Julia Madame

Meyer

Monsieur Michiel Wouters

A Input Output model in Polariton basis B Generalized Bogoliubov Transformation

Keywords: . . . . . . . . . . . . . . Polaritons excitoniques, systèmes entraînés-dissipatifs, équation de Gross-Pitaevskii, théorie de Bogoliubov, polaron de Bose, TMDC, graphène polaritonique Exciton Polaritons, Driven-Dissipative systems, Gross-Pitaevskii Equation, Bogoliubov theory, Bose Polaron, TMDC, Polaritonic Graphene Exciton Polaritons, Driven-Dissipative systems, Gross-Pitaevskii Equation

Resumé

Dans cette thèse, nous avons examiné théoriquement l'effet des interactions à deux corps dans différents systèmes de polariton-excitoniques en microcavité, en situation d'excitation laser résonante.

Dans le premier cas, nous avons étudié la dynamique d'une impureté mobile en interaction avec un bain de polaritons, qui constitue un problème de type "polaron Bosonique", dans un cadre hors-équilibre. L'impureté est ainsi soumise à une force d'entrainement cohérente exercée par le fluide de polaritons, dont ce dernier a la particularité d'etre dans une situation stationnaire gain/dissipation. Cette situation résulte en des caractéristiques uniques du spectre d'excitation du fluide de polariton, qui conduisent à différents régimes dynamiques pour le mouvement du polaron, dont en particulier un régime non-trivial d' "accélération à contre-courant".

Nous avons ensuite étudié les interactions excitoniques dans une monocouche de dichalcogénures de métaux de transition (TMDC) inséré dans une microcavité. Dans ces matériau d'épaisseur atomique, le potentiel d'interaction de Coulomb entre électron et trou est fortement influencée par la forte discontinuité de la fonction diélectrique selon l'axe perpendiculaire à la couche. Cette caractéristique donne lieu à des états excitoniques non-hydrogéniques, offrant une plus forte interaction à deux corps que dans le cas des états hydrogéniques conventionnelles. Pour étudier cette amélioration, noos avons analysé les résultats d'une expérience de spectroscopie de transmission optique d'une monocouche de MoSe2 en régime de couplage fort avec le mode d'une microcavité optique de manière quantitative, à l'aide de notre théorie entrée-sortie non-linéaire. Nous avons pu ainsi determiner que l'experience montre une amélioration de l'interaction exciton-exciton et de la saturation fermionique excitonique par rapport aux valeurs mesurées dans des systèmes où l'exciton est hydrogénique.

Dans une troisième partie, nous avons étudié l'effet des interactions polaritonpolariton lorsque ces derniers sont confinés dans un réseau artificiel en nid d'abeille, de même géométrie que celui du graphène. Nous avons analysé les excitations élémentaires d'un condensat de polaritons situé au point K de ce système en nous basant sur le théorie de Bogoliubov généralisée, et nous avons pu ainsi retrouver plusieurs caractéristiques observés dans les expériences.

l'émergence d'effets collectifs non triviaux tels que la superfluidité et la supraconductivité (L. [START_REF] Pitaevskii | Bose-Einstein condensation and superfluidity[END_REF][START_REF] Nozieres | The Theory of Quantum Liquids[END_REF]. La dynamique des fluides quantiques sous de faibles interactions entre particules peut souvent être étudiée en modélisant une équation différentielle partielle (EDP) non linéaire où la quantité de non-linéarité est directement proportionnelle à la force des interactions. L'EDP non linéaire décrit une équation de mouvement hydrodynamique qui a été utilisée pour étudier la formation de motifs dans les systèmes physiques. Dans le domaine de l'optique non linéaire, une telle non-linéarité peut être liée à la présence d'une interaction entre les photons, qui sont les particules quantiques de la lumière. En poussant l'analogie avec les fluides quantiques encore plus loin, un groupe de chercheurs actifs à la fin du vingtième siècle a commencé à poser la question suivante : peut-on traiter les photons dans un milieu non linéaire comme un fluide quantique de lumière ? Malheureusement, la quantité de non-linéarité offerte dans un tel milieu n'est pas assez forte pour atteindre la force d'interaction requise pour observer des phénomènes collectifs comme la superfluidité de la lumière. Pour surmonter cet obstacle, nous pouvons coupler les photons avec des excitations de matière et créer des particules hybrides lumière-matière appelées polaritons. Leur nature matérielle confère de fortes interactions à la partie photonique, créant ainsi des interactions suffisamment fortes entre les photons pour la traiter comme un fluide quantique de lumière (Carusotto and Ciuti, 2013). En effet, des effets collectifs ont été observés pour les polaritons excitoniques dans une microcavité : ils représentent notre principal système d'intérêt au cours de cette thèse.

Dans le chapitre 1, je commencerai par fournir un bref historique des polaritons excitoniques dans une microcavité, également connus sous le nom de polaritons de microcavité. Dans ce chapitre, je discuterai de la formation des polaritons exciton dans les semi-conducteurs afin d'avoir une idée conceptuelle de ce que sont ces particules, suivi d'une discussion sur le système qui nous intéresse : les polaritons de microcavité. Plus tard, je présenterai l'équation différentielle partielle non linéaireappelée Gross-Pitaevskii généralisée -que nous utiliserons pour étudier la dynamique des polaritons de microcavité, et je dériverai conceptuellement la relation de dispersion des petites fluctuations du système en utilisant la théorie de Bogoliubov des excitations élémentaires. Tous les outils théoriques nécessaires à la suite de la thèse étant prêts, nous allons maintenant commencer par décrire les principaux travaux de recherche effectués au cours de mon doctorat.

Le travail est divisé en trois parties correspondant à trois études de cas différentes d'interactions de particules dans un système de polariton de microcavité :

Partie I est consacrée à l'étude de l'interaction d'une impureté avec un bain de polaritons de microcavité qui, dans un régime plus large, sera traitée comme le problème d'un polaron de Bose dans un fluide quantique de lumière.

D ans le vide, la lumière peut échanger de la quantité de mouvement et de l'énergie avec des particules en mouvement libre via le mécanisme de la pression de radiation. Lorsqu'un photon heurte une particule (par exemple un atome, une bille de verre, une sonde interplanétaire), il peut être absorbé, réfracté et/ou diffusé, transférant ainsi une fraction de sa quantité de mouvement et de son énergie à la particule, qui est ainsi accélérée et dirigée par la lumière. Ce mécanisme est à la base de plusieurs applications et phénomènes clés tels que le refroidissement d'atomes ultrafroids [START_REF] Horak | Cavity-Induced Atom Cooling in the Strong Coupling Regime[END_REF][START_REF] Wieman | Atom Cooling, Trapping, and Quantum Manipulation[END_REF], l'optomécanique quantique en cavité [START_REF] Aspelmeyer | Cavity Optomechanics[END_REF], ainsi que dans la conception proposée de sondes interplanétaires [START_REF] Atwater | Materials challenges for the Starshot lightsail[END_REF].

Cette image devrait être profondément modifiée et enrichie pour les photons se propageant dans les solides, et d'autant plus pour les photons fortement habillés d'excitations électroniques, comme dans les microcavités semi-conductrices dans le régime de couplage fort entre excitons et photons de cavité. Expérimentalement, le fluide quantique polariton [START_REF] Byrnes | Exciton-Polariton Condensates[END_REF][START_REF] Schneider | Exciton-Polariton Trapping and Potential Landscape Engineering[END_REF] peut être généré dans un état de momentum bien défini et envoyé en collision contre un obstacle fixe (par exemple un potentiel de barrière localisé). Ce cas a été examiné à la fois théoriquement [START_REF] Carusotto | Probing Microcavity Polariton Superfluidity through Resonant Rayleigh Scattering[END_REF]Ciuti and Carusotto, 2005;[START_REF] Cancellieri | Superflow of resonantly driven polaritons against a defect[END_REF][START_REF] Amelio | Galilean Boosts and Superfluidity of Resonantly Driven Polariton Fluids in the Presence of an Incoherent Reservoir[END_REF] et expérimentalement [START_REF] Amo | Collective Fluid Dynamics of a Polariton Condensate in a Semiconductor Microcavity[END_REF][START_REF] Amo | Superfluidity of Polaritons in Semiconductor Microcavities[END_REF] : il a été montré que pour un ensemble approprié de paramètres expérimentaux, l'écoulement autour de l'obstacle présente de fortes signatures de superfluidité [START_REF] Leggett | Superfluidity[END_REF], mises en évidence par un écoulement sans frottement et l'absence de force de traînée, tandis que pour d'autres paramètres expérimentaux, la force de traînée est restaurée [START_REF] Van Regemortel | Negative drag in nonequilibrium polariton quantum fluids[END_REF], avec une magnitude et des directions non triviales [START_REF] Juggins | Coherently Driven Microcavity-Polaritons and the Question of Superfluidity[END_REF]. En particulier, comme cela est particulier aux fluides quantiques entraînés-dissipatifs, un régime de force de traînée négative a été signalé comme se produisant lorsque la relation de dispersion présente une partie imaginaire dispersive [START_REF] Van Regemortel | Negative drag in nonequilibrium polariton quantum fluids[END_REF]. Dans un tel régime, en raison de la dissipation dépendant du momentum, la diffusion vers l'avant est plus supprimée que celle vers l'arrière et le fluide ressent effectivement une force dans la direction opposée au mouvement de l'obstacle.

Dans cette partie, nous allons considérer le cas d'un obstacle de masse finie se déplaçant librement et soumis à la force de traînée exercée par un fluide quantique de polaritons entraîné-dissipatif. Ce problème peut être compris comme l'analogue de la détermination de la dynamique d'une particule soumise à une pression de radiation, lorsque la pression de radiation est remplacée par la force de traînée du fluide quantique de polaritons. Il s'agit également d'une classe de problème différente du cas de l'obstacle fixe mentionné ci-dessus : en effet, un obstacle fixe a une masse infinie et ne peut pas être excité par le fluide, et n'a donc pas de dynamique. D'un point de vue théorique, la situation que nous considérons est celle d'une impureté unique habillée par les excitations de son environnement à plusieurs corps, connue sous le nom de problème du polaron. Ce problème est d'une grande importance dans plusieurs contextes. En physique du solide, un état de polaron décrit des électrons habillés par des phonons de réseau [START_REF] Landau | Effective mass of a polaron[END_REF][START_REF] Fröhlich | XX. Properties of Slow Electrons in Polar Materials[END_REF][START_REF] Feynman | Slow Electrons in a Polar Crystal[END_REF]Alexandrov and Devreese, 2010b). Le polaron de Fermi, qui décrit une impureté mobile dans un gaz de Fermi dégénéré, a suscité un regain d'intérêt dans les gaz atomiques ultrafroids [START_REF] Chevy | Universal Phase Diagram of a Strongly Interacting Fermi Gas with Unbalanced Spin Populations[END_REF][START_REF] Lobo | Normal State of a Polarized Fermi Gas at Unitarity[END_REF][START_REF] Prokof'ev | Fermi-polaron problem: Diagrammatic Monte Carlo method for divergent sign-alternating series[END_REF][START_REF] Massignan | Polarons, Dressed Molecules and Itinerant Ferromagnetism in Ultracold Fermi Gases[END_REF][START_REF] Schirotzek | Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms[END_REF][START_REF] Nascimbène | Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass[END_REF][START_REF] Koschorreck | Attractive and Repulsive Fermi Polarons in Two Dimensions[END_REF][START_REF] Darkwah Oppong | Observation of Coherent Multiorbital Polarons in a Two-Dimensional Fermi Gas[END_REF][START_REF] Ness | Observation of a Smooth Polaron-Molecule Transition in a Degenerate Fermi Gas[END_REF] et a également été récemment observé dans des semi-conducteurs monocouches [START_REF] Sidler | Fermi Polaron-Polaritons in Charge-Tunable Atomically Thin Semiconductors[END_REF][START_REF] Tan | Interacting Polaron-Polaritons[END_REF] lorsque des excitons-polaritons se propagent dans un fluide d'électrons libres [START_REF] Bastarrachea-Magnani | Attractive and Repulsive Exciton-Polariton Interactions Mediated by an Electron Gas[END_REF][START_REF] Julku | Nonlinear Optical Response of Resonantly Driven Polaron-Polaritons[END_REF]. Le problème du polaron de Bose, c'est-à-dire le cas d'une impureté dans un fluide quantique bosonique a également attiré une attention significative [START_REF] Tempere | Feynman Path-Integral Treatment of the BEC-Impurity Polaron[END_REF][START_REF] Rath | Field-Theoretical Study of the Bose Polaron[END_REF][START_REF] Shashi | Radio-Frequency Spectroscopy of Polarons in Ultracold Bose Gases[END_REF] et a été démontré expérimentalement dans des atomes ultrafroids [START_REF] Catani | Quantum Dynamics of Impurities in a One-Dimensional Bose Gas[END_REF][START_REF] Hu | Bose Polarons in the Strongly Interacting Regime[END_REF][START_REF] Jørgensen | Observation of Attractive and Repulsive Polarons in a Bose-Einstein Condensate[END_REF][START_REF] Camargo | Creation of Rydberg Polarons in a Bose Gas[END_REF][START_REF] Yan | Bose Polarons near Quantum Criticality[END_REF][START_REF] Skou | Non-Equilibrium Quantum Dynamics and Formation of the Bose Polaron[END_REF]. Pour les polarons-polaritons de Bose à l'équilibre, les interactions médiées par Feshbach (Bastarrachea-Magnani, Camacho-Guardian, [START_REF] Bastard | Strong interactions and biexcitons in a polariton mixture[END_REF] et la formation d'états liés aux photons [START_REF] Camacho-Guardian | Mediated Interactions and Photon Bound States in an Exciton-Polariton Mixture[END_REF] ont été prédites. Les polaron-polaritons de Bose ont également été proposés comme sonde pour observer les états corrélés à plusieurs corps [START_REF] Levinsen | Spectroscopic Signatures of Quantum Many-Body Correlations in Polariton Microcavities[END_REF].

En nous concentrant sur la condition expérimentale pertinente de faible couplage entre l'impureté et le fluide, nous avons développé une approche Bogoliubov-Fröhlich en présence d'un bain externe et fourni des solutions, sous approximation markovienne, pour la masse effective du polaron et la force de traînée exercée par le fluide sur celui-ci. Nous avons considéré les deux cas du fluide au repos et en mouvement. Nous avons ensuite examiné le mouvement des impuretés induit par la traînée du fluide et identifié les signatures de superfluidité dans la dynamique du polaron. Je commencerai par donner un bref aperçu des polarons, en particulier des polarons dans les condensats de Bose Einstein (BEC) ultrafroids, dans le chapitre 2 et j'étendrai le concept des polarons de Fröhlich dans le système exciton-polariton entraîné-dissipatif dans le chapitre 3.

Partie II est consacrée à l'étude des interactions polariton-polariton dans une nouvelle classe de matériaux semi-conducteurs 2D : les monocouches de dichalcogénures de métaux de transition (TMDC).

A u cours des deux dernières décennies, l'une des ambitions les plus importantes et les plus stimulantes des communautés de la matière condensée et de l'optique a été de réaliser des nanostructures photoniques à l'état solide présentant une grande non-linéarité optique du troisième ordre -également connue sous le nom de nonlinéarité Kerr. L'intérêt est double : d'une part, les réseaux de microcavités optiques non linéaires couplées ou les treillis photoniques peuvent éventuellement constituer un puissant simulateur de la physique quantique à plusieurs corps sans équilibre [START_REF] Tomadin | Many-Body Phenomena in QED-Cavity Arrays[END_REF][START_REF] Noh | Quantum Simulations and Many-Body Physics with Light[END_REF] et, d'autre part, ils permettraient d'étendre le fonctionnement des portes logiques photoniques -ingrédient de base de la communication et du calcul optiques [START_REF] Andalib | All-Optical Ultracompact Photonic Crystal AND Gate Based on Nonlinear Ring Resonators[END_REF][START_REF] Liu | All-Optical Logic Gates Based on Two-Dimensional Low-Refractive-Index Nonlinear Photonic Crystal Slabs[END_REF][START_REF] Mabuchi | Nonlinear Interferometry Approach to Photonic Sequential Logic[END_REF][START_REF] Espinosa-Ortega | Complete Architecture of Integrated Photonic Circuits Based on and and Not Logic Gates of Exciton Polaritons in Semiconductor Microcavities[END_REF][START_REF] Salmanpour | Photonic Crystal Logic Gates: An Overview[END_REF] -vers le régime quantique, car une non-linéarité plus élevée permet de réduire l'énergie nécessaire à la commutation de la porte [START_REF] Flamini | Photonic Quantum Information Processing: A Review[END_REF]. La première intéresse une communauté croissante de physiciens qui s'efforcent de comprendre les phénomènes physiques qui se produisent dans le régime loin de l'équilibre, comme les états de Hall quantiques fractionnaires [START_REF] Cho | Fractional Quantum Hall State in Coupled Cavities[END_REF][START_REF] Umucalılar | Fractional Quantum Hall States of Photons in an Array of Dissipative Coupled Cavities[END_REF][START_REF] Hafezi | Non-Equilibrium Fractional Quantum Hall State of Light[END_REF], les états fermionisés [START_REF] Carusotto | Fermionized Photons in an Array of Driven Dissipative Nonlinear Cavities[END_REF], les phénomènes topologiques non-hermitiens [START_REF] Tangpanitanon | Topological Pumping of Photons in Nonlinear Resonator Arrays[END_REF] et une variété de transitions de phase hors équilibre, pour lesquels nous manquons de théories les expliquant de manière fiable ; tandis que la dernière est d'un intérêt appliqué pour les expérimentateurs et les ingénieurs quantiques qui construisent/construisent/fabriquent du matériel et des dispositifs quantiques basés sur les photons [START_REF] O'brien | Photonic Quantum Technologies[END_REF].

Pour réaliser cette ambition, l'une des possibilités est d'utiliser les états excitoniques des nanostructures semi-conductrices, qui peuvent induire un renforcement de la non-linéarité optique, car ils interagissent fortement avec la lumière. Au cours de la dernière décennie, de nombreuses recherches ont été menées sur l'utilisation de différents matériaux semi-conducteurs dans diverses géométries de nanostructures afin de porter leur valeur à des niveaux sans précédent. Dans les géométries à puits quantiques [START_REF] Schneider | Exciton-Polariton Trapping and Potential Landscape Engineering[END_REF] en particulier, une stratégie connue sous le nom de blocage quantique des polaritons [START_REF] Verger | Polariton Quantum Blockade in a Photonic Dot[END_REF] amplifie la non-linéarité excitonique effective, en principe jusqu'au régime quantique, en optimisant le couplage avec la lumière et en réduisant optiquement la fonction d'onde excitonique dans le plan. L'apparition de ce régime a été récemment démontrée dans des microcavitites à base de GaAs [START_REF] Muñoz-Matutano | Emergence of Quantum Correlations from Interacting Fibre-Cavity Polaritons[END_REF].

