# Search for vector dark matter in microwave cavities with Rydberg atoms

Jordan<EMAIL_ADDRESS>Aurélien Hees, Jérôme Lodewyck, Rodolphe Le
Targat, Peter Wolf  SYRTE, Observatoire de Paris, Université PSL, CNRS,
Sorbonne Université, LNE, 61 avenue de l’Observatoire 75014 Paris, France

###### Abstract

We propose a novel experiment to search for dark matter, based on the
application of an electric field inside a microwave cavity and electrometry
using Rydberg atoms. We show that this kind of experiment could be extremely
useful for detecting specific dark matter candidates, namely massive vector
fields coupled to the photon field, more commonly known as dark photons. Such
a massive vector field is a good candidate for dark matter. Using realistic
experimental parameters we show that such an experiment could improve the
current constraint on the coupling constant of the dark photons to Standard
Model photons in the 1 $\mu$eV to 10 $\mu$eV mass range, with the possibility
of tuning the maximum sensitivity via the cavity size. The main limiting
factors on the sensitivity of the experiment are the amplitude stability of
the applied field and the measurement uncertainty of the electric field by the
atoms.

##  I Introduction

While required to explain several astrophysical and cosmological observations,
the microscopic nature of dark matter (DM) is still to this day one of the
biggest mysteries in physics Bertone and Tait (2018). Among many other classes
of DM, ultra-light dark matter (ULDM) models have recently gained a lot of
attention in the scientific community, due to the absence of signals from
historically dominant models, such as weakly interacting massive particles
(WIMPs). These models are characterized by particles with low mass, from
$10^{-22}$ eV to $0.1$ eV, meaning that detection from recoiling in particle
accelerators is difficult or impossible. Thus, other kind of experiments have
to be considered for their detection.

One particularly well motivated model of ULDM is called the dark photon (DP),
a spin-1 field of mass $m$ which appears in many beyond the Standard Model
theories. Cosmologically speaking, the DP field is frozen after inflation and
starts to oscillate at its Compton frequency when $mc^{2}/\hbar\gg H$. It can
be shown that these oscillations scale as $a^{-3}(t)$, at late cosmological
times, behaving as cold dark matter (CDM). The production of CDM in this case
is done through non-thermal processes, such as the so-called misalignment
mechanism.

This additional $U(1)$ field is also well motivated on the particle physics
side, as the DP model is a particular case of the so-called $U$ boson, which
makes the minimal gauged Standard Model extension, through its coupling with
the B-L current of the Standard Model Fayet (2007).

For these reasons, DP is a very good light dark matter candidate, with many
new experiments being developed to hunt for it. In the following, we will be
interested in the coupling of the DP field with the electromagnetic (EM) field
with strength $\chi$. This coupling induces an electric field filling all
space, with amplitude directly proportional to $\chi$. In this paper, we
propose a new kind of experiment aiming at detecting DP, through this
additional electric field. The experiment is based on the use of a microwave
cavity in which, an applied external electric field acts as an amplifier for
the weak DP induced electric field, with the strength of amplification
directly proportional to the square root of the injected power. If these two
different electric fields have close frequencies the total field power will
oscillate at their frequency difference $\Delta\omega$ and become detectable
by atoms located at the center of the cavity through the Stark effect. We show
that using this setup, one could scan new regions of the m-$\chi$ parameter
space, in particular in the 1 $\mu$eV to 10 $\mu$eV mass range, with the
possibility of tuning the maximum sensitivity regions via the cavity size and
design. We use some simplifying assumptions, in particular for the model of
the cavity. The main aim of our paper is to obtain a rough estimate of the
experimental sensitivity using realistic parameters and taking into account
the main expected noise sources and systematic effects. Our results and
described methods will be useful for the design of a real-life experiment, in
which case more careful modeling should be carried out. Nonetheless, we do not
expect that to change the orders of magnitude of the sensitivity presented
here.

##  II Field equations

The Lagrangian describing the interaction between the EM field $A^{\mu}$ and a
DP field $\phi^{\mu}$ of mass $m$ is given by222In this manuscript, we use the
metric signature $(-,+,+,+)$ and the unit of the DP field $\phi^{\mu}$ is
Vs/m, as the usual EM vector potential $A^{\mu}$. Holdom (1986)

$\displaystyle\mathcal{L}=$
$\displaystyle-\frac{1}{4\mu_{0}}F^{\mu\nu}F_{\mu\nu}+j^{\mu}A_{\mu}-\frac{1}{4\mu_{0}}\phi^{\mu\nu}\phi_{\mu\nu}$
(1)
$\displaystyle\qquad-\frac{m^{2}c^{2}}{2\mu_{0}\hbar^{2}}\phi^{\mu}\phi_{\mu}-\frac{\chi}{2\mu_{0}}F_{\mu\nu}\phi^{\mu\nu}\,,$

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the usual
electromagnetic field strength tensor,
$\phi_{\mu\nu}=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}$ is the DP
field strength tensor, $\chi$ is the dimensionless kinetic mixing coupling
parameter which characterizes the coupling between the DP and the EM field,
and $j^{\mu}$ is the usual electromagnetic 4-current. A change in the EM
4-potential

$\bar{A}^{\mu}=A^{\mu}+\chi\phi^{\mu}$ (2)

allows one to redefine the fields in terms of mass eigenstates called massless
and massive photons (at the price of a non-standard interaction between the EM
field and standard matter), see the discussion in Nelson and Scholtz (2011).
Using this change of variable, Eq. (1) becomes, at first order in $\chi$,

$\displaystyle\mathcal{L}=$
$\displaystyle-\frac{1}{4\mu_{0}}\bar{F}^{\mu\nu}\bar{F}_{\mu\nu}+j^{\mu}\left(\bar{A}_{\mu}-\chi\phi_{\mu}\right)-\frac{1}{4\mu_{0}}\phi^{\mu\nu}\phi_{\mu\nu}$
(3)
$\displaystyle\qquad-\frac{m^{2}c^{2}}{2\mu_{0}\hbar^{2}}\phi^{\mu}\phi_{\mu}\,,$

with $F_{\mu\nu}=\bar{F}_{\mu\nu}-\chi\phi_{\mu\nu}$. The field equations read

$\displaystyle\bar{F}^{\beta\alpha}_{\,,\alpha}$
$\displaystyle=\mu_{0}j^{\beta}\,,$ (4a)
$\displaystyle\phi^{\beta\alpha}_{\,,\alpha}$
$\displaystyle=-\chi\mu_{0}j^{\beta}-\frac{m^{2}c^{2}}{\hbar^{2}}\phi^{\beta}\,.$
(4b) The antisymmetry of both strength tensors leads to the conservation of
the electromagnetic 4-current $\partial_{\mu}j^{\mu}=0$, and the continuity
equation for the DP field $\partial_{\mu}\phi^{\mu}=0$.

Using the Lorenz gauge for EM, $\partial^{\mu}\bar{A}_{\mu}=0$, Eqs. (4)
become

$\displaystyle\Box\bar{A}^{\beta}$ $\displaystyle=-\mu_{0}j^{\beta}\,,$ (5a)
$\displaystyle\Box\phi^{\beta}$
$\displaystyle=\frac{m^{2}c^{2}}{\hbar^{2}}\phi^{\beta}+\chi\mu_{0}j^{\beta}\,,$
(5b)

where
$\Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\equiv-\frac{1}{c^{2}}\partial^{2}_{t}+\nabla^{2}$.
These equations admit two classes of solutions in vacuum: a massless vector
field (standard EM) and a massive one, the latter being characterized by
solutions :

$\displaystyle\phi^{\beta}$ $\displaystyle=Y^{\beta}e^{ik_{\mu}x^{\mu}}\,,$
(6a) $\displaystyle\bar{A}^{\alpha}$ $\displaystyle=0\,,$ (6b)

where

$k_{\mu}k^{\mu}=-\frac{\omega^{2}}{c^{2}}+\left|\vec{k}\right|^{2}=-\left(\frac{mc}{\hbar}\right)^{2}.$
(7)

As can be noticed from Eq. (2), this solution induces an ordinary
electromagnetic field in a vacuum (see also Horns  _et al._ (2013))

$A^{\beta}=-\chi Y^{\beta}e^{ik_{\mu}x^{\mu}}\,.$ (8)

The stress-energy tensor from the DP field can be derived from Eq. (1) and
reads

$\displaystyle T_{\mu\nu}$
$\displaystyle=\frac{1}{\mu_{0}}\phi_{\mu\alpha}\phi_{\nu}^{\
\alpha}-\frac{1}{4\mu_{0}}\eta_{\mu\nu}\phi^{\alpha\beta}\phi_{\alpha\beta}$
(9)
$\displaystyle-\frac{m^{2}c^{2}}{\mu_{0}\hbar^{2}}\left(\frac{1}{2}\phi^{\alpha}\phi_{\alpha}\eta_{\mu\nu}-\phi_{\mu}\phi_{\nu}\right)\,.$

For the plane-wave solution derived in Eq. (6a), the time-averaged value of
the stress-energy tensor is given by

$\displaystyle\langle T_{\mu\nu}\rangle$
$\displaystyle=\frac{1}{2\mu_{0}}k_{\mu}k_{\nu}Y_{\alpha}Y^{\alpha}\,,$ (10)

which leads to an estimate of the energy density

$\rho=\langle T_{00}\rangle=\frac{\omega^{2}|\vec{Y}|^{2}}{2\mu_{0}c^{2}}\,,$
(11)

for $\vec{k}=0$. Similarly, the averaged pressure vanishes. This shows that
the DP can indeed be interpreted as cold and pressureless DM and, under this
assumption, the amplitude of oscillation of this vector field, i.e. $\vec{Y}$,
depends directly on the local DM energy density $\rho_{\mathrm{DM}}$. Its
experimental value has been considered for a while to be $\rho_{\mathrm{DM}}=$
0.4 GeV/cm3 McMillan (2011), but has recently been readjusted such that
$\rho_{\mathrm{DM}}\in\left[0.3;0.55\right]$ GeV/cm3 Evans  _et al._ (2019).

