Atteindre un régime où des photons peuvent interagir entre eux est un but important de l'optique quantique. Il s'agit d'une condition fondamentale pour de nombreux domaines, e.g. la mise en place d'un réseau de communication quantique, mais c'est aussi une manière de permettre l'étude et la simulation de la physique de la matière condensée.

Ce manuscrit présente le travail effectué durant trois ans pour assembler et tester une plateforme expérimentale visant à créer et à contrôler des interactions cohérentes entre photons optiques individuels et de franchir ainsi une étape importante pour le développement des technologies quantiques. Le montage combine des techniques venant de l'électrodynamique en cavité et des atomes froids: un ensemble d'atomes froids est transporté dans une cavité spécialement étudiée, permettant de transitoirement convertir les photons qui y circulent en polaritons de Rydberg atomique présentant de fortes interactions. Le contrôle de certains paramètres, relevant soit du nuage, de l'excitation ou bien de la cavité, doivent nous permettre de tester la validité de théorèmes fondamentaux en logique quantique, de générer des états non classiques et d'accéder à un régime dans lequel la simulation de système bidimensionnels propres à la matière condensée devient possible. Une première partie du manuscrit présente le montage et la manière dont les différents sous-systèmes requis ont été mis en place. Dans une deuxième partie les premières mesures d'EIT dans le nuage sont effectuées et décrites. Enfin, l'évolution à venir du montage est mentionnée.

a

CHAPTER 1

Introduction M any modern days technologies exist thanks to quantum physics, and this is likely not the end of it. Progress in the manipulation of quantum objects, such as quantum dots, ions, atoms, photons [START_REF] Ozawa | Topological photonics[END_REF] and superconducting circuit [2] leads to the development of new scientific and technological fields where systems and protocols based on quantum superposition, entanglement and teleportation could allow faster communications [3] [4] [5] [6], cryptography [START_REF] Pljonkin | Synchronization in quantum key distribution systems[END_REF], sensing [START_REF] Gilchrist | Schrödinger cats and their power for quantum information processing[END_REF] [9], simulations of complex systems [START_REF] Cirac | Quantum optics, what next?[END_REF] ... Quantum computation holds a massive importance in this evolution [START_REF] Preskill | Quantum computing in the NISQ era and beyond[END_REF]: this can be seen in the major efforts done not only in the academic world, but also from giants of industry. Unlike classical computation that is done with information stored in classical states, a quantum computer processes information that is stored in quantum states. Similarly to the revolution that lasers brought in many different fields, from medicine to commercial products, it is expected that quantum computation leads to a technological revolution of its own: it is not a substitute but rather a complement of classical computation. They could provide insight in the characterization and behaviour of microscopical systems and their engineering. What are the difficulties and challenges linked to the development of such a system? The DiVincenzo criteria [START_REF] Divincenzo | The physical implementation of quantum computation[END_REF] give a set of requirements that a quantum computer should fulfill. Reliable qubits with a long decoherence time are necessary, with a possibility to initialize the ensemble in a given state and to measure specific qubits. A universal set of logical gates should be implemented, and the system should be scalable. In addition, the ability faithfully transmit qubits them between specified locations is necessary. Many approaches exists, with different platforms using atomic systems, quantum dots [START_REF] Wolters | Simple Atomic Quantum Memory Suitable for Semiconductor Quantum Dot Single Photons[END_REF], superconducting qubits [START_REF] Kjaergaard | Superconducting Qubits: Current State of Play[END_REF], each system having its own strengths and limitations. Among these approaches, photons have the incomparable advantage of being able to travel over long distances while keeping to some extent their quantum properties. In addition, photon based systems can be implemented rather easily in current communication networks. Where photons are the least interesting is when it comes to interactions, as they do not spontaneously interact with each other. Other systems on the other hand can easily be used to store a qubit, but are restricted to a cryogenic system such as superconducting qubits, or to a vacuum chamber for ions and atoms. Thus, a promising path is a heterogeneous approach, with different subsystems working in parallel [START_REF] Schweickert | Electromagnetically Induced Transparency of On-demand Single Photons in a Hybrid Quantum Network[END_REF]: e.g. the combination of stationary qubits provided by atoms, linked by photons as flying qubits. Both are combined in an atom-photon interface: because photons do not naturally interact with one another, we then use atoms to mediate interactions.

Coherent interactions between single photon is a long-standing goal of quantum optics, and a non-linearity at the single photon level is necessary in order to reach photon-photon interactions. The most intuitive approach to reach this regime is to couple photons to a single atom, making use of its anharmonic energy level structure. In practice, reaching sufficiently strong couplings requires the use of very high finesse optical cavities, as limited coupling can be reached in free space [START_REF] Aljunid | Interaction of light with a single atom in the strong focusing regime[END_REF]. In the optical domain, this Cavity Quantum Electrodynamics (CQED) [START_REF] Yang | Photon-photon interaction in strong-coupling cavity-atom system[END_REF] approach has been extensively studied during the last three decades [START_REF] Thompson | Observation of normal-mode splitting for an atom in an optical cavity[END_REF]. A steady research effort allowed the observation of photon blockade [START_REF] Birnbaum | Photon blockade in an optical cavity with one trapped atom[END_REF], the generation of non-classical states of light [START_REF] Aspect | Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A new violation of Bell's inequalities[END_REF] [21], a prototype of a universal node for quantum communication [START_REF] Ritter | An elementary quantum network of single atoms in optical cavities[END_REF] and, very recently, a first proof-of-principle demonstration of a deterministic two-photon quantum gate [START_REF] Hacker | A photon-photon quantum gate based on a single atom in an optical resonator[END_REF]. This approach remains nevertheless very demanding technologically.

More recently, an alternative approach to deterministic photon-photon interactions emerged, strongly relaxing the technical requirements on the optical cavities or, in some cases, making them unnecessary. It consists in injecting the manipulated photons in an optically thick atomic ensemble and using auxiliary lasers to temporarily convert them into atomic polarization waves [START_REF] Gorshkov | Photon-photon interactions via Rydberg blockade[END_REF] known as polaritons, involving delocalized Rydberg excitations. The long lifetimes and long-range interactions of Rydberg atoms [START_REF] Carroll | Angular dependence of the dipole-dipole interaction in a nearly one-dimensional sample of Rydberg atoms[END_REF] lead to an Electromagnetically Induced Transparency (EIT) effect dependent on the number of Rydberg polaritons in the gas. In the beginning of the last decade, several groups reached a regime where a single Rydberg polariton could propagate through the gas with little losses but made the gas opaque to following ones [START_REF] Dudin | Strongly interacting Rydberg excitations of a cold atomic gas[END_REF] [27] [START_REF] Maxwell | Storage and control of optical photons using Rydberg polaritons[END_REF] [29] making the system act as a single-photon transistor [START_REF] Gorniaczyk | Single-photon transistor mediated by interstate Rydberg interactions[END_REF] [31] [START_REF] Paris-Mandoki | Free-space singlephoton transistor based on Rydberg interaction[END_REF]. By slightly detuning the photons from atomic resonances, the dispersion of the medium and not only its absorption can be made dependent on the number of Rydberg polaritons it contains, which led to the creation of photonic bound states [START_REF] Firstenberg | Attractive photons in a quantum nonlinear medium[END_REF] [START_REF] Liang | Observation of three-photon bound States in a quantum nonlinear medium[END_REF] and to a first demonstration of a two-photon gate using this mechanism [5]. Unfortunately, in this cavity-less approach, the dispersive photon-photon interactions cannot be made strong enough without introducing strong losses. Our team chose to hybridize the CQED and the Rydberg-based approaches, using the latter mechanism but placing the cloud in a specially designed optical cavity [START_REF] Guerlin | Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble[END_REF]. On one hand, using a cloud of a few hundreds of atoms instead of one allows one to reduce the finesse of the cavity and to decrease its length by a factor ∼ 1000 compared to the CQED regime. On the other hand, not only does the cavity relax the constrains on the cloud's optical thickness [START_REF] Lin | Strong photon blockade with intracavity electromagnetically induced transparency in a blockaded Rydberg ensemble[END_REF], but it qualitatively changes the effect it has on the impinging photons [START_REF] Lukin | Intracavity electromagnetically induced transparency[END_REF]: similarly to the CQED regime, its state-dependent absorption changes the optical impedance of the system, switching it from over-coupled to under-coupled and leading to a state-dependent phase shift on the reflected light as illustrated in Fig. 1.1. Strong single photon non-linearity have been demonstrated in previous years [START_REF] Jia | A strongly interacting polaritonic quantum dot[END_REF].

Besides making experiments on quantum logic [START_REF] Das | Photonic controlled-phase gates through Rydberg blockade in optical cavities[END_REF] and non-classical light generation [START_REF] Li | Entanglement between light and an optical atomic excitation[END_REF] more accessible, this approach allows one to use cavities with more exotic geometries than in CQED, where Rydberg-induced interactions make the photons form a strongly correlated quantum fluid, an ideal medium for the study and simulation of quantum matter [START_REF] Umucalllar | Generation and spectroscopic signatures of a fractional quantum Hall liquid of photons in an incoherently pumped optical cavity[END_REF] [START_REF] Carusotto | Photonic materials in circuit quantum electrodynamics[END_REF]. Our setup uses a four-mirror twisted resonator inducing a geometric rotation on the intracavity field, similar to the effect of magnetic field on charged particles [START_REF] Santos | Evolution of squeezed states under the Fock-Darwin hamiltonian[END_REF]. The parameters of the system can be adjusted for the photons to be driven in a Laughlin-like state, which describes the ground state of charged particles evolving in a plane under a uniform magnetic field, and underlies the fractional quantum Hall effect [START_REF] Schine | Synthetic Landau levels for photons[END_REF] [START_REF] Clark | Observation of Laughlin states made of light[END_REF].

Outline of the thesis

This thesis presents the work done on the assembly, characterization and first measurements of Rydberg EIT of a hybrid platform created for the study of deterministic photonic interactions in a non-linear medium. The system is composed of an atomic cloud of 87 Rb atoms placed in a moderate-finesse cavity mode of a specially designed resonator, and combines both the advantages of a increased interaction through CQED and Rydberg EIT.

• In the first part, we present briefly the theory linked to different part of the system: cold atom manipulation, interactions between an atomic sample and a cavity, and Rydberg interactions in a sample. The relevant parameters that we investigate are also specified.

• In a second part, the experimental setup subsystems are described, as well as the stabilization scheme and the optical resonator which is the heart of the experiment.

• In the fourth chapter, the characterization of the different subsystems that are the Magneto-Optical Trap (MOT), the transport of the sample and the coupling between the sample and the cavity are presented.

• In a fifth part the first EIT measurements are presented and analyzed.

• Ultimately, a brief presentation of the setup latest results is done, and an outlook of the future development of the experiment is presented.

CHAPTER 2

Theoretical Tools

Introduction

In this chapter, a theoretical description of the tools used in our project is presented. We discuss the manipulation of cold atoms, recall the manner in which interactions between an atomic sample and an optical resonator take place, and investigate Rydberg atoms.

Cold atom manipulation

In this thesis we work with the different states of the isotope 87 of Rubidium.

87

Rb atoms 87 Rb is an alkali atom that is naturally present on earth, under the shape of a soft, silvery metal. It oxidizes rapidly in an exothermic reaction. Alkali present one valence electron, and two isotopes of Rubidium exist, with respective proportion of 72% and 28%: 85 Rb which is stable and 87 Rb which has a half-life of 48.8 × 10 9 years, so with respect to the time scales in play in our experiment it can be considered stable. Alkali are popular atoms in laser cooling experiments since they contain closed cycling transition, which are useful as they allow the scattering of many photons. The laser frequencies associated with these transitions are also easily available, and producing a vapor or an atomic beam of such atoms is experimentally straightforward. The fine structure splitting comes from the spin orbit coupling: J=L+S, with L the orbital angular momentum (purely from the valence electron), S the spin angular momentum and J the total angular electron momentum. The associated quantum numbers range between

|𝐿 -𝑆| ≤ 𝐽 ≤ 𝐿 + 𝑆. (2.1)
For alkali atoms, S=1/2 and in the ground state L=0, so J=1/2. For the first excited state L=1 so J ∈ { 1 2 , 3 2 }. We work on the transition between the ground state and the first excited state whose degeneracy is lifted by the spin orbit coupling: the 𝐷 1 (respectively 𝐷 2 ) line corresponds to the two transitions 5 2 𝑆 1/2 → 5 2 𝑃 1/2 (respectively 5 2 𝑆 1/2 → 5 2 𝑃 3/2 ). Then taking into account the interaction of the electron with the nuclear spin I, we get the hyperfine splitting F=I+J: we have the range for the associated quantum numbers:

|𝐽 -𝐼| ≤ 𝐹 ≤ 𝐽 + 𝐼.
(2.2)

For 87 Rb, I=3/2 so the ground state is split in F ∈ [ [START_REF] Ozawa | Topological photonics[END_REF]2]] while the excited state of the D 1 line and the D 2 line respectively get split into F ∈ [ [START_REF] Ozawa | Topological photonics[END_REF]2]] and F ∈ [[0, 3]]. Each hyperfine state contains 2𝐹 + 1 degenerate magnetic sub-levels labeled 𝑚 𝐹 ∈ [[-𝐹,𝐹 ]]. These Zeeman levels determine the angular distribution of the electron wavefunction, and are degenerated in the absence of an external magnetic field. When a weak magnetic field is applied, the degeneracy is lifted, and the Zeeman levels split linearly, proportionally to the Landé g factor. Fig. 2.1 shows the resulting level structure. 87 Rb. The transitions between hyperfine splitting from the m F =0 Zeeman sub-level are given, as well as the approximate Landé 𝑔 𝐹 factors for the hyperfine splitting, with the corresponding Zeeman splittings. F (respectively F') are the hyperfine ground (respectively excited) states. The 'MOT' and 'repumper' show the transitions used for atomic trapping and cooling.

Trapping of neutral atoms

To reduce the velocity of an atom it is possible to use the exchange of momentum that takes place when atoms scatter photons from a laser beam. In one-dimension version, an optical molasses is made with two counter-propagating laser beams, the frequency of which is slightly red-detuned from an atomic absorption resonance. As a result, a motion of an atom in the direction of one of the beams will lead to a Doppler shift so that the absorption rate for the counter-propagating beam to the atom motion is increased, whereas the absorption rate for the opposite laser beam is reduced. The effective recoil results in a reduction of the atomic velocity, and can be seen effectively as a dissipative light force which is always directed opposite to the motion of the atom and therefore serves to reduce that motion. This cooling mechanism do not however prevent the atoms from diffusing and leaving the optical molasses region, so the creation of a trapping potential is required. By introducing a magnetic field gradient and changes to the polarization, this configuration can be turned into a MOT [START_REF] Hänsch | Cooling of gases by laser radiation[END_REF] [47] [START_REF] Stenholm | Laser cooling and trapping[END_REF]. With a linear magnetic field gradiant whose minimum is centered on the trap, the degeneracy of the Zeeman sublevels is lifted, and the gradiant gives rise to an energy splitting that is linearly dependant on the distance from the trap center. The Zeeman splitting is analogous to the Doppler shift in position space, so atoms located on one side of the trap will have a Zeeman splitting that makes them resonant with the counter-propagating bean: the atoms absorb with a higher probability the photons who have a momentum directed towards the trap center. This gives rise to a position dependant radiation pressure, and can be expanded in the two remaining direction by replacing the linear magnetic gradiant with coils in anti-Helmoltz configuration, thus creating a magnetic quadrupole field and by shining three pairs of counter-propagating beams in an orthogonal manner. Polarization-gradient -'Sisyphus' -cooling effects appearing in a MOT allow one to cool the atoms below the Doppler temperature 𝑇 𝐷 = 𝛤/(2𝑘 𝐵 ≈ 140 µK [START_REF] Lett | Observation of atoms laser cooled below the doppler limit[END_REF] down to approximately ten units of recoil temperature 𝑇 𝑅 = 2 𝑘 2 /(𝑚 𝑅𝑏 𝑘 𝐵 ) ≈ 360 nK [START_REF] Dalibard | Laser cooling below the Doppler limit by polarization gradients: simple theoretical models[END_REF] with 𝑘 𝐵 the Boltzmann constant. As density increases, the probability that an emitted photon leaves without interacting with a neighboring atom decreases, making the atomic density saturate at the level of a few 1 × 10 10 cm -3 .

Physical requirements: Typical timescales of our experiments, rarely exceeding a few hundreds of milliseconds, require an ultra-high-vacuum environment with a pressure of the order of 1 × 10 -9 mbar to avoid losing cold trapped atoms via collisions with the background gas. To reach a good atomic capture range, a 87 Rb MOT requires magnetic field gradients around 10 G cm -1 and three pairs of relatively large (∼ 1 cm) and relatively bright (∼ 10 mW) beams red-detuned by a few linewidths 𝛤 = 2𝜋 × 6 MHz from the 'cooling' transition |5 2 𝑆 1/2 ,𝐹 = 2⟩ → |5 2 𝑃 3/2 ,𝐹 = 3⟩. Due to polarization gradients this transition is not perfectly closed and an additional weak beam repumping the atoms out of the |5 2 𝑆 1/2 ,𝐹 = 1⟩ level is necessary.

Dipole trap

We briefly consider dipole traps [START_REF] Chu | Cooling and Trapping of Atoms with Laser Light[END_REF] [START_REF] Grimm | Optical Dipole Traps for Neutral Atoms[END_REF], as they are going to be used extensively in the setup. We will restrict the discussion to attractive traps formed by linearly-polarized Gaussian beams red-detuned from the D 2 transition in 87 Rb. For a single-beam trap the spatial intensity distribution is

𝐼 dip (𝑟,𝑧) = 𝐼 0 1 + ( 𝑧 𝑧 𝑅 ) 2 e ( -2 𝑟 2 𝑤(𝑧) 2 ) with 𝑤(𝑧) = 𝑤 0 √ 1 + ( 𝑧 𝑧 𝑅 ) 2 , 𝐼 0 = 2𝑃 𝜋𝜔 2 0 . (2.3)
where r the radial distance from the considered optical axis along z, P the beam power and 𝜔(𝑧) the 1/𝑒 2 radius, z the distance from the beam waist 𝑤 0

𝑈 𝑑𝑖𝑝 (𝑟,𝑧) = 𝑈 0 1 1 + ( 𝑧 𝑧 𝑅 ) 2 e ( -2𝑟 2 𝑤(𝑧) 2 ) with maximum potential depth ⏞ ⏟ 𝑈 0 = 𝛤 2 𝑃 2𝜋𝐼 𝑠𝑎𝑡 𝑤 2 0 ( 1 𝛿 1 + 2 1 𝛿 2 ) ⏟ ⏞
Effective laser detuning . (2.4) where I sat is the saturation intensity 𝐼 sat = 2𝜋ℎ𝑐𝛤 3𝜆 3 , 𝛿 𝑖 the detuning between the laser driving frequency and the atomic resonance on the D i line, 𝛤 the natural linewidth of the transition. We call 𝛥 the effective laser detuning.

The spatial intensity distribution of a Gaussian beam of power P propagating along the z-direction is expressed by

𝐼(𝑟,𝑧) = 2𝑃 𝜋𝑤(𝑧) 2 e ( -2 𝑟 2 𝑤(𝑧) 2 )
.

(2.5)

So we have for a two level atom with angular frequency 𝜔 0 and natural linewidth 𝛤 the following potential 𝑈 (𝑟,𝑧) and scattering rate 𝛤 𝑠𝑐 (𝑟,𝑧): The lasers used are red-detuned far from the atomic transitions (𝛥 ≫ 𝛤 ) in order to leave the atoms undisturbed. Since 𝛥 < 0 the atoms are attracted by the high intensity. Unlike the previously presented molasses where the force was dissipative, this dipole force is conservative. The potential scales as 𝛤/𝛥 while the scattering rate scales as 𝛤 2 /𝛥 2 . If the mechanical energy of the atom is lower than the trap depth, it will oscillate around the waist, whereas if the atoms are not slowly transferred to the trap, they will escape from the potential. When the atom temperature is low enough to allow trapping, we can expand the potential to the lowest -quadratic -order to find the results of a trapped particle of mass 𝑚 𝑅𝑏 in a harmonic potential, whose radial and longitudinal oscillation frequencies are:

𝑈 (𝑟,𝑧) = 3𝜋𝑐 2 2𝜔 3
𝜔 𝑟 = √ 4𝑈 0 𝑚 𝑅𝑏 𝑤 2 0 and 𝜔 𝑧 = √ 2𝑈 0 𝑚 𝑅𝑏 𝑧 2 𝑅 .
(2.8)

Except for very tightly focused beams, the axial confinement is approximately 𝑤 0 /𝑧 𝑅 ∼ 𝜆/𝑤 0 ≪ 1 weaker than the radial one. It can be enhanced either by using a crossed dipole trap where two non-interfering trapping beams intersect each other with an angle, or by forming a standing wave with two interfering counter-propagating beams where the relevant length scale for the axial confinement becomes 𝜆/2 ≪ 𝑧 𝑅 . Both techniques are used in our setup.

Transport using an optical moving lattice

A standing-wave optical trap can easily be turned into a running-wave one to transport atoms from one location to another [START_REF] Schrader | An optical conveyor belt for single neutral atoms[END_REF]. We place ourselves in the case of a red detuned optical lattices previously described: we have two counter-propagating laser beams of equal intensity, whose mutual detuning can be controlled and is not necessary zero:

𝛥𝜔 = 𝜔 1 -𝜔 2 ≪ 𝜔 1 , 𝜔 2 .
We consider the counter-propagating fields 𝐸 1 𝑒 𝑖(𝜔 1 𝑡-𝑘 1 𝑧) and 𝐸 2 𝑒 𝑖(𝜔 2 𝑡+𝑘 2 𝑧) , where the frequencies 𝜔 1,2 are close to each other.

𝐼 = ⃒ ⃒ ⃒𝐸 1 𝑒 𝑖(𝜔 1 𝑡-𝑘𝑧) + 𝐸 2 𝑒 𝑖(𝜔 2 𝑡+𝑘𝑧) ⃒ ⃒ ⃒ 2 (2.9) = 𝐸 2 1 + 𝐸 2 2 + 2𝐸 1 𝐸 2 cos (2𝑘𝑧 + 𝛥𝜔𝑡) with 𝛥𝜔 = 𝜔 1 -𝜔 2 .
(2.10)

For the atomic cloud to be at rest before and after the transport, the lattice must be accelerated then decelerated, which adds an acceleration term to the potential in the non-inertial frame of the transported atoms. The lattice potential becomes tilted and accelerations higher than 𝑎 𝑚𝑎𝑥 = 𝑈 0 𝑘/𝑚 𝑅𝑏 result in a spilling of the atoms from one lattice site to the next. Due to the finite atomic temperature and to the site-to-site tunneling, practically achievable accelerations are typically ten times lower but still remain in the 1 × 10 3 m s -2 to 1 × 10 4 m s -2 range, making the transport rather fast.

Physical requirements: Fluctuations of the relative phase between the two beams directly match a position fluctuation of the dipole trap potential, leading to heating of the trapped atoms, which could limit the lifetime in the trap: a good control of this phase is necessary. In order to put in place these trap we need ample available laser power at a frequency detuned from the atomic transition.