La non-linéarité obtenue par cette stratégie, bien que significative, est limitée aux températures cryogéniques, ce qui restreint son applicabilité aux dispositifs. Comme nous le verrons dans le chapitre 1, les principales contributions à la non-linéarité excitonique proviennent de l'interaction d'échange Coloumbic (de magnitude 𝑔 𝑥 ) entre excitons et de la saturation ferminonique des électrons et des trous impliqués (de magnitude 𝑔 𝑠 ). Pour que cette stratégie déploie tout son potentiel, il est essentiel de trouver un moyen de maximiser les constantes non linéaires et l'énergie de liaison, afin d'atteindre la stabilité à température ambiante. Mais ces quantités sont difficiles à obtenir dans les matériaux semi-conducteurs conventionnels utilisés jusqu'à présent (par exemple, arséniure, nitrure, tellurure, oxydes ou alliages de cuprate).

Les monocouches de dichalchogénures de métaux de transition semi-conducteurs (TMDC) [START_REF] Mak | Atomically Thin MoS 2 : A New Direct-Gap Semiconductor[END_REF] offrent une opportunité unique de manipuler la non-linéarité excitonique au-delà de ce compromis. En raison de leur épaisseur à l'échelle atomique, la constante diélectrique présente une forte discontinuité dans le plan du matériau, ce qui entraîne un potentiel de Coulomb fortement modifié [START_REF] Keldysh | Coulomb Interaction in Thin Semiconductor and Semimetal Films[END_REF] et une grande sensibilité à l'environnement diélectrique entourant la couche. Les états excitoniques qui en résultent sont uniquement non-hydrogéniques [START_REF] Berkelbach | Theory of Neutral and Charged Excitons in Monolayer Transition Metal Dichalcogenides[END_REF][START_REF] Chernikov | Exciton Binding Energy and Nonhydrogenic Rydberg Series in Monolayer WS 2[END_REF], hautement accordables via l'ingénierie des hétérostructures de van der Waals [START_REF] Novoselov | 2D Materials and van Der Waals Heterostructures[END_REF][START_REF] Liu | Van Der Waals Heterostructures and Devices[END_REF][START_REF] Jariwala | Mixed-Dimensional van Der Waals Heterostructures[END_REF], et leurs constantes non linéaires 𝑔 𝑥 et 𝑔 𝑠 devraient s'écarter de l'image hydrogénique. Une augmentation de 30% de 𝑔 𝑥 est par exemple prédite dans 𝑊 𝑆 2 [START_REF] Shahnazaryan | Exciton-Exciton Interaction in Transition-Metal Dichalcogenide Monolayers[END_REF]. Des indications de non-linéarités excitoniques importantes ont été observées dans les TMDCs [START_REF] Barachati | Interacting Polariton Fluids in a Monolayer of Tungsten Disulfide[END_REF], dans les états chargés [START_REF] Emmanuele | Highly Nonlinear Trion-Polaritons in a Monolayer Semiconductor[END_REF] et excités des excitons [START_REF] Scuri | Large Excitonic Reflectivity of MonolayerMoSe 2 Encapsulated in Hexagonal Boron Nitride[END_REF], et dans les polarons-polaritons [START_REF] Tan | Interacting Polaron-Polaritons[END_REF].

Nous avons profité de la force d'oscillateur géante des excitons TMDCs (G. [START_REF] Wang | Colloquium: Excitons in Atomically Thin Transition Metal Dichalcogenides[END_REF] pour mettre une monocouche MoSe 2 dans le régime de couplage fort avec la résonance d'une microcavité. Nos collaborateurs expérimentaux ont obtenu les spectres présentant les signatures d'une réponse non linéaire et nous avons développé un modèle d'entrée-sortie de champ moyen pour analyser les spectres. Nous avons ensuite résolu numériquement le modèle en utilisant les paramètres ajustés aux observations expérimentales pour obtenir une estimation quantitative de 𝑔 𝑠 et 𝑔 𝑥 . Dans les chapitres 4 et 5, je donnerai un petit aperçu des propriétés excitoniques des TMDC dans le premier et présenterai les résultats de notre travail, à savoir le développement du modèle de champ moyen pour obtenir une estimation des constantes non linéaires, dans le second chapitre.

Partie III est consacrée à l'étude des interactions polariton-polariton dans un réseau artificiel en nid d'abeille également connu sous le nom de graphène polari-tonique. Cette partie de la thèse représente un travail en cours qui est sur le point de s'achever.

L a structure du réseau en nid d'abeille a un effet profond sur la dynamique des particules dans cette géométrie de réseau. L'exemple le plus marquant est le mouvement des quasi-particules d'électrons sans masse dans le graphène [START_REF] Castro Neto | The Electronic Properties of Graphene[END_REF]. De nombreux phénomènes physiques exotiques dans le graphène, les TMDC et les isolants toppologiques [START_REF] Hasan | Colloquium: Topological Insulators[END_REF] tels que l'effet tunnel de Klein (Das Sarma et al., 2011) ou l'apparition d'états limites [START_REF] Delplace | Zak Phase and the Existence of Edge States in Graphene[END_REF] sont une conséquence directe de la structure en nid d'abeille du matériau. Il est intéressant de noter que tous ces phénomènes peuvent être étudiés en créant des modèles analytiques dans le cadre de l'approximation de liaison serrée.

Récemment, les chercheurs ont créé des configurations de nids d'abeille artificiels dans différents types de systèmes, comme des atomes froids dans des réseaux optiques [START_REF] Tarruell | Creating, Moving and Merging Dirac Points with a Fermi Gas in a Tunable Honeycomb Lattice[END_REF] et des cavités photoniques [START_REF] Ozawa | Topological Photonics[END_REF], afin de simuler la physique du graphène dans un environnement propre pour les particules autres que les électrons. Dans ces simulateurs, la structure du réseau et les fonctions d'onde des particules peuvent être modifiées à notre guise, ce qui ouvre la voie à l'étude des effets de la géométrie du réseau dans différents types de particules. Le fait que les photons puissent être contrôlés avec des niveaux de précision très élevés dans une configuration de microcavité a inspiré des efforts visant à émuler la physique du graphène pour les excitons-polaritons. En effet, un tel émulateur appelé graphène polaritonique a été réalisé en modifiant le paysage potentiel des polaritons [START_REF] Schneider | Exciton-Polariton Trapping and Potential Landscape Engineering[END_REF] pour simuler un réseau en nid d'abeille [START_REF] Jacqmin | Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons[END_REF]. La relation de dispersion ainsi obtenue ressemble à la structure de bande du graphène avec les points de Dirac caractéristiques aux six coins de la première zone de Brioullin.

Dans ce travail, nous étudierons l'effet des interactions des polaritons sur la relation de dispersion du graphène polaritonique en utilisant la théorie de Bogoliubov pour les particules sur un réseau. Dans le chapitre 6, je présenterai un bref historique du graphène polaritonique et je dériverai la relation de dispersion pour les particules dans un réseau en nid d'abeille dans l'approximation de liaison serrée. Je présenterai également l'effet du couplage spin-orbite sur cette relation de dispersion. Puis, à la fin du chapitre 7, nous dériverons la dispersion de Bogoliubov en présence d'interactions de polaritons et nous comparerons nos résultats avec les données expérimentales fournies par notre collaboration avec le groupe du Dr Maxime Richard à l'Institut N'eel.

Je conclurai cette thèse par un bref aperçu et les perspectives futures que nous avons tirées du travail présenté ici. Une liste d'articles publiés pendant cette période est présentée ci-dessous

Publications

• Vashisht, A., Richard M. & Minguzzi, A., Bose polaron in a quantum fluid of light, SciPost Phys. 12, 008 (2022) • Stepanov, P., Vashisht, A., Klaas, M., Lundt, N., Tongay, S., Blei, M., Höfling, S., Volz, T., Minguzzi, A.,

Introduction

I nteraction among particles in a quantum fluid often leads to the emergence of non-trivial collective effects such as superfluidity and superconductivity (L. [START_REF] Pitaevskii | Bose-Einstein condensation and superfluidity[END_REF][START_REF] Nozieres | The Theory of Quantum Liquids[END_REF]. Dynamics of quantum fluids under weak particle interactions can often be studied by modelling a nonlinear partial differential equation (PDE) where the amount of non-linearity is directly proportional to the strength of interactions. The nonlinear PDE describes a hydrodynamical equation of motion the likes of which have been used used to study pattern formation in physical systems [START_REF] Cross | Pattern Formation Outside of Equilibrium[END_REF].

In the field of nonlinear optics such a nonlinearity can be linked to the presence of interaction among photons, which are the quantum particles of light. Taking the analogy with quantum fluids even further an active field of researchers in late twentieth century started asking the question can we treat photons in a nonlinear medium as a quantum fluid of light? Sadly, the amount of nonlinearity offered in a such a medium is not strong enough to achieve the requisite interaction strength to observe collective phenomena like superfluidity of light. To overcome this hindrance we can couple photons with electronic excitations of a material and create hybrid light-matter particles called exciton polaritons wherein their material nature imparts strong interactions to the photonic part, effectively creating strong enough interactions among photons to treat it as a quantum fluid of light. Indeed, collective effects have been observed for exciton polaritons in a microcavity (Carusotto and Ciuti, 2013) and they will represent our primary system of interest during the course of this thesis.

In chapter 1, I'll start by providing a brief background on exciton polaritons in a microcavity, also known as microcavity polaritons. Therein, I'll discuss the formation of exciton polariton in bulk semiconductors to get a conceptual idea as to what these particles are, followed by a discussion of our system of focus: microcavity polaritons. Later I'll introduce the nonlinear partial differential equation -called generalized Gross-Pitaevskii -which we will use to study the dynamics of microcavity polaritons, and conceptually derive the dispersion relation of small fluctuations of the system using the Bogoliubov theory of elementary excitations. With all the theoretical tools required in the rest of thesis ready, we will start next with describing the main research work performed during the course of my PhD study.

The work is divided into three parts corresponding to three different case studies of particle interactions in a microcavity polariton system: Part I is devoted to the study of interaction of an impurity with a bath of TABLE OF CONTENTS microcavity polaritons which in a wider regime will be treated as the problem of a Bose polaron in a quantum fluid of light.

I n vacuum, light can exchange momentum and energy with free moving particles via the mechanism of radiation pressure. When a photon impinges a particle (e.g. an atom, a glass bead, an interplanetary probe) it can be absorbed, refracted and/or scattered, thereby transferring a fraction of its momentum and energy to the particle, which is thus accelerated and steered by light. This mechanism is at the basis of several key applications and phenomena such as ultracold atom cooling [START_REF] Horak | Cavity-Induced Atom Cooling in the Strong Coupling Regime[END_REF][START_REF] Wieman | Atom Cooling, Trapping, and Quantum Manipulation[END_REF], cavity quantum optomechanics [START_REF] Aspelmeyer | Cavity Optomechanics[END_REF], as well as in proposed interplanetary probes design [START_REF] Atwater | Materials challenges for the Starshot lightsail[END_REF]. This picture is expected to be profoundly modified and enriched for photons propagating in solids, and all the more so for photons strongly dressed with electronic excitations, such as in semiconductor microcavities in the strong coupling regime between excitons and cavity photons. Experimentally, the polariton quantum fluid [START_REF] Byrnes | Exciton-Polariton Condensates[END_REF][START_REF] Schneider | Exciton-Polariton Trapping and Potential Landscape Engineering[END_REF] can be generated in a well-defined momentum state and sent to collide against a fixed obstacle (e.g. a localized barrier potential). This case has been examined both theoretically [START_REF] Carusotto | Probing Microcavity Polariton Superfluidity through Resonant Rayleigh Scattering[END_REF]Ciuti and Carusotto, 2005;[START_REF] Cancellieri | Superflow of resonantly driven polaritons against a defect[END_REF][START_REF] Amelio | Galilean Boosts and Superfluidity of Resonantly Driven Polariton Fluids in the Presence of an Incoherent Reservoir[END_REF] and experimentally [START_REF] Amo | Collective Fluid Dynamics of a Polariton Condensate in a Semiconductor Microcavity[END_REF][START_REF] Amo | Superfluidity of Polaritons in Semiconductor Microcavities[END_REF]: it was shown that for a suitable set of experimental parameters the flow around the obstacle exhibits strong signatures of superfluidity [START_REF] Leggett | Superfluidity[END_REF], evidenced by frictionless flow and absence of drag force, while for other experimental parameters, the drag force is restored [START_REF] Van Regemortel | Negative drag in nonequilibrium polariton quantum fluids[END_REF], with a non-trivial magnitude and directions [START_REF] Juggins | Coherently Driven Microcavity-Polaritons and the Question of Superfluidity[END_REF]. In particular, as peculiar in driven-dissipative quantum fluids, a regime of negative drag force was reported to occur when the dispersion relation displays a dispersive imaginary part [START_REF] Van Regemortel | Negative drag in nonequilibrium polariton quantum fluids[END_REF]. In such a regime, due to momentum-dependent dissipation, the forward scattering is more suppressed than the backward one and the fluid effectively feels a force in the direction opposite to the motion of the obstacle.

In this part, we'll consider the case of a freely-moving finite-mass obstacle subject to the drag force exerted by a driven-dissipative quantum fluid of polaritons. This problem can be understood as the analogue of determining the dynamics of a particle subjected to radiation pressure, when radiation pressure is replaced by the polaritonic quantum fluid drag force. This is also a different class of problem than the fixed obstacle case mentioned above: indeed, a fixed obstacle has an infinite mass and cannot be excited by the fluid, and thus has no dynamics. From a theoretical point of view, the situation that we consider is that of a single impurity dressed by the excitations of its many-body environment, known as the polaron problem. This problem is of high importance in several contexts. In solid-state [START_REF] Landau | Effective mass of a polaron[END_REF][START_REF] Fröhlich | XX. Properties of Slow Electrons in Polar Materials[END_REF][START_REF] Feynman | Slow Electrons in a Polar Crystal[END_REF]Alexandrov and Devreese, 2010b). The Fermi polaron, describing a mobile impurity in a degenerate Fermi gas has attracted renewed attention in ultracold atomic gases [START_REF] Chevy | Universal Phase Diagram of a Strongly Interacting Fermi Gas with Unbalanced Spin Populations[END_REF][START_REF] Lobo | Normal State of a Polarized Fermi Gas at Unitarity[END_REF][START_REF] Prokof'ev | Fermi-polaron problem: Diagrammatic Monte Carlo method for divergent sign-alternating series[END_REF][START_REF] Massignan | Polarons, Dressed Molecules and Itinerant Ferromagnetism in Ultracold Fermi Gases[END_REF][START_REF] Schirotzek | Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms[END_REF][START_REF] Nascimbène | Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass[END_REF][START_REF] Koschorreck | Attractive and Repulsive Fermi Polarons in Two Dimensions[END_REF][START_REF] Darkwah Oppong | Observation of Coherent Multiorbital Polarons in a Two-Dimensional Fermi Gas[END_REF][START_REF] Ness | Observation of a Smooth Polaron-Molecule Transition in a Degenerate Fermi Gas[END_REF] and has also been recently observed in monolayer semiconductors [START_REF] Sidler | Fermi Polaron-Polaritons in Charge-Tunable Atomically Thin Semiconductors[END_REF][START_REF] Tan | Interacting Polaron-Polaritons[END_REF] when exciton-polaritons propagate in a fluid of free electrons [START_REF] Bastarrachea-Magnani | Attractive and Repulsive Exciton-Polariton Interactions Mediated by an Electron Gas[END_REF][START_REF] Julku | Nonlinear Optical Response of Resonantly Driven Polaron-Polaritons[END_REF]. The Bose polaron problem, i.e. the case of an impurity in a bosonic quantum fluid has also attracted significant attention [START_REF] Tempere | Feynman Path-Integral Treatment of the BEC-Impurity Polaron[END_REF][START_REF] Rath | Field-Theoretical Study of the Bose Polaron[END_REF][START_REF] Shashi | Radio-Frequency Spectroscopy of Polarons in Ultracold Bose Gases[END_REF] and was experimentally demonstrated in ultracold atoms [START_REF] Catani | Quantum Dynamics of Impurities in a One-Dimensional Bose Gas[END_REF][START_REF] Hu | Bose Polarons in the Strongly Interacting Regime[END_REF][START_REF] Jørgensen | Observation of Attractive and Repulsive Polarons in a Bose-Einstein Condensate[END_REF][START_REF] Camargo | Creation of Rydberg Polarons in a Bose Gas[END_REF][START_REF] Yan | Bose Polarons near Quantum Criticality[END_REF][START_REF] Skou | Non-Equilibrium Quantum Dynamics and Formation of the Bose Polaron[END_REF]. For Bose polaronpolaritons at equilibrium, Feshbach-mediated interactions (Bastarrachea-Magnani, Camacho-Guardian, [START_REF] Bastard | Strong interactions and biexcitons in a polariton mixture[END_REF] and formation of photon bound states [START_REF] Camacho-Guardian | Mediated Interactions and Photon Bound States in an Exciton-Polariton Mixture[END_REF] were predicted. Bose polaron-polaritons were also proposed as a probe to observe correlated many-body states [START_REF] Levinsen | Spectroscopic Signatures of Quantum Many-Body Correlations in Polariton Microcavities[END_REF].

Focusing on the experimentally relevant condition of weak coupling between the impurity and the fluid, we developed a Bogoliubov-Fröhlich approach in the presence of an external bath and provided solutions, under Markovian approximation, for the effective mass of the polaron and the drag force exerted by the fluid on it. We considered both the cases of fluid at rest and in motion. We then examined the impurity motion induced by the drag of fluid and identified the signatures of superfluidity in polaron dynamics. I'll start by giving a brief background to polarons, especially polarons in ultracold Bose Einstein condensates (BECs), in chapter 2 and extending the concept of Fröhlich polarons to the driven-dissipative excitonpolariton system in chapter 3.

Part II is devoted to the study of polariton-polariton interactions in a novel class of 2D semiconducting materials: monolayers of transition metal dichalcogenides (TMDC).