In conclusion, in vacuum, a DP will behave as a plane wave whose dispersion
relation is given by Eq. (7). On average, this DP will behave as cold DM and
its local energy density is directly related to the amplitude of the
oscillation and to its mass. Due to the coupling between the DP and the EM
field (coupling characterized by the mixing parameter $\chi$), the DP field
will induce a small electromagnetic field, whose strength is proportional to
$\chi$ and to the DP field amplitude, see Eq. (8). In particular, if one
considers that $\vec{k}=0$, (which is justified to leading order in
$v_{\mathrm{DM}}\sim 10^{-3}c$, where $v_{\mathrm{DM}}$ is the typical DM
galactic velocity, see Appendix A), then the induced electromagnetic field
consists mostly in an oscillating electric field the magnetic field component
is suppressed by a factor $v_{\mathrm{DM}}/c$.) of the form Horns  _et al._
(2013)

$E^{j}_{\mathrm{DM}}=-\frac{\partial A^{j}}{\partial t}=-i\chi\omega
Y^{j}e^{-i\omega t}\,.$ (12a) The amplitude of this oscillating electric field
is therefore directly related to the local DM density through
$\left|\vec{E}_{\mathrm{DM}}\right|=\chi
c\sqrt{2\mu_{0}\rho_{\mathrm{DM}}}\,.$ (12b)

The idea of several experiments searching for DP is to focus this small
electromagnetic field in order to enhance it and hopefully make it detectable
Brun  _et al._ (2019); Suzuki  _et al._ (2015); Knirck  _et al._ (2018);
Tomita  _et al._ (2020); Mizumoto  _et al._ (2013); Andrianavalomahefa  _et
al._ (2020).

##  III Theoretical modeling of the experiment

As mentioned in the previous section, an oscillating DP coupled to the
standard electromagnetic field will induce a small oscillating electric field
in a vacuum. One very peculiar feature of this electric field is that it does
not propagate, i.e. its wave vector vanishes $\vec{k}=0$ (to first order in
$v_{\mathrm{DM}}/c$). This is due to the fact that this electric field is
induced by a massive vector field and therefore its dispersion relation is
given by Eq. (7). In this section, we will show how an electromagnetic cavity
can be used to search for the electric field induced by a DP. In addition, we
will show how to use atoms as a tool to detect this electric field through the
Stark effect, i.e. the displacement of the energy levels of an atom under a
perturbation by a static electric field (or by an electric field whose
frequency is much lower than the transition frequency of the atom). There are
mainly two reasons to consider a cavity as an experiment to search for DP.

First of all, as for other DP experiments using resonators, the mirrors of the
cavity will enhance the electric field induced by the DP. Indeed, the electric
field parallel to the surface of a perfect conductor has to vanish. Therefore,
because of the presence of the oscillating DP-induced electric field, the
mirror will generate a standard electromagnetic field that will propagate
perpendicularly to the mirror and whose amplitude is such that it will cancel
the DP-induced field parallel to the surface. Physically, the DP-induced
electric field will induce an oscillation of the electrons within the mirror
which will create a standard electromagnetic field. Since a cavity consists in
two mirrors, this boundary conditions can, under some conditions, produce
resonances that will significantly enhance the small DP-induced electric
field.

The second reason to consider a cavity is related to the use of atoms to
measure the electric field inside the cavity through the Stark effect, which
is sensitive to the square of the electric field. If one applies a standard
electromagnetic wave inside the cavity (whose electric field will be denoted
by $\vec{E}_{a}$) the DP contribution to the square of the electric field
inside the cavity is $\sim\vec{E}_{\mathrm{DM}}\cdot\vec{E}_{a}$ (to first
order in $\chi$), which can also be enhanced by a resonant $\vec{E}_{a}$. In
addition to enhancing the amplitude of the signal to be measured, applying an
external field is also important to produce a signal at low, but non-zero,
frequency, where the Stark effect can be realistically measured. More
precisely, for a cavity whose length is of the order of a few cm, we will be
interested in searching for DP oscillating at a frequency $f=\omega/2\pi$ of
the order of a few GHz. The difficulty is that such a rapid oscillation of the
atomic transition frequency will be very hard to measure as the interrogation
cycle of the atoms is much longer. But, if one applies en external field at
angular frequency $\omega_{a}$ which is close to $\omega$, then the cross term
between the DP electric field and the applied electric field will have a
component oscillating at the low angular frequency
$\left|\omega-\omega_{a}\right|$, which can be measured by the atoms.

###  III.1 Expression of the total electric field inside the cavity

In this section, we derive the electric field induced by both the DP and the
applied field at the center of the cavity. Here, we only summarize the
methodology and discuss the main results. The detailed derivation can be found
in Appendix B. We consider a cavity consisting of two flat mirrors of
reflectivity $r$ separated by a distance $L$ 333For simplicity we assume that
the transverse size of the mirrors is $\gg L,\lambda$, where $\lambda$ is the
wavelength of the fields of interest.. The reflectivity and the cavity quality
factor $Q$ are related through (Q $\gg$ 1)Suter and Dietiker (2014)

$r\approx 1-\frac{\pi}{2Q}\,.$ (13)

where we considered only low resonance modes, such that $Q\sim\mathcal{F}$,
the finesse of the cavity. The principle of calculation is similar for both
the DP and the applied electric fields and can be found in Savalle (2020). The
idea is to propagate the electric field an infinite number of round trips
inside the cavity and to sum these infinite number of contributions at a given
location. To perform cavity round trips, the field is propagated along one
direction and when it reaches a mirror, its amplitude is multiplied by $-r$
and its wave vector is flipped. For $r<1$, the infinite sum converges and can
be calculated explicitly.

First, let us apply this procedure to the applied external field. We assume
that the external electric field is applied on the left edge of the cavity,
see Fig. 1. After entering the cavity, the wave will undergo an infinite
number of round trips inside the cavity and it will lead to the first
contribution to the total field, see Fig. 1.

Figure 1: An external field (in blue) is applied at the cavity edge. The
standing DP electric field (in yellow) generates a propagating electric field
inside the cavity (in orange). At the center of the cavity, the transition
frequency of Rydberg atoms is impacted by $|E|^{2}$ through the Stark effect
(see text.).

Ideally, the applied field can be parametrized as
$\vec{E}_{a}=\vec{X}_{a}\Re\left[e^{-i(\omega_{a}t-k_{a}(x+\frac{L}{2})+\phi})\right]$,
with its amplitude $X_{a}$, angular frequency $\omega_{a}$ and phase $\phi$.
Applying the mathematical procedure described above (see Appendix B.1), the
expression of the total applied field at the center of the cavity is given by

$\displaystyle\vec{E}^{\mathrm{tot}}_{a}=\vec{A}(\omega_{a})\cos(\omega_{a}t+\phi)+\vec{B}(\omega_{a})\sin(\omega_{a}t+\phi)\,,$
(14)

where $\vec{A}(\omega_{a}),\vec{B}(\omega_{a})$ are functions (given by Eqs.
(46)) depending on $X_{a}$, $r$ and $L$, respectively the applied field
amplitude, the reflectivity and the length of the cavity.

Let us now focus on the contribution from the DP field. From Eqs. (12), one
can, without loss of generality, write the expression of the electric field
related to the DP as
$\vec{E}_{\mathrm{DM}}=\vec{X}_{\mathrm{DM}}\Re\left[e^{-i\omega
t}\right]=X_{\mathrm{DM}}\hat{e}_{\mathrm{DM}}\Re\left[e^{-i\omega t}\right]$
where $\hat{e}_{\mathrm{DM}}$ is a unit vector characterizing the polarization
of the DP field and $X_{\mathrm{DM}}=\chi c\sqrt{2\mu_{0}\rho_{\mathrm{DM}}}$.
Because of this electric field, both mirrors will generate a propagating
standard electromagnetic field such that the total component of the electric
field parallel of the mirrors’ surface vanishes. These two electromagnetic
waves will follow an infinite number of round trips inside the cavity, loosing
some energy at each reflection with reflection coefficient $r<1$. The
calculation regarding this contribution is detailed in Appendix B and the
resulting total electric field induced by the DP at the center of the cavity
reads

$\displaystyle\vec{E}^{\mathrm{tot}}_{\mathrm{DM}}=\vec{C}(\omega)\cos(\omega
t)+\vec{D}(\omega)\sin(\omega t)\,,$ (15)

where $\vec{C}(\omega),\vec{D}(\omega)$ are functions (given by Eqs. (48)) of
$\vec{X}_{\mathrm{DM}}$, $\vec{X}_{\mathrm{DM,\parallel}}$, $r$ and $L$ with
the first two parameters being respectively the full amplitude of the DP field
and its component parallel to the cavity plates, both linearly dependent on
the coupling $\chi$.