Free-space EIT

We now discuss the model that is used later in this work: we consider an EIT measurement taking place in free space, with an intermediate level interacting with only one Rydberg level. The Hamiltonian can be written with the state operator σ𝑖𝑗 = |𝑖⟩ ⟨𝑗| as (2.11) with 𝛿 𝑒 the detuning from the excited intermediary level and 𝛿 𝑟 the two-photon detuning, 𝛺 𝑝 and 𝛺 𝑟 the Rabi frequencies of the probe and coupling lasers respectively. For the n-th atom, the Bloch equations can be written as follows, with the complex detunings 𝛥 𝑒 = 𝛿 𝑒 + 𝑖𝛤 from the intermediary level and 𝛥 𝑟 = 𝛿 𝑟 + 𝑖𝛾 𝑟𝑦𝑑 from the Rydberg level:

Ĥ = -𝛿 𝑒 𝑁 ∑ 𝑛=1 σ(𝑛) 𝑒𝑒 -𝛿 𝑟 𝑁 ∑ 𝑛=1 σ(𝑛) 𝑟𝑟 + 𝑁 ∑ 𝑛=1 𝛺 𝑝,𝑛 2 (σ (𝑛) 𝑒𝑔 + σ(𝑛) 𝑔𝑒 ) + 𝑁 ∑ 𝑛=1 𝛺 𝑟,𝑛 2 (σ (𝑛) 𝑒𝑟 + σ(𝑛) 𝑟𝑒 ),
𝑖 𝑑 ⟨σ (𝑛) 𝑔𝑒 ⟩ 𝑑𝑡 = -𝛥 𝑒 ⟨σ (𝑛) 𝑔𝑒 ⟩ + 𝛺 𝑝,𝑛 2 + 𝛺 𝑟,𝑛 2 ⟨σ (𝑛) 𝑔𝑟 ⟩ , (2.12) 𝑖 𝑑 ⟨σ (𝑛) 𝑔𝑟 ⟩ 𝑑𝑡 = -𝛥 𝑟 ⟨σ (𝑛) 𝑔𝑟 ⟩ + 𝛺 𝑟,𝑛 2 ⟨σ (𝑛) 𝑔𝑒 ⟩ . (2.13)
The steady state solutions are then

⟨σ (𝑛) 𝑔𝑒 ⟩ = ( 1 - 1 𝛥 𝑒 𝛺 2 𝑟,𝑛 4𝛥 𝑟 ) -1 𝛺 𝑝,𝑛 2𝛥 𝑒 , (2.14) ⟨σ (𝑛) 𝑔𝑟 ⟩ = ( 1 - 1 𝛥 𝑒 𝛺 2 𝑟,𝑛 4𝛥 𝑟 ) -1 𝛺 𝑟,𝑛 2𝛥 𝑟 𝛺 𝑝,𝑛 2𝛥 𝑒 . (2.15)
We define a local density 𝜌(⃗ 𝑟) and local Rabi frequencies 𝛺 𝑝 and 𝛺 𝑟 as

𝜌(⃗ 𝑟) = 𝜌 0 𝑒 -2((𝑥 2 +𝑦 2 )/𝑤 2 𝜌 +𝑧 2 /𝑤 2 𝑧 ) , (2.16
)

𝛺 𝑝 (⃗ 𝑟) = 𝛺 𝑝 (𝑧)𝑒 -(𝑥 2 +𝑦 2 )/𝑤 2 ,
(2.17)

𝛺 𝑟 (⃗ 𝑟) = 𝛺 𝑟 𝑒 -(𝑥 2 +𝑦 2 )/𝑤 2 𝛺 . (2.18)
We have in the fixed frame: 𝛺 𝑝 (⃗ 𝑟) = -𝑑 𝑔𝑒 𝐸 𝑝 (⃗ 𝑟)/ and d𝑔𝑒 = 𝑑 𝑔𝑒 (σ 𝑔𝑒 + σ𝑒𝑔 ). So in the rotating frame, the expectation value of the dipole operator can be expressed as

⟨ d𝑔𝑒 (⃗ 𝑟)⟩ = 2𝑑 𝑔𝑒 ⟨σ 𝑔𝑒 (⃗ 𝑟)⟩ = - 𝑑 2 𝑔𝑒 𝛾 𝑒 ⟨ σ𝑔𝑒 (⃗ 𝑟) 𝛺 𝑝 (⃗ 𝑟)/(2𝛾 𝑒 ) ⟩ 𝐸 𝑝 (⃗ 𝑟), (2.19)
and the susceptibility is expressed as

𝜒(⃗ 𝑟) = 𝜌(⃗ 𝑟)⟨ d𝑔𝑒 (⃗ 𝑟)⟩ 𝜀 0 𝐸(⃗ 𝑟) = - 𝜌(⃗ 𝑟)𝑑 2 𝑔𝑒 𝜀 0 𝛾 𝑒 ⟨ σ𝑔𝑒 (⃗ 𝑟) 𝛺 𝑝 (⃗ 𝑟)/(2𝛾 𝑒 ) ⟩ .
(2.20)

If 𝜒 ≪ 1, then the field and intensity of the probe are given by

𝑑𝐸 𝑝 𝑑𝑧 (⃗ 𝑟) = 𝑖𝑘 √ 1 + 𝜒(⃗ 𝑟)𝐸 𝑝 (⃗ 𝑟) ≈ 𝑖𝑘(1 + 𝜒(⃗ 𝑟)/2)𝐸 𝑝 (⃗ 𝑟) (2.21) ⇒ 𝑑𝐼 𝑝 𝑑𝑧 (⃗ 𝑟) ≈ -𝑘I (𝜒(⃗ 𝑟))𝐼 𝑝 (⃗ 𝑟). (2.22)
Using the transverse coordinate 𝑟 ⊥ = √ 𝑥 2 + 𝑦 2 the intensity can be written as

𝑑𝐼 𝑝 𝑑𝑧 (⃗ 𝑟) = 2𝜋𝜌 0 𝑑 2 𝑔𝑒 𝜀 0 𝛾 𝑒 𝜆 𝑔𝑒 𝑒 -2(𝑟 2 ⊥ /𝑤 2 𝜌 +𝑧 2 /𝑤 2 𝑧 ) I ( 𝛾 𝑒 𝛥 𝑒 [ 1 - 1 𝛥 𝑒 𝛺 2 𝑟 (𝑟 ⊥ ) 4𝛥 𝑟 ] -1 ) 𝐼 𝑝 (⃗ 𝑟). (2.23)
The integration along 𝑧 gives

𝐼 𝑝,𝑜𝑢𝑡 (𝑟 ⊥ ) = 𝐼 𝑝,𝑖𝑛 (𝑟 ⊥ ) exp ( 𝜌 0 2𝜋𝑑 2 𝑔𝑒 𝜀 0 𝛾 𝑒 𝜆 𝑔𝑒 √ 𝜋 2 𝑤 𝑧 𝑒 -2𝑟 2 ⊥ /𝑤 2 𝜌 I ( 𝛾 𝑒 𝛥 𝑒 [ 1 - 1 𝛥 𝑒 𝛺 2 𝑟 4𝛥 𝑟 𝑒 -2𝑟 2 ⊥ /𝑤 2 𝛺 ] -1 )) .
(2.24)

If the probe input intensity is 𝐼 𝑝,𝑖𝑛 (𝑟 ⊥ ) = 𝐼 0 exp(-2𝑟 2 ⊥ /𝑤 2 ) and the input probe power is

𝑃 𝑝,𝑖𝑛 = ˆ+∞ 0 2𝜋𝑟 ⊥ 𝑑𝑟 ⊥ 𝐼 𝑝,𝑖𝑛 (𝑟 ⊥ ) = 𝐼 0 ˆ+∞ 0 2𝜋𝑟 ⊥ 𝑑𝑟 ⊥ 𝑒 -2𝑟 2 ⊥ /𝑤 2 = 𝐼 0 𝜋𝑤 2 2 , ( 2.25)
we can simplify the integral giving the probe output power by defining 𝑢 = exp(-2𝑟 2 ⊥ /𝑤 2 ) to get

𝑃 𝑝,𝑜𝑢𝑡 = ˆ+∞ 0 2𝜋𝑟 ⊥ 𝑑𝑟 ⊥ 𝐼 𝑝,𝑜𝑢𝑡 (𝑟 ⊥ ) (2.26) = 𝑃 𝑝,𝑖𝑛 ˆ1 0 𝑑𝑢 exp ( 𝜌 0 2𝜋𝑑 2 𝑔𝑒 𝜀 0 𝛾 𝑒 𝜆 𝑔𝑒 √ 𝜋 2 𝑤 𝑧 𝑢 𝑤 2 /𝑤 2 𝜌 I ( 𝛾 𝑒 𝛥 𝑒 [ 1 - 1 𝛥 𝑒 𝛺 2 𝑟 4𝛥 𝑟 𝑢 𝑤 2 /𝑤 2 𝛺 ] -1 )) .
(2.27)

We call homogeneous the situation where the control laser power is homogeneous with respect to the probe laser and inhomogeneous the case where we consider the finite size of the control laser beam. The inhomogeneous case represents the physics of our system more accurately and does not overestimate the linewidth as in the homogeneous case. In the case of a homogeneous coupling beam, 𝑤 ≪ 𝑤 𝜌 ,𝑤 𝛺 and we can express the output power by

𝑃 𝑝,𝑜𝑢𝑡 𝑃 𝑝,𝑖𝑛 = exp ( 𝜌 0 2𝜋𝑑 2 𝑔𝑒 𝜀 0 𝛾 𝑒 𝜆 𝑔𝑒 √ 𝜋 2 𝑤 𝑧 I ( 𝛾 𝑒 𝛥 𝑒 -𝛺 2 𝑟 /(4𝛥 𝑟 ) ) ) (2.28)
For an inhomogeneous coupling beam where its size is taken into account, the output power can be expressed by

𝑃 𝑝,𝑜𝑢𝑡 𝑃 𝑝,𝑖𝑛 = ˆ1 0 𝑑𝑢 exp ( 𝜌 0 2𝜋𝑑 2 𝑔𝑒 𝜀 0 𝛾 𝑒 𝜆 𝑔𝑒 √ 𝜋 2 𝑤 𝑧 𝑢 𝑤 2 /𝑤 2 𝜌 I ( 𝛾 𝑒 𝛥 𝑒 -𝑢 𝑤 2 /𝑤 2 𝛺 𝛺 2 𝑟 /(4𝛥 𝑟 ) )) . (2.29)

Conclusion -free-space EIT

In the free-space configuration, we can expect an EIT signal on the sample absorption signal. We consider two approaches, where we have either an homogeneous control beam whose output power is given by Eq. 2.28, or a situation where the finite size of the control beam is modelled, whose output power is given by Eq. 2.29. From both equations, we can see that the ratio of the size of the probe beam 𝑤 and coupling beam 𝑤 𝛺 is going to play an important role. The remaining parameters are the probe and coupling lasers detunings.

Atoms in cavity

We start from a two-level atom and build up to the concepts that are necessary for the understanding of the experimental setup. In this section we discuss the coupling between atoms and an optical resonator, starting from one two level atom in a resonator and discussing important properties of the system.

Two level system in a single mode cavity

We consider an atom is in its excited state in an empty single mode cavity. We define the following state |𝑔,1⟩ (respectively |𝑒,0⟩) which describe the situation where the atom is in its ground state and a photon is running in the cavity (respectively where the atom is in its excited state and no photons are present in the cavity). The system is described by the Jaynes-Cumming Hamiltonian [START_REF] Jaynes | Comparison of Quantum and Semiclassical Radiation Theories with Application to the Beam Maser[END_REF], which is given in the rotating wave approximation by

𝐻 𝑡𝑜𝑡 = 𝐻 𝑐𝑎𝑣𝑖𝑡𝑦 + 𝐻 𝑎𝑡𝑜𝑚 + 𝐻 𝑖𝑛𝑡 (2.30) = 𝜔 𝑐 |𝑔,1⟩ ⟨𝑔,1| + 𝜔 𝑎 |𝑒,0⟩ ⟨𝑒,0| + 𝑔(|𝑔,1⟩ ⟨𝑒,0| + |𝑒,0⟩ ⟨𝑔,1|) (2.31)
with 𝜔 𝑐 the cavity photon frequency (respectively 𝜔 𝑎 the atomic transition frequency). The one photon Rabi frequency is given by 𝛺 = 2𝑔, with g the atom coupling rate. The atom-photon coupling strength is given by 𝑔 = 𝑑•𝐸, where d is the induced dipole moment of the atom and where the field E comes from

𝜔 𝑐 = 1 2 𝜀 0 𝐸 2 𝑉 (2.32)
with V the effective cavity volume. We can write the matrix representation |𝑔,1⟩ = ( 1 0

) and |𝑒,0⟩ = ( 0 1
)

and the Hamiltonian can be rewritten when setting to one and choosing 𝜔 𝑎 as the origin of energies as

𝐻 𝑡𝑜𝑡 = ( 𝛥 𝑐 𝑔 𝑔 0 ) with 𝛥 𝑐 = 𝜔 𝑐 -𝜔 𝑎 . (2.33)
Solving the energy eigenstates leads to

𝐸 ± = 1 2 (𝛥 𝑐 ± 𝛺 𝑅 ) with 𝛺 𝑅 = √ 𝛥 2 𝑐 + (2𝑔) 2 (2.34)
where 𝛺 𝑅 is the generalized Rabi frequency. The eigenstates depends on the sign of 𝛥𝜔: we set 𝜃 such that 𝑡𝑎𝑛(𝜃) = 2𝑔/𝛥𝜔 and they can then be written for 𝛥 𝑐 ≥ 0 as

|𝐸 + ⟩ = 𝑐𝑜𝑠(𝜃/2) |𝑔,1⟩ + 𝑠𝑖𝑛(𝜃/2) |𝑒,0⟩ (2.35
)

|𝐸 -⟩ = -𝑠𝑖𝑛(𝜃/2) |𝑔,1⟩ + 𝑐𝑜𝑠(𝜃/2) |𝑒,0⟩ , (2.36)
and for 𝛥 𝑐 < 0:

|𝐸 + ⟩ = 𝑠𝑖𝑛(𝜃/2) |𝑔,1⟩ + 𝑐𝑜𝑠(𝜃/2) |𝑒,0⟩ (2.37
)

|𝐸 -⟩ = -𝑐𝑜𝑠(𝜃/
|𝐸 -⟩ ≈ |𝑔,1⟩ -|𝑒,0⟩ . (2.40)
For small detunings, the dressed states are maximally entangled states, made up of equal parts of both bare eigenstates.

Three level system in cavity

The simple description above applies to a closed system. In reality, the cavity field decays with a constant 𝜅 and the atomic polarization with a constant 𝛾 = 𝛤/2 where 𝛤 is the decay constant of the atomic population in the state |𝑒⟩, assuming a homogeneous decay. Such a driven-dissipative system, where the number of excitation is not fixed, can be more appropriately described in the second quantization using optical Bloch equations. To derive these equations, let us first consider an empty cavity with a resonance angular frequency 𝜔 𝑐 and a field damping rate 𝜅. We will assume that the cavity field decays only via one of the cavity mirrors playing the role of an input-output coupler, so that 𝜅 = 𝑐𝑇 /(2𝑙) where T is the transmission of the coupler, 𝑙 is the round-trip cavity length and c is the speed of light. The cavity is driven by a classical probe beam with an angular frequency 𝜔. We consider that this beam is in a coherent state |𝛼𝑒 -𝑖𝜔𝑡 ⟩ with 𝛼 ∈ R such that 𝛼 2 is the photon flux. The density matrix ρ0 describing the state of the cavity evolves according to the master equation

𝑑ρ 0 𝑑𝑡 = -𝑖[𝜔 𝑐 â â + 𝑖𝛼 √ 2𝜅 0 ( â 𝑒 -𝑖𝜔𝑡 -â𝑒 𝑖𝜔𝑡 ) , ρ0 ] + L 𝑐 [ρ 0 ], (2.41)
L 𝑐 [ρ 0 ] = 𝜅(2âρ 0 â -â âρ 0 -ρ0 â â) (2.42)
where â is the photon annihilation operator describing the intracavity field. In the rotating frame of the probe, defining the complex detuning 𝛥 𝑐 = 𝛿 𝑐 + 𝑖𝜅 where 𝛿 𝑐 = 𝜔 -𝜔 𝑐 , the expectation value of this operator obeys

𝑖 ⟨â⟩ 𝑑𝑡 = -𝛥 𝑐 ⟨â⟩ + 𝑖𝛼 √ 2𝜅. (2.43)
In the steady state ⟨â⟩ = 𝑖𝛼 √ 2𝜅/𝛥 𝑐 , and the field â𝑟 reflected from the cavity obeys

â𝑟 = -â 𝑖 + √ 2𝜅â ⇒ ⟨â 𝑟 ⟩ = -𝛼 + √ 2𝜅 ⟨â⟩ = -𝛼 𝛿 𝑎 -𝑖𝜅 𝛿 𝑎 + 𝑖𝜅 . (2.44)
As expected for a perfect single-ended cavity, the intracavity field shows a Lorentzian peak as a function of 𝛿 𝑐 , and the reflection coefficient corresponds to a pure phase factor making a 2𝜋 rotation when 𝛿 𝑐 is scanned from -∞ to +∞.

We now add an ensemble of 𝑁 two-level atoms with an angular transition frequency 𝜔 𝑒 between a ground state |𝑔⟩ and an excited state |𝑒⟩ with a homogeneous dipole decay rate 𝛾. The atom 𝑛 is coupled to the cavity mode with a constant 𝑔 0 , which can be assumed equal for all atoms. We write the atomic state operators as σ𝑖𝑗 = |𝑖⟩ ⟨𝑗|. Like previously, we define the complex detuning 𝛥 𝑒 = 𝛿 𝑒 + 𝑖𝛾 with 𝛿 𝑒 = 𝜔 -𝜔 𝑒 to get, in the rotating frame,

𝑑ρ 𝑑𝑡 = -𝑖 [ -𝛿 𝑎 â â + 𝑖𝛼 √ 2𝜅 0 (â -â) -𝛿 𝑒 𝑁 ∑ 𝑛=1 σ(𝑛) 𝑒𝑒 + 𝑁 ∑ 𝑛=1 𝑔 0 (âσ (𝑛) 𝑒𝑔 + â σ(𝑛) 𝑔𝑒 ),ρ ] + L 𝑎 [ρ] + L 𝑒 [ρ]
(2.45)

L 𝑒 [ρ] = 𝛾 𝑁 ∑ 𝑛=1 (2σ (𝑛) 𝑔𝑒 ρ0 σ(𝑛) 𝑒𝑔 -σ(𝑛) 𝑒𝑒 ρ0 -ρ0 σ(𝑛) 𝑒𝑒 ). (2.46)
In the low-excitation regime, the steady-state expectation value of the cavity field becomes

⟨â⟩ = 𝑖𝛼 √ 2𝜅 𝛥 𝑐 -𝑔 2 /𝛥 𝑒 = 𝛼 √ 2 √ 𝜅 1 1 -𝑖𝛿 𝑎 /𝜅 + 2𝐶 0 /(1 -𝑖𝛿 𝑒 /𝛾) (2.47)
where we defined the collective coupling 𝑔 = 𝑔 0 √ 𝑁 and the cooperativity 𝐶 0 = 𝑁 𝑔 2 0 /(2𝜅𝛾) [START_REF] Gaëtan | Observation of collective excitation of two individual atoms in the Rydberg blockade regime[END_REF]. On resonance, when the collective coupling 𝑔 is strong (𝑔 2 ≫ 𝜅𝛾), the intracavity field is strongly attenuated. From Eq. 2.44, we see that in this case ⟨â 𝑟 ⟩ ≈ -𝛼: compared to an empty cavity, the reflected field keeps the same amplitude but its phase is rotated by 𝜋.

Let us now consider that the atomic state |𝑒⟩ is coupled to a third state |𝑟⟩ with an energy 𝜔 𝑟 and a linewidth 𝛾 𝑟 assumed to be homogeneous. The coupling is done by a laser with an angular frequency 𝜔 ′ and a Rabi frequency 𝛺 for each atom, taken as real without loss of generality. With the same approximations as before, we introduce the two-photon detuning 𝛿 𝑟 = 𝜔 + 𝜔 ′ -𝜔 𝑟 and the complex detuning 𝛥 𝑟 = 𝛿 𝑟 + 𝑖𝛾 𝑟 to get the steady-state solution

⟨â⟩ = 𝑖𝛼 √ 2𝜅 𝛥 𝑎 -𝑔 2 /[𝛥 𝑒 -𝛺 2 /(4𝛥 𝑟 )]
(2.48)

Conclusion -EIT in cavity

On resonance a sufficiently large Rabi frequency 𝛺 cancels the effect of 𝑔 and brings ⟨â⟩ and ⟨â 𝑟 ⟩ back to their empty-cavity values. We thus recover the EIT expected for such a three-level system.

Rydberg ensemble in cavity

In our experiments, the state |𝑟⟩ is a Rydberg state |𝑛𝑆 1/2 ⟩ with a principal quantum number n ranging between 46 and 101.

Rydberg atoms

Many properties of alkali atoms in Rydberg states can be deduced from a hydrogenoid model where the valence electron is bound to the nucleus screened by the inner electronic cloud, with an energy 𝐸 𝑛 = -𝑅 𝑦 /(𝑛 * ) 2 where 𝑅 𝑦 is the Rydberg constant and 𝑛 * = 𝑛 -𝛿(𝑛,𝑙,𝑗) is an effective principal quantum number corrected for the quantum defect 𝛿 [START_REF] Gallagher | Rydberg Atoms[END_REF]. This defect accounts for the deviation from the Hydrogen potential for states with low orbital quantum numbers, where the overlap between the wavefunction of the excited electron and the inner electronic cloud is not negligible. Table 2.1 shows the scaling in 𝑛 * of the properties of a low-angular momentum Rydberg state.

∼ 40 V cm -1 Orbital radius ≈ 1 2 (3(𝑛 * ) 2 -𝑙(𝑙 + 1)) (𝑛 * ) 2 ∼ 3000𝑎 0 dipole moment ⟨𝑛𝑙| 𝑒𝑟 |𝑛𝑙 + 1⟩ (𝑛 * ) 2 ∼ 1900𝑒𝑎 0 Radiative lifetime (𝑛 * ) 3 ∼ 100 µs Polarizability (𝑛 * ) 7 ∼ 30 MHz cm 2 V -2
Van der Waals coefficient 𝐶 6

(𝑛 * ) 11 2𝜋 × 50 THzµm 6

Rydberg blockade

Compared to low-lying states, Rydberg states have long lifetimes [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and PhotonsExploring the Quantum: Atoms, Cavities, and Photons[END_REF], in the hundreds of microseconds range for the states we are using, limited by the black-body radiation of the surrounding apparatus at 300 K. The corresponding linewidths 𝛾 𝑟 are generally inhomogeneous due to environmental noise, in a 2𝜋× 80 kHz to 100 kHz being typical numbers for our experiments. This inhomogeneity comes in part from the high polarizability of Rydberg atoms, very sensitive to stray electric fields [56] [58]. This can be problematic when the atoms are located close to surfaces [START_REF] Carter | Electric-field sensing near the surface microstructure of an atom chip using cold Rydberg atoms[END_REF], but this sensitivity also allows one to electrically tune the frequencies of Rydberg transitions. Above all, it makes Rydberg atoms very sensitive to the presence of other Rydberg atoms [60] [61]. For two Rydberg atoms separated by a distance 𝑅 in a state |𝑟,𝑟⟩ not degenerated with another pair of Rydberg states, the van-der-Waals interaction energy 𝑉 = 𝐶 6 /𝑅 6 will increase extremely fast with the principal quantum number, as the coefficient 𝐶 6 scales as (𝑛 * ) 11 . For n=100, 𝐶 6 / ∼ 2𝜋 ×50 THz.µm 6 : the energy of two Rydberg atoms located anywhere in a ∼ 10 µmsized cloud will be shifted by ∼ 50 MHz by the interactions. If this shift is larger that the effective width of the Rydberg state (determined in practice by the Rabi frequency 𝛺), one Rydberg atom excited in the cloud will 'block' the excitation of others.

A very simplistic way of describing this Rydberg blockade is by considering, in Eq. 2.48, that on the two-photon resonance |𝛥 𝑟 | = 𝛾 𝑟 ≪ 𝛺 for the first Rydberg excitation, so that the EIT makes the cloud transparent for a single photon, but the interactions lead to an effective |𝛥 𝑟 | ≫ 𝛺 for following ones, destroying the EIT and making the cloud opaque. One single Rydberg excitation can switch the response of the atoms present in the blockade sphere from EIT conditions to resonant two level scattering conditions [START_REF] Firstenberg | Nonlinear quantum optics mediated by Rydberg interactions[END_REF].

Rydberg ensemble in cavity

In free space, when a photon is stored in the Rydberg sample, this leads to high losses for the following photon. The presence of a Rydberg polariton within a cavity, as shown in Fig. 2.2 can lead to dramatic changes of the cavity response as was mentioned in the previous chapter. When no Rydberg polariton is present in the cavity, a photon impinging on the resonator is reflected with little losses. However the creation of a Rydberg polariton leads to a shift from an overcouple regime to an undercouple regime.

On resonance, when no photon is stored, EIT takes place: the losses are much less than the transmission in the cavity, which translates as no phase shift on the second incoming photon. When a photon is stored within the sample, a blockade sphere appears, the losses are then much more important than the transmission in the cavity, shifting the cavity out of resonance. This leads to a conditional 𝜋 phase shift on the impinging photon, inducing high reflectivity which also helps avoiding further losses. This mechanism is at the core of many experiments planned to be performed using this setup and briefly presented at the end of this manuscript.

The transmission of the in/out coupler is crucial for this switch of regime, as it is the parameter to take into consideration when comparing the intracavity losses from the atomic sample. We use a single-ended cavity, which is a cavity where the mirrors other than the incoupling mirrors are highly reflecting. An optical resonator is used in order to enhance the Rydberg non-linearity, whose incoupling mirror parameters are crucial to our experiment. The cavity has to be as close as possible to single-ended, and optical losses must be minimized. In addition, its finesse should not be too important as it leads to high constraints on the system. It should allow us to work within the collective strong coupling regime, and should offer enough optical access to the sample.

Experimental goal

During the course of this work blockade effects were not investigated, and all the results presented here can be described using the linear model of Eq. 2.48. Nevertheless, the considerations above lead to a set of experimental requirements:

• The cavity has to be as close as possible to single-ended, and optical losses must be minimized.

• The collective coupling 𝑔 of the cloud with the cavity must be strong: 𝑔 2 ≫ 𝜅𝛾. In other terms, the resonant absorption of the cloud must be much larger than the round-trip cavity losses.

• The cloud must be small enough to entirely fit within one Rydberg blockade sphere, and have a high enough density in order to lead to an increase of the Rydberg non-linearity.

• The control beam must be large enough in order to cover the entire cloud, and its intensity should be uniform over the sample.

• In the low-excitation, interaction-less regime, the EIT must make the cloud transparent, i.e. make its absorption much smaller than the round-trip cavity losses. This requires:

a sufficiently large Rabi frequency 𝛺 for Rydberg excitations, as uniform as possible at the scale of the atomic cloud, -a small Rydberg linewidth 𝛾 𝑟 , which in turn requires a noiseless atomic environment (in particular free of stray electric fields), a good control of the atomic internal states, a sufficiently cold atomic temperature, and a good stabilization of the laser frequencies.

Many of these requirements are contradictory and require compromises. For instance, a high Rabi frequency 𝛺 improves the EIT but decreases the Rydberg blockade range. Conversely, increasing the principal quantum number of the Rydberg state increases the blockade range but decreases 𝛺, which scales as (𝑛 * ) -3/2 . This manuscript presents the assembly of the experimental system and shows how and to what extent these requirements were satisfied.