F or the last couple of decades, one of the biggest and challenging ambition in the condensed matter and optics communities has been to realize solid state photonic nanostructures featuring a large third-order optical nonlinearity -also known as Kerr nonlinearity. The interest is twofold: on one hand, arrays of coupled nonlinear optical microcavities or photonic lattices can possibly constitute a powerful simulator of non-equilibrium quantum many-body physics [START_REF] Tomadin | Many-Body Phenomena in QED-Cavity Arrays[END_REF][START_REF] Noh | Quantum Simulations and Many-Body Physics with Light[END_REF] and on the other, it would help in extending the operation of photonic logic gates -basic ingredient in optical communication and computation [START_REF] Andalib | All-Optical Ultracompact Photonic Crystal AND Gate Based on Nonlinear Ring Resonators[END_REF][START_REF] Liu | All-Optical Logic Gates Based on Two-Dimensional Low-Refractive-Index Nonlinear Photonic Crystal Slabs[END_REF][START_REF] Mabuchi | Nonlinear Interferometry Approach to Photonic Sequential Logic[END_REF][START_REF] Espinosa-Ortega | Complete Architecture of Integrated Photonic Circuits Based on and and Not Logic Gates of Exciton Polaritons in Semiconductor Microcavities[END_REF][START_REF] Salmanpour | Photonic Crystal Logic Gates: An Overview[END_REF] -to the quantum regime as a higher nonlinearity helps in lowering the energy to switch the gate [START_REF] Flamini | Photonic Quantum Information Processing: A Review[END_REF]. The former interests a growing community of physicists who are working to understand physical phenomena happening in the far from equilibrium regime, such as fractional quantum Hall states [START_REF] Cho | Fractional Quantum Hall State in Coupled Cavities[END_REF][START_REF] Umucalılar | Fractional Quantum Hall States of Photons in an Array of Dissipative Coupled Cavities[END_REF][START_REF] Hafezi | Non-Equilibrium Fractional Quantum Hall State of Light[END_REF], fermionized states [START_REF] Carusotto | Fermionized Photons in an Array of Driven Dissipative Nonlinear Cavities[END_REF], nonhermitian topological phenomena [START_REF] Tangpanitanon | Topological Pumping of Photons in Nonlinear Resonator Arrays[END_REF] and a variety of non-equilibrium phase transitions, for which we have a dearth of theories reliably explaining them; while the latter is of applied interest to experimentalists and quantum engineers who are fabricating quantum hardware and devices based on photons [START_REF] O'brien | Photonic Quantum Technologies[END_REF].

To realize this ambition one of the possibilities is to utilize the excitonic states of the semiconductor nanostructures -as they can induce an enhancement of the optical nonlinearity -because they interact strongly with light. The previous decade has seen a lot of research activity, utilizing different semiconductor materials in various nanostructure geometries to boost its value to unprecedented levels. In quantum well geometries [START_REF] Schneider | Exciton-Polariton Trapping and Potential Landscape Engineering[END_REF] especially, a strategy known as polariton quantum blockade [START_REF] Verger | Polariton Quantum Blockade in a Photonic Dot[END_REF] amplifies the effective excitonic nonlinearity, in principle up to the quantum regime, by optimizing the coupling with light and optically narrowing down the in-plane excitonic wave function. The onset of this regime has been recently demonstrated in GaAs-based microcavitites [START_REF] Muñoz-Matutano | Emergence of Quantum Correlations from Interacting Fibre-Cavity Polaritons[END_REF].

The nonlinearity obtained using this strategy, though significant, is limited to cryogenic temperatures which restricts its device applicability. As we will see in Chapter 1 the main contributions to the excitonic nonlinearity comes from the Coulombic exchange interaction (of magnitude 𝑔 𝑥 ) between excitons and the fermifonic saturation of the involved electrons and holes (of magnitude 𝑔 𝑠 ). For this strategy to deliver its full potential, finding a way to maximize both the nonlinear constants and binding energy -to approach room temperature stability -is crucial. But these quantities are hard to engineer in the conventional semiconductor materials used so far (e.g. arsenide, nitride, telluride, oxides or cuprate alloys).

Monolayers of semiconductor transition metal dichalchogenides (TMDCs) [START_REF] Mak | Atomically Thin MoS 2 : A New Direct-Gap Semiconductor[END_REF] offer a unique opportunity to manipulate the excitonic nonlinearity beyond this trade-off. Owing to their atomic-scale thickness, the dielectric constant exhibits a sharp discontinuity across the material plane, that results in a strongly modified Coulomb potential [START_REF] Keldysh | Coulomb Interaction in Thin Semiconductor and Semimetal Films[END_REF], and a high sensitivity to the dielectric environment surrounding the layer. The resulting excitonic states are uniquely non-hydrogenic [START_REF] Berkelbach | Theory of Neutral and Charged Excitons in Monolayer Transition Metal Dichalcogenides[END_REF][START_REF] Chernikov | Exciton Binding Energy and Nonhydrogenic Rydberg Series in Monolayer WS 2[END_REF], highly tunable via van der Waals heterostructure engineering [START_REF] Novoselov | 2D Materials and van Der Waals Heterostructures[END_REF][START_REF] Liu | Van Der Waals Heterostructures and Devices[END_REF][START_REF] Jariwala | Mixed-Dimensional van Der Waals Heterostructures[END_REF],

TABLE OF CONTENTS

and their nonlinear constants 𝑔 𝑥 and 𝑔 𝑠 are expected to deviate from the hydrogenic picture. A 30% enhancement of 𝑔 𝑥 is for instance predicted in 𝑊 𝑆 2 [START_REF] Shahnazaryan | Exciton-Exciton Interaction in Transition-Metal Dichalcogenide Monolayers[END_REF]. Some indications of sizeable excitonic nonlinearities have been observed in TMDCs [START_REF] Barachati | Interacting Polariton Fluids in a Monolayer of Tungsten Disulfide[END_REF], in charged [START_REF] Emmanuele | Highly Nonlinear Trion-Polaritons in a Monolayer Semiconductor[END_REF] and excited states of excitons [START_REF] Scuri | Large Excitonic Reflectivity of MonolayerMoSe 2 Encapsulated in Hexagonal Boron Nitride[END_REF], and in polaron-polaritons [START_REF] Tan | Interacting Polaron-Polaritons[END_REF].

We took advantage of the giant oscillator strength of TMDCs excitons (G. [START_REF] Wang | Colloquium: Excitons in Atomically Thin Transition Metal Dichalcogenides[END_REF] to put a MoSe 2 monolayer in the strong coupling regime with the resonance of a microcavity. Our experimental collaborators obtained the spectra exhibiting signatures of a nonlinear response and we developed a mean field input output model to analyze the spectra. We then numerically numerically solved the model using the parameters fitted to the experimental observations to get a quantitative estimate of 𝑔 𝑠 and 𝑔 𝑥 . In Chapters 4 and 5 I'll give a small background on the excitonic properties of TMDCs in the former and present the results of our work, namely developing the mean-field model to get an estimate of the nonlinear constants, in the latter chapter.

Part III is devoted to the study of polariton-polariton interactions in an artificial honeycomb lattice also known as polaritonic graphene. This part of the thesis represents an ongoing work which is near its completion.

S tructure of the honeycomb lattice has profound effect on the dynamics of particles within this lattice geometry. The most prominent example being the motion of massless electron quasiparticles in graphene [START_REF] Castro Neto | The Electronic Properties of Graphene[END_REF]. A lot of exotic physical phenomena in graphene, TMDCs and topological insulators [START_REF] Hasan | Colloquium: Topological Insulators[END_REF] such as Klein tunneling [START_REF] Das Sarma | Electronic Transport in Two-Dimensional Graphene[END_REF] or appearance of edge states [START_REF] Delplace | Zak Phase and the Existence of Edge States in Graphene[END_REF] is a direct consequence of the honeycomb lattice structure of the material. Interestingly, all this phenomena can studied by creating analytical models within the tight-binding approximation.

Recently, researchers have created artificial honeycomb setups in various different kinds of systems like cold atoms in optical lattices [START_REF] Tarruell | Creating, Moving and Merging Dirac Points with a Fermi Gas in a Tunable Honeycomb Lattice[END_REF] and photonic cavities [START_REF] Ozawa | Topological Photonics[END_REF] to simulate the graphene-like physics in a clean environment for particles other than electrons. In such simulators the lattice structure and the particle wavefunctions can be engineered to our liking opening access to the study of lattice geometry effects in different kinds of particles. The fact that photons can be controlled with very high levels of precision in a microcavity setup has inspired efforts to emulate the physics of graphene for exciton-polaritons. Indeed, such an emulator called polaritonic graphene has been realised by engineering the potential landscape for polaritons [START_REF] Schneider | Exciton-Polariton Trapping and Potential Landscape Engineering[END_REF] to simulate a honeycomb lattice [START_REF] Jacqmin | Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons[END_REF]. The dispersion relation thus obtained resembles the band structure of graphene with the characteristic Dirac points at the In this work, we will study the effect of polariton interactions on the dispersion relation of polaritonic graphene using the Bogoliubov theory for particles on a lattice. In Chapter 6 I'll present a brief background on polaritonic graphene and derive the dispersion relation for particles in a honeycomb lattice within the tight-binding approximation. I'll also introduce the effect of spin-orbit coupling on such a dispersion relation. Then late in Chapter 7 we'll derive the Bogoliubov dispersion in the presence of polariton interactions and compare our results with the experimental data provided from our collaboration with the group of Dr.

Introduction to Exciton Polaritons

In this chapter, I'll present the basic concepts related to exciton polaritons in a semiconductor microcavity that will be used in later chapters. Starting with an introduction to the concept of excitons polaritons in bulk semiconductors, we'll later describe the case of microcavity polaritons as the normal modes of the coupling between excitons in a quantum well and a cavity photon mode. In the ensuing sections I'll describe the dynamics of the system in the presence of interactions in the system which will lead us a semi-classical nonlinear equation of motion known as the generalized Gross-Pitaevskii equation. The small low energy fluctuations of the system will then studied using the Bogoliubov theory of dilute Bose gases.

Concept of a polariton

Polaritons can be considered as hybrid (quasi)particles propagating inside a material medium under the influence of a light source [START_REF] Mills | Polaritons: The Electromagnetic Modes of Media[END_REF]. They are created as a result of mixing between the photon and electronic degrees of freedom in the so called strong light-matter coupling regime [START_REF] Frisk Kockum | Ultrastrong Coupling between Light and Matter[END_REF][START_REF] Forn-Díaz | Ultrastrong Coupling Regimes of Light-Matter Interaction[END_REF] and hence carry attributes of both their parent components. The electronic degrees of freedom are described by collective excitations (or modes) in a solid state material [START_REF] Mahan | Many-Particle Physics[END_REF] or superconducting circuits (S. [START_REF] Schmidt | Circuit QED Lattices: Towards Quantum Simulation with Superconducting Circuits[END_REF][START_REF] Blais | Circuit Quantum Electrodynamics[END_REF]. Different kinds of collective excitations in different materials give rise to various flavours of polaritons. For instance in Figure 1.1 we can see some of these polariton flavours which have been observed in 2D quantum materials recently [START_REF] Basov | Polaritons in van Der Waals Materials[END_REF][START_REF] Low | Polaritons in Layered Two-Dimensional Materials[END_REF]. The term polariton was first used by JJ Hopfield in the context of phonon polaritons in ionic crystals [START_REF] Hopfield | Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals[END_REF]. Other earlier contributions to the theory of polaritons -short of introducing this term -were made by [START_REF] Tolpygo | Physical properties of a rock salt lattice made up of deformable ions[END_REF][START_REF] Huang | Lattice Vibrations and Optical Waves in Ionic Crystals[END_REF][START_REF] Fano | Atomic Theory of Electromagnetic Interactions in Dense Materials[END_REF] in the same context. Indeed phonon polaritons were the first kind of polaritons to be observed experimentally [START_REF] Henry | Raman Scattering by Polaritons[END_REF]. At present, the panorama of polaritons consists of at least seventy different varieties [START_REF] Basov | Polariton Panorama[END_REF], most of which have been experimentally observed while the others are theoretically conjectured. [START_REF] Basov | Polaritons in van Der Waals Materials[END_REF] This thesis is devoted to the study of cavity exciton polaritons (or microcavity polaritons) which were first observed in [START_REF] Weisbuch | Observation of the Coupled Exciton-Photon Mode Splitting in a Semiconductor Quantum Microcavity[END_REF] Let us first, begin by introducing the theory of excitons in bulk semiconductors followed by the notion of exciton polaritons. After grasping this notion, we will extend it to the case 1.2. Exciton Polaritons in bulk semiconductors of excitons in semiconductor heterostructures -specifically quantum wells (QW)placed inside a semiconductor microcavity. Under this context, the hybridization between QW excitons and the photons trapped inside the cavity gives rise to the concept of microcavity polaritons. I address the reader to existing literature [START_REF] Savona | Linear Optical Properties of Semiconductor Microcavities with Embedded Quantum Wells[END_REF][START_REF] Savona | Optical Properties of Microcavity Polaritons[END_REF]Carusotto and Ciuti, 2013;[START_REF] Byrnes | Exciton-Polariton Condensates[END_REF][START_REF] Kavokin | Microcavities[END_REF] and the references therein for any further information on exciton polaritons and microcavity polaritons.

Exciton Polaritons in bulk semiconductors

Bulk semiconductor excitons

Excitons are quasi-particles which describe the lowest energy electronic excitations induced by the absorption of electromagnetic radiation in a semiconducting crystal [START_REF] Mahan | Many-Particle Physics[END_REF][START_REF] Koch | Semiconductor Excitons in New Light[END_REF]. When we shine light on a piece of pure semiconductor then an electron may jump from the valence band (VB) to the conduction band (CB) by absorbing a photon and leaving behind a hole in its initial state. The energy of the absorbed photon is given by the energy difference between the final CB state and the initial VB state. The minimum energy required to make this jump is known as the energy band gap in semiconductors (see Figure 1.2). However, when we look at the absorption spectrum of this process we also see some narrow resonance peaks below the band gap, 𝐸 𝑔 . These peaks correspond to the exciton states of the material.

Under the single or independent particle approximation, an exciton is simply described as a bound state of a CB electron 𝑒 -and a VB hole ℎ + formed as a consequence of electrostatic Coulomb interaction -screened by the dielectric medium of the material -between the two particles; similar to how an electron and a proton form the atomic bound states in a Hydrogen atom. Hence, borrowing terminology from atomic physics, the exciton states follow a hydrogen-like Rydberg series (1s, 2s, 2p, ...) which are characterized by a Bohr radius, 𝑎 B , and an effective binding energy 𝐸 binding which is calculated as

𝐸 binding = 𝐸 ground + 𝐸 𝑔 -𝐸 exc , (1.1)
where 𝐸 ground is the ground state energy of the system and 𝐸 exc is the energy of the exciton. A schematic illustration of exciton formation is shown in Figure 1.2. Also, the experimental observation of excitonic resonance peaks in the absorption spectrum of 𝐺𝑎𝐴𝑠 and 𝐶𝑢 2 𝑂 crystal is shown in Figure 1.3. In the hydrogenic model for exciton states, we use the screened Coulomb interac- 1.2. Exciton Polaritons in bulk semiconductors tion between the electron-hole pair [START_REF] Bassani | Electronic Impurity Levels in Semiconductors[END_REF]. This screening of the interaction is a consequence of polarization of the dielectric medium that shields the two charges, 𝑒 -and ℎ + from the presence of each other. The extent of screening -henceforth the strength of the interaction -depends on the dielectric constant of the material [START_REF] Kohn | Interaction of Charged Particles in a Dielectric[END_REF]. High values of dielectric constant leads to a highly reduced interaction between the two opposite charges which ultimately gives rise to an exciton Bohr radius which can be orders of magnitude larger than the primitive or elementary cell of the crystal. Such an excitonic state is known as a Wannier-Mott exciton [START_REF] Wannier | The Structure of Electronic Excitation Levels in Insulating Crystals[END_REF][START_REF] Mott | Conduction in Polar Crystals. I. Electrolytic Conduction in Solid Salts[END_REF]. Another member of the exciton family is the Frenkel exciton which is tightly bound or localized to a single elementary cell of the crystals. For our purposes we will consider semiconductors with large dielectric constants (typical of inorganic semiconductors) and hence will limit to the case of Wannier Utilizing the above simplified picture of a Wannier exciton as a Hydrogen-like bound state we may as well write a Hydrogen-like Schrödinger equation to calculate the exciton wavefunction and its eigenenergy. In the limit of small binding energy, small total momentum and large spatial extent of the exciton, we can write such an equation for the exciton under an effective mass approximation for the constituting electron and hole quasiparticles [START_REF] Bassani | Electronic States and Optical Transitions in Solids, 1st edition[END_REF]). However we have to remember that the presence of an exciton state in a semiconductor is strictly a many body phenomenon because of the fact that it is an excited state of a many-particle system: our semiconducting crystal. Therefore, in the next section I'll briefly write a derivation of the effective-mass Schorödinger equation for an exciton Chapter 1. Introduction to Exciton Polaritons from a many-body perspective -closely following [START_REF] Savona | Optical Properties of Microcavity Polaritons[END_REF] we'll see that it is equivalent to the real space Bethe-Salpeter equation [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF].

Effective-mass equation for an exciton I start by writing the full non-relativistic and time-independent quantum many-body Hamiltonian of electrons in a semiconductor crystal

ℋ crystal = - ∑ 𝐼 2 ∇ 2 𝐼 2𝑀 𝐼 - ∑ 𝑖 2 ∇ 2 𝑖 2𝑀 𝑖 + ∑ 𝐼<𝐽 𝑍 𝐼 𝑍 𝐽 𝑒 2 |R 𝐼 -R 𝐽 | - ∑ 𝐼,𝑖 𝑍 𝐼 𝑒 2 |R 𝐼 -r 𝑖 | + ∑ 𝑖<𝑗 𝑒 2 |r 𝑖 -r 𝑗 |
.