The calculations presented in this section have been carried out to leading
order in $\mathcal{O}(\chi)$. In particular, at each interaction between the
EM waves and the mirrors a small quantity of EM energy will be transformed
into DP. The amplitude of such a process is proportional to $\chi$ and
therefore neglected as it contributes terms of order $\mathcal{O}(\chi^{2})$.
Furthermore the corresponding energy loss is much smaller than the one coming
from the finite reflection coefficient $r$.

###  III.2 Modeling of the Stark effect

As mentioned in the introduction of this section, the main idea of the
experiment proposed here is to detect the hypothetical electric field induced
by the DP field by using atoms to measure it through the Stark effect. The
Stark effect consists in a shift in the energy levels of an atom under the
perturbation of a static (or slowly evolving444As long as the angular
frequency of oscillation is much smaller than the atomic transition angular
frequency from state k to i, $\Delta\omega<<\omega_{ik}$.) electric field, and
is given by Cohen-Tannoudji  _et al._ (1986)

$\Delta\nu=-\frac{\Delta\alpha}{2h}\left|\vec{E}\right|^{2}\,,$ (16)

where $h$ is the Planck constant, $\Delta\nu$ is the frequency shift induced
by the slowly evolving electric field $\vec{E}$ and $\Delta\alpha$ is the
differential polarizability of the atomic transition considered.

Taking into account both contributions from the applied electric field and the
DP field computed in the previous section, the total electric field power at
the center of the cavity is

$\displaystyle|\vec{E}(\omega,\omega_{a})|^{2}$
$\displaystyle=\left|\vec{E}^{\mathrm{tot}}_{a}+\vec{E}^{\mathrm{tot}}_{\mathrm{DM}}\right|^{2}\,$
$\displaystyle=\left(\vec{A}(\omega_{a})\cdot\vec{C}(\omega)+\vec{B}(\omega_{a})\cdot\vec{D}(\omega)\right)\cos(\Delta\omega
t+\phi)+\left(\vec{B}(\omega_{a})\cdot\vec{C}(\omega)-\vec{A}(\omega_{a})\cdot\vec{D}(\omega)\right)\sin(\Delta\omega
t+\phi)$ (17) $\displaystyle+\mathrm{constant\ and\ fast\ oscillating\
terms}\,,$

with $\Delta\omega=\omega_{a}-\omega$. In the following, we will not consider
the constant terms. Indeed, in the experimental scheme proposed here, we will
be interested in the oscillatory behavior of the atomic frequencies. Moreover,
we discarded the fast oscillating terms whose angular frequencies are
$2\omega_{a}$, $2\omega$ or $\omega_{a}+\omega$, with periods much shorter
than the atom interrogation time, such that on average, their impact vanishes.

Inserting the expressions of $A$, $B$, $C$ and $D$ from Eqs. (46) and (48) and
using the Eq. (12b), the amplitude of the oscillation can be simply written in
the form (See Appendix C.1)

$\frac{\chi\beta
c\sqrt{1-r^{2}}\sqrt{2\mu_{0}\rho_{\mathrm{DM}}}X_{a}}{\sqrt{1+2r\cos\left(\frac{\omega_{a}L}{c}\right)+r^{2}}}\sqrt{1+4\frac{1+(1+r)\cos\left(\frac{\omega
L}{2c}\right)}{1+2r\cos\left(\frac{\omega L}{c}\right)+r^{2}}}\equiv\chi
S(\omega,\omega_{a};\rho_{\mathrm{DM}},X_{a};L,r)\,,$ (18)

where

$\beta=\hat{e}_{\mathrm{DM}}\cdot\frac{\vec{X}_{a}}{X_{a}}\,,$ (19)

is the projection of the polarization of the DP field on the polarization of
the injected electric field. If the polarization is fixed and does not change
each coherence time, $\beta=\cos\theta$, with $\theta$ the angle between
$\vec{X}_{a}$ and $\vec{X}_{\mathrm{DM}}$. If the DP field is isotropically
distributed $\beta=1/\sqrt{3}$. To avoid any orthogonality between the two
polarizations, i.e $\beta=0$, one must run the experiment for a significant
time, i.e at least several days (see discussion in Caputo  _et al._
(2021))555Note that the signal Eq. (18) is the largest when $X_{a}$ is maximum
at the center of the cavity, which favours TE01 or TE10 modes. .

Figure 2: 3D plots (left: side view ; right: top view) of the signal
contribution Eq. (18) (arbitrary units) as function of $\omega,\omega_{a}$, in
units of $c/L$. Resonance peaks appear clearly when both frequencies
correspond to odd modes of the cavity.

In Fig. 2, 3D plots of the signal contribution Eq. (18) are shown as a
function of the angular frequencies of the DM field $\omega$ and the applied
field $\omega_{a}$. One can notice the various resonance peaks when both
frequencies correspond to an odd mode of the cavity. In the experiment scheme
presented here, we require the two frequencies to be close, hence we are only
interested in regions where $\omega\sim\omega_{a}$. This corresponds to the
diagonal of the plot seen from the top. However, in other types of experiment
where both frequencies could be separated by some hundreds of MHz or more,
other regions of this plot could be studied.

The amplitude of the signal is linear in $\chi$ and depends non-linearly on
the local DM density, on the amplitude and frequency of the applied external
field and on the mass ($mc^{2}=\hbar\omega$) of the DP field. It depends also
on some geometric factor of order unity that characterizes the DP field
polarization.

For DM and applied frequencies close to odd resonances of the cavity, i.e
$\omega L/c\approx\omega_{a}L/c\approx\pi+2n\pi$, with $n\in\mathbb{N}$, the
amplitude Eq. (18) becomes proportional to the quality factor $Q$, as
expected. Additionally, as it is also proportional to the injected amplitude
$X_{a}$, we directly see the interesting feature of looking at the field
squared amplitude, the applied field acting as an amplification for the DM
field.

Finally, the signal we look for is a slow oscillation in the transition
frequency of Rydberg atoms with respect to an unperturbed reference

$\nu(t)=\nu_{0}+\Delta\nu\cos\left(\Delta\omega t+\phi\right)\,,$ (20)

whose amplitude is given by

$\Delta\nu=-\chi\frac{\Delta\alpha
S(\omega,\omega_{a};\rho_{\mathrm{DM}},X_{a};L,r)}{2h}\,,$ (21)

which depends on both the applied external field and the DP field.

##  IV Experimental considerations

###  IV.1 Rydberg atoms

The transition frequency measurement can be performed using a regular atomic
clock, which allows very good uncertainty on the frequency measurement, but is
not very sensitive to the Stark effect e.g. for the $5s^{2}\,{}^{1}S_{0}\to
5s5p\,\,^{3}P_{0}$ clock transition in Sr the differential polarizability
$\Delta\alpha/2h\approx 3.1\times 10^{-6}$ Hz/(V/m)2 Middelmann  _et al._
(2012), thus requiring a strong applied field.

To overcome this lack of sensitivity to $E^{2}$, “regular” atoms can be
replaced by Rydberg atoms, which are in a quantum state with high principal
quantum number $n$ Gallagher (1988). The electrons are much further from the
nucleus thus the atom has much higher polarizability. For large $n$ the
corresponding polarizabilities in Sr can reach $\Delta\alpha/2h\approx 10^{5}$
Hz/(V/m)2 ($n>60$) Millen  _et al._ (2011).

In Bridge  _et al._ (2016) the measurement of the transition probability for a
given frequency lasts about 300 $\mu$s. At least three such measurements are
necessary to fully determine the resonance (amplitude, width and center
frequency), which implies a maximum sampling rate of about 1 kHz, which is the
maximum value we will assume. In principle the process could be faster with
higher laser power and/or nondestructive techniques.

Typically, Sr Rydberg measurements use laser cooled Sr atoms that are excited
to Rydberg states using two photon transitions Millen  _et al._ (2011); Bridge
_et al._ (2016) and direct spectroscopy of e.g. the $5s5p\,^{3}P_{1}\to
5sns\,^{3}S_{1}$ transition is performed, with $n$ up to 81 Millen  _et al._
(2011); Bridge  _et al._ (2016). In regular dispersive measurements, the
photon scattering rate per atom, i.e the rate at which an atom absorbs and re-
emits incident photons, is high implying that after a single detection, the
atom is too hot, and no longer trapped for a second detection. In that
situation, the measurement is said to be destructive. An alternative method is
a nondestructive measurement, based on a low photon scattering rate, meaning
that a single atom can be used for multiple measurement. This non-destructive
process has already been experimentally tested, and is based on a differential
dispersive measurement Vallet  _et al._ (2017). As a consequence, it is not
necessary to produce new Rydberg atoms for each frequency measurement,
implying that the 1 kHz sampling frequency is feasible.