CHAPTER 3

Experimental Apparatus

Introduction

We describe in this chapter the assembly of our experimental setup which aim, among others, is to allow us to create and examine the interactions between optical photons. The manner in which we aim to create the interaction is through the use of a delocalized Rydberg excitation over the entire cloud, which is located in an optical resonator of controlled geometry. The requirements of the setup are based of the properties of the Rydberg atoms, as well as the expected interaction. We have seen in the previous chapter different requirements concerning parts of the experimental setup, and we present here the technical solutions that were implemented in our experiment

The atoms initially comes from a 87 Rb ampule located far from the main vacuum chamber. After an efficient loading of the 3D MOT by a 2D MOT, the atomic sample is cooled to an optical molasses. The atoms are then loaded in a one-dimensional moving lattice, before being transported to the optical resonator where the Rydberg excitations will take place. The sample is transferred to a standing wave dipole trap and cooled down via degenerate Raman sideband cooling before the excitation takes place. We then describe the manner in which our experimental platform is frequency stabilized, and explain more thoroughly technical details concerning the science cavity and blue cavities that are fundamental parts of the setup. Last, we cover the manner in which acquisition and imaging is done.

Vacuum System

The vacuum chamber in which the experiment is taking place was commercially bought: it is an important part of the system as it provides a clean, low pressure environment for the experiment to take place. The different parts composing the vacuum chamber need to be well cleaned, else vaporization of water and organic matter happens. The vacuum chamber should of course be well sealed from the outside environment and be compact, while providing the required optical accesses for the different lasers beams needed. Another point of interest is to choose a material that would limit the influence of magnetic fields, i.e. the steel type. The final pressure reached in the vacuum chamber is the result of the interplay between pumping, leakage and outgasing of the material in the chamber. Our vacuum system is composed of two main parts that are the 2D MOT chamber and the main vacuum chamber, which are visible in Fig. 3.1.

The main vacuum chamber 1 is a hollowed sphere intersected by two parallel DN160CF

1 spherical square vacuum chamber, Kimball physics ports, with fours DN63CF and four DN40CF ports arranged along the edge and sixteen DN16CF ports surrounding the DN40CF ports. Each opening being surrounded with tapped bolt holes, and the chamber is a bit over 10 cm in thickness. The assembly process was mainly done by the current PhD student J. Vaneecloo, with the help of our current postdoctoral researcher S. Garcia, and thus will only be described briefly. The material used are chosen specifically for the purpose of being put under vacuum: the wires are insulated with Kapton, the Viton pads were heated and cleaned beforehand, non-magnetic stainless steel is used, and specific thoughts were given to avoid trapped volumes of gas in the vacuum volume, such as in blind tapped holes. The outgassing phenomena happens when gas trapped in a material is released. Water can be removed through the bakeout process, but grease stays much longer, and thus needs to be removed beforehand. In order to avoid outgassing, virtual leaks, and contamination of the vacuum, preparation of the different parts that compose the system must undergo specific cleaning processes. The parts, wires and optics where initially cleaned following direction from LIGO [START_REF] Coyne | LASER INTERFEROMETER GRAVITA-TIONAL WAVE OBSERVATORY (LIGO) Cleaning Methods and Qualification Procedures[END_REF] and baked at temperature ranging from 120 ∘ C to 300 ∘ C for several days individually, before being assembled and installed in the vacuum chamber. The bakeout process consists in pumping a closed system with a primary pump and a turbo pump to the lowest vacuum one can achieve, all the while heating slowly the system until the maximum wanted temperature is achieved, then wait for several days. It has to be done with a slow temperature gradient (between 0.5 ∘ C min -1 to 1 ∘ C min -1 ) in order to limit material stress, in particular at glass-metal interfaces. This process removes unwanted hydrogen from the system, thus improving long term stability of the achieved vacuum. Once the pressure has stabilized, we switched to an ionic pump and remove the heating apparatus. The different connections between the part of the setup are sealed through the use of copper gaskets and semi-flexible bellows when required. One of the last step of this process was the connection of the 2D MOT to the main chamber through a small bellow before heating it as illustrated in Picture 3.1. Once all of the above was done, what remained was the introduction of 87 Rb in our system. Our 87 Rb source is a 1 g ingot placed in a glass ampule 1 . This cell is then fixed in a titanium tube of 8 mm in diameter and 70 mm in length to a flange that is connected to the 2D MOT. After reaching a pressure of 8.2 × 10 -10 mbar in the main chamber, the 87 Rb cell was isolated from the rest of the system and broken open, leading to an increase in pressure. A heating home-made sleeve was placed around it in order to control the 87 Rb migration rate and by extension the atomic density in the chamber and we had to wait for the system to reach an equilibrium. What remains then to be done is to place the 2D MOT optics and witnessing fluorescence at 780 nm, thus indicating that the system is operational. After several months, and with only the ion pump 2 to keep the vacuum, we have a stable pressure of 𝑃 𝑐ℎ𝑎𝑚𝑏𝑒𝑟 = 2.5 × 10 -9 mbar.

A crucial point to consider when looking at the vacuum chamber was what optical accesses did it provide compared to the ones that we needed. In order to have a MOT, three pairs of counter-propagating beams are required, each pair being orthogonal to the other two. Once a MOT is present in the chamber, we want to transport and trap a small atomic cloud to the cavity that is placed above the MOT position (see Fig. 3.10): we then require access along the direction formed by the MOT and the science cavity. Once the atoms are transported, we want to cool them down, and this is done by transferring them to another lattice, which in turns implies more optical accesses. Only then can we excite the atoms, requiring access to the cavity. This list is not exhaustive, since we need to test, characterize and tune each of the specific subsystems, e.g. the properties of the MOT. Fig. 3.2 illustrate the vacuum chamber with some of the necessary optical accesses. Placed in the vacuum chamber are aspheric lenses of focal 𝑓 = 50 mm that are used for imaging the cloud of atom with a better collection efficiency than what can be done with a direct imaging from the side of the chamber.

MOT

In this section we discuss the configuration of the MOT used in our experiment. The atoms are trapped in a 2D MOT from a 87 Rb ampule, before being sent to the main vacuum chamber to a 3D MOT. We first consider the characteristics of the different lasers necessary for the preparation of the MOT, before discussing the implementation of the 2D MOT and 3D MOT in our setup. The cooling laser stabilization is discussed in Section 3.7.

MOT lasers

Main lasers -TA Pro 780

The main laser necessary for the MOT is the cooling laser. We use a Toptica TA pro, operating on the D 2 line of rubidium at a wavelength of 𝜆 = 780 nm. It has an output of 3 W which was reduced to supply approximately 670 mW on the tapered output in order to increase the laser diode lifetime, and a typical free-running linewidth in the range of 100 kHz according to the laser specification. It is stabilized in temperature and current by a DLC pro Digital Control from Toptica. Between the isolator and the tapered amplifier of the laser, an auxiliary beam is tapped off, providing 15 mW to 20 mW of power. The laser is locked in frequency using a Pound-Drever-Hall (PDH) scheme, using part of the auxiliary output sent to an ultra stable cavity, as described in Section 3.7: the stabilized laser present a linewidth of 3.3 kHz. Fig. 3.4 illustrates the beam path and the uses of this auxiliary probe beam: in addition to stabilizing the MOT laser, it is used to stabilize the repumping laser (PD Lock ), the Raman laser from Section 3.5 (To Raman setup), and to cross-check the MOT laser frequency using a Doppler-free saturated spectroscopy setup (PD Spectro ).

The main output of the laser is split between several paths shown in Fig. 3.3. This first optical table serves to provide light for the 2D and 3D MOT beams, respectively the couplers labeled 2D MOT and MOT. The 2D MOT coupler links to the repumping laser setup, described below, while the MOT coupler goes to a fiber splitter setup on a third table where the vacuum chamber is located, which then provides the six output beams used for the 3D MOT (see Subsection 3.3.3). For flexibility, part of the setup is shared with the DL Pro 795, the laser optimized for the D 1 line described in subsection 3.6.1. In particular, through the use of a flip mirror we have the possibility of using the optics set up for homodyne detection with the master laser, that is the beam paths that end at the couplers labeled Local oscillator, Probe 1 and Probe 2.

Repumping laser -DL Pro 780

As mentioned, 87 Rb presents a multi-level structure. In order to avoid losing the atoms to dark states, we need a repumping laser. Less power is required than the master laser, and in practice, we work with nearly a factor 20 in the output power of the repumping laser compared to the master laser. A Toptica DL Pro running at a wavelength of 𝜆 = 780 nm is used for the repumping light of the MOT. It has an output of 40 mW, a typical free-running linewidth of 100 kHz and is stabilized in current (respectively temperature) using the commercially available modules Toptica DTC 110 (respectively Toptica DCC 110). It provides hyperfine pumping on the |5 2 𝑆 1/2 , 𝐹 = 1⟩ → |5 2 𝑃 3/2 , 𝐹 ′ = 1⟩ transition. Fig. 3.17 shows the addressed transition with respect to the atom level structure. The optical table layout of the repumping laser is shown in Fig. 3.4. The repumper laser output is split between its lock, and three independent beam paths. A first path (To MOT ) is used for the 3D MOT: it is combined with the master laser in the fibered 2x6 splitter (see Section 3.3.3). The second path is used to repump both the transverse direction of the 2D MOT (MOT 2D y and MOT 2D z). The third beam (Rep add bench) is sent to the additional setup on the second optical table shown in Fig. 3.5. From there, part of the beam is used to provide hyperfine repumper in the vacuum chamber (Rep cav) while the remaining light is combined in another single-mode fiber (TA Cav combined) with the master laser and the Zeeman pump. A shared pick up with the master laser provides light to a wavemeter 1 (To wavemeter) in order to monitor both the master laser and repumping laser frequencies.

Secondary setup on the second optical table

In addition to the optical table where the previously mentioned lasers are located, a second optical table is used in order to provide Zeeman pumping, imaging and probing beams. Fig. 3.5 illustrate the relevant part of the optical setup: the input to this table is done through both couplers labeled To MOT table and To Zeeman pump in Fig. 3.3. From the first coupler labeled From TA pro 780, we get Acousto Optic Modulator (AOM)s set up in double pass that allow us to finely tune the frequencies of the probe and imaging beams. From the coupler labeled From Zeeman pump, we get light used for Zeeman pumping. The manner in which it was initially implemented on the first optical table 2 did not provide a large enough frequency detuning, hence its current setup on the second optical table 3 . The coupler labeled To MOT in Fig. 3.4 is sent to the coupler labeled From repumper, and is used to provide repumping light available to be used at will. All the beams couple to single-mode fibers that link the table to the vacuum chamber. This setup allows for a rapid and flexible control of the frequencies and powers for the pump and probe beams.

2D MOT

Loading directly the 3D MOT from the background pressure of 87 Rb can take a significant time, which in turns decreases the repetition rate of the experiment. The motivation behind the 2D MOT is that we can load the MOT significantly faster: loading the MOT from ambient pressure we can reach 8 × 10 5 atoms in 9 s, compared to 1.3 × 10 8 atoms in 100 ms using the 2D MOT to trap and cool the atoms beforehand. The 2D MOT used in our experimental setup consists of two pairs of magnetic coils in anti-Helmholtz configuration that create a quadrupole magnetic trap, combined with two transverse cooling beams. In order to shine laser light along a major part of the cloud, the 𝜎 + beams are outputted from homemade coupler (MOT y and MOT z in Fig. 3.10) that provides a large beam and go successively through a 70/30 and 50/50 beam splitters, the power thus being spread roughly in an equal manner along the cloud. The collimators are made from achromatic lens of focal 𝑓 = 100 mm with three quarter-waveplates located immediately before the glass cell and set to provide the correct polarization. The beams are then retro-reflected after passing a quarter waveplate and arrive on the atoms with a 𝜎 -polarization. Two pairs of copper wire coils along the y and z-direction respectively generate the quadrupole magnetic field. They are included between the cell and the outside frame on which the optics sit. The atoms are then cooled down along the laser beam in the transverse plane of the 2D MOT therefore getting arranged in a cylinder shape whose long axis (x-axis) is aligned along the 2D MOT long axis. At the end of the cell (left hand-side of each of the sub-figures in Fig. 3.6) a mirror and a quarter waveplate, each with a 2 mm hole drilled in their center are placed. A cooling beam along the x-axis pushes the atoms in front of the hole into the main vacuum chamber.

The 87 Rb cell, partially insulated and with a controllable temperature provides 87 Rb atoms to the 2D MOT. Due to the geometry of the link between the 2D MOT chamber and the main vacuum chamber, a difference in pressure exists and is maintained between the chambers: the flow of 87 Rb atoms has to go through a small opening in the quarter wave plate and mirror, before crossing the bellow and reaching the main chamber. We are in the molecular flow regime, where the conductance does not depend on pressure. The conductance can then be expressed for a molecular flow as 𝐶 = 12.1 𝑑 3 𝑙 , where d is the diameter of the opening and l the length of the connection. In our system, we have 𝐶 = 6.32 × 10 -6 l s -1 : here is approximately 10 cm between the 2D MOT beams and the output of the bellow, and then 20 cm until the area where the MOT is supposed to be. Knowing the partial pressure of 87 Rb, we can estimate that at a temperature of 49.5 ∘ C we had a pressure in the 2D MOT chamber 𝑃 2𝐷𝑀 𝑂𝑇 = 7 × 10 -7 mbar and a pressure in the main chamber of 𝑃 𝑐ℎ𝑎𝑚𝑏𝑒𝑟 = 1.3 × 10 -9 mbar. Now after several months, we decreased the temperature of the 87 Rb cell to 37 ∘ C, and we have 𝑃 2𝐷𝑀 𝑂𝑇 = 9.5 × 10 -8 mbar and 𝑃 𝑐ℎ𝑎𝑚𝑏𝑒𝑟 = 2.5 × 10 -9 mbar. A first trial for the 2D MOT cell was done by former post-doc S. Ćuk. The appropriated glass parts were glued to each other to reach the cell shown in Fig. 3.6(a). However some problems were encountered: during the bakeout process the structural integrity of the entire cell was compromised, since the glue softened when heated.

A revised version of the cell then consisted in a metallic rebar on which glass panels were glued. When baked, the previous issue did not appear on this second iteration of the cell. The typical power at the output of the coupler providing the transverse cooling is approximately 30 mW at the output of the couplers, and the typical power from the pushing beam coupler is approximately 13 mW. The couplers MOT y and MOT z from Fig. 3.10 come respectively from the first optical table (MOT 2D y MOT 2D z in Fig. 3.4), as well as the pushing beam (coupler MOT x in Fig. 3.4). For a single coil, we measure a gradient of 0.85 G cm -1 A -1 , so we should get four times as much with the two pairs: in order to get approximately 10 G cm -1 along the y and z-directions we need to run a current of 3 A per coil. Fig. 3.7 shows the magnetic field from one of the pair of coils of the 2D MOT.

3D MOT

We now focus on the 3D MOT. It was briefly described from a theoretical point of view in Section 2.2.2, while Fig. 3.17 depicts the atomic level addressed by the different cooling and repumping lasers previously described. We discuss here on the assembly of the MOT, as well as the magnetic field control. We want the atomic beam from the 2D MOT to actively feed the 3D MOT, leading to shorter loading times, and in a similar manner as the 2D MOT, we combine optical and magnetic field to achieve stable trapping of the atoms. In order to optimize the MOT, we must ensure that counter-propagating beams have the same power and that the zero of the magnetic field is centered on the crossing point of the beams. Stability in the power of the counter-propagating beams is necessary and the use of a fiber splitter guarantees a balance of power that does not change in time. We use a two inputs and six outputs fiber splitter 1 , whose schematic is presented in Fig. 3.8. One of the input of the fiber spitter comes from the master laser while the other comes from the repumping laser. Both inputs are then mixed and outputted towards the vacuum chamber through six homemade collimators in the manner shown in Fig. 3.10. The collimators are made from achromatic lens of focal 𝑓 = 80 with a quarter wave plate set such that the beams have the wanted polarization. The inputs taps lead to photodiodes, where the response is converted to a voltage and compared to a set-point: a Proportional-Integral-Derivative controller (PID) provided by a Red Pitaya is then used to lock the intensity of both the MOT input beam and repumper input beam. In many atomic physics experiment (e.g. Bose-Einstein Condensate (BEC)), the coils for the MOT are located outside of the vacuum chamber. This avoids out-gassing and cooling issues. However in our case, this would be impractical given the compactness of the setup, and keeping the MOT coils close to the anti-Helmholtz geometry for small coils would also imply that they should be inside the chamber. The two MOT coils located inside the chamber (see Picture: 3.9(a)) then provides the magnetic field gradient. They are made of 12 loops of copper wire, and the expected field gradient is 2 G cm -1 A -1 . Three pairs of compensation coils are present on the outside of the 1 Evanescent Optics Inc 2x6 array chamber to cancel stray magnetic fields at the center of the MOT, in order to ensure a zero magnetic field. The pairs are respectively used in a nearly Helmholtz configuration along the y, z+x and z-x-axis. Section 4.2.1 present the work done in order to achieve a zero magnetic field in the 3D MOT through microwave spectroscopy. These compensation coils are supplied by independent High Finesse current supply sources that allow greater control of the field: the bigger pair of coil along the y-axis which is going to provide the quantization axis gives a field of 8.5 G A -1 and details about the coils characteristics are given in Table : 3.1: we have with the resistance of the pair of coil a current of 1 A when the voltage is set to 5.5 V. The typical power of the cooling laser per MOT beam is approximately 12 mW, and 0.7 mW from the repumper laser. Above each of the MOT coils are placed two electrodes, and placed below each of the compensation coils along the z+x and z-x-axis are achromatic lenses that are used to collect light from imaging beams that are crossing the vacuum chamber in a diagonal manner. Fig. 3.10(a) and 3.10(b) show respectively top and side views of the vacuum chamber with an emphasis placed on the MOT optics, as well as the positions of the 3D MOT and science cavity with respect to each others.

Transport to the science cavity

In many cold atom experiment, the preparation of the atomic cloud does not take place where the end measurement is done. This implies that some sort of transport is to be used, and our setup does not deviate from that: we cannot implement the trapping and cooling process of our sample in the cavity where the interactions are taking place due to optical accesses constraints, and potential issue concerning the deposition of atoms on the cavity mirrors. In our setup the cloud of atoms is created in the 3D MOT, and it needs to be transported to the cavity where the interactions will take place. This transport takes place over a distance on the order of magnitude of 30 mm, and needs to be efficient and rapid, without heating too much the sample and ideally not be invasive towards the science cavity, i.e. have an atomic beam directed towards a mirror. Several approaches exist that allow for precise transport in a controlled manner: in our case where we only require a small atomic sample, the use of a moving optical lattice is appropriate, as it allows for a well controll transport to the optical cavity, with a precision on the order of magnitude of several hundred of nanometer after transport over a distance of several centimetres, with only two accesses above and below the setup. This moving array is created through use of a one-dimensional dipole trap where two counter-propagating laser beams are mutually detuned, which gives full control over the displacement. A theoretical description is done in Subection 2.2.2, and a similar atom transport using a One-Dimensional (1D) optical lattice has already been implemented in our laboratory on a parallel experiment [START_REF] Vitrant | Manipulating cold atoms through a high-resolution compact system based on a multimode fiber[END_REF]. We first describe the laser used in the creation of the lattice, followed by the manner in which we implemented this technique in our setup.

Laser and layout

We use a Toptica TA pro running at wavelength of 𝜆 = 783 nm which provides the power necessary for the one-dimensional moving array that we named lattice laser. It is stabilized in current and temperature using Toptica commercially available modules (respectively DTC 110 and DCC 110), and its main output provides 1.1 W of power of which approximately 350 mW is coupled in each of the arms of the moving lattice, due to diffraction efficiency of the AOM and the coupling into the optical fibers. This laser was set after installation to a wavelength of 783 nm far detuned from all atomic resonances and is left unlocked: the important parameter for this laser is the relative phase of the two arms of the lattice, since it translate into position fluctuation of the dipole trap potential, which in turns leads to heating of the atoms. Fig. 3.11 shows the optical setup located on the second optical table, used to create the optical moving lattice. The power is split between the two AOMs that each control one of the lattice beam frequency, and a shared wavemeter input is available for occasional measurement of the lattice laser wavelength.

In a perfect world, to keep the atoms trapped during the transport, the acceleration and deceleration should be adiabatic. The force applied to the lattice tilts the potential, thus reducing the effective trapping depth: if the acceleration and temperature are too high, the trapping along the axial direction will not be strong enough and the atoms will escape. The maximum acceleration that the lattice undergoes is of the order of 1 × 10 3 metre/s 2 : we transport the atoms in 16 ms over a distance of 33 mm. The transport is composed of two phases of linear acceleration and deceleration of 8 ms, and the typical transport velocity is 𝑣 = 2 mm ms -1 . Since the relative phase between the two beams is mapped directly on the spatial phase of the trap, phase fluctuations will lead to heating or losses of the atoms [START_REF] Gehm | Dynamics of noise-induced heating in atom traps[END_REF]. The control over the lattice beams comes from the two AOMs that are used to provide the amplitude and frequency shifts for each arms of the lattice (AOM 2 and 3 in Fig. 3.11). Since they are used in a double pass configuration, a frequency shift of 𝛥𝜈 over the Radio frequency (RF) carrier frequency will lead to a shift of 2𝛥𝜈 over the frequency of the lattice beam. Fig. 3.12 depicts the typical output of the channel of the Direct Digital Synthesizer (DDS) controlling the moving lattice beams in a typical transport sequence: during a typical transport sequence, there is a linear acceleration step, followed by a linear deceleration step. The control computer of the experiment communicates with a DDS whose RF outputs are amplified and sent to the AOMs. Fig. 3.13(a) represent the layout of the optics on the optical table below the vacuum chamber, and Fig. 3.13(b) shows the optics that are located on the two optical breadboard on top of the vacuum chamber. In Fig. 3.13(a) the coupler Lattice down provides power going up in the chamber: before reaching the periscope that sends it upwards, the beam goes through a wedged glass plate that allows the adjustment of the alignment of the arm of the lattice. Above the vacuum chamber, in a similar manner as before, in Fig. 3.13(b) the coupler Lattice up corresponds to the coupler Lattice up in Fig. 3.11, and provides power that goes down in the chamber. The couplers labeled coupler 1 in both the upper and lower breadboards can be used to repump the cloud, while the couplers labeled Pump HF 1 and 2 are used for the Zeeman pumping of the sample.

Size of the beam and waist position:

We want to transport the atoms from the MOT to the science cavity, over a distance 𝐿 = 33 mm. The waist of the moving lattice is placed halfway between the MOT and the science cavity lower mode, in 𝐿/2 in order to get a similar waist size in both the starting and final position. The ideal Rayleigh length 𝑧 𝑅 to ensure the smallest waist both at the MOT position and the cavity position is then 𝑧 𝑅 = 𝐿/2, which gives a waist 𝜔 𝑇,0 = 64 µm at mid-distance. Then at the MOT position and cavity position we have a waist of 𝜔 𝑇,-𝑧 = 𝜔 𝑇,𝑧 = 91 µm. From there, assuming a power in each of the arms of 350 mW, we have a trap depth 𝑈 𝑑𝑖𝑝 /𝑘 𝐵 ≈ -0.6 mK.

Cooling and trapping of the atomic sample

Once the atomic sample has been transported to the science cavity, it needs to be cooled down. We chose to use degenerate Raman sideband cooling of the atoms, since it is a technique that can be easily implemented on our setup, and which requires low laser power in order to efficiently cool the atoms [START_REF] Castin | Reabsorption of light by trapped atoms[END_REF] [67]: we present here its implementation, and we name the laser used Raman laser.

An optical lattice is used to confine the 87 Rb atoms. It is created by overlapping the two Raman beam which come from the same laser and hence have the same amplitude and same wavelength to form a standing wave. This standing wave creates a periodic potential which traps the atoms. We use a Toptica DL Pro running at a wavelength of 780 nm who outputs 62 mW for this Raman laser, providing 20 mW per fiber of power to the vacuum chamber. This laser layout can be seen in Fig. 3.11: it is phase-locked on the master laser, as described in Section 3.7.3. The optical bench schematic is shown in Fig. 3.11 and it is stabilized in current and temperature using previously mentioned DTC 110 and DCC 110 modules.

The beams used for the lattice are shown in Fig. 3.14 representing a horizontal cut in the vacuum chamber. The two beams coming from the Raman laser are placed in the plane of the atomic cloud and form an angle between them (dashed red beams). Due to space constraint, the coupler Raman 1 is set along the imaging path of camera 2 used to image the cloud of atoms during the transport phase, and is then dumped. The other arm from the coupler Raman 2 is collimated and retro-reflected after its passage through the science cavity. The Raman beams have waist of 150 µm at their waists, and respectively 170 µm and 160 µm at the atomic cloud position for the dumped beam and reflected beam. Intermediate breadboard details This figure also represent the different beam paths used in the excitation of the sample after transport (full red beams) mentioned further down in Section 3.6. In the dot and dash line are represented the MOT 5 and MOT 6 optical paths, as well as the imaging path used to do absorption imaging (camera 1). In blue are represented the beams coming from the two blue buildup cavities discussed in Section 3.9: the mirror labeled A (respectively B) is set up on a translation stages that can be used to spatially displace the power beam of the right (respectively left) blue cavity.

