RF signal. It worked from $8$ GHz to $9$ GHz
and provided a gain around $55$ dB and $30$ dB input return loss.

  * •

Single pole double through (SPDT) RF switch: This device allowed to visualize
the amplified RF signal at the test output port, or to check directly the down
converter circuit.

  * •

Down Converter: It down-converted the input signal from X band to a frequency
band around $140$ MHz. It was an image rejection mixer (IRM) which provided
$26$ dB of image band rejection.

  * •

Local Oscillator (LO): It provided the frequency signal to convert the RF
input signal to the $140$ MHz band. Its working power level was $0$ dBm
working from $7860$ MHz up to $8860$ MHz.

  * •

IF Filtering: This section filtered the desired IF signal from $134$ MHz to
$146$ MHz. It consisted of two filters in series: a low pass filter (LPF) to
reject the LO leakage from the IRM, and a surface acoustic wave (SAW) band
pass filter (BPF) to select the wanted IF frequency band.

  * •

IF Signal Conditioning: This module amplified or attenuated the IF signal to
obtain the desired IF output level. Its function was to bring the IF signal to
the best input level for the digital acquisition module. The amplification or
attenuation level depended on the input RF signal power.

  * •

Power supply unit: This unit fed all the modules that formed the analog
acquisition module.

The LO frequency was chosen to satisfy equation 18 taking into account the
radio-frequency (frequency of the first appearing mode of the cavity).

$\text{IF(MHz)}\,=\,\text{RF(MHz)}\,-\,\text{LO(MHz)}$ (18)

(for instance, $140$ MHz $=$ $8400$ MHz - $8260$ MHz)

Figure 15: Block diagram of the analog acquisition module.

The digital acquisition module represented in Figure 16 was especially
designed to process the IF signal obtaining the power spectra with adequate
dynamic range and frequency resolution. This is achieved by digitizing the
signal with a $14$ bits A/D converter and processing the data with a FPGA. The
unit analyzes a total instantaneous bandwidth of $12$ MHz around a central
frequency of $140$ MHz. This is performed by sampling with a clock frequency
of $37.5$ MHz and using an antialiasing filter centered in the eighth Nyquist
zone (bandpass sampling). Then, the FFT is calculated with $8192$ samples,
obtaining a frequency resolution of $4577.6$ Hz. The system is designed to
integrate (average) a number of the power spectra internally to avoid
unnecessary large data rates and file storage requirements in the external
computer. A pre-integrated spectrum is sent every $0.44739$ seconds to be
further integrated by the data acquisition software.

Figure 16: Block diagram of the digital acquisition module.

##  6 First results

In 2018 the 5-cavity haloscope described in section 4.1 and the associated
receiver were installed in the CAST magnet. The data-taking period went from
the 30th of October of 2018 to the 20th of December of 2018. During this time,
432 h of magnet-on and 141 h of magnet-off data at different LO frequencies
were taken. The reason for taking different LO frequencies will be explained
later on in this section.

A typical power spectrum consisted of data integrated for 90.37278 s and will
be labelled the 90s spectrum. For the first results, two sets of magnet-on and
two sets of magnet-off data were used. The magnet-on sets were taken at LO
frequencies $l_{1}$ = 8.240 GHz and $l_{2-\text{on}}$ = 8.247 GHz and
contained 4093 and 2446 spectra respectively. The magnet-off sets were taken
at LO frequencies $l_{1}$ and $l_{2-\text{off}}$ = 8.248 GHz and contained
1047 and 702 spectra respectively.

These 90s spectra (see figure 17(a)) were a combination of the electronic
background, the cavity resonance peak structure, thermal noise, and any
possible axion signal. The unwanted electronic background and the cavity
resonance structure were removed following these steps:

  1. 1.

Divide each 90s spectrum taken at $l_{1}$ by the average spectrum of
$l_{2-\text{on}}$. This was a first order correction to the electronic
background introduced by the DAQ (see figure 17(b)).

  2. 2.

Limiting the frequency region of the analysis to a range of $\sim$ 0.87 MHz
around the resonance peak (see figure 17(b)). This range covered more than the
full width at half maximum of the Lorentzian peak.

  3. 3.

Applying a Savitzky-Golay (SG) filter savitzky64 ; SG-Filters-2  to the
average spectrum of the 90s spectra taken at $l_{1}$. The filter produced a
fit labelled SG-fit (see figure 17(c)).

  4. 4.

Normalized spectra were created by dividing each 90s spectrum by the SG-fit
(see figure 17(d)).

  5. 5.

Due to drifts in the receiver chain, the normalized spectra still had unwanted
structure remaining within the spectra. A second SG-fit (SG∗) was created for
each normalized 90s spectrum. The spectra were divided by the SG∗-fits and the
result was subtracted by 1 to create unit-less normalized power spectra (see
figure 17(e)).