###  IV.2 Bandwidth of detection

In order to see the oscillations at $\Delta\omega$ in the total field power
Eq. (17), we require the DM and the applied fields to have different
frequencies. We also require the Nyquist frequency of the apparatus to be
higher than the angular frequency of the signal $\Delta\omega<\pi f_{s}$ to be
able to detect any oscillatory behavior in the transition frequency of the
atoms.

For a given applied angular frequency $\omega_{a}$, sampling frequency
$f_{s}$, Rydberg atoms perform the measurements of the electric field squared
during $T_{\mathrm{obs}}$ at an angular frequency $\Delta\omega$ (more
precisely, $T_{\mathrm{obs}}\times f_{s}$ measurements will be taken for each
$\omega_{a}$). The time $T_{\mathrm{obs}}$ is arbitrary; if it is longer than
the measurement process comprising excitation and ionisation, one has to
prepare again the atoms to their Rydberg state after deexcitation accordingly.
As detailed above, the experiment is sensitive to any $\Delta\omega$ such that
$2\pi/T_{\mathrm{obs}}\leq|\Delta\omega|\leq\pi f_{s}$, which, in terms of DM
angular frequency, is equivalent to

$\displaystyle\omega\in[\omega_{a}-\pi
f_{s};\omega_{a}-\frac{2\pi}{T_{\mathrm{obs}}}]\cup[\omega_{a}+\frac{2\pi}{T_{\mathrm{obs}}};\omega_{a}+\pi
f_{s}]\,.$ (22)

At the end of the individual measurement time $T_{\mathrm{obs}}$, the second
step would be to shift the applied angular frequency $\omega_{a}$ by the
sampling angular frequency, $2\pi f_{s}$ and make a second measurement of the
electric field squared for DM angular frequencies $\omega$ around this new
$\omega_{a}$, during $T_{\mathrm{obs}}$ again. This way, except at the exact
angular frequencies of the applied field $\omega_{a}$, all possible DM
frequencies are scanned.

This scheme can be repeated N times, as much as time allows. At the end of
this loop, the total experimental time is simply
$T_{\mathrm{tot}}=NT_{\mathrm{obs}}$. The corresponding total DM frequencies
band scanned is
$f_{\mathrm{tot}}=Nf_{s}=T_{\mathrm{tot}}f_{s}/T_{\mathrm{obs}}$. The larger
the total experimental time, the larger the band of scanned DM frequencies.
Additionally, the blind spots at exact $\omega_{a}$ can be avoided, and
sensitivity can be optimized (see below) by shifting $\omega_{a}$ by a little
less than $2\pi f_{s}$, at the expense of increasing the overall experimental
duration.

The coherence time of the DM field is inversely proportional to its Compton
frequency and follows Derevianko (2018)

$\displaystyle\tau(\omega)=\frac{c^{2}}{\sigma^{2}_{v}\omega}\approx\frac{10^{6}}{\omega}\,,$
(23)

where we considered the usual DM velocity distribution with dispersion
$\sigma_{v}\sim 10^{-3}c$. For DM frequencies in the GHz range, this coherence
time is much smaller than $T_{\mathrm{obs}}$, which needs to be taken into
account in the sensitivity estimate (see Eq. (24)).

##  V Sensitivity estimate

In this section, we will present the level to which a realistic experiment can
detect or constrain the coupling parameter $\chi$. First, we will focus on the
measurement noise and present some limits on $\chi$ reachable considering only
statistical noise. We will then discuss the main systematics related to this
experiment: the intensity variations of the applied field. Finally, by using
realistic values for the cavity, laser and various noise sources, we will
estimate the detection limit on $\chi$ reachable for a dedicated experiment.

###  V.1 Statistical measurement noise

The first source of noise limiting the sensitivity of the experiment is the
statistical noise related to the measurement of the frequency shift
experienced by Rydberg atoms under the perturbation from an external electric
field. In Bridge  _et al._ (2016) the reported resolution of the spectroscopy
of $n=56$ Rydberg states in Sr is of the order of a few kHz, at a maximum
possible sampling rate of 1 kHz (see Sec. IV.1).

We will thus consider two scenarios for our order of magnitude estimates of
the experimental sensitivity. One with a “modest” sampling rate of 100 Hz and
a second, more optimistic one, with higher sampling at $f_{s}=1$ kHz. In both
cases we will assume a single shot spectroscopic resolution of $\sim$1 kHz for
differential polarizabilities of $\Delta\alpha/2h\approx 10^{5}$ Hz/(V/m)2,
corresponding to Rydberg states with principle quantum numbers $n\sim 60-70$
Millen  _et al._ (2011).

We denote the power spectral density (PSD) measurement noise of $E^{2}$ as
$S_{E^{2}}$, which translates into a minimal detectable power of the total
field inside the cavity of

$\displaystyle
E^{2}_{\mathrm{min}}=\sqrt{\frac{2S_{E^{2}}}{\sqrt{T_{\mathrm{obs}}\tau(\omega)}}}\,,$
(24)

where $T_{\mathrm{obs}}$ is the individual integration time and $\tau(\omega)$
is the coherence time of the field such that
$T_{\mathrm{obs}}\gg\tau(\omega)$, as discussed in the previous section.

This means, that for a signal-to-noise ratio (SNR) of 1, the constraint on
$\chi$ for a mass of the DP $m$ corresponding to an angular frequency $\omega$
can be computed by equating Eqs. (18) and (24), which leads to

$\displaystyle\left[\chi(\omega)\right]_{\mathrm{stat}}=$
$\displaystyle\frac{E^{2}_{\mathrm{min}}}{S(\omega,\omega_{a};\rho_{\mathrm{DM}},X_{a};L,r)}\,$
$\displaystyle=$
$\displaystyle\frac{\sqrt{2S_{E^{2}}}}{(T_{\mathrm{obs}}\tau(\omega))^{\frac{1}{4}}S(\omega,\omega_{a};\rho_{\mathrm{DM}},X_{a};L,r)}\,.$
(25)

###  V.2  Amplitude fluctuation of the applied field

The main systematic identified for this experiment comes from fluctuations of
the amplitude of the applied electromagnetic field. Indeed, the principle of
the experiment consists in measuring oscillations of the electric field
intensity at the center of the cavity. Amplitude fluctuations of the injected
electromagnetic field will mimic such a signal and can jeopardize the results
of the experiment. In this section we assume that the main source of
fluctuations of the field inside the cavity are fluctuations of the power of
the signal that is fed into the cavity, i.e. relative intensity noise (RIN) of
the signal generator.

We model the amplitude of the injected electric field by including a
stochastic component, i.e. replacing the previously considered constant
$\vec{X}_{a}$ by

$\displaystyle\vec{X}_{a}\rightarrow\vec{X}_{a}\left[1+\int
d\omega_{0}\frac{\Delta
X_{a}(\omega_{0})}{X_{a}}\cos(\omega_{0}t+\phi_{0})\right]\,.$ (26)

In this expression, $\Delta X_{a}\left(\omega_{0}\right)$ is a stochastic
contribution modeling the spectral amplitude of the noise characterized by the
RIN PSD denoted $S_{\mathrm{RIN}}(\omega)$ and defined by

$\frac{\Delta
X_{a}(\omega_{0})}{X_{a}}\equiv\sqrt{\frac{S_{\mathrm{RIN}}(\omega_{0})}{2T_{\mathrm{obs}}}}\,.$
(27)

where the factor 2 in the denominator arises because the RIN is a fluctuation
of the laser power, not its amplitude. Typically, the RIN of frequency
generators in the microwave domain (GHz frequencies of interest here) is
characterized by a flicker noise, see e.g. Rubiola (2005), such that we can
parametrize its PSD as

$\displaystyle S_{\mathrm{RIN}}(\omega)=\frac{P_{\mathrm{RIN}}}{\omega}\,,$
(28)

where $P_{\mathrm{RIN}}$ is dimensionless.

Let us now show how the $\Delta X_{a}$ fluctuations can produce an harmonic
signal in $\left|\vec{E}\right|^{2}$ of angular frequency
$\Delta\omega=\omega_{a}-\omega$, i.e. mimick the searched signal of Eq. (20).