Vertical confinement

In order to provide vertical confinement, we use a vertical standing wave dipole trap. For this purpose, we use an 8 W fiber laser from AzurLight systems running at 𝜆 = 1064 nm. It has a manufacturer specified linewidth lower than 50 kHz in single mode and is used daily with an output of 3.5 W at the laser head, which become 2.8 W after the isolator. Lasers in this frequency range are widely available due to their uses in industry, usually rely on a well-known technology and thus are less costly per watt of power then their counterparts in the infrared. The schematic of the optical setup is shown in Fig. 3.15: at the laser output is placed an isolator in order to avoid feedback to the laser, and the output of the laser get split with a ratio of approximately 1000/1 into a high-power and low-power section. The high-power part of the setup prepares the four beams for the two dipole traps: the first one is a vertical crossed dipole trap along the same direction as the transport lattice (dipole trap up and down in Fig. 3.15), and the second one is a diagonal crossed dipole trap (dipole trap left and right in Fig. 3.15) discussed further down. The low-power part is used to stabilize the laser, to provide a lock for the science cavity as well as to monitor the laser wavelength. If this laser was only used for a dipole trap, then it would not be necessary to lock it, but since it is used to stabilize the science cavity it needs to be: an advantage of using this laser for this purpose is the fact that due to the large detuning with 87 Rb transitions, it will interact very weakly with the atomic sample in the cavity. After the Polarized Beam Splitter (PBS) that split the output power between the high and low-power section, the weaker beam is sent through an EOM 1 providing sidebands that are used to stabilize the laser using a PDH scheme on a secondary cavity as described in Section 3.7.3. The control of the laser is done via feedback on the current and temperature of the seeder2 : a ±5 volt input on the laser temperature control correspond to ±196 MHz in the laser frequency, while an input of ±5 volt on the current control corresponds to a frequency shift of ±900 MHz in the laser frequency. Initial plans were to set up this laser for the transport lattice given the amount of available power and the fact that it is red detuned from the atomic transitions, however due to mode jumps of the laser it was replaced with the previously described lattice laser (TA Pro 780). The issue was that the laser seeder was located inside the high power amplifier case, leading to mode jumps over several tens of megahertz and making this laser impossible to stabilize in frequency. Azurlight System fixed the laser by replacing and moving the seeder outside of the amplifier case, and we isolated the seeder box from external vibrations.

Trap details:

The vertical dipole trap is superposed to the transport lattice. In Fig. 3.13(a), the coupler Dipole trap down is superposed on the beam path of the lower arm of the moving lattice on a dichroic mirror, in while Fig. 3.13(b) we can see the symmetric situation with the coupler Dipole trap up. We aim for a waist of approximately 40 µm for at the position of the cavity mode. In order to reach it, we use homemade collimators for the two arms of the trap, with an achromatic lens of focal f=50 mm placed before the wedge window and the f=500 mm lens placed on the translation stage (lens 1 in Fig. 3.13(a) and lens 2 in 3.13(b)). The beam is then merged with the moving lattice beam on a dichroic plate. The use of a wedged plate on the upper and lower beams also allows for fine translation of the dipole trap.

Diagonal crossed dipole trap

Once the atoms are transported to the cavity mode, and after they are cooled down, we want to transfer them to a shallower dipole trap in order to store the atoms before we can perform measurement on the sample. A second dipole trap is then put in place: it has a smaller trap depths compared to the other crossed dipole trap in the setup in order not to heat the sample, and we aim at a waist of 8 µm for each of the beam of the crossed dipole trap at the sample position. It makes use of the two aspheric lenses present in cavity, and their layout is shown in Fig. 3.13(b): the two beams come from the couplers Dipole trap left and Dipole trap right, and they are combined with imaging on both sides. Each of these beams are sent in a diagonal manner in the vacuum chamber, alongside the MOT beams 1 and 4. After each of the couplers, a lens of the telescope is placed on a translation stage (lenses 3 and 4 ) in order to control the position of the waist. A dichroic plate is used to superpose imaging light at 780 nm towards cameras 1 and 2, as we can image the sample along each of the diagonals.

Excitation of the atomic cloud

We have in the previous sections described the motivation and layout of the system that allows us to transport the atomic sample to the main resonator and to cool it down. What remains is the cloud preparation, as well as the Rydberg excitation. Concerning the cloud preparation, Fig. 3.13 and Fig. 3.14 present the different fiber couplers that provide access to the chamber for Zeeman pumping and repumping light. We discuss here the two-photon excitation to a Rydberg level.

Main lasers -DL Pro 795

In order to gain flexibility in the excitation scheme to Rydberg levels and have the possibility of addressing the D 1 line transitions of 87 Rb either as a pump or a probe, another laser at a wavelength of 𝜆 = 795 nm is required. For this, we use a Toptica DL Pro set at 𝜆 = 795 nm, stabilized in current and temperature in a similar manner as the repumping laser. It has an output power of approximately 36 mW, and we label it as the 795 laser. Fig. 3.3 shows the corresponding layout: the laser can be either used as a probe or a pump on the D 1 line. Optics are also set up such a way that homodyne detection can be implemented, as will be mentioned in Section 3.10. Be it as a pump or a probe, the laser frequency needs to be stabilized. Recall that the secondary cavity is assumed to be stable, so sending light from the coupler Lock 795 to this cavity in a PDH scheme allows us to stabilize the laser (see Section 3.7). The probe is provided by the master laser if we choose to work on the D 2 line, and we recall that this laser is stabilized on a cavity. We then focus on the second leg of the two-photon excitation to a Rydberg level. The transition dipoles from the 5P to high-lying Rydberg nS states are small (typically 𝑑 0 ∼ 1.0 × 10 -32 C m -1 ), so in order to reach sufficient Rabi frequencies on this transition a high control power is required. In addition, these states are spectrally narrow, so the absolute frequencies of the probe and control lasers driving two-photon transitions to these states must be stable to a kHz level. In order to fulfill the aforementioned requirement for the experiment, we use two Titanium Sapphire laser (Ti:Sapphire) lasers for the control light of the Rydberg excitation.

Titanium Sapphire lasers

These lasers are used to create the second excitation for the two photon Rydberg excitation, in combination with either the master laser of the previously mentioned 795 laser as probe. They allow for a highly frequency-adjustable high intensity excitation of chosen Rydberg transitions, while providing a high power output of more than 1 W of doubled power for a Full Width at Half Maximum (FWHM) below 50 kHz with the etalon locked. In our experimental setup, two M squared Ti:Sapphire lasers 1 are used: they are both pumped by pump lasers producing nominally 18 W of power at 532 nm, then doubled and output 1.2 W of blue light at 480 nm. These numbers dropped to approximately 300 mW of power at 475 nm and 480 nm at the lasers outputs: the pump power started to drop due to an issue with the doubling crystal, and the pump laser is currently being fixed.

These lasers have an auxiliary output located before the doubling cavity, with approximately 70 mW of available power: we use them to lock the lasers and to monitor their wavelengths. The manner in which they are stabilized is covered in Section 3.7. This is done using a PDH lock scheme on either an ultra-stable cavity for the laser we label Tisaph SLS, or a secondary cavity, called transfer cavity for the laser labeled Tisaph transfer. The locking electronics of the laser is based on the Red Pitaya, and discussions on the Red Pitaya implementation is done in Section 3.7.5. Fig. 3.16 shows the third optical table, with the optical layout of the two Ti:Sapphire lasers. Also present on this table, but not shown here is the vacuum chamber and the breadboards surrounding it. Locking the Ti:Sapphire lasers to their respective cavities, we can measure the beat note between them on a fast photodiode, and we reach a relative stability of 3.4 kHz between them. Ti:Sapphire SLS This Ti:Sapphire lasers is called Ti:Sapphire SLS as the cavity on which it is locked is the SLS cavity. Approximately 60 mW of power at a wavelength of 960 nm is available from the auxiliary output of the Ti:Sapphire SLS, which are used to monitor its frequency (coupler To wavemeter), and to lock it to the ultra-stable cavity (see Subsection 3.7.1). Fig. 3.16 shows the optical setup of the SLS Ti:Sapphire lasers. We currently have 450 mW of power at a wavelength of 480 nm available at the main output of the laser, that gets split between three paths. The first one goes through AOM 2 and is sent in free-space to a breadboard placed above the optical table, located next to the vacuum chamber. It is the power beam that is injected in the left blue cavity described in Section 3.9. The second path goes through AOM 3 which is set in double pass, and is injected in the coupler Spectroscopy. It is available if we want to perform a spectroscopy measurement on the sample. The third path is used to provide the locking beam for the blue left cavity, presented in Section 3.9. It goes through AOM 4 which is set up in double pass, before being reflected into the EOM and sent to the coupler Lock Blue Left Cavity. A single-mode fiber bring the beam to the vacuum cavity: in Fig. 3.14 the couplers Lock Blue right Cavity and Lock Blue left Cavity are coming from the similarly named couplers in Fig. 3.16. These are the locking beams of the blue cavities.

Ti:Sapphire transfer The second Ti:Sapphire laser is called the Ti:Sapphire transfer. In a similar fashion as the Ti:Sapphire SLS, approximately 70 mW of power at a wavelength of 960 nm is available from the auxiliary output of the Ti:Sapphire transfer laser. It is used to monitor its frequency and to lock the laser to the transfer cavity as described in Subsection 3.7.1. Fig. 3.16 shows the optical setup of the transfer Ti:Sapphire, next to the SLS Ti:Sapphire. For this laser, we currently have about 200 mW of power at a wavelength of 475 nm available at the main output, which is split between two paths. The first one is reflected on the first PBS and goes through AOM 1 before being sent in free-space to the upper breadboard next to the vacuum chamber. This beam is the power beam that is injected in the blue cavity right (see Section 3.9). The second path is used to lock the blue cavity right. It first goes through an EOM, before passing in AOM 4 which is set in double pass, and going into the coupler Lock Blue Cavity Right. From there a single-mode fiber brings the locking beam to a coupler located close to the vacuum chamber.

Blue cavity locks:

The modulation for the locking beams of the blue cavity is provided by the free-space Electro Optic Modulator (EOM), each beam passing in it in a counterpropagating manner. They go through AOM 4 before (respectively after) the EOM: this AOM shifts the frequencies of the locking beams by 500 MHz to the next resonance of the confocal buildup cavities in order to avoid spurious Rydberg excitations. Table : 3.2 and table: 3.3 present respectively in a comparative manner to the target values a summary of the lasers powers and frequencies among other things. The induced transitions are depicted in Fig. 3.17. The full optical layout of the lasers optical tables that are in the laboratory can be seen in Appendix: A.

Setup frequency stabilization

In this section, we describe the manner in which the entire setup is stabilized. From the beginning it was assumed that the main laser was stabilized in frequency on a cavity, and both the Raman laser and the 795 laser are phase-locked to it. We also mentioned a secondary cavity that allowed for the stabilization of the remaining lasers. The manner in which all of these elements come together is covered in the following paragraphs, after a discussion of the stability required for our setup.

The stability of the laser setup that is used to interact with the atoms is fundamental for the experiment. As mentioned in the previous section, we aim at a two-photon excitation to a Rydberg level via either the 𝐷 1 or 𝐷 2 line doing two photons transitions. This implies a very good control of the absolute frequencies of the probe and control lasers. Experimentally, the MOT laser could simply be locked to the atomic transitions, however for it to be used as a probe in the two-photon excitation we need a better stability. For lasers that are slightly detuned from the master laser, a phase lock is sufficient, but if one needs a laser with a very large frequency detuning to the master laser then typically this laser would need its own spectroscopic lock method. In our setup, an ultra-stable cavity is used to lock the master laser, which in turns stabilize a secondary cavity called the transfer cavity. Remaining lasers and resonators are then locked on one of these elements.

SLS cavity

The cavity that is at the center of the locking scheme is a ATF-6010-4 ultra-stable cavity from Stable Laser Systems. It is a 100 mm long Fabry-Perot cavity, where a pair of planoconcave mirrors are optically contacted to a bar of ultra-low expansion glass, the concave mirror having a 500 mm Radius Of Curvature (ROC). It is enclosed in a vacuum housing 1 that allows for a controlled environment with respect to thermal and humidity variation, while also shielding the resonator from vibrations. The system is pumped 2 to a vacuum of 𝑃 𝑆𝐿𝑆 = 2 × 10 -6 mbar and maintained at a fixed temperature of 28 ∘ C, which is the temperature where the minimum in the coefficient of thermal expansion of the ULE glass bar is, thus leading to a stable length over long period of time. In this way a stable, well-defined cavity length is ensured. The cavity has a Free Spectral Range (FSR) of 1.5 GHz, and a Half Width at Half Maximum (HWHM) of 28 kHz at a frequency of 780 nm, with a targetted finesse F = 25000 at 780 nm-795 nm and 950 nm-960 nm. This resonator presents a frequency shift below 5 kHz per day if the temperature is kept at the zero of the coefficient of thermal expansion, thus providing high long term stability to the setup.

Master laser lock

The master laser was originally locked on the crossover between the |5 2 𝑃 3/2 , 𝐹 ′ = 1⟩ and |5 2 𝑃 3/2 , 𝐹 ′ = 3⟩ levels on the D 2 line, it is now and stabilized on the ultra-stable cavity using a PDH scheme at the same detuning. A pickup of a little over 7 mW of power coming from the auxiliary laser output (To SLS cavity in Fig. 3.4) is sent through a fibered EOM 3 to the ultra-stable cavity. A controlled central frequency is combined on a mixer with a frequency providing phase modulation at 20 MHz, and the resulting frequency is sent to the EOM. By controlling the central frequency, we can tune the shift between the laser and the cavity, thus having the possibility to lock it at a chosen frequency within three FSR of the cavity: the large bandwidth of the fibered EOM (5 GHz) provides tunability in the locking frequency. The master laser is then detuned by -211 MHz from the |5 2 𝑆 1/2 , 𝐹 = 2⟩ → |5 2 𝑃 3/2 , 𝐹 ′ = 3⟩ transition, and locked using a PDH scheme. Fig. 3.18 shows the optical layout of the ultra-stable cavity with the master laser output: coupler TA Pro 780 is split on a PBS between a low power section (approximately 0.5 mW) towards the ultra-stable cavity and a high power part towards the transfer cavity. The power 1 SLS vacuum housing 6010 2 Agilent 919-0520, 2 l s -1 3 Photline Modulator NIR-MPX800 towards the ultra-stable cavity provides a PDH error signal that is then sent to a Fast Analog Linewidth Controller (FALC) 110 which ensure feedback to the master laser: the fast output acts on the laser current, and the slow output on the laser's grating. From there, the 2D MOT, MOT, probe, pump and imaging beams frequencies can be finely adjusted using the appropriate AOM setup in double pass. Currently, the detuning between the SLS cavity and the master laser is 38 MHz and once the master laser is locked on the SLS, it has a linewidth of ∼ 3.3 kHz. Given the reflectivity requirements of the mirror of the SLS cavity, and the large range of wavelengths used in our setup (from 480 nm to 1064 nm), it was not possible to stabilize all of our lasers to it. We then actively stabilize the master laser on the SLS cavity, and then stabilize a secondary cavity called the transfer cavity on the master laser. In this manner, the stability from the SLS cavity is passed down to the transfer cavity.

Transfer cavity

The transfer cavity 1 is, as its name suggests, used to transfer stability to the lasers that are far detuned from the master laser. By locking the transfer cavity to the master laser, other lasers of very different wavelengths can be stabilized, with a better relative stability than independently locked setup for each lasers. In our case, we use it for wavelengths ranging from 780 nm to 1064 nm. The transfer cavity is a home-made design of a Fabry-Perot cavity, composed of a plane mirror as the entry mirror, and a concave mirror of ROC 5000 mm. Each of the mirrors rest in home-made supports that are maintained by ultra-low expansion glass rods, and the setup is placed in a vacuum chamber. A 3D schematic of the assembly can be seen in Fig. 3.19: viton pads are used to push on the cavity support so that it cannot move, they provide isolation from vibrations and are rigid enough to ensure that the optical axis of the cavity stays in place. The entire system is then placed under a vacuum of 𝑃 𝑡𝑟𝑎𝑛𝑠𝑓 𝑒𝑟 = 2.7 × 10 -8 mbar maintained by a ionic pump 1 . While not fundamentally necessary, placing this system under vacuum presents the advantage of making it insensitive to humidity and thermal variations, therefore ensuring long term stability on the order of weeks, with very occasional degassing leading to spikes in the pressure. The transfer cavity has a length of 750 mm and a FSR of 200 MHz, a theoretical finesse of 1.84 × 10 3 and a measured finesse of 1.8 × 10 3 at 780 nm (FWHM of 106 kHz). The injection of the different lasers is done on the flat mirror entry of the cavity, and has waist size of 670 µm for wavelengths of 780 nm to 795 nm, 740 µm for wavelengths of 950 nm to 960 nm and 780 µm for wavelengths of 1064 nm. The transfer cavity is locked on the master laser using a PDH scheme: on Fig. 3.18 one can see that part of the output of the master laser is to the transfer cavity. The error signal is then fed to a PID module which feedbacks on the cavity by controlling the cavity length via three piezoelectric elements 2 located behind the concave mirror.

Remaining lasers locks

As was already mentioned in the laser's descriptions, the different lasers used in the setup are also stabilized. We recall and describe here the manner in which they are locked. In various experiments, it is common for the lasers to be locked on a fundamental mode of the resonator, with the beam frequency tuned by an AOM set in double pass in order to reach a desired frequency (e.g. atomic transition frequency). Using an AOM is a cost-efficient solution that however is limited by the bandwidth of the AOM: for the SLS cavity, since we have a FSR of 1.5 GHz we cannot choose to use an AOM. As an alternative we use a dual sideband locking method on a fiber EOM. It is similar to the PDH technique, with the only difference of an extra function generator providing an offset frequency which is combined with the PDH modulation frequency before being sent to the EOM. Since the EOM has a large bandwidth of several GHz, we can change the laser frequency by the same order of magnitude by controlling the offset frequency. By using this method, we can lock the master laser as well as the Ti:Sapphire SLS at the same time on the SLS cavity, and easily tune them in an independent manner. Concerning the way the remaining lasers are locked on the transfer cavity, since it has a FSR of 200 MHz that can easily be accessed by an AOM set in double pass we set up the locks in the more common manner.

Repumping laser: The frequency between the repumper and the master laser is hard to separate optically but is still in the range of an electronic lock, hence the choice to phase-locked it to the master laser. Fig. 3.4 shows the auxiliary probe beam of the coupling laser, which is combined to a part of the output beam of the repumping laser on a fast photodiode 1 (𝑃 𝐷 𝐿𝑜𝑐𝑘 ). From there, a beat note of frequency 𝜈 𝑏𝑒𝑎𝑡 = 𝜈 𝑐𝑜𝑜𝑙 -𝜈 𝑟𝑒𝑝 is picked up. The signal generated by the photodiode is proportional to the intensity of the incident light:

𝐼(𝑡) ∝ |𝐸 𝑐𝑜𝑜𝑙 𝑐𝑜𝑠(2𝜋𝜈 𝑐𝑜𝑜𝑙 𝑡) + 𝐸 𝑟𝑒𝑝 𝑐𝑜𝑠(2𝜋𝜈 𝑟𝑒𝑝 𝑡)| 2 (3.1)
with 𝐸 𝑖 the amplitude of the electric field in beam i and 𝜈𝑖 the frequency of the beam i. We eliminate the DC component in the signal and by eliminating the higher frequency component which is outside of the photodiode bandwidth, we get the following signal since 𝜈 𝑏𝑒𝑎𝑡 is within the bandwidth of the detector:

𝐼 𝑅𝐹 ≈ 𝐸 𝑐𝑜𝑜𝑙 𝐸 𝑟𝑒𝑝 𝑐𝑜𝑠(2𝜋𝜈 𝑏𝑒𝑎𝑡 𝑡) (3.2)
The signal is then sent on a mixer 2 with the signal outputted by a signal generator 3 (LO) generating a frequency 𝜈 𝐿𝑂 , leading to the frequencies 𝜈 𝑡𝑜𝑡1 = 𝜈 𝐿𝑂 -𝜈 𝑏𝑒𝑎𝑡 and 𝜈 𝑡𝑜𝑡2 = 𝜈 𝐿𝑂 + 𝜈 𝑏𝑒𝑎𝑡 . The demodulated output provides the error signal ≈ 1 2 𝑐𝑜𝑠(2𝜋(𝛥 𝑏𝑒𝑎𝑡 -𝜈 𝐿𝑂 )𝑡) that is then sent to a FALC module whose slow output acts on the repumping laser grating, and fast output on the laser's current. The repumper is addressing the transition |5 2 𝑃 1/2 , 𝐹 = 1⟩ → |5 2 𝑃 3/2 , 𝐹 ′ = 1⟩, which is reached by setting the signal generator to a frequency of 6.543 GHz and considering the fact that the repumping beams go in single pass through AOM 1 and AOM 2 from Fig. 3.4, set at 80 MHz.

Raman laser:

The Raman laser is phase locked to the master laser. The situation is similar to that of the repumping laser that was described above. Part of the auxiliary output of the master laser is sent to the Raman laser table (coupler To Raman setup in 1 Hamamatsu G4176-03 2 Minicircuits ZX05-153-S+ 3 Windfreak Technologies SynthHD Fig. 3.4). A fast photodiode 1 (PD Lock in Fig. 3.11) measures the beat note between the master laser auxiliary output and the Raman laser, which is then mixed with the output of a signal generator 2 on a frequency mixer 3 . The demodulated result is sent to a FALC module whose fast output acts on the Raman laser current and slow output on the laser's grating. The Raman laser is detuned by -10 GHz from the D 2 line.

laser:

The 1064 laser is locked on the transfer cavity using a PDH scheme: the sidebands are provided by the EOM located at the beginning of the low power section of the laser as mentioned in Section 3.5. In Fig. 3.15, the coupler To Transfer cavity sends light passing through AOM 1 to the transfer cavity on which the laser is stabilized in a PDH manner. The resulting error signal is then sent to a FALC module which provides both a fast and slow feedbacks: the fast dynamic feedback is sent to a Voltage Controlled Oscillator (VCO) and acts on AOM 1 in order to compensate the noise, while the slow output acts on the laser current to compensate for slow drifts. In this manner, we bypass the low-bandwidth of the fine frequency control of the temperature of the seeder. AOM 2 of Fig. 3.15 is then used to control the frequency of the locking beam that stabilizes the resonator (coupler To science cavity).

laser:

The 795 laser is locked on the transfer cavity using a PDH scheme. Light is tapped from the laser and goes in double pass in AOM 4 in Fig. 3.3, which allows for shifts in the locking frequency. The beam is then sent towards the transfer cavity. In Fig. 3.18 this corresponds to the coupler labeled DL Pro 795. The phase modulation is provided by the EOM further down the optical path, and the PDH error signal is sent to a FALC module. The fast feedback output acts on the 795 laser current, and the slow output on the laser's grating.

Ti:Sapphire lasers These lasers have two similar piezoelectric inputs used for feedback. A slow piezoelectric input with a bandwidth ranging from DC to 50 Hz (roll off) gives a tunability of ±15 GHz with a sensibility of 1.5 GHz V -1 , while the fast piezoelectric input has a bandwidth ranging from DC to 50 kHz with a tunability of ±40 MHz with a sensibility of 4 MHz V -1 . This allows correction using fast and slow dynamics.

Ti:Sapphire SLS: The auxiliary output of the Ti:Sapphire SLS is sent to the ultra-stable cavity via a fibered EOM 4 whose output is the coupler Tisaph SLS located on the first optical table shown in Fig. 3.18. In a similar manner as the master laser, a controlled central frequency and a frequency providing the sidebands are mixed and drive the EOM: we can in this manner tune the shift between the laser and the ultra-stable cavity, and lock the laser using a PDH setup anywhere in the transfer cavity FSR. The resulting error signal is sent to a digital PID programmed on a Red Pitaya that then provides feedback to the two piezoelectric inputs.

Ti:Sapphire transfer: The auxiliary output of the Ti:Sapphire transfer is sent to the transfer cavity by the coupler From tisaph transfer in Fig. 3.18. AOM 1 in Fig. 3.18 is set in a double pass fashion on the Ti:Sapphire transfer optical path, allowing for a control over the lock frequency. It is then merged with the 795 laser on the same optical path using a dichroic plate, before being sent to the transfer cavity via an EOM 1 that provides the sidebands used to lock the laser with a PDH scheme. Given the fact that the transfer cavity FSR is 200 MHz, it can be fully reached by changing the frequency of AOM 1, thus allowing us to lock the Ti:Sapphire transfer anywhere in the transfer cavity. In a similar manner as the Ti:Sapphire SLS, the error signal is sent to a Red Pitaya (RP) and feedback takes place on the laser's piezoelectric inputs.

Locking sequence

The cavities described in Section 3.9 are stabilized on the Ti:Sapphire lasers, and the main resonator presented in Section 3.8 is stabilized by the 1064 laser. A schematic of the complete locking chain is shown in Fig. 3.20 and the locking sequence is shown in Fig. 3.21.

Red Pitaya

The Red Pitaya module consists in a Field Programmable Gate Array (FPGA) chip 2 whose hardware and programming software is provided by the manufacturer Xilinx, combined with a Linux single board computer 3 . The computer allows for a flexible implementation of reconfigurable digital circuits that would normally require specific and costly hardware. Choosing to use such modules for our locks not only allow us to work digitally, which is cost-wise more efficient, and it allows greater reproducibility of a system. The two fast analog outputs 4 of the Red Pitaya have a resolution of 14 bits which, given the finesses of our cavities, generally doesn't allow to scan a full FSR and at the same time provide a precise noise-free lock on the top of a fringe. To circumvent this issue, we used both analog outputs, one for low-resolution high-range scanning and drift compensation, the other for high-resolution low-range fast locking. For locking the Ti:Sapphire lasers these outputs were directly connected to the slow and fast piezoelectric inputs; for controlling cavities with a single input they were combined on dedicated analog boards.

Other cavities are used in the experiment. These are located in the main vacuum chamber, and are the science cavity and blue cavities. The science cavity is where the interactions between the atoms and the light are taking place, while the blue cavities are used to provide amplification for the control power required in our two-photon excitation to a Rydberg We have dedicated inputs to the wavemeter, with the lasers grouped by pairs of color. Also shown are the ultra-stable cavity, the transfer cavity, the science cavity and the two blue cavities. The numbers in the cavities represent respectively the FSR (in MHz) and the linewidth (in kHz) of the cavity. level as mentioned in Section 3.6. A Summary of the properties and characteristics of the different cavities used in the setup is presented in Table : 3.4.

