# Chapter 16

## Search for vacuum birefringence and dichroism in *LISA*

<figure><img src="image_27.png" /></figure>

Figure 16.1: Simplified optical benches in *LISA* spacecrafts. The two optical benches A and B are inside two different spacecrafts (shown as the light green cube) and exchange light between each other. The optical elements are shown in blue, the light polarization in red and the direction of propagation with black arrows. Inside each of these optical benches, an initial "horizontal" linear polarization is produced by the laser and the association of the optical elements inside each bench fulfill all the necessary requirements for the interferometers to work (see text.).

As we discussed it in Chapter 8.2.3, the axion-photon coupling can be probed by its effect on polarization of light. In this section, we study the sensitivity of *LISA* to such effects.

We first describe the light polarization in *LISA* optical benches. For our purposes, we are interested in the polarization of light produced and detected inside the spacecrafts and travelling in vacuum between the spacecrafts. The current (simplified) optical bench is shown in Fig. 16.1, from [32], where we only take into account the interference between laser beams of the local spacecraft with a distant one. We assume for simplicity that the light wave is produced with "horizontal" linear polarization along the x-axis, is travelling along the z direction and arrives on the detector with "vertical" polarization after passing through the half-wave plate. In terms of light polarization, several requirements are necessary for the detection of GW in *LISA*.

Each spacecraft contains two optical benches, as shown in Fig. 16.1. Inside one given
optical bench (say *A* in spacecraft 1), two beams are interfering: the one produced by the local laser of the given optical bench and the one which has been produced by one of the optical benches of another spacecraft (*B* in spacecraft 2) and which has been travelling in free space. We require the two beams to have the same polarization, when they enter the interferometer inside optical bench *A* (or at least that they do not have opposite linear polarizations), otherwise the interference between the two beams is zero.

As leaving one spacecraft, one small portion of the local beam can be reflected on it, come back inside and create an additional noise in the interferometer. Assuming that the local beam is at frequency $f_ℓ$ while the distant beam (from the second spacecraft) has slightly different frequency $f_ℓ + \delta f$, due to standard Doppler from S/C differential velocity, and frequency planning, see [33], the reflected component of the local beam of frequency $f_ℓ$ could interfere with itself but only creating a DC component. However, it could also interfere with the distant beam, and create a noise at $\delta f$. In addition, the motion of the reflecting component at the edge of the spacecraft would induce a time dependent phase on the reflected beam, which would mimic a Doppler effect induced by the GW, and therefore, this would greatly alter the measurement. To overcome this issue, we require the outgoing polarization to be different than the incoming one. Then, the polarized beam splitter selects only the polarization of interest, i.e the one of the incoming beam. It can be shown easily that the current optical benches in Fig. 16.1 fulfill those requirements. The *LISA* constellation is depicted in Fig. 16.2. Two optical benches are present in each spacecraft which permits six individual interference measurements at each time. Following our arbitrary choice on light polarization in Fig. 16.1, one can see that two different linear polarizations travel in space: vertical polarization travels clockwise while horizontal polarization travels counterclockwise.

<figure><img src="image_28.png" /><figcaption>Figure 16.2: *LISA* constellation involves three spacecrafts, denoted 1, 2, 3, each containing two optical benches as described in Fig. 16.1 denoted by "A" and "B". Six different light signals are sent between the different spacecrafts to form the interferometer; three vertically polarized beams are propagating clockwise and the other three horizontally polarized beams are propagating counterclockwise.</figcaption></figure>
16.1 Birefringence

We first study the possibility of LISA to detect axion-photon coupling through vacuum birefringence as described in Eq. (8.31). In the last section, we presented the standard design of LISA optical benches with linear polarized light travelling in free space. In this section, we consider a non standard design of the optical benches to check if the search of axion-photon coupling in LISA is possible.

16.1.1 Jones formalism

For this, in addition to the two previous requirements for LISA to work, we require the light
polarization to be circular in free space, i.e when travelling from one spacecraft to another,
in order to see any phase shift due to the long propagation between the spacecrafts. Indeed,
the axion field only impact the phase velocity of circularly polarized light and not linearly
polarized one. To do so, we will still consider an initial linear polarization but we will
introduce quarter-wave plates, or other optical elements, inside the original optical benches
shown in Fig. 16.1.

We can model the system using Jones formalism for polarization states of light and optical
elements which is very convenient when treating interference phenomena [34]. This formal-
ism discomposes the light polarization on a basis formed of the two linear polarizations. The
polarization state is modelled using a 2 × 1 matrix while the optical elements are represented
by 2 × 2 matrices.

As shown in Fig. 16.1, we assume the light beam to propagate along the z-axis with an
initial polarization along the x-axis, that we call "horizontal" polarization. Then, its Jones
representation is [34]

$$
|H\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}. \tag{16.1}
$$

The other linear polarization along y-axis, that we will call "vertical" is represented as

$$
|V\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \tag{16.2}
$$

A half-wave plate adds a phase $\phi = \pi$ to the light polarization parallel to its slow axis<fnref n="1" />. Its Jones matrix representation is [34]

$$
\hat{\lambda}_2(\theta) = e^{-i\frac{\pi}{2}} \begin{pmatrix} \cos^2 \theta - \sin^2 \theta & 2 \sin \theta \cos \theta \\ 2 \sin \theta \cos \theta & \sin^2 \theta - \cos^2 \theta \end{pmatrix} \equiv \begin{pmatrix} \cos(2\theta) & \sin(2\theta) \\ \sin(2\theta) & -\cos(2\theta) \end{pmatrix}, \quad (16.3)
$$

up to an irrelevant total phase, where θ is the angle between the horizontal axis (the x-axis)
and the fast axis of the plate. Similarly, a quarter-wave plate introduces a phase shift of π/2
between the two linear polarized states and can be represented as [34]

$$
\hat{\lambda}_4(\theta) = \begin{pmatrix} \cos^2(\theta) - i \sin^2(\theta) & (1+i) \sin(\theta) \cos(\theta) \\ (1+i) \sin(\theta) \cos(\theta) & \sin^2(\theta) - i \cos^2(\theta) \end{pmatrix}, \quad (16.4)
$$

<footnote n="1">¹The principle of a wave plate is that it presents two perpendicular directions with respective refractive index *n*<sub>*f*</sub>, *n*<sub>*s*</sub> with *n*<sub>*f*</sub> &lt; *n*<sub>*s*</sub>, that we call respectively fast and slow axis [35]. This leads to a phase shift between the light polarization along the fast axis and the one along the slow axis.</footnote>
up to a total phase. Note that if the half or quarter wave plate is rotated such that its fast or slow axis is along the initial linear horizontal or vertical polarization of light (i.e $\theta = 0$ or $\theta = \pi/2$), there is no effect, i.e the input and output polarization are the same (up to an irrelevant phase). Note also that the determinant of these matrices is 1, i.e they preserve energy, or in other words, they have no absorption. In the Jones formalism, right and left circular polarizations are represented as [34]

$$|R\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix} \qquad (16.5a)$$

$$|L\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}. \qquad (16.5b)$$

We now introduce horizontal, vertical, left and right polarizers, which respectively transmit only horizontal, vertical, left and right polarizations of the input, defined as [36]

$$\hat{C}_H = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \qquad (16.6a)$$

$$\hat{C}_V = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \qquad (16.6b)$$

$$\hat{C}_L = \frac{1}{2} \begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix} \qquad (16.6c)$$

$$\hat{C}_R = \frac{1}{2} \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}. \qquad (16.6d)$$

In practice, the right polarizer is made of the combination $\hat{\lambda}_4(\pi/4)\hat{C}_H\hat{\lambda}_4(-\pi/4)$ (and for the left polarizer, one replaces $\hat{C}_H$ by $\hat{C}_V$ or exchange the two quarter-wave plates). Finally, a mirror reverses the direction of one of the two linear polarizations while letting the other one unchanged. Its representation is [36]

$$\hat{M} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad (16.7)$$

therefore it is the "horizontal" polarization that gets flipped.

It can be shown easily that the quarter and half-wave plates commute, implying that a set of optical elements containing $n$ half-wave plates, $m$ quarter-wave plates can be described as the matrix multiplication $\hat{\lambda}_2^n \hat{\lambda}_4^m$. However, the mirror and polarizers do not commute with the other optical elements.

### 16.1.2 Requirements for both gravitational waves and axion detection

Between the polarized beam splitter and the end of the optical bench, we will model the set of optical elements of $n$ half-wave plates, $m$ quarter-wave plates and $l$ polarizers (of any kind) that we add by $\hat{S}(\theta, \theta')$, where $\theta(\theta')$ is the angle between the fast axis of the half-wave (quarter-wave) plates and the horizontal axis. We will denote this set of elements with superscript $A$ or $B$ for the optical bench $A$ or $B$, which are respectively associated with angles $\theta_A(\theta_B)$, $\theta'_A(\theta'_B)$, i.e we note $\hat{S}^{(A)}(\theta_A, \theta'_A)$ and $\hat{S}^{(B)}(\theta_B, \theta'_B)$.
The first requirement (noted $R_1$) is that the incoming polarization from space can interfere with the local beam, and is not fully rejected by the polarized beam splitter. For an initial $|H\rangle$ polarization state for the beam on both spacecrafts, the polarization of the beam after leaving the optical bench A and B is respectively $\hat{S}^{(A)}(\theta_A, \theta'_A)|H\rangle$ and $\hat{S}^{(B)}(\theta_B, \theta'_B)|H\rangle$. Then, after travelling in free space, each light beam enters the other spacecraft and goes through the set of optical elements but with opposite propagation direction. In such a case, light sees the fast axis with opposite angle compared to horizontal plane such that the total polarization of the beam becomes respectively

$$ \hat{S}^{(B)}(-\theta_B, -\theta'_B)\hat{S}^{(A)}(\theta_A, \theta'_A)|H\rangle \quad (16.8a) $$

$$ \hat{S}^{(A)}(-\theta_A, -\theta'_A)\hat{S}^{(B)}(\theta_B, \theta'_B)|H\rangle. \quad (16.8b) $$

$R_1$ implies that both polarizations Eq. (16.8a) are not a pure "horizontal" linear polarization (see Fig. 16.1), i.e

$$ R_1 \Rightarrow \begin{cases} \hat{S}^{(B)}(-\theta_B, -\theta'_B)\hat{S}^{(A)}(\theta_A, \theta'_A)|H\rangle \neq |H\rangle \\ \hat{S}^{(A)}(-\theta_A, -\theta'_A)\hat{S}^{(B)}(\theta_B, \theta'_B)|H\rangle \neq |H\rangle, \end{cases} \quad (16.9) $$

up to a constant phase.

The second requirement ($R_2$) for the optical interferometer to work is that the reflected component of light is the same as the initial one, such that it is rejected by the beam splitter, i.e we require that after travelling inside the set $\hat{S}(\theta, \theta')$, getting reflected on the mirror and travelling back inside $\hat{S}(\theta, \theta')$ (but with opposite propagation direction, as before), the polarization is "horizontal". This means

$$ R_2 \Rightarrow \begin{cases} \hat{S}^{(A)}(-\theta_A, -\theta'_A)\hat{M}\hat{S}^{(A)}(\theta_A, \theta'_A)|H\rangle = |H\rangle \\ \hat{S}^{(B)}(-\theta_B, -\theta'_B)\hat{M}\hat{S}^{(B)}(\theta_B, \theta'_B)|H\rangle = |H\rangle, \end{cases} \quad (16.10) $$

still up to a constant phase.

Finally, the additional requirement ($R_3$) to be sensitive to the axion field is that at least one of the two light polarizations in outer space has to be circularly polarized. Following the constellation representation of Fig. 16.2, if we assume the light propagating clockwise is right circularly polarized, then, the light propagating counterclockwise can be anything else, i.e linearly polarized or left polarized state<fnref n="2" />. Mathematically,

$$ R_3 \Rightarrow \begin{cases} \hat{S}^{(A)}(\theta_A, \theta'_A)|H\rangle = |R\rangle \\ \hat{S}^{(B)}(\theta_B, \theta'_B)|H\rangle \neq |R\rangle, \end{cases} \quad (16.11) $$

up to a constant phase.

### 16.1.3 Solution

One can show that a solution to all these requirements is

$$ \hat{S}^{(A)} = \hat{C}_R \quad (16.12a) $$

$$ \hat{S}^{(B)} = \mathbb{1}_2, \quad (16.12b) $$

<footnote n="2">²If it is linearly polarized or left polarized, the beatnote between the two beams will oscillate at the axion frequency, but the amplitude will be twice as big in the second case.</footnote>
where the latter is the identity matrix. The solution is therefore to replace the last $\hat{\lambda}_2$ in the optical bench A (see Fig. 16.1), by a $\hat{\lambda}_4(\pi/4)\hat{C}_H\hat{\lambda}_4(-\pi/4)$ combination, i.e a first quarter-wave plate oriented with $\theta = \pi/4$, then a horizontal linear polarizer and then a second quarter wave plate oriented with $\theta = -\pi/4$, as it is shown in Fig. 16.3. We show explicitly how it

<figure><img src="image_29.png" /><figcaption>Figure 16.3: Modified optical bench inside LISA spacecraft, compared to the original one presented in Fig. 16.1, in order to be sensitive to vacuum birefringence induced by the axion-photon coupling. The half-wave plate at the end of the optical bench A is now replaced with a right polarizer which is a combination of a quarter-wave plate, a horizontal polarizer and a second quarter-wave plate, which are oriented conveniently. This configuration produces a right polarized light at the output of optical bench A, and fulfill all the requirements for the light interferometer to work (see text).</figcaption></figure>

resolves the various requirements. Starting with $R_3$, we have

$$ \hat{S}^{(A)}|H\rangle = |R\rangle \qquad (16.13a) $$

$$ \hat{S}^{(B)}|H\rangle = |H\rangle \neq |R\rangle, \qquad (16.13b) $$

and therefore, in free space, one polarization is right circularly polarized and the other linearly polarized such that the axion field impacts only one of the two beams (see Fig. 16.3).

Then, looking at $R_1$, we have

$$ \hat{S}^{(B)}\hat{S}^{(A)}|H\rangle = |R\rangle \neq |H\rangle \qquad (16.14a) $$

$$ \hat{S}^{(A)}\hat{S}^{(B)}|H\rangle = |R\rangle \neq |H\rangle, \qquad (16.14b) $$

i.e the light coming from the optical bench A can interfere in the interferometer in optical bench B and vice-versa. Indeed, the light leaving optical bench A is right circularly polarized, arrives in optical bench B and since a circular polarization is a linear superposition of the two linear polarizations, the vertical component is reflected towards the interferometer, as shown in Fig. 16.3. The other beam coming from optical bench B is in the "horizontal" polarization state in vacuum, arrives on the right polarizer of optical bench A, gets right circularly polarized and similarly as above, its vertical component gets reflected by the polarized beam splitter inside the interferometer (see Fig. 16.3).
Finally, let us make sure that no parasitic component of light gets inside the interferometers after getting reflected at the edge of the spacecraft (requirement $R_2$). First, as we have not modified anything in the optical bench B, we have

$$ \hat{S}^{(B)} \hat{M} \hat{S}^{(B)} |H\rangle = |H\rangle, \qquad (16.15) $$

i.e the reflected component is in the "horizontal" polarization state, and therefore transmitted by the polarized beam splitter. Then, it is easy to show that

$$ \hat{C}_R \hat{M} \hat{C}_R = 0, \qquad (16.16a) $$

such that

$$ \hat{S}^{(A)} \hat{M} \hat{S}^{(A)} |H\rangle = 0, \qquad (16.16b) $$

i.e the entire power reflected at the edge of optical bench A is absorbed by the right polarizer and therefore does not jeopardize the functioning of the interferometer. This can be intuitively understood by the fact that after reflection, the right circularly polarization becomes left circularly polarized and goes back inside the right polarizer, whose role is to extract and transmit only the right polarization of the input. Therefore, nothing from the left polarized parasitic light goes through.

In this section, we have shown that a simple modification of LISA optical benches can make it sensitive to vacuum birefringence. Note that this modification leads to a decrease of power of light that arrives at each interferometer by a factor 4. In the next section, we will derive explicitly the signal induced in LISA TDI combinations, neglecting the reduction in power.

Note that other non standard designs of the optical benches exist. Indeed, while we only looked at the phase velocity difference between left and right circular polarizations of light, this birefringence effect can also be seen as the oscillation of the direction of a linear polarization around its nominal direction. Then, starting with a linear polarization (say $|H\rangle$) and adding a polarized beam-splitted such that the small $|V\rangle$ component is separated from the rest, one could detect such oscillation by photon-counting. This is essentially the detection idea in e.g. [19, 37, 38]. However, in the case of LISA, the laser power arriving at each interferometer from the distant spacecraft is small (of order of hundreds of picoWatt [6]), and since the oscillation of direction of polarization is $O(g_{ay}) \ll 1$, the number of photons arriving on the detection port will be extremely suppressed.

### 16.1.4 Expected Doppler effect in LISA

In this section, we derive the Doppler effect in LISA induced by the axion-photon coupling. As a reminder, this coupling modifies the phase velocity of left and right circularly polarized light as derived in Eq. (8.31). As derived in Section 9.3 of Chapter 9, this leads to a phase shift of a circularly polarized light travelling in vacuum for a length L compared to an unaffected
linearly polarized beam<fnref n="3" /> of the form<fnref n="4" />

$$ \Delta\phi(t) = \frac{\sqrt{16\pi G\rho_{\text{DM}}E_P g_{ay}}}{\omega_a c} \sin\left(\frac{\omega_a L}{2c}\right) \sin\left(\omega_a \left(t - \frac{L}{2c}\right) + \Phi\right) \equiv \Delta\phi_{\text{single link}} \quad (16.17) $$

where $\omega_a$ is the axion frequency, $g_{ay}$ is the axion-photon coupling and $E_P$ is the reduced Planck energy. Therefore, following Fig. 16.4, we have

$$ \Delta\phi_{12} = \Delta\phi_{23} = \Delta\phi_{31} = 0 \qquad (16.18a) $$

$$ \Delta\phi_{13} = \Delta\phi_{32} = \Delta\phi_{21} = \Delta\phi_{\text{single link}}, \qquad (16.18b) $$

where the ij subscript denote the arm pointing from spacecraft j to spacecraft i.

<figure><img src="image_30.png" /><figcaption>Figure 16.4: LISA proposed modified constellation for the search of vacuum birefringence, as a consequence of Fig. 16.3. In this framework, only the arms where light is circularly polarized are impacted by the axion field, and get phase shifted (see text).</figcaption></figure>

Then, using Eq. (14.6), we find the single-link Doppler shifts as

$$ y_{12} = y_{23} = y_{31} = 0 \qquad (16.19a) $$

$$ y_{13}(t) = y_{32}(t) = y_{21}(t) = \frac{\sqrt{16\pi G\rho_{\text{DM}}E_P g_{ay}}}{2\pi v_0 c} \times \sin\left(\frac{\omega_a L}{2c}\right) \cos\left(\omega_a \left(t - \frac{L}{2c}\right) + \Phi\right), \quad (16.19b) $$

where $v_0$ is the laser frequency. Note that, throughout this derivation, we neglect the propagation phase of the field, and it should be a valid approximation, because as it was discussed in Chapter 15, this would add a common phase to all spacecrafts plus a small contribution $\propto k_a L \ll 1$, for all axion frequencies of interest.

This configuration suggests that the optimized TDI combination to consider is the Sagnac $\alpha$ one, which, in the constant and equal arm length approximation, reads (using Eq. (14.10))

$$ \alpha_2 = - \left(1 - e^{-\frac{3i\omega_a L}{c}}\right) \left(y_{13} - y_{12} + e^{-\frac{i\omega_a L}{c}}(y_{32} - y_{23}) + e^{-\frac{2i\omega_a L}{c}}(y_{21} - y_{31})\right). \quad (16.20) $$

The amplitude of the signal at Fourier frequency $\omega$ is then

$$ |s_{\text{DM}}(\omega)| = \frac{2\sqrt{16\pi G\rho_{\text{DM}}E_P g_{ay}}}{2\pi v_0 c} \sin^2\left(\frac{3\omega L}{2c}\right). \qquad (16.21) $$

In Chapter 18, we will use this signal to infer the sensitivity of LISA to $g_{ay}$.

<footnote n="3">³In reality, this coupling affects the two linear polarization states, i.e a linear state slightly oscillates into the other one and vice-versa. Using e.g. an initial pure "horizontal" polarization state, one could see such effect using a vertical polarizer and counting the photons that go through due to the rotation of the polarization plane. However, different dispersion relation between the two linear states arises at second order in the perturbation.</footnote>

<footnote n="4">⁴The factor 2 difference with Eq. (9.10) comes from the fact that in Section 9.3, we assumed the amplitude of the phase shift of e.g. the right circularly polarized compared to the left one, while here we compare one circular polarization with a linear one.</footnote>
## 16.2 Dichroism

As discussed in Section 8.2.3, in presence of an external magnetic field, the polarisation of light parallel to the magnetic field gets absorbed due to the photon-axion conversion in this direction. This implies that for incoming linear polarized light, the polarization gets elliptical in presence of magnetic field.

The LISA constellation orbiting around the Sun, it will be immersed in the solar magnetic field, therefore we can wonder if one could see such dichroic effects using LISA.

As it was discussed in the previous section, each light beam is polarized linearly, and depending on the direction of propagation, the polarization is either along the x or y axis (see Fig. 16.2). Therefore, in the presence of axions and considering a constant magnetic field in the x-y plane, both polarizations get affected, i.e both components in the direction of the magnetic field get a relative phase shift compared to the other component (perpendicular to the magnetic field). Therefore, comparing the two polarization states together, the phase shift is twice as large compared to Eq. (8.35b), i.e

$$ \phi(L) = \frac{\hbar c g_{a\gamma}^2 B_0^2}{\mu_0 2q^2} (qL - \sin(qL)). \qquad (16.22) $$

For our estimates in Chapter 18, we will neglect the orbit of the constellation, and we will simply make the optimistic assumption that the magnetic field orientation is the same along the whole orbit. We will also make the assumption that the axion wind is parallel to the photon propagation, i.e $q = |\vec{k}_a - \vec{k}_\gamma| = |k_a - k_\gamma|$.

## 16.3 Conclusion

In this chapter, we have seen how to search for the axion-photon coupling in LISA using its effect on birefringence and dichroism. We have shown that a slight modification of the current optical benches inside LISA spacecrafts would make LISA an axion-photon coupling detector. In Chapter 18, we will show the expected sensitivity of LISA to this coupling is very competitive with existing bounds.

In terms of dichroism, one can make estimate of the phase shift that light acquires when moving in the solar magnetic field following Eq. (16.22), but as we shall see in Chapter 18, such phase shift is a constant of time, and therefore it will be invisible.
[1] Éanna É Flanagan and Scott A Hughes. "The basics of gravitational wave theory". In: *New Journal of Physics* 7.1 (Sept. 2005), p. 204. DOI: 10.1088/1367-2630/7/1/204. URL: https://dx.doi.org/10.1088/1367-2630/7/1/204.

[2] Peter Saulson. "If light waves are stretched by gravitational waves, how can we use light as a ruler to detect gravitational waves?" In: *American Journal of Physics* 65 (1997), pp. 501-505. URL: https://api.semanticscholar.org/CorpusID:42802456.

[3] Pau Amaro-Seoane et al. *Laser Interferometer Space Antenna*. 2017. arXiv: 1702.00786 [astro-ph.IM].

[4] Monica Colpi et al. *LISA Definition Study Report*. 2024. arXiv: 2402.07571 [astro-ph.CO].

[5] Travis Robson, Neil J Cornish, and Chang Liu. "The construction and use of LISA sensitivity curves". In: *Classical and Quantum Gravity* 36.10 (Apr. 2019), p. 105011. ISSN: 1361-6382. DOI: 10.1088/1361-6382/ab1101. URL: http://dx.doi.org/10.1088/1361-6382/ab1101.

[6] Antoine Petiteau. "DE LA SIMULATION DE LISA A L'ANALYSE DES DONNEES. Détection d'ondes gravitationnelles par interférométrie spatiale (LISA : Laser Interferometer Space Antenna)". Theses. Université Paris-Diderot - Paris VII, June 2008. URL: https://theses.hal.science/tel-00383222.

[7] J. W. Armstrong, F. B. Estabrook, and Massimo Tinto. "Time-Delay Interferometry for Space-based Gravitational Wave Searches". In: *The Astrophysical Journal* 527.2 (Dec. 1999), p. 814. DOI: 10.1086/308110. URL: https://dx.doi.org/10.1086/308110.

[8] Massimo Tinto and J. W. Armstrong. "Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation". In: *Phys. Rev. D* 59 (10 Apr. 1999), p. 102003. DOI: 10.1103/PhysRevD.59.102003. URL: https://link.aps.org/doi/10.1103/PhysRevD.59.102003.

[9] S. V. Dhurandhar, K. Rajesh Nayak, and J.-Y. Vinet. "Algebraic approach to time-delay data analysis for LISA". In: *Physical Review D* 65.10 (May 2002). ISSN: 1089-4918. DOI: 10.1103/physrevd.65.102002. URL: http://dx.doi.org/10.1103/PhysRevD.65.102002.

[10] Jean-Baptiste Bayle, Olaf Hartwig, and Martin Staab. "Adapting time-delay interferometry for LISA data in frequency". In: *Physical Review D* 104.2 (July 2021). ISSN: 2470-0029. DOI: 10.1103/physrevd.104.023006. URL: http://dx.doi.org/10.1103/PhysRevD.104.023006.

[11] Dam Quang Nam et al. "Time-delay interferometry noise transfer functions for LISA". In: *Phys. Rev. D* 108 (8 Oct. 2023), p. 082004. DOI: 10.1103/PhysRevD.108.082004. URL: https://link.aps.org/doi/10.1103/PhysRevD.108.082004.

[12] Olaf Hartwig et al. "Time-delay interferometry without clock synchronization". In: *Phys. Rev. D* 105 (12 June 2022), p. 122008. DOI: 10.1103/PhysRevD.105.122008. URL: https://link.aps.org/doi/10.1103/PhysRevD.105.122008.