We will work to leading order in $\Delta X_{a}/X_{a}$ and in particular
neglect terms that are $\mathcal{O}\left(\left(\frac{\Delta
X_{a}}{X_{a}}\right)^{2}\right)$ and $\mathcal{O}\left(\frac{\Delta
X_{a}}{X_{a}}\frac{X_{\mathrm{DM}}}{X_{a}}\right)$. Considering the
modification of applied field amplitude Eq. (26), the fluctuation $\Delta
X_{a}$ will only be considered at frequencies $\omega_{0}$ producing a noise
in the electric field squared at angular frequency $\Delta\omega$. These
angular frequencies are $\omega_{0}\simeq\\{\Delta\omega;2\omega_{a}\\}$.
Considering the RIN as a flicker noise characterized by a PSD of the form Eq.
(28), the fluctuation amplitude at $\omega_{0}=\Delta\omega$ will be multiple
order of magnitudes larger than its amplitude at $\omega_{0}=2\omega_{a}$. For
this reason, in the following, we will only consider the fluctuation at
frequency $|\Delta X_{a}(\omega_{0}=\Delta\omega)|$ such that the amplitude of
the applied field Eq. (26) becomes

$\displaystyle\left(\vec{X}_{a}+\Delta\vec{X}_{a}(\Delta\omega)\cos(\Delta\omega
t+\phi_{0})\right)\cos(\omega_{a}t+\phi)\,.$ (29)

We now use the same procedures as the one described in Sec III.1 to compute
the RIN contribution to the total electric field at the center of the cavity.
This calculation is presented in details in Appendix B.1 and leads to

$\displaystyle\left[\vec{E}^{\mathrm{tot}}_{a}\right]_{\mathrm{RIN}}$
$\displaystyle=\frac{\Delta
X_{a}(\Delta\omega)}{2X_{a}}\left(\vec{A}(2\omega_{a}-\omega)\cos([2\omega_{a}-\omega]t+\phi_{+})\right.$
$\displaystyle\left.+\vec{B}(2\omega_{a}-\omega)\sin([2\omega_{a}-\omega]t+\phi_{+})\right.\,$
(30) $\displaystyle\left.+\vec{A}(\omega)\cos(\omega
t+\phi_{-})+\vec{B}(\omega)\sin(\omega t+\phi_{-})\right)\,,$

in addition to the contribution from Eq. (14) and where
$\phi_{\pm}=\phi\pm\phi_{0}$.

Then, the RIN contribution to the total field power at the center of the
cavity can be obtained by multiplying the last equation with Eq. (14). Keeping
only the terms oscillating at angular frequency $\Delta\omega$, the RIN
contribution to $E^{2}$ is given by

$\displaystyle\left[E^{2}(\omega,\omega_{a})\right]_{\mathrm{RIN}}$
$\displaystyle=\frac{\Delta
X_{a}(\Delta\omega)}{2X_{a}}\left(\left(A(\omega_{a})\left[A(2\omega_{a}-\omega)+A(\omega)\right]+B(\omega_{a})\left[B(2\omega_{a}-\omega)+B(\omega)\right]\right)\cos(\Delta\omega
t+\phi_{0})\,\right.$
$\displaystyle\left.\qquad\quad+\left(A(\omega_{a})\left[B(2\omega_{a}-\omega)-B(\omega)\right]+B(\omega_{a})\left[A(\omega)-A(2\omega_{a}-\omega)\right]\right)\sin(\Delta\omega
t+\phi_{0})\right)$ (31a) $\displaystyle=\frac{\Delta
X_{a}(\Delta\omega)}{2X_{a}}\sqrt{N(\omega,\omega_{a})}\cos(\Delta\omega
t+\varphi)=\sqrt{\frac{P_{\mathrm{RIN}}N(\omega,\omega_{a})}{8T_{\mathrm{obs}}\Delta\omega}}\cos(\Delta\omega
t+\varphi)\,,$ (31b)

where we have used Eqs. (27) and (28). In this expression, the functions $A$
and $B$ are the norm of $\vec{A}$ and $\vec{B}$ whose complete expressions are
given by Eqs. (46). The function $\sqrt{N(\omega,\omega_{a})}$ correspond to
the noise amplification factor by the cavity (see Appendix C), quadratic in
$X_{a}$ and whose expression is given by Eq. (52d) and $\varphi$ is a phase.

The RIN contribution to $E^{2}$ from Eq. (31b) will limit the sensitivity of
the experiment to values of $\chi$ that makes the signal from Eq. (18) larger
than the systematic (i.e. larger than Eq. (31b). In other words, the RIN will
limit the sensitivity of the experiment to value of $\chi$ that are larger
than

$\left[\chi(\omega)\right]_{\mathrm{RIN}}=\frac{\sqrt{\frac{P_{\mathrm{RIN}}N(\omega,\omega_{a})}{8T_{\mathrm{obs}}\Delta\omega}}}{S(\omega,\omega_{a};\rho_{\mathrm{DM}},X_{a};L,r)}\,,$
(32)

where the function $S$ is defined in Eq. (18). Note that this limit is linear
in $X_{a}$, the amplitude of the applied electric field.

Note that vacuum fluctuations noise, i.e shot noise, can also be viewed as an
amplitude fluctuation of the electric field squared, as seen by Rydberg atoms,
thus is analogous to the RIN. However, it is negligible since its PSD
normalized by the number of photons $N$ $\sqrt{S_{\mathrm{SN}}}/N\sim
10^{-11}\>(\mathrm{/\sqrt{Hz})}\ll\sqrt{S_{\mathrm{RIN}}}$ (using experimental
parameters considered in Sec. V.4 and a typical mode radius of $\sim 0.01$ m).

###  V.3 Optimum choice of experimental parameters

The sensitivity of the experiment at a given angular frequency $\omega$ relies
on the signal amplitude Eq. (18) but also on the limiting noise. Combining Eq.
(25) and Eq. (32), this is simply

$\displaystyle\left[\chi(\omega)\right]_{\mathrm{limit}}$
$\displaystyle=\frac{\sqrt{\frac{2S_{E^{2}}}{\sqrt{\tau(\omega)}}+\frac{P_{\mathrm{RIN}}N(\omega,\omega_{a})}{8\Delta\omega\sqrt{T_{\mathrm{obs}}}}}}{T_{\mathrm{obs}}^{\frac{1}{4}}S(\omega,\omega_{a};\rho_{\mathrm{DM}},X_{a};L,r)}\,,$
(33)

where we used the usual quadratic sum of uncertainties since the two
contributions are uncorrelated.

The maximum angular frequency difference $\Delta\omega$ corresponds to, from
Sec. IV.2, half the sampling frequency ($=\pi f_{s}$). To better understand
the sensitivity of the experiment, we will simplify the expressions of
$S(\omega,\omega_{a};\rho_{\mathrm{DM}},X_{a};L,r)$ and $N(\omega,\omega_{a})$
in Eqs. (18), (31b) and (52d), considering $r=1-\epsilon$, $\epsilon\ll 1$ and
$\omega\sim\omega_{a}$666Even in the case where $\Delta\omega=\pi f_{s}$, we
still consider $\omega\sim\omega_{a}$ in this approximation, as
$f_{s}\lessapprox\frac{\omega}{Q}\approx\omega(1-r)\ll\omega$. For a sampling
frequency $\sim$ kHz and Compton frequency in the GHz domain, and
$r=1-10^{-6}\sim 1$. In that case,

$\displaystyle N(\omega_{a})$
$\displaystyle\approx\frac{X^{4}_{a}\epsilon^{2}}{\cos^{4}(\frac{\omega_{a}L}{2c})}\,,$
(34a) $\displaystyle S(\omega_{a})$ $\displaystyle\approx\frac{\beta
cX_{a}\sqrt{2\mu_{0}\rho_{\mathrm{DM}}}\sqrt{\epsilon}}{\sqrt{2}\cos^{2}(\frac{\omega_{a}L}{2c})}\left(1+\cos\left(\frac{\omega_{a}L}{2c}\right)\right)\,,$
(34b)

at lowest order in $\epsilon$. We see from (34a) that the two uncertainties in
the numerator of (33)) depend differently on $X_{a}$ (the statistical
uncertainty is independent of $X_{a}$, the RIN contribution is quadratic in
$X_{a}$), while the signal strength in the denominator is linear in $X_{a}$.
This suggests an ’optimum’ value of $X_{a}$ such that Eq. (33) is minimum,

$\displaystyle\frac{d\chi(\omega_{a})}{dX_{a}}$ $\displaystyle=0\,$ (35a)
$\displaystyle\Rightarrow X_{a}(\omega_{a})$
$\displaystyle\approx\sqrt[4]{\frac{16S_{E^{2}}\Delta\omega}{\epsilon^{2}P_{\mathrm{RIN}}}\sqrt{\frac{T_{\mathrm{obs}}}{\tau(\omega_{a})}}}\left|\cos\left(\frac{\omega_{a}L}{2c}\right)\right|\,$
(35b) $\displaystyle\approx\sqrt[4]{\frac{16\pi
f_{s}S_{E^{2}}\sqrt{\omega_{a}T_{\mathrm{obs}}}}{10^{3}\epsilon^{2}P_{\mathrm{RIN}}}}\left|\cos\left(\frac{\omega_{a}L}{2c}\right)\right|\,,$
(35c)

where we used Eqs.(34a),(34b). We considered the maximum angular frequency
shift between the DM and applied frequencies $\Delta\omega=\pi f_{s}$, in
order to have the best sensitivity on $\chi(\omega)$. Note that this equation
to approximate the optimum value of $X_{a}$ is not valid for angular
frequencies $\omega_{a}L/c=2\pi+4n\pi,n\in\mathbb{N}$ and
$\omega_{a}L/c=\pi+2n\pi,n\in\mathbb{N}$. In the first case, the signal
decreases significantly (see Eq. (34b)) and the experiment becomes insensitive
to DM, while in the second case, Eq. (35c) would indicate to apply $X_{a}=0$,
which would automatically set the signal to 0, at first order in $\chi$.