Science cavity

We call science cavity the cavity where the coupling between the 87 Rb atoms and the cavity light takes place: it is the central part of the experiment. It is a non-planar cavity composed of two pairs of mirrors, the first pair being concave and the second pair being plane, and schematic of the cavity can be seen in Fig. 3.22. This cavity is the heart of our experimental setup: we aim at a small, well controlled sample of 87 Rb interacting in the cavity mode. In this section we discuss the choice of geometry, as well as the technical details. We also consider the properties of the atomic sample in the cavity, and the difficulties that we encountered.

Cavity geometry

The choice of cavity geometry was motivated by a common ground between a combination of constraints such as the size of the mode waist, stability, tunability and usability. The most immediate limiting factor is the size of the vacuum chamber which limits us to a compact resonator which should still have large enough optical accesses for the different probing, imaging and pumping beams, as well as for the loading of the atomic sample.

Foremost, the size of the waist of the cavity mode (between the concave mirrors) is a critical parameter to consider in order to achieve interaction with the atomic sample. The interactions are mediated through Rydberg blockade, and as was explained in Section 2.5 the blockade radius for Rydberg atoms of principal quantum numbers of the order of 𝑛 ≈ 100 is of the order of 10 µm, so we need a waist of the same order of magnitude. In order to avoid having two Rydberg blockade sphere next to each other, we must restrict the size of the cavity waist to roughly the size of the blockade sphere. Another point is that the smaller the cavity is, the greater the coupling becomes between the cavity and the atoms, but the electrostatic fields becomes problematic as well as the optical access: 𝑔 ∝ 𝑑 𝑔𝑒 • 𝐸 𝑝ℎ𝑜𝑡𝑜𝑛 ∝ 𝑉 -1/2 . It also becomes harder to transport the sample and have all the required optical accesses for optical pumping of the atoms. After simulation and modeling, a non-planar four-mirror resonator was designed, allowing a small waist with reasonable cavity length and mirrors ROC, alongside a reasonable length and optical accesses. The chosen cavity geometry presents a pair of flat mirrors that do not refocus the beam, and a pair of concave mirror that allows for a small waist while still keeping the overall round trip short. Two waists are then present, the smallest between the two concave mirrors and the largest between the two flat mirrors, Fig. 3.22 shows a 3D model of the cavity. This cavity only allows for circular polarization, and induces a light twist that can be seen as a synthetic magnetic field [START_REF] Schine | Synthetic Landau levels for photons[END_REF]. This crucial property of the resonator is central to our experimental setup, and its use will be explained in Section 6.3.

There are however some disadvantages to such a running wave cavity. During a single round trip, a photon will encounter more high reflective surface than in a Fabry-Perot, so the resonator will be more lossy. The running wave cavity will also lead to some astigmatism, since the concave mirrors are not normal to the cavity mode, and the beam exiting the cavity will also present some astigmatism since it has the same mode than the cavity. In the final design the theoretical waist at the atoms position is approximately 21 µm, located between the concave mirror and the bigger waist between the mirrors further away is approximately of 103 µm between the plane mirrors. The theoretical finesse is F = 750, with a FWHM of 4 MHz and a FSR of 3 GHz. The points of interest that we have to address are the different optical accesses that we have accessible with the resonator geometry, the mirrors used, as well as the control of the length and shape of the resonator. We also need to look into the stabilization of the cavity.

Figure 3.22:

A schematic of the science cavity. In blue the mirrors pairs, in green the atomic sample located at the small cavity waist. The incident probe beam is shown in red, while the control beam is blue. The cavity locking beam is injected from one of the upper mirror.

Mirrors characteristics

The cavity mirrors should be large enough to support a few transverse mode, while being small enough to keep the resonator compact. The mirror pair closest to the atoms is chosen to be concave while the pair further from the atoms should be flat or convex to allow for a small waist between the concave mirrors. In practice, the ROC of the concave mirror is of 20 mm for a diameter of 6.3 mm, and the second pair of mirror present a similar diameter while being plane. Recall that the incoupling miror reflectivity is an important parameter for us, as it controls the limit at which we can switch between the overcoupled (losses much smaller than the incoupling mirror transmission) and undercoupled (losses much larger than the incoupling mirror transmission) regime for the science cavity (see Section 2.6). For the probe and pumping light at 780 nm and 795 nm, the input coupler should have a transmission 𝑇 ∼ 1% while being highly reflective (𝑇 ≪ 1) for other wavelengths. All other mirrors of the science cavity should be highly reflective to the probe and pumping light, and the coating run for the science cavity mirrors was custom made by Laseroptik. Concerning the blue laser (480 nm), ideally the injection in the cavity should be done from a different coupler: it should not be resonant in the cavity, and near the atom have a waist larger than the cavity mode so that the atomic sample is withing the control laser cross section. As mentioned in Subsection 3.7.4, we use the 1064 laser to lock the science cavity since it is far off resonant with 87 Rb atoms: for this reason it requires a linewidth of the same order of magnitude as the master laser linewidth. Piezoelectric elements located behind two mirrors allow us to lock the cavity. Furthermore, in order to control the shape and length of the resonator, one of the concave mirror and the flat mirror of the same side (i.e both in 𝑦 > 0 in Fig. 3.22) are set on a SmarAct stick-slip positioning stages1 . The nominal extension of the translation stage is approximately 8 mm, which make it possible to have the two mirrors move by ± 4 mm, leading to an independent control of the FSR and the transverse mode splitting. After receiving the mirrors, the science cavity was set up and tested: for a wavelength of 780 nm, a finesse of F = 760 was measured.

Coupling mirror transmission

We mentionned that the transmission of the incoupling mirror of the science cavity was crucial to our setup. At 780 nm we have at zero degree of incidence a theoretical transmission of 1% and a measured transmission of 0.8 ± 0.05%. At 795 nm we have at zero degree of incidence a theoretical transmission of 1% and a measured transmission of 0.9 ± 0.05%.

Cavity finesse degradation

The assembly of the different cavities in the vacuum chamber was done, and the bakeout process of the ensemble took place. Fig. 3.23 shows a measure of the linewidth of the science cavity several months after the first vacuum was achieved and into the characterization of the setup: a finesse of F = 591 is measured at 795 nm, and for a wavelength of 780 nm we measure a finesse of F = 621. As it turns out, the finesse of the science cavity dropped compared to the initial linewidth measurements that were done when putting the cavity in the vacuum chamber.

From [START_REF] Gangloff | Preventing and reversing vacuum-induced optical losses in high-finesse tantalum (V) oxide mirror coatings[END_REF], it appeared that the coatings of the cavity mirrors could degrade under vacuum, possibly from oxygen depletion from Ta2O5 layers. After inquiries with LASEROPTIK, it was confirmed that the last layer in the coatings was SiO2, with a thickness ranging from 20 nm to 214 nm. By doing testing with a similar set of coating, we confirmed that the oxygen depletion was not the issue. A Fabry-Perot cavity was assembled with similar mirrors: measuring the reflected power off while the test setup was in the air as a benchmark, it was put under a vacuum of 3.4 × 10 -8 mbar, and the cavity reflected power was still the same. The test setup was then heated to 170 ∘ C the pressure stabilizes at 2.3 × 10 -7 mbar after degassing, and neither the transmission nor the reflection changed over several days. Another possibility for this drop in the finesse is the deposition of 87 Rb on the mirrors: desorption with ultra violet light was also done. Ultra violet light was sent on the two lowest cavity mirrors, with the electrodes shut down. After different exposition times ranging from one to ten minutes, all the way to being used over twelve hours, no change in the pressure of the vacuum chamber was witnessed. A last method was tried by heating a mirror holder of the science cavity with 50 mW of power at 780 nm for up to two hours in order to potentially remove 87 Rb deposited on the mirror, yet no improvement in the finesse was seen. A last trial was done by S. Garcia by trying to provide a cool point in the vacuum chamber, where the Rb would likely migrate. The idea was to cool down the MOT coils with a chiller and a heat exchanger, however the heat transfer proved inefficient, thus making this solution lapsed. The science cavity has not degraded further, and we still have a finesse of F = 621 and F = 591 for wavelengths of 780 nm and 795 nm respectively (respectively a linewidth of 2𝜋 × 2.6 MHz and 2𝜋 × 2.89 MHz).

Rydberg atoms in electric fields

Furthermore, as mentioned in Section 2.5, Rydberg atoms are highly polarizable, and thus are extremely sensitive to stray electric fields. The Stark effect can induce large shifts in their energy levels, which leads to a shortened coherence time of Rydberg polaritons, so stray and inhomogeneous electric fields are to be avoided as possible. Thus, an important part of the design was to get the atoms as far away as possible from the mirrors, and more generally any surface where electric charges can occur. Passive shielding of electrostatic fields is done by placing the atoms 'far' from the mirrors (in practice, approximately 11 mm), with the metallic parts of the cavity linked to the ground. This is essentially important with respect to the piezoelectric elements that are located behind the mirrors, since they typically require high voltages. From an active point of view, we require a control of the electric field at the atomic cloud position: 87 Rb might deposit on any surfaces thus leading to stray electric fields that we should compensate. Eight electrodes were placed around the cavity waist in the vacuum chamber. Two electrodes per MOT coil are fixed to on the upper part towards the waist of the cavity, and four electrodes are located above the cavity waist, hanging from the top part of the cavity.

Atoms in cavity

In order to get the atoms in the cavity, we could simply create the 3D MOT directly in the lower cavity mode. This would be however impractical from an optical access point of view, given the size of the MOT beam and the size of the science cavity. The possibility of depositing atoms on the cavity mirror could also be an issue. In addition, fitting the MOT coils would have been difficult as the space around the science cavity is limited. Another way would be to have a 2D MOT directly guide atoms into a small MOT in the cavity. In [START_REF] Bruneau | Guided and focused slow atomic beam from a two-dimensional magneto optical trap[END_REF], by using a intense far detuned laser in combination with a 2D MOT they manage to focus an atomic beam to a size of 0.4 mm as far as 60 cm. Our vacuum chamber is smaller than this dimension, and we could create a small MOT in the cavity mode using a similar technique. It would loosen a bit the optical accesses required compared to making a full size MOT in the cavity, but would still require at least eight optical accesses, and does not solve the issue of 87 Rb deposition on the mirrors. The implemented solution is to use a 2D MOT that feeds into a 3D MOT located below the cavity, before transporting the atoms to the science cavity. In this manner, the constraints on the optical accesses on the 2D MOT and 3D MOT are extremely relaxed as we only require beams along the vertical direction, and we only transport a small sample of atom on which to perform the experiment, limiting potential degradation of the mirrors. It also decorrelates the atomic cloud production from its transport. One important issue however is the end temperature of the atomic cloud once it reached its final destination: by transporting the cloud we are heating it, and some form of recooling at the end of the transport phase is necessary. This is illustrated in Fig. 3.22, where we can see the science cavity position relative to the 3D MOT.

Atomic cloud requirements

The size of the atomic cloud in the cavity should be comparable to the Rydberg blockade range, so typically on the order of 10 µm for the states we are using. Its optical density should be as high as possible, but unwanted collisions between micron-sized Rydberg atoms and ground state atoms limit practically usable atomic densities to 1 × 10 12 cm -3 . Atoms should be cold enough to avoid the decoherence of Rydberg polaritons during the typical 0.1 µs to 1 µs timescales of our experiments, 10 µK being an acceptable value. In order to have the possibility to do multi-mode experiments using wider atomic clouds while maintaining sufficiently large Rabi frequencies for Rydberg excitation beams, these beams are amplified by two in-vacuum buildup cavities, called left and right blue cavities in the following. Fig. 3.25 shows their position relative to the science cavity: the left blue cavity has its input mirror in 𝑦 > 0 while the right blue cavity has its input coupler in 𝑦 < 0. These cavities are crossed and set roughly orthogonally to the plane the science cavity. These cavity are confocal Fabry-Perot cavities. The mirrors were coated by Layerteck and are highly reflective (> 99.7%) in the 360 nm to 490 nm range with a ROC of 150 mm. The idea is to use the axial mode (TEM 00 ) of the cavity to stabilize it, and to use the bowtie mode of the cavity to provide power for the excitation (running wave). The stabilization is done through a PDH scheme on the blue output of the corresponding Ti:Sapphire lasers (see Table : 3.4), and the retroaction takes place on a piezoelectric behind the front mirror of the cavity. As shown in Fig. 3.25, the alignment of the blue beam in the blue cavities is controlled through a translation stage. The running wave configuration is used to excite the atoms to Rydberg states. The finesse of the axial mode of the right (respectively left) cavity before being placed into vacuum was F = 477 (respectively F = 465), and after Considering the use of the blue cavities, we are interested in the enhancement of the power that they will provide. The enhancement of the incoming power can be written as 𝐺 = 𝑃 𝑐𝑖𝑟𝑐 /𝑃 𝑖𝑛 , with 𝑃 𝑖𝑛 the incident power on the cavity and 𝑃 𝑐𝑖𝑟𝑐 the power circulating inside the cavity. G can also be expressed via the finesse of the mode with 𝐺 ≈ F /𝜋: we have then 𝐺 𝑙𝑒𝑓 𝑡 ≈ 55 and 𝐺 𝑟𝑖𝑔ℎ𝑡 ≈ 60. Table : 3.4 presents the properties of the cavities used in the setup either as part of the locking scheme or as a main part of the experimental setup.

Blue cavities

Imaging Setup

In this part, we discuss the manner in which the images required for the characterization of the system are acquired and analyzed to extract physical parameters such as temperature measurements, the number of atoms in the cloud, the cloud size, etc. As shown in Fig. 3.13, there are two cameras that are positioned along the top beams of the crossed dipole trap. These are used to image the sample after the end of the transport, once it reaches the science cavity. These imaging paths make use of the two aspheric lenses that are present in the chamber, leading to more collected light and higher resolution imaging of the cloud in the science cavity. Dichroic mirrors are used to separate the trapping light from imaging light. Two other cameras are located on the side of the vacuum chamber, shown in Fig. 3.14: Camera 1 is used to image the MOT while camera 2 allows us to see the upper part of the 3D MOT, the atoms in the moving lattice and the atoms in the science cavity lower mode. We then have at our disposal imaging paths that allow us to perform absorption and fluorescence imaging in the 3D MOT, absorption imaging of the sample anywhere in between the MOT and its final position in the science cavity, and high resolution imaging along the two diagonal arms of the crossed dipole trap that focuses on the science cavity area.

Temperature: In order to characterize the temperature of the 3D MOT or our cold atom sample, we rely on Time of flight (TOF) data. By monitoring the free expansion of the atomic cloud when released from its trap, we can extract the initial temperature of the cloud. In practice, this is done by releasing the cloud of atom from the MOT and letting it expand for several different times, typically on the order of a few milliseconds. In a cloud initially in thermal equilibrium, the atoms are not subject to any force except the gravity, and they keep their final velocity. We then monitor the width of the Gaussian fit as a function of the TOF: the radius of the cloud after a time t can be expressed as 𝜎(𝑡):

𝜎(𝑡) = √ 𝜎(0) 2 + 𝑘 𝑏 𝑇 𝑚 𝑅𝑏 𝑡 2 (3.3)
so the temperature of the cloud is

𝑇 = 𝑚 𝑅𝑏 (𝜎(𝑡) 2 -𝜎(0) 2 ) 𝑡 2 𝑘 𝑏 . (3.4)
We can expect the temperature along the two axis to be different since the MOT is not perfect. The cloud expansion leads to a decrease of the optical depth of the sample, thus giving us a upper limit on the duration of the time we can wait before imaging the sample.

In addition to the characterization, we should monitor the system for measurement signals. The use of the cavity transmitted and reflected light allows for the monitoring of the coupling between the atoms and the cavity. The measurements and characterization done in this work used an Avalanche Photodiode (APD), but this is only a temporary setup.

Single photon counters are available and were being installed during the writing of this work, and a homodyne detection setup is also assembled. It is a crucial part of the setup, necessary to measure non linearity signals (photon bunching and antibunching), Rabi oscillations and changes in the science cavity transmission depending upon the presence of a Rydberg polariton.

CHAPTER 4

Characterization of the subsystems

Introduction

We now characterize the different parts of the setup that were presented in the previous chapter. We first discuss the optimization of the 3D MOT and the tuning of the compensation coils through the use of microwave spectroscopy. Then the transport lattice and the vertical dipole trap in which the atoms are transferred are characterized. The final section covers the atom-cavity coupling between the transported atomic sample and the science cavity mode. We start by reviewing the relevant characteristics of the 2D MOT and 3D MOT setup. Measurements relative to the number of atoms as well as the temperature and size of the MOT are done in order to optimize both traps with respect to loading times and densities.

MOT

2D MOT optimization

The 2D MOT optimization was done by measuring the number of atoms at the 3D MOT position while scanning the amplitude and frequency of the 2D MOT trapping beams in order to get the largest MOT as possible, the magnetic field was also scanned with the atom number being monitored. Fig. 4.1 shows the evolution of the number of atoms as well as the dimension of the sample as a function of the 2D MOT beam intensity and detuning, with a fixed repumper power of 3 mW in both the transverse directions. The frequency scan is done with respect to the cooling frequency which corresponds to the theoretical optimum of -3𝛤 = -18 MHz from the |5𝑆 1/2 , 𝐹 = 2⟩ → |5𝑃 3/2 , 𝐹 ′ = 3⟩ transition.

The maximum atom number that appear provides optimal values for these parameters, that we use from now on.

MOT sequence

Fig. 4.2(a) present a typical experimental sequence with the order of magnitude of the steps durations. We look into the preparation of the atomic cloud that is illustrated in Fig. 4.2(b). Three successive steps are done during the preparation that are respectively a loading step, a compression of the atomic sample and a molasses phase. The loading step consists in switching on both the 2D MOT and 3D MOT together, having the former feeding atoms to the latter. The two remaining steps are then used to cool down the cloud of atoms below the Doppler limit [START_REF] Phillips | Optical molasses[END_REF]. During the compression step, the MOT is compressed during a couple of milliseconds by ramping the current in the MOT coils and the 2D MOT is turned off. Then the optical molasses step follows: the MOT coils are turned off, and the trapping beams intensity is decreased, while its detuning is increased, in order to lower the temperature of the cloud. These compression and optical molasses step duration need to be optimized in order not to lose too many atoms, and to reach low temperature in the atomic sample. We first characterize the MOT before describing the compression and the optical molasses.

MOT characterization

We first measure the atom number as a function of the loading time using the 2D MOT. the pressure inside the vacuum chamber is 𝑃 𝑐ℎ𝑎𝑚𝑏𝑒𝑟 = 2.5 × 10 -9 mbar, and we have a loading of approximately 1 × 10 5 atoms per second. In comparison, Fig. 4.3(b) presents the loading of the MOT using the 2D MOT: in this regime we have a loading of 1.3 × 10 9 atoms per second, and we can see that MOT loading without the use of the 2D MOT is clearly less efficient. After further optimization of the MOT, the loading is now stopped after 100 ms as the cloud becomes too large, thus making the optical molasses less effective: if 80 ms are used for the loading process duration, we reach approximately 5 × 10 10 atoms in the MOT, which is too high a number of atoms to reach low temperatures. In practice, we usually set up the sequence such that 1 × 10 8 atoms are in the MOT at the end of the loading phase. We have four parameters on which to act in order to optimize the MOT: we can control the MOT beams intensity and detuning, as well as the current in the MOT coils and the duration of the different steps. Fig. 4.5(a) shows the evolution of the number of atoms with respect to the magnetic field and Fig. 4.5(b) the variation of the number of atoms as a function of the MOT beam detuning. These parameters are controlled by monitoring the atom number while respectively varying the current in the MOT coils and the detuning of the MOT laser. We can see an optimum in the number of atoms, from which we extract the optimal parameter values. In order to have the most optimized 3D MOT, we must ensure similar power in counter propagating beams and that the zero of the magnetic field is centered with respect to the optical molasses. So we looked at the MOT with low intensity in the beams while the current in the MOT coils was increased, allowing us to tune the compensation coils such that the MOT did not move with the varying magnetic field. In the end, this calibration of the compensation coils was not completely accurate, and in order to get a more precise calibration, spectroscopy of the MOT was done.

Microwave Spectroscopy

In order to calibrate the compensation coils of the 3D MOT, spectroscopy of the hyperfine transition of 87 Rb is done. The end goal is getting a zero of the magnetic field in the center of the MOT. In order to do so, a Micro-Wave (MW) generator was setup along one of the MOT beams, as shown in Fig. 3.10(b). We start by explaining briefly the working principle before describing the measurement itself.

Microwave excitation principle

The Zeeman sub-levels degeneracies can be lifted linearly via the Zeeman effect by applying a small magnetic field. For a given hyperfine level F, the energy difference is 𝛥𝑈 = 𝜇 𝐵 𝑔 𝐹 𝑚 𝐹 𝐵 𝑧 , where 𝜇 𝐵 is the Bohr magneton and 𝑔 𝐹 the Landé g-factor of the concerned transition. The shift 𝛿𝑓 𝑚,𝑚 ′ of the different transitions |𝐹 = 2, 𝑚 𝐹 = 𝑚⟩ → |𝐹 = 1, 𝑚 𝐹 = 𝑚 ′ ⟩ with respect to the clock transition F=2, 𝑚 𝐹 = 0 and F=1, 𝑚 𝐹 = 0 is given by

𝛿𝑓 𝑚,𝑚 ′ = 𝜇 𝐵 ℎ (𝑔 2 𝑚 -𝑔 1 𝑚 ′ )|B|. (4.1)
For 87 Rb, 𝑔 1 ≈ -𝑔 2 ≈ -1/2, so we can rewrite the previous expression as

𝛿𝑓 𝑚,𝑚 ′ = 𝜇 𝐵 2ℎ (𝑚 + 𝑚 ′ )|B|. (4.2)
For the transition

|5 2 𝑆 1/2 , 𝐹 = 2, 𝑚 𝐹 = 0⟩ → |5 2 𝑆 1/2 , 𝐹 = 1, 𝑚 𝐹 = 1⟩
, we get a frequency shift of approximately 0.7 MHz G -1 . If the compensation coils magnetic field does not compensate exactly at the center of the MOT, the residual fields then lift the degeneracy of the Zeeman sub-levels in a sub-level dependent manner. One can use this fact in order to characterize the field, thus allowing for a compensation using the compensation coils pairs. We can pump the atoms in the |52 𝑆 1/2 , 𝐹 = 2⟩ hyperfine level, from where the RF kick transfers the atoms to the |5 2 𝑆 1/2 , 𝐹 = 1⟩ hyperfine level. If before the MW pulse a magnetic field is applied by one of the pair of coils, then the Zeeman sub-levels degeneracies are lifted and the 𝜋-pulse only transfers atoms that are in the resonant sub-level. Remaining atoms in |5 2 𝑆 1/2 , 𝐹 = 2⟩ can be removed by exciting the transition |5 2 𝑆 1/2 ,𝐹 = 2⟩ → 5 2 𝑃 3/2 , 𝐹 = 3. By using the repumping beams, we can then image the transferred atoms. This allows us to observe the different Zeeman sub-levels transitions that are induced by the MW pulse: except from the atoms in the resonant Zeeman sub-level, no other atoms are transferred by the MW pulse in |5 2 𝑆 1/2 , 𝐹 = 1⟩ and can be detected while imaging the cloud. So by scanning the frequency of the RF pulse, we should see the full spectrum of the 5 2 𝑆 1/2 hyperfine structure. We can precisely measure the frequency shift of one of the transition, e.g. |5 2 𝑆 1/2 , 𝐹 = 2, 𝑚 𝐹 = 0⟩ → |5 2 𝑆 1/2 , 𝐹 = 1, 𝑚 𝐹 = 1⟩. We then maximize the number of atoms transferred by the 𝜋-pulse on the chosen transition by changing the pulse duration. Following that, by scanning the full spectrum of the 5 2 𝑆 1/2 hyperfine structure and changing the magnetic field, we can get the current value for which the splitting becomes minimal. This process is then repeated along the other compensation coils direction, and ultimately provides accurate values for the current of the different pairs of compensation coils.

Microwave setup

We use a MW generator 1 combined with the output of a computer controlled RF generator 2 on a mixer3 whose output is then amplified4 . All of the elements are linked with SMA cable whose losses are approximately of -3 dB at a frequency of 6.834 GHz for 2 m cables. The output RF is set to 6.834 GHz + 𝛿, with a varying 𝛿 in MHz provided by the Arduino that we can control from the lab's computer. The position of the MW generator relative to the other elements of the setup is shown in Fig. 3.10(b). It is placed along one of the lower MOT beam labeled MOT beam 2, and pointed roughly towards the atoms. -1.09 G to -0.54 G. By fitting these shifts to the magnetic field as in Fig. 4.8(b), we can deduce the magnetic field value at which the shift is minimal, which in turns allows us to get the correct calibration for the corresponding pair of coils. Iterating the process in all remaining directions with the two remaining pairs of coils leads to a situation where for each direction we have a minimum linewidth for the considered transition. After optimization, the scan of the 5 2 𝑆 1/2 hyperfine structure shows a nearly completely degenerate peak for a frequency of approximately 6.834 GHz + 700 kHz, giving a residual magnetic field on the order of 10 mG.

Compression step

The following measurements and characterizations of the compressed MOT and the optical molasses were done with an optimized MOT, using the newly obtained values for the compensation coils current. The characteristics of the compressed MOT are measured through properties of the cloud after the MOT loading step. The MOT repumping light is kept during the compression phase. The 2D MOT is switched off, and the current in the MOT coil is increased over a few milliseconds: it is ramped from 7 A to 20 A in the MOT coils, which correspond to a field gradient ranging from 14 G cm -1 to 40 G cm -1 . Fig. 4.9(a) shows the atom number in the MOT with respect to the duration of the compression: a compression of approximately 2 ms seems optimal in our case with respect to the number of atoms.

