where $v_j$ is the velocity of the emitting spacecraft. This leads to

$$y_{ij}^{\text{DM}}\left(t - \frac{L}{c}\right) \approx \mathfrak{R} \left[ \gamma_{ij}^{\text{DM}}(t) e^{i\phi_i(t)} e^{-\frac{i\omega_{\phi_i}L}{c}} \right] = \mathfrak{R} \left[ \mathcal{Y}_{ij}^{\text{DM}}(t) e^{-\frac{i\omega_{\phi_i}L}{c}} e^{2\pi i q_0 t} \right]. \quad (D.9)$$

Applying recursively this relationship leads to

$$D_{ijk} y_{kl}^{\text{DM}}(t) = \mathfrak{R} \left[ \mathcal{Y}_{kl}^{\text{DM}}(t) e^{-\frac{2i\omega_{\phi_i}L}{c}} e^{2\pi i q_0 t} \right], \quad (D.10)$$

and so on. Using this result, the first generation TDI combination can be written as

$$X_1^{\text{DM}}(t) = \mathfrak{R} \left[ \mathcal{X}_1^{\text{DM}}(t) e^{2\pi i q_0 t} \right], \quad (D.11a)$$

and $\mathcal{X}_1^{\text{DM}}$ is slowly evolving with time and is given by (using Eq. (14.7a))

$$\begin{aligned} \mathcal{X}_1^{\text{DM}} &= \left(1 - e^{\frac{-2i\omega_{\phi_i}L}{c}}\right) \left[ \mathcal{Y}_{13}^{\text{DM}} - \mathcal{Y}_{12}^{\text{DM}} + e^{\frac{-i\omega_{\phi_i}L}{c}} \left( \mathcal{Y}_{31}^{\text{DM}} - \mathcal{Y}_{21}^{\text{DM}} \right) \right] \\ &= 2i \sin \left( \frac{\omega_{\phi_i}L}{c} \right) e^{\frac{-i\omega_{\phi_i}L}{c}} \left[ \mathcal{Y}_{13}^{\text{DM}} - \mathcal{Y}_{12}^{\text{DM}} + e^{\frac{-i\omega_{\phi_i}L}{c}} \left( \mathcal{Y}_{31}^{\text{DM}} - \mathcal{Y}_{21}^{\text{DM}} \right) \right]. \end{aligned} \quad (D.11b)$$

Similarly the second generation TDI can be written as

$$X_2^{\text{DM}}(t) = \mathfrak{R} \left[ \mathcal{X}_2^{\text{DM}}(t) e^{2\pi i q_0 t} \right], \quad (D.12a)$$

with the slowly evolving part

$$\mathcal{X}_2^{\text{DM}} = \left(1 - e^{\frac{-2i\omega_{\phi_i}L}{c}} - e^{\frac{-4i\omega_{\phi_i}L}{c}} + e^{\frac{-6i\omega_{\phi_i}L}{c}}\right) \left[ \mathcal{Y}_{13}^{\text{DM}} - \mathcal{Y}_{12}^{\text{DM}} + e^{\frac{-i\omega_{\phi_i}L}{c}} \left( \mathcal{Y}_{31}^{\text{DM}} - \mathcal{Y}_{21}^{\text{DM}} \right) \right]. \quad (D.12b)$$

Since $1 - x - x^2 + x^3 = (1 + x)(1 - x)^2$, the previous expression becomes simply

$$\begin{aligned} \mathcal{X}_2^{\text{DM}} &= \left(1 + e^{\frac{-2i\omega_{\phi_i}L}{c}}\right) \left(1 - e^{\frac{-2i\omega_{\phi_i}L}{c}}\right)^2 \left[ \mathcal{Y}_{13}^{\text{DM}} - \mathcal{Y}_{12}^{\text{DM}} + e^{\frac{-i\omega_{\phi_i}L}{c}} \left( \mathcal{Y}_{31}^{\text{DM}} - \mathcal{Y}_{21}^{\text{DM}} \right) \right] \\ &= \left(1 + e^{\frac{-2i\omega_{\phi_i}L}{c}}\right) \left(1 - e^{\frac{-2i\omega_{\phi_i}L}{c}}\right) \mathcal{X}_1^{\text{DM}} = 2i \sin \left(\frac{2\omega_{\phi_i}L}{c}\right) e^{\frac{-2i\omega_{\phi_i}L}{c}} \mathcal{X}_1^{\text{DM}}. \end{aligned} \quad (D.12c)$$

