Keywords: Intensity correlation measurement

A single photon filter using Rydberg excitations . . . . . . .

encouraging me in my pursuit of an academic career. I am very thankful to the foundation of École Polytechnique for sponsoring my master studies and especially François Hache for being a great mentor during the course of two years. It was a tough decision to come to a non-Anglophonic country with a very knowledge. But it turned out to be one of the best decisions I have every made, I consider myself quite lucky to be trained under some of the best professors in the world. In addition, I've met some of the nicest people both inside and outside the university and made some very good friends. I am very grateful to Philippe Grangier for giving me an opportunity to work on such an interesting experiment. I am very honored to work under Philippe whose genuine enthusiasm and enormous experience was necessary to keep our spirits up during the course of the three years of my PhD. Also the weekly meetings with Philippe, Andrey and Etienne helped me a lot to keep up with different theoretical models as well as the current progress in the field. I would like to appreciate the time taken by Michel Brune, Aurélien Romain Dantan, Jean-François Roch and Alberto Bramati for being part of the jury. I am really grateful to them for evaluating my work and for their invaluable comments on my thesis. I am quite fortunate to be part of such well versed experimental physicists like Alexei Ourjourmtsev and Erwan Bimbard. With the determined guidance and knowledge of both of them, I understood a great deal of both theoretical and experimental aspects of quantum optics. When I came to work at the institute as an internship student in 2013, the experiment seemed daunting but I felt very much at home with their guidance and patience in welcoming me. There were a number of days we spent late in the evening enthusiastically trying to fix various problems on the experiment which motivated me to further continue working on it for my thesis.

During the final days of the departure of Erwan and Alexei, Imam Usmani joined our team as a postdoc and was of a great help to fill the void. His distinct approach to solving problems and his enormous enthusiasm for sports was a welcoming change. We spent countless days struggling in the lab to make everything work. I would also like to thank Senka, who joined our team about an year ago and made an extensive contribution for the installation of the new cavity system. Her hardworking nature and her organization skills were an important addition to our team. The experiment is now in the hands of Imam and Senka, I wish you both the very best in the future. I would also like to acknowledge the quantum optics team for making my stay at Institut d'Optique so much more enjoyable. I am very grateful to Florence Nogrette, her technical expertise on many things related to the experiment were outstanding.

In addition, her patience when it comes to listening to my terrible French. Merci Florence, Imam, Martin etc. Vous m'avez beaucoup aidé à apprendre la langue, la culture, l'histoire Française. I really appreciate the help of permanent members -Antoine Browaeys, Yvan Sortais, Thierry Lahaye et Rosa Tualle-Brouri who were always there to ansmer my questions about Physics or even about French culture.

I would also like to thank Martin and Sylvain for the motivation to play basketball, a welcoming break from continuously working (Even though we lost most of the matches but the very few times we won was pretty great); Stephen Jennewein for always being there whenever I need to rant about the French bureaucracy; Mauro Persechino for his discussions on how to cook Italian food; Henning for his ever smiling face in the group. In addition, I would like to thank various members of the team with whom I interacted during my time at the institute Daniel Barredo, Guillaume Boucher, Melissa Ziebell, Bashkar Kanseri, Ludovic Brossard, Vincent Lienhard, Nicolas Vitrant, Jean Etesse, Sylvain Ravets, Tom Peyrot and various other people at Institut d'Optique. I know that you are all capable of wonderful things and I will always cherish the time spent with you guys. I wish you all good luck for your future in science and may the force be with you.

I would like to extend my gratitude to André Guilbaud, Patrick Roth, André Villing and Frédéric Moron from the mechanical and the electrical workshop who were very welcoming and always available whenever we needed them. I would also like to thank the administrative department both at the Institut d'Optique and at the École doctorale -Martine Basset, Nathalie Baudry, Charline Joli. It was also quite fun to work on something completely unrelated to science with Nicole Bidoit and her team for the mission doctorale.

Finally, I would like to thank all my friends from Paris: Satya, Anirudh, Kanna, Carolyna, Sviatoslav, Inna. I really enjoyed the countless number of occasions discussing about broad range of topics both inside and outside science. The things I learned from them made me a better person today. My best friend, Mafalda, I have no words to tell you how much I appreciate your presence during the course of my PhD. I have to thank you from the bottom of my heart for all your love and patience. You are my inspiration and you complete me.

Contents Table of contents v List of Figures xi

List of Tables xiii 1 Introduction

I Theoretical tools Chapter 1

Introduction

According to Moore's law, the number of transistors per cm 2 would increase expo- nentially by doubling every 1 to 2 years [1]. Soon enough we will reach a fundamental boundary where the storage of information can no longer be explained by classical laws [2]. The strange effects in the physics of individual quantum objects will start to play role in how the electrons behaves. Hence, it's necessary to understand how we can take these effects into consideration and go beyond the classical systems.

Quantum computation is a way of exploiting the special properties of the quantum world to speed up computational process. Richard Feynman speculated that quantum systems can be used to build advanced simulators [3]. This idea has attained a significant interest since Shor developed a quantum algorithm to factorize a large number (N) [4]. Shor's quantum factorization method can run in polynomial time in log(N) whereas the best classical method (general number field sieve) scales exponentially in log(N) [5]. The experimental realization of quantum simulators or computers has motivated many research groups. Numerous interesting algorithms have been proposed but their experimental implementation in large scales is still a long standing goal.

Quantum Computation

In computers, the information is stored in the form of zeros and ones known as bits. The smallest units of a quantum computing machine are known as quantum bits, also referred to as qubits [6]. Unlike classical bits, each qubit can exist in superposition states and qubits can be entangled with one another. This enables quantum computers to perform calculations in a vast Hilbert space at higher speed.

These properties give quantum computers an unprecedented advantage over their classical counterpart.

A quantum computer cannot be built like a classical one as it requires a new technology which can enable us to store information in superposition states. There has been a considerable progress in the last decade to make quantum computation a reality. There are many quantum systems such as neutral or charged atoms, superconducting qubits, quantum dots etc., which have many characteristic traits to become a component of a quantum computer. Currently, it is possible to en-gineer quantum states like Schrödinger cat states using optical systems [START_REF] Ourjoumtsev | Generation of optical 'Schrodinger cats' from photon number states[END_REF][START_REF] Vlastakis | Deterministically Encoding Quantum Information Using 100-Photon Schrödinger Cat States[END_REF] and microwave photons [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF][START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF]. However, in most of these schemes realizing a system which has a strong isolation from environment and on demand 'on/off ' control over interactions is a technical challenge. Some companies like 'D-wave Systems' claim they invented the first quantum computer, but they use "quantum annealing" for adjusting interactions to shape the final collective quantum state. Upto now this type of approach cannot be scaled up efficiently without overcoming environmental noise.

Modular quantum networks

Scaling up quantum systems without losing their coherence is an ambitious task.

The conventional way of connecting every qubit to one another is likely to fail due to buildup of errors. Algorithms like quantum error correction require more qubits to correct for errors which in turn increases the complexity of the system.

A promising solution is distributed quantum information processing or modular networks, where the quantum nodes are connected by networks of channels [START_REF] Moehring | Quantum networking with photons and trapped atoms (Invited)[END_REF][START_REF] Monroe | Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects[END_REF].

Instead of entangling all the qubits with one another, few qubits form a node which makes it easier to manage them [START_REF] Monroe | Quantum Connections Scientific[END_REF]. Our group's approach is to use cavity enhanced neutral atoms with photons as carriers. In

Photons as flying qubits

For the past decade, optical fibers have dominated as information communication channels. Photons are robust against environmental noise which makes them ideal candidates for information transfer [START_REF] Caulfi | Why future supercomputing requires optics[END_REF]. They can be transported with ease over long distances and can be incorporated into existing communication channels. Photons are also relatively easy to produce using many systems like spontaneous parametric down-conversion or cold atomic memories. Several existing methods such as Hanbury Brown and Twiss effect or homodyne tomography help us to easily characterize the quality of these photons [START_REF] Lvovsky | Continuous-variable optical quantum-state tomography[END_REF][START_REF] Leonhardt | Quantum-state tomography and discrete Wigner function[END_REF].

In conventional optical fiber systems, it is common to use optical amplifiers to boost the signal to transport it over long distances. For quantum networks, a specific system, the quantum repeater, has been designed to counteract the decoherence effects introduced during transmission over long distances. One cannot simply detect or amplify the signal akin to classical communication. A quantum repeater system transports information without actually measuring it. The standard approach involves a source of entangled photons and a quantum repeater node where logical operations take place. In order to project the signal state onto one of the entangled photon, a projective measurement operation is carried out between the signal photon and the other entangled one. In addition, by using an entanglement purification process between different nodes, photonic states can be transferred over long distances with high fidelity [START_REF] Song | Heralded quantum repeater based on the scattering of photons off single emitters using parametric downconversion source[END_REF][START_REF] Sangouard | Quantum repeaters based on atomic ensembles and linear optics[END_REF].

Quantum computation using photons

The main building blocks of quantum networks are non-classical states and coherent manipulation techniques. A gate operation is necessary for the realization of a full scale computational system [START_REF] Divincenzo | Two-bit gates are universal for quantum computation[END_REF]. One commonly used multi-bit gate is a controlled phase gate where one of the input bits acts as a control. The operations of a controlled phase gate are illustrated in Figure 1. In 2001, Knill et. al. [START_REF] Knill | A scheme for efficient quantum computation with linear optics[END_REF] proposed a scheme where single photons coupled with linear elements (like beam splitters, phase shifters) can be used for quantum information protocols. However its experimental implementation demands highly efficient single photon sources and counters. In addition, they are inherently probabilistic [START_REF] Adami | Quantum Computation with Linear Optics[END_REF]. Therefore, we resort to photon-photon nonlinearities for realization of quantum gates.

Since photons interact weakly with one another, it is a challenging task to observe non-linearities at few photon level. To achieve deterministic quantum interactions, we need strongly non-linear systems [START_REF] Lukin | Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons[END_REF][START_REF] Lukin | Trapping and manipulating photon states in atomic ensembles[END_REF]. For example, if we need an infrared photon wave packet (with a bandwidth of 1 MHz) to attain an optical phase shift of 𝜋 by propagating in a medium of length equal to Rayleigh range of the photon then the medium's third order susceptibility(𝜒 (3) ) has to be greater than 10 -3 V -2 m 2 [START_REF] Pritchard | Nonlinear optics using cold Rydberg atoms Annual Review of Cold Atoms and Molecules[END_REF]. But conventional optical medias exhibit extremely weak optical susceptibilities and the phase shift obtained is twenty orders of magnitude lower than the desired value [START_REF] Pritchard | Nonlinear optics using cold Rydberg atoms Annual Review of Cold Atoms and Molecules[END_REF]. Strong Kerr non-linear effects can be realized by using resonant optical medias like cold atoms or trapped ions [START_REF] Häffner | Quantum computing with trapped ions[END_REF].

Cold atoms

Quantum optical non-linearities using atomic ensemble as an intermediary medium has gained a significant interest since the proposal by Jaksch et.al. [START_REF] Jaksch | Fast quantum gates for neutral atoms[END_REF]. Cold atoms have well established trapping techniques along with tremendous control over its states which makes them promising candidates for quantum computers.

Information can be stored in 'built-in' internal states of atoms which possess long coherence times. In addition, it is possible to use optical lattices to scale up neutral atomic systems [START_REF] Bloch | Ultracold quantum gases in optical lattices[END_REF]. Dipole-dipole interactions between atoms can influence the properties of neighboring atoms and can give rise to collective cooperative behavior like super-radiance [START_REF] Dicke | Coherence in Spontaneous Radiation Processes[END_REF]. These interactions can be only observable in a very dense atomic ensembles like BEC, etc. Our main objective is to enhance these interactions to achieve quantum optical non-linearity in atomic clouds. Until now, the two main promising approaches to strong photon-photon interactions are Electromagnetically Induced Transparency (EIT) and cavity Quantum Electro-Dynamics (cQED) [START_REF] Duan | Photonic Quantum Computation through Cavity-Assisted Interactions[END_REF].

To enhance the optical non-linearity one can couple a two-level system to a high finesse optical cavity. The study of this strong coupling between the atoms and cavity is referred to as cQED. Many interesting results have been obtained using cavity QED systems since the demonstration of quantum state manipulation using microwave cQED [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF][START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF].

Two level atomic media can exhibit strong non-linear response close to resonance, but the photons are lost by scattering. In 1990, Harris et al. proposed that by adding an additional coupling transition, the medium can be rendered transparent while enhancing the optical susceptibilities [START_REF] Harris | Imamoglu A Nonlinear optical processes using electromagnetically induced transparency[END_REF][START_REF] Schmidt | A Giant Kerr nonlinearities obtained by electromagnetically induced transparency[END_REF]. In addition, the group velocity of the light is strongly reduced which allows us to store light as polaritons [START_REF] Heinze | Stopped Light and Image Storage by Electromagnetically Induced Transparency up to the Regime of One Minute[END_REF]. The reported susceptibilities are still not sufficient to achieve a phase shift of 𝜋 at few photon level.

Our approach is to use long range dipole-dipole interactions between collective atomic excitations to achieve quantum optical non-linearities. In the next section, we present how one can tune the dipole-dipole interaction strength to achieve few photon non-linearities.

Strong photonic non-linearities via Rydberg EIT

The idea is to map the photons onto highly excited atomic states called Rydberg states and then use the long range dipole-dipole interactions between them to enable photon-photon interaction. The idea of using Rydberg atoms has generated substantial interest since the proposal of non-classical state generation using the 'dipole blockade' mechanism in mesoscopic ensembles [START_REF] Lukin | Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons[END_REF]. However, exciting them directly to Rydberg states using only optical light is inefficient because of their weak transition dipole moments. In 2005, Friedler et. al. proposed exciting to Rydberg states using EIT, where a two photon excitation scheme is used to render the medium transparent [START_REF] Friedler | Long-range interactions and entanglement of slow single-photon pulses[END_REF]. Prior to this, most EIT experiments were carried out using a second ground state as metastable state. They showed that two photon pulses can be converted to dark state polaritons which interact via dipole-dipole interactions and acquire a phase shift of 𝜋 to realize a photonic gate. Moreover, Rydberg atoms in optical lattices can be used as a simulator for many body interaction systems [START_REF] Weimer | Rydberg quantum simulator[END_REF].

Since then, a number of interesting experiments have been carried out to convert photonic excitations to Rydberg polaritons starting with the group of Charles Adams at University of Durham [START_REF] Pritchard | Optical nonlinearity in a dynamical Rydberg gas[END_REF] during 2010-13. They demonstrated optical non-linear effects in classical regime using free-space atomic systems. More recently, non-linearities at quantum level have been demonstrated by creating a single photon source [START_REF] Dudin | Strongly Interacting Rydberg Excitations of a Cold Atomic Gas[END_REF], a photon blockade where only single photons are transmitted [START_REF] Peyronel | Quantum nonlinear optics with single photons enabled by strongly interacting atoms[END_REF],

photon bunching in dispersive regime [START_REF] Firstenberg | Attractive photons in a quantum nonlinear medium[END_REF]. A Rydberg based photon switch where a π μ π

Thesis Layout

This thesis is a summary of the work carried out at the Institut d'Optique in an attempt to observe quantum optical non-linearities. We use a cold atomic cloud trapped in the mode of a low finesse optical cavity. This system has been chosen

for its ability to combine the advantages of cavity and atomic cloud systems. This manuscript is organized in the following manner

• In part 1, we establish a standard notation for the computation of various relevant parameters of our system. We show why optical interactions induced by two level and three level non-interacting EIT systems is not sufficient to move to the quantum regine. We also introduce two models to explain Rydberg induced optical non-linearities in the classical regime.

• In part 2, we describe various parts of our experimental setup, and present the necessary lasers and hardware required to trap atoms. We conclude with various atomic cloud characterization methods.

• In part 3, we begin with various detection methods available on our setup and go on to present the Rydberg non-linearity measurements for S and D states. We also describe higher order correlation effects predicted for the measured non-linearity. We conclude the chapter with second order correlation measurements using a Hanbury Brown Twiss setup and squeezing spectrum measurements using a homodyne setup.

• In part 4, we present how a new high finesse cavity with a small mode-waist designed to move towards the quantum regime. We describe the new cavity design and its properties. We include details on how the new cavity is mounted and characterized on our setup. We present the recent EIT measurements along with some numerical calculations of squeezing spectrum expected with the new setup. We also present some theoretical ideas which could be imple- In this chapter we will present the framework to describe our atom-cavity system.

We will establish basic notions of light propagation through a cavity enhanced cold atomic media by deriving the expressions for susceptibility. We will show how to quantify the non-linearity arising from two-level and multi level atoms. Finally, we show why one needs to go beyond these systems to have quantum optical nonlinearities.

Light propagation in a dielectric medium

Here we introduce expressions for a classical field propagating in a dielectric medium.

The optical waves propagate through a dielectric medium by continuous absorption and re-emission of wave energy by atoms in the medium. This light-matter interaction can be quantified by defining a parameter called polarizability of the medium.

It is related to the induced polarization ⃗ 𝑃 (⃗ 𝑟, 𝑡) by the propagating electromagnetic wave ⃗ 𝐸(⃗ 𝑟, 𝑡) [START_REF] Boyd | Quantum-Mechanical Theory of the Nonlinear Optical Susceptibility[END_REF] ⃗ 𝑃 (⃗ 𝑟, 𝑡)

= 𝑡 ∫ -∞ 𝑑𝑡 ′ 𝛼 1 (𝑡 -𝑡 ′ ) ⃗ 𝐸(⃗ 𝑟, 𝑡 ′ )𝑑𝑡 ′ + 𝑡 ∫ -∞ 𝑡 ∫ -∞ 𝑑𝑡 ′′ 𝛼 2 (𝑡 -𝑡 ′ , 𝑡 -𝑡 ′′ ) ⃗ 𝐸(⃗ 𝑟, 𝑡 ′ ) ⃗ 𝐸(⃗ 𝑟, 𝑡 ′′ ) + ... (2.1)
Where 𝛼 𝑖 is the polarizability of the medium of order i. This induced polarization in turn affects both the phase and the amplitude of the propagating electromagnetic waves. If we expand the incoming field in the frequency domain, then the polarization can be written as

𝑃 (𝜔) = 𝜀 0 𝜒(𝜔)𝐸(𝜔) (2.2) 𝜒(𝜔) = 𝜒 1 (𝜔) + 𝜒 2 (𝜔)𝐸(𝜔) + 𝜒 3 (𝜔 = 𝜔 1 + 𝜔 2 )𝐸 1 (𝜔 1 )𝐸 2 (𝜔 2 )... (2.
3)

The real part of the susceptibility contributes to the dispersion of light, and the imaginary part to the absorption of light. From the above expression we can evaluate the effect of matter on light using the the expression 𝑛 = √ 1 + 𝜒. Most materials and gases exhibit centro-symmetry and it can be shown that the second order term vanishes in these media. In linear systems, one can neglect all the higher order terms and the susceptibility term is independent of the field strength.

In non-linear systems, the third order term is the lowest contributing one and the susceptibility depends on the incoming field's intensity. In the following sections we will describe how one can evaluate the susceptibility term in atomic systems.

Semi-classical approach of the light-atom system

A monochromatic wave incident on an atomic medium close to the resonance of twolevels can be described by a semi-classical approach, where the atoms are considered as two level quantum systems. We denote the two levels of the atom as g (ground state) and e (excited state). The energy difference between the levels is represented by 𝐸 𝑒 -𝐸 𝑔 = 𝜔 𝑔𝑒 . We introduce atomic operators that are denoted by σ(𝑛) 𝑘𝑙 = |𝑘⟩⟨𝑙| for atom 𝑛. The atomic Hamiltonian for atom 'n' can then be denoted by ℋ

(𝑛) 𝑎 = 𝜔 𝑔𝑒 (σ (𝑛) 𝑒𝑔 σ(𝑛)
𝑔𝑒 ). The light-atom system can be described by evaluating

ℋ 𝑠 = ℋ 𝑎 + ℋ 𝑖 (2.4)
As we treat light classically ( ⃗ 𝐸(𝜔) = ℰ 0 ⃗ 𝑢 (𝑒 𝑖𝜔𝑡 + 𝑒 -𝑖𝜔𝑡 )), we consider that the only role of light is to create an interaction potential. An atom driven close to the resonance of an optical transition can be considered as an oscillating dipole. The transition moment operator of the atom 'n' for |𝑔⟩ → |𝑒⟩ is given by

⃗ 𝑑 (𝑛) = 𝑑 0 ⃗ 𝑢(σ (𝑛) 𝑔𝑒 + σ(𝑛) 𝑒𝑔 )
(2.5)

I 0 I ω ω ge χ |g〉 |e〉 γ Figure 2
.1: Two-level excitation scheme: Each atom is considered as a two-level system with ground state 'g' and excited state 'e'. An optical beam with intensity I 0 is incident on the atomic media with frequency 𝜔, and we denote the susceptibility of the atoms as 𝜒. The energy difference between the two levels is denoted by 𝜔 𝑔𝑒 . By using the rotating wave approximation 1 , one can neglect the rapidly oscil- lating terms and rewrite the interaction Hamiltonian as

ℋ 𝑖𝑛𝑡 = ∑ 𝑛 -⃗ 𝐷 (𝑛) . ⃗ 𝐸(𝑟 𝑎 ) = ∑ 𝑛 𝑑 0 ℰ 0 ( 𝑒 -𝑖𝜔𝑡 σ(𝑛) 𝑔𝑒 + 𝑒 𝑖𝜔𝑡 σ(𝑛) 𝑒𝑔 )
. The total Hamiltonian of the system can be expressed as

Ĥ𝑠 = ∆ 𝑒 𝑁 ∑ 𝑛 σ(𝑛) 𝑒𝑒 + ∑ 𝑛 Ω [ 𝑒 -𝑖𝜔𝑡 σ(𝑛) 𝑔𝑒 + 𝑒 𝑖𝜔𝑡 σ(𝑛) 𝑒𝑔 ] (2.6)
where Ω = -(𝑑 0 ℰ 0 )/ , 𝑁 is total number of atoms and ∆ 𝑒 = 𝜔 -𝜔 𝑔𝑒 . One can then use Heisenberg equations to evaluate the evolution of any operator  using [START_REF] Walls | Quantum Optics Walls and Milburn[END_REF].

𝑑 𝑑𝑡 Â = 𝑖 [ Ĥ𝑠 , Â] -𝛾 Â Â + 𝐹 𝑎 (2.7)
where 𝛾 𝑒 is the decay rate of the excited state and 𝐹 𝑎 is the Langevin noise operator associated with the operator Â. Using the steady state approximation, we obtain the following expression for the average value of the coherence term (σ 𝑔𝑒 ) ⟨ σ(𝑛)

𝑔𝑒 ⟩ = 𝑖Ω( ⟨ σ(𝑛) 𝑒𝑒 ⟩ - ⟨ σ(𝑛) 𝑔𝑔 ⟩ ) 2(𝛾 𝑒 + 𝑖∆ 𝑒 ) = - 𝑖Ω 2(𝛾 𝑒 + 𝑖∆ 𝑒 ) 1 (1 + Ω 2 2(𝛾 2 𝑒 +Δ 2 𝑒 ) ) ; = Ω 2 ∆ 𝑒 -𝑖𝛾 𝑒 𝛾 2 𝑒 + ∆ 2 𝑒 + Ω 2 /2 (2.8)
where

𝛾 𝑒 = 𝑑 2 0 𝑘 3 6𝜋𝜀 0
is half of the natural linewidth of the excited state.

We can define an average coherence term using σ𝑔𝑒 =

𝑃 (𝜔) = 𝜌⟨ D⟩ = 𝜌𝑑 0 ⟨ σ𝑔𝑒 𝑒 -𝑖𝜔𝑡 + σ𝑒𝑔 𝑒 𝑖𝜔𝑡 ⟩ = 1 2 𝜀 𝑜 ℰ 0 ( 𝜒(𝜔)𝑒 -𝑖𝜔𝑡 + 𝜒 * (𝜔)𝑒 𝑖𝜔𝑡 ) (2.9)
which gives the following expression for the atomic susceptibility

𝜒(𝜔) = 𝜌𝑑 2 0 𝜀 0 ⟨σ 𝑔𝑒 ⟩ Ω/2 = - 𝛼 0 𝛾 𝑒 𝑘 ⟨σ 𝑔𝑒 ⟩ Ω/2
(2.10)

1 Complete derivation can be found in many standard textbooks, for example [START_REF] Puri R R | The Electromagnetic Field Mathematical Methods of Quantum Optics[END_REF] Chapter 2. Theoretical tools for Atom-Light coupling

In the above equation we used the standard expressions for resonant optical crosssection for a two-level atom 𝜎 eff = 3𝜆 2

2𝜋

and 𝛼 0 = 𝜌𝜎 eff [START_REF] Keaveney | Collective Atom -Light Interactions in Dense Atomic Vapours[END_REF]. The optical susceptibility is proportional to the term ⟨σ 𝑔𝑒 ⟩ /Ω ∝ (∆ 𝑒 -𝑖𝛾 𝑒 )/2(𝛾 2 𝑒 + ∆ 2 𝑒 ) in the weak feeding approximation (𝛾 2

𝑒 + ∆ 2 𝑒 ≫ Ω 2 ).
It is worth noting that the susceptibility doesn't depend on input intensity, hence the response of the system is linear in this regime.