  6. 6.

The unit-less normalized power spectra were combined into a grand unified
spectrum.

  7. 7.

The previous steps were repeated for the magnet-off data sets.

  8. 8.

The final spectrum was the subtraction of the magnet-on and magnet-off grand
unified spectra (see figure 17(f)).

In step 1, by changing the LO frequency, the resonance peak was displaced in
the IF frequencies introduced in equation 18. However, the intrinsic
background structure introduced by the DAQ remained qualitatively similar. The
SG-fit of step 3 was constructed to remove large frequency scale features like
the resonance peak, while the SG∗-fits of step 5 removed time scale variations
like the one produced by gain drifts.

Spectra produced after step 5 should be a combination of thermal noise and a
possible axion signal. Thermal noise follows a Gaussian distribution and the
expected fluctuation of these spectra is given by:

$\sigma=\frac{1}{\sqrt{t\cdot\Delta\nu}},$ (19)

where $\Delta\nu$ is the resolution bandwidth of the DAQ (4577 Hz) and $t$ is
the integrated time of the power spectra. For $t=90$ s, the theoretical
$\sigma=0.001554$. The expected values of the pull distribution of a Gaussian
distributed value are a mean value $\mu=0$ and a width of $\sigma=1$. The
histogram of the pull distribution of the unit-less normalized power spectra
was created and a Gaussian distribution was fitted to it. The fit gave a mean
value $\mu=0.0002\pm 0.0009$ and a width of $\sigma=1.0048\pm 0.0009$, which
proved that at the 90 s fluctuation level the background structure was
removed. After combining the 103 hours of magnet-on together, the pull
distribution of the histogram of the grand unified spectrum showed a
unexpected non Gaussian structure with $\mu=0.0\pm 0.2$ and $\sigma=1.7\pm
0.2$.

This structure was identified as a systematic residual, which also appeared in
the magnet-off data. Step 8 was done to remove this structure without
affecting a possible axion signal. The fitted Gaussian function to the final
spectrum yielded $\mu=0.00\pm 0.08$ and $\sigma=1.07\pm 0.08$. An axion search
was done to this spectrum.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 17: Removal procedure of the electronic background. (a): The initial
raw spectra taken by the DAQ were a combination of electronic background, the
cavity resonance peak, white noise and a possible axion signal. The electronic
background and cavity resonance were removed by (b): dividing the spectra at
different LO frequencies and (c-d): applying two SG filters. (e): The unit-
less normalized spectra were combined into a grand unified spectrum. (f): The
magnet-on and off grand unified spectra were subtracted to remove the
systematic residual structure and create the final spectrum. Figures taken
from Alvarez (2021).

To perform the search, the axion line shape was created using the velocity
distribution based of the standard isothermal spherical halo model
Jimenez:2002vy . The two SG-fits were applied to the line shape to replicate
the effects of the background removal steps 3 to 5. A signal attenuation of 20
$\%$ and a distortion of the line shape were the results of applying the SG-
filters. The resulting distorted line shape (which had a range of 16 frequency
bins) was fitted through the 190 frequency bins of the final spectrum. The
amplitude $A$ given by the fit function represented the axion power deposited
at each frequency. After applying the look-elsewhere effect Lista:2017jsy  the
highest $A$ excess yielded a global significance of $\sigma=3.05$. Thus no
significant signal above statistical fluctuations was observed.

The Bayesian method Lista:2017jsy  was used to compute the upper limit (UL).
The fact that a possible axion signal would appear as a positive $A$ value on
the fit function was used as the prior for the posterior function. The
likelihood function of the normalized power excess values of the final
spectrum can be expressed with a $\chi^{2}$ distribution because the variable
was Gaussian distributed. The UL of $A$ ($A_{UL}$) was computed integrating
over the posterior function as shown in the following equation:

$\frac{1}{N}\int_{0}^{A_{\text{UL}}}\text{e}^{-(\chi^{2}/2)}dA=1-\alpha,$ (20)

where 1 - $\alpha$ is the credibility level (CL) of the UL. For these results
a 95 $\%$ CL was used. The unit-less $A_{UL}$ can be converted back to power
units with:

$P_{d}=A_{\text{UL}}\cdot P_{N}=A_{\text{UL}}k_{b}T_{\text{sys}}\Delta\nu,$
(21)

and with equation 5 one can compute the exclusion limit to axion-photon
coupling $g_{a\gamma}$ of this measurement.