[13] Thomas A. Prince et al. "LISA optimal sensitivity". In: *Phys. Rev. D* 66 (12 Dec. 2002), p. 122002. DOI: 10.1103/PhysRevD.66.122002. URL: https://link.aps.org/doi/10.1103/PhysRevD.66.122002.
[14] Olaf Hartwig et al. "Stochastic gravitational wave background reconstruction for a nonequilateral and unequal-noise LISA constellation". In: *Physical Review D* 107.12 (June 2023). ISSN: 2470-0029. DOI: 10.1103/physrevd.107.123531. URL: http://dx.doi.org/10.1103/PhysRevD.107.123531.

[15] R. et al. Abbott. "GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run". In: *Physical Review X* 13.4 (Dec. 2023). ISSN: 2160-3308. DOI: 10.1103/physrevx.13.041039. URL: http://dx.doi.org/10.1103/PhysRevX.13.041039.

[16] Stanislav Babak, Martin Hewitson, and Antoine Petiteau. *LISA Sensitivity and SNR Calculations*. 2021. arXiv: 2108.01167 [astro-ph.IM].

[17] H. Grote and Y. V. Stadnik. "Novel signatures of dark matter in laser-interferometric gravitational-wave detectors". In: *Physical Review Research* 1.3 (Dec. 2019). ISSN: 2643-1564. DOI: 10.1103/physrevresearch.1.033187. URL: http://dx.doi.org/10.1103/PhysRevResearch.1.033187.

[18] Alexandre S. Göttel et al. "Searching for Scalar Field Dark Matter with LIGO". In: *Phys. Rev. Lett.* 133 (10 Sept. 2024), p. 101001. DOI: 10.1103/PhysRevLett.133.101001. URL: https://link.aps.org/doi/10.1103/PhysRevLett.133.101001.

[19] Jiang-Chuan Yu et al. "Sensitivity of space-based gravitational-wave interferometers to ultralight bosonic fields and dark matter". In: *Phys. Rev. D* 108 (8 Oct. 2023), p. 083007. DOI: 10.1103/PhysRevD.108.083007. URL: https://link.aps.org/doi/10.1103/PhysRevD.108.083007.

[20] S. Babak et al. *LISA Data Challenge: Sangria*. Technical Report LISA-LCST-SGS-MAN-001. 2020.

[21] Neil J. Cornish and Louis J. Rubbo. "LISA response function". In: *Phys. Rev. D* 67 (2 Jan. 2003), p. 022001. DOI: 10.1103/PhysRevD.67.022001. URL: https://link.aps.org/doi/10.1103/PhysRevD.67.022001.

[22] John Skilling. "Nested Sampling". In: *Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 24th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering*. Ed. by Rainer Fischer, Roland Preuss, and Udo Von Toussaint. Vol. 735. American Institute of Physics Conference Series. AIP, Nov. 2004, pp. 395-405. DOI: 10.1063/1.1835238.

[23] Michael J. Williams. "nessai: Nested Sampling with Artificial Intelligence". Version latest. Feb. 2021. DOI: 10.5281/zenodo.4550693. URL: https://doi.org/10.5281/zenodo.4550693.

[24] Neil J. Cornish and Tyson B. Littenberg. "Tests of Bayesian model selection techniques for gravitational wave astronomy". In: *Physical Review D* 76.8 (Oct. 2007). ISSN: 1550-2368. DOI: 10.1103/physrevd.76.083006. URL: http://dx.doi.org/10.1103/PhysRevD.76.083006.

[25] Jean-Baptiste Bayle et al. *LISA Orbits*. Version 2.0. Apr. 2022. DOI: 10.5281/zenodo.6412992. URL: https://doi.org/10.5281/zenodo.6412992.
[26] Waldemar Martens and Eric Joffre. "Trajectory Design for the ESA LISA Mission". In: *The Journal of the Astronautical Sciences* 68.2 (June 2021), pp. 402–443. ISSN: 2195-0571. DOI: 10.1007/s40295-021-00263-2. URL: http://dx.doi.org/10.1007/s40295-021-00263-2.

[27] Andrei Derevianko. "Detecting dark-matter waves with a network of precision-measurement tools". In: *Phys. Rev. A* 97 (4 Apr. 2018), p. 042506. DOI: 10.1103/PhysRevA.97.042506. URL: https://link.aps.org/doi/10.1103/PhysRevA.97.042506.

[28] Etienne Savalle et al. "Searching for Dark Matter with an Optical Cavity and an Unequal-Delay Interferometer". In: *Phys. Rev. Lett.* 126 (5 Feb. 2021), p. 051301. DOI: 10.1103/PhysRevLett.126.051301. URL: https://link.aps.org/doi/10.1103/PhysRevLett.126.051301.

[29] Kentaro Miuchi et al. "CYGNUS". In: *Journal of Physics: Conference Series* 1468.1 (Feb. 2020), p. 012044. DOI: 10.1088/1742-6596/1468/1/012044. URL: https://dx.doi.org/10.1088/1742-6596/1468/1/012044.

[30] F. van Leeuwen. "Validation of the new Hipparcos reduction". In: *Astronomy & Astrophysics* 474.2 (Aug. 2007), pp. 653–664. DOI: 10.1051/0004-6361:20078357. URL: https://doi.org/10.1051/0004-6361:20078357.

[31] Leinert, Ch. et al. "The 1997 reference of diffuse night sky brightness". In: *Astron. Astrophys. Suppl. Ser.* 127.1 (1998), pp. 1–99. DOI: 10.1051/aas:1998105. URL: https://doi.org/10.1051/aas:1998105.

[32] Jean-Baptiste Bayle and Olaf Hartwig. "Unified model for the LISA measurements and instrument simulations". In: *Phys. Rev. D* 107 (8 Apr. 2023), p. 083019. DOI: 10.1103/PhysRevD.107.083019. URL: https://link.aps.org/doi/10.1103/PhysRevD.107.083019.

[33] Gerhard Heinzel et al. "Frequency planning for LISA". In: *Phys. Rev. D* 110 (4 Aug. 2024), p. 042002. DOI: 10.1103/PhysRevD.110.042002. URL: https://link.aps.org/doi/10.1103/PhysRevD.110.042002.

[34] E. Collett. *Field Guide to Polarization*. Field Guides. SPIE Press, 2005. ISBN: 978-0-8194-5868-1. URL: https://books.google.fr/books?id=5lJwcCsLbLsC.

[35] E. Hecht. *Optics*. Pearson education. Addison-Wesley, 2002. ISBN: 978-0-321-18878-6. URL: https://books.google.fr/books?id=T3ofAQAAMAAJ.

[36] A. L. Fymat. "Jones's Matrix Representation of Optical Instruments. 2: Fourier Interferometers (Spectrometers and Spectropolarimeters)". In: *Appl. Opt.* 10.12 (Dec. 1971), pp. 2711–2716. DOI: 10.1364/AO.10.002711. URL: https://opg.optica.org/ao/abstract.cfm?URI=ao-10-12-2711.

[37] Koji Nagano et al. "Axion Dark Matter Search with Interferometric Gravitational Wave Detectors". In: *Phys. Rev. Lett.* 123 (11 Sept. 2019), p. 111301. DOI: 10.1103/PhysRevLett.123.111301. URL: https://link.aps.org/doi/10.1103/PhysRevLett.123.111301.

[38] Aldo Ejlli et al. "Probing dark matter with polarimetry techniques". In: *Phys. Rev. D* 107.8 (2023), p. 083035. DOI: 10.1103/PhysRevD.107.083035. arXiv: 2211.09922 [hep-ph].
# Part VII

## Sensitivity estimates
# Chapter 17

## Experimental considerations

In this chapter, we will review the various experimental parameters that impact the theoretical signals presented in the previous chapters, and that for all experiments under consideration. In particular, for most of these experiments, we will estimate their main characteristics, their noise levels and their integration time. Then, this will allow us to compute their expected sensitivity to a given coupling, as shown in Eq. (5.13), which we will do in the next chapter (Chapter 18).

### 17.1 DAMNED and optical fibers

We first discuss the sensitivity of an already existing experiment, located at SYRTE, based on an optical cavity, *DArk Matter from Non Equal Delays (DAMNED)* [1]. In the dilatonic framework developed in Section 8.1, we have seen that the fine structure constant $\alpha$ and the electron mass $m_e$ oscillate at different space time locations, according to the phase value of the dilatonic field. Those oscillations imply a variation of the Bohr radius $a_0 = \hbar/cm_e\alpha$ and a variation of length of objects accordingly. DAMNED aims at detecting those oscillations by using an optical cavity and measuring the tiny variations of the length of the cavity through an additional phase shift of the laser.

As an experiment aiming at measuring length variations of an optical cavity, DAMNED could be sensitive to the vacuum birefringence effect from the axion-photon coupling on an optical cavity presented in Chapter 9, whose corresponding signal is shown in Eq. (9.7). Its sensitivity to such coupling will be obtained in Section 18.3.1.

For the experimental parameters of DAMNED, we follow [1]. An infrared laser of wavelength $\lambda = 1.542$ µm is sent inside an optical cavity of length $l = 15.42$ cm<fnref n="1" /> and of finesse $F = 8 \times 10^5$, such that $r \sim 1-2 \times 10^{-6}$. We assume the experimental noise to be white between roughly 10 kHz and 1 MHz with phase noise PSD $S_\phi(f) \sim 10^{-8}$ rad²/Hz [2]. This implies that the experiment will be sensitive at axion masses between approximately $4 \times 10^{-11}$ and $4 \times 10^{-9}$ eV. The experiment lasted twelve days [2]. For our estimates, we will assume that the experiment runs for 1 year, i.e $T_{obs} \sim 3.15 \times 10^7$ s.

For optical fibers, we will use measurements on a french 86 km urban link [3] whose phase noise PSD is estimated at $S_\phi \sim 10^{-4}$ rad²/Hz from 1 to $10^3$ Hz. We will also assume multiple measurements of the phase for a total integration time of one year. Similarly as DAMNED,

<footnote n="1">¹such that the condition $l = n\lambda, n \in \mathbb{N}$ is fulfilled.</footnote>
optical fibers could detect the axion-photon coupling through vacuum birefringence effect, as shown in Chapter 9, with its corresponding signal in Eq. (9.10). Its expected constraint on such coupling will also be obtained in Section 18.3.1.

## 17.2 Rydberg atoms in a microwave cavity

We now turn to the experiment involving Rydberg atoms inside a microwave cavity for the detection of DP-photon coupling $\chi$ presented in Chapter 10. We recall that in this experiment, we aim at measuring the quadratic Stark effect, i.e the change in the transition frequency of atoms $\Delta\nu$, which is proportional to the electric field squared $E^2$. We first discuss the main sources of noise in this experiment, and then the various numerical parameters that will be used for the estimation of the sensitivity of the experiment.

### 17.2.1 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. We denote the 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

$$E_{\min}^2 = \sqrt{\frac{2S_{E^2}}{\sqrt{T_{\text{obs}}\tau(\omega_U)}},} \qquad (17.1)$$

where $T_{\text{obs}}$ is the individual integration time and $\tau(\omega_U)$ is the coherence time of the field in the case where $T_{\text{obs}} \gg \tau(\omega_U)$, as discussed in the end of Chapter 10.

This means, that for SNR = 1, the sensitivity on $\chi$ for a mass of the DP $m_U$ corresponding to an angular frequency $\omega_U$ can be computed by equaling Eqs. (10.12a) and (17.1), which leads to

$$[\chi(\omega_U)]_{\text{stat}} = \frac{E_{\min}^2}{S(\omega_U, \omega_A; \rho_{\text{DM}}, X_A; L, r)} = \frac{\sqrt{2S_{E^2}}}{(T_{\text{obs}}\tau(\omega_U))^{\frac{1}{4}}S(\omega_U, \omega_A; \rho_{\text{DM}}, X_A; L, r)}. \qquad (17.2)$$

### 17.2.2 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

$$\vec{X}_A \rightarrow \vec{X}_A \left[ 1 + \int d\omega_0 \frac{\Delta X_A(\omega_0)}{X_A} \cos(\omega_0 t + \phi_0) \right]. \qquad (17.3)$$
In this expression, $\Delta X_A(\omega_0)$ is a stochastic contribution modeling the spectral amplitude of the noise characterized by the RIN PSD denoted $S_{\text{RIN}}(\omega_U)$. Since RIN is related to fluctuation of power inside the cavity, we must relate its PSD to the intensity fluctuation $\Delta X_a$. At first order in the fluctuation $\Delta X_A$, the injected power $P$ inside the cavity is

$$P(\omega_0) = (X_A + \Delta X_A(\omega_0))^2 = X_A^2 + 2X_A\Delta X_A(\omega_0) + O((\Delta X_A)^2) \equiv P_0 + \Delta P(\omega_0), \quad (17.4a)$$

and the RIN PSD is defined as

$$\frac{\Delta P(\omega_0)}{P_0} \propto \sqrt{\frac{2S_{\text{RIN}}(\omega_0)}{T_{\text{obs}}}} \rightarrow \frac{\Delta X_A(\omega_0)}{X_A} = \sqrt{\frac{S_{\text{RIN}}(\omega_0)}{2T_{\text{obs}}}} \quad (17.4b)$$

Typically, the RIN of frequency generators in the microwave domain (GHz frequencies of interest here) is characterized by a flicker noise, see e.g. [4], such that we can parametrize its PSD as

$$S_{\text{RIN}}(\omega) = \frac{P_{\text{RIN}}}{\omega}, \quad (17.4c)$$

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

Let us now show how the $\Delta X_A$ fluctuations can produce an harmonic signal in $|\vec{E}|^2$ of angular frequency $\Delta\omega = \omega_A - \omega_U$, i.e. mimic the searched signal of Eq. (10.13a).

We will work to leading order in $\Delta X_A/X_A$ and in particular neglect terms that are $O\left(\left(\frac{\Delta X_A}{X_A}\right)^2\right)$ and $O\left(\frac{\Delta X_A}{X_A} \frac{X_{\text{DM}}}{X_A}\right)$. Considering the modification of applied field amplitude Eq. (17.3), the fluctuation $\Delta X_A$ will only be considered at frequencies $\omega_0$ producing a noise in the electric field squared at angular frequency $\Delta\omega$. By simply computing the electric field squared component $\propto X_A\Delta X_A$, we find that these angular frequencies are $\omega_0 = \{\Delta\omega; 2\omega_A - \Delta\omega\}$. Considering the RIN as a flicker noise characterized by a PSD of the form Eq. (17.4c), the fluctuation amplitude at $\omega_0 = \Delta\omega$ will be multiple order of magnitudes larger than its amplitude at $\omega_0 = 2\omega_A - \Delta\omega \sim 2\omega_A (\omega_A \sim \text{GHz}, \text{ while } \Delta\omega \le \text{kHz})$. 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. (17.3) becomes

$$\left(\vec{X}_A + \Delta\vec{X}_A(\Delta\omega) \cos(\Delta\omega t + \phi_0)\right) \cos(\omega_A t + \phi_A). \quad (17.5)$$

In Appendix A.3, we show how to derive the RIN contribution to the total electric field oscillating at frequency $\Delta\omega$. This reads

$$[E^2(\omega_U, \omega_A)]_{\text{RIN}} = \sqrt{\frac{P_{\text{RIN}}N(\omega_U, \omega_A)}{8T_{\text{obs}}\Delta\omega}} \cos(\Delta\omega t + \varphi), \quad (17.6)$$

where $\sqrt{N(\omega_U, \omega_A)}$ corresponds to the noise amplification factor by the cavity, quadratic in $X_A$ and whose expression is also given in Appendix A.3.

The RIN contribution to $E^2$ from Eq. (17.6) will limit the sensitivity of the experiment to values of $\chi$ that makes the signal from Eq. (10.12a) larger than the systematic (i.e. larger than
Eq. (17.6)). In other words, the RIN will limit the sensitivity of the experiment to values of $\chi$ that are larger than

$$[\chi(\omega_U)]_{\text{RIN}} = \frac{\sqrt{\frac{P_{\text{RIN}}N(\omega_U, \omega_A)}{8T_{\text{obs}}\Delta\omega}}}{S(\omega_U, \omega_A; \rho_{\text{DM}}, X_A; L, r)}, \quad (17.7)$$

where the function $S$ is defined in Eq. (10.12a). Note that this limit is linear in $X_A$, the amplitude of the applied electric field.

Let us discuss shot noise. For a general detector, light is received as a random flux of photons whose probability density follows the Poisson distribution [5]. In our case, it can also be viewed as an amplitude fluctuation of the electric field squared, as seen by Rydberg atoms, thus is analogous to the RIN. Here, we compute an estimate of the shot noise PSD and we show that it is negligible compared to the RIN PSD. Let us start with the typical number of photons received by the atoms at interval $1/f_s$ as the ratio of laser energy and photon energy

$$N_{\gamma} = \frac{E_{\text{laser}}}{E_{\gamma}} = \frac{1}{\hbar\omega_{A}} \int dS_{\text{laser}} \frac{X_{A}^{2}\epsilon_{0}c}{2f_{s}}, \quad (17.8a)$$

where we integrate over the laser waist, given by the mode radius $r_L$. The variance on the number of photons received is also $N_\gamma$ (from Poisson distribution) and therefore the shot noise PSD normalized by the number of photons is

$$S_{\text{SN}} = \frac{\sigma_{N_{\gamma}}^2}{N_{\gamma}^2 f_s} = \frac{1}{N_{\gamma} f_s} = \frac{2\hbar\omega_A}{X_A^2 \epsilon_0 c \pi r_L^2} \quad (17.8b)$$

Using experimental parameters that we will discuss in the following (with $r_L = 0.01$ m), one can show easily that $S_{\text{SN}} \ll S_{\text{RIN}}$, therefore we will neglect this contribution.

### 17.2.3 Optimum choice for applied field amplitude $X_A$

While the signal Eq. (10.12a) is linear in applied field amplitude $X_A$, increasing $X_A$ would also increase the systematic effect $\sqrt{N(\omega_U, \omega_A)}$, as it is quadratic in $X_A$, while leaving the statistical effect unchanged. This suggests an optimum value of $X_A$ at each applied frequency, and this is what we want to find in this section.

The sensitivity of the experiment at a given angular frequency $\omega_U$ relies on the signal amplitude Eq. (10.12a) but also on the limiting noise. Combining Eq. (17.2) and Eq. (17.7), this is simply

$$[\chi(\omega_U)]_{\text{limit}} = \frac{\sqrt{\frac{2S_{E2}}{\sqrt{\tau(\omega_U)}} + \frac{P_{\text{RIN}}N(\omega_U, \omega_A)}{8\Delta\omega\sqrt{T_{\text{obs}}}}}}{T_{\text{obs}}^{\frac{1}{4}} S(\omega_U, \omega_A; \rho_{\text{DM}}, X_A; L, r)}, \quad (17.9)$$

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 Section 10.3.2, half the sampling frequency ($= \pi f_s$). To better understand the sensitivity of the experiment,
we will simplify the expressions of $S(\omega_U, \omega_A; \rho_{DM}, X_A; L, r)$ and $N(\omega_U, \omega_A)$ in Eqs. (10.12a), (17.6) and (A.17d), considering $r = 1 - \epsilon$, $\epsilon \ll 1$ and $\omega_U \sim \omega_A^2$. In that case,

$$N(\omega_U) \approx \frac{X_A^4 \epsilon^2}{\cos^4\left(\frac{\omega_U L}{2c}\right)}, \qquad (17.10a)$$

$$S(\omega_U) \approx \frac{\beta c X_A \sqrt{2\mu_0 \rho_{DM}} \sqrt{\epsilon}}{\sqrt{2} \cos^2\left(\frac{\omega_U L}{2c}\right)} \left(1 + \cos\left(\frac{\omega_U L}{2c}\right)\right), \qquad (17.10b)$$

at lowest order in $\epsilon$. As expected, we see from (17.10a) that the two uncertainties in the numerator of (17.9)) 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. (17.9) is minimum,

$$\frac{d\chi(\omega_U)}{dX_A} = 0 \qquad (17.11a)$$

$$\Rightarrow X_A(\omega_U) \approx \sqrt[4]{\frac{16S_{E^2}\Delta\omega}{\epsilon^2 P_{\text{RIN}}}} \sqrt{\frac{T_{\text{obs}}}{\tau(\omega_U)}} \left|\cos\left(\frac{\omega_U L}{2c}\right)\right| \approx \sqrt[4]{\frac{16\pi f_s S_{E^2} \sqrt{\omega_U T_{\text{obs}}}}{10^3 \epsilon^2 P_{\text{RIN}}}} \left|\cos\left(\frac{\omega_U L}{2c}\right)\right|, \qquad (17.11b)$$

where we used Eqs.(17.10a),(17.10b). 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 $\chi(\omega_U)$. Note that this equation to approximate the optimum value of $X_A$ is not valid for angular frequencies $\omega_U L/c = 2\pi + 4n\pi, n \in \mathbb{N}$ and $\omega_U L/c = \pi + 2n\pi, n \in \mathbb{N}$. In the first case, the signal decreases significantly (see Eq. (17.10b)) and the experiment becomes insensitive to DM, while in the second case, Eq. (17.11b) would indicate to apply $X_A = 0$, which would automatically set the signal to 0, at first order in $\chi$. In this case, one must relax the constraint $\omega_U = \omega_A$ and compute the exact value of $X_A$ with the exact expressions of noise and signals, when $\omega_A$ corresponds to a mode of the cavity.

From Eq. (17.11b), we can express the sensitivity of the experiment $\chi(\omega_U)$ as

$$\chi(\omega_U) \approx \frac{\sqrt{2} \left| \cos \left( \frac{\omega_U L}{2c} \right) \right|}{1 + \cos \left( \frac{\omega_U L}{2c} \right)} \left( \frac{P_{\text{RIN}} S_{E^2}}{10^3 \pi f_s} \right)^{\frac{1}{4}} \frac{\left( \omega_U T_{\text{obs}}^{-3} \right)^{\frac{1}{8}}}{\beta c \sqrt{2\mu_0 \rho_{\text{DM}}}}, \qquad (17.12)$$

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

When we will evaluate the sensitivity of the experiment in Section 18.4.1, 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. (17.11b) is always fulfilled. However, we will use the full expressions of $S(\omega_U, \omega_A; \rho_{DM}, X_A; L, r)$ and $N(\omega_U, \omega_A)$ in Eqs. (10.12a) and (17.6).

<footnote n="2">²Even in the case where $\Delta\omega = \pi f_s$, we can assume $\omega_U \sim \omega_A$ for our estimates, as we have $f_s \lesssim \frac{\omega_U}{Q} \approx \omega_U(1-r) \ll \omega_U$, for $f_s$ and $Q$ that we will choose.</footnote>
17.2.4 Experimental parameters

Now, we discuss in details the numerical values of the various experimental parameters.
They are all summarized in Table 17.1.

First, we consider the random polarized case for the DP polarization direction, i.e $\beta = 1/\sqrt{3}$ (see Eq. (10.12b)). Some already existing constraints come from experiments with this assumption [6–13].

Let us remind that the observational scheme considered here consists in electric signals of
angular frequency ωA. For each injected electric field we perform a measurement of duration
Tobs and then shift the angular frequency of the applied field by πfS. Each measurement of
duration Tobs provides constraints in the frequency range {ωA − πfS; ωA + πfS} in steps of
2π/Tobs. We will consider an individual measurement duration of Tobs = 60 s. This duration
is arbitrary, but we list two important considerations for this choice. We require Tobs to be
large for best sensitivity (see Eq. (17.9)), but short enough to allow scanning a large range of
DM frequencies in a reasonable amount of time (≤ O(month)).

Now, let us discuss the sampling frequency. In [14], the measurement of the transition
probability for a given frequency lasts about 300 µs. At least three such measurements
are necessary to fully determine the resonance (amplitude, width and centre 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
non-destructive techniques. 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 non-destructive 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
[15]. As a consequence, it is not necessary to produce new Rydberg atoms for each frequency
measurement, implying that high (> 1 kHz) sampling frequency is feasible. 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 fs = 1 kHz. In both cases we will assume a single shot spectroscopic resolution
of ~1 kHz for differential polarizabilities of Δα/2h ≈ 105 Hz/(V/m)2, corresponding to
Rydberg states with principle quantum numbers n ~ 60 – 70 [16]3.

In terms of statistical noise, in [14] 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 previous paragraph). Two different sampling rates will lead to different measurement noise PSD and directly affect the sensitivity of the experiment (cf Eq. (17.12)). To put some numbers, assuming a single shot spectroscopic resolution of ~1 kHz [14] for

<footnote n="3">³While we are in the regime $T_{\text{obs}} \gg \tau(f_{\text{U}})$, note that we did not take into account the stochastic nature of the field which would impact the sensitivity of the experiment. In particular, we are sampling a signal at $10^2 - 10^3$ Hz, which roughly corresponds to the inverse of the coherence time of the field at GHz frequency (see Eq. (5.11)). Expressing the field as a superposition of plane waves, see e.g. [17], and assuming an intrinsic frequency $f_{\text{U}} = 10^9$ Hz, a quick simulation reveals that this frequency broadening decreases the SNR by a factor $\mathcal{O}(1)$ for $f_s = 10^3$ Hz, and $\mathcal{O}(10)$ for $f_s = 10^2$ Hz. As expected, this suggests that one would need to increase the sampling frequency of the apparatus, in order to be in the regime where $f_s > 1/\tau(f_{\text{U}})$, to avoid substantial loss in sensitivity.</footnote>
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 $S_{E2} \approx 10^{-6}$ (V/m)$^4$/Hz for $f_s = 100$ Hz and $S_{E2} \approx 10^{-7}$ (V/m)$^4$/Hz for $f_s = 1$ kHz.