Classical field in an optical cavity

We consider a linear cavity of length 'L' with an enclosed atomic medium. We represent the reflectivities of its mirrors by 𝑅 1 = 1 -𝑇 1 ≈ 1 and 𝑅 2 = 1 -𝑇 2 and its resonant frequency by 𝜔 𝑐 . The normalized transmission of the cavity is given by

𝑇 1 𝑇 2 |(1- √ 𝑅 1 𝑅 2 𝑒 𝑖Δ𝑐/𝜔 𝑓 )| 2
where 𝜔 𝑓 = 𝑐 If we have a medium of length 𝑙 inside the cavity with refractive index 𝑛 = √ 1 + 𝜒 ≈ 1+𝜒/2, then the light acquires a complex phase 𝑒 𝑖 𝜋 𝜆 𝜒𝑙 per single pass. The transmission through the cavity in this case can be written as

𝑇 1 𝑇 2 |(1- √ 𝑅 1 𝑅 2 𝑒 𝑖Δ𝑐/𝜔 𝑓 𝑒 𝑖 2𝜋 𝜆 𝜒𝑙 )| 2
which can be approximated to

𝑇 = 𝐼 𝑡 𝐼 𝑖 ≈ 𝑇 1 𝑇 2 (1 - √ 𝑅 1 𝑅 2 + 2𝜋 𝜆 ℑ(𝜒)𝑙) 2 + 𝑅 1 𝑅 2 sin 2 (∆ 𝑐 /𝜔 𝑓 + 2𝜋ℜ(𝜒)/𝜆𝑙) ≈ 4𝑇 1 /𝑇 2 (1 + 4𝜋 𝑇 2 𝜆 ℑ(𝜒)𝑙) 2 + (∆ 𝑐 /𝛾 𝑐 + 4𝜋ℜ(𝜒)/𝑇 2 𝜆𝑙) 2 (2.11)
where 𝛾 𝑐 = 𝑇 2 𝜔 𝑓 /2 by using 𝑅 1 , 𝑅 2 ≈ 1 and |𝜒| ≪ 𝜆/2𝜋𝑙. In the case of atoms trapped in the cavity mode, by using the Eq. 2.10 the cavity transmission can be rewritten as

𝑇 = 𝑇 0 ( 1 -4𝛼 0 𝑙𝛾𝑒 𝑇 2 Ω ℑ(⟨σ 𝑔𝑒 ⟩) ) 2 + ( Δ𝑐 𝛾𝑐 -4𝛼 0 𝑙𝛾𝑒 𝑇 2 Ω ℜ(⟨σ 𝑔𝑒 ⟩) ) 2 = 𝑇 0 ( 1 -2𝐶 Ω/2𝛾𝑒 ℑ(⟨σ 𝑔𝑒 ⟩) ) 2 + ( Δ𝑐 𝛾𝑐 -2𝐶 Ω/2𝛾𝑒 ℜ(⟨σ 𝑔𝑒 ⟩) ) 2 = 𝑇 0 ( 1 - 2𝐶 1+Δ 2 𝑒 /𝛾 2 𝑒 +Ω 2 /2𝛾 2 𝑒 ) 2 + ( Δ𝑐 𝛾𝑐 - 2𝐶Δ𝑒/𝛾𝑒 1+Δ 2 𝑒 /𝛾 2 𝑒 +Ω 2 /2𝛾 2 𝑒 ) 2 (2.12)
where 𝑇 0 = 4𝑇 1 /𝑇 2 and 𝐶 = 𝛼 0 𝑙 𝑇 2 is the ratio of the absorption of the media (also known as optical depth) to the transmission of the lossy mirror. The term C is commonly referred to as the cooperativity of the medium and corresponds to a collective phenomena exhibited by the atoms.

As shown in Figure 2.2, the cavity transmission contains two normal modes when the detuning of the probe light is scanned around the atomic resonance. The position of the modes corresponds to those detunings for which the second term in the denominator goes to zero. In the weak feeding approximation, for large cooperativities the modes are separated by √ 8𝐶𝛾 𝑐 𝛾 𝑒 when the cavity is resonant with the atoms.

Quantum mechanical approach

In order to more accurately describe the system evolution we use a fully quantum mechanical approach to describe the light-atom coupling. We quantize the electromagnetic field in the cavity mode in the Schrödinger picture. By using the second quantization, we can describe the Hamiltonian of the single mode light field by

ℋ 𝑙 = 𝜔 𝑐 ( â † â + 1 2 ) (2.13)
where 𝜔 𝑐 is the cavity resonance frequency. We restrict ourselves to excitations only in the fundamental mode of the cavity. We can then write the quantized electromagnetic field operator in a cavity by using the boundary conditions for a cavity placed along the z axis [START_REF] Saleh | Photon Optics Fundamentals of Photonics[END_REF] Ê(𝑥, 𝑦, 𝑧) = ⃗ ℰ(𝑥, 𝑦, 𝑧)

[ â𝑒 𝑖𝑘𝑧 + â † 𝑒 -𝑖𝑘𝑧 ] = ⃗ ℰ(𝑥, 𝑦, 𝑧) ( cos(𝑘𝑧) [ â + â † ] + 𝑖 sin(𝑘𝑧) [ â -â † ]) (2.14)
where 𝑘 = 𝜔 𝑐 /𝑐. We also assume that all the atoms are restricted within the Rayleigh range of the cavity mode, and hence the waist is assumed to be constant 𝜔(𝑧) ≈ 𝜔 0 . The electric field term can be written as ⃗ ℰ(𝑥, 𝑦, 𝑧) = ℰ 0 ⃗ 𝑢𝑒

-𝑥 2 +𝑦 2 2𝑤 2 0 ,
where

ℰ 0 = √ 𝜔𝑐 2𝑉 𝜀 0
is the electric field per photon and the mode volume is 𝑉 = ∫ cos 2 (𝑘𝑧)𝑒

-𝑥 2 +𝑦 2 2𝑤 2 0 𝑑 3 𝑟 .
Our system is composed of N atoms interacting with a single cavity mode. One can describe such a system using the Dicke model [START_REF] Dicke | Coherence in Spontaneous Radiation Processes[END_REF], where the atomic dipoles are coherently interacting with the electromagnetic field. The cavity is pumped with a coherent feeding rate 𝛼. Here we work in the open cavity limit (𝜆 ≪ 𝐿), where the spontaneous emission rate (𝛾 𝑒 ) for an atom is close to its value in free space. The total Hamiltonian is given by

Ĥ𝑠 = Ĥ𝑙 + Ĥ𝑎 + Ĥ𝑖𝑛𝑡 + Ĥ𝑓 = ∆ 𝑐 â † â + ∑ 𝑛 ∆ 𝑒 σ(𝑛) 𝑒𝑒 - ∑ 𝑛 𝑔(𝑟 𝑛 ) [ âσ (𝑛) 𝑔𝑒 + â † σ(𝑛) 𝑒𝑔 ] + 𝛼(â + â † ) (2.15)
In the above expression, the position dependent atom-light coupling term is given by 𝑔(𝑟

𝑛 ) = 𝑔 0 𝒮(𝑟 𝑛 ) where 𝑔 0 = 𝑑 0 √ 𝜔𝑐 2 𝜀 0 𝑉
is the vacuum Rabi frequency term and the function 𝒮(𝑟 𝑛 ) = cos(𝑘𝑧)𝑒

-𝑥 2 +𝑦 2 2𝑤 2 0
depends on position of each atom 𝑟 𝑛 .

Optical Bloch equations

The dynamics of the system can be described by using the density matrix formalism (Schrödinger picture) or by Heisenberg-Langevin equations (Heisenberg picture).

As our system exhibits dissipation, it is easier to describe using master equation in the Lindblad form. We must take into account both the dissipation of the cavity (â)

and atoms σ𝑔𝑒 to establish the evolution of the system. The evolution of the density matrix(𝜌 𝑠 ) can be expressed using Liouvillian terms (ℒ( Â) 𝜌𝑠 ) in the following form

ρ𝑠 = - 𝑖 [ Ĥ𝑠 , ρ𝑠 ] -𝛾 𝑐 (â † âρ 𝑠 + ρ𝑠 â † â -2âρ 𝑠 â † )
-𝛾 𝑒 (σ 𝑔𝑒 σ𝑒𝑔 ρ𝑠 + ρ𝑠 σ𝑔𝑒 σ𝑒𝑔 -2σ 𝑒𝑔 ρ𝑠 σ𝑔𝑒 )

⟨ Â⟩ = 𝑖 ⟨[ Ĥ𝑠 , Â]⟩ -𝛾 ⟨ Â⟩ Â [51]
. The evolution of the intracavity field and the coherence terms can be calculated using

𝑑 𝑑𝑡 ⟨â⟩ = (𝑖∆ 𝑐 -𝛾 𝑐 )⟨â⟩ + ∑ 𝑛 𝑔(𝑟 𝑛 )⟨σ (𝑛) 𝑔𝑒 ⟩ -𝑖𝛼 (2.17) 𝑑 𝑑𝑡 ⟨σ (𝑛) 𝑔𝑒 ⟩ = (𝑖∆ 𝑒 -𝛾 𝑒 )⟨σ (𝑛) 𝑔𝑒 ⟩ + 𝑖𝑔(𝑟 𝑛 )⟨â(σ (𝑛) 𝑒𝑒 -σ(𝑛) 𝑔𝑔 )⟩ (2.18)
where the term 𝛼 denotes the feeding rate into the cavity. In the steady state condition, we observe that ⟨σ (𝑛)

𝑔𝑒 ⟩ ⟨â⟩ = 𝑖𝑔(𝑟 𝑛 ) (∆ 𝑒 + 𝑖𝛾 𝑒 ) 𝑓 (⟨â⟩ 2 ) (2.19) ⟨â⟩ = 𝛼 ( ∆ 𝑐 + 𝑖𝛾 𝑐 -𝑖 ∑ 𝑛 𝑔(𝑟 𝑛 ) ⟨σ (𝑛) 𝑔𝑒 ⟩ ⟨â⟩ ) = 𝛼 ( ∆ 𝑐 + 𝑖𝛾 𝑐 + ∑ 𝑛 𝑔 2 (𝑟𝑛) 𝛾𝑐𝛾𝑒 𝜒 𝑛 ) (2.20)
where 𝜒 𝑛 = 𝛾𝑒 𝑔(𝑟𝑛)

⟨σ (𝑛)
𝑔𝑒 ⟩ ⟨â⟩ is the susceptibility of each atom n. In a linear regime the function f(⟨â⟩

2
) is a constant and can be shown to be equivalent to the semi-classical description for coherent fields.

Quantum mechanical approach

Cavity transmission

The normalized transmission of the cavity is proportional to the intracavity intensity. In steady state, for a weak feeding rate the normalized cavity transmission can be evaluated from

𝑇 = ⟨â⟩ 2 𝛾 2 𝑐 𝛼 2 = 𝑇 0 ( 1 + ℑ( ∑ 𝑛 𝑔 2 (𝑟𝑛) 𝛾𝑐𝛾𝑒 𝜒 𝑛 ) ) 2 + ( Δ𝑐 𝛾𝑐 + ℜ( ∑ 𝑛 𝑔 2 (𝑟𝑛) 𝛾𝑐𝛾 𝐸 𝜒 𝑛 ) ) 2 (2.21) ≈ 𝑇 0 ( 1 -2𝐶 1+Δ 2 𝑒 /𝛾 2 𝑒 ) 2 + ( Δ𝑐 𝛾𝑐 -2𝐶Δ𝑒/𝛾𝑒 1+Δ 2 𝑒 /𝛾 2 𝑒 ) 2 (2.22)
where T 0 is a proportionality constant and 𝐶 = ∑ 𝑛 𝑔 2 (𝑟𝑛) 2𝛾𝑐𝛾𝑒 . It must be noted here that the cooperativity parameter is equivalent to the one we derived in section : 2.2.

It is an important figure of merit across various systems to quantify the coherent atom-light coupling. It is the ratio of the square of the coherent coupling to the product of the dissipation rates (cavity and atoms). The effective volume over an atomic medium of length 𝑙 can be evaluated by We can see that for cooperativities higher than 4

𝑉 = ∫ 𝑑 3 ⃗ 𝑟⟨ ⃗ 𝐸 + (⃗ 𝑟) 2 ⟩ ⟨ ⃗ 𝐸 𝑚𝑎𝑥 (⃗ 𝑟) 2 ⟩ = ∫ ∫ ∫ 𝑑𝑥𝑑𝑦𝑑𝑧 cos 2 (𝑘𝑧)𝑒 -2 𝑥 2 +𝑦 2 𝜔 2 0 = 𝜋 4 𝐿𝜔 2
there is a range of input intensities where the output light is multivalued.

Until now we have dealt with the situation where the susceptibility doesn't depend on the intensity of the incident light. It is a valid approximation for weak intensities, where the excited level population is not saturated. As soon as we reach a regime where

Ω 2 ∼ 𝛾 2 𝑒 + ∆ 2
𝑒 then the susceptibility depends on the intracavity field and vice versa.

We will show how the bistable behavior is originated in absorptive and dispersive regimes. The incident and transmitted light are normalized using saturation intensity (𝐼 𝑠𝑎𝑡 ) and expressed as 𝑋 = 𝐼𝑡 𝑇 2 𝐼𝑠𝑎𝑡 = Ω 2 2𝛾 2

𝑒

and 𝑌 = 𝐼 𝑖 𝐼𝑠𝑎𝑡 [START_REF] Lugiato | Theory of Optical Bistability[END_REF][START_REF] Rosenberger | Absorptive optical bistability in two-state atoms[END_REF]. We can rewrite the stationary transmission functions of the cavity as

𝑋 = 𝑌 [ ( 1 -2𝐶 ℑ⟨σ 𝑔𝑒 ⟩ Ω/2𝛾 𝑒 ) 2 + ( ∆ 𝑐 𝛾 𝑐 -2𝐶 ℜ⟨σ 𝑔𝑒 ⟩ Ω / 2𝛾 𝑒 ) 2 ] -1 (2.23)
where the term ⟨σ𝑔𝑒⟩ Ω/2𝛾𝑒 is a function of 𝑋. If both the cavity and the atoms are on resonance then we are in purely absorptive regime and we can evaluate the probe transmission from

𝑌 = 𝑋 [ ( 1 + 2𝐶 1 + 𝑋 ) 2 ] (2.24)
The system can exhibit bistability when the cooperativity 𝐶 > 4 as shown in Figure On the contrary, if the detunings are much larger than the incident probe intensity (∆ ≫ Ω), then we are in the dispersive regime.

In the latter case the transmission function can be simplified to the following expression when ∆ 𝑒 ≫ 1 and ∆ 𝑒 ≫ Δ𝑐

𝛾𝑐 𝑋 = 𝑌 1 + (𝜃 -2𝐶ℜ𝑓 (𝑋)) 2 (2.25) ≈ 𝑌 1 + (𝜃 -2𝐶 Δ𝑒 + 2𝐶 Δ 3 𝑒 𝑋) 2 (2.26)
where we replaced Δ𝑐 𝛾𝑐 by 𝜃. The above expression is in the form of 𝑌 = 𝑋(1 + (𝐵 -𝐴𝑋) 2 ) where 𝐴 = 2𝐶 Δ 3 𝑒 and 𝐵 = 2𝐶 Δ𝑒 -𝜃. It has more than one solution when 3𝐴 2 𝑋 2 + -4𝐴𝐵𝑋 + 𝐵 2 + 1 = 0. It exhibits bistability when the term B is greater than √ 3. In Figure 2.5(a) we see how the transmitted light varies with the incident probe intensity when 𝐵 = 3 and 𝐴 = 1.

In both cases of absorptive or dispersive bistability one needs to have X > 1 which corresponds to an intracavity intensity comparable to 𝐼 𝑠𝑎𝑡 . To achieve strong non-linearity for few photons, one needs to have a small mode volume and large cavity bandwidth. This can be achievable in the cavity-QED, regime but for our cavity atomic system where the mode-waist is of the order of 10 -100 microns and cavity decay rate of tens of MHz, one needs few thousands of photons to reach non-linear regimes.

Three level atoms

In order to further enhance the optical non-linearities one must move to multi-level systems where phenomena like Electromagnetically Induced Transparency allows us to move closer to resonance, without losing photons by scattering [START_REF] Harris | Imamoglu A Nonlinear optical processes using electromagnetically induced transparency[END_REF][START_REF] Fleischhauer | Electromagnetically induced transparency: Optics in coherent media[END_REF]. In addition to the attenuated absorption window, the steep dispersion curve increases the effective interaction time between the probe pulses and atoms [START_REF] Gea-Banacloche | Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment[END_REF][START_REF] Hau | Light speed reduction to 17 metres per second in an ultracold atomic gas[END_REF]. In the past two decades many groups have attempted, both theoretically and experimentally to lower the threshold to reach quantum regime [START_REF] Harshawardhan | Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences[END_REF][START_REF] Wang | Controlling light by light with three-level atoms inside an optical cavity[END_REF][START_REF] Wang | Nonlinearity via Atomic Coherence in a Three-Level Atomic System[END_REF]. The three level excitation scheme is presented here along with the probe and the control fields.

, Ω p  e  s , Ω s Δ s  ge |g〉 |e〉 |s〉  es  s Ω p Ω s (a) (b)
In this section we will describe the optical response of three level atoms in a cavity.

⟨σ (𝑛) 𝑔𝑒 ⟩ = 1 𝐷 𝑒 ( Ω 𝑠 2 ⟨σ (𝑛) 𝑔𝑠 ⟩ -𝑔⟨â(σ (𝑛) 𝑒𝑒 -σ(𝑛) 𝑔𝑔 )⟩ ) ⟨σ (𝑛) 𝑔𝑠 ⟩ = 1 𝐷 𝑠 ( Ω 𝑠 2 ⟨σ (𝑛) 𝑔𝑒 ⟩
)

(2.28)

From these equations we obtain the optical response of the system. In the next subsection we show some interesting effects which arise with three level system.

Electromagnetically Induced Transparency

In order to have a strong two level non-linear susceptibility one needs to move closer to resonance but the medium becomes opaque due to strong scattering by atoms. One way to circumvent this problem is by using Electromagnetically Induced Transparency, otherwise known as EIT. As the name indicates, the medium attains a narrow transparency band in the absorption spectrum and the dispersion spectra exhibits a steep slope when excited in certain conditions. Since the first observation of EIT [START_REF] Boller | Observation of electromagnetically induced transparency[END_REF], many groups have performed interesting experiments such as slow light [START_REF] Hau | Light speed reduction to 17 metres per second in an ultracold atomic gas[END_REF], lasing without inversion [START_REF] Zibrov | Experimental demonstration of laser oscillation without population inversion via quantum interference in Rb[END_REF], etc. To understand how the transparency works, lets consider a system of three level atoms which are weakly driven (Ω 𝑝 ≪ Ω 𝑠 , 𝛾 𝑒 ) and all the transitions are resonant (∆ 𝑖 = 0 for i = 𝑒, 𝑠, 𝑐 ) then the eigenstates of the system are (|𝑠⟩ ± |𝑒⟩)/ √ 2 and |𝑔⟩. The probe couples to the ±|𝑒⟩ components of the two eigenstates, which have equal magnitude and opposite sign. Hence it leads to destructive interference of both excitation pathways and therefore diminished absorption [START_REF] Harris | Imamoglu A Nonlinear optical processes using electromagnetically induced transparency[END_REF][START_REF] Fleischhauer | Electromagnetically induced transparency: Optics in coherent media[END_REF]. Electromagnetically induced transparency curves in the presence and the absence of the cavity. The transmission of the system is plotted with respect to probe detuning normalized to 𝛾 𝑒 . We have assumed a low finesse (100) one-sided cavity. The frequency of the probe beam is sweeped around the resonance and the control beam is incident on resonance with the secondary transition (∆ 𝑠 = 0). The value of control field Rabi frequency is assumed to be 2𝛾 𝑒 and the linewidth it taken to be 𝛾 𝑠 = 0.06𝛾 𝑒 .

Linear response

The response of three level atoms can be simplified using the Bloch equation under weak feeding approximation and assuming all the atoms have equal coupling. If we neglect the atomic excitation (⟨σ (𝑛)

𝑒𝑒 -σ(𝑛) 𝑔𝑔 ⟩ ≪ 1) then the coherence term for any atom n is

⟨σ (𝑛) 𝑔𝑒 ⟩ ≈ 1 𝐷 𝑒 ( Ω 𝑠 2 ⟨σ (𝑛) 𝑔𝑠 ⟩ + 𝑔⟨â⟩ ) = 𝑔𝛼 D𝑒 𝐷 𝑐 - ∑ 𝑛 𝑔 2 = 𝑔𝛼 D𝑒 𝐷 𝑐 -2𝛾 𝑒 𝛾 𝑐 𝐶 (2.29)
The above expression is the same as two level susceptibility except that the term

𝐷 𝑒 is replaced by D𝑒 = ∆ 𝑒 -Ω 2 𝑠 4𝐷𝑠
.

One can obtain an analytical expression to the transmission in steady state:

𝑇 = ⃒ ⃒ ⃒ ⃒ ⃒ 𝛾 𝑐 D𝑒 D𝑒 𝐷 𝑐 -2𝛾 𝑐 𝛾 𝑒 𝐶 ⃒ ⃒ ⃒ ⃒ ⃒ 2 (2.30)
In Figure 2.8 we compare the EIT in cavity systems with free space. We can observe that the EIT transmission peak is lower in cavity systems due to multiple reflections inside the cavity.

On resonance excitation

If the detuning of the control beam is zero (∆ 𝑠 = 0) and the probe is on resonance with atoms (∆ 𝑒 = 𝛿 = 0) then the imaginary part of the susceptibility goes to zero.

The probe is transmitted by the system as if the medium is transparent. In the absence of control field, even for moderate cooperativities (C ≈ 5) the transmission drops below 1%. When the control field is Ω 𝑠 ≈ 2𝛾 𝑒 and the linewidth of the state |𝑠⟩ 𝛾 𝑠 ∼ 0.06𝛾 𝑒 then the on-transmission is close to 30%.

In Figure 2.7, the dispersion curve in EIT exhibits a steep slope which leads to the phenomena of slow light [START_REF] Hau | Light speed reduction to 17 metres per second in an ultracold atomic gas[END_REF] and the imaginary part of the susceptibility drops to zero on resonance.

Off-resonance excitation

If we excite the atoms off-resonantly on the probe transition, then interesting features are observed near the two photon resonance [START_REF] Gheri | Quantum noise reduction close to an optically bistable dark resonance[END_REF][START_REF] Sautenkov | Switching between photonphoton correlations and Raman anticorrelations in a coherently prepared Rb vapor[END_REF]. The resonant absorption line remains almost unaffected while a second absorption/dispersion response is observed at ∆ 𝑒 ≈ -∆ 𝑐 . This corresponds to a two-photon resonance, where the atomic population is transfered from ground state to the meta stable one. of the susceptibilities of a system when the control field is detuned by -15𝛾 𝑒 , its Rabi frequency is set to 5𝛾 𝑒 and the linewidth is taken as 0.1𝛾 𝑒 . This creates a sharp medium response at two photon resonance.

Multilevel system

In degenerate energy systems like Rydberg D states where multiple magnetic sublevels are degenerate any weak electric fields can lead presence of more than three levels in the system refer Sec:7.5.2. Here we calculate how the transmission of the system is modified in the presence of more than one metastable state. We assume that we are in a linear excitation regime (where ⟨𝜎 (𝑖)

𝑔𝑔 ⟩ ≫ ⟨𝜎 (𝑖) 𝑒𝑒 ⟩).

4𝛿

. By evaluating the above expression we can evaluate the EIT spectrum for multilevel atoms.

Non-linearity

The presence of a third level modifies the optical response in a non-trivial way. The two photon resonance in off-resonant excitation scheme exhibits a narrow response.