From Eq. (35c), and still considering $\omega\sim\omega_{a}$, we can express
the sensitivity of the experiment $\chi(\omega)$ as

$\displaystyle\chi(\omega)$
$\displaystyle\approx\frac{\sqrt{2}\left|\cos\left(\frac{\omega
L}{2c}\right)\right|}{1+\cos\left(\frac{\omega
L}{2c}\right)}\left(\frac{P_{\mathrm{RIN}}S_{E^{2}}}{10^{3}\pi
f_{s}}\right)^{\frac{1}{4}}\frac{\left(\omega
T_{\mathrm{obs}}^{-3}\right)^{\frac{1}{8}}}{\beta
c\sqrt{2\mu_{0}\rho_{\mathrm{DM}}}}\,,$ (36)

for $\frac{\omega L}{c}\neq\pi+2\pi n$, $n\in\mathbb{N}$. This is a simplified
expression that provides an approximate evaluation of the optimal sensitivity
of the experiment.

In the following section, we will assume that the applied field amplitude
$X_{a}$ is modified each time its angular frequency $\omega_{a}$ is shifted,
so that the condition Eq. (35c) is always fulfilled, but we use the full
expressions of $S(\omega,\omega_{a};\rho_{\mathrm{DM}},X_{a};L,r)$ and
$N(\omega,\omega_{a})$ in Eqs. (18) and (31b) to evaluate the sensitivity.

###  V.4 Reachable sensitivity in a realistic experiment

Figure 3: Current constraints on dark photons through their kinetic mixing
coupling $\chi$ with photons (renormalized from O’Hare (2020); Caputo  _et
al._ (2021), see text). The expected sensitivity of this work, with
”optimistic” parameters of the system described in Table 1 is shown in orange,
denoted ”SYRTE (optimistic)”, while the sensitivity with the ”modest”
parameters of Table 1 is shown in purple, denoted ”SYRTE (modest)” (see text).
The bordering red line around the orange curve indicates the approximate
sensitivity of the experiment in the optimistic case, computed from Eq. (36).

We now estimate the sensitivity on $\chi$ of the experimental method defined
in Sec. IV by considering the two noise sources described in Sec.V. The
measurement noise will lead to a lower limit on $\chi$ given by Eq. (25) while
the main systematic effect, the RIN, will lead to a lower limit on $\chi$
given by Eq. (32).

First, we consider the random polarized case for the DP polarization
direction, i.e $\beta=1/\sqrt{3}$. Some already existing constraints come from
experiments with this assumption Brun  _et al._ (2019); Dixit  _et al._
(2021); Nguyen  _et al._ (2019); Suzuki  _et al._ (2015); Knirck  _et al._
(2018); Andrianavalomahefa  _et al._ (2020); Ramanathan  _et al._ (2023); An
_et al._ (2023). Other experiments Asztalos  _et al._ (2010); Du  _et al._
(2018); Braine  _et al._ (2020); Bartram  _et al._ (2021a); Boutan  _et al._
(2018); Bartram  _et al._ (2021b); Alesini  _et al._ (2019, 2021, 2022); Lee
_et al._ (2020); Jeong  _et al._ (2020); Kwon  _et al._ (2021); Lee  _et al._
(2022); Kim  _et al._ (2022); Yi  _et al._ (2022) considered the fixed
polarization scenario, and we have rescaled them accordingly (using Caputo
_et al._ (2021)). Finally, some of them assumed perfectly aligned
polarizations between DP and device, so they also have been rescaled Suzuki
_et al._ (2015); Phipps  _et al._ (2020)777For ORGAN Quiskamp  _et al._ (2022)
and its $\mathcal{O}$(month) of data taking, this correction factor is of
$\mathcal{O}$(1), as for long time experiment, the sensitivity in both
scenarios is equal (Sec. VI.6 of Caputo  _et al._ (2021)).

We consider the local DM energy density to be $\rho_{\mathrm{DM}}=$
$0.45\mathrm{\ GeV/cm}^{3}$ Caputo  _et al._ (2021).

Let us remind that the observational scheme considered here consists in
electric signals of angular frequency $\omega_{a}$. For each injected electric
field we perform a measurement of duration $T_{\mathrm{obs}}$ and then shift
the angular frequency of the applied field by $\pi f_{s}$. Each measurement of
duration $T_{\mathrm{obs}}$ provides constraints in the frequency range
$\\{\omega_{a}-\pi f_{s};\omega_{a}+\pi f_{s}\\}$ in steps of
$2\pi/T_{\mathrm{obs}}$.

Let us now consider some numbers for the experiment, which are all summarized
in Table 1. First of all, we consider an individual measurement duration of
$T_{\mathrm{obs}}=60$ s. This duration is arbitrary, but we list two important
considerations for this choice. We require $T_{\mathrm{obs}}$ to be large for
best sensitivity (see Eq. (33)), but short enough to allow scanning a large
range of DM frequencies in a reasonable amount of time
($\leq\mathcal{O}$(month)).

As described in Section V.1, we consider two different sampling rates which
lead to different measurement noise PSD and directly affect the sensitivity of
the experiment (cf Eq. (36)). To put some numbers, assuming a single shot
spectroscopic resolution of $\sim$1 kHz Bridge  _et al._ (2016) for
differential polarizabilities of $\Delta\alpha/2h\approx 10^{5}$ Hz/(V/m)2,
leads to a noise PSD of the measured electric field power of $S_{E^{2}}\approx
10^{-6}$ (V/m)4/Hz for $f_{s}=100$ Hz and $S_{E^{2}}\approx 10^{-7}$ (V/m)4/Hz
for $f_{s}=1$ kHz. Regarding the systematic effect, the amplitude of the
flicker noise can be considered to be $P_{\mathrm{RIN}}=10^{-13}$ (based on
“off the shelf” components studied in Rubiola (2005) more than a decade ago)
in the modest case, and we assume an improved RIN control for the optimistic
case, with an amplitude of $P_{\mathrm{RIN}}=10^{-15}$. The level of the
systematic effect PSD is also impacted by the sampling rate, following Eq.
(31b). Moreover, as derived in Sec. V.3, the approximate sensitivity of the
experiment Eq. (36) scales as $(P_{\mathrm{RIN}}S_{E^{2}}/f_{s})^{1/4}$,
implying that the modest and optimistic scenarios will differ in the
sensitivity by a factor $\sim$10\. The optimum value of the amplitude of the
applied field $X_{a}$ is then derived from Eqs. (33) (for the minimum
value888The smallest value of $X_{a}$ corresponds to DM/applied frequencies
close to odd modes, from Eq. (35c). In this regime, both approximate
amplitudes of noise Eq. (34a) and signal Eq. (34b) reach infinity, implying we
must consider the real expression of $\chi(\omega)$ derived in Eq. (33) and
check for which value of $X_{a}$ the sensitivity on $\chi$ is the highest.)
and (35c) and all other experimental parameters. Since it depends on the DM
Compton frequency, we provide the range of optimal $X_{a}$, for the modest
case

$\displaystyle 18.1\,\,\mathrm{V/m}$ $\displaystyle\lesssim X_{a}\lesssim
1.70\times 10^{5}\ \mathrm{V/m}\,.$ (37)

It is independent of the sampling frequency since, from Eq. (35c),
$X_{a}(\omega_{a})\propto(f_{s}S_{E^{2}})^{1/4}$ and $S_{E^{2}}\propto
f^{-1}_{s}$, but not of the systematic effect level $P_{\mathrm{RIN}}$.