We confirm that the cloud gets compressed with Fig. 4.9(b), which shows the evolution of the cloud size with respect to the magnetic field created by the MOT coils at the end of the compression step.

Optical molasses

Recall that we aim at an atomic sample of a cold enough temperature, typically 10 µK. By creating an optical molasses [START_REF] Dalibard | Laser cooling below the Doppler limit by polarization gradients: simple theoretical models[END_REF] the temperature can be cooled further down. This is done by switching off the magnetic field gradient from the MOT coils, and by decreasing the intensity of the MOT beam while detuning them. We first balance the power in the beams: by using a small fixed magnetic field and increasing the power in the beams, the center of the optical molasses should not shift if the beams are properly balanced. There are two important parameters in this process: the temperature decreases with the laser detuning for a fixed laser power, and for a fixed detuning the temperature increases the laser power. So a reduction of the intensity of the MOT beams is important in order to reach low temperatures: experimentally, we reach an almost null intensity at the end of the optical molasses phase. In order to avoid loosing too many atoms, the optical molasses phase is optimized as follows. The detuning passes in 1 ms from -18 MHz to -70 MHz, corresponding to approximately 12𝛤 , followed by a 1 ms step where the final value of the detuning is maintained on the MOT beams. At the same time the intensity of the MOT trapping beams is linearly decreased over a 2 ms step by approximately a factor 4 from its initial value to 1 mW cm -2 , and the magnetic field gradient is linearly decreased from 40 G cm -1 to 0 G cm -1 . The repumping laser intensity is also turned off during the optical molasses, except at the very end of the molasses step, where the atoms are briefly repumped before the transport.

The importance of the detuning and beams intensity on the end temperature can be seen in Fig. 4.10: both parameters have an important impact on the properties of the end cloud. Given the fact that the size of the MOT remains constant, the atom number is proportional to the density. Fig. 4.11(a) shows thus the evolution of the density in the optical molasses with respect to the detuning at the end of the molasses step, and Fig. 4.11(b) shows the evolution of the density with respect to the end beam intensity. As mentioned, in Fig. 4.11 the dimension X and Y of the cloud remain mostly constant.

Another thing to consider is the duration of the optical molasses. By increasing it, no gain in density and temperature was witnessed. What remains now to be done is to perform an imaging step in order to measure the sample temperature. Fig. 4.12 shows the temperature reached in the MOT at the end of the compression and optical molasses phases with optimized parameters. After a 100 ms loading of the MOT, we have 50 × 10 6 atoms, 𝜎 𝑥 = 500 µm and 𝜎 𝑦 = 750 µm. If we consider the third dimension as the average of the previous two, then 𝜎 𝑎𝑣 = 650 µm. This leads to a volume 𝑉 = 1.02 × 10 -3 cm 3 and a density after optical molasses of n av = 4.8 × 10 10 atoms per cm 3 , with a temperature T sample,x = 9.6 µK and T sample,y 8.6 µK.

In conclusion: The overall process takes about 50 ms to 100 ms for the loading of the MOT, 1 ms to 2 ms for the compression step, and about 2 ms for the optical molasses. During the MOT and compression steps, the repumper laser intensity stays the same, and during the optical molasses it is decreased by a factor 15. The sample is repumped for a short pulse before the transport takes place. Table : 4.2.3 presents a summary of the typical range for different parameters of the MOT and the 2D MOT.

Transport in moving array

Once the MOT has been prepared, and we performed the optical molasses phase to get the sample to a lower temperature, we need to transport the atoms to the cavity. This is done through the use of a one-dimensional moving lattice, aligned in the vertical direction on the MOT and the science cavity. While Section: 3.4 presents the layout of the moving lattice and subsection 2.2.2 gives a brief overview of the theory behind this transport technique, we characterize here the transport in terms of atom number, temperature and lifetime. The important parameter is the end temperature of the sample, since we aim for a small, cold cloud of atoms to be brought to the science cavity. The transport takes place over 33 mm between the MOT position and the science cavity, along the vertical direction. The waist of the moving lattice is located half-way between the starting and ending position of the transport, with a waist ∼ 60 µm. We have a coupling efficiency of 85% between the two lattice beams couplers, with the typical power of a lattice arm entering the chamber from the bottom and from the top being 250 mW at 783 nm.

Lattice timing and characterization

The temperature is initially measured after the optical molasses step. We are heating the atoms during the loading process, but there was no significant impact on the temperature of the ample from the overlap between the moment the lattice beams were turned on and the end of the molasses step. We currently have a configuration where the lattice is turned on during the last millisecond of the optical molasses step. Fig. 4.13 shows the temperature of the cloud at the initial, starting position of the transport with and without the lattice beams: we can see that during the loading process the atoms are heated by about 10 µK to 20 µK. The atom number seems to vary between the two situations: when no lattice beams are present, it decreases over the 19 ms of TOF, but when the lattice beams are present the number of atoms increases over the 14 ms of TOF. For short TOF the Optical Depth (OD) might be too large, which leads to saturation of the detection: once the atoms spread the OD decreases and the overall atom number is better evaluated. The decrease seen when the transport lattice is not present could come from the fact that some atoms fly away, compared to the situation where the lattice is here and can trap a portion of them. Once the atoms are loaded in the transport lattice, we can start to move them: important parameters to look at is the effect on the number of atoms of the transport time and transport distance. Fig. 4.14(b) shows absorption imaging of the sample transported away from the MOT, with the transport lattice along the y-dimension. The effect of the loading duration in the lattice can be seen in Fig. 4.14(a), where we decrease the temperature along the lattice long axis for longer loading time. Fig. 4. 15(a) shows the evolution of the atom number with respect to different transport duration between the MOT and the cavity and Fig. 4. 15(b) shows the evolution of the temperature for different transport time between the MOT and the science cavity. We can see that the transport duration has an important impact on the final atom number brought at the cavity position. Concerning the temperature, it has a relatively weak effect on the temperature along the x-direction, but it can decrease the temperature along the y-direction, which correlate with what we saw in Fig. 4.14(a). The acceleration being proportional to 1/𝑡 2 𝑡𝑟 with 𝑡 𝑡𝑟 the transport time, by decreasing the transport time we increase the temperature 𝑇 𝑦 along the transport direction. Typically, a transport duration of 10 ms or higher allows for an relatively fast and efficient transport that ensures a high number of atoms. In our experimental sequence, we transport the atoms in 16 ms over a distance of 33 mm: the first step of the transport is a linear acceleration of 8 ms, followed We are interested in the temperature the sample reaches at the end of the transport at the cavity position. For a transport duration of 10 ms, the temperature along the longitudinal direction at the end of the transport when we reach the cavity is approximately three time as high as the initial temperature after loading the sample in the lattice at the MOT position: we reach after a time of flight of 100 µs 𝑇 𝑥 = 160 µK and 𝑇 𝑦 = 60 µK. It turns out that the waist of the moving lattice was not located halfway between the starting and ending point. In order to correct the position of the waist, we measure the trap frequency at the initial and final position of the transport and compare them, in order to get a more accurate idea of the waist location. Following [START_REF] Gehm | Dynamics of noise-induced heating in atom traps[END_REF], we determined the oscillation frequency of the atoms in the lattice by modulating its intensity and looking at atom losses, which peaks when the modulation frequency is twice the oscillation frequency due to parametric heating. We have initially at the level of the MOT 2𝑓 ≈ 245 kHz and for the cavity 2𝑓 ≈ 650 kHz for oscillations along the lattice axis. Fig. 4. [START_REF] Aljunid | Interaction of light with a single atom in the strong focusing regime[END_REF] shows the evolution of the number of atoms trapped in the transport lattice at the MOT position and at the science cavity position before optimization of the lattice waist position. We shifted the waist position using the appropriate optics in order to get the same trapping frequency on both the initial and final position of the transport, ensuring in this manner that the waist of the lattice is located halfway between the MOT and the cavity mode: Fig. 4.17 shows the same measurements done after the waist optimization. This allows us to gain nearly a factor three on the temperature along both axis, reducing them to ∼ 50 µK from 160 µK along the x-axis, and to ∼ 20 µK from 60 µK along the y-axis. Fig. 4.18 shows the lifetime of the sample after optimization of the waist position once it reached the lower cavity mode. The apparent zero values for the cloud radii are errors in the size measurement, so these two data points where both dimensions go to zero should not be taken into account. Fig. 4.19 shows absorption imaging of the MOT and the moving lattice for loading time of the lattice ranging from 50 ms to 500 ms. The number of atoms transported to the lower cavity mode ranges from 0.93 to 1.9 million atoms.

Vertical Dipole Trap

Once the sample has been transported to the cavity, it is transferred to a vertical standingwave dipole trap whose waist is located on the lower cavity mode. This vertical dipole trap has been since then replaced by a crossed dipole trap presented briefly below in section 4.5, but in this section, we characterize the temperature of the atoms as well the lifetime of the sample in the vertical dipole trap. The layout of the optics for this dipole trap is described in subsection 3.5. The waist of the trap at the cavity position is ∼ 40 µm and we have a coupling of 70% between the two arms of the vertical dipole trap, with a typical power of 500 mW per arms before couplers. Its position is moved until it matches the position of the one-dimensional moving array, and its alignment is fine-tuned by monitoring the science cavity transmission and maximizing the Rabi splitting with the atoms trapped in the vertical dipole trap.

Transfer from moving lattice

Ideally, the sample should not be heated during the transfer. In order to do a passage that is as adiabatic as possible, we ramp down the beams of the moving lattice while ramping up the beams of the vertical dipole trap. At the end position of the moving lattice and once the atoms have been transferred to the dipole trap, a temperature of approximately 25 µK is reached along the x-direction and ∼ 12 µK along the y-direction.

Lifetime in vertical dipole trap

We also look at the lifetime of the sample once it is transferred. This is an important parameter to know, as we will then cool the atoms before performing the measurement. Fig. 4.20 shows the evolution of the atom number for up to 4 s: two dynamics are visible at short and long time. Thermal losses have a characteristic time of 32 ms, while losses induced by background pressure have a characteristic time of 1.5 s.

Raman sideband cooling and crossed dipole trap

In order to cool down further the sample after the transport to the science cavity, Raman cooling was implemented. This work was done by J. Vaneecloo and thus will be described in his thesis, we simply present the main results. Once the atoms are brought to the cavity mode, Raman sideband cooling of the sample is performed. The sample is then transferred to a crossed dipole trap, where it gets compressed in order to increase its density. After that compression step, a second Raman cooling step is applied: Fig. 4.21(a) presents the corresponding temperature measurement, and Fig. 4. 21(b) shows the size of the cloud after 50 ms of trapping. We have approximately 800 atoms in the trap at a temperature of ∼ 5 µK with a typical radius of 7 µm and a peak density of 5.4 × 10 11 atoms per cm 3 . This is within the appropriate range of parameters for our experiment.

Atoms-cavity interactions

In this last section, we focus on the atoms interacting with the science cavity. We first talk about manner the alignment of the cavity and the sample, before looking at the coupling between the atoms and the cavity.

Alignment

In order to optimize the cloud position we investigate the coupling between the atoms and the science cavity mode. This is done either by monitoring the transmission of a probe through the cavity, or by imaging the cloud of atom which allows us to see a 'hole' created by the losses from the probe beam. In order to align the dipole trap beams, we send resonant light through the dipole traps couplers, and we look at the losses creates in the cloud of atom. The blue cavities were aligned in a similar manner: the science cavity mode was pinpointed on the cloud, and resonant light was sent in their respective TEM 00 . The injection was aligned and the input beam was then slowly translated to the bow-tie mode such that the blue power beam would cross the science cavity mode. Fig.

Cooperativity measurement

The atoms are transported to the science cavity, and we measure the cooperativity after loading the atoms in the vertical dipole trap. During the measurement, the trap is turned off in order to avoid lightshift. We get 𝜅 from the fit of the cavity without atoms since it corresponds to the losses from the cavity. From there, we want to estimate the cooperativity C of the system:

𝐶 = 4𝑔 2 𝜅𝛤 (4.3)
with 𝛤 the atomic linewidth. When 𝐶 ≫ 1, the coupling g can be directly extracted from the Rabi splitting, which is a sign that the atoms are interacting with the cavity [START_REF] Sanchez-Mondragon | Erratum: Theory of spontaneous emission line shape in an ideal cavity (Physical Review Letters)[END_REF]. Fig. 4. 23(a) shows the measured Rabi splitting for a probe at 780 nm. We have 𝜅 𝑓 𝑖𝑡 = 2𝜋 × 2.9 MHz and 𝑔 𝑓 𝑖𝑡 = 2𝜋 × 15.5 MHz, which gives a cooperativity 𝐶 ≈ 55. Depending on the direction of the probe scan, we measured a variation in the coupling of approximately 4%. A characterization of the cooperativity was also done at 795 nm. Fig. 4. 23(b) shows the cooperativity measurement done at this frequency in the crossed dipole trap, after the Raman cooling and compression steps: we have 𝜅 𝑓 𝑖𝑡,795 = 2𝜋 × 2.9 MHz and 𝑔 𝑓 𝑖𝑡,795 = 2𝜋 × 13.5 MHz, which leads to a cooperativity 𝐶 ≈ 46 with a sample of density 1.8 × 10 11 atoms per cm 3 . Recall that the regime in which we work is the strong coupling regime (𝑔 ≫ 𝜅 ≫ 𝛾), and that we require system that allows for a tuning of different parameters in order to control the transmitted light behavior. This value of the cooperativity is small compared to traditional CQED experiments and is the result of a compromise between the two important aforementioned concepts that are the enhancement of the non-linearity from Rydberg polaritons by using a low finesse resonator, while avoiding the complexity and non-tunability of higher finesse resonators. CHAPTER 5

Results

Introduction

In this chapter we present the initial EIT measurements made with our setup. We start by presenting spectroscopy measurements that were made in order to tune the laser. Follows then a brief presentation of the initial EIT measurements and the technical issues we encountered. The following section presents the results obtained while performing measurement of EIT in the MOT. Most measurement presented in this work were done before the Raman cooling was added to the experiment, as well as the crossed dipole trap, however during the writing of this thesis improvements were made concerning the control of the system, and excitations to higher Rydberg levels were achieved, discussed in the following chapter.

Rydberg spectroscopy

The first step is to perform free-space spectroscopy with the Ti:Sapphire laser in order to tune it. A coupler is set up for the control laser on the side of the vacuum chamber, and 780 nm light was used to confirm the overall alignment with the atomic sample. Then the signal from the MOT is recorded during a two-photons excitation of a Rydberg state. We work on the D 2 line towards the Rydberg level 𝑛 = 46, and atoms are pumped in |5𝑆 1/2 , 𝐹 = 2⟩. The two-photon transition is then done by a fixed 780 nm probe detuned by -18 MHz from the |5𝑆 3/2 , 𝐹 ′ = 3⟩ level, and the control laser is on the |5𝑆 3/2 , 𝐹 ′ = 3⟩ → |𝑛 = 46⟩ transition that is scanned around the two-photon resonance on the coherent Rydberg transition. When the control laser frequency coincides with Rydberg transitions, it leads to a dip in the measured signal.

46S spectroscopy

The expected probe frequency for the |5𝑆 1/2 , 𝐹 = 2⟩ → |5𝑃 3/2 , 𝐹 ′ = 3⟩ transition is taken from [START_REF] Steck | Rubidium 87 D Line Data[END_REF] and the expected frequency for the transition |5𝑃, 𝐽 = 3/2⟩ → |46𝑆, 𝑚 𝑗 = 1/2⟩ Rydberg level is calculated using [START_REF] Šibalić | ARC: An open-source library for calculating properties of alkali Rydberg atoms[END_REF]. Fig. 5.1 shows the atom number with respect to the detuning of the control laser from the Rydberg level: from this, we are able to extract the correct value for the control laser frequency.

Using the science cavity

In this section, we discuss the different EIT measurements that were done in the science cavity. We assume that all the lasers and cavities are properly locked, as mentioned in Subsection 3.7.4. The atomic sample is prepared and transported to the lower mode of the science cavity as described in Section 4.6, where it is transferred to the vertical dipole trap. The control laser frequency is known from the spectroscopy measurements. We then looked at the Rydberg excitation at the position of the lower cavity waist while using either the blue cavities, or a free space control beam (Subsection 5.2.1).

With a free space control

We consider a different configuration for the EIT measurement: the control laser is injected in the science cavity counter-propagating to the probe as shown in Fig. 5.2(a). This configuration present the advantage of letting us access certain critical parameters more easily, as well as removing any potential issue related to the use of the blue cavities: on the downside, less control power is available. and a 𝜎 --polarized coupling beam. We monitor in transmission the science cavity which is set to resonance on the probe laser, where light is collected in a fiber and sent to the APD.

Using only the probe, the expected signal is a typical normal mode splitting signal. If however the coupling laser is present, then the EIT signal appears under the presence of a central peak, which will shift the normal mode splitting peaks.

Measurements

We now investigate the effect of different parameters on the signal. In the following measurements presented in this section the probe is set such that the power inside the science cavity is P probe in = 75 nW that is scanned for a duration of 500 µs. We have a magnetic field of 1.5 G, along the y-axis and a coupling power of 380 mW in free-space, measured before the cavity mirror.

Effect of the vertical dipole trap

We first want to investigate the effect of having the sample trapped in the vertical dipole trap or free from it during the measurement phase. Fig. 5.3 shows the two cases where the EIT measurement is done at the science cavity position, and the trap is present or not (Fig. 5.3(a) and Fig. 5.3(b) respectively). We can see that the signal linked to the control laser is shifted by approximately 4.5 MHz between the situation with and without the vertical dipole trap, and the peak amplitude increases when free from the trap. When the sample is trapped, we measure a cavity linewidth 𝜅 𝑓 𝑖𝑡,𝑡𝑟𝑎𝑝𝑝𝑒𝑑 = 2𝜋 × 2.70 MHz and coupling 𝑔 𝑓 𝑖𝑡,𝑡𝑟𝑎𝑝𝑝𝑒𝑑 = 2𝜋 × 12.19 MHz between the cavity and the atoms. For the case where the sample is free, the fit gives 𝜅 𝑓 𝑖𝑡,𝑓 𝑟𝑒𝑒 = 2𝜋 × 2.68 MHz and 𝑔 𝑓 𝑖𝑡,𝑓 𝑟𝑒𝑒 = 2𝜋 × 9.8 MHz. We can see that the signal from the EIT appears here as a damped central peak. In addition, this signal linked to the control laser is broader than what we would expect, and the Rabi peaks are not shifted enough by the coupling light. The dipole trap induces a light shift and from now on, measurements are made with the atoms not trapped. We turn ourselves towards the magnetic field applied during the measurement.

Effect of the magnetic field amplitude

We then measure the EIT with a larger magnetic field in order to split the Rydberg levels even more, and we also work with an increased coupling power. ) for the cavity linewidth and coupling with the sample in the case where 𝐵 = 0 G (respectively 𝐵 = 6.5 G). We can see that here, the difference between the case with and without a magnetic field is quite noticeable: the damped EIT peak is much more pronounced when an important magnetic field is applied.

Effect of the scanning duration

We now scan the resonance faster than in the previous measurements, with other conditions being similar to see whether the scanning speed could have an effect. Fig. 5.5 shows a measurement done in a similar setting as Fig. 5.4, with a scan duration of 100 µs instead of the previous 500 µs. We get from the fit a cavity linewidth 𝜅 𝑓 𝑖𝑡 = 2𝜋 × 2.56 MHz and atom cavity coupling 𝑔 𝑓 𝑖𝑡 = 2𝜋 × 6.01 MHz. It seems that the amplitude of the EIT signal is a bit more important, but apart from that has no noticeable effects.

Electrodes optimization

Stray electric and electric fields from the piezoelectric elements could enlarge the EIT peak, so we investigate the compensation provided by the electrodes. In order to calibrate the electrodes, we monitor the EIT response in a fixed magnetic field to different voltage values applied to the pair of electrodes along each axis. From this is extracted the optimal voltage value and by extension the optimal field such that the EIT shift becomes minimal. Fig. 5.6 shows the different EIT measurement done for voltage values ranging from -0.4 V to 0.4 V on the electrodes along the z-axis, as well as a fit of the value of the EIT detuning as a function of the applied voltage. Each of the electrodes are optimized in this manner, and Fig. 5.7 shows a measurement of the EIT spectrum obtained for different control powers ranging from 0 mW to 277 mW. We can see that the EIT peak shifts by approximately 3 MHz between the lowest and highest control power used, and that it definitively have an effect on the width of the peak, but that it does not solve this issue completely.

Rotation of magnetic field axis

The quantization axis is fixed by the pair of compensation coil along the y-axis. We can wonder whether there is a mismatch between the quantization axis and the axis along which the probe and coupling beams are polarized, which would lead to issues with the optical pumping and incidentally the EIT measurements. In this series of measurement we have a probe power inside the cavity P probe in =70 nW. We tested the rotation of the magnetic field in both directions: by applying a magnetic field on the x-axis, we can rotate the quantization axis around the z-axis, i.e. in plane. This is done by using the two remaining compensation coil pairs, along the x+z and x-z-directions. We apply a magnetic field ranging from ±1.6 G to ±2.7 G along the x axis. Fig. 5.8 shows the EIT spectrum for a fixed probe and coupling power, with negative and positive fields along the x-axis, with the 𝐵 = 0 G case and the normal mode splitting shown for reference. When a negative field is applied along the x-axis, the EIT peak increases, whereas when a positive field is applied along the same direction the EIT peak amplitude is decreased. Until now, both the repumping laser and pump laser addresses the hyperfine level F'=2. We can try to change the repumping beam to the transition F=1 → F'=1 in order to avoid mixing between both lasers. Fig. 5.9 shows a similar situation as in Fig. 5.8 with the repumper on the F=1 → F'=1 transition, and we can observe to some extent similar results concerning the evolution of the EIT signal depending on the rotation of the quantization axis. We can also see that the peak amplitude is larger than before, which implies that the cooperativity is a bit smaller than in the previous set of measurement. This confirms the fact that the magnetic field is a crucial parameter when it comes to EIT measurements.

Effect of the science cavity locking beam

We can also wonder if the locking beam used for the science cavity might have an effect, as the far detuned vertical dipole trap created a lightshift. Fig. 5.10 shows an EIT spectrum for the case where no magnetic field is present, for a situation where the locking beam is either present in the science cavity or not. We can see that the locking beam from the 1064 laser has no impact whatsoever on the measurement amplitude or frequency of the EIT resonance.

EIT characterization

We now work with the previously described optimizations. Looking at the EIT spectrum for different control powers, we can measure the evolution of the EIT peak width and EIT peak amplitude as a function of the control power. affected whether the peak is close to a Rabi resonance, so this comparison for different control power might not be fully relevant. In any case, the EIT peak amplitude increases with the power which is what we would expect, the width however is too important. A sample temperature too high could have a broadening effect on the EIT signal, as well as a laser frequency width too large: these measurement were indeed done before we noticed and corrected the issue related to the waist position of the one-dimensional moving array discussed in Section 4.3.

Conclusion

In summary, we have witnessed a hierarchy in the importance of different parameters on the EIT signal in the configuration where the science cavity is used. A overview of this ranking can be seen in Table : 5.1. The magnetic field is a crucial parameters in both its amplitude and direction: this makes sense, as it is required to lift the degeneracy of the Zeeman sublevels that are used in the initial optical preparation of the sample. It also appears that we may depump the sample with a duration of the scan of the probe that is too 'long' in duration. In addition, we witnessed that the lightshifths induced by the transport lattice beams and the science cavity locking beam have a relatively weak effect on the EIT signal. What was interesting the fact that that despite Rydberg atoms being extremely sensitive to stray electric fields, it appears here that this is not the limiting factor for us even though it was expected to be a limiting factor: it did shift the EIT signal around, but did not dampen it. Stray electric fields could have a more important impact for Rydberg with higher n. All of this led us to investigate more closely the effect of the control and probe lasers, as well as the optical pumping for the sample preparation.

EIT in the MOT

EIT in the MOT

Since the EIT measurement when the atomic sample was transported to the science cavity and the blue excitation was set in free-space proved inconclusive, it was decided to do EIT measurement directly at the MOT position, thus simplifying the problem by removing issues linked to the transport and atom trapping. We moved the free-space control coupler that was previously injected in the science cavity to the lower part of the vacuum chamber. We are now in a geometry where the blue beam and the probe beam are counter-propagating with respect to each other, each one entering the vacuum chamber from the opposite side. In order to get the proper waist for the beam, we add the required lenses to focus the probe beam (respectively the coupling beam) to have typical waists size of 30 µm (respectively 60 µm). A dichroic plate is used to superpose the control beam and the probe beam, and light is collected in transmission of the atomic cloud by a fiber sent to the APD. Fig. 5.12(a) shows the layout of the optics for this measurement set. By using the compensation coil pairs along the z+x and z-x-axis, we can provide a quantization axis along the probe direction. We are working with a similar two-photon ladder excitation scheme shown in

Optical pumping

The optical pumping might not be fully optimal, and this could be part of the reason the different EIT measurements did not yield any proper results. We then investigate different parameters concerning the optical pumping, such as the magnetic field, the pumping duration and the efficiency of said pumping. In order to check whether we pump the atoms to the |𝐹 = 2, 𝑚 𝐹 = 2⟩, we measure the ratio of the optical depth with Zeeman pumping in 𝜎 + and 𝜎 -, for a fixed 𝜎 + probe polarization. Given the fact that the OD is proportional to the resonant cross section, and that the resonant cross section is itself proportional to the dipole squared, the 𝜎 + /𝜎 -ratio of the two dipoles is approximately (

1/ √ 30 1/ √ 2 ) 2 = 1
15 [START_REF] Steck | Rubidium 87 D Line Data[END_REF]. This means that there should be a factor 15 between the optical depth measured in the two different pumping configurations. First the effect of the magnetic field on the optical pumping was investigated, with a fixed pump power of 15 mW and a pumping duration of 20 ms: a zero absorption in the 𝜎 -configuration was measured and could be due to a pumping power too important, a pumping duration too long, or the fact that the atoms are not prepared in the correct initial state. The latter turned out to be the right way to go, as a ratio of 5 was measured between the optical depth in 𝜎 ± pumping configurations.