This allows us to easily saturate the medium which gives rise to a non-linear optical response. Large non-linearities have been observed experimentally based on these schemes [START_REF] Wang | Controlling light by light with three-level atoms inside an optical cavity[END_REF][START_REF] Sautenkov | Switching between photonphoton correlations and Raman anticorrelations in a coherently prepared Rb vapor[END_REF][START_REF] Alzar | Super-Poissonian intensity fluctuations and correlations between pump and probe fields in Electromagnetically Induced Transparency EPL[END_REF].

If we look at the expression for susceptibility, we can observe that the intracavity intensity is rescaled by the factor 1 + Ω 2 𝑠 8𝛿 2 . One can observe large non-linearities with even weaker probe fields by having a strong control field Rabi frequency. However, the effects cannot be described as "true" non-linearities because they are manifested from saturation of |𝑠⟩. The photons are just converted into atomic excitations in the other ground state. Even though compared to two level systems the number of photons required to observe the non-linearity has decreased, the number of atomic excitations are still of the same order. So, in order to observe strong non-linear effects we need an extra feature which doesn't convert the photons into atomic excitations. As we will see in the next chapter, atoms excited to |𝑠⟩ must exhibit long range interactions, which will render the remaining atoms further detuned from the optical fields.

Introduction

As we have shown in the previous chapter that non-linearities obtained using multilevel or EIT schemes are not strong enough to be observed at single photon level. To achieve this we need some kind of interaction between the atoms which can modify the behavior of the cloud even with few polaritons in the system. The promising solution is to use highly excited Rydberg atoms. The remarkably properties exhibited by atoms excited to Rydberg states will help us in realizing strong interatomic interactions [70].

Atoms excited to a very high principal quantum number, higher than 30 are referred as Rydberg atoms. They have long lifetimes (few 100 𝜇s) and large dipole moments, which makes them sensitive to weak perturbations in their environments [START_REF] Comparat | Dipole blockade in a cold Rydberg atomic sample[END_REF]. This large dipole moment creates a dipole potential through which it can interact with neighbouring atoms over long distances (few microns). Rydberg-Rydberg interactions have several degrees of freedom, one can control the type of interactions, the distance and the direction by selecting a Rydberg state. By mapping photons onto Rydberg excitations one can achieve exotic states of photons or strong non-linear interactions.

The dipole-dipole interaction energy between neutral dipoles separated by a distance 'R' is given by

𝑉 𝑑 1 ,𝑑 2 (𝑅) = 1 4𝜋𝜀 0 [ 𝑑 1 .𝑑 2 |𝑅| 3 -3 (𝑑 1 .𝑅).(𝑑 2 .𝑅) |𝑅| 5 ] (3.1)
In order to evaluate the interacting two-atom system, one needs to take into account mainly the neighboring Rydberg energy levels which contribute the signifi-

r ~ n 2 a 0 -5 -4 -3 -2 -1 0 5S 1/2 5D 5/2 5P 1/2 5P 3/2 E (eV) L = 0 L = 2 L = 1
𝐸 ± = ∆ ± √ ∆ 2 + 4𝑉 (𝑅) 2 2
The eigenenergies of the total Hamiltonian can be described in two different regimes: the Van der Waals regime, and the dipole-dipole regime. One in which the interaction term is much smaller than the energy difference (𝑉 (𝑅) ≪ ∆) it is referred to as off-resonant Van der Waals interactions, and at the other extreme where the interaction term is much larger than the detuning (𝑉 (𝑅) ≫ ∆) are called as resonant dipole-dipole interactions.

In the Van der Waals regime the eigenenergies are shifted by (∆𝐸 ∝ 𝐶 6 /𝑅 6 ) and in the resonant dipole-dipole regime they are shifted by (∆𝐸 ∝ 𝐶 3 /𝑅 3 ). This dipole interaction strength 𝐶 3 and 𝐶 6 scales as (𝑛 * ) 4 and (𝑛 * ) 11 respectively.

These interactions lead to a Rydberg level detuned out of resonance and the control field is no longer resonant with the secondary transition, this effect is known as Rydberg blockade effect. The medium then acts as two-level atoms and the absorption dominates. They are currently one of the most actively researched candidates in quantum optics for generating non-classical states or quantum effects using dipole-dipole interactions [START_REF] Lukin | Dipole Blockade and Quantum Information Processing in Mesoscopic Atomic Ensembles[END_REF].

The Rydberg-Rydberg interaction Hamiltonian term between atoms in Rydberg state |𝑟⟩ is [START_REF] Grankin | Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: A perturbative approach[END_REF]

ℋ 𝑖𝑛𝑡 = 1 2 ∑ 𝑚̸ =𝑛 𝜅 𝑚,𝑛 σ(𝑚) 𝑟𝑟 σ(𝑛) 𝑟𝑟 (3.2)
3.2. Mean field approach If the excitation linewidth is lower than the V 𝑖𝑛𝑡 (𝑟) then it is not possible to reach the |𝑟, 𝑟⟩ state.

The Bloch equations for the coherence terms can be derived from the total Hamiltonian along with the interaction term, leading to the following equations.

𝑑 𝑑𝑡 σ(𝑛) 𝑔𝑒 = 𝑖(∆ 𝑒 + 𝑖𝛾 𝑒 )σ (𝑛) 𝑔𝑒 + 𝑖𝑔â(σ (𝑛) 𝑒𝑒 -σ(𝑛) 𝑔𝑔 ) -𝑖 Ω 𝑐𝑓 2 σ(𝑛) 𝑒𝑟 + F (𝑛) 𝑔𝑒 𝑑 𝑑𝑡 σ(𝑛) 𝑒𝑟 = 𝑖(𝛿 + 𝑖𝛾 𝑒𝑟 )σ (𝑛) 𝑒𝑟 + 𝑖𝑔𝑎 † σ(𝑛) 𝑔𝑟 + 𝑖 Ω 𝑐𝑓 2 (σ (𝑛) 𝑟𝑟 -σ(𝑛) 𝑒𝑒 ) -𝑖σ (𝑛) 𝑒𝑟 ∑ 𝑚̸ =𝑛 𝜅 𝑚,𝑛 σ(𝑚) 𝑟𝑟 + F (𝑛) 𝑒𝑟 𝑑 𝑑𝑡 σ(𝑛) 𝑔𝑟 = 𝑖(∆ 𝑟 + 𝑖𝛾 𝑟 )σ (𝑛) 𝑔𝑟 + 𝑖𝑔âσ (𝑛) 𝑒𝑟 -𝑖 Ω 𝑐𝑓 2 σ(𝑛) 𝑔𝑒 -𝑖σ (𝑛) 𝑔𝑟 ∑ 𝑚̸ =𝑛 𝜅 𝑚,𝑛 σ(𝑚) 𝑟𝑟 + F (𝑛) 𝑔𝑟 (3.3)
where 𝐹

(𝑛) 𝑖𝑗 is the Langevin noise operator corresponding to the operator σ(𝑛)

𝑖𝑗 , 𝛿 = ∆ 𝑒 + ∆ 𝑠 is two photon detuning term and Ω 𝑐𝑓 is the control field Rabi frequency (|𝑒⟩ → |𝑟⟩). The interaction term is restricted to the symmetric subspace, where states are invariant under the permutation of two particles and 𝜅 𝑚,𝑛 is interaction energy between atoms m and n in Rydberg state. This interaction term is a function of 𝐶 6 in the case of Van der Waals interaction and 𝐶 3 for resonant dipole-dipole interaction.

The interaction term in the Hamiltonian is an N body problem and it cannot be solved without making some approximations like the mean field approach. Here we will describe two such methods which we use to explain our Rydberg non-linearity measurements. The models have been developed by Andrey Grankin and Etienne Brion, they are well described in A. Grankin's thesis [START_REF] Grankin | Theoretical studies of optical non-linear effects in ultracold Rydberg gases[END_REF].

Mean field approach

The computation of the non-linear response of N body interactions scales exponentially and becomes intractable for even few atoms. One way to circumvent this problem is by using a mean field approach [START_REF] Tong | Blockade of Rydberg Excitation in an Ultracold Gas[END_REF][START_REF] Weimer | Quantum Critical Behavior in Strongly Interacting Rydberg Gases[END_REF] where each atom's response is modified by a mean field it experiences from its neighboring atoms.

We introduce two new collective atomic operators corresponding to the transi-

Chapter 3. Rydberg-Rydberg Interactions tions |𝑔⟩ → |𝑒⟩ and |𝑔⟩ → |𝑟⟩ b = 1 √ 𝑁 ∑ 𝑖 σ(𝑖) 𝑔𝑒 ĉ = 1 √ 𝑁 ∑ 𝑖 σ(𝑖)
𝑔𝑟 which are governed by the following dynamical equations:

𝑑 𝑑𝑡 ⟨â⟩ = (𝑖∆ 𝑐 -𝛾 𝑐 ) ⟨â⟩ -𝑖𝑔 √ 𝑁 ⟨ b⟩ -𝑖𝛼 𝑑 𝑑𝑡 ⟨ b⟩ = (𝑖∆ 𝑒 -𝛾 𝑒 )⟨ b⟩ -𝑖 √ 𝑁 𝑔 ⟨â⟩ -𝑖 Ω 𝑐𝑓 2 ⟨ĉ⟩ 𝑑 𝑑𝑡 ⟨ĉ⟩ = (𝑖∆ 𝑟 -𝛾 𝑟 ) ⟨ĉ⟩ -𝑖 Ω 𝑐𝑓 2 ⟨ b⟩ -𝑖𝜅⟨ĉ † ĉĉ⟩
The complex constant 𝜅 characterizes the effect of dipole-dipole Rydberg interactions, and was found to be

𝜅 = -2 ( 𝑉 𝑏 𝑉 -𝑉 𝑏 )( Ω 2 𝑐𝑓 4(𝐷 𝑒 + 𝐷 𝑟 - Ω 2 𝑐𝑓 4(𝐷𝑒) ) -(𝐷 𝑟 ) ) (3.4) 𝑉 𝑏 = √ 2𝜋 2 3 ⎯ ⎸ ⎸ ⎸ ⎷ 𝐶 6 𝐷 𝑒 - Ω 2 𝑐𝑓 4(𝐷𝑒+𝐷𝑟- Ω 2 𝑐𝑓 4𝐷𝑒 ) (3.5)
where 𝐷 𝑘 ≡ ∆ 𝑘 + 𝑖𝛾 𝑔𝑘 for 𝑘 = 𝑒, 𝑟, 𝑉 𝑏 is the blockaded volume of a single blockade, and 𝑉 is the total volume of the cloud.

In steady state conditions when all the transitions are resonant, the ratio 𝑉 𝑏 𝑉 ≪1.

Hence, we can use the mean field approximation where the state of the system is approximated to be coherent. We can replace the operators with complex numbers which leads to a simplified interaction term 𝑉 𝑖𝑛𝑡 = |⟨ĉ⟩| 2 𝜅⟨ĉ⟩. This simplifies the response of the system in steady state to

⟨â⟩ = -⟨ b⟩𝑔 √ 𝑁 Δ𝑐+𝑖𝛾𝑐 -𝛼 Δ𝑐+𝑖𝛾𝑐 ⟨ b⟩ = -⟨â⟩𝑔 √ 𝑁 Δ𝑒+𝑖𝛾𝑒 - ⟨ĉ⟩𝑖Ω 𝑐𝑓 2(Δ𝑒+𝑖𝛾𝑒) + 𝑔 √ 𝑁 ⟨â⟩ Δ𝑒+𝑖𝛾𝑒 ⟨ĉ⟩ = - ⟨ b⟩𝑖Ω 𝑐𝑓 2(Δ𝑟+𝑖𝛾𝑟) -|⟨ĉ⟩| 2 𝜅⟨ĉ⟩ *

Δ𝑟+𝑖𝛾𝑟

The average interaction term 𝜅 can be calculated in the large volume approximation (V ≫ 𝑉 𝑏 ). The cavity transmission is then evaluated from the cavity field ⟨â⟩ with the expression:

𝑇 = 𝛾 2 𝑐 .|⟨â⟩| 2 𝛼 2

Rydberg bubble model

For Rydberg interactions in D-states, one needs to take into account an additional dephasing mechanism due to the presence of degenerate magnetic sublevels [START_REF] Tresp | Dipolar Dephasing of Rydberg 𝐷-State Polaritons[END_REF].

Rydberg bubble model

This mechanism creates stray Rydberg excitations which are no longer coupled to control field. These long lived excitations inhibit creation of any new Rydberg excitations upto a certain distance (blockade radius 𝑟 𝑏 ) around them. This blockaded sphere is commonly referred to as Rydberg bubble, or Rydberg superatom, and is frequently used to explain the Rydberg blockade effect [START_REF] Liebisch | Atom Counting Statistics in Ensembles of Interacting Rydberg Atoms[END_REF]. This bubble picture puts an upper limit on the number of Rydberg excitations (𝒩 𝑏 ) one can have in a finite volume cloud which allows us to calculate the number of atoms per bubble from

𝑛 𝑏 = 𝑁 𝒩 𝑏 = 2𝜋 2 𝜌 𝑎𝑡 3 √ |𝐶 6 | ∆ 𝑟 -Ω 2 𝑐𝑓 /4∆ 𝑒 (3.6)
Here, we give details about the phenomenological model we used to account for the dynamical behavior of the cavity transmission, in the case of samples excited to Rydberg 𝐷 states. We assume that the Rydberg blockade phenomenon effectively splits the atomic sample into independent and equivalent "bubbles" which can at most accommodate for one Rydberg excitation. Accordingly, the two-photon transition towards the Rydberg level |𝑟⟩ essentially couples the two collective symmetric states

|𝐺⟩ ≡ |𝑔 • • • 𝑔⟩ |𝑅⟩ ≡ 1 √ 𝑛 𝑏 𝑛 𝑏 ∑ 𝑛=1 σ(𝑛) 𝑟𝑔 |𝐺⟩ = 1 √ 𝑛 𝑏 (|𝑟𝑔 • • • 𝑔⟩ + • • • |𝑔 • • • 𝑔𝑟⟩)
where 𝑛 𝑏 denotes the number of atoms in a Rydberg bubble, and the corresponding lowering operator is the Pauli-like matrix σ𝐺𝑅 ≡ |𝐺⟩ ⟨𝑅| (we also define σ𝑅𝑅 ≡ |𝑅⟩ ⟨𝑅| and σ𝑅𝐺 ≡ |𝑅⟩ ⟨𝐺| ). Note that 𝑛 𝑏 can be evaluated by

𝑛 𝑏 = 𝑁 × ⃒ ⃒ 𝑉 𝑏 𝑉 ⃒ ⃒
, where 𝑁 is the total number of atoms in the sample, 𝑉 is the total volume of the sample and 𝑉 𝑏 is the volume of a Rydberg bubble whose expression is given in Eq.

3.4.

In a Rydberg bubble, the intermediate state can, by contrast, be arbitrarily populated; we will further assume that we remain in the low excitation regime (corresponding to moderate cavity feeding rates) so that the transition to the intermediate state is never saturated. In this approximation scheme, the collective lowering operator

β ≡ 1 √ 𝑛 𝑏 𝑛 𝑏 ∑ 𝑛=1 σ(𝑛) 𝑔𝑒 (3.7)
can be considered bosonic, i.e.

[

β, β † ] ≈ 1.
In order to account for the dynamical behavior observed experimentally, we introduce an extra Rydberg state, denoted by |𝑠⟩, to which the state |𝑟⟩ decays: this implies that, in a bubble, the collective states |𝑅⟩ and |𝑆⟩ ≡ 1

√ 𝑛 𝑏 (|𝑠𝑔 • • • 𝑔⟩ + • • • |𝑔 • • • 𝑔𝑠⟩)
are coupled by a Lindblad-like operator.

To simplify the treatment, we furthermore assume the cavity mode to be classical, that is we replace 𝑎 by its expectation value ⟨𝑎⟩ whose time evolution is ruled by the equation

𝑑 𝑑𝑡 ⟨𝑎⟩ = i (∆ 𝑐 + i𝛾 𝑐 ) ⟨𝑎⟩ -i ( 𝑁 𝑛 𝑏 ) 𝑔 √ 𝑛 𝑏 ⟨ β⟩ -i𝛼
Note that the second term of this equation arises from the coupling of the cavity mode with the ensemble of (

𝑁 𝑛 𝑏

)

Rydberg bubbles with the magnified coupling strength 𝑔 √ 𝑛 𝑏 . The first term accounts for the detuning and decay of the cavity, while the last one results from the feeding by the probe field. In this semi-classical approximation, bubbles do not entangle with the cavity mode and therefore cannot get entangled with each other. The atomic sample can hence be described by the tensor product density matrix 𝜌 ⊗ • • • ⊗ 𝜌, where 𝜌 is the density matrix of any of the bubbles (they are all equivalent). The dynamic semi-classical equation for 𝜌

now writes 𝑑 𝑑𝑡 𝜌 = -i [𝐻, 𝜌] + 𝒟 𝑙 (𝜌) + 𝒟 𝑛𝑙 (𝜌) where 𝐻 = -∆ 𝑟 |𝑅⟩ ⟨𝑅| -∆ 𝑒 β † β + {( Ω 𝑐𝑓 2 σ𝑅𝐺 + 𝑔 √ 𝑛 𝑏 ⟨𝑎⟩ * ) β + h.c. } 𝒟 𝑙 (𝜌) = 𝛾 𝑒 ( 2 β𝜌 β † -β † β𝜌 -𝜌 β † β) +𝛾 𝑟 (2σ 𝐺𝑅 𝜌σ 𝑅𝐺 -σ𝑅𝑅 𝜌 -𝜌σ 𝑅𝑅 ) +𝛾 𝑠 (2σ 𝐺𝑆 𝜌σ 𝑆𝐺 -σ𝑆𝑆 𝜌 -𝜌σ 𝑆𝑆 ) 𝒟 𝑛𝑙 (𝜌) = 𝜉 ⟨σ 𝑅𝑅 ⟩ (2σ 𝑆𝑅 𝜌σ 𝑅𝑆 -σ𝑅𝑅 𝜌 -𝜌σ 𝑅𝑅 )
Note that the phenomenological extra nonlinear decay 𝒟 𝑛𝑙 (𝜌) we introduced is time dependent through ⟨σ 𝑅𝑅 ⟩ (𝑡); its rate is moreover governed by the ad hoc free parameter 𝜉, whose value can be tuned so as to reproduce the experimental results. This model will be used in Sec: 7.5.3.

Part II Experimental apparatus

Chapter 4 In this chapter, we will present various elements which constitute our experimental setup. The primary requirement of a cold atomic experiment is a Magneto-Optical trap (MOT), where the atoms are trapped using optical beams in magnetic field. The setup has been comprehensively described by Erwan Bimbard's thesis [START_REF] Bimbard | Production and interaction of photons using atomic polaritons and Rydberg interactions[END_REF]. Here we will recall the setup along with some additional characterization tests. We will describe the experiment in three parts -in the first part we describe the laser table, where we prepare the optical beams; in the second part we describe the acquisition/control setup, and in the final part we describe the main table ,   where we trap and manipulate the atoms.

Optical and control system

Trapping and excitation lasers Introduction

We use a dedicated optical table to prepare the laser beams necessary for our experiment. These beams are later coupled into various fibers and then sent to the main table where we trap our atoms. In this section we describe how we lock and prepare the optical beams for our experiment. We use a mixture of diode and Titanium-Sapphire lasers. The diode lasers are very compact and can be modified to have extremely narrow linewidths, while the Ti-Sa lasers have a large tuning range with high output power.

Master laser

The main reference laser on our system is a Toptica DL Pro diode laser which is used to trap, cool and probe the D2 transition of Rb 87 at 780 nm. The diode is mounted on an aluminum plate with a collimator lens and an output coupler. An anti-reflection coating at the back of the diode makes the system act like a resonator.

A grating is mounted on a piezo inside the resonator to tune the resonance. The frequency of the laser diode can also be adjusted by changing the driving current.

The temperature and the current of the diode is controlled by using the commercial control modules Toptica DTC110 and Toptica DCC110, respectively. The temperature of the diode is maintained at 20 ∘ 𝐶. The power of the diode is later amplified using a tapered amplifier and at the output we can achieve up to 1W. We split the output power into various arms for trapping and probing.

The master laser is frequency stabilized using a stabilization system which will be discussed further in section 4.2. We use dedicated double pass Acoustic Optic Modulators (AOMs), which enables us to shift the frequency of the laser without losing coupling to fibers.

Repumper and dipole trap lasers

Due to the hyperfine splitting of 5𝑆 1/2 in Rb 87 , the atoms are lost to a dark state when trapped using an open transition. To recapture the lost atoms, we employ a DL pro Toptica laser (referred to as repumper) with an output of 30 mW. It is frequency stabilized on a crossover transition 5𝑆 1/2 𝐹 = 1 → 5𝑃 3/2 𝐹 ′ = 1 -2 by using standard absorption saturation spectroscopy. The laser is then frequency shifted to be on resonance with atoms using an AOM.

Recently, we installed a new diode laser at 810 nm to optically trap the atoms and to lock the newly installed optical cavity around the atoms.

Laser Transition

Trapping

5𝑆 1/2 𝐹 = 2 → 5𝑃 3/2 𝐹 ′ = 3 Repumper 5𝑆 1/2 𝐹 = 1 → 5𝑃 3/2 𝐹 ′ = 2 Depumper 5𝑆 1/2 𝐹 = 2 → 5𝑃 1/2 𝐹 ′ = 2 Optical pumping 5𝑆 1/2 𝐹 = 2 → 5𝑃 1/2 𝐹 ′ = 2 Probe 5𝑆 1/2 𝐹 = 2 → 5𝑃 3/2 𝐹 ′ = 3

Titanium-Sapphire traps

We have two Coherent MBR-110 Titanium doped Sapphire lasers which can be tuned between wavelengths of 750 to 900 nm using an etalon in a built-in cavity.

The lasers are pumped by two Verdi V10 lasers at 532 nm with a maximum possible 4.2. Frequency stabilization system power of 16 W. The power at the output of these systems can reach upto 2 W. We use one of them at 810 nm for creating a dipole trap, as well as for other purposes where we need a beam far from all optical transitions. The second one at 795 nm is used to address D1 transition. Unlike the diode lasers, Ti-Sa lasers are stabilized to an internal reference cavity. The cavity length can be adjusted to fine tune the frequency. Our laser stabilization system has a Free Spectral Range (FSR) of 110 MHz which lets us to stabilize our lasers to any frequency we want with the help of double pass AOMs with a range of 50 MHz.

Blue laser

In order to excite the atoms to Rydberg state the transition frequencies are in ultraviolet regime. It is possible to coherently excite using a single photon transition [START_REF] Hankin | Two-atom Rydberg blockade using direct 6𝑆 to 𝑛𝑃 excitation[END_REF], however it is technically challenging to work with lasers at ∼297 nm. Hence we use a two photon transition, where the secondary atomic transition 5𝑃

Frequency stabilization system

It is not always possible to find a suitable atomic transition to lock a laser for e.g. dipole trap lasers. The most convenient way is to lock all the lasers on an optical cavity known as transfer cavity. This type of locking technique provides a convenient way to lock all the lasers and at the same time gives enough flexibility to tune the frequency to whatever value we need. One can either employ a very high finesse (> 10 4 ) passive optical cavity [START_REF] Ludlow | thermal-noise-limited optical cavity for diode laser stabilization at 1Ö10=15[END_REF] or an actively stabilized cavity locked onto an atomic reference [START_REF] Burghardt | Precise rf tuning for cw dye lasers[END_REF]. We use a transfer cavity stabilized to a crossover transition

5𝑆 1/2 𝐹 = 2 → 5𝑃 3/2 𝐹 ′ = 2 -3 in Rb 87 .
We extract a part of the master laser and modulate it using an Electro-Optic Modulator (EOM) at ≃ 21 MHz, which provides the sidebands necessary for the Pound Drever Hall error signals.

∼ ≈ 150 kHz 5S 1/2 F = 2 → 5P 3/2 F = 1 -F = 3

Characterization tests

In this section, we will describe the various characterization tests performed to measure the stability of the transfer cavity locking scheme. This method has been chosen and designed for its flexibility to freely lock at any required frequency simply by using AOMs. However, the linewidth of the transfer cavity is only 150 kHz which limits the precision we can achieve on the laser frequency.