To evaluate equations 5 and 21, the external parameters of the system were
measured. For the $T_{\text{sys}}$ the Y-method Y-method  was followed, the
loaded Q-value was computed using $Q_{l}=f/\Delta f$ where $f$ was the
resonant frequency of the cavity’s first mode and $\Delta f$ was the resonance
width. The cavity coupling was measured using the reflection coefficient
$S_{11}$111To measure the cavity coupling a RF switch was installed after the
data taking.The possibility to by-pass the LNA was thus not possible during
data-taking. The $Q_{l}$ measured after the installation of the switch was
similar to the one computed during the data-taking period. From this result it
can be assumed that the measured coupling before and after the data-taking
period was the same.. The geometric factor and volume were taken from
simulations. The resolution bandwidth was determined by the DAQ specifications
described in section 5, the standard dark matter density value used by other
axion haloscope experiment was used to enable comparisons with other results.
The magnetic field was provided by the CAST collaboration. Finally, an
attenuation factor ($\eta$) had to be included to account for the losses
between the cavity and the LNA.

Table 1 lists the values for all the parameters needed to compute
$g_{a\gamma}$. An exclusion limit with a 95% credibility level on the axion-
photon coupling constant of g${}_{a\gamma}\gtrsim 4\times 10^{-13}$ GeV-1 over
a mass range of 34.6738 $\mu$eV $<m_{a}<$ 34.6771 $\mu$eV was achieved. This
constitutes a significant improvement over the current strongest limit set by
CAST at this mass. A detailed description of the first results can be found in
Alvarez (2021). Figure 18 compares the CAST-RADES results with other
haloscopes experiments and CAST’s solar axion results.

Parameter | Value
---|---
$\Delta\nu$ | 4577 Hz
$T_{\text{sys}}$ | (7.8 $\pm$ 2.0) K
$Q_{l}$ | $11009\pm 483$
$\kappa$ | 0.33 $\pm$ 0.05
$B_{e}$ | (8.8 $\pm$ 0.0088) T
$\rho_{a}$ | 0.45 GeVcm-3
$C$ | 0.65
Volume | 0.03 l
$\eta$ | 0.83
Table 2:  Parameter used to compute the exclusion limit of the axion-photon
coupling. Figure 18:  Axion-photon coupling vs. axion mass phase-space. In red
the CAST-RADES axion-photon coupling exclusion limit compared to other
haloscope results and the CAST solar axion results. Inset: Zoom-in of the
parameter range probed in the first CAST-RADES results
($34.6738~{}\mu$eV$<m_{a}<34.6771~{}\mu$eV), where the green region represent
the uncertainty of the measurement. Figure taken from Alvarez (2021).

##  7 Envisaged work

After the encouraging results of these first years, the collaboration is
currently and will be in the next years focusing on increasing the sensitivity
of the experiment, on increasing the mass range by means of tuning mechanisms
or materials, on developing new haloscopes for other mass regions, and on
developing computational techniques which can efficiently obtain an accurate
solution for the electromagnetic field induced by the axion-photon
interaction. Next, these future goals are briefly described.

###  7.1 Increasing the experiment sensitivity

As said before, the experiment sensitivity, in terms of the haloscope
parameters, depends on its volume, its quality factor and its coupling to the
line which extracts the signal generated by the axion - photon conversion.

The volume can be greatly increased if high multi-cavities are used. Since our
operation mode is the $TE_{101}$, the detection frequency is independent of
the cavity height, depending only on its width and length. In this way, the
haloscope height can be increased until it reaches the magnet bore diameter.
However, mode clustering around the $TE_{101}$ mode will appear due to the new
resonant $TE_{1n1}$ and $TM_{1n1}$ modes (with $n=1,2,3,..$) for very high
cavities. We are now studying the proper combination of different ports of the
haloscope that can alleviate this clustering and make possible this increase
of volume without distorting the mode $TE_{101}$ resonance.

Moreover, the multi-cavity concept can be used in other dimension, creating so
2D multi-cavities, which not only increase their volume and exploit the magnet
bore room, but also are able to introduce transmission zeros into the
haloscope response for the rejection of unwanted cavity modes.

Regarding the improvement of the quality factor, a considerable increase of
the experiment sensitivity is expected from the use of superconductive
materials, and we recently manufactured a few cavities with the specific
purpose of verifying the potential of this technology. The design of these new
cavities is specifically tailored to allow soldering High-Temperature
Superconductor (HTS) coated tapes of commercial production on the cavity
surface. These tapes are optimized for magnetic flux pinning and thus ideal
for operation in a strong magnetic field. Measurement of a first prototype
covered with such HTS tapes has shown more than 50% improvement of quality
factor compared to copper, when tested at 4.2 K in a magnetic field up to 11.5
T Golm:2021ooj . Further improvements in the soldering process have already
been identified and are under test, which should allow reaching even higher
performance.

Finally, as remarked before, the sensitivity of the coaxial - cavity coupling
with the position, dimensions and shape of the probe makes advisable to
develop mechanical systems for modifying this coupling in situ and get the
desired critical coupling. This moving coupling system is mandatory when a
frequency range is explored by tuning the haloscope properly. The
collaboration is currently designing the moving coupling system with
nanopositioners. For the tuning system, both mechanical and electromagnetic
mechanisms, based on ferroelectric and ferromagnetic materials, are being
studied.