In order to characterize the state in which the atoms were prepared, we measured the Zeeman shift for 𝜎 ± and 𝜋-polarized Zeeman pumping. We have a 𝜎 + -polarized probe scanned in 100 µs at a fixed power of approximately 3 nW, and we vary the magnetic field from 0 G to 2 G. The measured shifts were not exactly what we expect, especially for the 𝜎 -case. This discrepancy between the calculated and measured values might come from the pumping power: by scanning the three different polarizations with no magnetic field for a fixed pump power, the resonances corresponding to the different pump polarizations moved by an average of 0.3 MHz. This can lead to a wrong estimation of the Zeeman shift, and could imply that the atoms are kicked for their position by the pump beam, or be a result of non-linearity in the pump power. The pump power was monitored from 0 µW to 48 µW and no non-linearity was visible. The OD was measured for 𝜎 ± and 𝜋-polarized pump in magnetic field ranging from 1 G to 4 G with a fixed 𝜎 + probe power of 3 nW, a pump power of 15 µW and a pump duration of 5 µs. Table : 5.2 gives the values of the Zeeman shifts for the different polarizations: we see that for both 𝜎 + and 𝜋-polarized pump the measured Zeeman shift is much closer to the expected value than for a 𝜎 -pump. In addition, if the atoms are indeed in the Zeeman sub-level 𝑚 𝐹 = -2 we should have an OD 15 times smaller than with the 𝜎 + polarization, which is not the case. After investigation of the system, we found that the timing between the repumper and the pump was not well controlled. Fixing this, we measure the OD for each polarization of the Zeeman pump in a magnetic field of 2 G, with a fixed probe power and a scan duration of 100 µs. The ratio for the OD of the pumping in 𝜎 + and 𝜎 --polarized configuration have a value of approximately 11, and the detunings measured are compared to the theoretical values in Table : 5.3: they are much closer to their theoretical value than before, but the low value of the optical depth of the 𝜎 -configuration makes it hard to extract a definitive value for the Zeeman shift.

The probe power was also investigated as it might affect the optical pumping in some manner: it was scanned over 10 nW with no magnetic field and no pump, and no visible non-linearity appeared in the spectrum, as the value of the OD for the higher probe powers vary only by a few percents when compared to the OD of the case of the weakest probe power. A potential issue is the fact that the probe pump the cloud of atoms during the scan of the resonance: we measured the Zeeman shift for 𝜎 ± and 𝜋-polarized pump in a magnetic field of 2 G, for a fixed probe power of 2 nW and a probe duration of 20 µs and 100 µs. The measured values of the Zeeman shifts are much closer to the expected theoretical values: for a 𝜎 --polarized pump measurement with the same parameters is not possible due to the weak OD, and for a 𝜋-polarized pump a value of -0.91 MHz G -1 was extracted. This confirms the fact that a long probe scan has a depumping effect.

Optical pumping: conclusion

We can see that the preparation of the sample via optical pumping is a crucial step, likely responsible for the initial failed EIT measurement. A good control of the magnetic field is necessary, and a shorter probing duration avoid effect where the probe might pump the sample. We now perform an EIT measurement, implementing the parameters found to be optimal during optical pumping.

Estimation of the Rydberg linewidth

The manner in which we extract the Rydberg linewidth 𝛾 𝑟𝑦𝑑 is as follows: a scan of the empty space with the probe is used as a normalization measure. Then the MOT is created, and cooled down to a molasses. The probe laser is scanned without the control laser, providing a measurement from which we can extract the density. The control laser is turned on, and the EIT measurement is done, iterating over the coupling light power. The model used for the fit describes free space EIT, where we consider one intermediate level interacting with only one Rydberg level, and it was described in Section 2.3. We first calculate the OD of the acquired data which is normalized on a run with no atoms before defining the zero detuning for our data set. This is done by fitting a Lorentzian to the run corresponding to the two-level system situation where we do not consider the Rydberg level, as no coupling light is present. From this, we can center the scan and get an estimated value of the initial density 𝜌 0 by fitting the OD. The measurement then is iterated over the power of the control laser, from which it is possible to extract the detuning of the coupling laser from the Rydberg level. This provides us with the value of the OD at the EIT resonance for different control powers. By fitting this data with either the homogeneous or inhomogeneous expression mentioned above for a fixed value of 𝛾 𝑟𝑦𝑑 , we can extract the ratio 𝛺 2 𝑏 /(𝛾 𝑟𝑦𝑑 𝑃 𝑏𝑙𝑢𝑒 ): recall that we call homogeneous the situation where the control laser is homogeneous with respect to the probe laser: 𝑤 𝑝𝑟𝑜𝑏𝑒 ≪ 𝑤 𝑏𝑙𝑢𝑒 : typically, 𝑤 𝑐𝑙𝑜𝑢𝑑 ≈ 600 µm, 𝑤 𝑏𝑙𝑢𝑒 ≈ 70 µm and 𝑤 𝑝𝑟𝑜𝑏𝑒 ≈ 30 µm so the approximation is valid. By comparison, the inhomogeneous case corresponds to a situation where the finite size of the control beam is considered. An independent measurement is then done to extract the value of 𝛾 𝑟𝑦𝑑 : the probe power is fixed and the Rydberg transition is scanned, with a fixed

𝜎 + 1.4 MHz G -1 1.38 MHz G -1 𝜎 - 0.47 MHz G -1 / 𝜋 0.93 MHz G -1 1.05 MHz G -1
control power. And once again, by fitting the OD corresponding to the EIT peak to the data we can extract the value of the Rydberg linewidth since we control the blue detuning and measured the previously mentioned ratio 𝛺 2 𝑏 /(𝛾 𝑟𝑦𝑑 𝑃 𝑏𝑙𝑢𝑒 ).

Results

The scan of the control laser was done with a coupling power of 10 µW over a range of ±4 MHz around the resonance with the Rydberg level. We get a value for both the homogeneous and the inhomogeneous case: we have a homogeneous linewidth 𝛾 ℎ𝑜𝑚𝑜 = 2𝜋 × 274 kHz (respectively an inhomogeneous linewidth 𝛾 𝑖𝑛ℎ𝑜𝑚𝑜 = 2𝜋 × 267 kHz). The Rabi frequencies obtained from the fits and the knowledge of the coupling beam intensity gives access to effective values for transition dipoles, giving approximately 𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,ℎ𝑜𝑚 /𝑑 𝑡ℎ𝑒𝑜 = 0.85 (respectively approximately 𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,𝑖𝑛ℎ𝑜𝑚 /𝑑 𝑡ℎ𝑒𝑜 = 0.92 for the inhomogeneous case). Fig. 5.13(a) shows a fit of the EIT data for each coupling powers done sequentially.

As we estimated already the different parameters that are the Rydberg linewidth, the detuning from the Rydberg level and the density, and since we know the control power for each run it is possible to fit directly the raw data, without passing through the process sus-mentioned. The two free parameters are the Rabi frequency of the coupling laser 𝛺 𝑏,𝑡𝑜𝑡 and the Rydberg linewidth 𝛾 𝑟𝑦𝑑,𝑡𝑜𝑡 . We have as initial values the previously fitted values, and the free parameters are bound to their respective intervals

𝛺 𝑏,𝑡𝑜𝑡 ∈ [𝛺 𝑏,𝑖𝑛ℎ𝑜𝑚𝑜 ±( 𝛺 𝑏,𝑅𝑀 𝑆 2 × 𝛺 𝑏,𝑖𝑛ℎ𝑜𝑚𝑜 +10%)], 𝛾 𝑟𝑦𝑑,𝑡𝑜𝑡 ∈ [2𝜋×𝛾 𝑟𝑦𝑑,𝑖𝑛ℎ𝑜𝑚𝑜 ±( 𝛾 𝑟𝑦𝑑,𝑖𝑛ℎ𝑜𝑚𝑜,𝑅𝑀 𝑆 𝛾 𝑖𝑛ℎ𝑜𝑚𝑜 )].
where 𝛺 𝑏,𝑅𝑀 𝑆 and 𝛾𝑟𝑦𝑑,𝑅𝑀 𝑆 are the variance of the estimated parameters. The bounds of the Rabi frequency are increased by 10% to take into account for potential errors in the measurement of the coupling power. Fig. 5.13(b) shows the evolution of the two fitted parameters as a function of the coupling power P blue . Concerning the Rabi frequency we find back the square root dependency on the power. Concerning the linewidth, it is hard to conclude anything as measurement related uncertainties prevent a correct extraction of the parameter in some regimes, specifically in the low-power (≤ 25 mW of coupling power). The values that should be considered as well extracted are the ones at intermediary coupling powers, and it should be power independant. The average linewidth is 𝛾 𝑟𝑦𝑑,𝑖𝑛ℎ𝑜𝑚𝑜 = 2𝜋 × 226 kHz, 18% less than what we estimated in the inhomogeneous case in the previous step by step analysis. Before the optimization of the optical pumping, the Rydberg linewidth was measured to be 𝛾 𝑟𝑦𝑑,ℎ𝑜𝑚𝑜 = 2𝜋 × 297 kHz, so we have approximately a 10% improvement over the linewidth.

Switch to the D 1 line

To simplify the preparation of the atomic sample and to more easily separate the probe used to measure the EIT signal from the cooling beams, we measured the EIT spectrum on the D1 line. The crystal of the Sapphire Ti:Sapphire was moved to shift the beam wavelength by approximately 15 nm. We then perform spectroscopy of the cloud in order to tune the control laser, before doing EIT measurements in the MOT.

Excitation scheme and measurement preparation

We work in a similar configuration as the one presented previously with the probe and coupling laser counter-propagating along the quantization axis. The only thing that changed is the 87 Rb line we work on: the two-photon excitation scheme is shown in Fig. We tested EIT measurements with both these configurations. No major issue occurred when investigating the pumping and sample preparation, and to ensure that the atoms were pumped in the right sub-level, we looked at the Zeeman shift on the D 1 line, and we measured a deviation from the theoretical values of less than 4% in the case of a 𝜎 -pump, and of approximately 10% for a 𝜎 + pump.

Estimation of Rydberg linewidth

The idea is to have the same data acquisition and analysis as what was previously described for the D 2 line: an iterative scan of the probe for increasing coupling laser power, and a scan of the coupling laser frequency around the resonance with the Rydberg level at a fixed power. From this data run, we get a homogeneous linewidth of 𝛾 ℎ𝑜𝑚𝑜 = 2𝜋 × 136 kHz and an inhomogeneous Rydberg linewidth of 𝛾 𝑖𝑛ℎ𝑜𝑚𝑜 = 2𝜋 × 137 kHz. We estimate the ratio of the measured dipole to the theoretical dipole of 𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,ℎ𝑜𝑚𝑜 /𝑑 𝑡ℎ𝑒𝑜 = 0.5 (respectively 𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,𝑖𝑛ℎ𝑜𝑚𝑜 /𝑑 𝑡ℎ𝑒𝑜 = 0.54 for the inhomogeneous case). Fig. 5.15(a) shows the fit of the raw data for each coupling power in an independent manner, done in a similar manner as what was done earlier: the fixed parameters are the density and the coupling laser detuning from the Rydberg level, which have the values estimated from the previous fits of the data. The free parameters are the Rydberg linewidth 𝛾 𝑟𝑦𝑑,𝑡𝑜𝑡 and the Rabi frequency 𝛺 𝑏,𝑡𝑜𝑡 . We can see that the Rabi frequency scales as expected, as 𝛺 𝑏 ∝ √ 𝑃 𝑏𝑙𝑢𝑒. The Rydberg linewidth also increases as the coupling power increases. We extract an inhomogeneous linewidth of 𝛾 𝑟𝑦𝑑,𝑡𝑜𝑡 = 2𝜋 × 177 kHz from the fit. This time the measurement uncertainties are less important at lower coupling powers, but seem to increase at higher powers. Given the low amount of coupling power available, it is hard to extrapolate to higher power and to conclude upon the efficiency of our model.

There are several points that could explain the discrepancy between the calculated and measured value of the dipole. As we saw earlier, an inefficient preparation of the sample in the initial state could lead to a larger linewidth. During the measurement we were also limited in the available blue power as the pump laser of the Ti:Sapphire was slowly getting misaligned, an issue that since has been fixed by the laser manufacturer: by comparison to the measurement that was done on the D 2 line we only had 35 % of the coupling power available for this measurement run, as typical output powers from the Ti:Sapphire laser should be on the order of 0.5 W to 1 W.

Introduction

In this section, we present a summary of the achieved results, as well as the evolution and planned work on the experimental setup. The different milestones that the setup should allow us to reach are also presented.

Experimental platform

Experimental setup

We have presented in this work the assembly of a system that merges many different technologies and ideas. My main contribution to the project was the assembly of the optical bench, the frequency stabilization of the lasers on which a large part of my PhD was spent, and alongside J. Vaneecloo. and S. Garcia the optimization of the atom trapping, transport and cooling of the sample, as well as the first EIT measurements.

Starting from a nearly empty optical table, the laser system required for the preparation of the sample and Rydberg excitations was built and stabilized to an ultra-stable cavity. The vacuum chamber was set up, cleaned and a vacuum of ∼ 1 × 10 -9 mbar was reached. A 2D MOT feeding a 3D MOT allows for an atomic sample that is compressed and cooled: during a loading of 100 ms, we reach a sample density of approximately 5 × 10 10 atoms per cm 3 , with a sample temperature of ∼ 10 µK. The atoms are then transported over 33 mm to the science cavity in 16 µs using a one-dimensional moving array: a sample temperature of approximately 25 µK is measured after the transport. The coupling between the sample and the cavity was monitored at 780 nm and 795 nm, with cooperativities 𝐶 780 ≈ 55 and 𝐶 795 ≈ 46 measured. Then Rydberg excitations were done in the science cavity, before performing EIT measurements in the MOT, showing a 𝛾 𝐷2 = 2𝜋 × 226 kHz on the D 2 line and 𝛾 ℎ𝑜𝑚𝑜 = 2𝜋 × 177 kHz on the D 1 line.

Optimization and setup improvement

Progress has been made during the writing of this thesis, and some issues have been solved by my coworkers by the time it was finished.

Raman sideband cooling: Resolved sideband Raman cooling of the atoms after transport and their loading in a crossed far-detuned dipole trap now leads to the preparation of an atomic cloud of ≈ 800 atoms at a temperature below 5 µK with a typical radius of 7 µm and a peak density higher than 1 × 10 11 atoms per cm 3 . Control of the system: A better control of the optical pumping, of the stray electrical field compensation and of the magnetic fields makes the atoms better approximate true closed three-level systems. Together with a reduced laser noise [START_REF] Léséleuc | Analysis of imperfections in the coherent optical excitation of single atoms to Rydberg states[END_REF], this allowed the team to bring the effective Rydberg linewidth to approximately 100 kHz for the Rydberg state 100S 1/2 .

Rydberg excitation scheme: The use of buildup cavities, providing waist-dependent power enhancement of the control field by a factor ∼ 60, greatly improves the Rydberg Rabi frequencies and allows EIT in excess of 80% of the empty cavity transmission for the 100S state, where the blockade range exceeds the size of the cloud.

These improvements recently allowed the team to observe strong photon blockade in the resonant regime, and photon bunching when the beams were tuned off-resonance with the intermediate 5P state. We can thus consider that the system reaches its goal specifications, at least as far as single-mode experiments are concerned.

Required investigations

We could investigate into the degradation of the finesse of the science cavity, as the trials where the mirror holders were heated proved inconclusive. Another important part of the system that still needs to be completed is the spectroscopy of the science cavity, to quantify the effect of aberrations in the regime where subsets of transverse modes are brought close to degeneracy [START_REF] Clark | Observation of Laughlin states made of light[END_REF].

Milestones

We now briefly discuss the different experiments that we aim for with our setup.

Deterministic preparation of photonic 'Schrödinger's cat' states

The first series of experiments focuses on the generation of non-classical states of light, more specifically of optical 'Schrödinger's cat' states defined as quantum superposition of coherent states with opposite phases. For this, we can adapt a protocol recently demonstrated in the context of single-atom cavity QED [START_REF] Hacker | Deterministic creation of entangled atom-light Schrödinger-cat states[END_REF]. A 𝜋/2 pulse will prepare the cloud in a quantum superposition 𝛹 𝑖𝑛𝑖 = |𝐺⟩ + |𝑅⟩ where |𝐺⟩ is the collective ground state and |𝑅⟩ corresponds to a single delocalized Rydberg excitation. Then a weak coherent pulse |𝛼⟩ resonant with a state |𝑒⟩ in the 5P manifold will be reflected off the cavity in presence of a laser coupling this state to another Rydberg state. The state-dependent cavity reflection coefficient will induce a state-dependent phase flip on the pulse, preparing the entangled state 𝛹 = (|𝐺⟩ |𝛼⟩ + |𝑅⟩ |-𝛼⟩). From there, another pulse followed by an optical measurement of the state of the cloud will project the optical part of this state into an 'even' or 'odd' cat state 𝛹 = |𝛼⟩ ± |-𝛼⟩, whose sign depends on the result of measurement.

Efficient two-photons quantum gate

The same mechanism can be used to implement a proposal for a two-photon quantum gate, a crucial building block of quantum computation. The proposal from [START_REF] Das | Photonic controlled-phase gates through Rydberg blockade in optical cavities[END_REF] should allow for the realization of a high-fidelity control-Z gate. A first photon impinging on the cavity is transiently stored as a superposition between ground and Rydberg polaritons, making a second impinging photon experience a state-dependent phase flip. Compared to the 'cat-state' experiment, the main additional step is the storage and retrieval of the first photon, whose efficiency needs to be investigated in our context [START_REF] Gorshkov | Photon storage in 𝛬 -type optically dense atomic media. I. Cavity model[END_REF] [76].

Strongly correlated quantum fluid

An important milestone would consist in moving towards a multi-photon and multi-mode regime, where the interacting intracavity photons would form a strongly correlated quantum fluid, thus making use of the fact that our non-planar cavity induces a twist on the photons inside the cavity.

Strongly correlated photon fluids can, under the proper conditions simulate solid-state physics: the specific non planar geometry of the resonator leads to a rotation of the polarization and field of the trapped photons. We have an equivalence between photons in a near-degenerate optical resonator with Rydberg ensemble and massive interacting particles in synthetic magnetic field trapped in 2D harmonic potential, Table : 6.1 provides the analogy between the two different systems. The underlying physics can be described by a Fock-Darwin Hamiltonian which describes a charge particle moving in the plane and confined by a harmonic potential under an external uniform magnetic field [START_REF] Santos | Evolution of squeezed states under the Fock-Darwin hamiltonian[END_REF]. It typically illustrates the behaviour of electrons in InAs/GaAs quantum dots, or the dynamics of trapped ions under a magnetic field. In our case, by tuning the cavity it is possible to remove the centrifugal force, leaving only the Lorentz force: the cavity modes become degenerate, and the system described by the Hamiltonian is one of massive particles in Landau levels. This implies that it is possible to simulate Landau levels with photons giving access to measures that were restricted to the field of solid state physics. In addition, it is possible to add a cloud of atoms in the cavity. By combining Rydberg polariton physics [START_REF] Ningyuan | Observation and characterization of cavity Rydberg polaritons[END_REF], [START_REF] Jia | A strongly interacting polaritonic quantum dot[END_REF] and the effect of the non planeity of the resonator, and following the same logic, the system simulates Laughlin states, which describe the ground state of charged particles evolving in a plane under a uniform magnetic field, and underlies fractional quantum Hall effect. Table 6.1: Analogy between our experimental system and a 2D gas of massive particles in a magnetic field. A recent pioneering experiment using a setup similar to ours succeeded in preparing a two-photon Laughlin state in a synthetic Landau level [44] [45], emulating the physics of the quantum Hall effect. This makes a convincing case for the ability of such platforms to reproduce effects typical to condensed-matter systems while offering the unique possibility to observe them in real time with a single-particle resolution. While several experimental parameters, and in particular the optical aberrations of our cavity, still need to be characterized, the general performance of our platform gives us good confidence that we will be able to push these experiments from a two-particle/three-mode to a truly collective and multimode regime.

MOTS CLÉS

Optique quantique, atomes froids, atomes de Rydberg, ingénierie quantique, non-linéarité optique RÉSUMÉ Atteindre un régime où des photons peuvent interagir entre eux est un but important de l'optique quantique. Il s'agit d'une condition fondamentale pour pour de nombreux domaines, e.g. la mise en place d'un réseau de communication quantique, mais c'est aussi une manière de permettre l'étude et la simulation de la physique de la matière condensée. Ce manuscrit présente le travail effectué durant trois ans pour assembler et tester une plateforme expérimentale visant à créer et à contrôler des interactions cohérentes entre photons optiques individuels et de franchir ainsi une étape importante pour le développement des technologies quantiques. Le montage combine des techniques venant de l'électrodynamique en cavité et des atomes froids: un ensemble d'atomes froids est transporté dans une cavité spécialement étudiée, permettant de transitoirement convertir les photons qui y circulent en polaritons de Rydberg atomique présentant de fortes interactions. Le contrôle de certains paramètres, relevant soit du nuage, de l'excitation ou bien de la cavité, doivent nous permettre de tester la validité de théorèmes fondamentaux en logique quantique, de générer des états non classiques et d'accéder à un régime dans lequel la simulation de système bidimensionnels propres à la matière condensée devient possible. Une première partie du manuscrit présente le montage et la manière dont les différents sous-systèmes requis ont été mis en place. Dans une deuxième partie les premières mesures d'EIT dans le nuage sont effectuées et décrites. Enfin, l'évolution à venir du montage est mentionnée.

ABSTRACT

A long-standing goal of quantum optics is to reach a regime where interactions between single photons can take place. This would lead to new perspectives in the domain of quantum communication and quantum simulation. This work presents the assembly and characterization of a hybrid experimental platform for the study and creation of deterministic interactions between single photons, an important milestone in quantum technologies. The experimental setup combines different technologies from cavity quantum electrodynamics and cold atoms techniques. A cold atom cloud is transported in a specially designed optical resonator, allowing the temporary conversion of intracavity photons into atomic Rydberg polaritons, creating a medium which enables strong interactions. This setup should allow us to deterministically entangle optical photons in order to generate non-classical light, to develop photonic quantum logic gates, and to emulate low-dimensional highly correlated condensed-matter systems. In a first part the setup is presented, including the manner in which the different subsystems work together. In a second part, the first EIT measurements are shown and analyzed, before describing the latest advancement made.

KEYWORDS

Quantum optics, cold atoms, Rydberg atoms, quantum engineering, optical non linearity

Figure 1 . 1 :

 11 Figure 1.1: The presence of a Rydberg polariton in the resonator translates to extra losses, which then induce a phase shift on a second incoming photon.

Figure 2 . 1 :

 21 Figure 2.1: D 1 and D 2 line of 87 Rb. The transitions between hyperfine splitting from the m F =0 Zeeman sub-level are given, as well as the approximate Landé 𝑔 𝐹 factors for the hyperfine splitting, with the corresponding Zeeman splittings. F (respectively F') are the hyperfine ground (respectively excited) states. The 'MOT' and 'repumper' show the transitions used for atomic trapping and cooling.

( a )

 a In the overcoupled case we have EIT, which implies that absorption by the sample is much less than the incoupling mirror transmission.(b) In the undercoupled regime we have a Rydberg blockade sphere, which leads to intracavity losses that are much greater than the incoupling mirror transmission.

Figure 2 . 2 :

 22 Figure 2.2: If no photon is stored in the sample (a), then no phase shift is induced on the photon. The presence of a Rydberg polariton in the resonator (b) translates to extra losses, which then induce a 𝜋 phase shift on a second incoming photon.

Figure 3 . 1 :

 31 Figure 3.1: A picture of the vacuum chamber connected to the Two-Dimensional (2D) MOT.We can see the titanium cell which the 87 Rb cartridge is placed along the 2D MOT, whose optics are not in place.

Figure 3 . 2 :

 32 Figure 3.2: A 3D model of the vacuum chamber. In dark yellow, the 2D MOT, with the optics holders in purple. The large red beams represent the Three-Dimensional (3D) MOT beams, while the vertical beams show the one-dimensional moving lattice. The two extra beam that are in the plane represent the Raman cooling beams.

Figure 3 . 3 :

 33 Figure 3.3: A plan of the TA Pro 780 and DL Pro 795 optical table.

Figure 3 . 4 :

 34 Figure 3.4: A plan of the optical table of the repumper laser where the auxiliary output of the master laser is shown.

1

  HighFinesse WSU-10. The use of an opto-mechanical switch (MEMS) allows us to have several input ports used for different laser on a single wavemeter. 2 AOM 3 on Fig. 3.3 3 AOM 1 and 2 on Fig. 3.5

Figure 3 . 5 :

 35 Figure 3.5: A plan of the MOT optical table.

Figure 3 . 6 :

 36 Figure 3.6: Iteration of the 2D MOT cell, from a simple glued panned cell to the rebar version of the cell currently implemented in the setup.

Figure 3 . 8 :

 38 Figure 3.8: Schematic of the fiber splitter used to mix the master laser and the repumping laser, before outputting the six MOT/repumper beams.