Stability of the transfer cavity

To measure the stability of the transfer cavity's length, we use a reference laser which is frequency stabilized on a very high finesse cavity. The reference laser has a linewidth of < 10 kHz and drifts about 10 kHz/day. We measure the reflection signal from our transfer cavity when its resonant frequency is about 100 kHz (half the transfer cavity's linewidth) away from the laser. The half power points on the The problem is caused due to the specific way we separate the reflected light from the incoming one. We use a quarter wave plate and a PBS to separate them, which couples a circularly polarized light to the cavity. To circumvent this issue, we replace the quarter wave plate with a Faraday isolator which rotates the polarization by 𝜋/4 but maintains the polarization linear. With the new system, the measured linewidth of the blue laser is 50 kHz. Since we use 960 nm to stabilize the frequency, the linewidth is larger after the doubling system.

Control/Acquisition setup

Cold atomic experiments require a precise control of the timing, frequency and power of various lasers. The preparation and manipulation of atoms take place at vastly different time scales, from few seconds for atom loading to few microseconds for pumping. In this section we will describe the control software and data acquisition system which are necessary to synchronize various elements of our setup.

Hardware components

The frequency and the power of lasers are controlled by acoustic optical modulators (AOMs) before being coupled to a fiber. Each optical beam typically passes twice through a dedicated AOM (MT- The software provides graphical user interface where the sequence can be configured in three main steps. The main steps in cold atomic experiments consists of atomic cloud trapping (using magneto-optical traps, dipole traps), cooling (molasses), preparation in a specific configuration and measurement.

1. In the first step, the digital/analog channels which are necessary for the experiment are selected. The physical addresses are mapped to virtual names using a configuration file which is loaded with the software. The sequence only runs the necessary channels which are required for the experiment. The main experimental requirement to observe strong non-linear effects with Rydberg EIT setup is to have large atomic density to achieve significant optical depth on photonic fields. In hot vapor cells it is possible to achieve high atomic densities but the blockade effect is significantly affected by rapidly moving atoms [START_REF] Kübler | Coherent excitation of Rydberg atoms in micrometre-sized atomic vapour cells[END_REF]. Hence, we chose a cold atomic system capable of going below the Doppler limit.

To have the frozen gas approximation valid, the Doppler broadening mechanism should be much smaller than the Rydberg induced energy shifts. To fulfil the above criteria, we trap a cold atomic cloud which can reach densities of 10 10 cm -3 at temperatures below 50 𝜇K. In this chapter, we will describe how we prepare and characterize our cold atomic cloud.

Atom cloud preparation

We work with Rubidium Rb 87 atoms because of easily available diode lasers for addressing its optical transitions. We create Rb vapor by heating a Rb salt upto 50 ∘ C. The atoms are first trapped using a magneto optical trap (MOT) and can be easily cooled down to ≈1 mK.

The experimental apparatus consists of two vacuum chambers -auxiliary and science chamber connected by a differential tube. This tube helps to maintain a pressure ratio of 100 between the two chambers. The auxiliary chamber acts as a source of low velocity atoms for the main trap in the science chamber and is Chapter 5. Atomic setup significantly smaller than the other one. The experimental chambers are illustrated in Figure 5.1. In typical MOT systems, where the atoms are loaded from background pressure, it can take upto few minutes to load a significant number of atoms. By using an auxiliary trap we can achieve upto 10 9 atoms in a few seconds.

LVIS with beam collimators

Cavity holder MOT coils

Magneto Optical Trap

Atoms are captured and trapped using a standard magneto optical trap (MOT) [START_REF] Raab | Trapping of Neutral Sodium Atoms with Radiation Pressure[END_REF][START_REF] Pritchard | Light Traps Using Spontaneous Forces[END_REF]. The atoms are laser cooled using a cooling laser tuned to near resonance on The trap consists of six large independent beams of ∼ 1 cm diameter created using beam collimators mounted on a Thorlab 30 mm optical cage system. The beams are produced using two stable Schäfter-Kirchoff variable fiber splitters with 5.3. Low Velocity Intense Source (LVIS)

D2 transition 5𝑆 1/2 𝐹 = 2 → 5𝑃 3/2 𝐹 ′ = 3 while the resonant repumper 5𝑆 1/2 𝐹 = 1 → 5𝑃 3/2 𝐹 ′ = 2 is
5S 1/2 5P 3/2 F =3 F =2 F =1 F =0 F =2 F =1 MOT Repumper

Low Velocity Intense Source (LVIS)

The experiments we performed are schematically described in Figure 5.1. The pressure in the LVIS chamber is maintained at 10 -8 mbar using a small ion pump of 2 l/s (Agilent Model 919-0520 ) later on replaced with a 3 l/s ion pump (Gamma vacuum model 3S). We load atoms into the auxiliary trap called Low Velocity Intense Source (LVIS) [START_REF] Lu | Low-Velocity Intense Source of Atoms from a Magneto-optical Trap[END_REF], where the atoms are trapped in a similar way as a Magneto-optical trap using three pairs of orthogonally polarized laser.

As shown in Figure 5. from resonance and then kept constant for the remaining time. In the last 3 ms, its power is reduced by a factor of 10 linearly. The final temperature after the optical molasses can be anywhere between 30-50 𝜇K depending on the loading parameters.

Imaging system

The trapped atoms are imaged by shining a near resonant beam on to the atoms.

The fluorescence from the atoms depends on various parameters like Intensity, detuning etc of the incident beam . The scattering rate of one atom is calculated using

𝑆 1 = 𝐼/𝐼 𝑠𝑎𝑡 1 + ∆ 2 /𝛾 2 𝑒 + 𝐼/𝐼 𝑠𝑎𝑡 𝛾 𝑒 (5.1)
Where the 𝐼 𝑠𝑎𝑡 is 1.64 mW/cm 2 , 𝛾 𝑒 is 2𝜋 3 MHz for the D2 transition of 87 Rb and ∆ is the detuning of the imaging laser. Now we can compute the number of atoms 𝑁 𝑎 from the photons detected by the camera (N 𝑑𝑒𝑡 ) from

𝑁 𝑎 = 1 𝜂 𝑐𝑜𝑙𝑙 𝑔 𝜂 𝐶𝐶𝐷 1 𝜂 𝑡𝑟𝑎𝑛𝑠 𝑆 1 1 𝑡 𝑒𝑥𝑝 𝑁 𝑑𝑒𝑡 (5.2)
Where 𝜂 𝑐𝑜𝑙𝑙 is the collection efficiency , 𝜂 𝑡𝑟𝑎𝑛𝑠 is the transmission efficiency and 𝜂 𝐶𝐶𝐷 is the detection efficiency of the camera, t 𝑒𝑥𝑝 is the exposure time and g MOT cycles which would allow us to further increase the density by minimizing the repulsive forces between the atoms due to rescattering [START_REF] Ketterle | High densities of cold atoms in a dark spontaneous-force optical trap[END_REF].

Chapter 6 Rydberg-Rydberg interactions in a resonator can be used to convert coherent states into non-classical ones and hence require sophisticated detection systems to characterize them [START_REF] Stanojevic | Generating non-Gaussian states using collisions between Rydberg polaritons[END_REF]. In this chapter we will describe how we measure the photon statistics, phase, and the squeezing spectrum of an optical beam.

Second order correlation measurement

To understand more details about the physics of the system it is necessary go beyond simple linear detection systems. A normal photo diode can only measure average intensity and cannot distinguish between different photon number components of a signal beam. To determine the quantum statistics of the light we measure the second order correlation function. If we assume that the incoming signal beam is invariant under time translation, then the normalized second order correlation function 𝑔 (2) (𝜏 ) is given by [START_REF] Walls | Quantum Optics Walls and Milburn[END_REF] 𝑔 (2)

(𝜏 ) = ⟨ â † (0)â † (𝜏 )â(𝜏 )𝑎(0) ⟩ ⟨â † (0)â(0)⟩ 2 (6.1)
where â(𝜏 ) is the annhilation operator on the signal mode at time 𝜏 . By using a number operator N = â † â we can rewrite the g (2) (0) by

𝑔 (2) = ⟨ N 2 ⟩ -⟨ N ⟩ ⟨ N ⟩ 2 = 1 + ∆ N 2 -⟨ N ⟩ ⟨ N ⟩ 2 (6.2)

Flippable mirror

Top probe

Bottom probe

Homodyne Tomography g (2) ( ) measurement

/4 PBS (2)
(2) the cavity is sent to a 50:50 beam splitter and two detectors are placed at the two output arms to measure the intensity correlations between them. It can be shown that measuring g

g (2) (τ ) = 1 g (2) (τ ) > 1 g (2) (τ ) < 1 g (2) (τ ) ≥ 1 g (2) (τ ) < 0
(2) (𝜏 ) is equivalent to ⟨𝑎 † 1 (0)𝑎 † 2 (𝜏 )𝑎 2 (𝜏 )𝑎 1 (0)⟩ ⟨𝑎 † 1 (0)𝑎 1 (0)⟩⟨𝑎 † 2 (0)𝑎 2 (0)⟩
where 1 and 2 denote the two output modes. To evaluate the correlations between the detectors at the same time one can use the following expression 𝑔 (2)

(0) = ⟨ N1 N2⟩ ⟨ N1⟩⟨ N2⟩
The light from the cavity is collimated and filtered to reject any background photons from sources like the trapping beam, etc. It is then incident on a 50/50 beam splitter and intensity correlation between the two output paths are measured using single photon counting modules (SPCMs; Perkin-Elmer SPCM-AQR-12-FC).

The counting modules creates a 20 ns pulse for every detected photon and a dark count rate of 50 counts/sec The specified quantum efficiency at 800 nm is 65 % and the dead time is 30 ns. It is important to note here that the quantum state measured using avalanche on/off photodiodes doesn't exactly give us the true g (2) value. It is only possible to distinguish between zero and 'one or more photons' event. It can be possible to design a photon-number resolving detectors using multiple on/off detectors [START_REF] Sperling | True photocounting statistics of multiple on-off detectors[END_REF]. But if the probability of having more than one photon in the same spatio-temporal mode is low then one can still use on/off detectors to accurately measure the g (2) function.

The total detection efficiency from the output of the cavity until the photons are detected is 37.5±2 %, it includes the transmission efficiency from cavity to counters of 57 % and detection efficiency of the counting modules (65 %).

While analyzing the data we ignore the first few seconds of the data as the MOT needs to attain quasi-steady state loading. The g can retrieve information about the intensity correlations but it is inherently insensitive to phase fluctuations. By using homodyne measurements one can measure many properties like mode purity, phase, etc., of the photonic state however it is technically challenging to implement it. In this section, we will show how one can measure the phase shift of the retrieved photons. The optical signal we want to measure is mixed with a mode-matched strong coherent beam called local oscillator (𝛼 𝐿𝑂 ≫ 1) on a 50:50 beam splitter as illustrated in Figure 6.3. The two outputs are measured using two high efficiency low noise photo diodes (PD). The difference between the detected signals is monitored and analyzed to extract the phase, the squeezing spectrum, and also to reconstruct the density matrix using maximum likelihood algorithms. In 2013, we demonstrated that our atomic cloud can be used as an on-demand single photon source with fully controlled quantum state by characterizing it with homodyne tomography [START_REF] Bimbard | Ourjoumtsev A and Grangier P Homodyne Tomography of a Single Photon Retrieved on Demand from a Cavity-Enhanced Cold Atom Memory[END_REF].

By assuming that the amplitude fluctuations of the LO are negligible compared its amplitude 𝛼 𝐿𝑂 , one can show that the difference of the photodiode currents is proportional to quadrature value of the signal field i.e.

Î1 -Î2 ∝ |𝛼 𝐿𝑂 |q 𝜃 (6.3)
where 𝑞 𝜃 is the quadrature value along any axis at an angle 𝜃 in phase space ( q𝜃=0 = x, q𝜃=𝜋/2 = p where the orthogonal quadratures x = 1 The larger the LO beam intensity the higher is the signal to electronic noise ratio however the detector efficiency is not linear for very large optical powers. So, we chose the maximum LO power where the detector is still linear, which corresponds to ≈ 7 mW. The bandwidth of the detectors is about 100 MHz, and the electronic noise is about 15 dB lower than the vacuum noise (refer Figure 6.4).

Phase locked experimental setup

In our experiment, we overlap a cavity mode-matched LO beam with the signal on From Andrey Grankin's calculation [START_REF] Grankin | Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: A perturbative approach[END_REF], we expect bunching/anti-bunching correlation in the probe beam reflected from the cavity depending on the parameters.

In order to lock the phase of this probe beam with respect to the LO beam, we use an auxiliary beam in the same path as the reflected probe sent to the cavity but detuned by 88 MHz. The locking beam and the probe beam pass through a single pass AOM, which is supplied by a signal 𝑉 0 (𝑠𝑖𝑛(𝜔 1 𝑡) + 𝑠𝑖𝑛(𝜔 2 𝑡)) where 𝜔 1 = 212 MHz and 𝜔 2 =300 MHz. The auxiliary beam is rejected by the cavity and then interfered with the LO beam.

Assuming the frequency of the probe, auxiliary and LO beam is 𝜔 0 , as they originate from the same laser. After the AOM one can denote the respective fields as 𝐸 𝑝 𝑒 (𝑖(𝜔 0 +𝜔 1 )𝑡) , 𝐸 𝑎 𝑒 (𝑖(𝜔 0 +𝜔 2 )𝑡) and 𝐸 𝐿𝑂 𝑒 (𝑖(𝜔 0 +𝜔 1 )𝑡+𝑖𝜑) . Both the probe and the auxiliary beam propagate in the same path for a distance 𝐿 1 and LO for a distance 𝐿 2 . The field output at the two arms of the homodyne setup can be written as

𝐸 ± = 𝐸 𝐿𝑂 𝑒 (𝑖𝜔𝑎(𝑡-𝐿 2 /𝑐)+𝜑) ± (𝐸 𝑝 𝑒 (𝑖𝜔𝑎(𝑡-𝐿 1 /𝑐)) + 𝐸 𝑎 𝑒 (𝑖𝜔 𝑏 (𝑡-𝐿 1 /𝑐)) ) (6.4)
where we replaced 𝜔 0 + 𝜔 1 by 𝜔 𝑎 and 𝜔 0 + 𝜔 2 by 𝜔 𝑏 . The output voltage of the homodyne signal is given by

𝑉 𝐻𝐷 = 𝐼 1 -𝐼 2 = 2𝐸 𝐿𝑂 𝐸 𝑝 cos ( 𝜔 𝑎 𝐿 2 -𝐿 1 𝑐 + 𝜑 ) 𝐸 𝐿𝑂 𝐸 𝑎 cos ( 𝜔 𝑏 𝐿 2 -𝐿 1 𝑐 + 𝛿𝜔 ( 𝑡 - 𝐿 1 𝑐 ) + 𝜑 ) = 2𝐸 𝐿𝑂 𝐸 𝑝 cos𝜃 + 𝐸 𝐿𝑂 𝐸 𝑎 cos ( 𝜃 -𝛿𝜔 ( 𝑡 - 𝐿 2 𝑐 )) (6.5)
where 𝛿𝜔 = 𝜔 𝑎 -𝜔 𝑏 and 𝜃 = (

𝜔 𝑎 𝐿 2 -𝐿 1 𝑐 + 𝜑 )
. This signal propagates through a cable of length 𝐿 3 and acquires an additional phase of 𝛿𝜔𝐿 3 /𝑐. It is then used by a FALC (Fast Analog Linewidth Control box from Toptica) to act on the input signal of the AOM to lock the relative phase (𝜃) between the LO and the probe beam.

The homodyne signal is mixed with an RF signal at 𝛿𝜔, which propagates through another cable of length 𝐿 4 . The DC signal at the output of FALC is then given by

𝑉 𝑚𝑖𝑥 = 𝑉 𝑅𝐹 𝐸 𝐿𝑂 𝐸 𝑎 cos ( 𝜃 + 𝛿𝜔 ( 𝐿 2 + 𝐿 3 -𝐿 4 𝑐 )) .
By locking the above signal to zero, we can ensure that the relative phase between the LO and the locking beam doesn't vary with time. In order to change the relative phase difference between the probe and the LO by 𝜋/2 for a 𝐿 2 + 𝐿 3 -𝐿 4 ≈ 20 m would require changing the frequency of the locking beam by 4 MHz. As the bandwidth of the phase lock is limited ∼ 100 kHz, the phase must be varied relatively slowly to measure the orthogonal quadrature.

Maximum phase lock power

Since the auxiliary beam is not very far from the atomic resonance, it entails specific constraints on the maximum allowable power one can use. There are many ways to estimate the effects of the locking beam on our system. If we start with the Bloch equations ( refer to equation:2.28) and using perturbative expansion, we can estimate the Rydberg atom population. In the lowest order of 𝛼 we can assume

𝜎 𝑟𝑟 = |𝜎 𝑔𝑟 | 2 [73]
. The number of Rydberg atoms can be estimated from

𝑁 𝑟𝑟 = 𝑁 𝜎 𝑟𝑟 = 𝑁 g𝑎 ( D𝑒 𝐷 𝑐 -2𝛾 𝑒 𝛾 𝑐 𝐶) , (6.6)
where g = Ω 𝑐𝑓 𝑔 2𝐷𝑠

, 𝑎 = √ 𝛾 𝑐 𝛼 𝑖𝑛 and for a given input power 𝑃 𝑖𝑛 , the photon rate

𝛼 𝑖𝑛 = 𝑃 𝑖𝑛 𝜔𝑝 .
For a typical experiment we have a C = 10, N = 10

4
, Ω 𝑐𝑓 = 5𝛾 𝑒 and ∆ 𝑒 = ∆ 𝑐 = 30𝛾 𝑒 . Hence, to create a single Rydberg excitation we require at least a power of ∼ 100 nW in locking beam. Since the lock of the relative phase is stable for a power of ≈ 50 pW, it limits the probability of Rydberg population to less than 10 -3 in the cavity mode. (green curve) compared to the empty cavity (red curve). When the coupling beam is switched on, the phase is flipped again by 0.9 𝜋 when the EIT peak appears (blue curve). The vertical lines are denoted to show the location of ideal phase flip of 𝜋.

The dashed lines are denoted to visualize the deviation from ideal phase shifts.

Phase measurements with cavity EIT

We know that when the cavity is empty and locked on resonance with the probe, the reflected light has the same phase as the incoming light. In the presence of atoms, the resonant probe is scattered and the light reflected from the bottom mirror acquires a phase of 𝜋. As we can observe in Figure 6.6, the cavity with atoms has almost the same amplitude (95 % corresponding to the bottom mirror reflectivity) but acquired a phase shift of 0.85𝜋. In the presence of EIT in the linear regime, the resonant probe can enter the cavity again and the phase is flipped by 𝜋 but the reflected intensity is weaker as the EIT transmission is not 100 %. We observe a phase shift of 0.9𝜋 compared to the situation with atoms. These measurements suggest that this setup is suitable to measure phase flips of the probe field and can be used to measure the fidelity of the Controlled phase gate.

Squeezing measurements in transmission

We setup a homodyne detection method which allows us to study the spectral components of the light retrieved from the cavity. We can extract the squeezing spectrum from the homodyne signal by spectral filtering. The noise power spectrum of the photo current is given by [START_REF] Walls | Quantum Optics Walls and Milburn[END_REF] 𝑆(𝜔, 𝜃) = Experimentally one has to fix the phase or know the phase for measuring the squeezing spectrum using the homodyne measurements. In the previously described method the locking beam and the probe beam are in the same mode, which allows us to lock the relative phase between the LO and the signal beam. However, we cannot use the same system for measuring the squeezing spectrum of the light transmitted through the cavity. For this set of experiments, we implemented a simpler scheme to measure the noise squeezing without having to actively control the relative phase

Part III

Absorptive Rydberg Non-linearities

A decade ago, single photon non-linearities had never been observed experimentally in the optical domain. Since the proposal of using Rydberg-Rydberg interactions as a means of observing quantum optical non-linearities many groups have investigated various schemes where the photons are mapped onto atomic excitations using the EIT mechanism. The first experiment using resonant scheme in cold atoms was performed by Charles Adam's group [START_REF] Pritchard | Cooperative Optical Non-linearity in a blockaded Rydberg Ensemble[END_REF]. In free space, many resonant EIT experiments have been designed to act as single photon filters [START_REF] Peyronel | Quantum nonlinear optics with single photons enabled by strongly interacting atoms[END_REF], controlling optical photons using microwave Rydberg excitations [START_REF] Maxwell | Control of Optical Photons Using Rydberg Polaritons[END_REF].

In this part, we will discuss the experimental observation of the optical non-linear response of a cloud of cold atoms trapped inside a cavity. We will first describe the expected non-classical correlations and squeezing using perturbative approaches in weak feeding regime. In chapter 7 and 8 we will present our experimental results with a cavity of finesse 100. In chapter 9 we will describe our recent efforts to move towards the observation of quantum effects.

The important factor determining the non-linearity is the number of atoms which are influenced by a single Rydberg excitation. The sphere of influence is commonly referred to as the Rydberg bubble and the number of atoms blockaded are denoted by 𝑛 𝑏 . During the last few years, there were several new experiments which started employing Rydberg ensembles for strong optical non-linearities. The proposed idea is to slow down light using EIT conditions and then to further enhance the nonlinearity using Rydberg-Rydberg interactions. This helps us to study many body interaction effects as well as the optical non-linearity at the few photon level.

To strongly increase the Rydberg effect, one needs to probe the system in a regime where even a single Rydberg excitation can change the cloud's optical response which can enable us to perform logical gate operations.

Dispersive non-linearities

As absorptive non-linearities in free space cause an irreversible loss of photons, one of the first experiments performed on this setup was to observe dispersive nonlinearities. In free space, the effects are weaker which makes them harder to detect.

Using an optical cavity one can amplify these effects into resonance shifts. In this section we will give a brief overview of dispersive non-linear measurements in classical regime carried out in our group during 2011-12. The experimental results and theoretical models were published in [START_REF] Parigi | Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms[END_REF] and [START_REF] Stanojevic | Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium[END_REF] respectively.

As we have seen in section 3, the Van der Waals interaction between Rydberg atoms shifts the resonance of nearby atoms and modifies its optical susceptibility.

To avoid any resonance absorption the atoms are excited off-resonantly to a Rydberg state. The goal of the experiment is to observe nonlinear dispersion by Rydberg excitations using the scheme illustrated in Figure 6.13.

g In order to measure the non-linear response, the cavity's length is scanned to probe the resonance position for various probe field intensities. Due to the Rydberg blockade effect, the optical susceptibility moves closer to the two level system for higher probe powers. The Rydberg induced shift of cavity resonance was measured as a function of the normalized probe intensity (Y), for different Rydberg levels.

Ω cf |5S 1/2 ,F=2,m F =+2〉 ∆ = -25γ δ=0.3γ
A more precise criteria, to determine if the system can be used for phase gate like operation is to measure the non-linear resonance shift equivalent to the cavity linewidth. For n=62 D 5/2 , the steady state photon number inside the cavity needed for shift of transmission peak by 𝛾 𝑐 is 30. One cannot take only photonic excitations in the system as a benchmark, as there are many atoms (∼ 5000) excited to the Rydberg state in the steady state. Nevertheless, the non-linearity is much stronger compared to the two level system where all the atoms(10 4 ) must be saturated to observe the expected behavior. The susceptibility was amplified in the presence of cavity of finesse ∼ 100. The expected 𝜒 (3) value for the system is 4.10 -9 m 2 .V -2 .

It is only two orders of magnitude lower than the resonant non-linearities observed in free space [START_REF] Pritchard | Optical nonlinearity in a dynamical Rydberg gas[END_REF], and the largest observed dispersive susceptibility at the time of measurement.

A large non-linear dispersion should pave the way for optical bi-stability over a certain intensity, but the lack of such observation can be explained by the Rydberg Chapter 7

Measurement of Absorptive Optical Non-linearities The main experimental requirement to observe a strong non-linear effect is a small dense cloud whose size is comparable to the size of a typical Rydberg blockade radius. An advantage of having a small cloud is that it allows us to focus our blue coupling beam to waist size comparable to the cloud size, which in turn gives us the ability to access high lying Rydberg states. In this chapter we will describe how we prepare our atomic cloud and demonstrate classical non-linearity measurements with Rydberg S and D states.

In the current experimental setup, the atoms are excited resonantly, hence the Rydberg-Rydberg interactions leads to optical losses in transmission. The objective of this experiment is to measure the scaling of non-linearity for different Rydberg levels.

Main cavity

A cavity plays a crucial role in defining the optical mode, which interacts with the atoms. It enhances the coupling strength between the atoms and the light, which is defined by the mode volume and the cavity finesse. However it brings in more constraints such as restriction of frequencies which can be used to excite the atoms, and complexity. The cavity systems require precise positioning with respect to atoms and length stabilization. We will show how we resolve the technical issues by using a system which is designed with external control while still staying under vacuum.