We have now expressed the second generation TDI combination in time domain induced by scalar DM as function of a slowly evolving part (the $\mathcal{X}$ factors) and a fast oscillating part, oscillating at the closest Fourier bin to the wave Compton frequency. As mentioned previously, this formulation is very convenient because in Fourier domain, the signal is simply the product of the Fourier transform of $\mathcal{X}$, which is numerically fast to obtain, and the Fourier transform of the $\exp(2\pi i q_0 t)$, which has an analytical solution (see [6]). These expressions are directly implemented in FastGB and FastDM softwares for the fit of both GB and scalar ULDM data.
# Appendix E

## Propagation of the electric field from the dish to the fictional plane in SHUKET

The calculation presented in Section 11.2 is based on the thin optical element approximation, which is valid only for low curvature dish $R \gg r$, see the discussion in Section 11.2.2. In this appendix, we want to check if Eq. (11.7) is valid in the SHUKET setup, in the case of emission from standing DP electric field. To do so, we propose the following "reverse engineering-like" method:

1. Consider the field on the dish using boundary conditions Eq. (11.1), i.e $\vec{U}_D(\vec{x}) = i\chi\omega_U\vec{Y}_{\parallel,D}(\vec{x})$,

2. Compute the field at each point $\vec{x}'$ on the plane closing the dish $\vec{U}_P(\vec{x}')$ from Eq. (11.7),

3. Compute numerically the field on the dish $\vec{U}_D^{\text{test}}(\vec{x})$ from $\vec{U}_P(\vec{x}')$ using Kirchhoff integral theorem Eq. (11.6),

4. Compute the relative error between $\vec{U}_D(\vec{x})$ and $\vec{U}_D^{\text{test}}(\vec{x})$.

We consider the plane located at $z' = R - a$, then the field at a point $(\rho', \phi', z')$ on the plane going towards the dish, located at $z \ge z'$, is given by

$$ \vec{U}_{P \to D}(\rho', \phi', z') = i\chi\omega_U e^{-ikf(\rho')} \vec{Y}_{\parallel,D}(\rho', \phi', f(\rho') + z'). \quad (E.1) $$

Notice the change of sign on the wavevector $k$, compared to Eq. (11.7) as we are now considering emission towards the dish (positive $z$ axis). In addition, we aim at computing the field reflected by the dish, while Eq. (11.7) gives the incident field. By boundary conditions, we assume that both incident and reflected are equal, up to a sign, hence the positive sign in front of Eq.(E.1).

Using Eqs. (11.6), (11.14) and (E.1), the field in a point on the receiving surface, i.e, the
dish, of coordinates $(\rho, \phi, z)$ is given by

$$ \vec{U}_D^{\text{test}}(\rho, \phi, f(\rho) + z') = -\frac{i\chi\omega_U f(\rho)}{2\pi} \int_0^r d\rho' \rho' e^{-ikf(\rho')} \int_0^{2\pi} d\phi' \frac{ikL-1}{L^3} e^{ikL} \vec{Y}_{\parallel,D}(\rho', \phi', f(\rho') + z') \quad (\text{E.2a}) $$

$$ \approx -\frac{i\chi\omega_U f(\rho)}{2\pi} \begin{pmatrix} Y_x \\ Y_y \\ 0 \end{pmatrix} \int_0^r d\rho' \rho' e^{-ikf(\rho')} \int_0^{2\pi} d\phi' \frac{ikL-1}{L^3} e^{ikL}, \quad (\text{E.2b}) $$

where the difference of z positions $\Delta z$ between the fictional plane and the dish from Eq. (11.14) is now positive and corresponds exactly to $f(\rho)$, $\Delta z = f(\rho) = (r^2 - \rho^2)/2R > 0$. Then, the relative error between $\vec{U}_D^{\text{test}}$ and $\vec{U}_D \approx i\chi\omega_U (Y_x, Y_y, 0)^T$ (from Eq. (17.14)) is