Table 1: Assumed experimental parameters Parameters | Numerical values
---|---
Quality factor $Q$ Guena  _et al._ (2012) | $10^{4}$
Mirrors reflectivity r | $\approx 1-2\times 10^{-4}$
Cavity length L | $7.5\mathrm{\ cm}$
Injected field strength $X_{a}(\omega)$ |  $[18.1,1.70\times 10^{5}]$ V/m
Sampling frequency $f_{s}$ |  $10^{2}\,;\,10^{3}$ Hz
Individual measurement time $T_{\mathrm{obs}}$ | 60 s
Range of $f_{a}=\omega_{a}/2\pi$ |  $[0.5,20.5]$ GHz
Range of $\Delta\omega$ |  $[2\pi/T_{\mathrm{obs}},\pi f_{s}]$ rad/s
Statistical noise PSD $S_{E^{2}}$ | $10^{-4}/f_{s}\,\,(\mathrm{V/m})^{4}/\mathrm{Hz}$
Systematic effect PSD $S_{\mathrm{RIN}}(\omega)$ | $10^{-13}/\omega\,;\,10^{-15}/\omega$

The sensitivity of the experiment obtained considering all experimental
parameters described in Table (1) and respectively with $\\{f_{s}=1$ kHz,
$P_{\mathrm{RIN}}=10^{-15}\\}$ and $\\{f_{s}=100$ Hz,
$P_{\mathrm{RIN}}=10^{-13}\\}$ is presented by the orange and purple curves of
Fig. 3. One can clearly see the sensitivity peaks arising from the cavity’s
odd resonances. This happens when the applied field amplitude $X_{a}$ is
small, as shown in Eq. (35c). This equation works well for frequencies far
from odd resonances. However, on those odd resonances, this approximate
equation cannot be used as discussed previously. Instead one should use the
exact expressions of signal and noise to optimize Eqs. (33). As an example,
when the applied field frequency corresponds exactly to the first odd
resonance of the cavity, i.e $\omega_{a}L=\pi c$ and $\omega=\omega_{a}+\pi
f_{s}$ the optimum amplitude of $X_{a}$ is $\sim 18.1$ V/m, whose
corresponding experimental sensitivity is $\chi\sim 10^{-13}$ in the modest
scenario, as shown in Fig. 3. Additionally, one can notice the presence of
specific frequencies where this sensitivity decreases significantly, the
experiment is almost insensitive to these DM frequencies. As discussed in the
previous section, from the approximate expression of the signal contribution
Eq. (34b), we have $S(\omega,X_{a})\simeq 0$ for $\frac{\omega L}{c}=4\pi+2\pi
n$, $n\in\mathbb{N}$ accounting for the loss of sensitivity. In both scenarios
presented here (modest and optimistic), one can see from Fig. 3 that the
experiment setup proposed here would improve the current constraint on the
coupling $\chi$ compared to cosmological and astrophysical observations and
other already existing laboratory experiments.

##  VI Discussion and conclusion

If one decides to run this experiment aiming at unconstrained regions of the
exclusion plot, it would take approximately five days of data-taking to cover
the mass range from 7 $\mu$eV to 10 $\mu$eV, while around 35 days would be
needed to cover the mass range from 35 $\mu$eV to 60 $\mu$eV, assuming no dead
time between the $T_{\mathrm{obs}}=60$ s observation runs. More realistically,
reserving say 50% of the total experimental time for manipulation of the atoms
and applied field, the total duration increases by a factor two, which is
still very reasonable.

With the appropriate set of parameters, in particular the applied field
amplitude $X_{a}$ following Eq. (35c), both sources of noise, systematic and
statistical, are equal in amplitude. This means that, in the search for high
sensitivity of the experiment, the optimum choice of $X_{a}$ is not to
increase it as much as possible to maximize the signal. Even though the signal
is linear in $X_{a}$, the systematic uncertainty is quadratic in $X_{a}$, as
stated at the end of Sec. V.2, implying a loss of sensitivity if the
experimenter decides to apply too much power inside the cavity.

If the level of intensity fluctuations (RIN) of the applied field could be
reduced e.g. by stabilizing the power using low noise intensity measurements
Cervantes  _et al._ (2022), the applied field and/or the quality factor of the
cavity could be increased leading to an increase of the signal whilst keeping
the contribution from the RIN below that of the measurement noise in Eq. (33).
This way, the optimistic curve presented in Fig. 3 would be achievable.

Some experiments use curved mirrors (Brun  _et al._ (2019), Suzuki  _et al._
(2015), Knirck  _et al._ (2018)) to focus the DM induced electric field into a
reduced region, to be able to detect more power. This method is not considered
here, but with the appropriate curvature, it may improve the sensitivity of
the experiment.

In conclusion, we propose a novel experiment that uses Rydberg atoms inside a
microwave cavity to search for DP through its mixing with standard
electromagnetism. Our proposal features an optimized applied electric field
inside the cavity and Rydberg atoms as highly sensitive probes of $E^{2}$.
This latter feature allows searching for the cross-term between the applied
and DP-induced fields, thus allowing best sensitivity with relatively low
sampling frequencies.

Using realistic experimental parameters we show (see Fig. 3) that such an
experiment has the potential to significantly improve on existing laboratory
experiments in terms of the sensitivity, and more importantly, in terms of the
explored DP-mass regions. The latter can be more specifically targeted by
tuning the cavity size such that resonances match the least explored regions.
Finally, we note that around the resonances the experimental sensitivity is
also better than indirect bounds coming from cosmological considerations (blue
region in Fig. 3).

###### Acknowledgements.

The authors acknowledge Sébastien Bize, Luca Lorini and Giorgio Santarelli for
very helpful discussions. This work was supported by the Programme National
GRAM of CNRS/INSU with INP and IN2P3 cofunded by CNES.

##  Appendix A Does the DP field propagate or not?

Let us assume the dark photon vector field is an oscillating field which does
not propagate. In this case,

$\vec{E}_{\mathrm{DM}}=\vec{X}_{\mathrm{DM}}\cos(\omega t)\,.$ (38)

This statement is true in the rest frame of dark matter with constant mode k =
0. If we consider it moving with respect to Earth at a velocity $v=10^{-3}$ c,
which is the typical mean velocity of the dark matter halo, we have

$\vec{E}_{\mathrm{DM}}=\vec{X}_{\mathrm{DM}}\cos(\omega^{\prime}t-\vec{k^{\prime}}.\vec{x})\,,$
(39)

and from special relativity, we have

$\displaystyle\omega^{\prime}$
$\displaystyle=\gamma(\omega-\vec{v}.\vec{k})\approx\omega\,,$ (40a)
$\displaystyle\vec{k}^{\prime}$
$\displaystyle=\vec{k}+\frac{1}{v^{2}}(\gamma-1)(\vec{v}.\vec{k})\vec{v}-\frac{1}{c^{2}}\gamma\omega\vec{v}\approx-\frac{1}{c^{2}}\omega\vec{v}\,,$
(40b)

since $\gamma=1-\frac{v^{2}}{2c^{2}}\sim 1$. To neglect this propagation term,
we require $2\pi\gg k^{\prime}L$, with $L$ the length of the cavity, or
equivalently, the de Broglie wavelength of the field has to be much bigger
than the size of the cavity. Considering a length $\mathcal{O}$(10 cm), the
corresponding constraint on $k^{\prime}$ is

$\displaystyle k^{\prime}\ll 20\pi\mathrm{\ m}^{-1}\,.$ (41)

In this paper, we are interested in DP masses in microwave domain,
$10^{-6}<mc^{2}\mathrm{\ (eV)}<10^{-4}$, or

$\displaystyle k^{\prime}\in[0.05,5]\times 10^{-1}\mathrm{\ m}^{-1}\,,$ (42)

implying we can safely neglect the propagation term. Note that if we were to
work in the optical frequency range, this propagation term should be kept, as
in this case, $k^{\prime}\in[5,25]\times 10^{6}\mathrm{\ m}^{-1}\gg 2\pi/L$.

##  Appendix B Full calculation of field amplitude in the cavity

###  B.1 Applied field

The experimental setup (Fig. 1) gathers an applied field $E_{a}$ with
amplitude $X_{a}$, angular frequency $\omega_{a}$ and a time varying amplitude
fluctuation $\Delta X_{a}$ with angular frequency $\omega_{0}$, considered
much smaller than $\omega_{a}$. This field has random phase $\phi$ with
respect to the DM induced oscillating field $\vec{E}_{\mathrm{DM}}$.
Considering application of this field at the left edge of the cavity, and
assuming a transmission coefficient of the mirror being $t=\sqrt{1-r^{2}}$,
the first contribution at the center of the cavity reads

$\displaystyle\vec{E}^{0}_{a}(x=0,t)$
$\displaystyle=\Re\left[t\left(\vec{X}_{a}e^{-i(\omega_{a}t-k_{a}\frac{L}{2}+\phi)}+\frac{\Delta\vec{X}_{a}}{2}\left(e^{-i(\omega_{+}t-k_{+}\frac{L}{2}+\phi_{+})}+e^{-i(\omega_{-}t-k_{-}\frac{L}{2}+\phi_{-})}\right)\right)\right]\,,$
(43)

where $k_{\pm}=k_{a}\pm k_{0}$. This contribution propagates until the other
cavity boundaries, gets reflected once with coefficient $-r$ such that
boundary conditions are respected, then comes back to the center, implying
that the second contribution reads

$\displaystyle\vec{E}^{1}_{a}(x=0,t)$
$\displaystyle=\Re\left[-tre^{ik_{a}L}\left(\vec{X}_{a}e^{-i(\omega_{a}t-k_{a}\frac{L}{2}+\phi)}+\frac{\Delta\vec{X}_{a}}{2}\left(e^{-i(\omega_{+}t-k_{+}\frac{L}{2}+\phi_{+})}+e^{-i(\omega_{-}t-k_{-}\frac{L}{2}+\phi_{-})}\right)\right)\right]\,,$
(44)

the additional phase $e^{ik_{a}L}$ shows the time delay of $E^{1}$ compared to
$E^{0}$ after half a round trip. This occurs several times and after an
infinite number of round trips N, the full contribution of the external
applied field inside the cavity is