Regarding the systematic effect, the amplitude of the flicker noise can be considered to be $P_{RIN} = 10^{-13}$ (based on "off the shelf" components studied in [4] more than a decade ago) in the modest case, and we assume an improved RIN control for the optimistic case, with an amplitude of $P_{RIN} = 10^{-15}$. The level of the systematic effect PSD is also impacted by the sampling rate, following Eq. (17.6). Moreover, as derived in Section 17.2.3, the approximate sensitivity of the experiment Eq. (17.12) scales as $(P_{RIN}S_{E2}/f_s)^{1/4}$, implying that the modest and optimistic scenarios will differ in the sensitivity by a factor ~10. The optimum value of the amplitude of the applied field $X_A$ is then derived from Eqs. (17.9) (for the minimum value$^4$) and (17.11b) (for the maximum value, where we will consider DP frequencies until ~2 × 10$^{10}$ Hz) 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

$$18 \text{ V/m} \le X_A \le 2.4 \times 10^5 \text{ V/m}. \qquad (17.13)$$

It is independent of the sampling frequency since, from Eq. (17.11b), $X_A(\omega_A) \propto (f_s S_{E2})^{1/4}$ and $S_{E2} \propto f_s^{-1}$, but not of the systematic effect level $P_{RIN}$. In Section 18.4.1, we will obtain the

<table><thead><tr><th>Parameters</th><th>Numerical values</th></tr></thead><tbody><tr><td>Quality factor Q [18]</td><td>10<sup>4</sup></td></tr><tr><td>Mirrors reflectivity r</td><td>1 - 2 × 10<sup>-4</sup></td></tr><tr><td>Cavity length L</td><td>7.5 cm</td></tr><tr><td>Injected field strength X<sub>A</sub>(ω)</td><td>[18, 7.6 × 10<sup>5</sup>] V/m</td></tr><tr><td>Sampling frequency f<sub>s</sub></td><td>10<sup>2</sup>; 10<sup>3</sup> Hz</td></tr><tr><td>Individual measurement time T<sub>obs</sub></td><td>60 s</td></tr><tr><td>Range of f<sub>A</sub> = ω<sub>A</sub>/2π</td><td>[0.5, 20.5] GHz</td></tr><tr><td>Range of Δω</td><td>[2π/T<sub>obs</sub>, πf<sub>s</sub>] rad/s</td></tr><tr><td>Statistical noise PSD S<sub>E2</sub></td><td>10<sup>-4</sup>/f<sub>s</sub> (V/m)<sup>4</sup>/Hz</td></tr><tr><td>Systematic effect PSD S<sub>RIN</sub>(ω)</td><td>10<sup>-13</sup>/ω; 10<sup>-15</sup>/ω</td></tr></tbody></table>

Table 17.1: Assumed experimental parameters for the experiment involving Rydberg atoms in a microwave cavity for the detection of the DP-photon coupling. Two different numerical values for the same parameter means that one is used for the modest scenario while the other is used for the optimistic one (see text).

expected sensitivity of this experiment to the DP-photon kinetic mixing coupling $\chi$, in both modest and optimistic case, and using all the experimental parameters described above.

<footnote>$^4$The smallest value of $X_A$ corresponds to DM/applied frequencies close to odd modes, from Eq. (17.11b). In this regime, both approximate amplitudes of noise Eq. (17.10a) and signal Eq. (17.10b) reach infinity, implying that we must consider the real expression of $\chi(\omega)$ derived in Eq. (17.9) and check for which value of $X_A$, the sensitivity on $\chi$ is the highest.</footnote>
17.3 SHUKET

We now turn to dish antenna experiments for the search of DP-photon $\chi$ coupling, as described in Chapter 11. We will focus on a practical example to illustrate how the Kirchhoff integral theorem and the overlap of modes affect the expected power received by an antenna from a spherical dish emitter. We consider the setup of the SHUKET experiment [6], which has been used to search for a DP in the frequency range 5 – 6.8 GHz using a spherical dish and a horn antenna for the detection. In the analysis of this experiment, following [19], it has been assumed that all the power emitted by the dish is received by the antenna. In this section, we will present the experimental parameters, in particular we will assume the mean DP frequency $f_U \sim 6$ GHz, and show that the calculations done in Chapter 11 are valid in that case. Then, in the next chapter, Section 18.4.2, we will estimate the power received by the antenna and estimate its implication on a more realistic sensitivity of the experiment on $\chi$.

The spherical dish used in [6] has a curvature radius of $R = 32$ m and an area of $A_{\text{dish}} = 1.2$ m², implying a radius of $r \approx 0.618$ m (see Fig. 11.1). Then, $R \gg r$ (or equivalently the dish-fictional plane distance $a \sim 6 \times 10^{-3}$ m, as shown in Fig. 11.2, in the case of SHUKET is way smaller than $r$, i.e. $a \ll r$) which ensures that the low curvature approximation is valid. The expression of the electric field emitted by the dish at its surface is given by Eq. (11.1) where $\vec{Y}_{\parallel,D}$ has the form

$$ \vec{Y}_{\parallel,D}(\theta, \phi) = \vec{Y} - (\vec{Y} \cdot \hat{e}_r) \hat{e}_r \approx \begin{pmatrix} Y_x \\ Y_y \\ 0 \end{pmatrix} + \mathcal{O}(\theta), \qquad (17.14) $$

with $\theta, \phi$ corresponding to a spherical coordinate system centered in dish's curvature center and where we have used the notation $\vec{Y} = (Y_x, Y_y, Y_z)$ in the cartesian coordinate system depicted in Fig. 11.1 and where we used the low curvature approximation at the last step<fnref n="5" />. This means that at leading order in $\theta$, the polarization of the emitted electric field does not depend on the dish coordinate, and is only polarized in the $x-y$ plane.

For this configuration, the power emitted by the dish is independent of the mass of the DP and is given by Eq. (11.2)<fnref n="6" />

$$ p_{\text{dish}}^{\text{SHUKET}} = 1.73 \times 10^{-20} \left( \frac{\chi}{10^{-12}} \right)^2 \text{W}. \qquad (17.15) $$

In Appendix E, we show how with these experimental parameters, the procedure using the thin optical element approximation, described in Chapter 11 to compute the electric field from the dish to the fictional plane is valid for SHUKET. Indeed, we find that this approximation leads to a relative error on the electric field of order $10^{-4}$. Additionally, the calculation in Appendix E shows that the small curvature approximation and Eq. (17.14) are valid in our regime.

In the SHUKET experiment, the detection is made by the Schwarzbeck BBHA-9120-D antenna, which is sensitive to electric field frequencies from 0.8 to 18 GHz. The datasheet [20] contains several pieces of information that will be necessary for our calculations on the

<footnote n="5">⁵The relative error induced by the neglect of $O(\theta)$ is $\sim 1\%$.</footnote>

<footnote n="6">⁶Note there is a factor 2 discrepancy with what [6] indicated, due to a calculation error.</footnote>
expected amount of power received by the antenna wires, namely the gain and the antenna factor of the antenna as function of those various field frequencies. These quantities have been measured experimentally by the manufacturer. The antenna consists in a rectangular surface of area $S_{phys} = 0.25 \times 0.142 \text{ m}^2$, with $A = 0.25 \text{ m}$ the long length and $B = 0.142 \text{ m}$ the small length. This implies that for the fundamental mode of the antenna at frequency $\omega_c$, the effective size of the antenna corresponds to its physical size (in other words $A_{eff}(\omega_c) = A$ and $B_{eff}(\omega_c) = B$) and therefore the ratio between the long and small sides of the antenna is $R_{AB}(\omega_c) = A/B \approx 1.76$. We will make the assumption that the ratio $R_{AB}$ is independent of the frequency, i.e $R_{AB}(\omega_c) \equiv R_{AB} = A_{eff}(\omega)/B_{eff}(\omega)$, for any frequency $\omega > \omega_c$, such that the effective surface of the antenna at frequency $\omega$ is simply $S_{eff}(\omega) = A_{eff}(\omega)B_{eff}(\omega) = A_{eff}^2(\omega)/R_{AB}$. Then, using Eq. (11.18), the effective long and small lengths of the antenna at the frequency $\omega_U$ are simply given by

$$ \frac{A_{eff}^2(\omega_U)}{R_{AB}} = \frac{e_r \pi G(\omega_U) c^2}{\omega^2} ; \quad B_{eff}(\omega_U) = \frac{A_{eff}(\omega_U)}{R_{AB}}. \qquad (17.16) $$

These two dimensions define the effective surface area over which we need to integrate Eq. (11.24) to estimate the output of the antenna.

Therefore, we will use the electric field at the surface of the fictional plane Eq. (11.15) to compute the field power received by the antenna. This will be done in details in Chapter 18 on sensitivities.

## 17.4 MICROSCOPE

We now discuss the experimental parameters of MICROSCOPE for the detection of ULDM candidates that produce a violation of the UFF, as discussed in Chapter 12. As mentioned in Section 12.1, MICROSCOPE is a space mission that was launched to test the UFF in space [21] with a final result on the $\eta$ parameter Eq. (12.2). The full MICROSCOPE dataset used to constrain the UFF is made of 17 sessions [22]. The noise PSD of the differential acceleration of the session 404 is given by [23]

$$ S_a(f) = 2.2 \times 10^{-24} f^{-1} + 2.3 \times 10^{-17} f^4 (\text{m/s}^2)^2 / \text{Hz}, \qquad (17.17) $$

for frequencies between $10^{-5}$ Hz and 0.3 Hz. The $f^{-1}$ slope noise comes from thermal effects of the gold wire connecting the test masses to the cage, while the high frequency noise in $f^4$ is the second derivative of the position measurement white noise [21].

The UFF signal is modulated by spinning the satellite at a frequency chosen to minimize the noise, $f_{EP} \sim 3$ mHz. Note that $f_{EP}$ is in reality a linear combination of the spin frequency $f_{spin}$ and orbital frequency $f_{orb}$, with $f_{spin} \gg f_{orb} \sim 1.5 \times 10^{-4}$ Hz [21] for session 404. Three different spin frequencies exist depending on the session [24], such that one should take both contributions into account for the determination of $f_{EP}$ for other sessions. The bucket frequency of this acceleration noise corresponds approximately to $f_{bucket} \sim 30$ mHz [21, 23].

To take into account the contribution of the full experiment, i.e the 17 different sessions, we make the hypothesis that all of them have the same level of acceleration noise Eq. (17.17) while we consider the orbital period to be constant $T_{single orbit} \approx 5946$ s [22]. This hypothesis
is sufficient for the rough sensitivity analysis we do here, but will have to be revisited when
doing a complete MICROSCOPE data analysis in search for DM candidates.

Using the total number of orbits of the experiment, i.e $N_{\text{orbits}} = 1362$ [22], we find that the
total integration time is $T_{\text{int}} = T_{\text{single orbit}} \times N_{\text{orbits}} \equiv 8.1 \times 10^6$ s ($\approx 94$ days).

As an experiment measuring the differential acceleration between two test masses, the
corresponding theoretical signal is given by Eq. (12.5) projected onto the sensitive axis of the
experiment (see Eq. (12.1)). In Appendix C, we compute the projection of the galactic velocity
onto the sensitive axis of MICROSCOPE, and we find that $\hat{e}_v \cdot \hat{e}_{\text{meas}} \approx 0.85 \cos(\omega_{\text{spin}}t + \psi)$,
where $\omega_{\text{spin}} = 2\pi f_{\text{spin}}$ and $\psi$ is an irrelevant phase.

## 17.5 Atom interferometers

Similarly as for MICROSCOPE, we have shown in Chapter 13 that the sensitivity of some
AI experiments to ULDM fields are proportional to the dot product between the laser pulse
direction and the galactic velocity (see e.g. Eqs. (13.21) and (13.27)). In Appendix C, we
derive the various projections of the galactic velocity onto the laser pulses directions of those
experiments, which depend on their location on Earth, but also on their integration time.
The AI experiments under consideration are located at Stanford, USA, Fermilab, USA and
Oxford, UK. In Appendix C, we find $\hat{e}_v \cdot \hat{e}_{\text{kick}}|_{\text{Stanford}} \approx 0.43$, $\hat{e}_v \cdot \hat{e}_{\text{kick}}|_{\text{Oxford}} \approx 0.99$ and
$\hat{e}_v \cdot \hat{e}_{\text{kick}}|_{\text{Fermilab}} \approx 1.00$.

### 17.5.1 Stanford Tower

The most stringent constraint on the Eötvös parameter $\eta$ obtained from atom interferometry
experiment is achieved using a Bragg atom interferometry experiment in the Stanford Tower.
In this experiment, the relative acceleration of freely falling clouds of two isotopes of Rubid-
ium ($^{85}$Rb and $^{87}$Rb) is measured [25]. The differential interferometric phase measurement
between these two interferometers leads to a constraint on $\eta$ given by [25]

$$ \eta = (1.6 \pm 5.2) \times 10^{-12}. \qquad (17.18a) $$

Both atoms are launched upwards inside the Stanford Tower for an interferometric sequence
of total duration $2T = 1.91$ s, corresponding to the total time for the atoms to fall back [25].

Double diffraction interferometry is performed using two lasers both resonant with the
$|5^2S_{1/2}\rangle \rightarrow |5^2P_{3/2}\rangle$ $^{87}$Rb transition, i.e. their wavevectors are $k_1 \approx k_2 \approx 2\pi/\lambda$, with $\lambda = 780$ nm. Three different momentum splittings have been used: $\{4\hbar k, 8\hbar k, 12\hbar k\}$. All are in agreement with no EP violation [25]. In the following, we will consider the highest momentum transfer since this will enhance the signal searched for, which corresponds to using an effective wavevector $k_{\text{eff}} = 24\pi/\lambda$ in Eq. (13.21).

The resolution per shot on the differential acceleration is $\sigma_{\Delta a} = 1.4 \times 10^{-11} g$ [25], with $g = 9.81 \text{ m/s}^2$ the Earth gravitational acceleration on ground. We assume white noise (see below) with a corresponding acceleration noise PSD

$$ S_{\Delta a} = \frac{2\sigma_{\Delta a}^2}{f_s} \approx (7.7 \times 10^{-11} g)^2 \text{ Hz}^{-1}, \qquad (17.18b) $$
with $f_s$ the sampling frequency of the experiment, defined as $f_s = 1/T_{cycle}$ where $T_{cycle} = 15$ s is the duration of one interferometric sequence, including atom preparation, launch and free fall [25]. The raw measurement in this experiment is actually a differential phase shift $\Delta\Phi$ which is related to the differential acceleration between the atoms $\Delta a$ through $\Delta\Phi = k_{eff}T^2\Delta a$ [26]. Therefore, we can infer the PSD of the phase shift using

$$ \sqrt{S_{\Phi}} = k_{eff}T^2\sqrt{S_{\Delta a}}. \qquad (17.18c) $$

The final uncertainty of this differential acceleration measurement is mostly limited by electromagnetic effects, coming from the Bragg lasers and non homogeneous magnetic field [25]. Knowing the final uncertainty of the differential measurement (17.18a) we derive the total “effective” experiment time $T_{int}$ under our white noise assumption i.e. assuming that individual experimental cycles are uncorrelated. Then, the number of cycles $N$ can be derived as

$$ N = \frac{1}{g^2} \left( \frac{\sigma_{\Delta a}}{\sigma_{\eta}} \right)^2 \approx 77, \qquad (17.18d) $$

and

$$ T_{int} = N \times T_{cycle} \approx 1148 \text{ s.} \qquad (17.18e) $$

As a Bragg differential AI, the Stanford experiment is sensitive to ULDM fields that produce oscillations of rest mass and transition frequency of atoms, whose corresponding signal is presented in Eq. (13.21).

### 17.5.2 AION-10

The Atom Interferometer Observatory and Network (AION) is an experimental program to search for ULDM and gravitational waves in the $10^{-1}-10$ Hz range using atom interferometry [27]. AION-10 is a 10 meter-long single-photon atom gradiometer instrument which will use $^{87}$Sr atoms and that will be built in Oxford [28]. Contrary to the other experiments previously introduced, AION-10 will operate in a distant future, assuming a much better control on noise than current experiments. As a gradiometer, AION-10 will be sensitive to the signal presented in Eq. (13.26). Following [28], we will use the following experimental parameters ultimately envisaged for AION-10:

$$ T_{int} = 10^8 \text{ s}, \qquad (17.19a) $$

$$ T = 0.74 \text{ s}, \qquad (17.19b) $$

$$ S_{\Phi}^{\text{AION}}(f) = 10^{-8} \text{ rad}^2/\text{Hz}, \qquad (17.19c) $$

$$ n = 1000, \qquad (17.19d) $$

$$ \omega_{\text{Sr}} = 2.697 \times 10^{15} \text{ rad/s}, \qquad (17.19e) $$

$$ \Delta r = 4.86 \text{ m}, \qquad (17.19f) $$

$$ L = 10 \text{ m}, \qquad (17.19g) $$

respectively the interrogation time, the free evolution time, the gradiometer phase noise PSD, the number of LMT kicks, the optical transition frequency used, the gradiometer separation,
and the total size of the baseline. Note that, while a 10 m facility is used, which would imply a free fall time of $2T = \sqrt{8L/g} \sim 2.86$ s, i.e $T \sim 1.43$ s, [28] computed the optimum free evolution time as function of the other experimental parameters, in particular considering a non-zero separation between the two interferometers $\Delta r \neq 0$, reason for the difference.

### 17.5.3 SPID

We now focus on the SPID variation, which can also probe oscillations of rest mass and transition frequency of atoms through the signal presented in Eq. (13.27) (Section 13.4).

We consider first the experimental parameters from one running mode of MAGIS-100. They are listed in [29], see their Fig. 3. Their noise levels are much lower than current experiments, similar to AION-10. MAGIS-100 will use two isotopes of Strontium ($^{87}$Sr and $^{88}$Sr) in a 100-meter baseline. The transition under consideration in [29] is $|5^1S_0\rangle \rightarrow |5^3P_1\rangle$ with frequency $\omega_{Sr} = 2.73 \times 10^{15}$ rad/s. In terms of acceleration noise level and order of LMT, we consider the upgraded parameters, i.e

$$ \sqrt{S_a^{\text{MAGIS}}(f)} = 6 \times 10^{-17} \text{g}/\sqrt{\text{Hz}}, \qquad (17.20a) $$

$$ n = 1000. \qquad (17.20b) $$

In addition, for a $L = 100$ m high tower, the free-fall time of atoms is given by $2T = \sqrt{8L/g}$, which implies $T \sim 4.5$ s. Following Eq. (17.18c), the phase noise PSD is

$$ S_{\Phi}^{\text{MAGIS}}(f) = \left( \frac{n\omega_{\text{Sr}}T^2}{c} \right)^2 S_a(f) \sim 10^{-8} \text{ rad}^2/\text{Hz}, \qquad (17.20c) $$

i.e a similar phase noise PSD as AION-10. Finally, the full integration time of the experiment corresponds to 1 year of observation, $T_{int} \approx 3.16 \times 10^7$ s.

Another reason why we consider the SPID setup is to directly compare the expected sensitivity on axion and dilaton couplings of two experiments with the exact same experimental parameters but operating a gradiometer on one hand and the SPID variation on the other hand. For this matter, we will compare the sensitivity of the current version of AION-10, i.e a gradiometer, with a SPID setup using the same noise levels. In this case, in addition to the phase noise, we also include the EOM noise (which we show to be negligible in the following), which is used to shift the frequency of the laser to account for the isotope shift (~ 1 GHz) as discussed in Sec. 13.4 and depicted in Fig. 13.6. An alternative scheme would be to phase-lock two lasers with a ~ 1 GHz frequency offset. In either case, this setup generates an additional frequency fluctuation $\sigma_f(t)$ in the second beam, which comes from the EOM internal noise and from the phase noise of the GHz reference frequency. Whilst the raw EOM phase noise PSD can be approximated as $S_{\Phi}^{\text{GHz}}(f) \approx 10^{-13} (f/\text{Hz})^{-2} \text{ rad}^2/\text{Hz}$ [30–32], it enters only differentially in the SPID setup at times separated typically by $L/c$. The resulting PSD is altered by a factor $2(1 - \cos(2\pi fL/c)) \approx (2\pi fL/c)^2$, since $2\pi fL/c \ll 1$ for all frequencies $f$ of interest. Since this noise is uncorrelated with the gradiometer phase noise, the total phase noise PSD of the SPID setup is simply

$$ S_{\Phi}^{\text{SPID}}(f) = S_{\Phi}^{\text{GHz}}(f) \left( \frac{2\pi fL}{c} \right)^2 + S_{\Phi}^{\text{AION}}(f) \approx S_{\Phi}^{\text{AION}}(f), \qquad (17.21) $$
For the remaining experimental parameters, such as the total time of experiment, the flight time of atoms and the number of LMT kicks, we will assume the same values as AION-10, see Eq. (17.19). We will also consider the use of four different pairs of isotopes of alkaline-Earth neutral atoms commonly used as optical clocks (see Table 8.2) and we will assess the impact of the choice of the atoms on the ULDM sensitivity.

## 17.6 LISA

In Chapter 15, we already discussed most of the experimental parameters affecting LISA sensitivity to DM fields. LISA one-link phase noise PSD is given in Eq. (14.11). The sensitivity of a given TDI O combination to a coupling $\epsilon$ at frequency $f$ is given by [33]

$$S_O(f) = \frac{N_O(f)}{\mathcal{T}_O^2(f)}. \qquad (17.22)$$

The amplitude of the transfer functions of TDI $X_2$ combination is (at leading order in $v_{DM}/c$ and using Eq. (15.8))

$$|\mathcal{T}_X^{\text{DM}}(f)| \approx 16 \sin\left(\frac{2\pi f L}{c}\right) \sin\left(\frac{4\pi f L}{c}\right) (\hat{n}_{23} \cdot \hat{e}_v) \sin^2\left(\frac{\pi f L}{c}\right). \qquad (17.23\text{a})$$

From this, one can compute both A, E transfer functions (using Eq. (14.9))

$$|\mathcal{T}_A^{\text{DM}}(f)| \approx \frac{16}{\sqrt{2}} \left| \sin\left(\frac{2\pi f L}{c}\right) \sin\left(\frac{4\pi f L}{c}\right) \left( (\hat{n}_{12} - \hat{n}_{23}) \cdot \hat{e}_v \right) \right| \sin^2\left(\frac{\pi f L}{c}\right) \qquad (17.23\text{b})$$

$$|\mathcal{T}_E^{\text{DM}}(f)| \approx \frac{48}{\sqrt{6}} \left| \sin\left(\frac{2\pi f L}{c}\right) \sin\left(\frac{4\pi f L}{c}\right) (\hat{n}_{13} \cdot \hat{e}_v) \right| \sin^2\left(\frac{\pi f L}{c}\right) \qquad (17.23\text{c})$$

$$|\mathcal{T}_{AE}^{\text{DM}}(f)| \approx \frac{16}{\sqrt{2}} \left| \sin\left(\frac{2\pi f L}{c}\right) \sin\left(\frac{4\pi f L}{c}\right) \right| \sqrt{\left( (\hat{n}_{12} - \hat{n}_{23}) \cdot \hat{e}_v \right)^2 + \left( \sqrt{3}\hat{n}_{13} \cdot \hat{e}_v \right)^2} \sin^2\left(\frac{\pi f L}{c}\right), \qquad (17.23\text{d})$$

where at the last line, we define the A, E joint transfer function, which we used for our analysis of the realistic limit on sensitivity of LISA, in Section 15.3. The noise transfer functions of the A, E combinations are defined in Eq. (15.25), and the Sagnac $\alpha$ noise transfer function is [34]$^7$

$$N_{\alpha}(f) = 4 \sin^2\left(\frac{3\pi f L}{c}\right) \left( 12S_{\text{oms}}(f) + 4 \left( 3 - 2 \cos\left(\frac{2\pi f L}{c}\right) - \cos\left(\frac{6\pi f L}{c}\right) \right) S_{\text{acc}}(f) \right). \qquad (17.24)$$

<footnote>$^7$In [34], only the first generation noise transfer functions are presented, so in order to compute the second generation noise transfer function for the Sagnac $\alpha$ combination, one needs to add a factor $(2 \sin(3\omega L/2c))^2$, using Eq. (14.10). In addition, the modification of the optical benches of LISA that we propose to be able to be sensitive to vacuum birefringence implies a decrease in the amplitude of light (or equivalently in the number of photons $N_\gamma$) that reaches each interferometer by a factor 2 because light gets partially absorbed by the right polarizer, see Fig. 16.3. Since the OMS noise includes a contribution from shot noise whose PSD $\propto 1/N_\gamma$, we have conservatively added a factor 2 to the whole OMS contribution.</footnote>
In the following, we will make our estimates assuming a fixed arm length $L = 2.5 \times 10^9$ m, and two different integration times. For the estimates related to Chapter 15 on searches for time oscillating rest mass, we will be using the parameters used in the simulation : we will use one year of observation time $T_{obs} \sim 3.15 \times 10^7$ s, and we will compute the sensitivity of the A, E TDI combinations together, i.e we will be using the joint transfer function Eq. (17.23d) (the noise transfer functions are the same for both combinations, see Eq. (15.25)). For the estimates related to Chapter 16 on searches for vacuum birefringence, we will be assuming four years of integration time, $T_{obs} \sim 1.26 \times 10^8$ s, which is the expected mission duration [35] and we will compute the sensitivity of the Sagnac combination, as it is the optimized combination to use for this matter, as we discussed it in Chapter 16.
# Chapter 18

## Sensitivity results on ultralight dark matter couplings

In this chapter, we will present the main results of this thesis, i.e the sensitivity estimates of the experiments presented in Chapters 9, 10, 11, 12, 13, 15 and 16 with the experimental parameters and noises presented in Chapter 17.

### 18.1 Dilaton-SM couplings $d'_i$s

In Fig. 18.1, we show the existing most stringent laboratory constraints on the dilatonic couplings: Torsion balances [37] and MICROSCOPE<fnref n="1" /> [36], which are denoted as "Fifth force exp.", and hyperfine and optical clocks [38, 39] denoted as "Lab-Atomic clocks".

#### 18.1.1 MICROSCOPE

MICROSCOPE's sensitivity to the oscillating dilaton field is given by the projection of the differential acceleration Eq. (12.5a) to the sensitive axis of measurement, i.e

$$|\Delta\vec{a}(t)| = \frac{\sqrt{16\pi G \rho_{DM} v_{DM}}}{c} \left| \left( [Q_M^{Pt}]_d - [Q_M^{Ti}]_d \right) \hat{e}_v \cdot \hat{e}_{meas.}(t) \right|_{\mu\text{SCOPE}} . \quad (18.1)$$

The dot product is computed in Appendix C and the mass charges $[Q_M^{Pt}]_d$, $[Q_M^{Ti}]_d$ are defined in Eq. (8.6a) and the various partial dilatonic charges are shown in Table 8.1. The sensitivity is then computed using Eq. (5.13) and the experimental parameters of Section 17.4. The associated sensitivity of MICROSCOPE to the various dilatonic couplings is shown in black dotted line in Fig. 18.1 (denoted "MICROSCOPE (osc.)"). As mentioned above, this sensitivity is not competitive with MICROSCOPE's ability to detect a fifth force induced by the dilaton field, as it is shown in black full line in Fig. 18.1.