$$ \epsilon(\rho, \phi) \approx \left| \frac{\vec{U}_D^{\text{test}} - \vec{U}_D}{\vec{U}_D} \right| \approx \left| -\frac{f(\rho)}{2\pi} \left( \int_0^r d\rho' \rho' e^{-ikf(\rho')} \int_0^{2\pi} d\phi' \frac{ikL-1}{L^3} e^{ikL} \right) - 1 \right| \quad (\text{E.3a}) $$

Fig. E.1 shows the numerical computation of this relative error as function of the position

<figure><img src="image_39.png" /><figcaption>Figure E.1: Numerical computation of $\epsilon(\rho, \phi)$ from Eq. (E.3a) for various dish positions $(\rho, \phi)$. Over the full surface, it is of the order of the ‰. The dish antenna is located inside the non-radiative near field region of the fictional plane (i.e the distance $L < 0.62\sqrt{(2r)^3/\lambda}$). One can notice the interference pattern with a typical period ~ 0.05 m, which corresponds to the wavelength of emission.</figcaption></figure>

($\rho, \phi$) on the dish. One can notice that the error is less than 1%, implying that Eq. (11.7) can be safely used for the propagation of the electric field from the dish in the SHUKET system, with negligible error. Note that the error $\epsilon(\rho, \phi)$ is computed only for points on the dish with radial coordinates $\rho < r$, as the circle of points belonging to the dish with coordinates $(r, \phi, z_{\text{plane}}), \phi \in [0, 2\pi]$ belongs to the fictional plane as well, and Kirchhoff integral is only valid for the computation of the field at reception points outside the emission surface. Therefore, one can deduce that the calculations presented in [7] are valid for SHUKET.
<figure><img src="image_40.png" /></figure>
<figure><img src="image_41.png" /></figure>
# RÉSUMÉ

La matière noire ultralégère (avec pour acronyme anglais ULDM), classe de candidats de matière noire de faible masse (< 1 eV), constitue une alternative pertinente pour des modèles historiquement dominants, tels que les WIMPs, et a récemment fortement attiré l'attention de la communauté scientifique. Dans cette thèse, nous étudions de nombreuses stratégies expérimentales pour la détection directe d'ULDM, au sol et dans l'espace. Plus précisément, nous proposons une modélisation théorique d'expériences actuelles et futures, et nous dérivons une estimation de leurs sensibilités respectives. Nous nous concentrons principalement sur trois types de phénoménologies. La première consiste en une interaction entre un candidat de matière noire vectoriel, plus communément appelé photon noir (DP), et l'électromagnétisme, qui induit un faible champ électrique oscillant à la fréquence de Compton du DP. Nous proposons d'abord une méthode innovante pour détecter ce petit champ électrique, qui consiste à mesurer l'effet Stark quadratique sur des atomes de Rydberg, piégés au centre d'une cavité micro-onde. Nous montrons qu'une telle expérience aurait une sensibilité lui permettant d'être compétitive avec les expériences de laboratoire existantes. Une autre façon de détecter ce champ électrique consiste en l'utilisation d'une antenne parabolique qui le réfléchit et concentre ainsi la puissance électromagnétique en son centre de courbure, où l'on place une antenne cornet (par exemple, l'expérience SHUKET au CEA Saclay). Nous investiguons analytiquement les effets de diffraction et de couplage de mode dans ce type de configuration, et nous montrons que l'intensité du signal attendu peut être significativement réduite comparativement aux estimations habituelles. Dans cette étude, nous proposons aussi une optimisation des paramètres expérimentaux pour augmenter le signal. La section phénoménologie d'intérêt pour cette thèse est l'oscillation de la masse et de la fréquence de transition d'atomes ou de masses tests. Ces oscillations pourraient être produites par le couplage non universel entre les particules du Modèle Standard et un candidat scalaire de matière noire, comme le dilaton ou l'axion. Nous étudions en détail l'impact de ces oscillations sur plusieurs variations d'interféromètres atomiques et sur des expériences de tests classiques de l'universalité de la chute libre, et nous démontrons comment ces différentes expériences pourraient sonder des régions inexplorées de l'espace des paramètres. Ces oscillations de masse au repos pourraient aussi être observées à l'aide de détecteurs spatiaux d'ondes gravitationnelles, comme LISA, et nous investiguons la possibilité d'une telle détection en utilisant des orbites réalistes pour le détecteur. En particulier, en utilisant des méthodes bayésiennes, nous montrons que LISA pourra en effet séparer un signal de matière noire d'un signal d'ondes gravitationnelles. Nous montrons aussi que la faible vitesse de l'onde de matière noire n'est pas résoluble sur la majorité de la bande de fréquences de LISA, ce qui induit une diminution de la sensibilité de l'expérience aux constantes de couplage de la matière noire scalaire par rapport aux estimations précédentes. Enfin, nous étudions la birefringence du vide induite par l'interaction entre les photons et les axions, et nous nous intéressons à sa possible détection à l'aide de cavités optiques, de fibres optiques et de LISA. En particulier, nous montrons qu'une légère modification des bancs optiques de LISA lui permettrait de devenir l'expérience la plus sensible à ce couplage à faibles masses.