MOT Coils

MOT Trapping beams

Cavity holder The cavity is positioned vertically parallel to both the MOT coils and its length is set to 6.6 cm which allows us to excite Rb 87 atoms from both the hyperfine levels of the ground state 5 𝑆 1/2 (which are separated by 6.8 GHz). To have maximum overlap

R 1 > ±0.1 R 2 ≈ T 2 T 1 γ c = c 4L T 2 = 2π 10 ± 1 MHz ≈ 2π 11 MHz

Calibration of photon flux

In this part we will describe how to evaluate the number of photons in the cavity when it is excited with a continuous optical field. The number of photons inside the resonator depends on many different parameters like the cavity resonance, the losses, etc. To convert the detected photon rate into number of photons present in the system, we use input/output relations of a two sided cavity.

In general, if we consider a cavity with decay rates 𝛾 1 = 𝑐 2𝐿 𝑇 1 and 𝛾 2 = 𝑐 2𝐿 𝑇 2 for the top and the bottom mirrors and the cavity is locked on resonance then in steady state

𝑎 𝑜𝑢𝑡 + 𝑎 𝑖𝑛 = √ 𝛾 1 𝑎 𝑏 𝑜𝑢𝑡 + 𝑏 𝑖𝑛 = √ 𝛾 2 𝑎 (7.1)
where a and b refer to the fields on the top and bottom side of the cavity respectively as illustrated in from the following equations:

𝑎 𝑜𝑢𝑡 = 𝛾 1 -𝛾 2 𝛾 1 + 𝛾 2 𝑎 𝑖𝑛 𝑎 = 2 √ 𝛾 1 𝛾 2 + 𝛾 2 𝑎 𝑖𝑛 𝑏 𝑜𝑢𝑡 = √ 𝛾 1 𝑎 = 2 √ 𝛾 1 𝛾 2 𝛾 1 + 𝛾 2 𝑎 𝑖𝑛 (7.2)
As the decay rate of the bottom side mirror is much higher than the top one then

HWHM of cavity 𝛾 𝑐 = 𝛾 1 +𝛾 2 2 ≈ 𝛾 2 2
. If we feed the cavity with a coherent state 𝛼 then under steady state condition the intracavity field is given by 𝑎 = -𝑖𝛼

𝛾𝑐

. Hence the average photon number inside the cavity is given by 𝛼 2 /𝛾 2 𝑐 , and the output flux of the cavity is given by 2𝛼 2 /𝛾 𝑐 .

Experimental calibration

To calibrate the photon flux, we measure the output of the cavity in the absence of atoms and the probe is incident on resonance with the cavity. We measure the photon count on the SPCMs.

Preparation of a small cloud

In order to generate a cloud of few tens of microns in size, a very usual employed method is to load the atoms from the MOT into a crossed dipole trap. In the following section, we will describe the loading and preparation of a small dense atomic cloud.

Dipole traps

Atoms can be trapped by using radiative forces from an optical beam [START_REF] Grimm | Optical Dipole Traps for Neutral Atoms[END_REF]. This can be achieved by using lasers far from any atomic resonance. In practice, it is easier to work with red-detuned gaussian beams, in which the atoms are pushed towards its center. But any dipole trap with large Rayleigh range can only confine the atoms in directions orthogonal to the beam propagation. Hence, To achieve a three dimensional confinement we use more than one dipole trap.

On our experiment, we use a red detuned beam from D1 and D2 transitions at 810 nm from Ti-Sa lasers to create dipole traps; one along the cavity axis and two orthogonal to it. The dipole trap along the cavity is created from the same beam that is used to lock the cavity's length.

MOT 5S 1/2 5P 3/2 F =3 F =2 F =1 F =0 F =2 F =1 Repumper 5P 1/2
Dipole traps Depumper

1/2 1/2 F 87 ≈ 8 mm μ U dip = Γ 2 e 8Δ I Isat

Cloud preparation from dipole traps

As the cavity dipole trap has a Rayleigh range larger than 1 mm, the atoms a trapped in a cigar trap. In addition, the atoms outside the horizontal dipole traps require upto few 100 ms to escape but if one waits too long it will result in lower atomic densities. In order to only use the atoms which are in the crossed dipole traps we use a selective depump-repump scheme which allows us to achieve small clouds in reasonably short time.

We load the atoms from the MOT into crossed dipole traps by overlapping both the traps during 20 ms (upto 80 ms) of molasses stage. As the repumping power is set to low value, most of the atoms are already depumped to a dark state (5 𝑆 1/2 𝐹 = 1) where the atoms are invisible to the probe transition (5 𝑆 1/2 𝐹 = 2 → 5 𝑃 1/2 𝐹 = 3).

We use an additional depumper pulse on the D1 transition to depump any left over atoms to the dark state. Then for 100 𝜇s we shine a repumper pulse (5 𝑆 1/2 𝐹 = 1 → 5 𝑆 1/2 𝐹 = 3) along the horizontal traps and a circularly polarized optical pumping pulse (5 𝑆 1/2 𝐹 = 1 → 5 𝑃 1/2 𝐹 = 1) along the cavity dipole trap. This allows us to spatially isolate the atoms that are of interest and prepare a small dense cloud in the state 5 𝑆 1/2 𝐹 = 2 𝑚 𝐹 = 2. In addition, it improves the duty cycle of the experiment as we don't have to wait until all the atoms are lost.

The final loading sequence is shown in Figure 7.8. After loading the trap, we image the trapped atoms for 1 ms. From is 25 𝜇m. The orientation of the cavity trap is not vertical as the imaging axis is not orthogonal to any of the traps. Finally, the size of the small cloud is 35 𝜇m.

Cooperativity measurement

The trapped small cloud is characterized by its cooperativity. The cooperativity of the system can be probed by measuring the dispersive shift or resonant absorption.

As the cooperativities obtained with our small cloud are less than 50, we cannot obtain a precise value using dispersive shift measurements.

We use a 780 nm optical beam close to the atomic transition (5 S 1/2 F = 2 → 5 P 3/2 F' = 3) to probe the atoms. The cavity is locked on resonance with atoms while the probe frequency is scanned around the resonance of the atoms from -

Empty cavity Cooperativity

Atomic inhomogeneities

It is important to account for various inhomogeneous effects which contribute to the deviation of experimental observations from the ideal behavior. Due to the selective preparation of the small atomic cloud it creates many atoms trapped in dark state (F = 1) in the cavity mode. In small numbers these atoms wouldn't affect the probe transmission but once we reach cooperativities of few hundreds for atoms in F = 1 we see a shift in the cavity's resonance. This shift leads to the observation of asymmetric normal modes even when the cavity is locked on resonance with the atoms. We measure the cavity transmission in the presence of depumped atoms and no atoms and the shift corresponds to a cooperativity of ≈250 for atoms in F=1.

Finally, the probe creates a standing wave inside the cavity which makes it impossible to have uniform coupling strength for all the atoms. The probability distribution of the intensity seen by the atoms is shown in Figure 7.11. We assume an average coupling parameter for all the atoms and the approximation is valid for short time scales when the atoms don't move.

EIT in a cavity

The small cloud allows to have a strong uniform blue beam to couple to the Rydberg states. But the power we have is still not sufficient to reach high lying Rydberg states. Therefore, a blue cavity was designed to have a large optical powers while minimizing the effects of standing wave.

Blue cavity

The weak dipole moment of the coupling transition requires a large amount of coupling power on the Rydberg transition. In order to enhance the Rabi frequency of the blue laser we use a cavity to increase the effective power of the coupling beam on the atoms. The easiest way to design a cavity that wouldn't require breaking the vacuum is to build a cavity outside the chamber. In order to minimize the standing wave effect on atoms, a confocal cavity is chosen with minimum overlap of incident and reflected beams. We use viewports which are anti-reflection coated for blue frequencies to construct the blue cavity with a length of 1 m and curved mirrors of f = 500±5 mm radius. To match the losses of the viewport the input coupler of the cavity was chosen with a reflectivity of 2.9% and the other mirror has a reflectivity (≫99.9%) from Layertec. By scanning the length of the cavity we measure the finesse of the cavity to be 90. The enhancement factor between the intra-cavity power and the incident beam is 14. This was measured from the scattering on one of the viewports. The waist of the blue beam can be set using the optics which couple the light to the cavity.

Stabilization of cavity length

Since the cavity is 1 m long and the mirrors rest on two independent breadboards situated 25 cm above the optical table, it has numerous sources of noise, making it a challenging task to stabilize its length.

We mount each mirror on two different kinds of piezos to cancel oscillations in different frequency range. One of the mirrors is mounted on a standard lens holder which is fixed to a translating piezo (P-611.10) with travel range of 120 𝜇m. The other one is glued onto a cylindrical piezo (PICA P-025.10H, 15 𝜇m range).

An auxiliary beam at 810 nm was used to stabilize the cavity length. The locking scheme was designed by taking advantage of the inherent birefringence created by the viewports present inside the cavity. We used a Hansch-Couillaud method to obtain an error signal to lock the cavity [START_REF] Hansch | Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity Optics[END_REF]. After optimization we couldn't attain a stable lock which was attributed to the stability of the breadboards.

The mirrors are placed on a 1.3 cm thick plain aluminum breadboards (Thorlabs MB4545) at a height of 25 cm, mounted on 4 legs of 1" diameter. We replaced them with 450 x 600 mm honeycomb breadboards (Thorlabs B4560A) with 6 cm thickness and mounted on legs of 1.5" diameter. However, this did not resolve the issue but helped to eliminate the mechanical noise. We observed that even though the optical path was stabilized for 810 nm, it wasn't for 480 nm. The reason was due to large

𝑑𝑡 = 𝑖(𝐷 𝑟 -𝐷 𝑒 - Ω 2 𝑐𝑓 4𝐷 𝑟 )𝜎 𝑒𝑟 + 𝑖 Ω 𝑐𝑓 2 (𝜎 𝑟𝑟 -𝜎 𝑒𝑒 ) + 𝑖𝑔𝑎 † 𝜎 𝑔𝑟 𝑑𝜎 𝑒𝑒 𝑑𝑡 = -2𝛾 𝑒 𝜎 𝑒𝑒 + 2𝛾 𝑟𝑟 𝜎 𝑟𝑟 -𝑖 Ω 𝑐𝑓 2 (𝜎 𝑒𝑟 -𝜎 𝑟𝑒 ) + 𝑖𝑔(𝑎 † 𝜎 𝑔𝑒 -𝑎𝜎 𝑒𝑔 ) 𝑑𝜎 𝑟𝑟 𝑑𝑡 = -2𝛾 𝑟𝑟 𝜎 𝑟𝑟 + 𝑖 Ω 𝑐𝑓 𝑓 2 (𝜎 𝑒𝑟 -𝜎 𝑟𝑒 )
where 𝛾 𝑟𝑟 is the population decay rate of the Rydberg state. The number of Rydberg excitation in the cavity can be evaluated from

𝑁 𝑟𝑟 = 𝑁 𝜎 𝑟𝑟 = 𝑁 g𝛼 ( D𝑒 𝐷 𝑐 -2𝛾 𝑒 𝛾 𝑐 𝐶) (7.4)
where g =

Ω 𝑐𝑓 𝑔 2𝐷𝑟 and 𝛼 = 2𝛾𝑐𝑃 𝑖 𝜔𝑝 .
For our typical experimental conditions, i.e. C = 10, ∆ 𝑒 = 0, ∆ 𝑐 = 50𝛾 𝑒 the locking beam Rabi frequency of Ω 𝑐𝑓 = 0.1𝛾 𝑒 limits the probability of Rydberg population to less than 10 -4 in the atomic cloud. By measuring the scattered power, we estimate that the intra-cavity power is stable within 10% of the maximum value.

Parameter Value

Blue cavity alignment and inhomogeneities

The cavity length was adjusted to confocal by using the separation between the transverse modes in the reflected light when the cavity's length is scanned. By carefully merging we can obtain a confocal cavity but a clean symmetric signal is obtained only when its length is exactly equal to 1 m.

In order to align the beam with the atoms, we shine on-resonant probe and look for the EIT signal in transmission by using only 3% of the transmitted light. Once the beam's position over the atoms is optimized, while having the reflection overlap with the incoming beam, its position is fixed and we only adjust the rear mirror to align the cavity with the incoming beam. This allows us to optimize the cavity mode on the atoms.

When coupling the blue beam to the atoms, its waist plays a crucial role in determining the quality of the EIT signal. A non-uniform blue results in asymmetric absorption across the cloud. At the same time we cannot use a very large blue beam waist compared to the atomic waist, as it will reduce the blue beam intensity. In addition due to interference between the counter propagating fields is not completely negligible. For a small waist (∼ 40 𝜇𝑚), we can obtain more homogeneous high intense beam but most of the atoms see zero intensity, while for large beam waists (∼ 180 𝜇𝑚) most atoms see non-zero blue intensities but the average intensity is low. To reach a compromise between the two extreme situations we choose 90 𝜇𝑚 (refer Figure 7.14). Experimentally, in order to determine the suitable waist we optimize the EIT transmission by adjusting the waist size. We observe that the maximum EIT signal was observed for waist size twice the cloud size, i.e. 90 𝜇m,

where the average intensity observed by the atoms is still higher with not many atoms seeing zero intensity.

Linear regime

The probe is coupled to the fundamental mode of the cavity and the coupling light is sent orthogonal to the cavity mode as shown in Figure 7.12. As discussed in Section: 2.4, when we couple a coupling laser between the excited level to a long lived state, the medium can be rendered transparent. In this section we will describe the EIT measurements performed using our cavity.

To avoid, at first, any nonlinear effects arising from a Rydberg blockade, we couple the atoms to a level with a low principal quantum number using a control field resonant with the 5 𝑃 3/2 F = 3 → 37 𝐷 5/2 F = 4 transition. Since the Rydberg blockade effect is negligible, the response is identical for S and D states. In Figure 7.15, we show the cavity transmission as a function of the probe detuning. In the presence of the control field which couples the atoms to the 37 D Rydberg level, a transparency window is created at the two-photon resonance frequency. We note that such intracavity linear EIT was observed in many setups [START_REF] Mücke | Electromagnetically induced transparency with single atoms in a cavity[END_REF][START_REF] Albert | Cavity electromagnetically induced transparency and all-optical switching using ion Coulomb crystals[END_REF] and more recently in a similar setup [START_REF] Ningyuan | Observation and characterization of cavity Rydberg polaritons[END_REF]. We note that the centre of the two normal modes doesn't coincide exactly with the empty cavity resonance. This effect, taken into account in the fitting curves, is due to the presence of background atoms in a dark state (5 S 1/2 𝐹 = 1) which shifts the cavity line.

Probe detuning (MHz

Rydberg linewidth

Due to weak dipole moments the natural Rydberg linewidths are extremely narrow but due to experimental noise the linewidth can be broadened. Several sources for these noise are oscillating weak electric fields, blackbody radiation, laser linewidth etc. Due to the presence of a weak static electric field in an unknown direction (created by stray charges) our quantization axis is not fixed. This allows us to couple to all the m 𝐹 sub-levels in D states. To quantify the Rabi frequency and linewidth we measure how the cloud transmission varies for different blue powers.

By using the fact that the Rabi frequency of the control field is proportional to √ 𝐼 𝑐 , where 𝐼 𝑐 we can scale it accordingly.

We fit the cavity transmission for one linewidth and then extract the Rabi frequency from it. Then we compare how well the parameters agree when we scale the Rabi frequency with the measured intensity. As we can see in Figure 7.16 the fit for n = 37 𝐷 5/2 agrees well when the linewidth = 0.14 𝛾 𝑒 which gives ≈ 450 kHz. The natural radiative lifetimes are much lower and vary with the principal quantum number. The absence of this variation suggests that the linewidth is limited by different experimental noise sources. The laser linewidth even after optimization couldn't be lower than 50 kHz due to experimental constraints. The Doppler effect is the main source for the broadening, it has been estimated that in the worst case that it could lead to halfwidth at half maximum of ∼ 250 kHz for Rb atoms at 40 𝜇K. In addition, AC fields can also broaden the linewidth, as we will show in Section 7.5.3 there is a static electric field of ∼ 100 mV/cm.

Non-linearity measurements

Once we increase the power of the probe beam it creates many Rydberg atoms in the cloud which lead to the creation of blockaded atoms. Thus Rydberg-Rydberg interaction scatter the probe light and the transmitted light starts to attenuate with increasing probe power.

Resonant nonlinearity with Rydberg S state

For S states, the nonlinear effect is induced by a long range and isotropic potential of the form 𝑉 (𝑟) = -𝐶 6 /𝑟 6 , where 𝑟 is the interatomic distance and the 𝐶 6 coefficient for 87 Rb atoms is given by [START_REF] Singer | Long-range interactions between alkali Rydberg atom pairs correlated to the ns-ns,np-np and nd-nd asymptotes[END_REF]:

𝐶 6 ≈ ( 63 -267 ( 𝑛 60
) + 64 ( 𝑛 60
) 2 ) ( 𝑛 60
) 11 GHz • 𝜇m 6
where n is the principal quantum number. This Van der Waals type of interactions creates a blockade effect in EIT conditions [START_REF] Pritchard | Nonlinear optics using cold Rydberg atoms Annual Review of Cold Atoms and Molecules[END_REF], so that the transmission of the medium decreases with increasing probe field strength. This translates, in our case, to a cavity transmission that reduces with the number of photons. We measured this transmission by scanning the probe for various photon rates (Figure 7.17a). To bring out more clearly the nonlinear absorption, we take the transmission at zero detuning in Figure 7.17a and plot it as a function of the probe photon rate measured with the same probe power for an empty cavity (Figure 7.17b). This procedure has been repeated for various principal quantum numbers, since we expect a stronger nonlinearity for higher n.

We used a mean field approximation to calculate the expected transmission.

More details can be found in Section 3.2. The experimental parameters such as the cooperativity, Rydberg state linewidth or the Rabi frequency of control field have been measured (section 7.4.2), leaving no free parameters in the model. The experimental data appears to be in a relatively good agreement with the theory for any probe detuning or Rydberg level (Figure 7.17), at least for small photon rate, i.e. as long as the fraction of blockaded atoms remains small. This is the first experimental test of the model provided in [START_REF] Grankin | Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: A perturbative approach[END_REF] and it seems reliable to predict the nonlinear transmission of the system. Unfortunately due to limited blue intensities we couldn't access higher Rydberg S states. The maximum cooperativity per blockade for the 79 S 1/2 state is only 0.1 which is not sufficient to observe strong quantum non-linear effects.

Resonant non-linearity using Rydberg D states

Rydberg D-states have higher coupling strength which would allow us to reach high lying states for same blue powers. The eigen modes of the blue cavity determine the polarization of the control field so it restricts our freedom to choose a specific magnetic sub-level. Nevertheless, we measure the Rydberg D state non-linearity by starting with a probe scan in linear regime. We observed that instead of observing a single Rydberg D transition we see multiple transmission lines in EIT spectrum.

This can be caused by the stark effect of Rydberg D states.

Stark shift for Rydberg levels

Rydberg states are so sensitive to weak external fields that they are used as electric field probes for very weak fields of the order of tens of micro volts per centimeter [START_REF] Neukammer | Spectroscopy of Rydberg Atoms at 𝑛 ≈ 500: Observation of Quasi-Landau Resonances in Low Magnetic Fields[END_REF][START_REF] Holloway | Broadband Rydberg Atom-Based Electric-Field Probe for SI-Traceable[END_REF]. However it can be problematic if any stray electric fields around the atoms are not properly controlled.

In the presence of weak electric field the degeneracy of hyperfine sub-levels is lifted due to the stark effect. This effect can be quantified by considering the hamiltonian of the system where the field is along the 𝑧 axis. We estimate the splitting for various field strengths using the model in Appendix C .

In our case, a small stray electric field was sufficient to lift the degeneracy as shown in a spectroscopic scan for a high Rydberg level (Figure 7.18a). The observed splittings are in good agreement with a simple Stark shift model [START_REF] Beguin | Measurement of the van der Waals interaction between Rydberg atoms[END_REF], see Figure Electric eld along the quantum axis Fz (V/cm) respect to the electric field in quantization axis. By comparing it to the measured splittings in (a), we can estimate the stray electric field in the quantization axis to be about 90 mV/cm. In order to be in linear regime, the probe photon rate is less than 10 photons/𝜇s.

x x x |m J | = 1/2 |m J | = 3/2 |m J | = 1/2 D 3/2 D 5/2 D 5/2 (a) (b)

Dephasing in Rydberg D-States

Compared to S states, Rydberg D states have larger dipole couplings to the intermediate 5 𝑃 3/2 state, which allows one to reach states with higher principal quantum numbers for the same coupling light intensity, and thus to get higher nonlinearities due to the associated increase of the blockade volume. However, no quantum effects such as anti-bunching have been observed for D states even though several experiments have been conducted to observe a nonlinearity in such systems [START_REF] Pritchard | Optical nonlinearity in a dynamical Rydberg gas[END_REF][START_REF] Tresp | Dipolar Dephasing of Rydberg 𝐷-State Polaritons[END_REF].

One must note two differences with S states which might complicate the blockade effect. Primarily, the Rydberg interactions for D states are anisotropic, they can even vanish for some orientations [START_REF] Saffman | Quantum information with Rydberg atoms[END_REF]. This angular dependance has been directly measured for two atoms in the 82 D 3/2 state [110]. In our model, we considered however an isotropic interaction [START_REF] Stanojevic | Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium[END_REF]: (keeping the probe and control field on resonance), we observe a transient decrease of the transmission during ≈10 𝜇s (Figure 7.19(b)). This dynamical behaviour was also observed in [START_REF] Tresp | Dipolar Dephasing of Rydberg 𝐷-State Polaritons[END_REF] and is neither present for S states (Figure 7.19(a)), nor predicted by the steady-state model we used in section 7.5.1. In addition, we have also observed a decrease in transmission even after switching off the excitation lasers for 10 𝜇𝑠 and probing it with a weak probe beam (as shown in figure 7.20). The decay time of 10 𝜇𝑠 is far from any timescales present in our steady-state model. red pulse (and both blue) contain less than 10 photons/𝜇s. We observe a transmission decay over time for D states for strong probe fields. This transmission decay also affects the transmission of the weak probe after switching off all the excitation lasers for 10 𝜇s.

𝐶
62S 1/2 60D 5/2 (a) (b)
To explain the transient behavior, we consider that, because of the anisotropy of D states and their multilevel structure, a Rydberg atom can decay into another level that is uncoupled from the control field, a dark Rydberg level. This process creates long-lived blockading Rydberg atoms which decrease the total transmission. This transient decay, slower than all the timescales predicted by our simple blockade model, is thus the sign of the creation of long-lived blockading atoms. In [START_REF] Tresp | Dipolar Dephasing of Rydberg 𝐷-State Polaritons[END_REF],

the experiments performed using Rydberg D states with degenerate Zeeman sublevels exhibited strong dephasing phenomenon. In their model, the decay was also explained by the evolution of Rydberg atoms into such levels. It is therefore interesting to notice that the dynamical behaviour is still present in our case, even when an electric field has lifted the degeneracy.

To account for this process, we included a decay term 𝜉 in the bubble model (see 3.3), to account for the decay rate into "dark" Rydberg states. This gives us a free parameter to fit the curves in Figure 7. 19(b). The good agreement with the data -for short times, i.e. when the medium is not saturated with blockade spheres -indicates that it is a good phenomenological model.

Even while scanning the probe rapidly around the two-photon resonance, we observed a nonlinear transmission similar to S states. The nonlinearity appears to be stronger, although the 𝐶 6 coefficient for D states are of the same order. The reason is that the transmission is affected by the creation of long-lived blockading atoms during the scan. Including the decay to dark Rydberg levels into our model allows us to fit the data with good confidenc (Figure 7.21) for all detunings and various principal quantum numbers. We obtained a decay parameter for each principal quantum number: 𝜉 = {1.8 ± 1.7, 2.2 ± 1.7, 2.3 ± 1.7, 1.1 ± 0.66} MHz for 𝑛 = {60, 66, 77, 85}, where the error bars contain a 95% confidence interval. It must be noted that once we fix the value of 𝜉, we observe that it fits very well for different cavity feeding rates. While this parameter allows to fit the data with good confidence, the large incertitude doesn't permit to conclude about its dependence on the principal quantum number. We believe that the creation of such long-lived Rydberg atoms is detrimental to the production of non-classical light states via the blockade-induced optical nonlinearity of the atomic cloud. Indeed, after the medium is saturated with long-lived blockading atoms, the medium has a low transmission, but it has a linear response to an incoming field. This may explain why effects such as photonic anti-bunching have never been observed in experiments where atoms were driven to Rydberg D states.