#### 18.1.2 Atom interferometers

For the various atom interferometric schemes, we use Eqs. (13.21), (13.26) and (13.27) and we express respectively the amplitude of the phase shift between atomic species A and B in

<footnote n="1">¹The best MICROSCOPE sensitivity comes from the static term of the field, i.e when considering a fifth force generated by Earth on the test masses, while we only focused on the oscillatory term in this thesis.</footnote>
<figure><img src="image_31.png" /><figcaption>Figure 18.1: Current constraints on all the dilatonic couplings of interest in this thesis: $d_e$ (top left), $d_{m_e} - d_g$ (bottom left), $d_{\delta\hat{m}} - d_g$ (top right), $d_{\hat{m}} - d_g$ (bottom right) from [36–39], with 95% confidence level (shown in light grey background). All these existing constraints are shown in solid lines. The expected sensitivity of all the experiments considered in this thesis are shown in dashed lines (with 68% detection threshold). The sensitivity of MICROSCOPE is shown in black, and denoted "MICROSCOPE (osc.)". The expected sensitivity of AION-10 is shown in orange dashed line while the expected sensitivity of AION-10 using the SPID variation, noted "AION-10-SPID" are respectively shown in green, light blue, red and gold dashed lines, depending on the isotope pair used. The expected sensitivity of MAGIS-100 is shown in dark blue. Finally, the expected sensitivity of LISA is shown in pink and purple respectively when the DM velocity is assumed as a fixed and free parameter (see text).</figcaption></figure>

differential two-photon transition AI, gradiometers and the SPID setup, with respectively AION-10 and MAGIS-100 experimental parameters. The experimental parameters of these various experiments are discussed in Sections 17.5.1, 17.5.2, 17.5.3. Then, the different phase shifts read

$$ \Delta\Phi_{^{87}\text{Rb},^{85}\text{Rb}}^{\text{Stanford}} \approx \frac{4\sqrt{16\pi G\rho_{\text{DM}}v_{\text{DM}}k_{\text{eff}}}}{\omega_{\phi}^2 c} \left| ([Q_M^{^{87}\text{Rb}}]_d - [Q_M^{^{85}\text{Rb}}]_d) \hat{e}_v \cdot \hat{e}_{\text{kick}} \right|_{\text{Stanford}} \sin^2\left(\frac{\omega_{\phi}T}{2}\right), \qquad (18.2a) $$

$$ \Delta\Phi_{^{87}\text{Sr}}^{\text{AION-10}} \approx \frac{4n\omega_{^{87}\text{Sr}}^0 \Delta r \sqrt{16\pi G\rho_{\text{DM}}[Q_{\omega}^{^{87}\text{Sr}}]_d}}{\omega_{\phi}c} \sin^2\left(\frac{\omega_{\phi}T}{2}\right), \qquad (18.2b) $$

$$ \Delta\Phi_{^{88}\text{Sr},^{87}\text{Sr}}^{\text{MAGIS-100}} \approx \frac{4n\omega_0\sqrt{16\pi G\rho_{\text{DM}}v_{\text{DM}}}}{\omega_{\phi}^2 c^2} \left| ([Q_M^{^{88}\text{Sr}}]_d - [Q_M^{^{87}\text{Sr}}]_d) \hat{e}_v \cdot \hat{e}_{\text{kick}} \right|_{\text{Fermilab}} \sin^2\left(\frac{\omega_{\phi}T}{2}\right), \qquad (18.2c) $$

$$ \Delta\Phi_{\text{AB}}^{\text{AION-10-SPID}} \approx \frac{4n\omega_0\sqrt{16\pi G\rho_{\text{DM}}v_{\text{DM}}}}{\omega_{\phi}^2 c^2} \left| ([Q_M^A]_d - [Q_M^B]_d) \hat{e}_v \cdot \hat{e}_{\text{kick}} \right|_{\text{Oxford}} \sin^2\left(\frac{\omega_{\phi}T}{2}\right), \qquad (18.2d) $$

where the $[Q]_d$ charges contain partial dilatonic mass and/or frequency charges defined in Eqs. (8.6a) and (8.7). Note that for the SPID variation, we let the atomic species A, B free
because we will derive expected sensitivities for various isotope pairs.

In Fig. 18.1, the sensitivity of the Stanford tower experiment to the various dilatonic couplings is not visible, because it is less sensitive than the best existing constraints (for example, it reaches $d_e \sim 2 \times 10^3$ at best, around $m_\phi c^2 \sim 10^{-16}$ eV, almost 8 orders of magnitude above the constraint from MICROSCOPE). As it was discussed in [28], AION-10 would improve the current constraint on the $d_{m_e} - d_g$ coupling, over a mass range approximately between $7 \times 10^{-16}$ eV and $4 \times 10^{-15}$ eV.

With the same experimental parameters as AION-10, operating the SPID setup (denoted "AION-10-SPID" in Fig. 18.1) would improve the current constraints on various dilatonic couplings. By respectively using Ca and Hg isotopes pairs, the constraint on $d_e$ would be improved by a factor ~2 and 3 respectively, over a mass range covering four orders of magnitude (approximately from $10^{-19}$ eV to $10^{-15}$ eV), compared to MICROSCOPE, the current best constraint in this mass range. Regarding the $d_{m_e} - d_g$ coupling, the use of Ca and Hg isotopes respectively would improve the best constraints, by a factor 2.5 and 1.5 respectively over the same mass range. Finally, all pairs of isotopes presented would improve the current best constraint on $d_{\delta m} - d_g$ by one order of magnitude, depending on the isotope pair used, over a mass range covering more than 4 orders of magnitude (from $10^{-19}$ eV to $3 \times 10^{-15}$ eV).

One can notice also that operating both gradiometers and SPID at AION-10 would give complementary sensitivities for the search of the $d_{m_e} - d_g$ coupling of the dilaton. Indeed, while the gradiometer would improve the current constrain in the ~$7 \times 10^{-16} - 3 \times 10^{-15}$ eV/$c^2$ mass range, the sensitivity of the SPID setup at lower mass is better, as described in the last paragraph. In addition, SPID would be more sensitive than AION-10 to the $d_e, d_{\hat{m}} - d_g$ and $d_{\delta m} - d_g$ couplings in the full range of masses of interest (i.e lower than $10^{-14}$ eV/$c^2$).

Regarding the sensitivity of MAGIS-100 operating the SPID setup, one can notice that it would overall give the best constraint at low masses. One can notice that MAGIS-100 would reach better sensitivity on most couplings, compared to the AION-10-SPID variation. The reason is that for low masses, such that $\omega_\phi T \ll 1$, the signal Eq. (18.2d) is quadratic in the free fall time $T$, and MAGIS-100 using a much longer baseline than AION-10, the free fall time is roughly 6 times longer for MAGIS-100 (as mentioned previously, the free fall time in the case of AION-10 is smaller than what the facility size allows), resulting in increased signal. However, note that we made the calculations assuming three years of integration time for AION-10 (from [28]) and the SPID variation, but only one year for MAGIS-100 (from [29]). This is why the difference in sensitivity is roughly one order of magnitude between the two dual-Sr configurations, and not more. Note that compared to the sensitivity curves on dilaton couplings presented in [29], but using the gradiometer setup of MAGIS-100, the curve presented in this paper would constrain a larger DM mass range. Indeed, while the former has a peak sensitivity around a mass $10^{-15}$ eV and quickly loses sensitivity for lower masses, the SPID-like setup of MAGIS-100 would have a constant sensitivity of the same order of magnitude for 3 orders of magnitude of mass ($10^{-19} - 10^{-16}$ eV).

### 18.1.3 LISA

In Chapter 15, we have seen that despite the fact that LISA can probe dilaton-SM couplings through oscillation of the rest mass of the freely falling test masses, the DM velocity cannot be constrained accurately and this leads to large uncertainty on such couplings. Now, we
<figure><img src="image_32.png" /><figcaption>Figure 18.2: On the left, we show the sensitivity of LISA to the general coupling $\epsilon_{min}$ when the velocity is assumed fixed (blue curve) and when it is a free parameter (orange curve). One can notice a deterioration of the sensitivity when the velocity is a free parameter of the Bayesian analysis. On the right, we show the ratio of the orange to the blue curves. At low frequency, the ratio is ~ 2.5, and when the frequency becomes $f > 10^{-1}$ Hz, this ratio starts decreasing because the argument of the cosine in Eq. (18.4) becomes important (see text.).</figcaption></figure>

might wonder what would be the effect of such uncertainty in the regions of the parameter space where LISA can probe the smallest couplings, i.e when SNR=1.

Following Eq. (15.17b), the sensitivity of the combination A, E on the coupling $\epsilon$ without correlation with other parameters is (at SNR=1)

$$ \epsilon_{\min}(f) = \frac{1.5 \times 2\pi f c^2}{\sqrt{16\pi G \rho_{DM} v_{DM}}} \sqrt{\frac{S_{A,E}(f)}{T_{\text{obs}}}} \quad \text{or} \quad \epsilon_{\min}(f) = \frac{2\pi f c^2}{\sqrt{16\pi G \rho_{DM} v_{DM}}} \sqrt{\frac{S_{A,E}(f)}{T_{\text{obs}} \tau(f)}}, \quad (18.3) $$

depending on if $T_{\text{obs}}$ is smaller or larger than $\tau(f)$ (see Eq. (5.13)). Now, we estimate the sensitivity of the experiment, taking into account the correlation between the coupling and the DM velocity (see Section 15.3). When SNR=1, the likelihood Eq. (15.47b) reduces to<fnref n="2" />

$$ \log \mathcal{L}(f, \epsilon^M) = -\frac{1}{2} \left( 1 + \left( \frac{v_{\text{DM}}^M \epsilon^M}{v_{\text{DM}}^D \epsilon_{\min}(f)} \right)^2 - 2 \frac{v_{\text{DM}}^M \epsilon^M}{v_{\text{DM}}^D \epsilon_{\min}(f)} \cos \left( \frac{2\pi f |\vec{x}_{\text{AU}}| \cos(\beta)}{c^2} (v_{\text{DM}}^D - v_{\text{DM}}^M) \right) \right), \quad (18.4) $$

where the parameters with superscript D mean true values injected in the simulation, and the ones with superscript M mean model values. Similarly as what was done in Section 15.3, we will now compute numerically the standard deviation on the posterior distribution of $\epsilon^M$, in order to have a new sensitivity estimate of LISA.

In the following, we consider the same specific values of $\beta^D$, $v_{\text{DM}}^D$ as in Chapter 15 (as a reminder, we assume the mean values of velocity and sky localization from the velocity distribution Eq. (5.7a), i.e. ($v_{\text{DM}}^D, \beta^D$) = (10$^{-3}$c, 1.046 rad)). In order to infer $\sigma_{\epsilon_{\min}}(f)$, the uncertainty on the posterior distribution of $\epsilon_{\min}$ at a given frequency $f$, we numerically

<footnote n="2">²At frequencies where $T_{\text{obs}} < \tau(f)$, the noise in Eq. (15.47b) scales as $\sqrt{N_A/\sqrt{T_{\text{obs}}\tau(f)}}$, and we will obtain the same result.</footnote>
integrate the marginalized likelihood Eq. (18.4) over $v_{DM}^M$, following Eq. (15.49) and using an uniform prior on $\epsilon^M$ with (arbitrary) bounds $[0, 100 \epsilon_{min}(f)]$. This gives us the posterior distribution $P(\epsilon^M(f))$ which we again integrate over $\epsilon^M$ (as in Eq. (15.50)) to obtain $\sigma_{\epsilon_{min}}(f)$.

In the left panel of Fig. 18.2, we show sensitivities of LISA to $\epsilon$ when $v_{DM}$ is fixed, i.e from Eq. (18.3) and when $v_{DM}$ is a free parameter, i.e from the width of the posterior distribution Eq. (15.49) (but with the likelihood given in Eq. (18.4)). One can notice that at low masses, letting $v_{DM}$ free decreases the sensitivity by a factor ~ 2.5 compared to the fixed $v_{DM}$ case (right panel of Fig. 18.2). This difference is entirely determined by the cosine term in Eq. (18.4). Indeed, as mentioned in Chapter 15, it is the accuracy on the determination of the velocity $v_{DM}^M$ which impacts the width of the coupling $\epsilon^M$ posterior distribution. In particular, in order to maximize the likelihood, one needs to get the log $\mathcal{L}$ Eq. (18.4) as close to 0 as possible, i.e one needs the cosine term to be close to 1. This means that the argument of the cosine

$$ \frac{2\pi f |\vec{x}_{AU}| \cos(\beta)}{c^2} (v_{DM}^D - v_{DM}^M) \sim \frac{10^3 \Delta v}{c} \left(\frac{f}{1 \text{ Hz}}\right), \qquad (18.5a) $$

must be close to 0, where $\Delta v = v_{DM}^D - v_{DM}^M$. We can consider that the accuracy on the velocity parameter starts to increase when $\Delta v \le \sigma_v$, where $\sigma_v$ is the width of the velocity prior distribution. This implies that the accuracy on the coupling starts to increase for frequencies

$$ \frac{10^3 \sigma_v f}{c} \ll 2\pi \Rightarrow f \ll 10 \text{ Hz}, \qquad (18.5b) $$

i.e when $f \gtrsim 0.1$ Hz. This is what is shown on the right panel of Fig. 18.2.

Since $\epsilon = [Q_M^{TM}]_d$, and $[Q_M^{TM}]_d = \sum_i [Q_{M,i}^{TM}]_d d_i / \sqrt{2}$ (from Eq. (8.6a)), the sensitivity of LISA to the coupling $d_i$ reads

$$ d_i = \frac{\sqrt{2}\sigma_{\epsilon_{min}}}{[Q_{M,i}^{TM}]_d}. \qquad (18.6) $$

Using Eq. (17.22) together with Eqs. (15.25) and (17.23d), we can now derive the sensitivity curves. In Fig. 18.1, we show respectively in pink and purple the sensitivity of LISA to the different dilatonic $d_i$ couplings when the velocity is assumed as a fixed or free parameter. Similarly as for the $\epsilon$ coupling shown in Fig. 18.2, there is an approximate factor ~ 2.5 difference. One can notice that LISA will be competitive with the other futuristic experiments for all dilatonic couplings. In particular on the $d_{m_e} - d_g$, $d_{\hat{m}} - d_g$ couplings, it will reach unconstrained regions of the parameter space, improving respectively fifth force constraints by a factor ~ 10, ~ 5 over one order of magnitude of mass (~ $10^{-17}$ eV - $10^{-16}$ eV), in the case of fixed velocity. The reason it is much more sensitive to this coupling than AI experiments comes from the fact that AI experiments mainly use isotopes $I1, I2$, while LISA makes use of the finite light travel time with a single atomic species. Isotopes have very close quark contribution to their rest mass, and therefore the differential mass charge $\Delta[Q_{M,\hat{m}}]_d = [Q_{M,\hat{m}}^{I1}]_d - [Q_{M,\hat{m}}^{I2}]_d$ is very small, and the corresponding sensitivity decreases significantly.
<figure><img src="image_33.png" /><figcaption>Figure 18.3: Current lab constraints on $1/f_a$ axion coupling [40–43] (from [44]) (the constraint from [42] has been rescaled for consistent value of local DM energy density) (shown in solid lines). The new sensitivity estimates resulting from this thesis are shown in dashed lines, i.e from Stanford, AION-10 and MAGIS-100 atom interferometry experiments, EP classical test from MICROSCOPE and LISA. The expected sensitivity of the AION-10-SPID-like experiment is shown in four different colors, each using four different pairs of isotopes, denoted "AION-10-SPID".</figcaption></figure>

## 18.2 Axion-gluon coupling $f_a^{-1}$

The current best laboratory constraints on the axion-gluon coupling are from [40–43] and are shown in solid red in Fig. 18.3.

### 18.2.1 MICROSCOPE

Similarly as in the previous section, we can express the amplitude of the differential acceleration between two test-masses A and B as (using Eqs. (12.5b))

$$|\Delta\vec{a}(t)| = \frac{16\pi G \rho_{\text{DM}} v_{\text{DM}} E_P^2}{f_a^2 \omega_a c^2} \left| ([Q_M^{\text{Pt}}]_a - [Q_M^{\text{Ti}}]_a) \hat{e}_v \cdot \hat{e}_{\text{meas.}}(t) \right|_{\mu\text{SCOPE}}. \quad (18.7)$$

The axionic mass charges $[Q_M^{\text{Pt}}]_a$ and $[Q_M^{\text{Ti}}]_a$ can be found in Table 8.3. Similarly as in the previous section, the sensitivity to the axion-gluon coupling is then computed using Eq. (5.13)
and the experimental parameters of Section 17.4. In Fig. 18.3, the expected sensitivity of MICROSCOPE is shown by the orange full line. This curve presents two breaking point frequencies. The first one $f \sim 1$ mHz corresponds to half of $f_{\text{spin}} \sim 3$ mHz. As discussed in Appendix C, we consider two different frequency regimes, depending on whether the signal frequency is higher or lower than $f_{\text{spin}}$. In the axion case, the signal Eq. (12.5b) oscillates at twice the axion field frequency, therefore the breaking point is $f_{\text{spin}}/2$. The second breaking point arising at $f \sim 15$ mHz corresponds to half of the bucket frequency of the acceleration noise PSD, for the same reason as above. Note that there is no breaking point frequency associated with the coherence time of the field because it would arise at a frequency larger than the bandwidth of the noise PSD, at around 500 mHz. As it can be seen from this curve, a complete re-analysis of MICROSCOPE's data would enable to constrain a new region of the parameter space, over approximately two orders of magnitude in mass, compared to existing laboratory experiments.

### 18.2.2 Atom interferometers

For the atom interferometry experiments, we do the same procedure as in Section 18.1.2 and we find

$$ \Delta\Phi^{\text{Stanford}}_{^{87}\text{Rb},^{85}\text{Rb}} \approx \frac{16\pi G \rho_{\text{DM}} E_p^2 v_{\text{DM}} k_{\text{eff}}}{\omega_a^3 f_a^2 c^2} \left| ([Q_M^{87\text{Rb}}]_a - [Q_M^{85\text{Rb}}]_a) \hat{e}_v \cdot \hat{e}_{\text{kick}} \right|_{\text{Stanford}} \sin^2(\omega_a T), \quad (18.8a) $$

$$ \Delta\Phi_{^{87}\text{Sr}}^{\text{AION}-10} \approx \frac{32\pi G \rho_{\text{DM}} E_p^2 n \omega_a^0 \Delta r [Q_\omega^{87\text{Sr}}]_a}{\omega_a^2 f_a^2 c^3} \sin^2(\omega_a T), \quad (18.8b) $$

$$ \Delta\Phi_{^{88}\text{Sr},^{87}\text{Sr}}^{\text{MAGIS}-100} \approx \frac{16\pi G \rho_{\text{DM}} E_p^2 n \omega_0 v_{\text{DM}}}{\omega_a^3 f_a^2 c^3} \left| ([Q_M^{88\text{Sr}}]_a - [Q_M^{87\text{Sr}}]_a) \hat{e}_v \cdot \hat{e}_{\text{kick}} \right|_{\text{Fermilab}} \sin^2(\omega_a T), \quad (18.8c) $$

$$ \Delta\Phi_{\text{AB}}^{\text{AION}-10-\text{SPID}} \approx \frac{16\pi G \rho_{\text{DM}} E_p^2 n \omega_0 v_{\text{DM}}}{\omega_a^3 f_a^2 c^3} \left| ([Q_M^A]_a - [Q_M^B]_a) \hat{e}_v \cdot \hat{e}_{\text{kick}} \right|_{\text{Oxford}} \sin^2(\omega_a T). \quad (18.8d) $$

In Fig. 18.3, we present the sensitivity of the Stanford Tower [25], AION-10 experiment [28], MAGIS-100 experiment [29] and the SPID AI setup with AION-10 experimental parameters, denoted "AION-10-SPID".

One can notice that MICROSCOPE is approximately two to three orders of magnitude more sensitive than Stanford. This is consistent considering that the signal is quadratic in $1/f_a$, that MICROSCOPE constrains the Eötvös parameter $\eta$ three orders of magnitude better than the Stanford experiment and that the difference in axionic mass charges of species used in the experiments is 2 orders of magnitude larger for MICROSCOPE.

Contrary to the gradiometer setup of AION-10 which is almost insensitive to the axion-gluon coupling (at leading order, the sensitivity is proportional to the axionic frequency charge of the optical transition of Sr, which is 0), the SPID variation or AION-10 would have the largest sensitivity to this coupling compared to existing laboratory experiments. Such an experiment would improve the current lab constraint [40] by 2 orders of magnitude over a mass range of 4 orders of magnitude ($10^{-17} - 10^{-13}$ eV). In this mass range, the MAGIS-100
experiment, which uses the same setup has also a very interesting sensitivity on the coupling, which is comparable to the one of AION-10 in the SPID setup.

### 18.2.3 LISA

Similarly as for the dilatonic couplings, LISA will be able to probe the axion-gluon coupling through Doppler effects Eq. (15.17b). In the same way as what we did for the dilatonic couplings in Section 18.1.3, we will derive two sensitivity curves. The first one is computed assuming no correlations between parameters, where the constraint on the dimensionless coupling $E_P/f_a$ is given by

$$ \left(\frac{E_P}{f_a}\right)_{\min}^2(f) = \frac{1.5 \times (2\pi f)^2 c^3}{8\pi G \rho_{\text{DM}} v_{\text{DM}}} \sqrt{\frac{S_{A,E}(2f)}{T_{\text{obs}}}} \quad \text{or} \quad \left(\frac{E_P}{f_a}\right)_{\min}^2(f) = \frac{(2\pi f)^2 c^3}{8\pi G \rho_{\text{DM}} v_{\text{DM}}} \sqrt{\frac{S_{A,E}(2f)}{\sqrt{T_{\text{obs}}\tau(f)}}}, \qquad (18.9) $$

taking into account the fact that the signal has a different normalization amplitude (from Eq. (15.3c)) and that it oscillates at twice the axion frequency. The second sensitivity curve will take into account the correlations between parameters. The signal being quadratic in $1/f_a$, the lack of precision on the galactic velocity induces an uncertainty

$$ \sigma_{\epsilon} \equiv \sigma_{E_P^2/f_a^2}(f). \qquad (18.10a) $$

Note that here, we choose a dimensionless random variable parameter $(E_P/f_a)^2$, for consistency with our analysis involving the dimensionless parameter $\epsilon$. As we are interested in the uncertainty on $E_P/f_a$, we make the assumption that, at the frequency $f$, the random variable $E_P/f_a(f)$ follows a normal distribution with zero mean<fnref n="3" /> and deviation $\sigma_{E_P/f_a}(f)$ such that we have

$$ \sigma_{E_P^2/f_a^2}(f) = \sqrt{2}\sigma_{E_P/f_a}^2(f) = \sqrt{2}E_P^2\sigma_{1/f_a}^2(f) \qquad (18.10b) $$

$$ \rightarrow \sigma_{1/f_a}(f) = \frac{1}{2^{1/4}} \frac{\sqrt{\sigma_{E_P^2/f_a^2}(f)}}{E_P}, \qquad (18.10c) $$

where we used the fact that $E_P$ is constant. To get a curve of $\sigma_{1/f_a}(f)$ as function of the frequency, we perform the same iteration as with the dilatonic couplings, i.e we compute the posterior distribution of the $\epsilon_{\min} = (E_P/f_a)_{\min}^2$ variable, where $\epsilon_{\min}$ is computed following (18.3).

In Fig. 18.3, we show the sensitivity of LISA to $1/f_a$ coupling when the DM velocity is fixed at $v_{\text{DM}}$ (LISA, fixed $v_{\text{DM}}$) and when the velocity is taken as a free parameter, i.e for $\sigma_{1/f_a}$ as given by Eq. (18.10c) (LISA, free $v_{\text{DM}}$). One can notice a slight difference in sensitivity between the two curves, compared to the dilaton case, and this is because the axion signal is quadratic in the coupling while the dilaton signal is linear. At axion masses between $10^{-17}$ and $10^{-14}$ eV, LISA would be the most sensitive experiment to the axion-gluon coupling, in particular around $10^{-16}$ eV, where the sensitivity is improved by a factor ~3 compared to the best AI experiments (MAGIS and AION-10-SPID).

<footnote n="3">³Here, we are interested in computing an upper limit on the coupling in case of no detection, therefore, we can assume that the mean of the distribution of such coupling is 0.</footnote>
## 18.3 Axion-photon coupling $g_{a\gamma}$

In Fig. 18.4, we show in red the current best laboratory constraints on the axion-photon coupling, from [45–80].

<figure><img src="image_34.png" /><figcaption>Figure 18.4: Current laboratory constraints on the axion-photon coupling $g_{a\gamma}$ (from [44]). The expected sensitivity of LISA through vacuum birefringence is shown in black.</figcaption></figure>

### 18.3.1 DAMNED and optical fibers

The respective phase shift induced by the axion-photon coupling in *DAMNED* or the optical fiber can be found in Eqs. (9.7) and (9.10) respectively. All the experimental parameters of interest are described in Section 17.1. As for the other sensitivity curves, using Eq. (5.13), we compute the corresponding sensitivity on the axion-photon coupling $g_{a\gamma}$ of *DAMNED* and the optical fiber, which are respectively presented in red and blue in Fig. 18.5. As it can be noticed, the smallest coupling that they can reach is $O(10^{-1}, 1)$ GeV$^{-1}$ respectively, which is very far away from the best existing constraints in this range (which reach $\sim 10^{-8}$ GeV$^{-1}$, see e.g. Fig. 18.4).
Therefore, both experiments in their current form are not competitive. For DAMNED, this is because the signal resonances happen at modes of the cavity, where $\omega_a l/c = n\pi$, $n \in \mathbb{N}$, which correspond to GHz frequencies, while the apparatus is sensitive to much lower frequencies, from 10 kHz to 1 MHz, as we described in Chapter 17. One could overcome this by modulating the signal with another periodic signal of frequency comparable to the axion one, e.g. with a rotating waveplate. Note that even in the case where the resonances of the cavity would be visible, the associated sensitivity would reach $g_{a\gamma} \sim 10^{-5} \text{ GeV}^{-1}$, which is still far from the best existing constraints in this mass range (see Fig. 18.4).