(1.2)

Here R 𝐼 and r 𝑖 correspond to the nuclear and electronic coordinates respectively. Also I've neglected any spin dependent contributions for simplicity. Using the Born-Oppenheimer approximation results in the separation of nuclear and electronic motions. After dropping the kinetic energy of nucleons because of the large mass difference between the nuclei and the electrons, we see that the electrons feel a mutual Coulombic repulsion due to the presence of other electrons plus a Coulombic attraction to the static lattice formed by the nuclei whose coordinates are fixed at the lattice sites. Solving the resulting Hamiltonian is still a tedious task and is hardly possible, hence we have to make further approximations to actually calculate some meaningful ground state. First, we approximate the many-body wavefunction as a Slater determinant of 𝑁 single particle wavefunctions where 𝑁 is the number of electrons. These single particle wavefunctions are assumed to be Bloch wavefunctions, 𝜓 𝑛kwith energy 𝐸 𝑛 (k) -where 𝑛 is the band index and k is a quantum number associated with the wavevector in the first Brillouin zone of the crystal [START_REF] Marder | Condensed matter physics[END_REF]. The mathematical form of a Bloch wavefunction in real space is given as

𝜓 𝑛k (r) = 1 √ 𝑉 𝑢 𝑛k (r) exp(𝑖k • r) (1.3)
where 𝑢 𝑛k (r) is the periodic Bloch function and 𝑉 is the normalization volume. In writing the Bloch wavefunction I have also assumed that only one valence electron -for each lattice site atom -contributes to the Bloch electron states. Hence, the total number of k values in the first Brillouin zone is also equal to 𝑁 . Secondly, we restrict ourselves to a two band model consisting of the valence (𝑛 = v) and conduction (𝑛 = c) bands. Finally, under these approximations we may write our ground state of the semiconductor as a fully filled valence band,

Ψ(r 1 , ..., r 𝑁 ) = 1 √ 𝑁 ! Det { 𝜓 𝑣k 1 (r 1 ) ... 𝜓 𝑣k 𝑁 (r 𝑁 ) } .
(1.4)

Exciton Polaritons in bulk semiconductors

The total wavevector in the ground state is zero and the expectation value of the crystal Hamiltonian over the approximate ground state defined above gives us the ground state energy, 𝐸 ground = ⟨Ψ(r 1 , ..., r 𝑁 )|ℋ crystal |Ψ(r 1 , ..., r 𝑁 )⟩.

(1.5) Now, excitons -as I stated above -are the lowest energy excited state of the crystal, which means that one electron from the valence band jumps to the conduction band. The many particle state defining this jump to the first excited state can be written as

Ψ k𝑛,k ′ 𝑛 (r 1 , ..., r 𝑁 ) = 1 √ 𝑁 ! Det { 𝜓 𝑣k 1 (r 1 ) ... 𝜓 𝑐k ′ 𝑛 (r 𝑛 ) ... 𝜓 𝑣k 𝑁 (r 𝑁 ) } .
(1.6)

The above defined excited state has a total wavevector, k = k ′ 𝑛 , -k 𝑛 and its energy is given as 𝐸 ground -𝐸 𝑣 (k 𝑛 ) + 𝐸 𝑐 (k ′ 𝑛 ), where 𝐸 𝑣 (k 𝑛 ) and 𝐸 𝑐 (k ′ 𝑛 ) are the valence band and conduction band Bloch energies. We know that there are 𝑁 allowed values of the wavevector in a Bloch state hence we can have can have 𝑁 equivalent first excited states which give the same value of total wavevector, hence a better approximation for the first excited state of the crystal (exciton) -is actually a linear combination of these equivalent states of type given in Eq. (1.6) and is given as

Φ k (r 1 , ..., r 𝑁 ) = ∑ k ′ 𝐹 k (k ′ ) Ψ k ′ -(𝑚 ℎ /𝑀 )k , k ′ +(𝑚𝑒/𝑀 )k = ∑ k ′ 𝐹 k (k ′ ) Ψ k ℎ ,k𝑒 , (1.7)
where I used conservation of energy and the fact that k = k ′ 𝑛 , -k 𝑛 to write the subscripts of Ψ in a different form. Here, 𝑀 = 𝑚 𝑒 + 𝑚 ℎ which we'll later see defines the mass of the exciton. Since there is only one effective particle jump from valence to conduction band, the probability of all the contributions should be equal to one. Hence, the normalization of Eq. (1.7) should be given as

∑ k ′ |𝐹 k (k ′ )| 2 = 1.
(1.8)

𝐹 k (k ′ ) can be interpreted as the Fourier transform of the real space wavefunction of the excited state with the center of mass wavevector k and can be calculated by minimizing the expectation value of ℋ crystal over the state defined in Eq. (1.7) subject to the normalization constraint defined above in Eq. (1.8). Hence, we can use the method of Lagrange multipliers where the Lagrange parameter, 𝐸 exc , will provide us with the energy of the exciton state. After the full minimization procedure (see [START_REF] Bassani | Electronic Impurity Levels in Semiconductors[END_REF] we obtain the following set of equations,

[ 𝐸 ground -𝐸 𝑣 ( k ′ - 𝑚 ℎ k 𝑀 ) + 𝐸 𝑐 ( k ′ + 𝑚 𝑒 k 𝑀 ) -𝐸 exc ] 𝐹 k (k ′ ) + ∑ k ′′ 𝐹 k (k ′′ )× Chapter 1. Introduction to Exciton Polaritons ∫ 𝑑r 1 𝑑r 2 𝜓 * 𝑣,k ′ -(𝑚 ℎ /𝑀 )k (r 1 ) 𝜓 * 𝑐,k ′′ +(𝑚𝑒/𝑀 )k (r 2 ) 𝑒 2 |r 𝑖 -r 𝑗 | 𝜓 𝑣,k ′′ -(𝑚 ℎ /𝑀 )k (r 1 ) 𝜓 𝑐,k ′ +(𝑚𝑒/𝑀 )k (r 2 ) = 0
(1.9)

In order to simplify the above equation -and arrive at an hydrogen-like equation for the exciton -we have to make one last assumption that only a small range of k ′ values around zero contribute to the excited state defined in Eq. (1.7) because of large spatial extent of this Wannier-Mott exciton state. Under this assumption its fair to introduce the effective mass approximation for the electron and hole quasiparticle Bloch energies and states which can be written in the following simplified way,

𝐸 𝑐 (k) = 𝐸 𝑔 + 2 𝑘 2 2𝑚 𝑒 𝜓 𝑐k (r) = 1 √ 𝑉 𝑢 𝑐 0 (r) exp(𝑖k • r) (1.10a) 𝐸 𝑣 (k) = - 2 𝑘 2 2𝑚 ℎ 𝜓 𝑣k (r) = 1 √ 𝑉 𝑢 𝑣 0 (r) exp(𝑖k • r) (1.10b)
Substituting the above Bloch energies and wavefunctions in Eq. (1.9) and noticing the fact that all the terms in the integral are slowly varying with respect to r except for the periodic Bloch functions 𝑢 𝑐/𝑣,0 (r), we may take all the slowly varying parts as constants inside each elementary cell leaving only the Bloch functions to be integrated over an integral region corresponding to the elementary cell. Since the Bloch functions are normalized we can finally arrive to

[ 𝐸 ground + 𝐸 𝑔 + 2 𝑘 ′2 2𝜇 + 2 𝑘 2 2𝑀 -𝐸 exc ] 𝐹 k (k ′ ) + ∑ k ′′ 𝐹 k (k ′′ ) 𝑒 2 |k ′′ -k ′ | = 0, (1.11)
which is the momentum space Schrödinger equation for an exciton with total wavevector k. Here, 𝜇 -1 = 𝑚 -1 𝑒 + 𝑚 -1 ℎ is the reduced mass of the electron and hole quasiparticles. The Fourier transform of this equation in k ′ finally gives us the hydrogenlike Schrödinger equation for the energies and the wavefunction of an exciton which is written as

[ 𝐸 ground + 𝐸 𝑔 + 2 𝑘 2 2𝑀 - 2 2𝜇 ∇ 2 r - 𝑒 2 |r| ] 𝜁 k (r) = 𝐸 exc 𝜁 k (r).
(1.12)

Here 𝜁 k (r) is the Fourier transform of 𝐹 k (k ′ ) and r is the relative distance between the electron and hole.

In the last subsection I had mentioned that the Coulomb interaction between an electron and a hole is screened due to the presence of other charges in the semiconducting crystal. Again, using our assumption of large spatial extension of the exciton, the effect of screening can be included by introducing the static dielectric constant 𝜖 in the Coulomb potential as 𝑒 2 /𝜖|r|. The proof of this inclusion is pre-1.2. Exciton Polaritons in bulk semiconductors sented in [START_REF] Sham | Many-Particle Derivation of the Effective-Mass Equation for the Wannier Exciton[END_REF], where they utilize the two particle Green's function of the semiconducting system to arrive at the homogeneous Bethe Salpeter equation which under the assumptions of a Wannier-Mott like state -small total momentum and large spatial extent of the exciton -finally results in

[ 𝐸 ground + 𝐸 𝑔 + 2 𝑘 2 2𝑀 - 2 2𝜇 ∇ 2 r - 𝑒 2 𝜖|r| ] 𝜁 k (r) = 𝐸 exc 𝜁 k (r).
(1.13)

Borrowing the nomenclature from atomic physics we can define the Bohr radius and the binding energy of 1𝑠 Exciton state1 as:

𝑎 𝐵 = 2 𝜖 𝜇𝑒 2 𝐸 binding = 𝜇𝑒 4 2𝜖 2 2 𝐸 exc = 𝐸 ground + 𝐸 𝑔 -𝐸 binding . (1.14)
Including the exciton center of mass motion we get a parabolic dispersion for exciton energies given as

𝐸 exc (k) = 𝐸 exc + 2 k 2 2𝑀 (1.15)

Exciton Polaritons in bulk

Exciton Polaritons are the normal modes of the coupled exciton-photon system.

Here, I'll obtain the polariton modes of this coupled system using a second quantized description of exciton and photon fields, and their interaction. Let's start by defining the exciton creation, b † k , and annihilation, bk , operators defined as

b † k |0⟩ = ∑ k ′ 𝐹 k (k ′ ) ê † k ′ +(𝑚𝑒/𝑀 )k ĥ † -k ′ +(𝑚 ℎ /𝑀 )k |0⟩ = |Φ k ′ ⟩ (1.16a) bk |Φ k ′ ⟩ = 𝛿 k,k ′ |0⟩, (1.16b)
where |0⟩ is the vacuum state defined by a fully filled valence band, ê † k is the electron creation operator in the conduction band with wavevector k, ĥ † k ′ is the hole creation operator in the valence band with wavevector k ′ , and 𝛿 k,k ′ is the Kronecker delta which is equal to one when k ′ = k and zero otherwise. The real space representation of exciton state |Φ k ′ ⟩ is given by Eq. (1.7) from where we can also see that real space representation of the state ê † k ĥ † k ′ |0⟩ is given by the Slater determinant in Eq.(1.6). Note that under the assumption of low exciton density -few excitations of the crystal -the exciton creation and annihilation operators follow the bosonic commutation relations [START_REF] Hawton | Quasibosonic Exciton Dynamics near the Semiconductor Band Edge[END_REF][START_REF] Rochat | Excitonic Bloch Equations for a Two-Dimensional System of Interacting Excitons[END_REF]. This can be intuitively understood as follows: the number of available levels for an electron to jump in the conduction band which have the same wavevector is equal to the number of elementary cells in the crystal which is equal to 𝑁 , hence we can pile up many electrons in same wavevector state until we reach the max occupancy of 𝑁 particles at which point we adding more particles would violate the Pauli exclusion principle. Therefore, as long as we stay well below 𝑁 i.e. less number of excitations, the use of exciton operators as bosonic operators is justified. Now we have the exciton operators and using them we may write the exciton Hamiltonian operator as

Ĥexciton = ∑ k 𝐸 exc,k b † k bk , (1.17)
where 𝐸 exc,k is the energy of the 1𝑠 exciton state (see Eq. (1.14)) plus its center of mass motion. Likewise, the Hamiltonian operator for the free electromagnetic field is,

Ĥem = ∑ k,𝜆 𝑐|k| â † k𝜆 âk𝜆 , (1.18)
where â † k𝜆 and âk𝜆 are the photon creation and annihilation operator respectively with the wavevector k and polarization 𝜆, and 𝑐 is the velocity of light. The photon operators also satisfy the bosonic commutation relations.

The interaction Hamiltonian between the exciton and electromagnetic fields is given as,

Ĥint = - 𝑒 𝑚𝑐 ∑ 𝑖 A • p 𝑖 + 𝑒 2 𝑚𝑐 2 ∑ 𝑖 A 2 , (1.19)
where the index 𝑖 goes through all the electrons in the semiconducting crystal, while

A describes the electromagnetic vector potential and p 𝑖 is the momentum of the 𝑖 th electron. The interaction Hamiltonian in the second quantized form with the exciton and photon operators is given as [START_REF] Savona | Linear Optical Properties of Semiconductor Microcavities with Embedded Quantum Wells[END_REF])

Ĥint = 𝑖 ∑ k,𝜆 Ω k𝜆 ( bk𝜆 â-k𝜆 + bk𝜆 â † k𝜆 -b † k𝜆 âk𝜆 -b † -k𝜆 â † k𝜆 ) + ∑ k,𝜆 𝐷 k𝜆 ( âk𝜆 â-k𝜆 + âk𝜆 â † k𝜆 + â † k𝜆 âk𝜆 + â † -k𝜆 â † k𝜆 ) . (1.20)
Here

Ω k𝜆 = ( 𝐸 exc,k 𝑐 ) ( 2𝜋 𝑐 𝑉 |k| )1 2 ⟨Φ k |𝑒r𝑒 𝑖k • r |0⟩ = ( 𝐸 exc,k 𝑐 ) ( 2𝜋 𝑐 |k| ) 1 2 𝜂 𝑐𝑣 𝜁 k (0) (1.21) 𝐷 k𝜆 = |Ω k𝜆 | 2 𝐸 exc,k , (1.22)
where 𝜂 𝑐𝑣 is the dipole matrix element between the conduction and valence band states. Note here that I have introduced a polarization index for the exciton op-1.2. Exciton Polaritons in bulk semiconductors erators also because in correspondence with the polarization of electromagnetic radiation only those excitons interact with photons whose dipole moment lies in the plane perpendicular to the propagation of the light beam; such excitons are called transverse excitons and the ones along the direction of propagation of light are called longitudinal excitons. Hence, the index 𝜆 has two values for the two transverse polarizations and the polariton modes are only formed from mixing of the transverse excitons and photons. Now we may diagonalize the total Hamiltonian, Ĥtotal = Ĥexciton + Ĥem + Ĥint since it is quadratic in nature. It will gives us an exact solution for our problem. The full diagonalization was first done by JJ Hopfield [START_REF] Hopfield | Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals[END_REF], but I'll make one further assumption to simplify the diagonalization procedure. If the frequency of the electromagnetic radiation is very close to resonance with the exciton frequency and the light-matter coupling interaction energy is very small compared to the exction energy [START_REF] Ciuti | Quantum Vacuum Properties of the Intersubband Cavity Polariton Field[END_REF] then we can make the well known rotating wave approximation to neglect the anti-resonant terms -the ones with double creation or annihilation operators -and further we can also neglect the terms with double photon operators as they just result in blueshift of the photon energy. Ultimately, our total Hamiltonian has the simple form,

Ĥtotal = ∑ k𝜆 [ 𝐸 exc,k b † k𝜆 bk𝜆 + 𝑐|k| â † k𝜆 âk𝜆 + 𝑖Ω k𝜆 ( bk𝜆 â † k𝜆 -b † k𝜆 âk𝜆 ) ] (1.23)
The above equation can be diagonalized by introducing two new operators lk𝜆 and ûk𝜆 which are linear superpositions of the excitonic and photonic operators with real valued coefficients known as Hopfield coefficients. The new operators lk𝜆 and ûk𝜆 correspond to the two exciton polariton eigenmodes, called lower and upper polaritons respectively and are defined as

lk𝜆 = 𝑋 k bk𝜆 + 𝐶 k âk𝜆 (1.24a) ûk𝜆 = -𝐶 k bk𝜆 + 𝑋 k âk𝜆 . (1.24b)
Using this transformation in Eq. (1.23) we finally get,

Ĥtotal = ∑ k𝜆 [ 𝐸 𝑙,k l † k𝜆 lk𝜆 + 𝐸 𝑢,k û † k𝜆 ûk𝜆 ] , (1.25)
where we have defined

𝐸 𝑢/𝑙,k = 𝐸 exc,k + 𝑐|k| 2 ± [ Ω 2 k + ( 𝐸 exc,k -𝑐|k| 2 ) 2 ] 1/2 (1.26)
as the energies of the upper and lower polaritons. The Hopfield coefficients 𝐶 k and Chapter 1. Introduction to Exciton Polaritons 𝑋 k quantify the fraction of photonic and excitonic content in the two polariton modes and are given as

|𝐶 k | 2 = 1 2 ⎛ ⎝ 1 - 𝐸 exc,k -𝑐|k| √ (𝐸 exc,k -𝑐|k|) 2 + 4Ω 2 k ⎞ ⎠ , |𝑋 k | 2 = 1 -|𝐶 k | 2 . ( 1

Exciton Polaritons in a semiconductor microcavity

Quantum Well Excitons

Quantum wells are semiconductor nanostructures with quantum confinement in one direction which are fabricated by sandwiching a thin layer of a semiconducting material between two thick layers of another semiconducting material which has a bigger band gap compared to the sandwiched layer [START_REF] Harrison | Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures[END_REF]. This leads to a breaking of translation symmetry along the growth direction which implies that 1.3. Exciton Polaritons in a semiconductor microcavity conservation of momentum is restricted to the two dimensional (2D) plane of the sandwiched layer: the quantum well. The in-plane conserved momentum is denoted by k || and the momentum perpendicular to the plane by 𝑞 (which is not conserved).

The valence and conduction band states of the small gap material become confined along the growth direction and if the sandwiched layer is thin enough it can gain access to the regime of quantum confinement. It results in the splitting of the valence and conduction bands into discrete subbands called electron and hole subbands labeled by quantum numbers 𝑛 𝑒 and 𝑛 ℎ respectively. The minimum energy -labeled by 𝐸 2D 𝑔 -needed for an electron to jump to the conduction band is equivalent to the band gap of the sandwiched layer 𝐸 𝑔 plus the electron confinement energy 𝐸 𝑒 conf and the hole confinement energy 𝐸 ℎ conf of the subbands with quantum numbers 𝑛 𝑒 and 𝑛 ℎ equal to one. For our purposes we'll only consider these particular subbands onwards. In Figure 1.6 we show an illustration of a quantum well structure formed by sandwiching a layer of InGaAs between two higher band gap GaAs substrates and resulting energy structure along the growth direction. The steps in the derivation of an effective mass equation for the case of excitons in a semiconductor quantum well are similar to the one derived in Sec. 1.2.1 for the bulk case with the only caveat that the single-particle electron wavefunctions are a product of a 2D Bloch wavefunction and an envelope wavefunction which stems from the quantum confinement along the growth direction [START_REF] Bastard | Strong interactions and biexcitons in a polariton mixture[END_REF][START_REF] Savona | Linear Optical Properties of Semiconductor Microcavities with Embedded Quantum Wells[END_REF]. The single-particle electron wavefunction in the valence or conduction band is given as

𝜓 𝑣/𝑐,k || (𝜌, 𝑧) = 1 √ 𝐴 𝑢 𝑣/𝑐, 0 (𝜌) exp(𝑖k || • 𝜌) 𝜒 𝑐/𝑣 (𝑧), (1.28)
where 𝜌 describes a point in the 2D plane of the quantum well, 𝑧 the growth direction, Chapter 1. Introduction to Exciton Polaritons 𝐴 is the surface area for normalization and 𝜒(𝑧) is the envelope wavefunction in the growth direction.