Figure 3 . 7 :

 37 Figure 3.7: Measure of the magnetic field from the coil along the y-direction of the 2D MOT as a function of the current.

  (a) A picture of the vacuum chamber, where the two MOT coils can be seen, with the four lowest electrodes used to control the electric field near the atoms location. (b) Trapped atoms in the 3D MOT, loaded from background pressure.

Figure 3 . 9 :

 39 Figure 3.9: Picture: 3.9(a) illustrate the MOT coils with the four lowest electrodes used to control the electric field near the atoms location. Picture: 3.9(b) shows atoms loaded in the MOT from the background pressure of 87 Rb.

( a )

 a Top view of the experimental setup vacuum chamber. (b) Side view of the experimental setup vacuum chamber.

Figure 3 . 10 :

 310 Figure 3.10: A schematic of the optics around the magnetic trap section: the 87 Rb cell is linked to the 2D MOT where the atoms are first captured and pushed towards the 3D MOT located in the main vacuum chamber. Next to the MOT beam 2 coupler a microwave antenna is placed, used to calibrate the compensation coils through microwave spectroscopy (see Section 4.2.1).

Figure 3 . 11 :

 311 Figure 3.11:A plan of the optical table for the Raman laser (DL Pro 780) and lattice laser (TA Pro 780). While the Raman laser is phase-locked to the master laser, the lattice lasers is free-running.

Figure 3 . 12 :

 312 Figure 3.12: Frequency output of one of the channel of the DDS that controls the AOMs of the moving lattice.

( a )

 a Schematic of the optics below the vacuum chamber.(b) Schematic of the top optical table. The dipole trap and lattice couplers can be seen, as well as the imaging part of the setup.

Figure 3 . 13 :

 313 Figure 3.13: We can see the optics layout above and below the vacuum chamber. Lenses 1,2,3 and 4 are place on translation stages in order to easily adjust the waists of the vertical dipole trap and crossed dipole traps. For the vertical dipole trap, the use of wedged glass allows for a convenient translation of the beams.

Figure 3 . 14 :

 314 Figure3.14: Schematic of the intermediary breadboard around the vacuum chamber. The optics are laid out on several levels: the MOT optics, then optics allowing the injection of beams in the science cavity, and on a third levels the optics that are related to the blue cavities.

Figure 3 .

 3 Figure 3.15: A plan of the optical table for the 1064 laser.

Figure 3 . 16 :

 316 Figure 3.16: A plan of the Ti:Sapphire lasers optical table layout.

𝐷 2 ,𝐷 1 ,Figure 3 . 17 :

 21317 Figure 3.17: Schematic of the relevant levels for the laser system.

Figure 3 . 18 :

 318 Figure 3.18:A plan of the optical table with the two locking cavities and the different lasers beam paths.

Figure 3 . 19 :

 319 Figure 3.19: Schematic of the transfer cavity structure, with a mirror at each end. In blue are represented the three ultra-low expansion glass bars, sitting on steel supports. Viton pads (in green) are used to decouple the cavity from the chamber, and the beam path is represented in red.

Figure 3 . 20 :

 320 Figure3.20: This is a summary of the laser system, where we can see their locking arrangement. We have dedicated inputs to the wavemeter, with the lasers grouped by pairs of color. Also shown are the ultra-stable cavity, the transfer cavity, the science cavity and the two blue cavities. The numbers in the cavities represent respectively the FSR (in MHz) and the linewidth (in kHz) of the cavity.

Figure 3 . 21 :

 321 Figure 3.21: Locking sequence for the different lasers and cavities of the system.

  Free spectral range of the science cavity. Transmission of the science cavity.

Figure 3 . 23 :

 323 Figure 3.23: Characterisation of the science cavity at a wavelength of 795 nm, acquisition averaged 32 times.

Figure 3 . 24 :

 324 Figure 3.24: Schematic of the optics allowing the on and off axis injection in a blue cavity.

Figure 3 . 25 :

 325 Figure 3.25: A schematic of the science cavity surrounded by the two blue cavities. The blue cavities each present a locking beam, as well as a blue high power beam.

Figure 3 . 26 :

 326 Figure 3.26: Characterisation of the axial mode of the blue left cavity.

  Blue right cavity bowtie mode transmission.

Figure 3 . 27 :

 327 Figure 3.27: Characterisation of the bowtie mode of the blue right cavity.

( a )

 a Atom number and cloud dimension as a function of the 2D MOT beam intensity. (b) Atom number and cloud dimension as a function of the 2D MOT trapping beams frequency.

Figure 4 . 1 :

 41 Figure 4.1: 2D MOT optimization: atom number with respect to (a) the 2D MOT beams intensity and (b) frequency in MHz.

Fig. 4 .

 4 3(a) shows the loading of the MOT from background 87 Rb pressure, with the 2D MOT coils being off and beams on. We have a characteristic time of approximately 5 s: (a) Typical experimental sequence. (b) Sequence for the preparation of the atomic cloud.

Figure 4 . 2 :

 42 Figure 4.2: Timeline of a typical experiential sequence.

( a )

 a Loading of the 3D MOT from ambient 87 Rb pressure without the use of the 2D MOT. (b) Atoms loading in the 3D MOT from the 2D MOT, loading of 1.3 × 10 9 atoms per second, before the lateste MOT optimizations.

Figure 4 . 3 :

 43 Figure 4.3: MOT loading from ambiant pressure and using the 2D MOT The lifetime of the MOT can be extracted by first loading it, and then imaging it with a variable delay. Fig. 4.4(a) shows the corresponding measurement: we have a characteristic time of 4.5 s.

  Lifetime of the MOT (b) Absorption imaging of the MOT during the lifetime measurement, i.e. during the time of flight measurement.

Figure 4 . 4 :

 44 Figure 4.4: MOT lifetime, with a corresponding absorption imaging of the cloud.

( a )

 a Scan of the MOT magnetic field via the current in the coils. (b) Evolution of the atom number with respect to the MOT laser detuning

Figure 4 . 5 :

 45 Figure 4.5: Optimization of the atom number in the MOT with respect to the magnetic field and the MOT beams detuning.

Fig. 4 .

 4 Fig.4.6(a) shows the evolution of the atom number during a scan of the intensity of the MOT beams. We can see that more atoms are loaded in the MOT, without significant heating of the sample. The temperature of the MOT is measured through a TOF measurement shown in Fig.4.6(b): the end temperature is of the order of magnitude of 50 µK, which is a temperature higher than what we would want. This is the reason behind the use of optical molasses, which allows for a cooling the sample before it is transported to the cavity mode.

( a )

 a MOT beam intensity scan. (b) Temperature measurement in the MOT.

Figure 4 . 6 :

 46 Figure 4.6: The evolution of the temperature and the number of atoms in the MOT with respect to variations of the MOT beam intensity can be seen in (a), while (b) shows a temperature of the MOT through a time of flight measurement.

  Scan of the |5 2 𝑆 1/2 ⟩ hyperfine structure in the atomic cloud, with the compensation coils not optimized. Scan of the |5 2 𝑆 1/2 ⟩ hyperfine structure in the atomic cloud, with the compensation coils optimized.

Figure 4 . 7 :

 47 Figure 4.7: Spectroscopy of the 5 2 𝑆 1/2 structure before (a) and after (b) optimization of the compensation coils current.

  Hyperfine structure shift measurement for five different magnetic fields ranging from -1.09 G to -0.54 G. Calibration curves for the compensation coil pair along the y-axis.

Figure 4 . 8 :

 48 Figure 4.8: Calibration of the y-oriented coil for the compensation of the residual magnetic field. In (a) the frequency shift of the Zeeman sublevels for varying fields of the calibration oriented along y, in (b) the calibration for the pair of coil.

( a )

 a Atom number in the MOT with respect to duration of the compression step. (b) Evolution of the size of the atomic cloud during the compression step.

Figure 4 . 9 :

 49 Figure 4.9: Compression step of the MOT: in (a) the evolution of the atom number with respect to the duration of the compression step, and in (b) the size of the cloud with respect to an increase of the magnetic field from the MOT coils.

Fig. 4 .

 4 10(a) shows the variation of the atom number and temperature as a function of the detuning at the end of the molasses step after a 1 ms frequency sweep and 3 ms constant detuning. In a similar manner, Fig.4.10(a) shows the evolution of the atom number and the temperature as a function of the MOT beam intensity at the end of the molasses step.

( a )

 a Evolution of the temperature of the cloud with respect to the end detuning of the trapping beams during the optical molasses. (b) Evolution of the temperature of the cloud with respect to the intensity of the trapping beams during the optical molasses.

Figure 4 . 10 :

 410 Figure 4.10: Temperature evolution with respect to (a) detuning of the MOT beams and (b) intensity of the MOT beams during the optical molasses.

( a )

 a Evolution of the cloud size and atom number during the variation of the MOT beam detuning in the optical molasses. (b) Evolution of the cloud size and atom number during the variation of the MOT beam intensity in the optical molasses.

Figure 4 . 11 :

 411 Figure 4.11: Density variation with respect to (a) detuning of the MOT beams and (b) intensity of the MOT beams in the optical molasses.

Figure 4 . 12 :

 412 Figure 4.12: Temperature in the MOT after the compression step and optical molasses.

( a )

 a Temperature measurement at the end of the molasses step without lattice beams being present. (b) Temperature measurement with lattice beam during the last millisecond of the molasses phase.

Figure 4 . 13 :

 413 Figure 4.13: Temperature measurement of the atoms at the initial position before transport without (a) and with (b) the transport lattice beams during the last millisecond of the optical molasses step.

( a )

 a Evolution of temperature as a function of the loading time. (b) Absorption imaging of the MOT and the atomic sample after transport in the moving lattice over a distance 𝑑 = 33 mm

Figure 4 . 14 :

 414 Figure 4.14: Atoms number with respect to the transport distance.

Figure 4 . 15 :

 415 Figure 4.15: Effect of the transport duration on a fixed distance 𝑦 = 33 mm on the number of atoms and the temperature of the sample in both the transport direction and transverse direction.

  Number of atom in the lattice at the MOT position before optimization of the lattice waist position. Number of atom in the lattice at the science cavity position before optimization of the lattice waist position.

Figure 4 . 16 :

 416 Figure 4.16: Evolution of number of atom as a function of the modulation frequency, before optimization.

  Number of atom in the lattice at the MOT position after optimization of the lattice waist position. Number of atom in the lattice at the science cavity position after optimization of the lattice waist position.

Figure 4 . 17 :

 417 Figure 4.17: Evolution of number of atom as a function of the modulation frequency, optimized situation.

Figure 4 . 18 :

 418 Figure 4.18: Lifetime of 51.4 ms at the cavity position after optimization of the waist position.

(a) 50

 50 ms loading, 9.3 × 10 5 atoms. (b) 100 msloading, 1.3 × atoms. (c) 250 msloading, 1.6 × 10 6 atoms.(d) 500 msloading, 1.9 × atoms.

Figure 4 . 19 :

 419 Figure 4.19: Moving lattice in the MOT for loading times ranging from 50 ms to 500 ms.

Figure 4 . 20 :

 420 Figure 4.20: Two different dynamics are present in the vertical dipole trap. In (a), thermal losses with a characteristic time of 32 ms, while in (b) background pressure losses with characteristic time of 1.5 s.

  Gaussian sigma) [μm] Tx = 4.4 μK Ty = 4.6 μK (a) Temperature in the diagonal dipole trap after the cooling and compression step. (b) Imaging of the atomic cloud in the crossed dipole trap after a loading of 50 ms.

Figure 4 . 21 :

 421 Figure 4.21: Characterization of the sample after degenerate Raman sideband cooling and compression.

4 .

 4 [START_REF] Ritter | An elementary quantum network of single atoms in optical cavities[END_REF] shows the TEM 00 of the blue left cavity on the sample. Later on, fine alignment is done by monitoring Rydberg excitation signals. (a) Reference image of the sample. (b) Image of the atomic cloud with 780 nm light injected in the right blue cavity, TEM00 mode.

Figure 4 . 22 :

 422 Figure 4.22: Alignment of the right blue cavity with respect to the atomic cloud.

  Normal mode splitting in the science cavity. Normal mode splitting in the science cavity for 795 nm probe.

Figure 4 . 23 :

 423 Figure 4.23: Rabi splitting at 780 nm (a) and 795 nm (b).

Figure 5 . 1 :

 51 Figure 5.1: Atom number as a function of the detuning from the 46S Rydberg level.

Fig. 5 .

 5 2(b) shows the relevant level scheme on the D 2 line for the Rydberg excitation in this new configuration: atoms are initially pumped in |𝐹 = 2, 𝑚 𝐹 = 2⟩ by the repumping laser set on the transition F=1 → F'=2, and by a 𝜎 + Zeeman pump provided by the 795 laser set on the F=2 → F'=2 transition of the D 1 line. The atoms are then excited to the Rydberg state |46𝑆, 𝑚 𝑗 = 1/2⟩ with a 𝜎 + -polarized probe detuned by 𝛥 = -8 MHz from the intermediate level |𝐹 ′ = 3, 𝑚 𝐹 = 3⟩

( a )

 a Probe and coupling beam configuration for Rydberg excitation in free-space using the science cavity.(b) Rydberg excitation scheme on the D2 line for a free space, counter-propagating coupling beam. On the left side is shown the two-photon transition, while the right side shows the magnetic sub-levels with the respective Zeeman levels.

Figure 5 . 2 :

 52 Figure 5.2: Geometry and excitation scheme for an EIT measurement without the blue cavities.

  EIT measurement after transport with the atomic sample in the vertical dipole trap, as a function of the probe detuning. EIT measurement after transport with the vertical dipole trap turned off, as a function of the probe detuning.

Figure 5 . 3 :

 53 Figure 5.3: EIT measurements after transport, (a) with and (b) without the sample being trapped in the vertical dipole trap, control beam in free space.

Fig. 5 .

 5 4 shows EIT measurements done in a similar configuration as what was done before in Fig.5.3, but with a magnetic field of 6.5 G instead of 1.5 G and a control power of 480 mW measured before the science cavity. Fig. 5.4(a) shows the case where no magnetic field is applied during the measurement and Fig. 5.4(b) corresponds to the situation where the magnetic field is set to 6.5 G. We have 𝜅 𝑓 𝑖𝑡,𝐵=0 = 2𝜋 × 2.85 MHz and 𝑔 𝑓 𝑖𝑡,𝐵=0 = 2𝜋 × 10.01 MHz (respectively 𝜅 𝑓 𝑖𝑡,𝐵=6.5 = 2𝜋 × 2.82 MHz and 𝑔 𝑓 𝑖𝑡,𝐵=6.5 = 2𝜋 × 7.76 MHz.

  500 µs scan of the resonance with B=0 G. 500 µs scan of the resonance with B=6.5 G.

Figure 5 . 4 :Figure 5 . 5 :

 5455 Figure 5.4: EIT measurements after transport at the science cavity position, in a magnetic field of (a), 𝐵 = 0 G and (b) 𝐵 = 6.5 G and a control power of 480 mW.

( a )

 a Shift of the EIT resonance as a function of the applied voltage to the electrodes along the z-axis. (b) Detuning of the EIT peak as a function of the applied voltage on the electrodes along the zaxis. We have a minimum detuning for a voltage of 0.04 V on the z-electrode pair.

Figure 5 . 6 :

 56 Figure 5.6: Effect of the electrode voltage along the z-axis as a function of the EIT detuning.

Figure 5 . 7 :

 57 Figure 5.7: EIT spectrum for coupling powers ranging from 0 mW to 277 mW.

Figure 5 . 9 :

 59 Figure 5.9: EIT scan done with the magnetic field rotated, repumper on the F=1 → F'=1 transition.

Fig. 5 .

 5 11(a) shows the relevant fitted data, and Fig.5.11(b) shows the evolution of the EIT resonance width and amplitude as a function of the control laser power. We have to keep in mind that these values are

Figure 5 . 10 :

 510 Figure 5.10: EIT spectrum for with and without the 1064 laser.

( a )

 a EIT spectrum for control powers ranging from 46 mW to 255 mW. Evolution of the amplitude and width of the EIT resonance as a function of the coupling power.

Figure 5 . 11 :

 511 Figure 5.11: Spectrum and evolution of the amplitude and width of the EIT resonance as a function of the coupling power.

Fig. 5 .

 5 12(b): atoms are initially pumped in |𝐹 = 2, 𝑚 𝐹 = 2⟩, and the two-photon transition is |𝐹 = 2, 𝑚 𝐹 = 2⟩ → |𝐹 ′ = 3, 𝑚 𝐹 = 3⟩ → |46𝑆, 𝑚 𝑗 = 1/2⟩ with a probe polarized in 𝜎 + and a control laser polarized in 𝜎 -. The repumper is along the |𝐹 = 1, 𝑚 𝐹 = 1⟩ → |𝐹 = 1, 𝑚 𝐹 = 1⟩ in 𝜋 on the D 2 line, and the Zeeman pump is along the |𝐹 = 2, 𝑚 𝐹 = 1⟩ → |𝐹 = 2, 𝑚 𝐹 = 2⟩ in 𝜎 + on the D 1 line. The probe laser is detuned from the intermediate state by 𝛥 = 11 MHz. One thing to note is that previously we were collecting light in transmission of the science cavity, whereas now we probe the MOT and collect the transmitted light towards the APD. This change in the configuration of the measurement implies that the expected EIT signal is not the same as what we had previously when working in the science cavity. The coupling laser creates a dip in the OD. (a) Layout of the optics and beam paths for Rydberg excitation in the MOT. (b) Ladder scheme for the Rydberg excitation taking place in the MOT. On the left the two-photon excitation can be seen, while on the right the hyperfine levels are shown.

Figure 5 . 12 :

 512 Figure 5.12: (a) Optical layout for EIT measurement in the MOT, (b) two-photon excitation scheme.

( a )

 a Independent fit of the EIT data for different coupling powers, with the Rydberg linewidth and the Rabi frequency as free parameters.(b) Evolution of the free parameters as a function of the different coupling powers.

Figure 5 . 13 :

 513 Figure 5.13: Independent fit of the EIT spectrum: in (a) the data along with the fits, while in (b) is shown the evolution of the fitted parameters as a function of the coupling power.

  5.14. The probe laser is detuned from the intermediate state by 𝛥 = 11 MHz. Atoms are initially pumped in |𝐹 = 1, 𝑚 𝐹 = ±1⟩. Both the repumper and the Zeeman pump are on the D 2 line. The repumper is 𝜋-polarized on the transition |𝐹 = 2, 𝑚 𝐹 = 1⟩ → |𝐹 = 3, 𝑚 𝐹 = 1⟩, and the Zeeman pump is 𝜎 ± -polarized on the |𝐹 = 1⟩ → |𝐹 = 1⟩ transition. There are two path for an excitation to the Rydberg level: with a 𝜎 ± probe on the |𝐹 = 1⟩ → |𝐹 = 2, 𝑚 𝐹 = ±2⟩ transition and a 𝜎 ∓ control laser along the |𝐹 = 2⟩ → |46𝑆⟩ transition.

Figure 5 . 14 :

 514 Figure 5.14: Two two-photons Rydberg excitation schemes on the D 1 line, for a counterpropagating probe (respectively coupling beam) that is 𝜎 ± (respectively 𝜎 ∓ polarized).

  Independent fits of the EIT spectrum for different coupling powers, with the Rydberg linewidth and the Rabi frequency as free parameters. Fitted values of the Rabi frequency and the Rydberg linewidth as a function of the coupling power.

Figure 5 . 15 :

 515 Figure 5.15: Independent fit of the EIT spectrum: in (a) the data along with the fits, while in (b) is shown the evolution of the fitted parameters as a function of the coupling power.

  geometry of the science cavity light twist → synthetic magnetic field mirror curvature transverse confinement: harmonic potential Rydberg blockade photon-photon interactions → contact interactions

Figure A. 1 :

 1 Figure A.1: Schematic of the first optical table, on which are placed the master laser, repumping laser, 1064 laset, 795 laser as well as the two cavities used for the locking process.

Figure A. 2 :

 2 Figure A.2: Schematic of the MOT optical table, with the conveyor and Raman part.

  If |𝛥 𝑐 | ≫ 𝑔, the detuning is much larger than the coupling strength and the eigenstates can be approximated by the initial states. If |𝛥 𝑐 | ≪ 𝑔, 𝜃 ∼ 𝜋/2 and

	2) |𝑔,1⟩ + 𝑠𝑖𝑛(𝜃/2) |𝑒,0⟩ .	(2.38)
	|𝐸 + ⟩ ≈ |𝑔,1⟩ + |𝑒,0⟩	(2.39)

Table 2 .

 2

	Property	Expression	𝑛 * scaling typical value ( 87 Rb at 46S)
	Binding energy	-	𝑅𝑦 (𝑛 * ) 2	(𝑛 * )	-2	6.5 meV
	Level spacing	𝐸 𝑛 -𝐸 𝑛-1	(𝑛 * )	-3	100 GHz
	Ionizing field			(𝑛

1: Table summarizing the scaling properties of Rydberg atoms with respect to the effective quantum number 𝑛 * , from [56]. * ) -4

Table 3 . 1 :

 31 Physical properties of the compensation coil pairs: N being the number of loops, R the coil radius, L the separation between the coils.

	Coils axis	N	L [mm] R [mm] Calibration
	y	144	184	105	1.552 G/V
	z + x	114	240	60	0.388 G/V
	z -x	144	240	60	0.330 G/V

Table 3 . 2 :

 32 Summary of the lasers -Power requirements

	Laser	𝜆 [nm]	Power required [mW] Power available [mW] Use
	Master laser	780 nm	∼ 500 mW	680 mW	MOT
	Repumping laser	780 nm	∼ 50 mW	40 mW	Repumper
	Raman laser	780 nm	∼ 50 mW	62 mW	Raman
	Moving lattice laser	780 nm	500 mW	1.1 W	Transport
	795 laser	795 nm	∼ 30 mW	36 mW	⎧ ⎨ ⎩ Hyperfine pump Probe
	Ti:Sapphire transfer 475 nm	1 W	410 mW	Rydberg Excitation
	Ti:Sapphire SLS	480 nm	1 W	560 mW	Rydberg Excitation
	ALS 1064	1064 nm	∼ 4 W	8 W	Trapping

Table 3 . 3 :

 33 Summary of the lasers -Frequency requirements

	Laser	FWHM required [kHz] FWHM reached [kHz]	Frequency
	Master laser	< 5 kHz	∼ 3.3 kHz	𝐷 2 , 𝐹 = 2 → 𝐹 ′ =
	Repumping laser	< 50 kHz		𝐷 2 , 𝐹 = 1 → 𝐹 ′ =
	Ramans laser	< 50 kHz	< 5 kHz

Table 4 . 1 :

 41 Summary of the typical 2D MOT and 3D MOT parameters.

	2D MOT pressure	9.5 × 10 -8 mbar
	2D MOT beam intensity	0 mW cm -2 to 3 mW cm -2 per
		beam
	2D MOT magnetic field gradient	8.75 G cm -1
	Chamber pressure	2.5 × 10 -9 mbar
	MOT beam intensity	0 mW cm -2 to 4 mW cm -2 per
		beam
	MOT beam detuning after optical molasses	12𝛤
	MOT magnetic field gradient on axis	0 G cm -1 to 40 G cm -1
	Characteristic MOT loading rate	1.3 × 10 9 atoms per second
	Characteristic steady-state number of atoms in	5 × 10 10 atoms
	MOT
	MOT size before transport (𝜎 𝑎𝑣 )	∼ 500 µm
	Temperature before optical molasses	40 µK to 60 µK
	Temperature after optical molasses	6 µK to 10 µK

Table 5 . 1 :

 51 Hierarchy of the importance of experimental parameters on the EIT signal.

		Parameter	Importance of parameter effect
		Magnetic field amplitude and orientation	+++
	Done	Stray electric fields Scanning duration	+ +
		Locking beam lightshift	-
		Transport-related issue	-
	To do	Optical pumping Control Laser power and duration	+++ ++
		Probe Laser power and duration	++

Table 5 . 2 :

 52 Theoretical and measured Zeeman shift for 𝜎 + , 𝜎 -and 𝜋 polarized pumping.Magnetic field \Pump𝜎 + polarization 𝜎 -polarization 𝜋 polarization OD 𝜎 + /OD 𝜎 -

	B=1 G	1.07 MHz G -1	0.79 MHz G -1	0.94 MHz G -1	3.73
	B=2 G	1.07 MHz G -1	0.86 MHz G -1	0.91 MHz G -1	4.17
	B=4 G	1.02 MHz G -1	0.86 MHz G -1	0.86 MHz G -1	5.65
	Theoretical expected value	1.4 MHz G -1	0.47 MHz G -1	0.93 MHz G -1	15

Table 5 . 3 :

 53 Theoretical and measured Zeeman shift for 𝜎 + , 𝜎 -and 𝜋 polarized pumping.

	Zeeman pump polarization Theoretical shift [MHz] Measured shift [MHz]

Sigma Aldrich

2 Agilent Technology Vaclon Plus 55, 50 l s -1

(a) Schematic of the initial version of the 2D MOT glass cell. (b) Cut-out of the current 2D MOT glass cell.The frame holding the glass windows is shown in plain grey color, while the frame used to hold the optics is semi-transparent grey.

Qubig, EO-F100L3

control box ALS-CB-ACC

SolsTis ECD-X

The assembly of the cavity described in this part was done by J. Vaneecloo during the span of his Master thesis project.

Nanometer linear positioner SLC-1720

Windfreaktech SynthHD, 15 dBm at 6.7 GHz

Adafruit ADA2045 driven by an Arduino Uno, 8 dBm at 134MHz

MiniCircuits ZX05-153+

MiniCircuits ZX60-8008E+

A Schematics of Setup