To use such transitions for quantum optics experiments, this decay mechanism must be circumvented. The above results can be found in the published article [START_REF] Boddeda | Rydberg-induced optical nonlinearities from a cold atomic ensemble trapped inside a cavity[END_REF].

7.5.4 𝜒 (3) determination

One can quantify the non-linearity by evaluating how many photons one require to observe the bistability in resonator systems. But as the Rydberg atoms evolve in the same time scales as our excitation pulses it is hinders us from observing the bistable behavior. Instead we evaluate the 𝜒 (3) value in the resonant regime using the following expression [START_REF] Sevincli | Nonlocal Nonlinear Optics in Cold Rydberg Gases[END_REF] 𝜒 𝑅 = -

4 √ 2𝜋 3 𝛾 𝑒 Ω 4 𝑐𝑓 𝐶 6 √ |𝐶 6 | 𝑘 3 √ 𝛾 𝑒 [𝛾 𝑒 𝛾 𝑟 + Ω 2 𝑐𝑓 ] 7/2 (7.5)
The susceptibility for our system at 90 D 5/2 is 7.5×10

-7 m 2 .V -2 " which is of the same order of magnitude as measured in the free space regime [START_REF] Pritchard | Optical nonlinearity in a dynamical Rydberg gas[END_REF]. The main limitations are the capability to reach high lying Rydberg states and the atomic density of our cloud. Even though the susceptibility is not much higher than the free space case, one must take into account the fact that the effects like phase shift induced by the media are amplified by the finesse of the cavity (∼ 100).

Conclusion

In this chapter, we described the two cavities present on our experimental setup. We presented the preparation cycle for a small atomic cloud using a selective depumprepump sequence. Here, we measured the Rydberg resonant non-linearity for both S and D states. We also showed that a semi-classical model can be used to reproduce the Rydberg S state non-linearity and a dynamical dephasing model to describe the non-linear effects for Rydberg D states using long lived dark states. The reasonably good fit suggests that the models are valid for any system where the cooperativity per blockade is smaller than 1.

Chapter 8

Second order correlation effects In order to observe strong resonant non-linear effects one would need the cooperativity per blockade sphere to be larger than one. In our experiment, however, the maximum value we could obtain was 0.1 and 0.2 for the 79 S and 92 D states respectively. For the observed non-linearity, weak correlation effects were predicted using theoretical models.

Rydberg-Rydberg interactions in an atomic cloud is a many body problems and cannot be solved analytically. It is possible to solve the system by perturbatively expanding the system's response in the lowest order of the feeding rate (𝛼) of the system. The average probe field component (⟨â⟩) at the lowest order depends only on the first or the third order of the feeding rate. To evaluate the second order effects, we measure the field correlation functions.

After preparing an optically pumped cold atomic ensemble, we study the nonlinear response of our system by characterizing the retrieved light using photon counting statistics and homodyne tomography. The calculation of the g (2) function is well described in the articles [START_REF] Grankin | Quantum statistics of light transmitted through an intracavity EIT Rydberg medium[END_REF][START_REF] Grankin | Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: A perturbative approach[END_REF] and in Andrey Grankin's thesis [START_REF] Grankin | Theoretical studies of optical non-linear effects in ultracold Rydberg gases[END_REF]. Here, we will numerically evaluate the function for the parameters of our system. We will also describe the squeezing spectrum measurements of the light transmitted through a cavity.

PBS

e |5S 1/2 |5P 3/2 |nD 5/2 (b) (a) (2) (2)
(2) g (2) (0) Characterized using:

g (2) ( √ 2π × 35) 3 μm 3 S 1/2 6 ≈ -3.6 × 10 3 • μ 6 Ω cf Ω cf = 2 γ e γ r (2C -
Ω cf /γ e Δ e = Δ c = Δ r = 0 Ω cf = 3 Δ c (2) (0)

Squeezing measurements using theoretical model

We can therefore expect to see, g (2) (0) > 1 for a resonant probe field when the impedance matching condition is set for single-photon component.

Photon counting measurements

The cavity is locked on resonance with atoms and the atomic cloud is prepared in the similar manner as the non-linearity measurements. According to numerical calculations, the correlation effects are observed both in transmission as well as in reflection. But the range of parameters necessary to observe non-linear effects in reflection correspond to the impedance matching condition where almost all the light is absorbed. Due to fluctuating cavity resonance and the oscillating control field Rabi frequency (Ω 𝑐𝑓 ) the system is pushed out of the impedance matching condition.

In transmission

Once the cloud is prepared, we shine a resonant probe on the cavity and look at the transmitted signal using two SPCMs. We look at the correlation effects between the two SPCM signal output under resonant EIT conditions (as shown in Figure 6.1).

According to numerical simulations, we must observe a weak anti-bunching close to 0.98 at resonance. Even after averaging over several cycles, the instability of cavity resonance didn't allows us to observe the expected anti-bunching effects on the transmitted light. By going Rydberg D-states one could have higher cooperativity per blockade which in turn can lead to stronger correlations, but the dephasing effect inhibits us from the observation of any higher order correlation effects.

-

(τ)

Phase noise at 700kHz

Relative phase Δ c = -0.5γ e Chapter 8. Second order correlation effects ment is not to observe squeezing which can be observed even simpler systems, we concentrated our efforts on moving towards quantum non-linearity.

Conclusion

In this chapter, we show that the second order effects can be used to study the nonlinearity of the system. We used numerical models developed by Andrey Grankin and studied how our system behaves under various resonant excitation schemes.

We measured the second order correlation function of the transmitted light but the fluctuations in the cavity resonance and the control field stamped out the correlation effects. In reflection, the bunching/anti-bunching correlations are observed only near the impedance matching condition where almost all the light is scattered.

In transmission, the 𝑔 (2) (𝜏 ) measurement was found to be the same as a coherent beam. This led us to conclude that the weak non-linear effects coupled with the experimental noise can damp out the correlation effects.

We showed that even with a laser with inherent phase noise, we can still study the physics of the Rydberg-EIT system. The observed rotation of the squeezing spectra was reproduced using a linear EIT model.

Conclusion

In this chapter we demonstrated how a small atomic cloud of few tens of microns size is loaded into three crossed dipole traps. The selective spatial preparation enables us to load atoms with very short duty cycles and large number of atoms. A build up cavity was installed to increase the power of the coupling beam such that it allows us to excite atoms to n = 95 for D states.

Experimentally, we couple to Rydberg states using resonant EIT transition.

Absorptive non-linearities were measured for both Rydberg S and D states. We developed theoretical models which explains the observed non-linearity with very good precision. The stark splitting created by weak electric field has been characterized and agrees reasonably well for different Rydberg D states. The cooperativity per blockade of the current system even when excited to highest possible Rydberg states is limited to 0.2 for n = 90 D 5/2 states and 0.1 for n = 79 S 1/2 states.

In addition, we also worked on observation of second order effects which helps us in giving a better understanding of the physics of our system. The correlation function of the transmitted or reflected light was evaluated perturbatively using second order expansion of feeding light. Unfortunately, the fluctuations of cavity moves the system out of resonance and averages the correlation effects.

The squeezing spectra gives more information about the higher order effects in the system. We measured and characterized the phase noise of the laser. In addition, we also explained the increased anti-squeezing observed through our cavity system using a linear model. We showed that these detection methods can help us understand more in details the physics of Rydberg-Rydberg interactions.

Part IV

Introduction

The observation of large classical non-linearities have encouraged us to find ways to improve the experimental setup even further. In free space, large non-linearities require a large single Rydberg sphere of the same order of magnitude of the atomic cloud [START_REF] Gorniaczyk | Single-Photon Transistor Mediated by Interstate Rydberg Interactions[END_REF]. In our system by using the cavity new schemes can be imagined where the Rydberg blockade sphere doesn't have to be very large. We will show that we can engineer a photonic gate using atomic systems without moving towards the cavity QED regime. This chapter will give an overview of our current efforts to increase the optical non-linearity of our system. In addition, we will also present multiple proposals through which we can probe the quantum optical non-linearities using correlation measurements and homodyne tomography.

An important parameter for observation of strong non-linear effects is the cooperativity per blockade. As we have observed in the previous experiment, the highest cooperativity per blockade we could achieve was about 0.2. To increase this value one must either have very large atomic densities (> 10 11 atoms/ cm 3 ) [START_REF] Hofmann | An experimental approach for investigating many-body phenomena in Rydberg-interacting quantum systems[END_REF] or higher finesse cavities.

Naively, one might assume that we can simply increase the repumper power in the preparation cycle which would efficiently repump all the atoms in the crossed dipole trap. Unfortunately, as we increase the repumper power, the size of the atomic cloud also increases as the atoms are repumped by its gaussian tail. Since, it leads to the same or even smaller atomic density, we didn't resort to this method.

Without increasing the finesse or decreasing the cloud size the optical non-linearity we can achieve is limited.

Quantum regime

If we use a small interacting mode-volume then it is easier to saturate the optical response by even a single Rydberg bubble. Hence, one must resort to higher finesse cavity with smaller mode waist to move to quantum regime. We therefore decided to change the cavity mirrors while still having the same length such that we could have a mode waist close to 15 𝜇m with a finesse of 1000 as proposed in [START_REF] Grankin | Quantum statistics of light transmitted through an intracavity EIT Rydberg medium[END_REF].

Rydberg bubble estimation

The cloud's non-linearity depends on the cooperativity of the single Rydberg sphere.

In order to have an estimate of the cooperativity per blockade sphere in the new cavity setup we consider the expression of the blockade volume on resonance [START_REF] Stanojevic | Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium[END_REF]:

𝑉 𝑏 = √ 2𝜋 2 3 ⎯ ⎸ ⎸ ⎷ 𝑖𝐶 6 𝛾 𝑟 - Ω 2 𝑐𝑓 /4 𝛾𝑟+𝛾𝑒-Ω 2 𝑐𝑓 /4𝛾𝑒 (9.1)
where 𝐶 6 is the van der Waal's interaction coefficient. The value of 𝐶 6 was measured using dispersive/absorptive non-linear measurements and its value is given by 𝐶 6 /𝛾 𝑒 = 35700 𝜇𝑚 6 for n = 61. For any Rydberg state n it scales as 𝐶 6 (𝑛) ≈ 35700 × ( 𝑛-1 60 ) 11 Starting with the parameters from the previous system, where the cloud is 30 𝜇𝑚 and the cooperativities are between 8-10, Since we need enough blue Rabi frequency to observe EIT at low power, we are somewhere on this curve where the value of 𝑉 𝑏 does not depend on 𝛾 𝑟 , and is of the order of 10000 𝜇𝑚 3 . Typically, the cloud has a measured cooperativity of ≈ 8, and an estimated density profile is proportional to If we have a cavity with a finesse 10 times higher and waist 5.67 times lower than the current cavity then each atom's cooperativity (∝ ℱ/𝜔 2 0 ) will increase by 320, for the same number of blockaded atoms, we can have a 𝐶 𝑏 ∼ 64. But by going to very high 𝐶 𝑏 , the cooperativity also increases and we lose the EIT transparency window since the control field power is limited. To circumvent this problem, we can lower the atomic density to decrease the cooperativity of the new system upto 100.

If we assume that only the atoms which are in the cavity mode of 15 𝜇𝑚 waist take part then we can have an effective volume

V = ( √ 2𝜋15) 2 ( √ 2𝜋30) ≈ 100000 𝜇𝑚 3 .
By using approximately the same blockaded volume, as we have now, the cloud can contain upto 10 blockaded spheres and the "cooperativity per blockade sphere" is still reasonably high. Even with a 𝐶 𝑏 ∼ 10, we are in a good range of parameters where quantum non-linearities should be observed. Another alternative would be to explore the dispersive regime where we can work with very high cooperativities [START_REF] Parigi | Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms[END_REF].

A new cavity system

The design of a one-sided cavity with a very narrow mode-waist is challenging because of the already existing constraints. As mentioned in the previous part, the distance between the mirrors must be set close to 6.6 cm to be able to address atoms present in both hyperfine levels 5S 1/2 F = 1 and F = 2. In addition, the MOT must be situated about 22 mm from the top mirror to have the maximum overlap between the two different standing waves.

Design

In order to obtain the smallest gaussian waist size one must move closer to the concentric configuration [START_REF] Hodgson | Laser Resonators and Beam Propagation[END_REF]. Given the restrictions with the MOT position, the inner radius of curvature of the top mirror and the bottom mirrors are fixed to r 𝑡 = 2.2 cm and 𝑟 𝑏 = 4.4 cm respectively. To achieve a finesse of 1200 at 780 nm, the bottom and top mirrors have a special coating to have a reflectivity of 𝑅 1 = 99.5 ± 0.15% and 𝑅 2 > 99.9% at 780 nm. In addition, to be able to couple blue light to the atoms along the cavity axis, the mirrors are specially coated to have

𝑅 1 , 𝑅 2 < 2% at 480 nm. 15μm 85μm R 1 ≈ 1 R 2 ≈ 0.95 R 1 ≈ 1 R 2 ≈ 99.

Mode waist calculation

The stability of a cavity depends on the distance between the mirrors (l) and their inner radius of curvature. By convention, we define 𝑔 𝑖 = 1 -𝑙 𝑟 𝑖 and the stability criterion is given as 0 < 𝑔 𝑡 𝑔 𝑏 < 1. The mode waist can be written as

𝑤 0 = √ 𝑙𝜆/𝜋 [ 𝑔 𝑡 𝑔 𝑏 (1 -𝑔 𝑡 𝑔 𝑏 ) (𝑔 𝑡 + 𝑔 𝑏 -2𝑔 𝑡 𝑔 𝑏 ) 2 ] 0.25 (9.2)
To obtain a mode waist of 15 𝜇m, the cavity length has to be set to 57 𝜇m shorter than concentric case. This gives us the beam waist at the top mirror using 𝜔 𝑡 = √ The minimum waist attainable for mirrors with fixed radius of curvature depends on the distance between them. The minimum waist occurs when the separation between the mirrors is equal to the sum of radius of curvatures (𝑅 1 + 𝑅 2 ). For a waist of 15 𝜇m the cavity length should be 57 𝜇m shorter than 66 mm.

In addition, we also took into account that the beam size at the viewports of the vacuum chamber are not bigger than them and any optical element cannot be placed at least 30 cm away from the mirrors.

Length Stabilization

As the top three piezos have to move the whole cavity holder setup from outside the vacuum, the length fluctuations at high frequencies (>50 Hz) couldn't be compensated. In order to circumvent this problem, a new piezo was installed to stabilize the cavity length. The only possibility was to install it under vacuum on the bottom mirror, which brought certain constraints on its shape and access. We use a ring type piezo with an inner diameter of 9 mm and outer diameter 15 mm, from Noliac, to allow light to be freely coupled in and out of the cavity. With a travel range of 3 𝜇m it gives us enough room to scan over 3 FSR. The piezo can be placed under UHV conditions and is glued to Kapton wires (with a SMA connector). We use a DN40 CF flange with a SMA feed through to supply voltage to the piezo.

In addition, the stabilization system we established for the previous cavity doesn't work because of the Ti-Sa broad linewidth even after locking its frequency.

The 810 nm Ti-Sa laser has a modulation at 100 kHz on its output which inhibits us from using it to stabilize the cavity. The only solution is to use a new diode at 810 nm to lock the cavity. In the previous resonator, we needed a locking beam of 10 mW to create a 300 𝜇K trap but in the new cavity we only need about 1 mW because of the finesse improvement.

Bottom mirror holder

Initially, the bottom mirror is attached to the cavity holder using a hollow cylinder and a cap with a spring. In order to eliminate all sources of instability, a new single piece is designed to hold the bottom mirror to minimize cavity length fluctuations.

Piezo

Installation and alignment of the cavity

Opening a vacuum chamber requires extreme care as it brings the inner parts in contact with external gases and possibly oils. To avoid that, we used two turbo pumps one connected to the main chamber and the other to the LVIS chamber to pump the system rapidly. Both of them are pumped down to 10 -7 mbar before opening the valve, which connects them to the vacuum chambers. Once we reach an equilibrium, dry nitrogen gas is slowly pumped into the system without letting any moisture in. After the vacuum chambers reach room pressure, we unscrew the cavity mount from the system while still pumping in dry N 2 gas. The cavity mirrors were replaced and the bottom mirror was glued to the holder to minimize vibrations. The pieces were baked at 120 ∘ C to remove any trapped gases from the surface.

Cavity finesse measurement

We determine the resonator lifetime by scanning the cavity length linearly. We create two side bands on a beam at a known frequency to allow us to determine rate of the scan. By measuring the reflection from the cavity we determine that the HWHM is ≈ 2 MHz, it corresponds to a finesse ℱ ≈ 500 (as shown in Figure 9.7).

It is considerably lower than what we initially expected and it can be attributed to 0.4% intracavity losses caused by some dust particle. The reflectivity of the bottom mirror was found to be ≈ 99.35%. By using the depth and the linewidth we estimate the intracavity losses and the reflectivity of the bottom mirror.

We spatially repositioned the beam to avoid any dust crept onto the surface while installing the system. Despite our best efforts, we couldn't achieve a higher finesse.

As the estimated intra-cavity losses are lower than the reflectivity of the bottom mirror, hence it should allow us to observe the impedance matching condition in reflection. To start with, we stabilize the length using only the fast piezo, but over a few hours the cavity drifts longer than the range over which we can act on. We use a low pass filter on the error signal to allow frequencies only below 50 Hz and use this to compensate for large drifts using the slow piezos. In order to optimally stabilize the cavity, we look at the probe transmission when it is detuned by 𝛾 𝑐 from resonance where it is most sensitive to fluctuations.

We measured that the cavity lock frequency has a standard deviation of 0.35 MHz, less than 10 % of the full width at half maximum.

Cooperativity and EIT measurements

After the installation of the cavity, we were unable to trap the same number of atoms as we used to before opening the vacuum chamber. Given the specific configuration through which we load our atoms, it is not very straight forward to determine if the atoms are not loading in the LVIS trap or in the main trap. The only possible explanation after realigning all the MOT beams and adjusting the LVIS atomic flux was that we ran out of Rb atoms.

Installation of a new Rb dispenser

As Rubidium is highly reactive to water, we use a special technique to avoid any contact with atmospheric moisture. The Rb dispenser was developed in the SYRTE laboratory in Paris, and is available in glass cells in solid form with an approximate weight of 2 gms. The glass cell is placed inside a malleable tube and attached to the chamber via a flange. After the removal of the old Rb cell, the new dispenser is installed without breaking the glass cell. Once we pump the system down to ∼10 -8 mbar, we squeeze the malleable tube using a special tool until we observe a spike in the pressure. After about a week of heating the dispenser at 50

∘ 𝐶, we managed to load the atoms in our MOT.

Cooperativity measurement

After the cavity mirrors were changed, the fundamental mode of the cavity no longer overlaps with the atomic cloud. In order to position the cavity mode onto the atomic cloud, we prepare a large cloud by increasing the atomic flow using a large B field gradient in the LVIS trap. We move the trap position by using the external magnetic field compensation coils. Once we see the normal mode splitting, we position the cloud more precisely using a smaller atomic cloud.

The cooperativity is measured using the same sequence as the previous chapter.

We prepare the cloud using 40ms of MOT cycle but without any preparation cycle.

The shape of the cloud in the current system is cigar shaped with a waist of 15 𝜇m.

The cavity transmission is measured with the probe detuning scanned around the resonance of the atoms. The cooperativity for the same number of atoms in the new cavity is enhanced by the increase in finesse i.e. 5.

EIT measurement

We avoid the Blue cavity to minimize the fluctuations in control field Rabi frequency.

In order to couple to Rydberg states, we can either incident the blue beam along the cavity mode or we can also couple it perpendicular to the cavity axis. The main advantage of the former setup is to have the freedom to select the polarization of the blue beam which will allow us to select the magnetic sub-level to excite to in Rydberg D-states.

Since the Blue is coupled in the same path as the probe, we can no longer have the enhancement cavity. In its absence, the control field power near the atoms is lowered by a factor ≈ 15 if we use the same waist for the same incoming control field power. But its waist can be reduced by 3 times (from 90 𝜇𝑚 to 30 𝜇𝑚) as the cavity mode is only 15 𝜇𝑚. Effectively, the Rabi frequency will be at least 0.6 times the one with the cavity and can be easily higher since we can channel all the power to excite to only one specific sub-level. We measure a control field Rabi frequency of 5 𝛾 𝑒 for a power of 50 mW as shown in Figure 9.9. We should be able to achieve at least twice the current Rabi frequency by replacing the amplifier of the blue laser. From the above measurements we observed that indeed with the new setup, the control field is stronger than what we have estimated.

Increasing the MOT density

With the new narrow dipole trap, we are not able to efficiently load the atoms into a 15 𝜇𝑚 trap. The most likely explanation is that we are limited by radiative losses due to the presence of repumper during the molasses which limits the loading efficiency. Currently, we are exploiting various schemes to make even larger atomic densities by using dark MOT cycle which enables us to achieve small samples with high density.

Possible schemes to exploit

After characterizing the non-linearity of the new atom-cavity system, there are many schemes that we can explore. One of our current goal is to be able to generate deterministically non-classical fields for quantum information processing tasks.

Off-resonant excitation

An interesting experiment that can be carried out with the new setup would be the theoretical scheme proposed in [START_REF] Grankin | Quantum statistics of light transmitted through an intracavity EIT Rydberg medium[END_REF]. In this configuration the statistics of the transmitted light is measured when excited off-resonantly to an intermediate state but close to the resonance of the two photon transition. This approximation is only valid when the number of Rydberg excitations is much smaller than the number of atoms.

As the intermediate state is far-detuned, one can adiabatically eliminate the population and coherence term as it mostly stays unpopulated [START_REF] Brion | Adiabatic elimination in a lambda system[END_REF]. Effectively, the system behaves as two-level superatoms and using a slight two-photon detuning will allows us to observe dispersive effects. The strong Rydberg-Rydberg interactions divide the cloud into N 𝑏 spheres (with n 𝑏 atoms) where each one cannot have more than one Rydberg excitation [START_REF] Vuletic | Quantum networks: When superatoms talk photons[END_REF][START_REF] Guerlin | Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble[END_REF]. The number of atoms per bubble is given by Equation 3. √ 𝑛 𝑏 . This is similar to the well known Tavis-Cummings model [START_REF] Tavis | Exact solution for an N-molecule-radiation-field Hamiltonian[END_REF].

The correlation function of the transmitted light can be shown to be equal to that of the intracavity field [START_REF] Walls | Quantum Optics Walls and Milburn[END_REF]. The g (2) function is evaluated from the coupled spin-field evolution equations by assuming that, at maximum, only the N 𝑏 Rydberg excitation can survive in the medium.

Physical interpretation

The non-linearity arises from the coupling of the nonlinear spin operator to the cavity. The presence of a finite number of atoms in each blockade sphere breaks the harmonicity of the excitation space. The excitation to doubly-excited subspace is detuned by a parameter κ compared to the single excitation subspace. Hence one can tune the parameters to go on resonance with either one or the other, and therefore one can observe bunching or anti-bunching correlations in the transmitted light. The numerical simulations for the new cavity using perturbative approach does agree with this physical picture (see Figure 9.11). All these parameters are within the current experimental reach.

On-resonance excitation

Going to an all-resonant case will increase the non-linearity, but at the cost of large optical losses. The current 'low' finesse cavity will help us to circumvent this problem without resorting to the cavity QED regime.

Cavity phase shift

Using an optical resonator one can convert losses into phase shifts. To achieve that, an unbalanced cavity with one mirror reflection much higher than the other is required. By using a single ended cavity in reflection one can move between two extreme regimes by tuning the intracavity losses only by the transmission of the 'lossy mirror'. A possible implementation scheme is illustrated in in Figure 9.12. In the absence of intra-cavity losses the incident field on the 'lossy mirror' enters into the cavity and is reflected with the same phase. If the losses are much higher compared to the transmission of the mirror then 'almost' all the incident field acquires a phase 𝜋 in reflection. We showed in Section 6.2.1 that our detection can be used to detect these phase shifts. The goal is to create strong enough losses by using a single Rydberg excitation in the atomic cloud.

Quantum scissors experiment

In the blockade model, we assume that in every Rydberg sphere there is only one gets dephased due the interaction potential. In retrieval process, one can extract only the excitations phase matched with the initial state and hence the photonic component retrieved will be of the form [START_REF] Stanojevic | Generating non-Gaussian states using collisions between Rydberg polaritons[END_REF][START_REF] Ourjoumtsev | Coherent Interactions between optical photons using single atoms and atomic vapors Memoire d'habilitation à diriger les recherches Institut d'Optique[END_REF] |𝑎

𝑜𝑢𝑡 ⟩ = |0⟩ + 𝛼|1⟩ √ 1 + 𝛼 2 (9.5)
where 𝛼 is the amplitude of the input coherent state. In addition to the creation of non-gaussian states, it allows us to study Rydberg-Rydberg collisional physics.