Utilizing the same methodology as in the bulk case and just replacing the single particle wavefunctions in the Slater determinants with the one just defined in Eq. (1.28), we can arrive to a 2D hydrogen-like Schrödinger equation in the center of mass frame of the QW exciton, which is written as

[ 𝐸 ground + 𝐸 2D 𝑔 - 2 2𝜇 ∇ 2 𝜌 - ∫ 𝑑𝑧 1 𝑑𝑧 2 𝑒 2 𝜖|r 1 -r 2 | |𝜒 𝑣 (𝑧 1 )| 2 |𝜒 𝑐 (𝑧 2 )| 2 ] 𝜁 k || (𝜌) = 𝐸 exc 𝜁 k || (𝜌).
(1.29) In the limit when Bohr radius of the 2D exciton far exceeds the confinement lengththickness of the sandwiched semiconductor layer -we can replace the envelope functions 𝜒 𝑐/𝑣 (𝑧) by delta functions 𝛿 𝑐/𝑣 (𝑧-𝑧 QW ) centered at the position of quantum well along the growth direction. We can now replace the integral in the equation above with a pure 2D Coulomb potential to finally achieve a 2D hydrogen-like Schrödinger equation,

[ 𝐸 ground + 𝐸 2D 𝑔 - 2 2𝜇 ∇ 2 𝜌 - 𝑒 2 𝜖𝜌 ] 𝜁 k || (𝜌) = 𝐸 exc 𝜁 k || (𝜌).
(1.30)

When we solve the 2D Hydrogen equation we find that the Bohr radius for the 1𝑠 exciton in the 2D Hydrogen case is half of the bulk (3D) case -defined in Eq. (1.14) -which also implies that its Binding energy is four times its bulk counterpart [START_REF] Yang | Analytic Solution of a Two-Dimensional Hydrogen Atom. I. Nonrelativistic Theory[END_REF] and are given as

𝑎 2𝐷 𝐵 = ( 1 2 ) 2 𝜖 𝜇𝑒 2 𝐸 2𝐷 binding = 4 𝜇𝑒 4 2𝜖 2 2 𝐸 exc = 𝐸 ground + 𝐸 𝑔 -𝐸 2𝐷 binding (1.31)

Cavity Photons

Cavity photons are defined as the quantized modes of the electromagnetic field inside a microcavity. A microcavity in essence is just a Fabry-Pérot resonator with special mirrors known as Distributed Bragg Reflectors (DBR) [START_REF] Kavokin | Microcavities[END_REF]. For our purposes we assume it to be a simple planar Fabry-Pérot resonator made up of two parallel planar mirrors separated by a dielectric slab of thickness 𝐿 𝑐 (see Figure 1.7). Cavity modes are the allowed stationary electromagnetic modes which are obtained by constructive interference between successive passes of a propagating wave. Hence, the phase change between a round trip along the growth direction inside the cavity must be a multiple of 2𝜋 which implies that

𝐿 𝑐 𝑞 𝑛 = 𝐿 𝑐 √ |k| 2 -|k || | 2 = 𝐿 𝑐 √ ( 𝑛 cav 𝜔 𝑐 ) 2 -|k || | 2 = 𝑛𝜋 (1.32)
where 𝑞 𝑛 is the component of the wavevector k orthogonal to the mirror plane labeled by a natural number 𝑛 which describes the 𝑛 th harmonic mode of the cavity, 1.3. Exciton Polaritons in a semiconductor microcavity 𝜔 is frequency of the electromagnetic wave, and 𝑛 cav is the refractive index of the dielectric.

Figure 1.7: Schematic of a microcavity with a semiconductor quantum well. Figure from [START_REF] Kasprzak | Bose-Einstein Condensation of Exciton Polaritons[END_REF] Although, the microcavities are nearly ideal i.e have almost perfectly reflecting mirrors, a very small percentage of light leaks out. Hence, the cavity photon modes are lossy with finite mirror transmission values. The coupling of the discrete or quantized cavity photon modes inside with a continuum of photon modes outside the cavity leads to an emergence of cavity photon linewidth, 𝛾 c [START_REF] Gardiner | Quantum noise[END_REF]. This ultimately leads to the dissipation of the cavity photon field. Hence, in order to maintain a steady flow of cavity photons inside the cavity we have to continuously pump photons from an external source of electromagnetic radiation such as a laser.

The input-output coupling of electromagnetic field inside and outside the microcavity conserves the in-plane wavevector k || and hence it is a good quantum number for the free cavity photon dynamics in the mirror plane, which can be described by the Hamiltonian Ĥcavity = ∑

k || 𝜆 𝜔 cav (k || )â † k𝜆 âk𝜆 , (1.33)
where â † k𝜆 and âk𝜆 are the photon creation and annihilation operators of a single cavity mode with polarization 𝜆, and

𝜔 cav (k || ) = 𝑐 𝑛 cav √ 𝑞 2 𝑛 + 𝑘 2 || ≈ 𝜔 0 cav + 𝑘 2 || 2𝑚 cav .
(1.34)

𝑚 cav and 𝜔 0 cav = 𝑐𝑞 𝑛 /𝑛 cav describes the effective mass of the photon and the cutoff frequency respectively. They are related by the relativistic expression

𝑚 cav = 𝑛 cav 𝑞 𝑐 = 𝜔 0 cav 𝑛 2 cav 𝑐 2 (1.35) Chapter 1. Introduction to Exciton Polaritons
The Hamitonian of the external pump can be described as

Ĥpump = ∑ k || ,𝜆 [ 𝜂 𝜆 Ẽ𝜆 (k || , 𝑡)â † k ||,𝜆 + ℎ.𝑐. ] , (1.36)
where Ẽ𝜆 (k || , 𝑡) is the Fourier transform of 𝐸 𝜆 (𝜌, 𝑡) -amplitude of the coherent incident electromagnetic field -and 𝜂 𝜆 quantifies the input-output coupling and is proportional to the transmittivity of the cavity mirrors.

The photons leaking out of the cavity can be captured to perform spectroscopic measurements. These measurements can then be used to analyze the dynamics of exciton field (present inside the cavity) which are coupled to the cavity field modes. The matter fields can be introduced inside the cavity by placing a material such as a semiconductor quantum well inside the cavity.

Microcavity Polaritons

Microcavity polaritons are the mixed or normal modes of the 2D exciton state and cavity photons. More rigorously, they represent the linear superposition of the cavity photon field with the quantum well excitonic field. The derivation of microcavity polariton Hamiltonian and its dispersion relation is analogous to the bulk case (see Eq. 1.23) just with replacing the free electromagnetic Hamiltonian with the cavity photon (1.33) one and the bulk exciton energy with the 2D exciton energy defined in Eq. (1.31) which gives us

Ĥtotal = ∑ k || 𝜆 [ 𝐸 exc (k || ) b † k || 𝜆 bk || 𝜆 + 𝜔 cav (k || ) â † k || 𝜆 âk || 𝜆 + 𝑖Ω k || ( bk || 𝜆 â † k || 𝜆 -b † k || 𝜆 âk || 𝜆 ) ] (1.37) = ∑ k || 𝜆 [ 𝐸 𝑙,k || l † k || 𝜆 lk || 𝜆 + 𝐸 𝑢,k || û † k || 𝜆 ûk𝜆 ] , (1.38)
where the 𝑙 and 𝑢 operators are introduced by using Hopfield transformation (1.24a).

Even the form of the polariton energies is the same -as defined in Eq. (1.26) -with the 2D energies replacing the bulk ones. The Rabi coupling Ω k || for exciton-photon coupling in high finesse cavities can be approximated as a constant given as [START_REF] Savona | Linear Optical Properties of Semiconductor Microcavities with Embedded Quantum Wells[END_REF][START_REF] Savona | Optical Properties of Microcavity Polaritons[END_REF])

Ω 2 k || ≡ Ω 2 = 2𝑐 Γ 0 𝑛 cav 𝐿 𝑐 , (1.39)
where Γ 0 is the bare quantum well exciton radiative rate at k || = 0. Γ 0 is typically of the order of 3.5 meV for a single quantum well in GaAs based semiconductor microcavities.

In Figure 1.8 we show the exciton polariton dispersion relations and their cor-1.3. Exciton Polaritons in a semiconductor microcavity responding excitonic and photonic fractions for the three cases where the detuning, 𝛿, between the exciton and photon modes at k || = 0 given as 𝛿 = 𝜔 0 cav -𝐸 exc is negative (a), zero (c) or positive (e). We see that at resonance both the polariton branches have equal photonic and excitonic content. Around this resonance the exciton-photon coupling opens an energy gap equal to twice of Rabi energy which is seen as the characteristic anti-crossing signature for formation of polariton modes in experiments. In the case of positive detuning, the upper polariton is highly photonic whereas the lower polariton is highly excitonic. Further, increasing the magnitude of detuning in this case leads to the two hybridized modes getting uncoupled and leaving behind bare excitonic and photonic modes.

For zero or negative detuning, the region close to zero wavevector upper polaritons can be considered to be far detuned to have any kind of affect on the low energy dynamics of lower polaritons. Hence, the lower polaritons can be approximated as free particles with a parabolic dispersion and an effective mass 𝑚 LP given as

1 𝑚 LP = |𝑋 k || | 2 𝑚 exc + |𝐶 k || | 2 𝑚 cav , (1.40)
where 𝑚 exc is the effective exciton mass which is equal to the sum of the electron and hole quasiparticles and hence is of the order of electron mass, 𝑚 𝑒 , and 𝑚 cav is given by Eq. (1.35) and is of the order of 10 -4 𝑚 𝑒 . Therefore, close to the exciton and photon resonance we can approximate the lower polariton mass as 2𝑚 cav .

Since, the dispersion relations are the same for bulk and microcavity polaritons, we can treat a semiconductor quantum well embedded in a microcavity as the 2D analog of a bulk semiconductor, albeit with the fact that the exciton and cavity photon modes in the latter case have finite energy linewidths. The broadening of linewidths are caused as a result of radiative losses which are the consequence of the coupling of the cavity photon field inside and free photon field outside the microcavity. This can be studied using an input-output theory [START_REF] Verger | Polariton Quantum Blockade in a Photonic Dot[END_REF] borrowed from the field of quantum optics . The so called strong light-matter coupling -which is necessary to observe the polariton dispersion relations -is obtained when the value of Rabi coupling energy Ω exceeds the losses in the system. Non-radiative losses such as the exciton-phonon coupling and disorder in the semiconductor system can also contribute to the broadening of the energy linewidths for the exciton modes and hence can be a further contribution to the polariton linewidth. The losses in the system are proportional to the observed linewidth of polariton resonances and give an estimate of the polariton lifetime which is generally of the order of picoseconds.

Note that we have not considered any polarization or spin dependent effects in the formation of exciton polaritons and have treated the Hamiltonians for different polarizations as independent of each other. But, polarization or spin mixing can Chapter 1. Introduction to Exciton Polaritons occur as an effective spin-orbit coupling of the polariton modes [START_REF] Shelykh | Polariton Polarization-Sensitive Phenomena in Planar Semiconductor Microcavities[END_REF]. The excitons polarized perpendicular to k || in the QW plane are called transverse or T excitons and they only couple to transverse electric (TE) modes of the cavity, while the excitons whose polarization is parallel to k || are called longitudinal or L excitons and they couple to only transverse magnetic (TM) modes of the cavity. Any kind of mixing between the two different polarizations leads to a TE-TM splitting of the polariton modes. We will look at this point in Part III of this thesis.

E/E 0 |k || |/k 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -1 -0.5 0 0.5 1 |k || |/k 0 -0.1 0.0 0.1 0.2 -1 -0.5 0 0.5 1 2Ω E/E 0 |k || |/k 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -1 -0.5 0 0.5 1 |k || |/k 0 -0.1 0.0 0.1 0.2 -1 -0.5 0 0.5 1 E/E 0 |k || |/k 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -1 -0.5 0 0.5 1 a) b) c) d) e) f) |k || |/k 0

Generalized Gross-Pitaevskii equation

Polariton-Polariton interactions

In the last section we got a physical description of our system of microcavity polaritons which arise due to the coupled field dynamics of cavity photons and quantum well excitons. Until now, we have limited ourselves to the optical linear properties of quantum well excitons in a microcavity. Our interest in the rest of thesis will be on the nonlinear optical response of such a system, which results from interactions among the microcavity polariton states. Since the playground of polariton dynamics is two dimensional, in the discussion that follows we'll remove the subscript || from our description of wavevectors and any mention would imply 2D character unless explicitly stated. Similar change in notation will be made for spatial coordinates with r replacing 𝜌.

The interactions among polaritons are solely the consequence of interactions among the different exciton states2 which contribute to the excitonic fraction of our polariton states (Carusotto and Ciuti, 2013). We can describe the interactions among the excitons using an effective Hamiltonian wherein we don't have to worry about the complicated Coulomb interactions between the constituent electron and hole particle. We can replace them by effective interaction potentials that mimic their behaviour utilizing only the exciton wavefunctions as long as the overlap between two exciton wavefunctions is negligible. In the Born approximation, low-energy scattering at length scales much larger than the exciton Bohr radius can be approximated by a contact two-body interaction potential term (Ciuti, Savona, 1.4. Generalized Gross-Pitaevskii equation et al., 1998;Carusotto and Ciuti, 2013) of the form

Ĥexc-exc = ∑ 𝜆 1 𝜆 2 𝑔 exc 𝜆 1 ,𝜆 2 2 ∑ k 1 ,k 2 ,k ′ b † k 1 ,𝜆 1 b † k 2 ,𝜆 2 bk 2 +k ′ ,𝜆 2 bk 1 -k ′ ,𝜆 1 (1.41)
where the spin indices 𝜆 1 , 𝜆 2 run over the circular polarization basis +,-. Rotational invariance for a contact interaction potential imposes that total exciton spin is conserved and that 𝑔 exc +,+ = 𝑔 exc -,-= 𝑔 triplet and 𝑔 exc +,-= 𝑔 exc -,+ = 𝑔 singlet . The main contribution to the 𝑔 triplet interaction channel comes from the exchange interaction between the constituent electron and hole quasi-particles and under the Born approximation it is equal to 6𝑒 2 𝑎 2𝐷 𝐵 /𝜖 [START_REF] Ciuti | Role of the Exchange of Carriers in Elastic Exciton-Exciton Scattering in Quantum Wells[END_REF]. In the vicinity of a scattering resonance -such as a molecular biexciton state -the value of the effective interaction constant 𝑔 singlet can change its sign and show a strong enhancement of its magnitude [START_REF] Carusotto | Feshbach Blockade: Single-Photon Nonlinear Optics Using Resonantly Enhanced Cavity Polariton Scattering from Biexciton States[END_REF][START_REF] Takemura | Spin Anisotropic Interactions of Lower Polaritons in the Vicinity of Polaritonic Feshbach Resonance[END_REF].

An additional interaction channel originates from saturation of the exciton oscillator strength [START_REF] Tassone | Exciton-Exciton Scattering Dynamics in a Semiconductor Microcavity and Stimulated Scattering into Polaritons[END_REF][START_REF] Rochat | Excitonic Bloch Equations for a Two-Dimensional System of Interacting Excitons[END_REF][START_REF] Glazov | Polariton-Polariton Scattering in Microcavities: A Microscopic Theory[END_REF]: The Pauli exclusion principle for electrons and holes forbids that another exciton be created at a distance shorter than the Bohr radius from an existing exciton with the same spin. At the lowest order in the exciton density, this can be modeled as an effective quartic Hamiltonian term of the form

Ĥsat = 𝑔 sat 2 ∑ k 1 ,k 2 ,k ′ b † k 1 b † k 2 bk 2 +k ′ âk 1 -k ′ + ℎ.𝑐. (1.42)
The saturation interaction is independent of the spin polarization and 𝑔 sat = Ω/𝑛 𝑠𝑎𝑡 , where 𝑛 𝑠𝑎𝑡 ∝ 1/𝑎 2𝐷 𝐵 and Ω is the Rabi coupling. In many relevant experimental conditions we can restrict ourselves to the bottom of the lower polariton (LP) branch and approximate its dispersion by a parabola around the minima of LP branch as was stated in last section. This restriction is valid as long as Ω is much larger than all the other relevant energy scales of the system. Under such considerations we can rewrite the interaction -both the exciton-exciton and saturation -energy in terms of the lower polariton operators only as

ĤLP-LP = 𝑔 LP 2 ∑ k 1 ,k 2 ,k ′ l † k 1 l † k 2 lk 2 +k ′ lk 1 -k ′ , (1.43)
where 𝑔 LP = 𝑋 4 𝑔 exc 𝜆 1 ,𝜆 2 + 2𝑋 3 𝐶𝑔 sat and 𝑋, 𝐶 are Hopfield coefficients.

Mean-field approximation

The interaction terms -introduced above -describing two-body interactions among excitons make the dynamics of the resulting polariton states nontrivial. They are responsible for numerous interesting nonlinear and collective phenomena (Richard et Chapter 1. Introduction to Exciton Polaritons al., 2005;[START_REF] Amo | Collective Fluid Dynamics of a Polariton Condensate in a Semiconductor Microcavity[END_REF] such as Bose-Einstein condensation [START_REF] Kasprzak | Bose-Einstein Condensation of Exciton Polaritons[END_REF] superfluidity [START_REF] Amo | Superfluidity of Polaritons in Semiconductor Microcavities[END_REF][START_REF] Kohnle | From Single Particle to Superfluid Excitations in a Dissipative Polariton Gas[END_REF][START_REF] Lerario | Room-Temperature Superfluidity in a Polariton Condensate[END_REF], appearance of quantized vortices [START_REF] Lagoudakis | Quantized Vortices in an Exciton-Polariton Condensate[END_REF][START_REF] Sanvitto | Persistent Currents and Quantized Vortices in a Polariton Superfluid[END_REF][START_REF] Krizhanovskii | Effect of Interactions on Vortices in a Nonequilibrium Polariton Condensate[END_REF][START_REF] Sanvitto | All-Optical Control of the Quantum Flow of a Polariton Condensate[END_REF] and presence of soliton states [START_REF] Amo | Polariton Superfluids Reveal Quantum Hydrodynamic Solitons[END_REF][START_REF] Sich | Observation of Bright Polariton Solitons in a Semiconductor Microcavity[END_REF]. A review can be found in Carusotto and Ciuti, 2013.