$\displaystyle\vec{E}^{\mathrm{tot}}_{a}(x=0,t)=\sum_{n=0}^{N=+\infty}\vec{E}^{n}_{a}(x=0,t)\,$
(45a)
$\displaystyle=t\vec{X}_{a}\Re\left[e^{-i(\omega_{a}t+\phi)}\frac{e^{i\frac{k_{a}L}{2}}}{1+re^{ik_{a}L}}\right]+t\Delta\vec{X}_{a}\Re\left[e^{-i(\omega_{+}t+\phi_{+})}\frac{e^{i\frac{k_{+}L}{2}}}{1+re^{ik_{+}L}}\right]+t\Delta\vec{X}_{a}\Re\left[e^{-i(\omega_{-}t+\phi_{-})}\frac{e^{i\frac{k_{-}L}{2}}}{1+re^{ik_{-}L}}\right]\,$
(45b)
$\displaystyle\equiv\vec{A}(\omega_{a})\cos(\omega_{a}t+\phi)+\vec{B}(\omega_{a})\sin(\omega_{a}t+\phi)+\frac{\Delta
X_{a}}{2X_{a}}\sum_{i=\pm}\left(\vec{A}(\omega_{i})\cos(\omega_{i}t+\phi_{i})+\vec{B}(\omega_{i})\sin(\omega_{i}t+\phi_{i})\right)\,,$
(45c)

where we used the fact that $r<1$ such that $r^{N}\rightarrow 0$ and with

$\displaystyle\vec{A}(\omega_{a})$
$\displaystyle=\frac{t\vec{X}_{a}\left(1+r\right)\cos\left(\frac{\omega_{a}L}{2c}\right)}{1+2r\cos(\frac{\omega_{a}L}{c})+r^{2}}\,,$
(46a) $\displaystyle\vec{B}(\omega_{a})$
$\displaystyle=\frac{t\vec{X}_{a}\left(1-r\right)\sin\left(\frac{\omega_{a}L}{2c}\right)}{1+2r\cos(\frac{\omega_{a}L}{c})+r^{2}}\,.$
(46b)

One can notice a constructive interference at the center of the cavity for
even modes of the cavity, as expected.

###  B.2 DM field

Starting from the DM induced electric field with unknown polarization
direction $\vec{X}_{\mathrm{DM}}$ and angular frequency $\omega$, the same
procedure as above can be realized to know the DM electric field amplitude at
the center of the cavity. The subtleties of this calculation are that : 1) the
field is emitted by the mirrors towards the center of the cavity, therefore
the transmission coefficient $t$ factor is not present ; 2) only the DM
polarization transverse to the mirror is reemitted, noted
$\vec{X}_{\mathrm{DM,\parallel}}$; and 3) there are two different
contributions, in phase, each being emitted from one of the edges of the
cavity. The total DM contribution at the center is then

$\displaystyle\vec{E}^{\mathrm{tot}}_{\mathrm{DM}}(x=0,t)=\Re\left[\vec{X}_{\mathrm{DM}}e^{-i\omega
t}+2\vec{X}_{\mathrm{DM,\parallel}}e^{-i(\omega
t-\frac{kL}{2})}\frac{1}{1+re^{ikL}}\right]\,$ (47a)
$\displaystyle\equiv\vec{C}(\omega)\cos(\omega t)+\vec{D}(\omega)\sin(\omega
t)\,,$ (47b)

with

$\displaystyle\vec{C}(\omega)$
$\displaystyle=\vec{X}_{\mathrm{DM}}+\frac{2\vec{X}_{\mathrm{DM,\parallel}}(1+r)\cos(\frac{\omega
L}{2c})}{1+2r\cos(\frac{\omega L}{c})+r^{2}}\,,$ (48a)
$\displaystyle\vec{D}(\omega)$
$\displaystyle=\frac{2\vec{X}_{\mathrm{DM,\parallel}}(1-r)\sin(\frac{\omega
L}{2c})}{1+2r\cos(\frac{\omega L}{c})+r^{2}}\,.$ (48b)

The first term of Eq. (47a) corresponds to the background oscillating DM field
at the center, which is always present, even without cavity. The second term
is the DM contribution from the cavity, which is almost equivalent, in its
form, to the total contribution of the applied field, with an additional
factor two, due to the emission of a field from both edges of the cavity
(instead of only one for the applied field).

##  Appendix C Signal and noise amplitudes

In this section, we wish to derive the expressions of the signal contribution
inside the cavity in Eq. (18) and of the noise in Eq. (31b).

###  C.1 Signal contribution

From Eq. (17), and using Eqs. (46),(48), we can write the signal contribution
as

$\displaystyle P(\omega,\omega_{a})$
$\displaystyle=\sqrt{\left(\vec{A}(\omega_{a})\cdot\vec{C}(\omega)+\vec{B}(\omega_{a})\cdot\vec{D}(\omega)\right)^{2}+{\left(\vec{B}(\omega_{a})\cdot\vec{C}(\omega)-\vec{A}(\omega_{a})\cdot\vec{D}(\omega)\right)^{2}}}\,$
(49a)
$\displaystyle=X_{a}X_{\mathrm{DM}}\beta\sqrt{\left(A^{\prime}(\omega_{a})^{2}+B^{\prime}(\omega_{a})^{2}\right)\times\left(C^{\prime}(\omega)^{2}+D^{\prime}(\omega)^{2}\right)}$
(49b)

where we used
$\vec{X}_{a}\cdot\vec{X}_{\mathrm{DM}}=\vec{X}_{a}\cdot\vec{X}_{\mathrm{DM,\parallel}}=X_{a}X_{\mathrm{DM}}\beta$
and where the prime quantities are defined as

$\displaystyle A^{\prime}(\omega_{a})$
$\displaystyle\equiv\frac{t\left(1+r\right)\cos\left(\frac{\omega_{a}L}{2c}\right)}{1+2r\cos(\frac{\omega_{a}L}{c})+r^{2}}\,,$
(50a) $\displaystyle B^{\prime}(\omega_{a})$
$\displaystyle\equiv\frac{t\left(1-r\right)\sin\left(\frac{\omega_{a}L}{2c}\right)}{1+2r\cos(\frac{\omega_{a}L}{c})+r^{2}}\,,$
(50b) $\displaystyle C^{\prime}(\omega)$ $\displaystyle\equiv
1+\frac{2(1+r)\cos(\frac{\omega L}{2c})}{1+2r\cos(\frac{\omega
L}{c})+r^{2}}\,,$ (50c) $\displaystyle D^{\prime}(\omega)$
$\displaystyle\equiv\frac{2(1-r)\sin(\frac{\omega
L}{2c})}{1+2r\cos(\frac{\omega L}{c})+r^{2}}\,,$ (50d)

i.e the polarizations are factorized from the amplitude functions
$\\{\vec{A},\vec{B},\vec{C},\vec{D}\\}$. The signal amplitude can be easily
simplified to

$\displaystyle
P(\omega,\omega_{a})=\frac{tX_{a}X_{\mathrm{DM}}\beta}{\sqrt{1+2r\cos(\frac{\omega_{a}L}{c})+r^{2}}}\sqrt{1+4\frac{1+(1+r)\cos(\frac{\omega
L}{2c})}{1+2r\cos(\frac{\omega L}{c})+r^{2}}}\,,$ (51a)

and $X_{\mathrm{DM}}=\chi c\sqrt{2\mu_{0}\rho_{\mathrm{DM}}}$ from Eq. (12b).

###  C.2 Noise contribution

Starting from the full expression of the systematic effect Eq. (31a), we can
derive the noise amplitude amplified by the cavity as the sum of three
different contributions, i.e

$\displaystyle\mathrm{N}_{1}(\omega,\omega_{a})$ $\displaystyle\equiv
A(\omega_{a})^{2}\left(\left(A(2\omega_{a}-\omega)+A(\omega)\right)^{2}+\left(B(2\omega_{a}-\omega)-B(\omega)\right)^{2}\right)\,,$
(52a) $\displaystyle\mathrm{N}_{2}(\omega,\omega_{a})$ $\displaystyle\equiv
B(\omega_{a})^{2}\left(\left(A(2\omega_{a}-\omega)-A(\omega)\right)^{2}+\left(B(2\omega_{a}-\omega)+B(\omega)\right)^{2}\right)\,,$
(52b) $\displaystyle\mathrm{N}_{3}(\omega,\omega_{a})$ $\displaystyle\equiv
4A(\omega_{a})B(\omega_{a})\left(A(2\omega_{a}-\omega)B(\omega)+B(2\omega_{a}-\omega)A(\omega)\right)\,,$
(52c) $\displaystyle\mathrm{N}(\omega,\omega_{a})$
$\displaystyle\equiv\mathrm{N}_{1}(\omega,\omega_{a})+\mathrm{N}_{2}(\omega,\omega_{a})+\mathrm{N}_{3}(\omega,\omega_{a})\,.$
(52d)

One can notice $N_{i}(\omega)$’s, and by extension $N(\omega)$, are of order 4
in $X_{a}$, making $\sqrt{N(\omega,\omega_{a})}\propto X^{2}_{a}$, as
expected.

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