In free space systems, the photon packets get distorted by propagating via highly non-linear medium which renders them unusable for quantum operations but using the cavity systems can avoid this distortion.

Photonic controlled-phase gate proposal

In order to estimate the above effect more precisely, we use a pulsed scheme to implement the controlled phase gate. It is one of the universal gates and is a building blocks of quantum computation. One of the input qubits act as a control and imprints a phase of 𝜋 on the other one depending on its incoming state.

As we have seen earlier, in free space locality and causality prevent us from achieving a perfect phase shift without destroying the purity of photons. We can overcome this by using a cavity scheme where one of the photon is stored in the cloud as a Rydberg excitation. The proposal and the calculations are performed in collaboration with Anders Sørensen's group [START_REF] Das | Photonic controlled-phase gates through Rydberg blockade in optical cavities[END_REF].

Gate implementation

In a single-rail system, the information is encoded in the presence or absence of a photon. We denote a qubit with no photons in the pulse as |0⟩, and with a single photon as |1⟩. The first photon pulse ('control') is stored in the atomic cloud via EIT to a state |𝑟⟩ using a control field Ω 𝑐𝑓 . This excitation is then transfered to another Rydberg state |𝑟 ′ ⟩. A second pulse ('target') is then incident onto the cavity. If the cloud doesn't contain a Rydberg excitation (|𝑟 ′ ⟩), then the second pulse will be reflected with the same phase. Conversely, if there is a Rydberg atom then it will detune the state |𝑟⟩ out of resonance for the remaining atoms. In this case, the photon field entering the cavity is lost and the field reflected acquires a phase shift of 𝜋 as shown in Figure 9.13.

In dual-rail systems, the information is encoded in the state of the photon rather than its presence. For eg: the logical states |0⟩ and |1⟩ can be encoded as two orthogonal polarizations of a single photon. As opposed to single rail implementation, the dual rail systems are conditioned by the detection of both photons.

Choi-Jamilowski Fidelity

To evaluate the performance of the gate operation, the Choi-Jamiolkowski fidelity has been calculated [START_REF] Choi | Completely positive linear maps on complex matrices Linear Algebra and Its[END_REF][START_REF]Jamiołkowski A Linear transformations which preserve trace and positive semidefiniteness of operators[END_REF]. The imperfections of storage/ retrieval process can be ignored as the fidelity is mainly affected by the target photon scattering. In free space, the Rydberg non-linearity is characterized by the optical depth per blockade d 𝑏 while in the cavity systems with a finesse (ℱ ) the parameter characterizing the system is cooperativity per blockade 𝒞 𝑏 ∼ ℱ𝑑 𝑏 .

The fidelity is estimated by evaluating how the real gate operation deviates from the ideal gate operation. The operation has no effect when the control pulse is in state |0⟩. Hence the fidelity is affected in the presence of a single photon in the control pulse and it is estimated to be [START_REF] Grankin | Theoretical studies of optical non-linear effects in ultracold Rydberg gases[END_REF][START_REF] Das | Photonic controlled-phase gates through Rydberg blockade in optical cavities[END_REF] :

𝐹 𝑐𝑗 = |⟨𝜑 𝑖𝑑𝑒𝑎𝑙 |𝜑 𝑟𝑒𝑎𝑙 ⟩| = 1 16 ⃒ ⃒ ⃒ ⃒ 3 - 1 -2𝒞 𝑏 1 + 2𝒞 𝑏 ⃒ ⃒ ⃒ ⃒ 2 (9.6)
where 𝒞 𝑏 is the cooperativity per blockade. As we can see from the above equation, the fidelity approaches one for large 𝐶 𝑏 . As the main source of error in single rail systems is photon scattering, the dualrail implementation is less likely to fail. In Figure 9.14, we compare the fidelities of single rail and dual rail systems and we can observe that for a reasonable 𝒞 𝑏 of 10 one can achieve very high fidelities (post selected) greater than 0.99. In addition, for practical application the proposed gate has shown to improve the communication rate of quantum repeaters [START_REF] Das | Photonic controlled-phase gates through Rydberg blockade in optical cavities[END_REF].

Conclusion

In this chapter we described our efforts to move to the quantum regime with the help of a new cavity. We showed how the designed resonator fulfills the necessary conditions to create strong optical non-linear effects. We also described its installation and the EIT measurements performed with this setup. Currently, we are in the process of characterizing the non-linearity of the system. Finally, we concluded the chapter with possible schemes we can exploit and a proposal for photonic phase gate.

General conclusion and outlook

In this manuscript, we studied classical optical non-linearities using dipole-dipole interactions between Rydberg atoms in a cavity. This work was motivated by the fact that Rydberg interactions in an atomic cloud have been theoretically proposed to be viable candidates for efficient manipulation of photons. We combined the advantages of EIT along with the interaction of Rydberg states, to enhance the optical non-linearity induced by a cold atomic system.

We demonstrated how one can prepare a small cold atomic ensemble, trapped from a relatively large cloud of 1 mm in diameter, using selective depumping and repumping cycles. By exciting the atoms resonantly to Rydberg states using EIT, we measured the effects of Rydberg-Rydberg interactions using both S and D states for n = 37-90. The resonant non-linear effects appeared as intensity dependent losses that increase as we move to high-lying Rydberg states. A semi-classical model was developed to explain the observed non-linear behavior arising from Rydberg-Rydberg interactions for S states. We also showed that we must move beyond the three level model to explain the non-linearity observed in Rydberg D states, where the collisional effects lead to the creation of long-lived excitations in the cloud.

The cooperativity per blockade sphere was enhanced by increasing the cavity finesse. This should, in principle, allow us to probe quantum optical non-linearities.

Based on the EIT signal observed at n = 37, we are confident that the available coupling power will allow us to excite to atleast n= 90-100 Rydberg states.

As we move away from the coherent classical picture like the Rydberg bubble model, dephasing mechanisms give way to non-linear losses. A cavity can help us take advantage of these irreversibly lossy photonic channels. Following this idea, we described a theoretical proposal for a very high fidelity controlled photonic phase gate. This proposal allowed us to circumvent the Shapiro's limitation, where large Kerr non-linearities cannot be used for quantum optical communication [START_REF] Shapiro | Single-photon Kerr nonlinearities do not help quantum computation[END_REF].

Finally, we showed that cavities can be used in novel ways to modify the light at the quantum level, such as the creation of higher order correlations.

In addition, with the help of Andrey Grankin, we observed that we can use the Schwinger-Keldish formalism to evaluate the correlation effects beyond the lowest order. This allowed us to calculate the elastic and inelastic components of the squeezing spectrum up to any given order in feeding rate.

It has been shown that dissipative propagation through Rydberg EIT ensembles, like single photon filters or subtractors irreversibly distort the shape of optical photons [START_REF] Gorshkov | Dissipative Many-Body Quantum Optics in Rydberg Media[END_REF]. Since such distortions will limit the fidelity of the operation, one Conclusion needs to characterize the purity of the manipulated photons. To achieve that, we use a powerful technique called optical homodyne tomography to measure more intricate details of the optical photons. We developed a setup which allows us to measure the Wigner function, phase dependent squeezing, etc., of light. We showed that our setup can be used to detect phase changes, by measuring the phase flips induced by a linear system. We also studied the squeezing spectrum of light along with the effects caused by the cavity and the EIT.

The experimental results and the theoretical ideas described in this thesis brings us one step closer to the quantum optical manipulation using cavity enhanced Rydberg-EIT systems. Moreover, the detection methods developed will help us study the manipulated photons more elaborately.

Outlook

Simple models, such as the perturbative approach and the bubble model, can be used to estimate the resonant classical non-linearity of Rydberg-EIT. Using the Schwinger-Keldish formalism, we observed that by evaluating the cavity transmission in third order, it reproduces the semi-classical non-linearity. Hence, we can use this formalism to calculate the non-linear effects observed beyond the semi-classical model by moving to higher orders.

The goal for the next few months is to investigate the resonant non-linearity of the current experimental setup, and study the non-linear evolution of few-photon excitations in the Rydberg-EIT cloud. In the long term, the objective is to create more exotic states such as "quantum fluids of light" [START_REF] Carusotto | Quantum fluids of light[END_REF]. This could be achieved by using the Rydberg-induced coupling between different but nearly-degenerate transverse modes of the cavity [START_REF] Schine | Synthetic Landau levels for photons[END_REF].

Part V

Appendices

The radial term in the numerator scales as (𝑛 * ) 2 and the energy difference between the neighboring energy levels scales as (𝑛 * ) -3 . The polarizability term scales as (𝑛 * ) 7 , hence the Rybderg states especially the high lying ones, are extremely sensitive to weak fields.

To estimate the shifts more precisely, its necessary to diagonlize the total Hamiltonian. We consider the atomic basis which includes all the neighboring states with a given |𝑚 𝑗 |. All the radial components are calculated using the Numerov algorithm [START_REF] Zimmerman | Stark structure of the Rydberg states of alkali-metal atoms[END_REF].

We show the stark map of Rydberg 85 D state along with the neighboring levels in

Figure 1 .Figure 1 . 1 :

 111 Figure 1.1: Modular computing: A quantum network of nodes A, B and C are connected by channels like fiber optic links. Each node consists of cavity systems. In distributed networks, nodes process information and perform quantum operations like quantum gates, etc.

Figure 1 . 2 :

 12 Figure 1.2: Quantum Control Z gate: Control phase or Z gate is a two qubit gate where one of the input qubits acts as a control (|𝑐⟩). The other input qubit is represented as |𝑠⟩. The output state of the gate is the same as the input state except when both the signal and the control qubit are in state '1', then it acquires a phase of 𝜋.

Figure 1 . 3 :

 13 Figure 1.3: Rydberg-Rydberg interaction energy shift vs Interatomic distance: When an atom (b) is far from a Rydberg atom then it can be excited to a Rydberg level using resonant lasers. If an atom (a) moves closer then its energy levels are perturbed and it is no longer possible to excite it to Rydberg state. The minimum distance upto which a Rydberg atom can influence another is defined as Rydberg blockade radius (𝑟 𝑏 ). A typical blockade radius is of the order of ∼ 5 𝜇m.

𝐿Figure 2 . 2 :

 22 Figure 2.2: Comparison of free space and cavity transmission. On the left: We observe the transmission of a probe beam close to atomic resonance. We assume the cavity to be resonant with the atoms and the transmission coefficients of input and output mirrors are 𝑇 1 and 𝑇 2 respectively. On the right: The normalized transmission as a function of probe detuning when scanned around the atomic resonance. The blue line denotes the transmission of atoms in free space, whereas the red line represents the transmission through a cavity finesse of 100 and a cooperativity of 8.

(2. 16 )

 16 From the above master equation we can extract the expectation values of the operators using the expression ⟨ Â⟩ = Tr( Âρ 𝑠 ). The evolution of operators can be evaluated from 𝑑 𝑑𝑡 ⟨ Â⟩ = Tr( ρ Â) which gives the Heisenberg equations for the operators 𝑑 𝑑𝑡

Figure 2 . 3 :

 23 Figure 2.3: Cooperativity measurement in absorptive and dispersive regime: The normalized cavity transmission versus probe detuning normalized to 𝛾 𝑒 for a cavity with a decay rate 𝛾 𝑐 = 3.3𝛾 𝑒 (a) When the cavity is on resonance with atoms the cavity transmission is split into two normal modes where the splitting is given by √ 8𝐶𝛾 𝑐 𝛾 𝑒 . (b) When the cavity is detuned from the resonance of the atoms by ∆ 𝑒 = 15𝛾 𝑒 . The cavity transmission curve is shifted with a shift proportional to the cooperativity.

Chapter 2 .

 2 Theoretical tools for Atom-Light coupling

Figure 2 . 4 :

 24 Figure 2.4: Absorptive non-linearity: The transmitted light (X) is plotted as a function of incident light (Y). We can see that for cooperativities higher than 4

Figure 2 . 5 :

 25 Figure 2.5: Dispersive non-linearity: (a) The transmitted light vs incident intensity when 𝐴 = 1 and 𝐵 = 3 (b) Transmitted light vs the detuning of the probe from the cavity in a linear case when 𝑋 = 0.1 and non-linear when 𝑋 = 4.

Figure 2 . 6 :

 26 Figure 2.6: Illustration of the excitation scheme for three level atoms in a cavity: (a) The atoms are enclosed in a cavity with the probe coupled to the fundamental mode of the cavity and the control field incident from the side. (b)

Figure 2 . 7 :

 27 Figure 2.7: Comparision of susceptibility of two level (red) and three level systems (blue): We assume that the linewidth 𝛾 𝑠 = 0.1𝛾 𝑒 (a) The real part of the susceptibility shows the dispersion spectra. It shows a sharp response for the medium at zero detuning. (b) The absorption peak at resonance splits into two peaks creating a transparency window.

  Figure 2.8: Electromagnetically induced transparency curves in the presence and the absence of the cavity. The transmission of the system is plotted

Figure 2 . 9 :

 29 Figure 2.9: Two-photon resonance response: The real and the imaginary part

Figure 3 . 1 :

 31 Figure 3.1: Quantized energy levels of Hydrogen-like atoms: The electronic energy levels for low angular momentum states are plotted. The valence electrons for low angular momentum states spend more time closer to the nucleus, hence need more energy to ionize. In the inset: The Rydberg atomic radius increases with the square of principal quantum number (n 2 ). The valence electron can move up to few microns away from the nucleus for n = 140.

Figure 3 . 2 :

 32 Figure 3.2: Rydberg-Rydberg Interactions: (a) Two interacting Rydberg atoms separated by a distance r. (b) The energy levels of the combined two atom system.

Figure 3 . 3 :

 33 Figure 3.3: Rydberg bubble: The atomic cloud is divided into spheres where each of them can at most have a single Rydberg excitation.

Contents

4. 1

 1 Trapping and excitation lasers . . . . . . . . . . . . . . . 33 4.2 Frequency stabilization system . . . . . . . . . . . . . . 35 4.2.1 Transfer cavity . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.2 Characterization tests . . . . . . . . . . . . . . . . . . . . 38 4.2.3 Stability of lasers . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Control/Acquisition setup . . . . . . . . . . . . . . . . . 41 4.3.1 Hardware components . . . . . . . . . . . . . . . . . . . . 41 4.3.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 4 . 3 :

 43 Figure 4.3: The locking schemes for transfer cavity and master laser (a) The master laser is locked using both the transfer cavity and the saturated absorption signal to compensate for high frequency and low frequency drifts, respectively. (b)The master laser is locked only on the transfer cavity, which is stabilized with saturated absorption signal.

Chapter 4 .

 4 side of resonance are most sensitive to any fluctuations in resonator's length. The histogram of the reflected intensities fits well for a cavity whose resonance has a Optical and control system of transfer cavity. In the Figure4.5 (a) we plot the spectrum of the beating signal between our blue laser and a reference one. We observed that the frequency of the blue laser was oscillating between two values. The origin of the problem can be seen in Figure4.5 (b), where we can see that the reflected light (blue) from the transfer cavity exhibits bimodal reflection.

Figure 5 . 1 :

 51 Figure 5.1: An illustration of LVIS and the main chamber: The LVIS chamber consists of auxiliary MOT where the atoms are trapped and transfered to the main chamber trap. Here we illustrate the LVIS trap beam collimators along with the position of the cavity holder and MOT coils in the main chamber.

  used to bring the atoms back to 5𝑆 1/2 𝐹 = 2. Both the beams are locked on crossover resonances and then frequency shifted close to resonance using AOMs (The schematics of the locking transitions are illustrated in Appendix A). All the cooling beams are red-detuned by 18 MHz (3Γ 𝑒 ) and each one of them has an intensity of 5 mW/cm 2 at the center.

Figure 5 . 2 :

 52 Figure 5.2: Excitation scheme for MOT: The trapping beams are excited on the D2 transition at 780 nm. The MOT trapping beam is red detuned by 18 MHz on transition 5S 1/2 F= 2 → 5P 3/2 F=3. The repumper beam is on resonance with the transition 5S 1/2 F= 1 → 5P 3/2 F=2.

10 5. 4 . 2 :

 1042 3 one pair of trapping beams are created using retroreflection from a mirror (and a quarter wave plate) with a small orifice. The atoms are not trapped in the cylindrical hole of the retro-reflected beam and are pushed through the differential tube on the chamber wall. The former beam acts as a pusher to create a steady stream of atoms. The position of the LVIS trap is set in such a way that the atoms are pushed towards the main optical trap region. This way of creating atomic beam has several advantages over conventional background loading such as fast loading and easy tunability of atom number. In typical MOT system one needs to change the power in the trapping beams or the gradient of the magnetic field to increase the loading. But using this setup, we can control the number of atoms by keeping the main trap parameters constant by increasing the magnetic field gradient of the LVIS trap. Main atomic cloud By optimizing the position of the LVIS cloud, we can upto 3*10 9 atoms per second. This allows us to load about 10 9 atoms in a characteristic time of ∼ 4 seconds. Main chamber parameters: The laser trap parameters are similar to LVIS trap 5.4.1 Optical molasses After the atoms are trapped in MOT, the temperature of the cloud is measured to be around the Doppler temperature of 150 𝜇K. The atoms are further cooled down using the radiation pressure of MOT beams, this technique is known as optical molasses. Using this method, sub-Doppler temperatures can be achieved because of spatially varying radiation fields experienced by the atoms [86, 87]. A typical optical molasses sequence requires the magnetic traps to be turned off and the cooling lasers are reduced while they are moved away from the resonance. The final sequence is optimized for maximum trapping efficiency and lasts for 6 ms. During the first 3 ms the cooling laser's frequency is ramped from -18 MHz to -48 MHz

  1 magnification. Then we use a negative doublet at -75 mm to further magnify the image. The final lens is positioned to achieve a magnification of ≈ 2.74± 0.05 times while minimizing the aberrations. The new setup has a resolution limit of only 8 microns while still having the capability to image 1-2 mm clouds. The imaging setup has been designed with the help of Tikai while he was an internship student and is described in more detail in his report.5.4.3 Temperature measurementsTo estimate the temperature of the cloud, we use the standard time of flight measurements[START_REF] Lett | Observation of Atoms Laser Cooled below the Doppler Limit[END_REF]. The radius of the cloud at any given time depends on the time of expansion and initial equilibrium temperature of the MOT. By considering the Boltzmann distribution for the velocity of atoms the cloud size can be estimated using the following equation[START_REF] Lett | Observation of Atoms Laser Cooled below the Doppler Limit[END_REF]

Figure 5 . 6 :

 56 Figure 5.6: The variation of cloud size with time: By fitting the cloud expansion we can estimate the temperature of the cloud.

Figure 6 . 2 :

 62 Figure 6.2: Hanbury Brown Twiss type Interferometer to measure Intensity correlations: The signal beam is incident on a 50/50 beam splitter and we measure the cross-correlation between the two output arms using two single photon counting modules (SPCMs).

2 ) 1 :

 21 measurement gives an accurate value as long as the photon count is below 20 counts/𝜇s. The typical sequence consists of probe detection for 120 𝜇s with counts between 40-100, which is well below the threshold. The measurements are insensitive to the optical losses between the cavity and the detector. Comparision of the g(2) measurement and the Homodyne detection6.2 Homodyne TomographyUnlike classical particles, quantum objects are not represented by a single point in phase space but with a probability distribution called Wigner function (for more information refer to the Appendix:D). It gives access to the precise quantum state of the measured particles. The Wigner function can be measured using a technique called homodyne tomography[START_REF] Lvovsky | Continuous-variable optical quantum-state tomography[END_REF]. From photon-counting measurements one

Figure 6 . 3 :

 63 Figure 6.3: Homodyne Tomography: The optical signal is superposed with a Local Oscillator (LO) beam. Two low noise photo diodes are placed at the two output arms. The difference in the intensity is used to measured the tomography of the quantum state.

√ 2 ( 1 √ 2 Figure 6 . 4 :

 21264 Figure 6.4: Measurement of electronic noise level: (a) The variance of the electronic noise is plotted against the local oscillator (LO) power. We can see that the variance increases linearly until it starts diverging above 7 mW of LO power. (b) The noise spectrum of both electronic and vacuum noise are plotted for 7 mW of LO power. The electronic noise level is 15 dB below the vacuum noise.

a 50 :Figure 6 . 5 :

 5065 Figure 6.5: Homodyne phase locking scheme in reflection: A locking beam detuned from the probe by 88 MHz is used to lock the phase of the local oscillator relative to the probe. The locking beam is detuned far enough to not perturb the atoms and close enough such that it is within the bandwidth of the detector.

Figure 6 . 6 :

 66 Figure 6.6: Phase measurement using a resonant probe in reflection. As relative phase between the LO and the locking beam is varied with time, we can observe that the phase is flipped by approximately 0.85 𝜋 when the atoms are switched on

Figure 6 . 7 :

 67 Figure 6.7: Squeezing spectrum in transmission: The probe phase is transmitted through the cavity and its relative phase is measured by changing its value by 𝜋/2. The measurements are repeated several times to acquire enough statistics for each value of the phase.

  kHz (a.u)

Figure 6 . 13 :

 613 Figure 6.13: Dispersive non-linearities using Rydberg excitations: The atoms are excited using two photon excitation scheme where the probe light is red, and the coupling beam is blue detuned. This allows for the light shifts to move further from resonance and due to Rydberg-Rydberg interactions, the D states moves away from two photon resonance. The probe light and the coupling light are coupled to their fundamental modes from the top of the cavity using a dichroic mirror.

Figure 7 . 1 : 1 :

 711 Figure 7.1: Main cavity setup: A schematic drawing of the MOT coils along with the position of the cavity holder in the system.

Figure 7 . 4 :

 74 Figure 7.4: Cavity lock stability: We measure the transmission of the probe when it is detuned by the cavity linewidth from resonance (i.e ∆ 𝑒 = 𝛾 𝑐 ) to probe the stability of the cavity. The probability distribution of the normalized transmission was shown along with a model (red curve) where the cavity's resonance frequency has a standard deviation of 2 MHz. Parameter Value Auxilary beam wavelength 810 nm Locking beam power

Figure 7 . 5 .Figure 7 . 5 :

 7575 Figure 7.5: Cavity with incident and reflected fields: A cavity which is excited using fields 𝑎 𝑖𝑛 and 𝑏 𝑖𝑛 . The intra-cavity field is denoted by 𝑎. The decay rates of the top and the bottom mirrors are indicated by 𝛾 𝑡 and 𝛾 𝑏 respectively.

Figure 7 . 6 :

 76 Figure 7.6: Experimental Setup: The probe light is coupled from the top side of the cavity and we measure the cavity transmission using two single photon counting modules. This setup was designed for the g (2) measurement but we also use if for photon counting experiments. To avoid detection of any stray light especially from the auxilary cavity locking beam, we use different filters at 780 nm.

Figure 7 . 8 :

 78 Figure 7.8: Experimental Sequence: After the cloud is trapped using the MOT for 45 ms, it is cooled down for 6.1 ms using polarization gradient cooling. During the molasses stage, the dipole traps are turned on and the atoms are allowed to load for about 21 ms. Then the whole cloud is depumped to 5 𝑆 1/2 𝐹 = 1 and only the atoms in the crossed dipole trap are repumped (and optically pumped) to 5 𝑆 1/2 𝐹 = 2𝑚 𝐹 = 2. The atoms are probed during 120 𝜇s during which the EIT transmission curves are measured.

Figure 7 . 9 :

 79 Figure 7.9: Images of atomic clouds during different stages (a)Image of complete MOT without dipole traps (b) Image of atoms trapped in cavity dipole trap (c) Image of atoms trapped in side dipole trap (d) Image of the final small cloud at the end of the preparation stage.

50

  MHz to +50 MHz. The dipole traps are switched off during the measurement to avoid any light shifts. The cavity transmission for different probe intensities is monitored either with an Avalanche Photo Diode (APD) or SPCMs. The measurement are performed over 120 𝜇s. When the cavity is empty we obtain the linewidth 7

Figure 7 . 10 :

 710 Figure 7.10: Cooperativity measurement: The black line denotes the empty cavity transmission and the red line is the cavity transmission in the presence of atoms. The cavity locked on resonance with atomic transition and the probe frequency is scanned across the resonance frequency. The atoms split the cavity mode into two normal modes separated by√8𝐶𝛾 𝑐 𝛾 𝑒 . For this measurement, the cooperativity of the system is 8.5 and the HWHM (half width at half maximum) of the cavity is measured to be 11 MHz.