Our first line of attack to analytically tackle these phenomena is to use the mean field approximation which entails writing the dynamical equations of motion for the expectation value of the LP polariton field operator, ΨLP (r) (remember we are neglecting the upper polariton state). The polariton field operator is the 2D Fourier transform of the LP operator lk . A semi-classical equation can thus be obtained by replacing every instance of the polariton field operator with its expectation value when we write the its Heisenberg equation of motion, which is given as

𝑖 𝜕 𝜕 𝑡 ΨLP (r, 𝑡) = [ ΨLP (r, 𝑡), ( Ĥtotal + ĤLP-LP + Ĥpump + Ĥbath ) ] (1.44)
where Ĥtotal is given in Eq. (1.38), ĤLP-LP in Eq. (1.43), Ĥpump in Eq. (1.36), and Ĥbath is a bath Hamiltonian describing the interaction of the discrete polariton mode with a continuum of bath modes -photon modes outside the cavity -which results in a finite linewidth of the polariton mode and introduction of dissipation in our system [START_REF] Verger | Polariton Quantum Blockade in a Photonic Dot[END_REF] 3 . To counteract the dissipation in the system, an external pump Hamiltonian has to be added in the Heisenberg equation of motion. Here we use a coherent pump which is resonant with a lower polariton mode. Note that other pumping schemes also exist like addition of an incoherent pump whose dynamics differ from the one we are discussing here (see Carusotto and Ciuti, 2013 and the references therein).

The resulting nonlinear equation goes by the name of driven-dissipative or generalized Gross-Pitaevskii equation (GPE) and is given as

𝑖 𝜕 𝜕 𝑡 Ψ LP (r, 𝑡) = [ 𝐸 0 𝑙 - 2 2𝑚 LP ∇ 2 r + 𝑔 LP |Ψ LP (r, 𝑡)| 2 - 𝑖𝛾 LP 2 ] Ψ LP (r, 𝑡) + 𝜂 LP 𝐸 inc (r, 𝑡).
(1.45) Here 𝑚 LP ≈ 𝑚 cav /𝐶 2 is the lower polariton mass, 𝛾 LP is the dissipation energy and 𝜂 LP = 𝐶𝜂 𝜆 quantifies the coupling of incident coherent laser field with the polariton field inside the microcavity. Gross-Pitaevskii was first written is the context of dilute Bose gases such as atomic BECs in equilibrium where Ψ LP (r, 𝑡) describes a semiclassical field known as the condensate wavefunction. Using this terminology for our generalized GPE the semiclassical polariton field operator is often referred 1.5. Bogoliubov dispersion of elementary excitations to as a condensate wavefunction and the system as a polariton fluid.

Bogoliubov dispersion of elementary excitations

In the last section, we were introduced to the generalized Gross-Pitaevskii equation (GPE) that describes the dynamics of the polariton states under the mean-field approximation. Next, we'll solve the generalized GPE to get a stationary state of our system under a coherent drive and the elementary excitations over this state using the Bogoliubov theory.

Steady state under a coherent drive

The stationary state solution of generalized GPE describes a steady state maintained by a delicate balance of pumping and losses in the system. It is necessarily an excited state of the system under a coherent drive and dissipation. Using a continuous coherent pump with a plane wave spatial profile of the form 𝐸 inc (r, 𝑡) = 𝐸 inc 0 𝑒 𝑖(k𝑝.r-𝜔𝑝𝑡) -where 𝑘 𝑝 and 𝜔 𝑝 are the pump wavevector and frequency respectively -in Eq. (1.45) and looking for stationary solutions of the form Ψ LP (r, 𝑡) = Ψ ss LP 𝑒 𝑖(k𝑝.r-𝜔𝑝𝑡) we obtain a polaritonic equation of state written as

( -𝜔 𝑝 + 𝐸 0 𝑙 + 2 2𝑚 LP k 2 𝑝 + 𝑔 LP |Ψ ss LP | 2 - 𝑖𝛾 LP 2 ) Ψ ss LP = -𝜂 LP 𝐸 inc 0 .
(1.46)

The solution of this nonlinear equation of state gives us the amplitude of the polariton condensate wavefunction labeled by Ψ ss LP and the polariton density is given by the modulus square of this amplitude 𝑛 LP = |Ψ ss LP | 2 . As it turns out, polariton density shows different behaviours as a function of the pump intensity, 𝐼 𝑝 = |𝐸 inc 0 | 2 , depending on whether the pump energy, 𝜔 𝑝 , is less than or greater than the polariton energy, 𝐸 𝑙,k𝑝 = 𝐸 0 𝑙 -2 k 2 𝑝 /2𝑚 LP , (as shown in Figure 1.10). The two different behaviours are:

• Optical limiter. When 𝜔 𝑝 < 𝐸 𝑙,k𝑝 , the nonlinear interaction term in Eq. (1.46) shifts the renormalized polariton energy -𝐸 𝑙,k𝑝 + 𝑔 LP |Ψ ss LP | 2 -even further away from resonance with the pump energy. The polariton density is hence, a continuous monotonically growing function of 𝐼 𝑝 with a sublinear behaviour.

• Optical bistability. When 𝜔 𝑝 > 𝐸 𝑙,k𝑝 , the polariton density shows a hysteretic behaviour as the growing interaction energy pushes the renormalized polariton energy closer to resonance with the the pump energy [START_REF] Baas | Optical Bistability in Semiconductor Microcavities[END_REF].

Chapter 1. Introduction to Exciton Polaritons

Increasing the pump intensity further still finally pushes the polariton energy above the pump energy and we recover the optical limiter behaviour4 .

Elementary excitations over steady state

We found the steady state solution in the last subsection, and now we can look for elementary excitations over this solution. The elementary excitations -also sometimes called collective excitations -are the small low energy fluctuations around the steady state and can be studied using the Bogoliubov theory of dilute Bose gases (L. [START_REF] Pitaevskii | Bose-Einstein condensation and superfluidity[END_REF]. It was first studied under the context of equilibrium Bose gases such as atomic Bose Einstein condensates [START_REF] Dalfovo | Theory of Bose-Einstein Condensation in Trapped Gases[END_REF] but has been extended to the case of non-equilibrium Bose gases like our driven-dissipative polariton fluids [START_REF] Carusotto | Probing Microcavity Polariton Superfluidity through Resonant Rayleigh Scattering[END_REF]Ciuti and Carusotto, 2005) and have been experimentally observed in [START_REF] Utsunomiya | Observation of Bogoliubov Excitations in Exciton-Polariton Condensates[END_REF][START_REF] Kohnle | From Single Particle to Superfluid Excitations in a Dissipative Polariton Gas[END_REF][START_REF] Stepanov | Dispersion Relation of the Collective Excitations in a Resonantly Driven Polariton Fluid[END_REF].

The low energy collective excitations can be analytically deduced by linearizing the generalized GPE using the ansatz

Ψ LP (r, 𝑡) = [ Ψ ss LP + ∑ k 𝛿Ψ k (𝑡) 𝑒 𝑖k.r + ∑ k 𝛿Ψ * k (𝑡) 𝑒 -𝑖k.r
] 𝑒 𝑖(k𝑝.r-𝜔𝑝𝑡) , (1.47)

where 𝛿Ψ k (𝑡) are the small plane wave fluctuations around the steady state. Using this ansatz in Eq. 1.45 and utilizing the polaritonic equation of state (1.46) we obtain a pair of dynamical linear equations for each fluctuation mode 𝛿Ψ k (𝑡) and its conjugate which are written in matrix form as

𝑖 𝜕 𝜕 𝑡 ⎛ ⎝ 𝛿Ψ k (𝑡) 𝛿Ψ * -k (𝑡) ⎞ ⎠ = ℒ Bog (k) ⎛ ⎝ 𝛿Ψ k (𝑡) 𝛿Ψ * -k (𝑡) ⎞ ⎠ . (1.48)
Here ℒ Bog (k) is known as the Bogoliubov matrix and has the form

ℒ Bog (k) = ⎛ ⎜ ⎝ 𝐸 𝑙,k𝑝+k -𝜔 𝑝 + 2𝑔 LP 𝑛 LP -𝑖𝛾 LP 2 𝑔 LP Ψ ss LP 2 -𝑔 LP Ψ ss LP *2 -𝐸 𝑙,k𝑝+k + 𝜔 𝑝 -2𝑔 LP 𝑛 LP -𝑖𝛾 LP 2 ⎞ ⎟ ⎠ .
(1.49)

The eigenvalues of this matrix finally gives us the Bogoliubov dispersion of elemen-

-1.5 0 1.5 Imag(E Bog )/ γ k ξ 0 B -1.5 0 1.5 k ξ 0 C -1.5 0 1.5 k ξ 0 D -1.5 0 1.5 k ξ 0 E -1.5 0 1.5 k ξ 0 F -1.5 0 1.5 k ξ 0
𝐸 Bog = 2 k.k 𝑝 2𝑚 LP -𝑖 𝛾 2 ± [( 2 𝑘 2 2𝑚 LP -∆ ) ( 2 𝑘 2 2𝑚 LP -∆ + 2𝑔 LP 𝑛 LP )] 1/2 , (1.50)
where we have defined a renormalized detuning parameter ∆ as

∆ = 𝜔 𝑝 - ( 𝐸 0 𝑙 + 2 k 2 𝑝 /2𝑚 LP + 𝑔 LP 𝑛 LP ) = ∆ 0 + 𝑔 LP 𝑛 LP .
(1.51)

The ± sign refers to the two branches of the spectrum with the plus (+) branch called as the normal branch and minus (-) branch as the ghost branch.

The presence of drive and dissipation in our system gives us access to novel regimes of Bogoliubov spectrum which are not accessible in equilibrium Bose gases. In general three different regimes can be accessed by tuning the parameter ∆. First, let's analyse these three regimes for the case of k 𝑝 = 0 (when the pump laser is perpendicular to the plane of the microcavity) as shown in the bottom panel of Figure 1.10.

• Gapped spectrum. When ∆ < 0 the value under the square root in (1.50) is positive for all values of wavevector k. It leads to an opening of an energy gap between the two spectral branches creating an effective barrier for Bogoliubov excitations that increases with growing pump intensity. Hence, only excitations with energy higher than this barrier can exist in the system. This is always the case for the optical delimiter regime as shown by the spectrum at point F in Figure 1.10. In the bistable regime gapped spectrum is present on the upper branch of the hysteresis curve as shown by point E in Figure 1.10.

• Sonic spectrum. When ∆ = 0, the two branches touch at k = 0 and the spectrum is analogous to the one for equilibrium atomic condensates (L. [START_REF] Pitaevskii | Bose-Einstein condensation and superfluidity[END_REF] albeit with a negative imaginary part accounting for dissipation in the system (as shown by point D in Figure 1.10). Close to 𝑘 = 0 the spectrum shows a linear phononic dispersion with an effective velocity of sound 𝑐 = √ 𝑔 LP 𝑛 LP /𝑚 LP , hence the name sonic spectrum. For wavevector values greater than the inverse of an effective length -which is equal to Chapter 1. Introduction to Exciton Polaritons the unstable regime5 signaled by the onset of a Kerr single-mode instability [START_REF] Drummond | Quantum Theory of Optical Bistability. I. Nonlinear Polarisability Model[END_REF]Ciuti and Carusotto, 2005) as shown by point C in Figure 1.10. For 𝑔 LP 𝑛 LP < ∆ < 2𝑔 LP 𝑛 LP imaginary values start to split around k = 0 signaling the onset of a parametric oscillator regime [START_REF] Savvidis | Angle-Resonant Stimulated Polariton Amplifier[END_REF][START_REF] Wouters | Parametric Oscillation Threshold of Semiconductor Microcavities in the Strong Coupling Regime[END_REF], where polaritons are scattered by into a signal and an idler mode which lie around the two maxima of the imaginary values. Increasing even further, when ∆ > 2𝑔 LP 𝑛 LP , a circle around k = 0 becomes real valued again and the bifurcation is shifted to a wavevector region resembling a ring with an inner radius √ 2𝑚∆ and an outer radius √ 2𝑚(∆ -𝑔 LP 𝑛 LP ) as shown by points A and B in Figure 1.10.

𝜉 0 = √ 2 /2𝑚 LP 𝑔 LP 𝑛 LP -
In Figure 1.11 we analyse these three regimes for the case of k 𝑝 ̸ = 0. We see that the real part of the dispersion relation gets tilted with respect to the wavevector axis. This can be linked to the fact that a finite k 𝑝 provides a finite velocity to the polariton fluid equal to k 𝑝 /𝑚 LP . As a result the tilting can be viewed as resulting from Doppler effect caused due to the fluid motion.

2

Background on Polarons

In this chapter, I'll present a brief background on Bose polarons in quantum gases. I'll start by describing the concept of a polaron followed up with a discussion of Fröhlich Hamiltonian which is a paradigmatic model usually utilized to describe a Bose polaron under the assumption of weak interaction between the impurity and the surroundings.

What are Polarons?

Interaction with an environment -or a bath -can lead to a drastic modification in the properties of quantum systems. One of the earliest systems in which this modification was identified is a simple system of an electron transiting through a distorted crystal. It was initially pointed out, in a series of papers in mid twentieth century by Landau and Pekar [START_REF] Landau | Effective mass of a polaron[END_REF], Fröhlich [START_REF] Fröhlich | XX. Properties of Slow Electrons in Polar Materials[END_REF][START_REF] Fröhlich | Electrons in Lattice Fields[END_REF], Feynman [START_REF] Feynman | Slow Electrons in a Polar Crystal[END_REF], and Holstein (Holstein, 1959a;Holstein, 1959b), that even a single electron can create a distortion field -phonons -inside an ionic lattice which is strong enough to cause a dramatic change in the motion of the electron. Qualitatively it may be understood as an electron moving in conjunction with a cloud of phonons which shields its presence to the rest of the system. The resulting "dressed" electron -particle plus screening cloud -is called a polaron quasiparticle. This dressing of electrons by phonons is important in comprehending properties of many condensed matter materials [START_REF] Franchini | Polarons in Materials[END_REF] such as polar semiconductors [START_REF] Alexandrov | Polarons in Advanced Materials[END_REF]Alexandrov and Devreese, 2010b;Alexandrov and Devreese, 2010a;[START_REF] Devreese | Fröhlich Polaron and Bipolaron: Recent Developments[END_REF], and high temperature superconductors (A. S. [START_REF] Mishchenko | Polaronic Metal in Lightly Doped High-T c Cuprates[END_REF]. At present, the concept of a polaron quasiparticle is not only limited to electronphonon systems but has expanded to other quantum systems and has become an important paradigm in condensed matter physics. Looking naively, the idea of a particle being dressed by the quantum particles of the traversed or host field is quite a universal one as whenever any particle moves through any interacting field it will create fluctuations which will engulf the particle leading to the formation 52 2.1. What are Polarons? of a polaron. Since there is an abundance of elementary particles and fields with which they can interact we have many different flavours of polarons. For instance magnetic polarons can be described as charge carriers surrounded by a cloud of magnetic fluctuations in systems such as such as holes doped in antiferromagnetic Mott insulators [START_REF] Dagotto | Correlated Electrons in High-Temperature Superconductors[END_REF][START_REF] Nagaev | Ground State and Anomalous Magnetic Moment of Conduction Electrons in an Antiferromagnetic Semiconductor[END_REF] or electrons in magnetic semiconductors [START_REF] Kaminski | Polaron Percolation in Diluted Magnetic Semiconductors[END_REF]. Even in the Standard Model of high energy physics, the formation of polaron can be linked with the mechanism of mass generation in elementary particles as a consequence of interaction with the Higgs field [START_REF] Higgs | Broken Symmetries and the Masses of Gauge Bosons[END_REF].

Polarons form a part of quantum impurity systems when taken in a wider perspective. These systems in general consist of a single impurity which introduces non-trivial correlations in the many-particle system which is hosting it. Since the early hay days of solid state physics, they have provided a test-bed to study analytical, field-theoretical, and numerical methods in quantum many-particle systems. Quantum impurity problems have been at the heart of some of the fundamental concepts in condensed matter physics. For instance, a two level quantum system interacting with a bath of harmonic oscillators -Caldeira-Legget model [START_REF] Caldeira | Influence of Dissipation on Quantum Tunneling in Macroscopic Systems[END_REF]Caldeira and Leggett, 1983;[START_REF] Leggett | Dynamics of the Dissipative Two-State System[END_REF] -is extensively used to describe dissipation in open quantum systems [START_REF] Breuer | The theory of open quantum systems[END_REF]; U. [START_REF] Weiss | Quantum dissipative systems[END_REF] such as the interaction of a qubit with the outside environment in superconducting circuits. Similarly, the study of Kondo effect [START_REF] Kondo | Resistance Minimum in Dilute Magnetic Alloys[END_REF] lead to the application of renormalization group (RG) approach to in the study condensed matter systems [START_REF] Wilson | The Renormalization Group: Critical Phenomena and the Kondo Problem[END_REF].

The environment or bath hosting the quantum impurity can be made up of either fermionic or bosonic particles and hence the problem could be of a Fermi polaron or a Bose polaron. Recent revival of interest in these class of problems comes from the technological advancements in the field of ultracold atoms which have emerged in the last two decades as a fertile test ground for quantum simulation of many particle phenomena in condensed matter systems. As we saw in the introduction to this part both the problem of a Fermi and Bose polaron have been extensively studied and experimentally demonstrated in the context of ultracold atomic gases. In this chapter, we'll introduce the problem of a Bose polaron in equilibrium ultracold bosonic quantum fluids in the so called Fröhlich regime -or large polaron1 -and will extend this concept to non-equilibrium bosonic quantum fluids of light in the next chapter.

Fröhlich Polaron in an atomic BEC

The Fröhlich polaron model describes an extensive class of problems concerned with understanding properties of a single mobile particle interacting with a bosonic bath: Bose polaron. It was originally introduced in the context of electrons interacting with phonons in crystals and has since been applied in the study of concepts ranging from strongly correlated electron systems to quantum information and high energy physics. In the last few years this model has been applied to describe impurity atoms immersed in Bose-Einstein condensates (BECs) of ultracold atoms.