<figure><img src="image_35.png" /><figcaption>Figure 18.5: Expected sensitivity of DAMNED (red curve) and a 86 km long fiber (blue curve) to $g_{a\gamma}$. Compared to already existing experiments (see Fig. 18.4), they are not competitive.</figcaption></figure>

### 18.3.2 LISA

As mentioned in Chapter 16, LISA could be sensitive to vacuum birefringence induced by the axion-photon coupling, by a slight modification of its optical benches. Here, we estimate the sensitivity of LISA to such coupling using the signal of the Sagnac $\alpha$ TDI combination $|s_{\text{DM}}(f)|$ from Eq. (16.21). We first compute the signal one-sided PSD at frequency $f$

$$S_s(f) = \left( \frac{\sqrt{8\pi G\rho_{\text{DM}}E_P g_{a\gamma}}}{\pi\nu_0 c} \right)^2 \sin^4 \left( \frac{3\pi f L}{c} \right), \quad (18.11)$$

bearing in mind that the signal is oscillating at frequency $f$, such that the amplitude of the signal power is $|s_{\text{DM}}(f)|^2/2$. Using the Sagnac $\alpha$ noise transfer function defined in Eq. (17.24) and Eq. (5.13), we can derive the expected sensitivity of the Sagnac combination to $g_{a\gamma}$. This is what is shown in black in Fig. 18.4 with four years of data. One can notice that LISA would improve the current laboratory constrain on $g_{a\gamma}$ for axion masses between $\sim 4 \times 10^{-19}$ to $4 \times 10^{-15}$ eV, by several orders of magnitude. In particular, between $10^{-17}$ and $10^{-16}$ eV, LISA would reach $g_{a\gamma} \sim 10^{-12} \text{ GeV}^{-1}$, improving by more than 4 orders of magnitude the current best laboratory constraint at these masses.

We now discuss quickly dichroism effects. As a reminder, in the case of LISA, the phase shift (between the elliptical polarization and the initial linear one) induced by the axion-photon coupling $g_{a\gamma}$ depends on the travelled distance $L$ inside a constant magnetic field $B_0$ as

$$\phi(L) = \frac{\hbar c g_{a\gamma}^2 B_0^2}{\mu_0 2q^2} (qL - \sin(qL)). \quad (18.12)$$

In LISA, the laser beams are operating with a wavelength $\lambda = 1064$ nm [81], which means the photon energy is $E_\gamma \approx 1$ eV. As it was discussed in Chapter 16, the axion and photon have equal energy for the photon-axion conversion to work in a static magnetic field, i.e $E_\gamma = \hbar k_\gamma c = \sqrt{(k_a c)^2 + (m_a c^2/\hbar)^2} = \hbar\omega_a$. This means that if the axion field is DM, the mass of the field is roughly equal to its Compton frequency (using Eq. (7.29b)), implying that LISA
is essentially only sensitive to DM mass of 1 eV. In addition, the $q$ parameter reduces to

$$q = |k_a - k_\gamma| = \frac{\omega_a}{c} \left| \frac{v_{\text{DM}}}{c} - 1 \right| \approx \frac{\omega_a}{c} \equiv \frac{m_a c}{\hbar}, \qquad (18.13)$$

i.e it corresponds to the wavevector of light, and we can approximate the phase shift by

$$\phi(L)_{\text{DM}} \approx \frac{\hbar c}{\mu_0} \frac{g_{ay}^2 B_0^2 L}{2q} \approx 2.5 \times 10^{-35} \left( \frac{g_{ay}}{10^{-10} \text{GeV}^{-1}} \right)^2 \text{rad}, \qquad (18.14)$$

since $qL \gg 1$ and where we used a solar magnetic field of $B_0 \sim 3.5$ nT at 1 astronomical unit [82]. For demonstration, let us relax the constraint that axion is DM, i.e we only permit the existence of the axion-photon coupling. In that case, axion can be relativistic, which implies that $\hbar\omega_a \gg m_a c^2$ (we are now sensitive to any axion mass much below 1 eV), and

$$q = \left| \sqrt{\left(\frac{\omega_a}{c}\right)^2 - \left(\frac{m_a c}{\hbar}\right)^2} - k_\gamma \right| \approx \frac{(m_a c^2)^2 k_\gamma}{2E_\gamma^2}, \qquad (18.15)$$

where we expanded the square root in small $m_a c^2/\hbar\omega_a$. For comparison, assuming an axion mass $m_a c^2 = 1$ neV, such that $qL \ll 1$, the phase shift becomes

$$\phi(L)_{\overline{\text{DM}}} \approx \frac{\hbar c}{12\mu_0} g_{ay}^2 B_0^2 qL^3 \approx 3.4 \times 10^{-22} \left( \frac{g_{ay}}{10^{-10} \text{GeV}^{-1}} \right)^2 \text{rad}. \qquad (18.16)$$

This phase shift does not oscillate with time, and therefore this effect will a priori not be visible by LISA.

## 18.4 Dark photon-photon coupling $\chi$

### 18.4.1 Rydberg atoms in microwave cavity

The sensitivity of the experiment involving Rydberg atoms inside a microwave cavity for the detection of the kinetic mixing coupling $\chi$, obtained considering all experimental parameters described in Table 17.1 and respectively with $\{f_s = 1 \text{ kHz}, P_{\text{RIN}} = 10^{-15}\}$ and $\{f_s = 100 \text{ Hz}, P_{\text{RIN}} = 10^{-13}\}$ is presented by the blue and orange curves of Fig. 18.6. One can clearly see the sensitivity peaks arising from the cavity’s odd resonances. At these frequencies, the applied field amplitude $X_A$ takes the smallest value (see Eq. (17.13)), as shown in Eq. (17.11b), in order to minimize Eq. (17.6). Eq. (17.11b) 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. (17.9). 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 $\omega_U = \omega_A + \pi f_s$ the optimum amplitude of $X_A$ is $\sim 18 \text{ V/m}$, whose corresponding experimental sensitivity is $\chi \sim 10^{-13}$ in the modest scenario, as shown in Fig. 18.6. 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 chapter, from the approximate expression of the signal contribution Eq. (17.10b), we have $S(\omega_U, X_A) \approx 0$ for $\frac{\omega_U L}{c} = 4\pi + 2\pi n, n \in \mathbb{N}$ accounting
<figure><img src="image_36.png" /><figcaption>Figure 18.6: Dark photons-photons kinetic mixing coupling $\chi$ parameter space, with cosmological constraints from CMB (from [44, 83]). The expected sensitivity of the experiment involving Rydberg atoms in a microwave cavity, with "optimistic" parameters of the system described in Table 17.1 is shown in blue, denoted "RYDBERG (optimistic)", while the sensitivity with the "modest" parameters of Table 17.1 is shown in orange, denoted "RYDBERG (modest)" (see text). In green is the original constraint on the kinetic mixing parameter from the SHUKET experiment obtained in [6]. In red is shown the updated constraint using the same experimental parameters and data but considering a realistic modeling of the experiment in the analysis, following Eq. (18.28). The sensitivity curve with optimized experimental parameters is shown in dashed orange.</figcaption></figure>

for the loss of sensitivity. In both scenarios presented here (modest and optimistic), one can see from Fig. 18.6 that the experiment setup proposed here would improve the current constraint on the coupling $\chi$ compared to cosmological and astrophysical observations (light grey curve, from CMB [84]).

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 $T_{obs} = 60$ 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. (17.11b), 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 maximise 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 Section 17.2.2, 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 [85], 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. (17.9). This way, the optimistic curve presented in Fig. 18.6 would be achievable.

### 18.4.2 SHUKET

In the previous chapter, we have shown that one can use the results of Chapter 11 for the SHUKET experiment, in order to infer an updated sensitivity curve. This is what we do in this section. We remind that the idea here is to model more accurately the experiment, taking into account diffraction and mode-matching effects.

#### Propagation of the field from the fictional plane to the antenna

The electric field induced by the dish at the location of the antenna is provided by Eq. (11.15), which is not solvable analytically. In order to simplify it and get an analytical expression, we will make different approximations:

* The distance between the fictional plane and the antenna needs to be much larger than the typical size of the dish, i.e. $|\Delta z| \gg \rho' \le r$ (For SHUKET, $|\Delta z| = R - a \sim R = 32$ m, while $r \sim 0.618$ m).

* The distance between the fictional plane and the antenna needs to be much larger than the typical size of the antenna, i.e. $|\Delta z| \gg \rho$ (For SHUKET, the largest dimension of the antenna is $A = 0.25$ m $\ll R$).

* The last approximation is known as the far field approximation (FFA). For the DM Compton frequency under consideration, i.e $f_U = 6$ GHz, $kL \sim k|\Delta z| \gg 1$ and we can safely neglect the factor $-1$ in Eq. (11.15).

The first two approximations simplify the distance $L$ between any point on the fictional plane $(\rho', \phi', R - a)$ and any point on the antenna (whose center is located at the origin of our coordinate system) $(\rho, \phi, 0)$ to

$$L \approx |\Delta z| + \frac{\rho^2 + \rho'^2 - 2\rho\rho' \cos(\phi - \phi')}{2|\Delta z|} \quad (18.17)$$

Using this expression of $L$ as well as the FFA, we can express Eq. (11.15) as

$$\vec{U}_{\text{dish}}^{\text{SHUKET}}(\rho, \Delta z) \approx \frac{\omega_{U}^{2} \chi e^{i\Phi(\rho, \Delta z)}}{\Delta z c} \begin{pmatrix} Y_{x} \\ Y_{y} \\ 0 \end{pmatrix} \int_{0}^{r} d\rho' \rho' e^{-i\varphi(\rho', \Delta z)} J_{0}\left(\frac{k\rho\rho'}{|\Delta z|}\right), \quad (18.18a)$$

where $J_0$ is the Bessel function of the first kind of order 0 and where the integral is performed over the radius of the fictional plane which closes the dish, where the dependence on the angle $\phi$ disappeared by spherical symmetry and with

$$\varphi(\rho', \Delta z) = \frac{k\rho'^2}{2} \left( \frac{1}{R} + \frac{1}{\Delta z} \right) \quad (18.18b)$$

$$\Phi(\rho, \Delta z) = k \left( \frac{r^2}{2R} - \Delta z - \frac{\rho^2}{2\Delta z} \right). \quad (18.18c)$$
One can find an analytical solution for the last integral in the case where $z_{\text{ant}} = 0$. We expand the exponential inside the integral of Eq. (18.18a), since the argument is much smaller than 1. Indeed, for the parameters of SHUKET, we have $\Delta z = -R + a$ (i.e $z_{\text{ant}} = 0$) and

$$ \frac{k\rho'^2}{2} \left( \frac{1}{R} + \frac{1}{\Delta z} \right) \approx \frac{k\rho'^2 a}{2R^2} < \frac{kr^2 a}{2R^2} = 1.4 \times 10^{-4} \qquad (18.19) $$

Then, the integrand of Eq. (18.18a) becomes

$$ \rho'(1 - i\epsilon\rho'^2)J_0\left(\frac{k\rho\rho'}{|\Delta z|}\right) \qquad (18.20) $$

at first order in the small parameter $\epsilon\rho'^2$, where $\epsilon = ka/2R^2$. Then, the integral is analytically calculable and the electric field reads

$$ \vec{U}_{\text{dish}}^{\text{SHUKET}}(\rho, \Delta z) \approx \frac{r\omega_U\chi}{\rho} \begin{pmatrix} Y_x \\ Y_y \\ 0 \end{pmatrix} e^{i\Phi(\rho)} \left( J_1(x) - i \frac{ar\Delta z}{2\rho R^2} (2J_2(x) - xJ_3(x)) \right), \qquad (18.21a) $$

where $x = rk\rho/|\Delta z|$. One can verify easily that, with the set of parameters considered, the second term in Eq. (18.21a) containing the Bessel functions of order 2 and 3 is smaller by a factor ~ $10^6$ compared to the other term $\propto$ the Bessel function of order 1. Therefore, we can simplify the expression of the field as

$$ \vec{U}_{\text{dish}}^{\text{SHUKET}}(\rho, \Delta z) \approx \frac{r\omega_U\chi}{\rho} e^{i\Phi(\rho)} \begin{pmatrix} Y_x \\ Y_y \\ 0 \end{pmatrix} J_1\left(\frac{rk\rho}{|\Delta z|}\right), \qquad (18.21b) $$

with $J_1$ the Bessel function of the first kind of order 1.

At the center of the curvature radius of the dish is located a polarized horn-antenna of physical surface $S_{\text{phys}} = 0.25 \times 0.142 \text{ m}^2$. Numerical integration of the power from the electric field from Eq. (18.21b) over the physical antenna surface leads to

$$ P_{\text{int}} = \int dS_{\text{phys}} \frac{\epsilon_0 |U_{\text{dish}}^{\text{SHUKET}}(\sqrt{x^2 + y^2})|^2 c}{2} \approx 2.85 \times 10^{-22} \left( \frac{\chi}{10^{-12}} \right)^2 \text{W}, \qquad (18.22) $$

where we assumed an emission from a plane surface in the random polarization scenario, as in Eq. (17.15). The ratio of the power emitted by the dish that crosses physically the antenna and the total power emitted by the dish in the SHUKET experiment (Eq. (17.15)) is

$$ Q = \frac{P_{\text{int}}}{P_{\text{dish}}^{\text{SHUKET}}} \approx \frac{2.85 \times 10^{-22}}{1.73 \times 10^{-20}} \approx 1.6 \%. \qquad (18.23) $$

Note that, in the geometrical optics approximation, i.e where diffraction effects are neglected, one would obtain $Q = 100 \%$. It is interesting to note that the ratio of the physical antenna's surface to the dish's surface is ~ 3%, meaning that there is actually no focus of the field generated by the dish on the antenna.

This result means that, considering simply the propagation of the field from the dish to the antenna using Kirchhoff integral, the majority of the electromagnetic power is lost through
diffraction, and the antenna is only able to detect a small amount of energy emitted by the dish. One can note that the usual criteria for diffraction effects to be negligible $d_{\text{dish}} = 2r \gg \lambda$ is not fulfilled in this system, as the proportionality factor between the two parameters is ~ 25, which explains this lack of focus.

As a cross-check of our calculations, integrating Eq. (18.22) over the infinite antenna plane gives

$$P_{\text{int,full}} \approx 1.73 \times 10^{-20} \left(\frac{\chi}{10^{-12}}\right)^2 \text{W}, \qquad (18.24)$$

which corresponds to the full power emitted $P_{\text{dish}}^{\text{SHUKET}}$ from Eq. (17.15). This provides some confidence in our set of approximations, made throughout the derivation.

## Detection of the field using a horn antenna

In the previous section, we showed that most of the power emitted by the dish is already lost through propagation of the field from the dish to the antenna. As described in Sec. 11.3, there is still a second step to consider before predicting the exact amount of energy generated by the antenna: overlap integral between incident and antenna modes.

### Computation using the modes overlap
As mentioned in Sec. 11.3, to predict the energy generated by the horn antenna, i.e the overlap of modes, we need to consider the effective surface of the antenna $S_{\text{eff}}$ at the frequency we are interested in ($f_U = 6$ GHz). Furthermore, we also need to find an analytical expression for the mode of the field emitted by the dish $\vec{M}_{\text{dish}}(x, y)$, such that it is possible to perform the mode overlap integral Eq. (11.24).

To be able to compute the integral overlap of modes, we need an analytic expression of the field at coordinate $(\rho, z)$, which is provided by Eq. (18.21b). Then, from Eqs. (11.22a) and (18.21b), we can separate the mode of the dish $\vec{M}_{\text{dish}}$ from the constant amplitude $V_{\text{dish}}$ expressed in Eq. (11.23), such that the mode of the dish at coordinates $(x, y, 0)$ is

$$\vec{M}_{\text{dish}}(x, y) = \sqrt{\frac{3}{2\pi}} \frac{1}{\rho |\vec{Y}|} J_1 \left( \frac{rk\rho}{R-a} \begin{pmatrix} Y_x \\ Y_y \\ 0 \end{pmatrix} \right), \qquad (18.25)$$

with $\rho = \sqrt{x^2 + y^2}$. Then, using Eqs. (11.17a), (11.25) and (18.25) and assuming the polarization of the DP to be randomly distributed, the ratio of receiving to emitted powers is simply

$$\gamma_{\text{Overlap}} = \left( \int dS_{\text{eff}} \vec{M}_{\text{ant}} \cdot \vec{M}_{\text{dish}} \right)^2 \approx 5.8 \times 10^{-4} e_r, \qquad (18.26)$$

with $m_{TE_{10}} \approx 25.6 \text{ m}^{-1}$ has been estimated from Eqs. (11.17b) and (11.18) using the antenna gain $G(f_U) = 11.86$ dBi at $f_U = 6$ GHz provided in the antenna datasheet and which has been measured experimentally by the manufacturer. For this numerical value of ratio, we assumed an axial detection, with $Y_x = Y_y$ in Eq. (18.25). Assuming $e_r = 1$, i.e no loss inside the antenna, this result means that only 0.06% of the emitted power is actually transmitted to the antenna wires, and therefore detectable.
**Computation using the antenna factor** As explained in Sec. 11.3.2, if the electric field emitted by the dish is seen as a plane wave by the antenna or equivalently if the mode of this electric field is approximately constant over the effective surface of the antenna, we can use another method to derive the output of the experiment.

From Eq. (18.25), it can be shown that the dish polarization mode is approximately constant over the effective long length of the antenna (~ 7 cm) (with a deviation of ~ 0.3%), with a value of $M_{\text{dish}}^y(x, y = 0) \approx 0.48 \text{ m}^{-1}$.

For a Horn antenna with internal $R_0 = 50 \Omega$ resistance (which is typically the case for the Schwarzbeck BBHA-9120-D antenna), the antenna factor at frequency $f_U$ is given by [86]

$$AF(f_U) = \frac{9.73 f_U}{c\sqrt{G(f_U)}} = 49.7 \text{ m}^{-1}, \qquad (18.27)$$

which is consistent with the value given in the antenna datasheet [20]<fnref n="4" /> Then, using Eq. (11.28), one can compute the ratio of received to emitted powers as

$$\gamma_{\text{AF}} \approx 3.6 \times 10^{-4}. \qquad (18.28)$$

Considering that $e_r = 1$, Eqs. (18.26) and (18.28) disagree by a factor of approximately 1.5. This means that our assumption of no loss inside the antenna is most likely wrong, and we can artificially consider $e_r \sim 0.62$ such that both methods coincide. Therefore, the second result, obtained using the antenna factor, Eq. (11.28) is probably more realistic, as the different parameters have been experimentally measured.

Combining the results from Section 17.3, we can now reevaluate the constraints on $\chi$ obtained in the SHUKET experiment [6]. In Fig. 18.6, we show how both effects (diffraction and mode overlap) affect the sensitivity of the SHUKET experiment. In green, denoted "SHUKET (original)" is the original sensitivity curve presented in [6]. Since the power received by the antenna wires is quadratic in $\chi$ coupling, the ratio Eq. (18.28) leads to a loss in $\chi$ of an approximate factor 53. Making the assumption that this loss factor is roughly the same over all DM frequencies to which SHUKET is sensitive, this leads to an updated sensitivity curve, shown in red in Fig. 18.6, denoted "SHUKET (updated)".

### 18.4.3 Optimized run of SHUKET

Following the full derivation described above, it is possible to find some optimized experimental configuration such that the power received by the antenna is maximized. The experimental parameters that can easily be modified are the emitter-detector distance $L$ and the optimized DM frequency $f_U$ to search for with this setup<fnref n="5" />.

The frequency of the background DM field has several impacts on the power received by the antenna. First, following Eq. (11.15), the electric field amplitude received at coordinates $(\rho, \Delta z)$ (obtained using the Kirchhoff integral) depends highly on the frequency of the field. It can be shown numerically from Eq. (18.18a) that the focusing of the electric field improves with the DM frequency, which is a consequence of diffraction effects that become non negligible for low frequency, i.e. when $f_U d_{\text{dish}}/c = 2f_U r/c \gg 1$. In particular, as already discussed in Chapter 11, for the size of dish and the frequency bandwidth of the horn antenna used in

<footnote n="4">⁴Since AF(dB)=20 log₁₀(AF) [87].</footnote>

<footnote n="5">⁵This requires an antenna appropriate for this frequency.</footnote>
SHUKET, diffraction effects are always non negligible (i.e. $f_{ul}d_{dish}/c \gg 1$). As a consequence, the optimal location for the horn antenna is not the center of curvature of the dish and this location becomes frequency dependent. In addition, the antenna factor (or equivalently the antenna gain) is also highly dependent on the frequency of the measured electric field. As it was shown in the previous section, the overlap integral of polarization modes contribute almost equally to the total power loss than the one from the propagation of the field. Therefore, the optimal distance $L$ between the dish emitter and the horn antenna detector where the maximum field power is transmitted is non trivial and depends on the frequency.

The goal of this section is to explore the parameter space to find the optimal frequency $f$ and distance $L$ such that the efficiency coefficient $\gamma_{AF}$ from Eq. (11.28) is maximized. To do so, we use Eq. (18.18a) with unknown parameter $\Delta z^6$ and Eq. (11.23) to find the mode of the field emitted by the dish.

We first consider only the value of the mode at $\rho = 0$, and then we show that for the optimized parameters, the mode is indeed constant over the effective size of the antenna, such that the method can be used. Additionally, we interpolated the antenna datasheet [20] to infer the value of the antenna factor as function of the frequency $AF(f)$. Then, Eq. (11.28) is used to estimate numerically the efficiency coefficient

$$ \gamma(f, \Delta z)_{AF} = \frac{Z_0 M_{\text{dish}}^2 (\rho = 0, \Delta z, f)}{2 R_0 A F(f)^2} . \quad (18.29) $$

The behavior of this efficiency coefficient as a function of the DP frequency and of the distance between the dish and the receiver is shown in Fig. 18.7. One can notice the increase of $\gamma$ for small frequencies and short distances, which is mainly driven by the behavior of the antenna factor. Indeed, even though the loss through diffraction effects is larger at lower frequencies, the $\gamma$ parameter also takes into account the mode matching of the antenna, which is larger at those frequencies.

The expected output signal from the SHUKET experiment using the optimized experimental parameters can be computed using the same procedure and approximations as the ones presented in Sec. 18.4.2 at the exception of one approximation that is no longer valid. Indeed, Eq. (18.21b) is obtained by assuming that $\Delta z \sim R = 32$ m, which is no longer the case for the optimized distances, as it is shown in Fig. 18.7. We now derive an analytic expression of the electric field emitted by the dish in the case of such optimized distances.

<figure><img src="image_37.png" /><figcaption>Figure 18.7: Efficiency coefficient $\gamma(f, \Delta z)_{AF}$ as function of the frequency $f$ and $\Delta z$, in the frequency range $f \in [6, 18]$ GHz, of the horn antenna [20] and for distances $|\Delta z| \gg r$. The efficiency coefficient increases for low frequencies and short distances and presents a local maximum around ($f \sim 16.5$ GHz, $\Delta z \sim -15$ m).</figcaption></figure>

<footnote>$^6$Note that in order to use this equation, we must restrict ourselves to dish-horn distance $|\Delta z| \gg r$.</footnote>
We will concentrate on one particular optimized frequency $f_O$. We consider the optimized parameters for DM frequency $f_O = 16.5$ GHz and dish-antenna distance $\Delta z_O \approx -14.8$ m, the exponent in the integrand of Eq. (18.18a) can be comparable to 1, therefore the same Taylor expansion as previously is not possible. Instead, we expand the Bessel function as

$$J_0(x) = \sum_{m=0}^{\infty} \frac{(-1)^m x^{2m}}{m! 4^m \Gamma(m+1)} \qquad (18.30)$$

In our case, $x = |k_O \rho \rho'/\Delta z_O|$, $k_O = 2\pi f_O/c$. The maximum value of $\rho$ depends on the effective size of the antenna at frequency $\omega_O = 2\pi f_O$ following Eq. (11.18). From the antenna gain $G(\omega_O) = 16.87$ dBi, the maximum value of $x$ is $x_{max} \approx 0.30$. In the range $[0, x_{max}]$, one can easily show that the relative error on $J_0$ by only taking the first two terms of the sum Eq. (18.30) is very small $\sim 10^{-4}$, which indicates that these two terms are sufficient to describe the Bessel function in our system. Therefore, the integrand of Eq. (18.18a) becomes

$$\rho' e^{-i\varphi(\rho', \Delta z_O)} \left(1 - \left(\frac{k_O \rho \rho'}{2\Delta z_O}\right)^2\right), \qquad (18.31a)$$

since

$$\sum_{m=0}^{1} \frac{(-1)^m x^{2m}}{m! 4^m \Gamma(m+1)} = 1 - \frac{x^2}{4}. \qquad (18.31b)$$

Then, the analytic integration is doable and gives

$$\vec{U}_{\text{dish}}^{\text{Opti}}(\omega_O, \Delta z_O) = \frac{\chi \omega_O R e^{ik_0 \Delta z_O}}{2(\Delta z_O)(R + \Delta z_O)^2} \begin{pmatrix} Y_x \\ Y_y \\ 0 \end{pmatrix} \left( (e^{i\phi'} - 1)(k_O \rho^2 R - 2i\Delta z_O(R + \Delta z_O)) - \frac{ir^2 k_O^2 \rho^2}{2} \left(1 + \frac{R}{\Delta z_O}\right) \right), \qquad (18.32)$$

with $\Phi' = k_O r^2 (R + \Delta z_O) / 2R \Delta z_O$. Note that from this expression, by setting $\Delta z_O = -R+a$, we recover the first order expression of the field Eq. (18.21b), as expected. From this expression, one can show that the mode associated to this electric field is roughly constant over the effective size of the antenna ($\sim 1\%$ variation), therefore both methods presented in the Sec. 11.3 can work to compute the relative power received by the antenna. We find

$$\gamma_{\text{Ov.}} \approx 5.1 \times 10^{-3} \qquad (18.33a)$$

$$\gamma_{\text{AF}} \approx 3.2 \times 10^{-3}. \qquad (18.33b)$$

Notice that in the same way as for SHUKET parameters, the two results differ by an approximate factor 1.5 difference, as expected. Comparing both values with the ones presented in Sec. 18.4.2, one finds an approximate factor 9 improvement in power received. Note that this order of magnitude agrees with the one reached at the local maximum in Fig. 18.7. The efficiency coefficient obtained for this optimized experimental setup is one order of magnitude larger than the one obtained using the initial set of parameters, Eq. (18.28). Compared to
the original 53 loss factor in $\chi$, this optimization leads to a loss of ~ 17 on the coupling $\chi$ compared to the initial approximation of $P_{\text{rec}} = P_{\text{dish}}$.