Figure 7 . 11 :

 711 Figure 7.11: Probe intensity as seen by atoms: The atoms don't see a uniform probe light because of the standing wave formed in the cavity. (a) Probability distribution of the intensities seen by the atoms. A significant number of atoms see zero intensity. (b) Contour plot of standing wave formed by the probe light.

Figure 7 .

 7 Figure 7.12: A bird's eye view of the blue cavity along with the viewports of the vacuum chamber: The cavity's position with respect to the main cavity and atoms is illustrated. Both cavity mirrors are mounted on two different piezos for stabilization.

5 :

 5 Blue locking specifications

Figure 7 . 14 :

 714 Figure 7.14: Blue inhomogeneous effects: Top: Contour plot of the blue intensity distribution at the waist of the cavity as a function of longitudinal and radial components for different values of waist. In the left most contour plot the standing wave effects are minimal but waist is comparable to atomic cloud size. In the right most contourplot the standing wave is prominent even though it is much bigger than the cloud size. Bottom: The probability distribution of the blue intensity as seen by the atoms. The units on the x-axis correspond to arbitrary units of blue intensity.

2 Figure 7 . 15 :

 2715 Figure 7.15: EIT measurements in linear regime: Cavity transmission as a function of the probe laser detuning for an empty cavity (black curve), with an intracavity atomic cloud (red curve) and with the presence of a control field resonant with the 5 𝑃 3/2 𝐹 = 3 → 37 𝐷 5/2 𝐹 = 4 transition (blue curve). A fit to the experimental observations allows us to extract various parameters of the system.

5 ΩFigure 7 . 16 :

 5716 Figure 7.16: EIT transmission for n =37 𝐷 5/2 as a function of probe detuning: Cavity transmission as a function of the probe laser detuning for various control field powers. A fit for linewidth 0.14 𝛾 𝑒 agrees well with different Rabi frequencies scaled according to the incident power.

Figure 7 . 17 :

 717 Figure 7.17: Non-linearity using Rydberg S states: (a) The cavity transmission spectrum is plotted for various probe photon rates for 70S 1/2 Rydberg state. (b) Cavity transmission at the center of the transparency window as a function of the probe photon rate, for various Rydberg levels. Each curve has been normalized to help their comparison and the inset gives a closer look of the transmission at low photon rates. One can observe the reduction of the transparency with higher photon rate due to the Rydberg blockade effect. The theory is in quite good agreement with the data at any detuning or Rydberg state.

Figure 7 . 18 :

 718 Figure 7.18: Stark effect on Rydberg D states: (a) A stray electric field lifts the degeneracy of the Zeeman sublevels of the 90 D states, which creates a transparency window for each transition. (b) Theoretical detuning of the degenerate levels with

6 = 45 •

 645 (𝑛/56) 11 GHz•𝜇𝑚6 , averaging the potential in all directions. This simplification allows one to keep analytical calculations tractable, but it prevents one from correctly modeling beyond-mean-field effects related to the anisotropy of the interactions. By exciting atoms to D states Time (µs)

Figure 7 . 19 :

 719 Figure 7.19: Transient measurements: Cavity transmission in EIT conditions (probe and control fields on resonance) for S(a) and D(b) states. In addition to a decrease in transmission with the probe photon rate due to the Rydberg blockade, we observe a transmission decay over time for D states. The dashed lines are a model which includes a possible decay of atoms into other Rydberg states that are uncoupled to the control field.

Figure 7 . 20 :

 720 Figure 7.20: Probing long lived Rydberg D states: On-resonance cavity transmission in EIT conditions (probe and control fields on resonance) for D states. The first pulse (only red) consists of more than 50 photons/𝜇s while the second pulse

Figure 7 . 21 :

 721 Photon rate(µs -1 ) 0 100 200 300 400 500 600

1 )

 1

2 g ( 2 )

 22 (0) Δ e /γ e Δ c = Δ r = 0 γ e

Figure 8 . 5 :/ 2 - 3

 8523 Figure 8.5: Intensity correlation measurement: The g (2) function of the transmitted light is measured for the Rydberg state 79 S 1/2 . The stability of the cavity lock and small cooperativity per blockade prohibit us from observing strong nonclassical effects like anti-bunching.

Figure 9 .

 9 1 shows the real part of 𝑉 𝑏 as a function of Ω 𝑐𝑓 , for several different possible values of 𝛾 𝑟 for n = 95. It might seem that the blockade size is large for smaller control field Rabi frequency, but it manifests from the fact that the Rydberg excitation linewidth is not broadened by the coupling powers lower than 1.

Figure 9 . 1 :

 91 Figure 9.1: Rydberg blockade volume dependence on the control field Rabi frequencies: The blockade volume depends strongly on the Rydberg linewidth. The blue curve corresponds to the measured EIT linewidth of ≈ 450 kHz

35

 35

Figure 9 . 2 :

 92 Figure 9.2: (a) Old cavity setup with a finesse of 120 and waist of 85 𝜇m. (b) New cavity setup with a finesse of 1200 and waist of 15 𝜇m

  𝑏 𝑔𝑡(1-𝑔𝑡𝑔 𝑏 )) 0.[START_REF] Häffner | Quantum computing with trapped ions[END_REF] = 364 𝜇𝑚, and the bottom mirror 𝜔 𝑏 = 728 𝜇𝑚. As the nearest viewports to the mirrors are atleast 30 cm away from them, it is better to curve the external surfaces of the mirrors to minimize the size of the beam outside the chamber.Spherical aberrationIn order to avoid the spherical aberration due to the edges of the curved surfaces, we fix the outer radius of curvature where the incident beams are orthogonal to the concave side of the mirror. If we assume the mirror has a thickness t and refractive index n then the r 𝑜𝑢𝑡 = 𝑛 𝑛+1 (𝑡 + 𝑟 𝑖𝑛 ). This means the outer radius of curvature has to be set to ≈ 16 mm and ≈ 29 mm to minimize spherical aberrations. With this configuration the main contribution for the aberration arises from the external optics.

Figure 9 . 3 :

 93 Figure 9.3: Minimum waist attainable vs. distance to concentricity: The

Figure 9 . 4 :

 94 Figure 9.4: Configuration of the bottom mirror: (a) To minimize any fluctuations, the mirror and the piezo are glued using a triangular ring adapter to maximize the surface contact. A non-conducting glue is used to minimize any potential contacts between the electrodes of piezo and other surfaces. (b) Bottom mirror holder: A single piece was designed without any springs.

Figure 9 . 6 :

 96 Figure 9.6: Different cavity modes excited: The cavity's length is adjusted to excite various TEM modes (a) TEM00 (b) TEM01 (c) TEM11

Figure 9 . 7 :

 97 Figure 9.7: Cavity linewidth measurement: The reflection signal from the 'lossy side' of the cavity when the probe frequency is varied around the cavity resonance.

Figure 9 . 8 :

 98 Figure 9.8: Cavity drifts: The stability of the cavity can be inferred by monitoring the transmission of a probe detuned by 𝛾 𝑐 from cavity resonance. We measure the histogram of the transmitted powers which corresponds to a standard deviation of 0.35 MHz in cavity resonance frequency

Figure 9 . 9 :

 99 Figure 9.9:

Figure 9 . 10 :

 910 Figure 9.10: Proposed excitation scheme in dispersive regime: Atoms are excited to Rydberg states off-resonantly. In this scheme, the interactions divide the atomic cloud into "bubbles" which can at most have only one excitation. The green bubbles have a single excitation and the blue ones have none. These spheres act like two-level superatoms. The statistical properties of the transmitted probe light is measured using correlation measurement.

6 .

 6 Each bubble '𝛼' can either have no excitations (|-𝛼 ⟩), or one collective Rydberg excitation (|+ 𝛼 ⟩) (as shown in Figure 9.10). The interaction term is simplified to the following form Ĥ𝑖𝑛𝑡 = 𝑔 𝑒𝑓 𝑓 𝑁 𝑏 (â Ĵ+ + â † Ĵ-), (9.3) where g 𝑒𝑓 𝑓 = 𝑔Ω 𝑐𝑓 2𝛿 and a collective spin lowering operator Ĵ = ∑ 𝑁 𝑏 𝛼=1 |-𝛼 ⟩⟨+ 𝛼 | corresponding to the spin 1/2 Rydberg bubbles introduced in [46]. The effective spin Ĵ interacts with the cavity mode â and evolves the system with an enhanced coupling factor

Figure 9 . 11 :

 911 Figure 9.11: The statistics of the transmitted light in off-resonant excitation scheme using a continuous probe: (a) The auto correlation function of the transmitted light as a function of normalized cavity detuning. It exhibits strong bunching/anti-bunching features depending on the cavity detuning which favors either photon emission in pairs or one by one. (b) The average number of photons ⟨𝑎 † 𝑎⟩ 2 (solid) and the average number of photon pairs ⟨𝑎 † 𝑎 † 𝑎𝑎⟩ 2 in the transmitted light exhibit a different resonant frequency. The experimental parameters chosen are 𝛾 𝑐 = 0.3 𝛾 𝑒 , Rydberg level n = 95, C = 300, 𝜌 = 0.1 atoms/𝜇𝑚 3 , 𝛿 = 0.5 𝛾 𝑒 and ∆ 𝑐 = -16 𝛾 𝑒 .

Figure 9 . 12 :

 912 Figure 9.12: Phase gate: A possible scheme to implement a phase gate with a single Rydberg excitation. (a) The Reflected field acquires no phase shift when incident under EIT conditions (b) If a single Rydberg atom creates losses much higher than 0.5% then the reflected field acquires a phase shift with reasonable losses.

  excitation. In reality, there is a finite probability of exciting two Rydberg excitations which evolve with the interaction potential. If we consider an initial ground state of the N-atom system |𝜑 𝑖 ⟩ = ∑ 𝑁 𝑛=1 |𝑔 1 , 𝑔 2 ....𝑔 𝑁 ⟩ evolving with term evolves conservatively and only the terms with n>1

Figure 9 . 13 :

 913 Figure 9.13: Scheme for photonic phase gate: A single photon stored as a Rydberg excitation acts a blockade for the second photon pulse and induces a phase shift of 𝜋 on it. On the right The excitation scheme for the first photon and the second photon is illustrated. The first photon is stored in a different Rydberg state using microwave transition. The second photon can no longer be excited to Rydberg state |𝑟⟩ due to interaction potential 𝒱 𝑅𝑦𝑑-𝑅𝑦𝑑 .

Figure 9 . 14 :

 914 Figure 9.14: Fidelity as a function of cooperativity per blockade:

  For each manifold |𝑚 𝑗 | we truncate the atomic basis with ∆𝑛 = 5.

FigureFigure C. 1 :

 1 Figure C.1.

  Using the expressions for 𝑔 0 , 𝛾 𝑐 , 𝛾 𝑒 , we can write cooperativity as 𝐶 = 𝑁 ′

	0
	where the ⃗ 𝐸 + is one of the propagating field components. If we assume the medium
	has a uniform density 𝜌 then we can write 𝜌𝜋𝑙𝜔 2 0 /4 is the effective number of atoms coupled to the cavity. ∑ 𝑛 𝑔 2 (𝑟 𝑛 ) = 𝑔 2 0 𝜌𝜋𝑙𝜔 2 0 /4. 𝑇 1 3𝜆 2 where 𝑁 ′ = 2𝜋𝜔 2 0

  To increase the cooperativity one needs to have a very high atomic density or a very high finesse cavity. In this thesis, we are going to concentrate on scenarios where the atom-light coupling 𝑔 0 ≪ 𝛾 𝑐 , 𝛾 𝑒 and √ 𝑁 𝑔 0 > 𝛾 𝑐 , 𝛾 𝑒 .

	2.3.2 Optical bistability using two level atoms
	Optical non-linearity has acquired tremendous interest since the prediction of op-
	tical bistability in resonator systems with non-linear media in the 1970s and has
	been well studied by many groups [54-56]. Optical bistability or multistability is
	the presence of more than one possible value for transmission for a given input
	intensity. This phenomenon can be observed only when the imaginary part or the
	real part of the susceptibility is a function of the intensity. This phenomena can be
	used to create an optical switch where one can alternate between two stable points
	deterministically [51, 57].

  Consider an atom with a ground state (g), intermediate state (e), and a metastable state (s) separated by 𝜔 𝑔𝑒 and 𝜔 𝑒𝑠 respectively. The two atomic transitions |𝑔⟩ → |𝑒⟩ and |𝑒⟩ → |𝑠⟩ are driven by a probe beam at 𝜔 and a control coherent beam at 𝜔 𝑠 respectively. The detunings of probe and control beam are denoted by ∆ 𝑒 = 𝜔 -𝜔 𝑔𝑒 , ∆ 𝑠 = 𝜔 𝑠 + 𝜔 -𝜔 𝑔𝑠 . The additional Hamiltonian term for

	feeding rate of 𝛼 are
	⟨â⟩ =	1 𝐷 𝑐	(𝑔	𝑛 ∑	⟨σ (𝑛) 𝑔𝑒 ⟩ + 𝛼)
	three level atoms is
	Ĥ𝑠 = -∆ 𝑠	∑ 𝑛	σ(𝑛) 𝑠𝑠 +	Ω 𝑠 2	𝑛 ∑	(σ (𝑛) 𝑠𝑒 + σ(𝑛) 𝑒𝑠 )	(2.27)
	Here we introduce a complex detuning term 𝐷 𝑖 = ∆ 𝑖 + 𝑖𝛾 𝑖 . The optical Bloch
	equations for the system at steady state when the cavity is coupled with coherent

  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Mean field approach . . . . . . . . . . . . . . . . . . . . . 27 3.3 Rydberg bubble model . . . . . . . . . . . . . . . . . . . 28

	Chapter 3
	Rydberg-Rydberg Interactions
	Contents
	3.1

  We assume that the pair states |𝑟 2 𝑟 2 ⟩ and |𝑟 1 𝑟 3 ⟩ are separated by ∆ and the state |𝑟 2 𝑟 2 ⟩ is coupled to |𝑟 1 𝑟 3 ⟩

by 𝑉 (𝑅). The eigenenergies of pair system can be written as

Table 4 .

 4

1: Summary of all required atomic transitions.

Table 4 .

 4 3/2 → 𝑛𝑆 or 𝑛𝐷 is used to excite to Rydberg S or D states. For this purpose, we need a tunable laser at 480 nm which is generated using a Toptica SHG pro laser of ≈

	Laser	Power
	Master laser	1 W 780 nm
	Repumper	30 mW at 780 nm
	Blue laser	400 mW at 480 nm
	TiSa 1	2 W at 810 nm
	TiSa 2	2 W at 795 nm

400 mW output. It consists of a diode at 960 nm which is amplified to 1 W. It is later up-converted to 480 nm using a resonant doubling cavity. We extract a part of 960 nm to stabilize the frequency. 2: Summary of all lasers.

Table 4 .

 4

3: Transfer cavity

  As the single photon detectors cannot be switched on/off at time scales smaller than few ms, the virtual digital channels provide us with necessary control with minimum noise from the background signals.

	Chapter 5
		Atomic setup
	Trapping	Molasses	Preparation	Acq. data
	~ 100ms	~ 20ms	∼ 100µs	∼ 120µs

2. A typical experimental sequence may not necessarily be unidirectional for e.g. some measurement steps might need to be executed multiple times once the atoms are prepared. The atomic cloud preparation can take upto 100 milliseconds and doesn't require further loading until the lifetime of the cloud (upto 10 ms). During this time the measurement and preparation steps might have to be repeated a large number of times like as illustrated in Figure 4.7. x10 x1000 Figure 4.7: Experimental sequence: A typical experimental sequence which can be programmed to run where certain parts are executed under logical conditions, and others are executed a certain number of times before the whole cycle is repeated 3. Finally each channel has to be configured for each step individually. The digital channels can be programmed using Time High (TH), Time Low (TB), delay time (dt) and Number of times (Nb) settings. In each step the following sequence is generated for 'Nb' times (high for TH and low (TB+dt) ...). The analog channels are equipped with ramp, constant and waveform inputs which can be used to create arbitrary functions easily.

All the operations are then converted into FPGA code which can then handle all channels simultaneously with 100 MHz update rate. This enables us to create logical operations which are otherwise hard to implement. For e.g. it would allow us to implement a sequence which must be executed upon a trigger like a detection event on a photon counter. In addition, there are some necessary features for proper functioning of the experiment such as virtual digital controllers, which are important when one needs to perform logical gating operations like triggering data acquisition upon detection of certain event. Contents 5.1 Atom cloud preparation . . . . . . . . . . . . . . . . . . 45 5.2 Magneto Optical Trap . . . . . . . . . . . . . . . . . . . . 46 5.3 Low Velocity Intense Source (LVIS) . . . . . . . . . . . 47 5.4 Main atomic cloud . . . . . . . . . . . . . . . . . . . . . . 48 5.4.1 Optical molasses . . . . . . . . . . . . . . . . . . . . . . . 49 5.4.2 Imaging system . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4.3 Temperature measurements . . . . . . . . . . . . . . . . . 51

Table 7 .

 7

2: Main cavity lock parameters

  The efficiency of the detector is 67% at 780 nm. The optical losses from the output of the cavity until the detectors (including the coupling efficiency) is 50% for both the detectors. If we detect N photons in t 𝜇s, then the photon flux at the the value of intermediate state linewidth 𝛾 𝑒 = 2𝜋 3 MHz and cavity's HWHM 𝛾 𝑐 = 2𝜋 10 MHz. The average number of photons inside the cavity for a typical detected photon rate of 40 per cycle corresponds to only 0.015 in steady state.

	output of the cavity is given by
	Flux =	𝑁 𝑡 × 0.335	=	2𝛼 2 𝛾 𝑐	(7.3)
	In a typical experimental scenario we have n photons detected on each SPCM in
	120 𝜇s, then
	2𝛼 2 𝛾 𝑐 𝛼 = 0.07 = 2.5 × 10 -3 𝑛𝛾 𝑒 √ 𝑛𝛾 𝑒
	where we use

Table 7 .

 7

	Parameter	Value
	Length	1 m
	FSR	150 MHz
	Mode waist	90 𝜇m
	Finesse	50
	Power enhancement factor	14
	4: Blue cavity paramters

  Equation 2.28) corresponding to terms 𝜎 𝑒𝑟 , 𝜎 𝑒𝑒 , 𝜎 𝑟𝑟 𝑑𝜎 𝑒𝑟

			x
			Lock	~20MHz
			Control
	R > 0.998	Atomic cloud	R = 0.971

(

  Second order correlation function . . . . . . . . . . . . . 98 8.2 Photon statistics in continuous excitation limit . . . . 100 8.2.1 Experimental implementation . . . . . . . . . . . . . . . . 100 8.2.2 Photon counting measurements . . . . . . . . . . . . . . . 101 8.3 Squeezing measurements using theoretical model . . . 101 8.4 Squeezing measurements on experiment . . . . . . . . . 102 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

	Contents
	8.1

  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 109 9.2 Quantum regime . . . . . . . . . . . . . . . . . . . . . . . 110 9.3 A new cavity system . . . . . . . . . . . . . . . . . . . . 111 9.3.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Installation and alignment of the cavity . . . . . . . . . . 9.3.3 Characterization of the cavity . . . . . . . . . . . . . . . . 9.4 Cooperativity and EIT measurements . . . . . . . . . . 118 9.4.1 Cooperativity measurement . . . . . . . . . . . . . . . . . 9.4.2 EIT measurement . . . . . . . . . . . . . . . . . . . . . . 9.5 Possible schemes to exploit . . . . . . . . . . . . . . . . . 119 9.5.1 Off-resonant excitation . . . . . . . . . . . . . . . . . . . . 9.5.2 On-resonance excitation . . . . . . . . . . . . . . . . . . . 9.5.3 Cavity phase shift . . . . . . . . . . . . . . . . . . . . . . 9.6 Photonic controlled-phase gate proposal . . . . . . . . . 123 9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

	Chapter 9
	Towards quantum optical non-linearities
	Contents
	9.1
	Towards Quantum Optical Non-linearities

  Even in the presence of dust, we could couple efficiently to the fundamental mode without scattering light into other modes. The locking scheme is similar to the previous cavity, we use a 810 nm auxiliary beam with side bands at 100 MHz. Instead of acting only on the top mirror, we act on the top one for slow large drifts and on the bottom one to compensate noise at high frequencies (above 50 Hz). We can use the same error signal for both the locking loops of top and the bottom mirrors, but any sharp movements disturbs the slow piezos enough to move it out of resonance.

	Parameter	Value
	Linewidth -FWHM	≈ 3.9 MHz
	Mode waist	15 𝜇m
	Finesse	555
	Table 9.1: Main cavity parameters
	Cavity stability

Acknowledgments

First of all, I would like to thank my family, particularly my brother, for always

List of Tables 4.1 Summary of all required atomic transitions. . . . . . . . . . . system which helps us to create large uniform Rabi-frequency to excite to highlying Rydberg states. We will also report the resonant optical non-linearity created by Rydberg-Rydberg interactions and characterize the system. In addition, we will present some numerical models which enable us to evaluate higher order nonlinear effects and determine the interesting range of parameters necessary to observe quantum non-linear effects. Finally, we conclude this part with the correlation and the squeezing measurements using Rydberg S states.

Fundamental constants and Energy levels of 87 Rb

We list some fundamental constants and various properties of 87 Rb that complement this thesis. Most of this information can be found in the NIST website.

A. A.2 87

Rubidium reference

We recall some constants we use in our calculations. A more detailed review of these constants can be found in [START_REF] Steck | Rubidium 87 D Line Data[END_REF]. We use only the D 2 for trapping and cooling of 87 Rb atoms. In The MOT trapping light is locked using the crossover transition between 5P 3/2 F = 1 and F = 3. It is then moved closer to resonance using a double pass AOM.

The repumper beam is locked on the crossover transition between the 5P 3/2 F = 1 and F = 3 and then shifted to on-resonance to repump atoms in 5 S 1/2

Appendix B

Rydberg atoms

In this appendix we will briefly recall the properties of Rydberg atoms and their interaction strengths. Rydberg electrons are so far from the nucleus they are extremely sensitive to weak fields. We are going to look at the properties of Rydberg states in alkali atoms, where only one valence electron is excited to high principle quantum numbers.

Due to the shielding effect of the non-valence electrons in an alkali atom, it acts like a Hydrogen atom and hence referred to as Hydrogenoid atom. Unlike the Hydrogen atom, where there are no other electrons in inner orbitals, Rydberg states are affected by the presence of non-valence electrons. This effect is very prominent in low orbital angular momentum states (l), where the electron orbits very close to the nucleus. These Hydrogenoid atoms can be accurately described by the quantum defect theory [70], especially for the low angular momentum levels (𝑙 ≤ 3). The physical properties of Rydberg atoms can be described by a formula similar to the one of the Hydrogen atom, with an effective principal quantum number 𝑛 * = 𝑛-𝛿 𝑛,𝑙,𝑗 where 𝛿 𝑛,𝑙,𝑗 is referred to as quantum defect.

The energy of any Rydberg level can then be evaluated using

where 𝑅 𝑦 = 13.6eV is the Rydberg constant.

Calculation of Stark shifts for D states

In this appendix, we will describe how to calculate the Stark structure of alkali atoms. As the principal quantum number increases, the electron moves away from the nucleus and the coupling to external electric field increases. The excitation to D states involves a large number of Zeeman sublevels, which are highly sensitive to the electric field especially at high principal quantum numbers. Therefore, the atomic level structure is perturbed by the presence of even weak electric fields [START_REF] Zimmerman | Stark structure of the Rydberg states of alkali-metal atoms[END_REF].

We assume that the electric field is in the direction of the quantization axis ⟨𝑛, 𝑙, 𝑗,

Here we consider the selection rules to select only the levels which are coupled by the electric field.

For weak fields, the Stark splitting for non-hydrogenic states (𝑙 ≤ 3) can be calculated using a perturbative expansion and the shift can be calculated using

where 𝛼 is the static polarizability of the energy level and is calculated using

Appendix D

Wigner Function

The g (2) measurement is inherently insensitive to phase fluctuations inside a quan- tum state, hence it doesn't give full information about the state. The quantum state of light can be fully described by using a density matrix. Usually, this density matrix is written in a Fock state basis, where each state is an eigenstate of the number operator ( N = â † â). One can also describe it in the optical phase space using quadratures of quantized electric field ( q = √ 2𝜔 (â + â † ) and p = 𝑖 Author's Publications μ μ