Our discussion in this section will be concentrated on an impurity atom interacting with a Bose-Einstein Condensate (BEC) of ultracold atoms which can be described by a Fröhlich Hamiltonian of the form

ℋ = p2 2𝑀 + ∑ k 𝜔 k â † k âk + ∑ k 𝑉 k 𝑒 𝑖k • r( â † k + âk ) , (2.1)
where r and p are the impurity position and momentum operators respectively; 𝑀 is the mass of the impurity; â † k and âk are the creation and annihilation operators of low energy excitations of the BEC system (colloquially known as Bogoliubov phonons or Bogolons); 𝜔 k describes the Bogolon dispersion relation; and 𝑉 k describes the scattering amplitude of the impurity-Bogoliubov phonon interaction at momentum k. While this model was originally introduced in the context of electron-phonon systems, it has been shown to describe the interaction between impurity atoms and Bogoliubov modes of a BEC [START_REF] Tempere | Feynman Path-Integral Treatment of the BEC-Impurity Polaron[END_REF], in the regime where the quantum depletion of the condensate around the impurity is small and scattering of phonons at finite momentum can be neglected. This model is characterized by two important parameters: the impurity mass 𝑀 2.2. Fröhlich Polaron in an atomic BEC and the impurity-phonon dimensionless coupling strength 𝛼. In Figure 2.2 we can have a look at the quantitative phase diagram describing the different approaches to solving the polaron problem depending on these two parameters. In the weak coupling regime, the polaron can be thought of as a quasi free impurity carrying a loosely bound screening cloud of phonons. In the strong coupling regime, on the other hand, the screening cloud is large and restricts the motion of the impurity, effectively trapping it inside the resulting potential. These two regimes have been previously described by a weak coupling mean-field theory (T. D. [START_REF] Lee | The Motion of Slow Electrons in a Polar Crystal[END_REF] and a strong coupling approach based on the adiabatic approximation [START_REF] Landau | Effective mass of a polaron[END_REF] respectively. While in the intermediate coupling case, the picture of a polaron as a bare particle engulfed by phonons may break down as the relatively light impurity has an additional role of acting as the mediator of interactions among the phonons, as a result of which the polaron state develops strong phononic correlations. Here, we won't concern ourselves with the intermediate or the strongly coupled regime but will rather stay in the weakly interacting regime which is more relevant in our case. Next, I'll provide a derivation of Fröhlich Hamiltonian.

Derivation of Bogoliubov-Fröhlich Hamiltonian

We start by considering an impurity in immersed in a 3𝐷 atomic condensate2 of density 𝑛 0 . We'll restrict our analysis to the case of dilute or low concentration of impurities. This allows us to consider a single impurity interacting with the condensate under the assumption that the average inter-particle distance between the impurities is much larger than the polaron size and the average distance between the condensate particles (of the order 𝑛 -1 0 ). We can write the many-body Hamiltoninan of our impurity-condensate system as follows:

ℋ = ∫ dr ψ † (r) ( -∇ 2 2𝑚 + 𝑔 2 ψ † (r) ψ(r) + 𝑔 IB 𝛿(r -r 0 ) ) ψ(r) + p0 2 2𝑀 . (2.2)
Here, ψ(r) is the bosonic field operator for the atoms, 𝑚 is the mass of the atomic particles, 𝑔 and 𝑔 IB describe the atom-atom and impurity-atom interaction constant respectively, p 0 and r 0 are the impurity momentum and position operators respectively, and 𝑀 is the impurity mass. The interactions are modelled as density-density interactions among the particle species and the interaction constants are modelled as contact (pseudo)potentials. Also, we consider here a uniform system neglecting any external potentials on either the impurity or the atoms. Hence, the field operators can be Fourier expanded -under periodic boundary conditions -as discrete plane

Chapter 2. Background on Polarons wave bosonic modes, âk ,

ψ(r) = 1 √ 𝐿 ∑ k 𝑒 𝑖k • r âk , (2.3)
and similarly for the annihilation field. The commutation relation between the modes is the usual bosonic one which is

[â k , â † k ′ ] = 𝛿 k,k ′ .
Using this expansion in our microscopic Hamiltonian in Eq. (2.2) we get

ℋ = ∑ k k 2 2𝑚 â † k âk + ∑ k,k ′ ,q 𝑔 2𝐿 â † k â † k ′ âk ′ +q âk-q + ∑ k,k ′ 𝑒 𝑖(k-k ′ ) • r0 â † k âk ′ + p2 2𝑀 . (2.4)
Next, we have to simplify the above equation to get it into a more malleable form as a quadratic Hamiltonian. For this purpose we utilize the Bogoliubov theory and assume a macroscopic occupation of the zero momentum mode -the condensatewhich can be done by introducing a c-number √ 𝑁 0 to replace the zero momentum operators âk and â † k , where 𝑁 0 is the number of atoms inside the condensate. This simplifies the atomic part of the Hamiltoninan into a quadratic Hamiltonian which can then be diagonalized by transforming 𝑎 operators to Bogoliubov 𝑏 operators using the following relations,

âk = 𝑢 k bk -𝑣 k b † -k (2.5a) â † k = 𝑢 k b † -k -𝑣 k bk , (2.5b)
where

𝑢 k , 𝑣 k = [ 1 2 ( 𝑘 2 /2𝑚 + 𝑔𝑛 0 𝜔 𝑘 ± 1
)]1

2

(2.6)

The atomic part now has a simple quadratic form

ℋ a = ∑ k̸ =0 𝜔 k b † k bk , (2.7)
where 𝜔 k is the energy of the elementary excitations or Bogolons (see appendix for full derivation).

Similarly the impurity-boson interaction in the Hamiltonian (2.4) can be simplified using Bogoliubov approximation as

ℋ IB = 𝑔 IB 𝐿 [ 𝑁 0 + ∑ k̸ =0 ( 𝑒 𝑖k • r0 √ 𝑁 0 â † k + ℎ.𝑐. ) + ∑ k,k ′ ̸ =0 𝑒 𝑖(k-k ′ ) • r0 â † k âk ′ ] (2.8)
We disregard the last term considering that the density of excitations is much less than the condensate density. Next, we transform the 𝑎 operators to 𝑏 operators. Neglecting the mean energy shifts terms in the constituent Hamiltonian ℋ a and ℋ IB 56 2.2. Fröhlich Polaron in an atomic BEC the simplified impurity-boson Hamiltonian is written as

ℋ = ∑ k̸ =0 [ 𝜔 k b † k bk + 𝑉 k 𝑒 𝑖k • r0 ( b † -k + bk )] + p0 2 2𝑀 , (2.9)
where

𝑉 k = 𝑔 IB √ 𝑁 0 /𝐿 (𝑢 k -𝑣 k )
is the scattering amplitude of impurity-boson interaction. This simplified Hamiltoninan has the same form as the Fröhlich Hamiltoninan and is termed Bogoliubov-Fröhlich Hamiltonian.

In the next chapter we'll utilize this concept of Bogoliubov-Fröhlich Hamiltonian in the context of driven-dissipative bosonic fluids as realized by exciton-polaritons in a semiconductor microcavity.

3

Bose Polaron in a quantum fluid of light

In this chapter, we developed the theory of a Bose polaron in the quantum fluids of light as realised by exciton polaritons in a semiconductor microcavity. I'll start by giving an analytical description of the system followed by realistic assumptions that we took to simplify our system Hamiltonian. Next, we'll derive the dynamical equations of motion describing the motion of the polaron followed by the results that we got from their solutions. I'll finish with conclusions to this part and a future outlook.

Physical system and model Hamiltonian

We consider an exciton-polariton fluid (called polariton onwards for brevity) coherently driven by a resonant pump and interacting with dilute impurities. As we saw in Chapter 1, the polariton modes come in two branches: the lower polariton and upper polariton. Here we restrict only to the lower polariton branch neglecting the upper branch since it is detuned far out of the relevant energy scales of our system. A schematic of the system setup considered in our approach is shown in Figure 3.1. We describe the polariton losses through the mirrors by their coupling to the continuum of photons outside the cavity, which thus acts as a bath. The total Hamiltonian of the system reads Ĥ = ĤP + ĤI + ĤB .

(3.1)

Here ĤP describes the polariton fluid coherently driven by a continuous wave laser pump of frequency 𝜔 𝑝 and wavevector k 𝑝 (Carusotto and Ciuti, 2013) and is given as

ĤP = ∑ k ( 2 k 2 2𝑚 -𝜔 𝑝 ) â † k âk + 𝑔 2𝐴 ∑ k,k ′ ,q â † k+q â † k ′ -q âk ′ âk + âk𝑝 𝐹 * 0 + â † k𝑝 𝐹 0 , (3.2)
where â † k , âk are the creation and annihilation operators of polaritons, 𝑔 is the polariton-polariton interaction constant, 𝐴 is the pumped area in the plane of the microcavity and 𝐹 0 is the plane-wave incident laser field with wavevector k 𝑝 .

Since we consider dilute impurities, we neglect any interactions among them and hence can limit ourselves to the case of a single impurity. ĤI describes the impurity weakly interacting with the polariton fluid, and is given as

Ĥ𝐼 = p2 2𝑀 + 𝑔 IB 𝐴 ∑ k,k ′ 𝑒 𝑖(k-k ′ ) • xâ † k ′ âk , (3.3)
where 𝑀 is the mass of the impurity and the second term describes the impuritypolariton interaction, characterized by coupling constant 𝑔 IB . Finally, ĤB describes the external radiation bath as harmonic excitations, linearly interacting -in the spirit of Caldeira-Legget model [START_REF] Caldeira | Influence of Dissipation on Quantum Tunneling in Macroscopic Systems[END_REF][START_REF] Gardiner | Quantum noise[END_REF][START_REF] Verger | Polariton Quantum Blockade in a Photonic Dot[END_REF] -with the polariton fluid:

ĤB = ∫ 𝑑𝑞 ∑ k 𝜔 𝑞,k α † 𝑞,k 𝛼 𝑞,k + ∫ 𝑑𝑞 ∑ k [ 𝜅 * 𝑞,k α † 𝑞,k âk + 𝜅 𝑞,k â † k α𝑞,k ] , (3.4)
where 𝜔 𝑞,k is frequency of the bath mode with in-plane wavevector k and a con-3.1. Physical system and model Hamiltonian tinuous wavevector 𝑞 in the orthogonal direction; 𝛼 † 𝑞,k , 𝛼 𝑞,k are the creation and annihilation operators of the bosonic bath modes and 𝜅 𝑞,k quantifies the coupling of the polariton modes with the external bath. We neglect the coupling of the impurity with the bath for simplicity.

In the next two subsections, we simplify this Hamiltonian; first by assuming a weak excitation density on top of the condensate at all times, and thus taking the Bogoliubov approximation, and then by moving to the impurity reference frame using the Lee, Low and Pines (LLP) transformation (T. D. [START_REF] Lee | The Motion of Slow Electrons in a Polar Crystal[END_REF].

Bogoliubov approximation

We will consider the three subsystems separately, starting with the polariton Hamiltonian in equation (3.2), which, following a standard convention, can be conveniently rewritten in terms of the excitation wavevectors k shifted with respect to the pump wavevector k 𝑝 :

ĤP = ∑ k ( 2 (k 𝑝 + k) 2 2𝑚 -𝜔 𝑝 ) â † k𝑝+k âk𝑝+k + 𝑔 2𝐴 ∑ k,k ′ ,q â † k𝑝+k+q â † k𝑝+k ′ -q âk𝑝+k ′ âk𝑝+k + âk𝑝 𝐹 * 0 + â † k𝑝 𝐹 0 . (3.5)
We now use the Bogoliubov approximation -which assumes a macroscopic occupation of k 𝑝 mode âk𝑝 (â † k𝑝 ) ≈ √ 𝑁 0 -in (3.5) to approximate it as a quadratic Hamiltonian in the creation and annihilation operators. It is then diagonalized with Chapter 3. Bose Polaron in a quantum fluid of light the help of Bogoliubov transformation (see Appendix B for details)

âk𝑝+k = 𝑢 k bk𝑝+k -𝑣 * -k b † k𝑝-k (3.6a) â † k𝑝-k = -𝑣 k bk𝑝+k + 𝑢 * -k b † k𝑝-k . (3.6b)
The polariton Hamiltonian in the Bogoliubov approximation then reads

ĤP = 𝐸 𝑠𝑠 + ∑ k̸ =0 𝐸 b b † k𝑝+k bk𝑝+k , (3.7)
where 𝐸 𝑠𝑠 is the energy of the macroscopically occupied steady state at k 𝑝 ; it provides a constant energy shift and since it doesn't contribute to the dynamics of the problem we can safely neglect this term moving forward. 𝐸 b is the Bogoliubov spectrum of polaritons and is given as

𝐸 b = 2 k • k 𝑝 𝑚 + 𝜔 k (3.8)
where

𝜔 k = [ 𝜖 k (𝜖 k + 2𝑔𝑛) ] 1 2 , 𝜖 k = 2 k 2 2𝑚 -∆ and ∆ = 𝜔 𝑝 - 2 k 2 𝑝 2𝑚
-𝑔𝑛, with 𝑔𝑛 the blueshift due to polariton-polariton interactions and 𝑛 = 𝑁 0 /𝐴 the density of polariton condensate. The parameters of Bogoliubov transformation are given by

𝑢 k , 𝑣 k = [ 1 2 ( 𝜖 k + 𝑔𝑛 𝜔 𝑘 ± 1 )]1 2 . (3.9) Note that if ∆ ≤ 0 we have 𝑢 k (𝑣 k ) = 𝑢 -k (𝑣 -k ) = 𝑢 * k (𝑣 * k ) but this is not true for ∆ > 0 where 𝑢 k (𝑣 k ) ̸ = 𝑢 * k (𝑣 * k )
and it calls for a special diagonalization treatment which is detailed in Appendix B. For the purpose of legibility, we will work with the former case and use 𝑢 k (𝑣 k ) for all the the instances of 𝑢 -k (𝑣 -k ) and 𝑢 * k (𝑣 * k ) in all subsequent derivations. The extension to the case ∆ > 0 can be easily made.

The nature of excitation spectrum at low momenta is controlled by the detuning ∆: depending on its value, the excitation spectrum can be gapless, gapped, or exhibit a non-dispersive region at small wavevectors (see panel b in Figure 3.2).

Using the Bogoliubov approximation in the impurity Hamiltonian (3.3) and neglecting the term describing the interaction between two Bogoliubov excitations of the condensate which are of order 1/ √ 𝑁 0 smaller than the interaction of an excitation with the impurity, we obtain

Ĥ𝐼 = p2 2𝑀 + 𝑔 IB 𝑛 + ∑ k̸ =0 𝑒 𝑖k • x ( 𝑉 k bk𝑝+k + 𝑉 k b † k𝑝-k ) , (3.10)
where 𝑉 k is the scattering amplitude of Bogoliubov excitation with the impurity, 3.1. Physical system and model Hamiltonian given as

𝑉 k = 𝑔 IB √ 𝑁 0 𝐴 (𝑢 k -𝑣 k ) . (3.11)
Notice that 𝑉 k is associated to the polariton fluid density fluctuations through the term (𝑢 k -𝑣 k ). In the case of a gapless phononic dispersion, this term is suppressed at low momenta as phase fluctuations dominate the Bogoliubov excitation spectrum.

Our analysis hence shows that the impurity dynamics can probe superfluid properties of the quantum fluid of light.

We finally apply the Bogoliubov transformation to the bath Hamiltonian (3.4), which results in

ĤB = ∫ 𝑑𝑞 ∑ k 𝜔 𝑞,k α † 𝑞,k 𝛼 𝑞,k + ∫ 𝑑𝑞 ∑ k [ 𝜅 * 𝑞,k α † 𝑞,k ( 𝑢 k bk𝑝+k -𝑣 k b † k𝑝-k ) + 𝜅 𝑞,k α𝑞,k ( 𝑢 k b † k𝑝+k -𝑣 k bk𝑝-k ) ] . (3.12)
The sum of equations (3.7), (3.10) and (3.12) provides an approximate description of our system, that takes the form of a Bogoliubov-Fröhlich Hamiltonian [START_REF] Tempere | Feynman Path-Integral Treatment of the BEC-Impurity Polaron[END_REF][START_REF] Shashi | Radio-Frequency Spectroscopy of Polarons in Ultracold Bose Gases[END_REF][START_REF] Grusdt | All-Coupling Theory for the Fröhlich Polaron[END_REF] with a dissipative bath term.

Lee Low Pines transformation

In order to further simplify our model we use the Lee Low Pines (LLP) transformation (T. D. [START_REF] Lee | The Motion of Slow Electrons in a Polar Crystal[END_REF] to remove the impurity degree of freedom in the impurity-polariton interaction term. This is achieved by applying the operator

𝑇 LLP = exp [ -𝑖 ( ∑ k̸ =0 k b † k𝑝+k bk𝑝+k ) • x] (3.13)
on the full system Hamiltonian, to move to the impurity reference frame. Denoting the momentum of the Bogoliubov excitations as Π = ∑ k̸ =0 k b † k𝑝+k bk𝑝+k , the resulting total Hamiltonian reads

ĤLLP = ∑ k̸ =0 𝐸 b b † k𝑝+k bk𝑝+k + 1 2𝑀 ( p - Π) 2 + ∑ k̸ =0 ( 𝑉 k bk𝑝+k + 𝑉 k b † k𝑝-k ) + ∫ 𝑑𝑞 ∑ k 𝜔 𝑞,k α † 𝑞,k 𝛼 𝑞,k + ∫ 𝑑𝑞 ∑ k [ 𝜅 * 𝑞,k α † 𝑞,k ( 𝑢 k bk𝑝+k -𝑣 k b † k𝑝-k ) 𝑒 -𝑖k • x + 𝜅 𝑞,k α𝑞,k ( 𝑢 k b † k𝑝+k -𝑣 k bk𝑝-k ) 𝑒 𝑖k • x] . (3.14)
Chapter 3. Bose Polaron in a quantum fluid of light

Quantum dynamical equations and observables

In order to obtain the impurity dynamics, we derive the Heisenberg equation of motion for x, p, bk𝑝+k and α𝑞,k operators. The resulting equations of motion read:

𝑑x 𝑑𝑡 = ( p - Π) 𝑀 , (3.15) 𝑑p 𝑑𝑡 = ∑ k 𝑖k𝑒 -𝑖k • x ∫ 𝑑𝑞 [ 𝜅 * 𝑞,k α † 𝑞,k ( 𝑢 k bk𝑝+k -𝑣 k b † k𝑝-k ) + 𝜅 𝑞,-k α𝑞,-k ( 𝑢 -k b † k𝑝-k -𝑣 k bk𝑝+k ) ] , (3.16) 𝑖 𝑑 𝑑𝑡 bk𝑝+k = ( 𝐸 b + 2 k 2 2𝑀 + k • Π 𝑀 - k.p 𝑀 ) bk𝑝+k + 𝑉 k + ∫ 𝑑𝑞 ( 𝜅 𝑞,k α𝑞,k 𝑢 k -𝜅 * 𝑞,k α † 𝑞,k 𝑣 k ) 𝑒 𝑖k • x , (3.17) 𝑖 𝑑 𝑑𝑡 α𝑞,k = 𝜔 𝑞,k 𝛼 𝑞,k + 𝜅 * 𝑞,k ( 𝑢 k bk𝑝+k -𝑣 k b † k𝑝-k ) 𝑒 -𝑖k • x. (3.18)
Notice that the impurity momentum is not conserved due to the presence of the bath. Now we proceed to obtain a solution for the four coupled equation of motions. We derive a self-consistent equation for Π in the next subsection, using which we can calculate the trajectory of polarons.