We use the same procedure to obtain a curve of efficiency coefficient as function of the frequency, where, for each frequency $f$, we always assume the optimized dish-horn distance $\Delta z_O$ such that $\gamma(f, \Delta z_O)$ is maximized. Then, we apply this frequency-dependent efficiency coefficient to the original constrain on the mixing parameter $\chi$ set by SHUKET [6], in order to obtain an estimate of the sensitivity of a new SHUKET-like experiment using these optimized frequencies and distances. However, as can be seen in [6], the SHUKET constraint on $\chi$ depends on the frequency, even though the assumed signal is frequency independent. As explained in [6], this is due to the frequency dependent gain of the power amplifier connecting the horn antenna to the spectrum analyzer. Since we can assume that a new run of SHUKET would operate another amplifier with higher gain, we will consider the highest constraint of the original experiment as a basis for our new estimate.

Such projection is shown by the magenta hashed curve in Fig. 18.6. One can notice that over the 15.5 – 17.3 GHz DM frequency range, this new run with optimized parameters would improve the current constraint on the kinetic mixing parameter $\chi$, compared to CMB, shown in light grey.

### 18.4.4 New experimental run of SHUKET

Motivated by the optimization procedure presented in the previous section, a new run of the experiment was performed at IRFU, CEA Saclay, with the same dish and horn antennas used for the original measurement campaign [6]. In order to ease the data taking, it was decided that the dish-horn distance would not be modified for each frequency, but would stay fixed. In addition, the signal searched for is contained in the frequency range $f \in [8; 18]$ GHz, which corresponds to the high frequency band of sensitivity of the horn antenna. Then, one needs to find the optimized single distance $\Delta z_O$ such that for all frequencies $f \in [8; 18]$ GHz,

$$ \sum_f \gamma(f, \Delta z_O), \qquad (18.34) $$

is maximized. Numerically, we find $\Delta z_O \sim 12.5$ m. Keeping the distance fixed decreases a little the maximum potential sensitivity of the experiment, but keeping in mind that the final constraint on $\chi$ is $\propto \sqrt{\gamma(f, \Delta z)}$, this loss of sensitivity is marginal. However, this eases greatly the experimental process, as the antenna should not be re calibrated at each measurement.

The full data acquisition covers 8–18 GHz frequency band, and is done following [83]. In short, the setup is said to be "axial", i.e it is only sensitive to one polarization direction, which corresponds to the antenna mode direction $\vec{M}_{\text{ant}}$, which in practice is pointing towards the Zenith. In order to maximize the signal in case of fixed polarization scenario$^7$, it is found that

<footnote>$^7$In the random polarization scenario, the field has a random polarization at each coherence time, such that for experiment lasting much longer, the average is simply $\beta_{\text{random}} = \langle \cos^2(\theta) \rangle$ over a sphere, where $\theta$ is the angle between the polarization direction and the axis of measurement, as it was computed in e.g. Eq. (17.15) from Eq. (11.2), with $\beta_{\text{random}} = 2/3$. In the fixed polarization scenario, the polarization direction is unknown but stays fixed at all times. In this case, it is the daily modulation of the signal due to Earth rotation that can help increasing the signal. Indeed, for a very short experimental time, the experiment probes only one specific direction and a conservative estimate yields $\beta_{\text{fixed}} \ll 1$ for most experimental setups [83]. However, if the experimental time increases significantly ($O(1 \text{ day})$), the experiment effectively averages over many different polarizations and therefore $\beta_{\text{fixed}} \rightarrow \beta_{\text{random}}$ [83].</footnote>
the optimized data-taking for such experiment is to make 3 individual measurements and wait for a third of a sidereal day between each measurement [83]. In practice, each acquisition lasts for $T = 200$s, and corresponds to a frequency window of 10 MHz. Then, the three acquisitions are summed to be able to average over the polarization. Since the total frequency interval is 10 GHz (from 8 to 18 GHz), the total acquisition time is $600 \times 1000 = 6 \times 10^5$s.

Next, the background noise in the data is estimated using a Savitzky-Golay filter with a 1 MHz bandwidth (which is much larger than the expected width of the signal ~ 10 kHz, from Eq. (5.10b) and using the fact that we are looking at DP frequencies around tens of GHz.). In short, this filter smooths out the data by using moving averages [88]. The ratio of the original data (potential signal + background noise) with the noise, given by the filter, follows a Gaussian distribution centered around 1. Then, each individual Fourier peak which is above $5\sigma$ in this distribution is identified and analyzed individually to see if it looks like a DM signal. A preliminary analysis was done, where the DM signal is modelled with a Gaussian distribution with a fixed width which is ~ $10^6$ smaller than the Fourier frequency of the peak (still following Eq. (5.10b)), but with variable amplitude and offset. No DM signal is found in the data, and therefore, the uncertainty on the amplitude is converted into an upper limit on the power received from the hypothetical DP. Then, using Eqs. (17.15) and (18.29), this is converted to an upper limit on the kinetic mixing coupling $\chi$. The preliminary constraint on $\chi$ from this new run is shown in Fig. 18.8. The sensitivity estimate around 16 GHz corresponds quite accurately to the theoretical prediction shown in magenta in Fig. 18.6. Over a large frequency band (~ 10 – 18 GHz), this preliminary curve reaches unconstrained regions of the parameter space, compared to CMB, see Fig. 18.6. One can notice some modulation of the sensitivity in Fig. 18.8, which comes from the value of the gain amplifier which depends on the frequency.

<figure><img src="image_38.png" /><figcaption>Figure 18.8: Preliminary constraint (95% confidence level) of a second run of SHUKET to the kinetic mixing coupling $\chi$.</figcaption></figure>
18.5 Discussion

In this chapter, we have derived the expected sensitivities of various experiments, both
already existing and futuristic, to various ULDM couplings to Standard Model fields. In par-
ticular, following this theoretical work, a new experimental run of *SHUKET* was performed,
with preliminary competitive results. We have shown that despite the existing strong con-
straints on those couplings, many experiments that will be conducted in the next few years
will be able to reach unconstrained regions of the parameter space, and therefore have a
chance to detect a positive signal, which will, without any doubt, be a major discovery.
[1] Etienne Savalle et al. "Searching for Dark Matter with an Optical Cavity and an Unequal-Delay Interferometer". In: *Phys. Rev. Lett.* 126 (5 Feb. 2021), p. 051301. DOI: 10.1103/PhysRevLett.126.051301. URL: https://link.aps.org/doi/10.1103/PhysRevLett.126.051301.

[2] Etienne Savalle et al. *Novel approaches to dark-matter detection using space-time separated clocks*. 2019. DOI: 10.48550/ARXIV.1902.07192. URL: https://arxiv.org/abs/1902.07192.

[3] H. Jiang et al. "Long-distance frequency transfer over an urban fiber link using optical phase stabilization". In: *Journal of the Optical Society of America B* 25.12 (Nov. 2008), p. 2029. ISSN: 1520-8540. DOI: 10.1364/josab.25.002029. URL: http://dx.doi.org/10.1364/JOSAB.25.002029.

[4] Enrico Rubiola. *The Measurement of AM noise of Oscillators*. 2005. DOI: 10.48550/ARXIV.PHYSICS/0512082. URL: https://doi.org/10.48550/ARXIV.PHYSICS/0512082.

[5] J-Y Vinet. *The Virgo Physics Book: Optics and related Topics*. 2020.

[6] Pierre Brun, Laurent Chevalier, and Christophe Flouzat. "Direct Searches for Hidden-Photon Dark Matter with the SHUKET Experiment". In: *Phys. Rev. Lett.* 122 (20 May 2019), p. 201801. DOI: 10.1103/PhysRevLett.122.201801. URL: https://link.aps.org/doi/10.1103/PhysRevLett.122.201801.

[7] Akash V. Dixit et al. "Searching for Dark Matter with a Superconducting Qubit". In: *Phys. Rev. Lett.* 126 (14 Apr. 2021), p. 141302. DOI: 10.1103/PhysRevLett.126.141302. URL: https://link.aps.org/doi/10.1103/PhysRevLett.126.141302.

[8] Le Hoang Nguyen, Andrei Lobanov, and Dieter Horns. "First results from the WIS-PDMX radio frequency cavity searches for hidden photon dark matter". In: *Journal of Cosmology and Astroparticle Physics* 2019.10 (Oct. 2019), p. 014. DOI: 10.1088/1475-7516/2019/10/014. URL: https://dx.doi.org/10.1088/1475-7516/2019/10/014.

[9] J. Suzuki et al. "Experimental search for hidden photon CDM in the eV mass range with a dish antenna". In: *Journal of Cosmology and Astroparticle Physics* 2015.09 (Sept. 2015), p. 042. DOI: 10.1088/1475-7516/2015/09/042. URL: https://dx.doi.org/10.1088/1475-7516/2015/09/042.

[10] Stefan Knirck et al. "First results from a hidden photon dark matter search in the meV sector using a plane-parabolic mirror system". In: *Journal of Cosmology and Astroparticle Physics* 2018.11 (Nov. 2018), p. 031. DOI: 10.1088/1475-7516/2018/11/031. URL: https://dx.doi.org/10.1088/1475-7516/2018/11/031.

[11] Arnaud Andrianavalomahefa et al. "Limits from the FUNK experiment on the mixing strength of hidden-photon dark matter in the visible and near-ultraviolet wavelength range". In: *Phys. Rev. D* 102 (4 Aug. 2020), p. 042001. DOI: 10.1103/PhysRevD.102.042001. URL: https://link.aps.org/doi/10.1103/PhysRevD.102.042001.

[12] K. Ramanathan et al. "Wideband Direct Detection Constraints on Hidden Photon Dark Matter with the QUALIPHIDE Experiment". In: *Phys. Rev. Lett.* 130 (23 June 2023), p. 231001. DOI: 10.1103/PhysRevLett.130.231001. URL: https://link.aps.org/doi/10.1103/PhysRevLett.130.231001.

[13] Haipeng An et al. *Direct detection of dark photon dark matter using radio telescopes*. 2022. DOI: 10.48550/ARXIV.2207.05767. URL: https://arxiv.org/abs/2207.05767.
[14] Elizabeth M. Bridge et al. "Tunable cw UV laser with &\#x0003C;35 kHz absolute frequency instability for precision spectroscopy of Sr Rydberg states". In: *Opt. Express* 24.3 (Feb. 2016), pp. 2281–2292. DOI: 10.1364/OE.24.002281. URL: https://opg.optica.org/oe/abstract.cfm?URI=oe-24-3-2281.

[15] G Vallet et al. "A noise-immune cavity-assisted non-destructive detection for an optical lattice clock in the quantum regime". en. In: *New Journal of Physics* 19.8 (Aug. 2017), p. 083002. ISSN: 1367-2630. DOI: 10.1088/1367-2630/aa7c84.

[16] J Millen et al. "Spectroscopy of a cold strontium Rydberg gas". In: *Journal of Physics B: Atomic, Molecular and Optical Physics* 44.18 (Sept. 2011), p. 184001. DOI: 10.1088/0953-4075/44/18/184001. URL: https://dx.doi.org/10.1088/0953-4075/44/18/184001.

[17] J. W. Foster, N. L. Rodd, and B. R. Safdi. "Revealing the dark matter halo with axion direct detection". In: *Physical Review D* 97.12, 123006 (2018), p. 123006. DOI: 10.1103/PhysRevD.97.123006. arXiv: 1711.10489.

[18] J. Guena et al. "Progress in atomic fountains at LNE-SYRTE". In: *IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control* 59.3 (Mar. 2012), pp. 391–409. DOI: 10.1109/tuffc.2012.2208. URL: https://doi.org/10.1109%5C%2Ftuffc.2012.2208.

[19] Dieter Horns et al. "Searching for WISPy cold dark matter with a dish antenna". In: *Journal of Cosmology and Astroparticle Physics* 2013.04 (Apr. 2013), pp. 016–016. ISSN: 1475-7516. DOI: 10.1088/1475-7516/2013/04/016. URL: http://dx.doi.org/10.1088/1475-7516/2013/04/016.

[20] *Double Ridged Broadband Horn Antenna*. BBHA 9120 D. Schwarzbeck. 2023. URL: https://www.schwarzbeck.de/en/antennas/broadband-horn-antennas/double-ridged-horn-antenna/404-bbha-9120-d-double-ridged-broadband-horn-antenna.html.

[21] Pierre Touboul et al. "MICROSCOPE Mission: First Results of a Space Test of the Equivalence Principle". In: *Phys. Rev. Lett.* 119 (23 Dec. 2017), p. 231101. DOI: 10.1103/PhysRevLett.119.231101. URL: https://link.aps.org/doi/10.1103/PhysRevLett.119.231101.

[22] Manuel Rodrigues et al. "MICROSCOPE mission scenario, ground segment and data processing". In: *Classical and Quantum Gravity* 39.20 (Sept. 2022), p. 204004. DOI: 10.1088/1361-6382/ac4b9a. URL: https://dx.doi.org/10.1088/1361-6382/ac4b9a.

[23] Hélène Pihan-Le Bars et al. "New Test of Lorentz Invariance Using the MICROSCOPE Space Mission". In: *Phys. Rev. Lett.* 123 (23 Dec. 2019), p. 231102. DOI: 10.1103/PhysRevLett.123.231102. URL: https://link.aps.org/doi/10.1103/PhysRevLett.123.231102.

[24] Joel Bergé. "MICROSCOPE's view at gravitation". In: *Reports on Progress in Physics* 86.6 (May 2023), p. 066901. DOI: 10.1088/1361-6633/acd203. URL: https://dx.doi.org/10.1088/1361-6633/acd203.

[25] Peter Asenbaum et al. "Atom-Interferometric Test of the Equivalence Principle at the 10⁻¹² Level". In: *Phys. Rev. Lett.* 125 (19 Nov. 2020), p. 191101. DOI: 10.1103/PhysRevLett.125.191101. URL: https://link.aps.org/doi/10.1103/PhysRevLett.125.191101.
[26] Pippa Storey and Claude Cohen-Tannoudji. "The Feynman path integral approach to atomic interferometry. A tutorial". In: *Journal de Physique II* 4.11 (1994), pp. 1999–2027. DOI: 10.1051/jp2:1994103. URL: https://hal.archives-ouvertes.fr/jpa-00248106.

[27] L. Badurina et al. "AION: an atom interferometer observatory and network". In: *Journal of Cosmology and Astroparticle Physics* 2020.05 (May 2020), p. 011. DOI: 10.1088/1475-7516/2020/05/011. URL: https://dx.doi.org/10.1088/1475-7516/2020/05/011.

[28] Leonardo Badurina, Diego Blas, and Christopher McCabe. "Refined ultralight scalar dark matter searches with compact atom gradiometers". In: *Phys. Rev. D* 105 (2 Jan. 2022), p. 023006. DOI: 10.1103/PhysRevD.105.023006. URL: https://link.aps.org/doi/10.1103/PhysRevD.105.023006.

[29] Mahiro Abe et al. "Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100)". In: *Quantum Science and Technology* 6.4 (July 2021), p. 044003. DOI: 10.1088/2058-9565/abf719. URL: https://dx.doi.org/10.1088/2058-9565/abf719.

[30] S. Barke. "Inter-spacecraft frequency distribution for future gravitational wave observatories". PhD thesis. Leibniz U., 2015.

[31] John G. Hartnett, Nitin R. Nand, and Chuan Lu. "Ultra-low-phase-noise cryocooled microwave dielectric-sapphire-resonator oscillators". In: *Applied Physics Letters* 100.18 (Apr. 2012), p. 183501. ISSN: 0003-6951. DOI: 10.1063/1.4709479. eprint: https://pubs.aip.org/aip/apl/article-pdf/doi/10.1063/1.4709479/14246326/183501\_1\_online.pdf. URL: https://doi.org/10.1063/1.4709479.

[32] Xiaopeng Xie et al. "Photonic microwave signals with zeptosecond-level absolute timing noise". In: *Nature Photonics* 11.1 (2017), pp. 44–47. ISSN: 1749-4893. URL: https://doi.org/10.1038/nphoton.2016.215.

[33] Stanislav Babak, Martin Hewitson, and Antoine Petiteau. *LISA Sensitivity and SNR Calculations*. 2021. arXiv: 2108.01167 [astro-ph.IM].

[34] Olaf Hartwig et al. "Stochastic gravitational wave background reconstruction for a nonequilateral and unequal-noise LISA constellation". In: *Physical Review D* 107.12 (June 2023). ISSN: 2470-0029. DOI: 10.1103/physrevd.107.123531. URL: http://dx.doi.org/10.1103/PhysRevD.107.123531.

[35] Monica Colpi et al. *LISA Definition Study Report*. 2024. arXiv: 2402.07571 [astro-ph.CO].

[36] Pierre Touboul et al. "MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle". In: *Phys. Rev. Lett.* 129 (12 Sept. 2022), p. 121102. DOI: 10.1103/PhysRevLett.129.121102. URL: https://link.aps.org/doi/10.1103/PhysRevLett.129.121102.

[37] T A Wagner et al. "Torsion-balance tests of the weak equivalence principle". In: *Classical and Quantum Gravity* 29.18 (Aug. 2012), p. 184002. DOI: 10.1088/0264-9381/29/18/184002. URL: https://dx.doi.org/10.1088/0264-9381/29/18/184002.

[38] A. Hees et al. "Searching for an Oscillating Massive Scalar Field as a Dark Matter Candidate Using Atomic Hyperfine Frequency Comparisons". In: *Phys. Rev. Lett.* 117 (6 Aug. 2016), p. 061301. DOI: 10.1103/PhysRevLett.117.061301. URL: https://link.aps.org/doi/10.1103/PhysRevLett.117.061301.
[39] Colin J. Kennedy et al. "Precision Metrology Meets Cosmology: Improved Constraints on Ultralight Dark Matter from Atom-Cavity Frequency Comparisons". In: *Phys. Rev. Lett.* 125 (20 Nov. 2020), p. 201302. doi: 10.1103/PhysRevLett.125.201302. URL: https://link.aps.org/doi/10.1103/PhysRevLett.125.201302.

[40] Ivo Schulthess et al. "New Limit on Axionlike Dark Matter Using Cold Neutrons". In: *Phys. Rev. Lett.* 129.19 (2022), p. 191801. doi: 10.1103/PhysRevLett.129.191801. arXiv: 2204.01454 [hep-ex].

[41] Tanya S. Roussy et al. "Experimental Constraint on Axionlike Particles over Seven Orders of Magnitude in Mass". In: *Phys. Rev. Lett.* 126.17 (2021), p. 171301. doi: 10.1103/PhysRevLett.126.171301. arXiv: 2006.15787 [hep-ph].

[42] Xue Zhang et al. "Search for ultralight dark matter with spectroscopy of radio-frequency atomic transitions". In: (Dec. 2022). arXiv: 2212.04413 [physics.atom-ph].

[43] C. Abel et al. "Search for Axionlike Dark Matter through Nuclear Spin Precession in Electric and Magnetic Fields". In: *Phys. Rev. X* 7 (4 Nov. 2017), p. 041034. doi: 10.1103/PhysRevX.7.041034. URL: https://link.aps.org/doi/10.1103/PhysRevX.7.041034.

[44] Ciaran O'Hare. *cajohare/AxionLimits: AxionLimits*. https://cajohare.github.io/AxionLimits/. Version v1.0. July 2020. doi: 10.5281/zenodo.3932430.

[45] S. J. Asztalos et al. "SQUID-Based Microwave Cavity Search for Dark-Matter Axions". In: *Phys. Rev. Lett.* 104.4, 041301 (Jan. 2010), p. 041301. doi: 10.1103/PhysRevLett.104.041301. arXiv: 0910.5914 [astro-ph.CO].

[46] N. Du et al. "A Search for Invisible Axion Dark Matter with the Axion Dark Matter Experiment". In: *Phys. Rev. Lett.* 120.15 (2018), p. 151301. doi: 10.1103/PhysRevLett.120.151301. arXiv: 1804.05750 [hep-ex].

[47] T. Braine et al. "Extended Search for the Invisible Axion with the Axion Dark Matter Experiment". In: *Phys. Rev. Lett.* 124.10 (2020), p. 101303. doi: 10.1103/PhysRevLett.124.101303. arXiv: 1910.08638 [hep-ex].

[48] C. Bartram et al. "Search for Invisible Axion Dark Matter in the 3.3-4.2 μeV Mass Range". In: *Phys. Rev. Lett.* 127.26 (2021), p. 261803. doi: 10.1103/PhysRevLett.127.261803. arXiv: 2110.06096 [hep-ex].

[49] C. Boutan et al. "Piezoelectrically Tuned Multimode Cavity Search for Axion Dark Matter". In: *Phys. Rev. Lett.* 121.26 (2018), p. 261302. doi: 10.1103/PhysRevLett.121.261302. arXiv: 1901.00920 [hep-ex].

[50] S. Lee et al. "Axion Dark Matter Search around 6.7 μeV". In: *Phys. Rev. Lett.* 124.10 (2020), p. 101802. doi: 10.1103/PhysRevLett.124.101802. arXiv: 2001.05102 [hep-ex].

[51] Junu Jeong et al. "Search for Invisible Axion Dark Matter with a Multiple-Cell Halo-scope". In: *Phys. Rev. Lett.* 125.22 (2020), p. 221302. doi: 10.1103/PhysRevLett.125.221302. arXiv: 2008.10141 [hep-ex].

[52] Youngjae Lee et al. "Searching for Invisible Axion Dark Matter with an 18 T Magnet Haloscope". In: *Phys. Rev. Lett.* 128.24 (2022), p. 241805. doi: 10.1103/PhysRevLett.128.241805. arXiv: 2206.08845 [hep-ex].
[53] Ohjoon Kwon et al. "First Results from an Axion Haloscope at CAPP around 10.7 µeV". In: *Phys. Rev. Lett.* 126.19 (2021), p. 191802. doi: **10.1103/PhysRevLett.126.191802**. arXiv: **2012.10764 [hep-ex]**.

[54] Jinsu Kim et al. "Near-Quantum-Noise Axion Dark Matter Search at CAPP around 9.5 µeV". In: (July 2022). arXiv: **2207.13597 [hep-ex]**.

[55] Andrew K. Yi et al. "DFSZ Axion Dark Matter Search around 4.55 µeV". In: (Oct. 2022). arXiv: **2210.10961 [hep-ex]**.

[56] L. Zhong et al. "Results from phase 1 of the HAYSTAC microwave cavity axion experiment". In: *Phys. Rev. D* 97.9 (2018), p. 092001. doi: **10.1103/PhysRevD.97.092001**. arXiv: **1803.03690 [hep-ex]**.

[57] K. M. Backes et al. "A quantum-enhanced search for dark matter axions". In: *Nature* 590.7845 (2021), pp. 238-242. doi: **10.1038/s41586-021-03226-7**. arXiv: **2008.01853 [quant-ph]**.

[58] Aaron Quiskamp et al. "Direct search for dark matter axions excluding ALP cogenesis in the 63- to 67-µeV range with the ORGAN experiment". In: *Science Advances* 8.27 (July 2022). doi: **10.1126/sciadv.abq3765**. URL: https://doi.org/10.1126/5C%2Fsciadv.abq3765.

[59] D. Alesini et al. "Galactic axions search with a superconducting resonant cavity". In: *Phys. Rev. D* 99.10 (2019), p. 101101. doi: **10.1103/PhysRevD.99.101101**. arXiv: **1903.06547 [physics.ins-det]**.

[60] D. Alesini et al. "Search for invisible axion dark matter of mass $m_a = 43$ µeV with the QUAX-αγ experiment". In: *Phys. Rev. D* 103.10 (2021), p. 102004. doi: **10.1103/PhysRevD.103.102004**. arXiv: **2012.09498 [hep-ex]**.

[61] D. Alesini et al. "Search for Galactic axions with a high-Q dielectric cavity". In: *Phys. Rev. D* 106.5 (2022), p. 052007. doi: **10.1103/PhysRevD.106.052007**. arXiv: **2208.12670 [hep-ex]**.

[62] Jonathan L. Ouellet et al. "First Results from ABRACADABRA-10 cm: A Search for Sub-µeV Axion Dark Matter". In: *Phys. Rev. Lett.* 122.12 (2019), p. 121802. doi: **10.1103/PhysRevLett.122.121802**. arXiv: **1810.12257 [hep-ex]**.

[63] Chiara P. Salemi et al. "Search for Low-Mass Axion Dark Matter with ABRACADABRA-10 cm". In: *Phys. Rev. Lett.* 127.8 (2021), p. 081801. doi: **10.1103/PhysRevLett.127.081801**. arXiv: **2102.06722 [hep-ex]**.

[64] C. Bartram et al. "Search for Invisible Axion Dark Matter in the 3.3-4.2 µeV Mass Range". In: *Phys. Rev. Lett.* 127.26 (2021), p. 261803. doi: **10.1103/PhysRevLett.127.261803**. arXiv: **2110.06096 [hep-ex]**.

[65] N. Crisosto et al. "ADMX SLIC: Results from a Superconducting LC Circuit Investigating Cold Axions". In: *Phys. Rev. Lett.* 124.24 (2020), p. 241101. doi: **10.1103/PhysRevLett.124.241101**. arXiv: **1911.05772 [astro-ph.CO]**.

[66] C. M. Adair et al. "Search for Dark Matter Axions with CAST-CAPP". In: *Nature Commun.* 13.1 (2022), p. 6180. doi: **10.1038/s41467-022-33913-6**. arXiv: **2211.02902 [hep-ex]**.
[67] Younggeun Kim et al. "Experimental search for invisible axions as a test of axion cosmology around 22 ueV". In: (Dec. 2023). arXiv: 2312.11003 [hep-ex].

[68] Byeongsu Yang et al. "Extended Axion Dark Matter Search Using the CAPP18T Halo-scope". In: (Aug. 2023). arXiv: 2308.09077 [hep-ex].

[69] Saebyeok Ahn et al. "Extensive search for axion dark matter over 1 GHz with CAPP's Main Axion eXperiment". In: (Feb. 2024). arXiv: 2402.12892 [hep-ex].

[70] Hojin Yoon et al. "Axion haloscope using an 18 T high temperature superconducting magnet". In: Phys. Rev. D 106.9 (2022), p. 092007. doi: 10.1103/PhysRevD.106.092007. arXiv: 2206.12271 [hep-ex].