# MOTS CLÉS

Matière noire ultralégère, axion, dilaton, photon noir, capteur quantiques, horloge atomique, interferométrie (atomique, optique), LISA, principe d'équivalence, physique fondamentale

# ABSTRACT

Ultralight dark matter (ULDM), as a class of low mass (< 1 eV) dark matter (DM) candidates, is a compelling alternative to historically dominant models such as WIMPs and has recently gained significant attention in the scientific community. In this thesis, we study various experimental schemes for the direct detection of ULDM, both on ground and in space. More precisely, we propose a theoretical modeling of current and futuristic experiments, and we derive an estimation of their respective sensitivity. We mainly concentrate on three distinct phenomenologies. The first one is the coupling between a DM $U(1)$ field, known as the dark photon (DP), and electromagnetism, which induces a small electric field oscillating at the DP Compton frequency. We first propose an innovative way of detecting this small electric field by measuring the quadratic Stark shift of Rydberg atoms inside a microwave cavity, and we show that such an experiment could reach competitive constraints compared to existing laboratory experiments. Another possibility of detecting this electric field is to use a spherical mirror, which reflects it and focuses the electromagnetic power at its center of curvature, where a horn antenna is located (e.g. *SHUKET* experiment at CEA Saclay). We analytically investigate the effects of diffraction and mode matching in this type of experiment, and we show that the expected signal intensity can be significantly reduced compared to usual estimates. In this study, we also propose an optimization of the experimental parameters in order to increase the signal. The second main phenomenology considered in this thesis is the oscillation of rest mass and transition frequencies of atoms and test masses. These oscillations could be produced by the non-universal coupling of standard matter with a scalar ULDM candidate (dilaton or axion-like particle). We extensively study the impact of such oscillations on various atom interferometer schemes and classical tests of the universality of free fall, and we demonstrate how these different experiments could probe unconstrained regions of the parameter space. The oscillation of rest mass could also be observed in space-based gravitational wave (GW) detectors, such as LISA, and we investigate the possibility of such detection using more realistic orbits of spacecraft compared to previous studies. In particular, using Bayesian methods, we show that LISA could disentangle scalar ULDM signals from monochromatic GWs. We also show that the small velocity of the DM wave is not resolvable for most frequencies in the LISA band, which induces a decrease in sensitivity to scalar ULDM couplings, with respect to previous studies. Finally, we study the effect of vacuum birefringence and dichroism induced by the coupling between axions and photons, and how it could be detected with optical cavities, fibers, and LISA. In particular, we show that a slight modification of LISA's optical benches would make LISA the most sensitive experiment to the axion-photon coupling at low axion masses.

# KEYWORDS

Ultralight dark matter, axion, dilaton, dark photon, quantum sensors, atomic clock, interferometry (atomic, optical), LISA, equivalence principle, fundamental physics