###  7.2 Haloscopes for other frequencies

As described above, the first four years of the RADES collaboration have been
focused in developments for the 34 – 35 $\mu eV$ mass range (8.2 – 10.6 GHz
frequency range). But the possibility of operation in future magnets has
motivated that part of the work of the collaboration is nowadays devoted to
the design of haloscopes in two far regions regarding this.

First, the future BabyIAXO magnet IAXO (2021); Dafni (2021), a first step of
the IAXO experiment Armengaud (2019), with two bores of 70 cm in diameter and
10 meter in length, allows to test new bi-dimensional multi-cavity concepts in
order to increase the volume in other dimensions. And, even more important,
this magnet has enough volume to even host single cavities for searching in
the 1 – 2 $\mu eV$ mass range (240 – 480 MHz, the lower limit of UHF band).
These masses are just below the region explored by ADMX currently. With a 2T
magnetic field, BabyIAXO will be able not only to develop helioscope searches,
but also haloscope experiments with a figure of merit ($B^{2}V$) of 15.5
$T^{2}m^{3}$, a great leap from 0.014 $T^{2}m^{3}$ in CAST. Currently, RADES
is designing these new cavities that take profit from the large volume in
BabyIAXO.

Second, the Canfranc Underground Laboratory (LSC) will have at the end of 2022
a dilution cryostat working at 10 mK with a 10 T solenoid magnet, and this
opens the possibilities of continuing with the development of multicavities in
X band, adapted to a solenoid magnetic field direction, with an improved noise
temperature, and, additionally, working on haloscopes in a theoretically well
motivated region for searching the axion, the 310 – 455 $\mu eV$ (75 – 110
GHz, W band). Both experiments will need detection devices that exploit the
ultra-low temperature and go beyond the standard quantum limit, such as
Josephson parametric amplifiers (JPA), superconducting qubit-based single
photon counters or, for higher frequencies, kinetic inductor devices (KID).

###  7.3 Developments in numerical simulation: application of the BI-RME 3D
method to axion-photon coupling in resonant cavities

The Boundary Integral Resonant Mode Expansion 3D (BI-RME 3D) birme3Dcavities ;
conciauro  is a full-wave modal method based on classical numerical techniques
used for electromagnetic analysis. The BI-RME 3D method, developed during
eighties and nineties, provides an exact formulation which allows to calculate
the electromagnetic field existing in a lossy microwave resonator in terms of
the electric and magnetic charge and current densities.

Applying the BI-RME 3D method to the axion field, it is possible to obtain the
complete information of the extracted signal from the cavity taking into
account not only the cavity shape, but also metallic and dielectric
materials.This knowledge could be useful for further experiments or analysis
in which the axion phase plays an important role, in particular when several
waveguide ports are included in the analysis, avoiding the typical Cauchy-
Lorentz approximation kim_CAPP_2019 .

In summary, it is envisaged a new approach to calculate the extracted and
dissipated power using a semi-analytical solution through the BI-RME 3D
method, obtaining the frequency spectrum of the electromagnetic axion field
generated inside the cavity. This rigorous broadband result could be of
special interest in the cases where neighbouring modes are quite close to the
axion resonant mode and might interfere.

##  8 Conclusions

The RADES collaboration joined in 2016 the international quest for the dark
matter axion detection, and during the last five years has developed a new
kind of resonant haloscope based on the multi-cavity concept, which allows to
increase the volume of the haloscope and, so, the detected power and the
sensitivity of the experiment, without modifying the resonant frequency, that
is, the exploration mass. This new concept has been manufactured and tested
for both inductive and alternated internal coupling, showing and improvement
in the isolation of the axion mode in the second case. Moreover, the first
data-taking campaign in the CAST magnet at CERN took place in 2018 with the
5-cavities inductive coupling haloscope. The future is full of new research
lines for increasing the sensitivity of the experiment as well as for
designing new haloscopes in lower (1-2 $\mu$eV) and higher (310-455 $\mu$eV)
mass ranges, or for developing new computational techniques for obtaining
rigorous electromagnetic solutions for the axion - photon coupling in resonant
cavities.

This work has been funded by the Spanish Agencia Estatal de Investigación
(AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under project
FPA-2016-76978-C3-2-P (supported by the grant FPI BES-2017-079787) and
PID2019-108122GB-C33, and was supported by the CERN Doctoral Studentship
programme. The research leading to these results has received funding from the
European Research Council under grant ERC-2018-StG-802836 (AxScale project).
IGI acknowledges also support from the European Research Council (ERC) under
grant ERC-2017-AdG-788781 (IAXO+ project).

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