[71] Jack A. Devlin et al. "Constraints on the Coupling between Axionlike Dark Matter and Photons Using an Antiproton Superconducting Tuned Detection Circuit in a Cryogenic Penning Trap". In: Phys. Rev. Lett. 126.4 (2021), p. 041301. doi: 10.1103/PhysRevLett.126.041301. arXiv: 2101.11290 [astro-ph.CO].

[72] Aaron Quiskamp et al. "Exclusion of ALP Cogenesis Dark Matter in a Mass Window Above 100 µeV". In: (Oct. 2023). arXiv: 2310.00904 [hep-ex].

[73] Ben T. McAllister et al. "The ORGAN Experiment: An axion haloscope above 15 GHz". In: Phys. Dark Univ. 18 (2017), pp. 67-72. doi: 10.1016/j.dark.2017.09.010. arXiv: 1706.00209 [physics.ins-det].

[74] M. J. Jewell et al. "New Results from HAYSTAC's Phase II Operation with a Squeezed State Receiver". In: (Jan. 2023). arXiv: 2301.09721 [hep-ex].

[75] R. Di Vora et al. "Search for galactic axions with a traveling wave parametric amplifier". In: Phys. Rev. D 108.6 (2023), p. 062005. doi: 10.1103/PhysRevD.108.062005. arXiv: 2304.07505 [hep-ex].

[76] Alexander V. Gramolin et al. "Search for axion-like dark matter with ferromagnets". In: Nature Phys. 17.1 (2021), pp. 79-84. doi: 10.1038/s41567-020-1006-6. arXiv: 2003.03348 [hep-ex].

[77] S. DePanfilis et al. "Limits on the abundance and coupling of cosmic axions at $4.5<m_a<5.0$ µeV". In: Phys. Rev. Lett. 59 (7 Aug. 1987), pp. 839-842. doi: 10.1103/PhysRevLett.59.839. URL: https://link.aps.org/doi/10.1103/PhysRevLett.59.839.

[78] Walter Wuensch et al. "Results of a Laboratory Search for Cosmic Axions and Other Weakly Coupled Light Particles". In: Phys. Rev. D 40 (1989), p. 3153. doi: 10.1103/PhysRevD.40.3153.

[79] C. Hagmann et al. "First results from a second generation galactic axion experiment". In: Nucl. Phys. B Proc. Suppl. 51 (1996). Ed. by D. B. Cline, pp. 209-212. doi: 10.1016/S0920-5632(96)00516-6. arXiv: astro-ph/9607022.

[80] C. Hagmann et al. "Results from a search for cosmic axions". In: Phys. Rev. D 42 (4 Aug. 1990), pp. 1297-1300. doi: 10.1103/PhysRevD.42.1297. URL: https://link.aps.org/doi/10.1103/PhysRevD.42.1297.
[81] Antoine Petiteau. "DE LA SIMULATION DE LISA A L'ANALYSE DES DONNEES. Détection d'ondes gravitationnelles par interférométrie spatiale (LISA : Laser Interferometer Space Antenna)". Theses. Université Paris-Diderot - Paris VII, June 2008. URL: https://theses.hal.science/tel-00383222.

[82] P. K. F. Grieder. *Cosmic rays at earth: Researcher's reference, manual and data book*. Amsterdam: Elsevier, 2001. ISBN: 978-0-444-50710-5.

[83] Andrea Caputo et al. "Dark photon limits: A handbook". In: *Phys. Rev. D* 104 (9 Nov. 2021), p. 095029. DOI: 10.1103/PhysRevD.104.095029. URL: https://link.aps.org/doi/10.1103/PhysRevD.104.095029.

[84] Paola Arias et al. "WISPy cold dark matter". In: *Journal of Cosmology and Astroparticle Physics* 2012.06 (June 2012), p. 013. DOI: 10.1088/1475-7516/2012/06/013. URL: https://dx.doi.org/10.1088/1475-7516/2012/06/013.

[85] R. Cervantes et al. "ADMX-Orpheus first search for 70 μeV dark photon dark matter: Detailed design, operations, and analysis". In: *Phys. Rev. D* 106 (10 Nov. 2022), p. 102002. DOI: 10.1103/PhysRevD.106.102002. URL: https://link.aps.org/doi/10.1103/PhysRevD.106.102002.

[86] James S. Mclean, Robert Sutton, and Robert William Hoffman. "Interpreting Antenna Performance Parameters for EMC Applications: Part 3: Antenna Factor". In: 2002. URL: https://api.semanticscholar.org/CorpusID:1097498.

[87] Constantine. A. Balanis. *Antenna theory: Analysis and Design, 3rd edition*. John Wiley and Sons, 2005.

[88] "Smoothing and differentiation of data by simplified least squares procedures." In: *Analytical Chemistry* 36.8 (1964), pp. 1627-1639.
# Part VIII
## Conclusion
General relativity and the Standard Model of particle physics are regarded today as the most successful theories in modern physics. Despite their incredible success, their inherent formalism differences make their unification into a single theory, the *theory of everything*, currently impossible. Many believe that these theories are therefore incomplete, and theorists around the world are determined to find a quantum theory of gravity. This would certainly provide answers to unresolved mysteries arising in both theories, such as the existence (or not) of dark matter and its nature. Indeed, while dark matter was first revealed almost a hundred years ago, its true microscopic nature is still unknown, making it one of the biggest puzzles in modern physics. However, one must remember that it could still be an artifact of an incomplete theory of gravitation.

Among many other candidates, ultralight fields present an undeniably elegant solution to the dark matter problem. As discussed in Chapter 8, they induce various effects on matter, such as variation of constants of Nature, modification of classical Maxwell's equations, appearance of equivalence-principle violating accelerations, or production of electromagnetic fields. This kind of phenomenology can be probed accurately with quantum precision technologies, which, over the past ~ thirty years, have reached unprecedented progress in the control of matter and light. These quantum sensors include, for example, atomic clocks, electromagnetic cavities, atom interferometers, and optical interferometers. We have seen in different parts of this thesis how each of these probes can be used to detect ultralight dark matter fields.

SYRTE, as one of the world's leading laboratories in time metrology, possesses some of the most accurate atomic clocks and atom interferometers (see e.g. [1–4]), and is therefore greatly involved in ultralight dark matter searches [5, 6]. SYRTE also has significant expertise in data analysis of dark matter and gravitational wave laboratory experiments. It is in this environment that this thesis took place, making possible the investigation of many different experiments for the search of ultralight dark matter.

In terms of phenomenology, the main focus of this thesis is on 1) the violation of the equivalence principle induced by couplings between a (pseudo-)scalar dark matter field, the dilaton and the axion, and various Standard Model sectors [7], 2) the modification of Maxwell's equations via the introduction of a pseudo-scalar field, the axion, coupled to electromagnetism, and 3) the production of an electromagnetic field in vacuum from the coupling of a vector dark matter candidate, the dark photon, to electromagnetism [8, 9].

The first phenomenological aspect is related to the spacetime variation of constants of Nature, such as the fine structure constant or the electron mass, which, as a consequence, induces an oscillation of intrinsic properties of bodies, such as their rest mass, and the transition frequency of atoms. The amplitude of such oscillations being atomic species-dependent, two bodies of distinct composition would accelerate at different rates, leading to a violation of the weak equivalence principle. We extensively studied how *MICROSCOPE*, as a classical test of the equivalence principle, and atom interferometers, as its quantum equivalent, can probe dark matter fields [7]. We found that *MICROSCOPE* could reach unprecedented sensitivity to axion-gluon coupling compared to already existing laboratory experiments, which warrants a corresponding detailed analysis of *MICROSCOPE* data to search for a potential signal. Additionally, we showed that diverse atom interferometric setups, despite being similar in terms of noise levels, have contrasting sensitivities to dark matter fields. In particular, we showed that dual-isotope interferometers, where both interferometers are spatially overlapped, have much more potential than gradiometers, where the two interferometers
are stacked at different altitudes. Even though it is natural to consider experiments involving different atomic species to detect oscillations that are species-dependent, experiments using a single species can still be competitive due to the finite speed of light. This is the case with large space-based gravitational wave detectors, such as LISA. Indeed, the movement of test masses caused by dark matter would generate Doppler shifts on light signals exchanged between the spacecrafts. We investigated the possibility of the degeneracy between this signal and a monochromatic signal from a galactic binary, and if it is realistic to expect making the distinction between the two, using a Bayesian approach and a full year of realistic orbits of the spacecraft. We found that the detector will be able to distinguish signals from scalar dark matter and monochromatic gravitational waves. Moreover, due to the non-relativistic nature of the dark matter field, the detector cannot probe accurately its propagation velocity. As a result, by allowing this velocity to be a free parameter in the Bayesian model, the sensitivity of the experiment to dark matter couplings decreases due to correlations.

The second phenomenological aspect of interest in this thesis is vacuum birefringence induced by the coupling between axions and photons. Considering an electromagnetic plane wave, the axion field couples to the magnetic polarization of light and, by its inherent oscillation, causes the light polarization to oscillate as well. If the light polarization is further decomposed into the two circular polarization states, one finds that the two states have different phase velocities, i.e., vacuum becomes birefringent. Considering an electromagnetic cavity in which circularly polarized light is sent in, we showed that this change in phase velocity of light is, mathematically speaking, equivalent to a change in the cavity's length, leading to a phase shift of light at the output of the cavity. We derived the sensitivity of an optical cavity and a tens-of-kilometer-long fiber link to such an effect, and we showed it is not competitive with current bounds. We then turned to LISA again. The current version of LISA's optical benches operates with linearly polarized light only. We showed that a slight modification of these benches would 1) produce circularly polarized light and 2) be permitted in the setup for the search of gravitational waves. We showed that such a modification would make LISA the most sensitive laboratory experiment to the axion-photon coupling at low masses, by five orders of magnitude.

Finally, we concentrated on laboratory probes of a small electromagnetic field induced by the kinetic mixing coupling between dark photons, a vector dark matter candidate, and photons. This electric field oscillates at the Compton frequency of the DM field, and we have been interested in experiments probing the GHz frequency region. We first proposed an innovative way of detecting this electric field by using Rydberg atoms inside a microwave cavity. In short, the idea of the experiment is 1) to use the cavity as a resonator for this small electric field; 2) to inject a strong electromagnetic field inside the cavity at a close frequency from the DM Compton one, in order to produce a slowly oscillating beatnote between the two fields; 3) to use Rydberg atoms to measure this slow component by the quadratic Stark effect it produces on them. The innovative concept of this experiment is twofold: first, the signal in our scheme is only linear in the small kinetic mixing coupling, while most experiments aim at detecting the field power, which is quadratic in the coupling; second, instead of using antennas for the measurement of the signal, we proposed to use atoms. We showed that such an experiment would reach competitive constraints on the kinetic mixing coupling compared to already existing experiments [8]. We also studied another kind of experiment to detect this electric field, which involves a dish antenna, which reflects the small electric field and focuses it onto a horn antenna located at the curvature center of the dish [10, 11]. On one
hand, we investigated the effects of diffraction in this type of experiment from an analytical
standpoint, making use of the Kirchhoff integral theorem in the low-curvature dish limit.
On the other hand, we estimated the impact of mode-matching inside the horn antenna. We
showed that the expected signal intensity can be significantly reduced compared to usual
estimates. Our method was applied to the re-interpretation of the *SHUKET* experiment
data, the results of which were shown to be degraded by a factor of approximately fifty
due to both diffraction and mode-matching [9]. This analytical method allowed optimizing
some experimental parameters to gain sensitivity in future runs. In particular, following this
theoretical modeling, a second run of *SHUKET* was recently performed, with a competitive
constraint on the kinetic mixing coupling.

This thesis combines several studies that can be useful to the dark matter search com-
munity in the near future: proposals of innovative experimental schemes (Chapter 10),
improved modeling of already existing and future experiments (Chapters 11, 13, and 15),
and demonstration of the potential of existing and future experiments to probe unpredicted
couplings (Chapters 12 and 16). It focuses on the theoretical modeling of such experiments
and their corresponding sensitivity estimates, but not on the analysis of data produced by
such experiments. As a simplifying assumption, the ultralight dark matter fields are consid-
ered monochromatic, while in reality, they have a finite coherence time. In situations where
the time of observation of the experiment falls below this coherence time, they should be
modeled as a stochastic superposition of plane waves oscillating at slightly different frequen-
cies. We believe that this should not significantly change the various sensitivity estimates
made in this thesis, but as soon as data analysis is concerned, one should take this stochastic
effect into account.

The outlooks opened by this thesis are large and extend from the re-analysis of existing
data (such as *MICROSCOPE* or *SHUKET*), the proposal of new experiments to search for
ultralight dark matter, and the development of international and futuristic projects (*AION-10*,
*MAGIS-100* and *LISA*) that will search for ultralight dark matter.
Acknowledgments

I genuinely enjoyed these years of PhD, and I owe it to many people I would like to thank. I will start by thanking deeply my two PhD advisors, Peter Wolf and Aurélien Hees. I am extremely grateful to have been their PhD student and to have enjoyed their supervision. I really appreciated their pedagogy, open-mindedness and humility, despite their experience. Like many other scientists who know them personally, I am still amazed by their intuition, their rigor and the (very) broad spectrum of knowledge they have. One day, I hope to become a scientist as complete as they are.

There are many other people who have had a very positive impact on me throughout these three years. First, I would like to offer a special thanks to Marc Lilley and Robin Corgier for helping me in the preparation of my defense, but more generally for their guidance all along my PhD. I am also thankful to Adrien Bourgoin, Samy Aoulad-Lafkih and Martin Staab for all their pieces of advice, scientifically, but not only. Then, I would like to thank all the members of the theory and metrology team of SYRTE for their support: Pacôme Delva, Christophe Le Poncin-Lafitte, Marie-Christine Angonin, Miltos Chatzinikos and Christopher Aykroyd. In addition, I really appreciated the help from all the people I collaborated with, and I thank them for their physical insights: Jérôme Lodewyck and Rodolphe Le Targat from SYRTE, and Etienne Savalle, Laurent Chevalier and Pierre Brun from CEA Saclay. More generally, I thank all the people from SYRTE with whom I exchanged directly or indirectly during these three years.

I would like to express my gratitude to all the members of my PhD jury, Franck Pereira Dos Santos, Diego Blas, Hartmut Grote, Clare Burrage, Antoine Petiteau and Marianna Safronova, for having accepted reading the manuscript and attending the defense. I am very glad to have a jury of such quality and I can only hope to have the opportunity to discuss more with them on those dark matter searches in the future.

Finally, I am grateful to all my friends and family, especially my parents for their unfailing support in my quest of becoming a researcher in fundamental physics. It has been a long journey, full of obstacles, but they always believed in me, and for this, I thank them with all my heart.
# Part IX
## Appendix
# Appendix A

## Electric fields inside a cavity

### A.1 Phase shift induced by vacuum birefringence

In this section, we explicitly derive Eq. (9.7), the phase of light measured at the output of a cavity, in the framework of vacuum birefringence induced by the axion-photon coupling. We follow closely the derivation of the phase explicit in [1]. We start by initializing the input electric field

$$ \vec{\epsilon}(t) = \vec{\epsilon}_0 e^{-i\omega_0 t}, \qquad (A.1) $$

with amplitude $\vec{\epsilon}_0$ and frequency $\omega_0$. We assume this field enters the cavity by one of the sides of the cavity. Note that in the following, all measurable electric fields are implicitly given by the real part only. When entering the cavity, the field gets transmitted by the cavity with a coefficient $\sqrt{1-r^2}$, where r is the mirror reflectivity of the cavity and becomes

$$ \vec{E}_0(t) = \sqrt{1-r^2} \vec{\epsilon}_0 e^{-i\omega_0 t}. \qquad (A.2) $$

After two reflections on the mirrors' surfaces, the electric field at the location of the first mirror is

$$ \vec{E}_1(t) = r^2 \vec{E}_0(t - \tau_{\leftrightarrow}) = r^2 \vec{E}_0(t) e^{2i \frac{\omega_0}{c} \ell} e^{2i \frac{\omega_0}{c} \delta l(t - \frac{\ell}{c})}, \qquad (A.3) $$

where we used Eq. (9.4a) to express the time variation of length of the cavity from the round trip $\tau_{\leftrightarrow}$. After n round trips, the electric field inside the cavity can be written as

$$ \vec{E}_n(t) = r^{2n} \vec{E}_0(t) e^{2in \frac{\omega_0}{c} \ell} e^{2i \frac{\omega_0}{c} \sum_{j=1}^{n} \delta \ell(t - (2j-1)\frac{\ell}{c})} \qquad (A.4) $$

at first order in the perturbation $\delta\ell(t)$. The total electric field transmitted outside the cavity (by the cavity's opposite wall, compared to where the input electric field enters) is then

$$ \vec{E}_{\text{tot}}^{\text{transmit}}(t) = \sum_{n=0}^{\infty} E_n^{\text{transmit}}(t) = \frac{\sqrt{1-r^2} \vec{E}_0(t)}{r^2} \sum_{n=1}^{\infty} r^{2n} e^{2in \frac{\omega_0}{c} \ell} e^{2i \frac{\omega_0}{c} \sum_{j=1}^{n} \delta \ell^{(j)}}. \qquad (A.5) $$

Since we are interested in the field transmitted to the second mirror (not the one from which the field initially enters the cavity), we explicitly add a factor $\sqrt{1-r^2}$ to account for the transmission to the second mirror and we remove a factor $r^2$, since in that case, the field is not reflected on the second mirror and then on the first one on his way back. This is also
why we sum over the fields from $n = 1$. We also defined $\delta\ell^{(j)} = \delta\ell(t - (2j - 1)\frac{\ell}{c})$. We now use the fact that the parameter $\delta\ell^{(j)}$ is small ($O(g_{a\gamma})$), allowing us to expand the exponential at first order in $\delta\ell^{(j)}$, i.e

$$ \sum_{n=1}^{\infty} r^{2n} e^{2i\frac{\omega_0}{c}\ell} e^{2i\frac{\omega_0}{c}\sum_{j=1}^{n} \delta\ell^{(j)}} = \sum_{n=1}^{\infty} r^{2n} e^{2ink_0\ell} + 2ik_0 \sum_{n=1}^{\infty} r^{2n} e^{2ink_0\ell} \sum_{j=1}^{n} \delta\ell^{(j)} \quad (A.6a) $$

$$ = \sum_{n=1}^{\infty} r^{2n} e^{2ink_0\ell} + 2ik_0 \left( \delta\ell^{(1)} \sum_{n=1}^{\infty} r^{2n} e^{2ink_0\ell} + \delta\ell^{(2)} \sum_{n=2}^{\infty} r^{2n} e^{2ink_0\ell} + \dots \right) \quad (A.6b) $$

$$ = \left( r^2 e^{2ik_0\ell} + 2ik_0 r^2 e^{2ik_0\ell} \delta\ell^{(1)} + 2ik_0 r^4 e^{4ik_0\ell} \delta\ell^{(2)} + \dots \right) \sum_{n=0}^{\infty} r^{2n} e^{2ink_0\ell} \quad (A.6c) $$

$$ = r^2 e^{2ik_0\ell} \left( 1 + \frac{2ik_0}{r^2} \sum_{n=1}^{\infty} \delta\ell^{(n)} r^{2n} \right) \sum_{n=0}^{\infty} r^{2n} e^{2ink_0\ell} \quad (A.6d) $$

$$ = \frac{r^2 e^{2ik_0\ell}}{1 - e^{2ik_0\ell r^2}} e^{\frac{2ik_0}{r^2} \sum_{n=1}^{\infty} \delta\ell^{(n)} r^{2n}} \quad (A.6e) $$

where $k_0 = \omega_0/c$ and where at the last line, we inverted the Taylor expansion, such that the total transmitted electric field becomes

$$ \vec{E}_{\text{tot}}^{\text{transmit}}(t) = \frac{\sqrt{1-r^2} \vec{E}_0(t) e^{2ik_0\ell} e^{\frac{2ik_0}{r^2} \sum_{n=1}^{\infty} \delta\ell^{(n)} r^{2n}}}{1 - e^{2ik_0\ell r^2}} \quad (A.7) $$

We recover the usual form of electromagnetic plane wave where the amplitude $\vec{A}(t)$ and phase $\phi(t)$ are respectively given by

$$ \vec{A}(t) = \frac{(1 - r^2)e^{2ik_0\ell}}{1 - e^{2ik_0\ell r^2}} \vec{\epsilon}_0 \quad (A.8a) $$

$$ \phi(t) = \omega_0 t - \frac{2k_0}{r^2} \sum_{n=1}^{\infty} \delta\ell^{(n)} r^{2n}. \quad (A.8b) $$

When the light source is locked on a mode of the cavity, i.e $k_0\ell = \pi m, m \in \mathbb{N}$, the transmitted amplitude is equal to the input amplitude. Using Eq. (9.6), the phase shift between the transmitted field and the input oscillates at the axion frequency $\omega_a$ whose amplitude is

$$ |\Delta\phi(t)| = \left| \frac{2k_0}{r^2} \sum_{n=1}^{\infty} r^{2n} \delta\ell^{(n)} \right| \quad (A.9a) $$

$$ = \frac{2\sqrt{16\pi G\rho_{\text{DM}}E_P g_{a\gamma}}}{\omega_a c} \sin^2\left(\frac{\omega_a \ell}{2c}\right) \frac{\left|\cos\left(\omega_a t - \frac{2\omega_a \ell}{c} + \Phi\right) - r^2 \cos\left(\omega_a t + \Phi\right)\right|}{1 - 2r^2 \cos\left(\frac{2\omega_a \ell}{c}\right) + r^4} \quad (A.9b) $$

$$ \approx \frac{2\sqrt{16\pi G\rho_{\text{DM}}E_P g_{a\gamma}}}{\omega_a c\sqrt{1 - 2r^2 \cos\left(\frac{2\omega_a \ell}{c}\right) + r^4}} \sin^2\left(\frac{\omega_a \ell}{2c}\right). \quad (A.9c) $$
## A.2 Amplitude of the square of the total electric field inside a cavity

In this appendix, we derive the amplitude of the electric field power inside a cavity Eq. (10.12a) in the context of the experiment presented in Chapter 10. From Eq. (10.11), we can write the signal amplitude as

$$ \sqrt{\left(\vec{A}(\omega_A) \cdot \vec{C}(\omega_U) + \vec{B}(\omega_A) \cdot \vec{D}(\omega_U)\right)^2 + \left(\vec{B}(\omega_A) \cdot \vec{C}(\omega_U) - \vec{A}(\omega_A) \cdot \vec{D}(\omega_U)\right)^2} \quad (A.10a) $$

$$ \equiv X_A X_{DM} \beta \sqrt{(A'(\omega_A)^2 + B'(\omega_A)^2) \times (C'(\omega_U)^2 + D'(\omega_U)^2)} \quad (A.10b) $$

where

$$ \beta = \hat{e}_{DM} \cdot \frac{\vec{X}_A}{X_A}, \quad (A.10c) $$

is the projection of the polarization of the DP field on the polarization of the injected electric field, such that $\vec{X}_A \cdot \vec{X}_{DM} = \vec{X}_A \cdot \vec{X}_{DM,\parallel} = X_A X_{DM} \beta$ and where the prime quantities are defined as

$$ A'(\omega_A) \equiv \frac{\sqrt{1-r^2} (1+r) \cos\left(\frac{\omega_A L}{2c}\right)}{1+2r \cos\left(\frac{\omega_A L}{c}\right)+r^2}, \quad (A.11a) $$

$$ B'(\omega_A) \equiv \frac{\sqrt{1-r^2} (1-r) \sin\left(\frac{\omega_A L}{2c}\right)}{1+2r \cos\left(\frac{\omega_A L}{c}\right)+r^2}, \quad (A.11b) $$

$$ C'(\omega_U) \equiv 1 + \frac{2(1+r) \cos\left(\frac{\omega_U L}{2c}\right)}{1+2r \cos\left(\frac{\omega_U L}{c}\right)+r^2}, \quad (A.11c) $$

$$ D'(\omega_U) \equiv \frac{2(1-r) \sin\left(\frac{\omega_U L}{2c}\right)}{1+2r \cos\left(\frac{\omega_U L}{c}\right)+r^2}, \quad (A.11d) $$

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

$$ \frac{\sqrt{1-r^2} X_A X_{DM} \beta}{\sqrt{1+2r \cos\left(\frac{\omega_A L}{c}\right)+r^2}} \sqrt{1+4 \frac{1+(1+r) \cos\left(\frac{\omega_U L}{2c}\right)}{1+2r \cos\left(\frac{\omega_U L}{c}\right)+r^2}} \quad (A.12) $$

## A.3 Amplitude fluctuation inside a cavity

In this section, we show how to express the contribution in field power from the amplitude fluctuation of the applied field $\Delta X_A$ Eq. (17.6).
We use the same procedure as the one described in Section 10.2 to compute the RIN contribution to the total electric field at the center of the cavity. Starting from its expression in Eq. (17.5), the first contribution of noise reads

$$[\vec{E}_A^0]_{\text{RIN}}(x=0,t) = \mathfrak{R} \left[ \sqrt{1-r^2} \left( \frac{\Delta \vec{X}_A(\Delta\omega)}{2} \left( e^{-i(\omega_+ t - k_+ \frac{L}{2} + \phi_+)} + e^{-i(\omega_- t - k_- \frac{L}{2} + \phi_-)} \right) \right) \right], \quad (A.13)$$

where we used the trigonometric identity $2\cos(A)\cos(B) = \cos(A+B) + \cos(A-B)$. Here, the only angular frequency noise we are interested in is $\omega_0 = \Delta\omega$, hence $k_{\pm} = k_A \pm \Delta k$, $\Delta k = \Delta\omega/c$ and $\Delta\vec{X}_A$ is evaluated at $\Delta\omega$. After getting reflected on the other boundary, the second contribution is

$$[\vec{E}_A^1]_{\text{RIN}}(x=0,t) = \mathfrak{R} \left[ -\sqrt{1-r^2} re^{ik_A L} \left( \frac{\Delta \vec{X}_A(\Delta\omega)}{2} \left( e^{-i(\omega_+ t - k_+ \frac{L}{2} + \phi_+)} + e^{-i(\omega_- t - k_- \frac{L}{2} + \phi_-)} \right) \right) \right], \quad (A.14)$$

and after an infinite number of round trips N, the full RIN contribution on the electric field inside the cavity is

$$[\vec{E}_A^{\text{tot}}]_{\text{RIN}}(x=0,t) = \sum_{n=0}^{N=+\infty} \vec{E}_{\text{RIN}}^n(x=0,t) \quad (A.15a)$$

$$= \sqrt{1-r^2} \Delta \vec{X}_A(\Delta\omega) \left( \mathfrak{R} \left[ \frac{e^{-i(\omega_+ t + \phi_+)} e^{i \frac{k_+ L}{2}}}{1 + re^{ik_+ L}} + \frac{e^{-i(\omega_- t + \phi_-)} e^{i \frac{k_- L}{2}}}{1 + re^{ik_- L}} \right] \right) \quad (A.15b)$$

$$= \frac{\Delta X_A(\Delta\omega)}{2X_A} \sum_{i=\pm} \left( \vec{A}(\omega_i) \cos(\omega_i t + \phi_i) + \vec{B}(\omega_i) \sin(\omega_i t + \phi_i) \right) \quad (A.15c)$$

$$= \frac{\Delta X_a(\Delta\omega)}{2X_A} \left( \vec{A}(2\omega_A - \omega_U) \cos([2\omega_A - \omega_U]t + \phi_+) + \vec{A}(\omega_U) \cos(\omega_U t + \phi_-) + \vec{B}(2\omega_A - \omega_U) \sin([2\omega_A - \omega_U]t + \phi_+) + \vec{B}(\omega_U) \sin(\omega_U t + \phi_-) \right) \quad (A.15d)$$

where the functions $\vec{A}$ and $\vec{B}$ have already been defined in Eqs. (10.7) and where we used $\omega_{\pm} = \omega_A \pm \Delta\omega$ at the last line, and where $\phi_{\pm} = \phi_A \pm \phi_0$. This expression has to be directly added to the noise-less contribution from Eq. (10.6c).

Then, the RIN contribution to the square of the total electric field at the center of the cavity can be obtained by multiplying the last equation with Eq. (10.6c), as we work at first order in $\Delta X_A/X_A$. Keeping only the terms oscillating at angular frequency $\Delta\omega$, the RIN contribution to $E^2$ is given by

$$[E^2(\omega_U, \omega_A)]_{\text{RIN}} = \frac{\Delta X_A(\Delta\omega)}{2X_A} \left( (A(\omega_A)[A(2\omega_A - \omega_U) + A(\omega_U)] + B(\omega_A)[B(2\omega_A - \omega_U) + B(\omega_U)]) \cos(\Delta\omega t + \phi_0) + (A(\omega_A)[B(2\omega_A - \omega_U) - B(\omega_U)] + B(\omega_A)[A(\omega_U) - A(2\omega_A - \omega_U)]) \sin(\Delta\omega t + \phi_0) \right) \quad (A.16a)$$

$$= \frac{\Delta X_A(\Delta\omega)}{2X_A} \sqrt{N(\omega_U, \omega_A)} \cos(\Delta\omega t + \varphi) = \sqrt{\frac{P_{\text{RIN}}N(\omega_U, \omega_A)}{8T_{\text{obs}}\Delta\omega}} \cos(\Delta\omega t + \varphi), \quad (A.16b)$$
where we have used Eqs. (17.4b) and (17.4c). In this expression, the functions A and B are the norm of $\vec{A}$ and $\vec{B}$ and $\varphi$ is an irrelevant phase dependent on A and B.

The function $\sqrt{N(\omega_U, \omega_A)}$ correspond to the noise amplification factor by the cavity, quadratic in $X_A$ and whose expression is given by

$$N(\omega_U, \omega_A) \equiv N_1(\omega_U, \omega_A) + N_2(\omega_U, \omega_A) + N_3(\omega_U, \omega_A), \quad (A.17a)$$

where

$$N_1(\omega_U, \omega_A) \equiv A(\omega_A)^2 \left( (A(2\omega_A - \omega_U) + A(\omega_U))^2 + (B(2\omega_A - \omega_U) - B(\omega_U))^2 \right), \quad (A.17b)$$

$$N_2(\omega_U, \omega_A) \equiv B(\omega_A)^2 \left( (A(2\omega_A - \omega_U) - A(\omega_U))^2 + (B(2\omega_A - \omega_U) + B(\omega_U))^2 \right), \quad (A.17c)$$

$$N_3(\omega_U, \omega_A) \equiv 4A(\omega_A)B(\omega_A) (A(2\omega_A - \omega_U)B(\omega_U) + B(2\omega_A - \omega_U)A(\omega_U)). \quad (A.17d)$$
# Appendix B

## Motion of atomic wavepackets inside an atom interferometer

In this appendix, we compute rigorously the wavepackets position and velocity along the $\pi/2 - \pi - \pi/2$ interferometer, which will be used for the computation of the phase shift observable in Chapter 13.

We start by the calculation of the position and velocity of the atom at the end of the trajectory portion 1 in Fig. 13.1. From Eq. (13.6), the atom does not undergo any kick velocities from the laser pulses, hence after a time T, its equations of motion read

$$ \vec{x}_A^{(1)}(t=T) = \vec{v}_{\text{DM}} \left( T - \frac{[Q_M^A]_d X_{\text{DM}}}{\omega_\phi} [\sin(\omega_\phi T + \Phi) - \sin(\Phi) - \omega_\phi T \cos(\Phi)] \right) \quad (\text{B.1a}) $$

$$ \vec{v}_A^{(1)}(t=T) = \vec{v}_{\text{DM}} \left( 1 - [Q_M^A]_d X_{\text{DM}} [\cos(\omega_\phi T + \Phi) - \cos(\Phi)] \right). \quad (\text{B.1b}) $$

On the other hand, the wavepacket on portion 2 in Fig. 13.1 has undergone a kick velocity with amplitude $v_{\text{kick},0} + \delta v_{\text{kick}}(t=0)$, following Eq. (13.7), with

$$ \delta \vec{v}_{\text{kick}}(t=0) = \vec{v}_{\text{kick},0} ([Q_\omega^L]_d - [Q_M^A]_d) X_{\text{DM}} \cos(\Phi), \quad (\text{B.2}) $$

with $\vec{v}_{\text{kick},0}$ along an arbitrary direction $\hat{x}$, hence

$$ \vec{x}_A^{(2)}(t=T) = (\vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0})T - X_{\text{DM}} \left( 2 \frac{[Q_M^A]_d}{\omega_\phi} \sin\left(\frac{\omega_\phi T}{2}\right) \cos\left(\frac{\omega_\phi T}{2} + \Phi\right) (\vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0}) + T([Q_M^A]_d \vec{v}_{\text{DM}} - [Q_\omega^L]_d \vec{v}_{\text{kick},0}) \cos(\Phi) \right), \quad (\text{B.3a}) $$

$$ \vec{v}_A^{(2)}(t=T) = \vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0} - X_{\text{DM}} ([Q_M^A]_d \cos(\omega_\phi T + \Phi)(\vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0}) + ([Q_M^A]_d \vec{v}_{\text{DM}} + [Q_\omega^L]_d \vec{v}_{\text{kick},0}) \cos(\Phi)), \quad (\text{B.3b}) $$

At time T, both wavepackets undergo a kick velocity with opposite direction, such that their momenta states are exchanged. Following Eq. (13.7), this means that at the end of portion 1, the atom undergoes a kick velocity of amplitude $v_{\text{kick},0} + \delta v_{\text{kick}}(t=T)$, with

$$ \delta \vec{v}_{\text{kick}}(t=T) = \vec{v}_{\text{kick},0} ([Q_\omega^L]_d - [Q_M^A]_d) X_{\text{DM}} \cos(\omega_\phi T + \Phi), \quad (\text{B.4}) $$
with same direction $\hat{x}$, hence, at the end of the portion 3, its coordinates read

$$ \vec{x}_A^{(3)}(t = 2T) = (2\vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0})T + X_{\text{DM}} \left( \frac{[Q_M^A]_d}{\omega_\phi} (\vec{v}_{\text{DM}}(\sin(\Phi) + 2\omega_\phi T \cos(\Phi)) + \right. \quad (B.5a) $$

$$ \vec{v}_{\text{kick},0} \sin(\omega_{\phi} T + \Phi) - (\vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0}) \sin(2\omega_{\phi} T + \Phi) + [Q_{\omega}^L]_d \vec{v}_{\text{kick},0} T \cos(\omega_{\phi} T + \Phi) $$

$$ \vec{v}_A^{(3)}(t = 2T) = \vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0} - X_{\text{DM}} \left( [Q_M^A]_d \left( (\vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0}) \cos(2\omega_\phi T + \Phi) - \vec{v}_{\text{DM}} \cos(\Phi) \right) + \right. \\ \left. [Q_\omega^L]_d \vec{v}_{\text{kick},0} \cos(\omega_\phi T + \Phi) \right). \quad (B.5b) $$

The atom at the end of the portion 2 undergoes a kick velocity in the other direction, i.e of amplitude $v_{\text{kick},0} + \delta v_{\text{kick}}(t = T)$, but with opposite direction $(-\hat{x})$, compared to the previous laser kicks, hence

$$ \vec{x}_A^{(4)}(t = 2T) = (2\vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0})T - X_{\text{DM}} \left( \frac{[Q_M^A]_d}{\omega_\phi} (\vec{v}_{\text{kick},0} \sin(\omega_\phi T + \Phi) + \vec{v}_{\text{DM}} \sin(2\omega_\phi T + \Phi) - \right. \\ \left. (\vec{v}_{\text{DM}} + \vec{v}_{\text{kick},0}) \sin(\Phi) - 2\vec{v}_{\text{DM}} \omega_\phi T \cos(\Phi) - [Q_\omega^L]_d \vec{v}_{\text{kick},0} T (\cos(\omega_\phi T + \Phi) - 2 \cos(\Phi)) \right), \quad (B.6a) $$

$$ \vec{v}_A^{(4)}(t = 2T) = \vec{v}_{\text{DM}} \left( 1 + 2X_{\text{DM}}[Q_M^A]_d \sin(\omega_\phi T) \right) \sin(\omega_\phi T + \Phi) + 2X_{\text{DM}}[Q_\omega^L]_d \vec{v}_{\text{kick},0} \sin\left(\frac{\omega_\phi T}{2}\right) \\ \sin\left(\frac{\omega_\phi T}{2} + \Phi\right), \quad (B.6b) $$

at the end of portion 4.

Since at the end of portion 3, the wavepacket is in the excited state, we must take into account an additional kick at time $t = 2T$ of amplitude $v_{\text{kick},0} + \delta v_{\text{kick}}(t = 2T)$ and direction $-\hat{x}$ to this wavepacket, in order to put it back to the ground state. Since, we assume the detection to be immediately after the kick, the position of the wavepacket will not be impacted by this additional kick. The final velocity of this wavepacket read

$$ \vec{v}_A^{(4)}(2T) = \vec{v}_{\text{DM}} \left( (1 - X_{\text{DM}}[Q_M^A]_d (\cos(2\omega_\phi T + \Phi) - \cos(\Phi))) - \right. \\ \left. X_{\text{DM}}[Q_\omega^L]_d \vec{v}_{\text{kick},0} (\cos(2\omega_\phi T + \Phi) - \cos(\omega_\phi T + \Phi)) \right), \quad (B.7) $$

resulting in a different velocity compared to the other wavepacket $(\vec{v}_A^{(4)}(2T))$.
# Appendix C

## Projection of galactic velocity onto sensitive axes of various experiments

In this appendix, we will compute the factors that are orientation dependent in the various acceleration or phase shifts observables, presented in Chapters 12 and 13.

The observable signatures depend on the orientation of the experimental setup with respect to the initial velocity of the atoms or tests masses, which corresponds at leading order to the galactic velocity $v_{DM}$. Indeed, for classical tests of the universality of free fall between two macroscopic bodies, one measures $\Delta\vec{a} \propto \vec{v}_{DM}$ (see Eq.(12.5)) projected onto the sensitive axis of the instrument. Therefore, the signal depends on the projection of the velocity of the bodies in the galactocentric frame with the sensitive axis of the experiment. Similarly, all the AI experiments depend on the scalar product between the initial galactocentric velocity of the atomic clouds and the direction of the velocity kick undergone in the interferometric schemes, i.e. on $\hat{e}_v \cdot \hat{e}_{kick}$, see Eqs. (13.21) and (13.27).

As mentioned in Chapter 15, the direction of the DM velocity in the galactic halo points towards $\alpha_{DM}$, $\delta_{DM} = 310.36^\circ\text{E}, 45.28^\circ\text{N}$ [2] in the equatorial frame.

### C.1 Atom interferometers

All the AI-based experiments operate at constant location, loc, on Earth with longitude $\lambda_{loc}$ and latitude $\phi_{loc}$. We assume the velocity kick to be directed vertically, i.e. in the Earth's reference frame $\hat{e}_{kick} = (\cos(\lambda_{loc}) \cos(\phi_{loc}), \sin(\lambda_{loc}) \cos(\phi_{loc}), \sin(\phi_{loc}))$. As first approximation, we consider the declination as equivalent to the terrestrial latitude, i.e $\delta_{DM} \approx \phi_{DM}$, such that the dot product $\hat{e}_v \cdot \hat{e}_{kick}$ is simply given by $\cos(\phi_{loc}) \cos(\phi_{DM}) \cos(\lambda_{loc} - \lambda_{DM}(t)) + \sin(\phi_{loc}) \sin(\phi_{DM})$. $\lambda_{DM}(t)$ is the longitude of $\alpha$ Cygni at the time $t$ of the experiment. Indeed, while $\alpha_{DM}$ is fixed, the former follows the Earth rotation with frequency $\omega_E \sim 7 \times 10^{-5}$ Hz and is therefore time dependent

$$ \lambda_{DM}(t) = \omega_E t + \varphi, \qquad (C.1a) $$

where the phase $\varphi$ corresponds to the longitude of $\alpha$ Cygni at the origin of time reference considered. For short experiments with an experimental time much smaller than a day (i.e. relevant for Stanford's experiment, see Sec 17.5.1), the dot product is roughly constant and depends therefore on the exact time of the day when the experiment was conducted. In
order to infer a sensitivity estimate, we will only consider the mean value of the dot product which is given by $sin(\phi_{loc}) sin(\phi_{DM})$, i.e

$$ \hat{e}_v \cdot \hat{e}_{\text{kick}}|_{\text{Stanford}} \approx 0.43. \qquad (\text{C.1b}) $$

For longer time experiments, like the SPID variation, the dot product evolves with time, such that the signal evolves as

$$ s(t) \propto \hat{e}_v \cdot \hat{e}_{\text{kick}} \cos(\omega_{\text{DM}}t + \Phi) = [\cos(\phi_{\text{loc}}) \cos(\phi_{\text{DM}}) \cos(\omega_E t + \varphi) + \sin(\phi_{\text{loc}}) \sin(\phi_{\text{DM}})] \cos(\omega_{\text{DM}}t + \Phi), \qquad (\text{C.1c}) $$

i.e the Earth rotation modulates the signal at frequency $f_E = \omega_E/2\pi$. We will make two approximations for our estimates. For all the DM frequencies of interest $\omega_{DM}$, the Earth rotation is a slowly varying function, because $\omega_E < \omega_{DM}$. Therefore, we will assume that the signal will manifest itself by a single peak at frequency $f_{DM} = \omega_{DM}/2\pi$ in Fourier space. In addition, we will be interested only by the maximum value of the dot product, which is only a function of the different latitudes.

$$ \hat{e}_v \cdot \hat{e}_{\text{kick}} = \text{Max}(|\cos(\phi_{\text{loc}} \pm \phi_{\text{DM}})|). \qquad (\text{C.1d}) $$

For the various locations under consideration in this thesis, we have

$$ \hat{e}_v \cdot \hat{e}_{\text{kick}}|_{\text{Oxford}} \approx 0.99, \qquad (\text{C.1e}) $$

$$ \hat{e}_v \cdot \hat{e}_{\text{kick}}|_{\text{Fermilab}} \approx 1.00, \qquad (\text{C.1f}) $$

where we assume the SPID variation operates at the same location as AION-10, i.e Oxford [3] for consistent comparison.

## C.2 MICROSCOPE

For MICROSCOPE, the axis of measurement is alongside the test masses cylinders' longitudinal symmetry axis [4]. The orbital motion of the satellite around Earth is sun-synchronous, which means that the orientation of the orbital plane evolves with time with an annual period. As it was mentioned in Section 17.4, the measurements are distributed on 17 different sessions, each of them lasting $T_{\text{int}}/17 \sim 5$ days on average. For such duration, we can assume the orbital plane to be fixed during each session. In addition, the satellite spins around an axis that is orthogonal to the axis of measurement with angular frequency $\omega_{\text{spin}}$. In total, this means the dot product can be written as

$$ \hat{e}_v \cdot \hat{e}_{\text{meas.}}(t)|_{\text{\mu SCOPE}} = A(t) \cos(\omega_{\text{spin}} t + \psi), \qquad (\text{C.1g}) $$

where $A(t)$ depends on the orientation of the orbital plane, i.e is fixed for one session but changes from one session to another, and with $\psi$ an irrelevant phase. Using MICROSCOPE's publicly available data, we estimated numerically the coefficient $A(t)$ for every session and we find that its value oscillates between 0.71 and 1. For our estimates, we will consider
its mean value, i.e. $|\hat{e}_v \cdot \hat{e}_{meas}| \approx 0.85$. In conclusion, the signal presented in Eq. (12.5) is modulated by the factor

$$ \hat{e}_v \cdot \hat{e}_{meas}(t)|_{\mu\text{SCOPE}} \approx 0.85 \cos(\omega_{\text{spin}}t + \psi). \qquad (\text{C.1h}) $$

Thus, the observable signal in the *MICROSCOPE* experiment is oscillating at the combination of the DM frequency and the spin frequency, i.e. has harmonics at the two frequencies $\omega_{DM} \pm \omega_{spin}$. Therefore, for DM frequencies in Eq. (12.5) such that $\omega_{DM} \ll \omega_{spin}$, we will consider that the signal in *MICROSCOPE* oscillates at $\omega_{spin}$, whereas if $\omega_{DM} \gg \omega_{spin}$, we will make our estimate with the signal oscillating at $\omega_{DM}$. As explained in Section 17.4, $\omega_{spin}$ depends on the measurement session since three different spinning frequencies have been used during the full mission. They differ by a factor 5 roughly [5]. For our rough sensitivity estimates, we use the data from session 404 as a basis, therefore we will assume a constant $\omega_{spin} \sim 18.4$ mrad/s.
# Appendix D

## Fast likelihood for scalar ultralight dark matter

In this appendix, we derive (in time domain) the slowly oscillating component of TDI X combination of a scalar ULDM signal, that will be used for the DM model in *FastDM* in Section 15.2.

We will make the detailed calculations for a (pure scalar) DM signal, and then we generalize to GW. Let us start again with the one-link response function induced by scalar DM Eq. (15.3b) at time of reception of the photon of the receiver $r$ $t_r$, when it is emitted at time $t_e = t_r - L/c$ by emitter $e$

$$y_{re}^{\text{DM}}(t_r) = (\hat{n}_{re} \cdot \hat{e}_v) \frac{\sqrt{16\pi G \rho_{\text{DM}} v_{\text{DM}} [Q_M]}_d}{\omega_{\phi} c^2} \mathfrak{R} [e^{i\phi_r} - e^{i\phi_e}] \quad (\text{D.1a})$$

$$= (\hat{n}_{re} \cdot \hat{e}_v) \frac{\sqrt{16\pi G \rho_{\text{DM}} v_{\text{DM}} [Q_M]}_d}{\omega_{\phi} c^2} \mathfrak{R} [e^{i\phi_r} (1 - e^{-2i\alpha_{re}^{\text{DM}}})] \quad (\text{D.1b})$$

$$= 2 (\hat{n}_{re} \cdot \hat{e}_v) \frac{\sqrt{16\pi G \rho_{\text{DM}} v_{\text{DM}} [Q_M]}_d}{\omega_{\phi} c^2} \sin(\alpha_{re}^{\text{DM}}) \mathfrak{R} [i e^{i\phi_r} e^{-i\alpha_{re}^{\text{DM}}}] \quad (\text{D.1c})$$

$$\equiv Y_{re}(t_r) e^{i\phi_r}, \quad (\text{D.1d})$$

where

$$\phi_r = \omega_{\phi} t_r - \vec{k}_{\phi} \cdot \vec{x}_r(t_r) + \Phi \quad (\text{D.2a})$$

$$\phi_e = \omega_{\phi} t_e - \vec{k}_{\phi} \cdot \vec{x}_e(t_e) + \Phi, \quad (\text{D.2b})$$

and therefore

$$2\alpha_{re}^{\text{DM}}(t_r) = \phi_r - \phi_e = \omega_{\phi} \left( (t_r - t_e) - \frac{\hat{k} \cdot \hat{n}_{re} |\vec{v}_{\text{DM}}|}{c^2} \left| \vec{x}_r(t_r) - \vec{x}_e(t_e) \right| \right) \quad (\text{D.2c})$$

$$= \frac{\omega_{\phi} |\vec{x}_r(t_r) - \vec{x}_e(t_e)|}{c} \left( 1 - \frac{\hat{k} \cdot \hat{n}_{re} |\vec{v}_{\text{DM}}|}{c} \right) + \mathcal{O}(\text{Shapiro}), \quad (\text{D.2d})$$

where we neglect the impact of the Shapiro delay on the signal. The expression of $\alpha_{re}^{\text{DM}}$ is therefore given by

$$\alpha_{re}^{\text{DM}}(t_r) = \frac{\omega_{\phi} |\vec{x}_r(t_r) - \vec{x}_e(t_e)|}{2c} \left( 1 - \frac{\hat{k} \cdot \hat{n}_{re} |\vec{v}_{\text{DM}}|}{c} \right). \quad (\text{D.3})$$
Let us now decompose the one-link response function into a slowly evolving part and a fast oscillating part. Let us consider the observation time baseline $T_{\text{obs}}$ such that the corresponding Fourier frequencies are given by integer times $1/T_{\text{obs}}$. Let us consider the Fourier frequency $q_0$ which is the closest to $f_{\phi}$, i.e.

$$q_0 = \text{round} \left[ f_{\phi} T_{\text{obs}} \right] / T_{\text{obs}} \qquad (\text{D.4})$$

where round $[x]$ provides the closest integer to $x$. One can rewrite Eq. (D.1) as

$$y_{re}^{\text{DM}}(t_r) = \mathfrak{R} [\mathcal{Y}_{re}^{\text{DM}}(t_r) e^{2\pi i q_0 t_r}], \qquad (\text{D.5a})$$

where

$$\mathcal{Y}_{re}^{\text{DM}} = 2i (\hat{n}_{re} \cdot \hat{e}_v) \frac{\sqrt{16\pi G \rho_{\text{DM}} v_{\text{DM}} [Q_M]_d}}{\omega_{\phi} c^2} \sin(\alpha_{re}^{\text{DM}}) e^{-i\beta_{re}^{\text{DM}}}, \qquad (\text{D.5b})$$

with

$$\begin{aligned} \beta_{re}^{\text{DM}} &= \alpha_{re}^{\text{DM}} + 2\pi q_0 t_r - \phi_r \\ &= \alpha_{re}^{\text{DM}} - \Phi + 2\pi(q_0 - f_{\phi})t_r + \frac{2\pi f_{\phi}}{c} \frac{|\vec{v}_{\text{DM}}|}{c} \hat{k} \cdot \vec{x}_r(t_r). \end{aligned} \qquad (\text{D.5c})$$

One can do the exact same job for the GW signals starting from Eq. (15.13b) to find

$$\mathcal{Y}_{re}^{\text{GW}} = \frac{\mathcal{A}}{2(1 - \hat{n}_{re} \cdot \hat{k}_{\text{GW}})} \sin(\alpha_{re}^{\text{GW}}) \mathfrak{R} \left[ i \hat{n}_{ij}^{\text{SSB}}(\iota, \Psi) \hat{n}_{re}^i \hat{n}_{re}^j e^{-i\beta_{re}^{\text{GW}}} \right] \qquad (\text{D.6a})$$

$$\alpha_{re}^{\text{GW}} = \pi \left( f_{\text{GW}} + \dot{f}_{\text{GW}} \xi_r \right) |\vec{x}_r(t_r) - \vec{x}_e(t_e)| \left( 1 - \hat{k} \cdot \hat{n}_{re}(t_r) \right) \qquad (\text{D.6b})$$

$$\beta_{re}^{\text{GW}} = \alpha_{re}^{\text{GW}} + \Phi_{\text{GW}} + 2\pi(q_0 - f_{\text{GW}})t_r + \frac{2\pi f_{\text{GW}}}{c} \hat{k} \cdot \vec{x}_r(t_r) - \pi \dot{f}_{\text{GW}} \xi_r^2, \qquad (\text{D.6c})$$

where $\xi_r = t_r - \hat{k} \cdot \vec{x}_r / c$.

We now quickly show how to express TDI combinations in this framework. We will focus on the case of constant and equal arm-length $L$. In TDI, we encounter quantities such as

$$y_{ij} \left( t - \frac{L}{c} \right) = Y_{ij} \left( t - \frac{L}{c} \right) e^{i \phi_i \left( t - \frac{L}{c} \right)}. \qquad (\text{D.7})$$

The quantity $Y_{ij}$ are slowly evolving with time such that $Y_{ij}(t - L/c) \approx Y_{ij}(t)$. For GW, this assumption holds if the GW amplitude slowly evolves with time, if the frequency drift of the GW is small and if the spacecraft velocity is small compared to the speed of light. For DM, this assumption holds for the same reasons, in particular, the amplitude of the wave is constant (in the approximation where the time of integration is smaller than the coherence time), and $Y_{ij}$ depends on the spacecraft velocity, which is small compared to $c$. Taking DM as an example, we can write

$$\phi_i \left( t - \frac{L}{c} \right) = \omega_{\phi} t - \frac{\omega_{\phi} L}{c} - \vec{k}_{\phi} \cdot \vec{x}_i \left( t - \frac{L}{c} \right) + \Phi \qquad (\text{D.8a})$$

$$= \omega_{\phi} t - \frac{\omega_{\phi} L}{c} - \vec{k}_{\phi} \cdot \vec{x}_i(t) + \Phi + \mathcal{O}(v_j/c \times L/x_i) \qquad (\text{D.8b})$$

$$\approx \phi_i(t) - \frac{\omega_{\phi} L}{c}, \qquad (\text{D.8c})$$