Dynamics of the fluid excitations

We commence by tracing out the bath degrees of freedom by substituting the solution of (3.18) for α𝑞,k in (3.17). The general solution of (3.18) is given as

α𝑞,k = 𝑒 -𝑖𝜔 𝑞,k (𝑡-𝑡 0 ) 𝛼 𝑞,k (𝑡 0 ) -𝑖𝜅 * 𝑞,k ∫ 𝑡 𝑡 0 𝑑𝑡 ′ 𝑒 -𝑖𝜔 𝑞,k (𝑡-𝑡 ′ ) ( 𝑢 k bk𝑝+k (𝑡 ′ ) -𝑣 k b † k𝑝-k (𝑡 ′ ) ) 𝑒 -𝑖k • x(𝑡 ′ ) .
(3.19)

Using (3.19) in (3.17) we obtain

𝑖 𝑑 𝑑𝑡 bk𝑝+k = ( 𝐸 b + 2 k 2 2𝑀 + k • Π 𝑀 - k.p 𝑀 ) bk𝑝+k + 𝑉 k -𝑖 ∫ ∞ -∞ 𝑑𝑡 ′ Γ (1) k (𝑡 -𝑡 ′ ) bk𝑝+k (𝑡 ′ ) -𝑖 ∫ ∞ -∞ 𝑑𝑡 ′ Γ (2) k (𝑡 -𝑡 ′ ) b † k𝑝-k (𝑡 ′ ) + 𝐹 sto k (3.20) 64
3.2. Quantum dynamical equations and observables where

Γ (1) k (𝑡 -𝑡 ′ ) = Θ(𝑡 -𝑡 ′ ) ∫ 𝑑𝑞 ( |𝜅 𝑞,k | 2 𝑢 2 k 𝑒 -𝑖𝜔 𝑞,k (𝑡-𝑡 ′ ) -|𝜅 𝑞,-k | 2 𝑣 2 k 𝑒 𝑖𝜔 𝑞,-k (𝑡-𝑡 ′ )
) 𝑒 -𝑖k(x(𝑡 ′ )-x(𝑡))

(3.21a)

Γ (2) k (𝑡 -𝑡 ′ ) = Θ(𝑡 -𝑡 ′ ) ∫ 𝑑𝑞 𝑢 k 𝑣 k ( |𝜅 𝑞,k | 2 𝑒 -𝑖𝜔 𝑞,k (𝑡-𝑡 ′ ) -|𝜅 𝑞,-k | 2 𝑒 𝑖𝜔 𝑞,-k (𝑡-𝑡 ′ )
) 𝑒 -𝑖k(x(𝑡 ′ )-x(𝑡))

(3.21b)

𝐹 sto k = ∫ 𝑑𝑞 ( 𝜅 𝑞,k 𝑢 k 𝑒 -𝑖𝜔 𝑞,k 𝑡 α𝑞,k -𝜅 * 𝑞,-k 𝑣 k 𝑒 𝑖𝜔 𝑞,-k 𝑡 α † 𝑞,k ) . (3.21c)
Here, Γ

(1) k and Γ

(2) k are the memory kernels of the integro-differential equation (3.20), and 𝐹 sto k describes stochastic fluctuations due to the coupling with the bath. This will not enter in the calculation of the average trajectories and henceforth will be neglected. We next employ the Markovian approximation, which assumes no memory of past times, in (3.20) and under this approximation the integral kernels

Γ (1) k ≈ Γ k 𝛿(𝑡 ′ -𝑡) and Γ (2)
k → 0. The value 2𝜋 /Γ k gives the radiative lifetime of the polaritons. In the following we will assume that Γ k = Γ is a constant in k, since the coupling constant 𝜅 𝑞,k can be well approximated as a constant for the range of wavevectors of interest in polariton experiments. Hence, the equation of motion for bk𝑝+k is simplified as

𝑖 𝑑 𝑑𝑡 bk𝑝+k = ⎛ ⎝ 𝐸 b + 2 k 2 2𝑀 - k • ( p - Π) 𝑀 -𝑖Γ ⎞ ⎠ bk𝑝+k + 𝑉 k . (3.22)
The steady-state solution of (3.22) in the mean-field approximation is a coherent state of the Bogoliubov excitations. This follows from the fact that there are no interactions between the excitations within the Bogoliubov approximation. Under this assumption we replace the bk𝑝+k quantum operator with a classical mean field 𝛽 k𝑝+k = ⟨ bk𝑝+k ⟩ in (3.22) and obtain the stationary solution of the field as

𝛽 k𝑝+k = -𝑉 k ( 𝐸 b + 2 k 2 2𝑀 -k • (p-Π) 𝑀 -𝑖Γ ) (3.23)
Similarly the stationary solution for βk𝑝+k

= ⟨ b † k𝑝+k ⟩ reads βk𝑝+k = -𝑉 k ( 𝐸 b + 2 k 2 2𝑀 -k • (p-Π) 𝑀 + 𝑖Γ ). (3.24)
Notice that βk𝑝+k ̸ = 𝛽 * k𝑝+k if 𝐸 𝑏 takes imaginary values. A detailed discussion of this regime is provided in Appendix B.

Effective mass of polarons

We next proceed in determining the observables characterizing the polaron in a driven-dissipative fluid of light. In the mean field approximation Π reads

Π = ∑ k̸ =0 (k + k 𝑝 ) βk𝑝+k 𝛽 k𝑝+k -k 𝑝 ∑ k̸ =0 βk𝑝+k 𝛽 k𝑝+k (3.25)
The first term on the r.h.s. of (3.25) describes the Bogoliubov excitations mean field momentum, while the second term describes an effective drift of the impurity due to the flow of polariton fluid. We derive a self-consistent equation for Π by substituting Eqs. (3.23) and (3.24) in (3.25), leading to

Π = ∑ k̸ =0 k 𝑉 2 k ( 𝐸 b + 2 k 2 2𝑀 -k • (p-Π) 𝑀 ) 2 + Γ 2 . (3.26)
In the case when 𝑝 → 0 the excitation momentum can be approximated as having a linear dependence in p coming from the first term in its Taylor expansion around p = 0, corresponding to Π ≈ 𝜂p. In this case an effective mass of the polaron can be defined as

p 𝑀 eff = p(1 -𝜂) 𝑀 (3.27)
where is 𝜂 is the fraction of excitation momentum in the direction of p.

Drag force

Drag force estimates the response of a quantum fluid to a moving obstacle (Astrakharchik and L. P. [START_REF] Astrakharchik | Motion of a Heavy Impurity through a Bose-Einstein Condensate[END_REF]. In particular, the vanishing of drag force provides an indication of superfluidity. The drag force experienced by an impurity is obtained, using Ehrenfest theorem, as the gradient of the impurity-fluid interaction potential:

F drag = - ∫ dx⟨𝜓 † (x) ∇ [ 𝑔 IB nimp (𝑥) ] 𝜓(x)⟩ (3.28)
where 𝑛 imp (𝑥) = 𝛿(xx) is the density operator for a single particle (impurity).

Using the Bogoliubov approximation for the fluid we obtain

F drag = - ∑ k̸ =0 𝑖k𝑉 k ⟨ 𝑒 𝑖k • x ( bk𝑝+k + b † k𝑝-k )⟩ . ( 3
F drag = ∑ k̸ =0 2𝑖 k𝑉 2 k ( 2 k 2 2𝑀 + 𝜔 k ) ⎛ ⎜ ⎝ 1 ( 2 k 2 2𝑀 + 𝜔 k ) 2 - ( 2 k • k𝑝 𝑚 -k • (p-Π) 𝑀 -𝑖Γ ) 2 ⎞ ⎟ ⎠ .
(3.31)

In the limit when 𝑀 → ∞ we recover the expression derived in (Van Regemortel and Wouters, 2014).

Polaron trajectory

We begin deriving an equation for the polaron trajectory, by substituting Eq. (3.19) in (3.16), that is, by tracing out the bath degrees of freedom. Within the Markovian and semiclassical approximations we get

𝑑p 𝑑𝑡 = - ∑ k̸ =0 2k [ 𝑢 2 k ⟨ bk𝑝+k (𝑡) b † k𝑝+k (𝑡) ⟩ + 𝑣 2 k ⟨ b † k𝑝-k (𝑡) bk𝑝-k (𝑡) ⟩ -𝑢 k 𝑣 k ⟨ bk𝑝+k (𝑡) bk𝑝-k (𝑡) ⟩ -𝑢 k 𝑣 k ⟨ b † k𝑝+k (𝑡) b † k𝑝-k (𝑡) ⟩ ] . (3.32)
We then use the solution of Eq. (3.22) for Bogoliubov operators to finally obtain a set of coupled semiclassical equations providing the trajectory of the polaron :

𝑑x 𝑑𝑡 = (p -Π) 𝑀 (3.33) 𝑑p 𝑑𝑡 = -2Γ ∑ k̸ =0 k𝑉 2 k ( 𝐸 b + 2 k 2 2𝑀 -k • (p-Π) 𝑀 ) 2 + Γ 2 { 1 -2 cos [( 𝐸 b + 2 k 2 2𝑀 + k • (Π -p) 𝑀 ) 𝑡 ] 𝑒 -Γ𝑡 + 𝑒 -2Γ𝑡 } .
(3.34) 3.3. Results

Results

Effective mass of the polaron

We present, first, the results for the polaron effective mass. They are obtained by using the numerically evaluated self-consistent solution of Eq. ( 3.26) at small values of impurity momentum 𝑝, from which the effective mass is extracted according to its definition in Eq. (3.27)1 . In the current and following subsections we present our results in a quasi-one dimensional geometry, as illustrated in Figure 3.1, but the equations that we derived are also valid in higher dimensions. Also, for all further results we will take the value of interaction constant, (1/2𝜋𝑛𝜉)(𝑔 IB /𝑔) 2 , with 𝜉 = / √ 𝑚𝑔𝑛 being the healing length, equal to 0.2. This ensures we stay in the regime of weak impurity-fluid interactions, as assumed in Sec.3.1. As shown in Figure 3.2a), upon increasing the detuning ∆ we observe a nonmonotonous behaviour of the effective mass. This can be understood as being related to the different nature of the Bogoliubov excitations at varying ∆. The various possible regimes are illustrated in Figure 3.2b). When ∆ ≤ 0 (region I in the figure) the excitation spectrum is gapped and becomes gapless and phononic at ∆ = 0. The latter regime is reminiscent of the Bogoliubov spectrum for equilibrium condensates albeit with dissipation. When ∆ > 0 the excitation spectrum is diffusive with purely imaginary eigenenergies in some range of wavevectors that depends on the ratio ∆/𝑔𝑛. If ∆/𝑔𝑛 < 1 (region II) the imaginary values lie in the region |k| < √ 2𝑚∆ with the maximum of the imaginary part at k = 0, while if ∆/𝑔𝑛 > 1 (region III) the diffusive regime is at finite k wavevectors.

In all the above cases, the polariton condensate is dynamically stable as long as the imaginary part of its excitation spectrum is smaller than zero [START_REF] Wouters | Parametric Oscillation Threshold of Semiconductor Microcavities in the Strong Coupling Regime[END_REF]Carusotto and Ciuti, 2013), since the presence of a positive imaginary part leads to a dynamical instability, occurring when Im[E b ] > 0. This condition corresponds to an exponential growth in time of the population of excitation modes and yields a rapid depletion of the polariton condensate. The presence of dissipation protects the system against such instabilities as long as

Γ > Im[E b ].
The failure to meet this condition leads to the breakdown of Bogoliubov theory, and our theory is not valid for such instances (as depicted by the shaded gray area in Figure 3.5). Hence, it is remarkable that driven-dissipative quantum fluids can reach a wealth of dynamical regimes not accessible by their equilibrium counterpart. This has a direct impact on the polaron effective mass: the way the fluid excitations dress the impurity depends on the dynamical regime. We find that the largest effective mass occurs close to the sonic case, where a large density of excitations can be generated by the impurity for vanishing energetic costs. Another peculiar regime found in this driven-dissipative quantum fluids is the one where the effective mass Chapter 3. Bose Polaron in a quantum fluid of light is smaller than the bare one: this occurs when the spectrum is diffusive at finite wavevectors. In this case the impurity feels a negative drag (see next subsection) and behaves as an effectively lighter particle. For positive values of ∆ we observe a region of negative drag force. This extends the results predicted by [START_REF] Van Regemortel | Negative drag in nonequilibrium polariton quantum fluids[END_REF] to the case of an impurity with finite mass. This regime corresponds to the case when the effective mass is smaller than bare one. For negative values of ∆ the drag force is non-zero and positive, showing an important increase starting from the case where the fluid velocity exceeds the Landau critical velocity, estimated as 𝑣 𝑐 = min k 𝐸 𝑏 (k)/|k| and marked by the red line in Figure 3.3. Our analysis hence shows that the impurity acts as a 'test particle' in the fluid to probe its superfluid properties, which strongly depend on the dynamical regime of the fluid.

Drag force

Results

Polaron dynamics

We finally follow the semiclassical dynamics of the polaron moving with a finite initial momentum in a fluid at rest, obtained by numerically solving the coupled differential equations in Eqs. (3.33) and (3.34). The results are summarized in Figure 3.4.

The key quantity to follow in order to determine the impurity dynamics is the excitation momentum Π. When Π reaches zero, it means that the impurity has reached its terminal velocity: it is not dressed by excitations anymore, and its effective mass equals the bare mass. Different kind of trajectories are shown in Figure 3.4 a). The arrows indicate the impurity momentum evolution in time, till it reaches a terminal value (black dots in the figure). Note that at long times, we have checked that the relation 𝑑𝑝/𝑑𝑡 = -𝐹 𝑑𝑟𝑎𝑔 (𝑝) is valid and helps understand these trajectories. For negative ∆ the drag force is positive, the impurity decelerates and the terminal momentum is always zero regardless of the initial momentum. For ∆ > 0, the situation is more exotic: above a certain positive ∆, the terminal momentum 𝑝 = 0 becomes unstable with respect to fluctuations of Π, and two nonzero terminal momenta of opposite sign become possible (cf. the bifurcation in the inset of Figure 3.4 a)). These nonzero terminal momenta result from the negative drag regime in which the impurity is accelerated. Note that this trajectory does not violate energy conservation in this driven-dissipative situation: the energy flux constantly traversing the polariton fluid (constituted by the drive and losses) provides the energy that such trajectories require.

In Figure 3.4b) and Figure 3.4c) we show the polaron momentum as a function of time 𝑡 and detuning ∆ for fixed initial momentum, smaller or larger than the terminal value. Two regimes clearly emerge from this analysis: for negative detuning the polaron decelerates till a final rest position, while for positive detuning, in the regime of negative drag, the impurity of given initial momentum accelerates till a terminal momentum is reached.

Finally, we analyse the influence of losses on the polaron dynamics for positive ∆. The results are summarized in Figure 3.5, where we show the terminal velocity reached by polarons as a function of the detuning ∆ and of the dissipation constant Γ. We see that the region of non-zero terminal velocity caused by the negative drag regime disappears upon increasing Γ. For increasing Γ we also see that the lower boundary of this region increases to compensate for the increased losses.

Conclusions and outlook

We have studied the motion of an impurity in a polariton fluid under drive and dissipation, assuming a weak coupling between the impurity and the fluid. The presence of Bogoliubov excitations lying on top of the coherent steady state of the polariton fluid dress the impurity particle giving rise to a Bose polaron in the Fröhlich regime. We have determined the polaron effective mass, the drag force acting on the impurity, as well as polaron trajectories at semiclassical level.

We have found different dynamical regimes, originating from the unique features of the excitation spectrum of driven-dissipative polariton fluids. We have shown that it is possible to tune polaron effective mass to values both smaller and larger than the bare one by adjusting the detuning ∆. In the ∆ > 0 regime of diffusive excitation spectrum, for specific ∆ values corresponding to the case of finite terminal momentum in Figure 3.5, the impurity is subjected to an instantaneous negative drag force: as a result, the impurity rest position is unstable and counter-intuitively, it starts accelerating against the flow until it reaches a non-zero terminal velocity. This work shows that the impurity dynamics can be used as a test particle to probe the different regimes of non-equilibrium quantum flow, including superfluidity, in quantum fluids of light.

As indicated by our analysis of the coupling with the electromagnetic vacuum bath outside the cavity, corrections beyond the Markov approximation (see [START_REF] Khan | Quantum Dynamics of a Bose Polaron in a 𝑑-Dimensional Bose-Einstein Condensate[END_REF] for an approximate treatment) could lead to experimentally relevant non-trivial corrections of the dynamics, that would be interesting to examine. Another open direction is to go beyond the semiclassical description of the impurity trajectories, and beyond the weak impurity-fluid interaction regime.

4

Background on TMDC monolayers I present a small background on the properties of monolayers of Transition Metal Dichalcogenide (TMDC), focusing especially on their optical properties and the related excitonic effects. In the first section, TMDC monolayers are introduced giving a glimpse of their crystal structure and electronic bandstructure followed by description of the excitonic effects in the next section. Finally, in the last section I briefly note the first experiments carried in the strong light-matter coupling regime of TMDC excitons and cavity photons.

What are TMDC monolayers?

The recently emerged class of bulk crystals which are made up of stacks of two dimensional (2D) layers of atomic thickness and bonded together by weakly attractive van der Waals interaction [START_REF] Novoselov | Two-Dimensional Atomic Crystals[END_REF] can be separated into individual layers -monolayers -using mechanical exfoliation [START_REF] Frindt | Single Crystals of MoS 2 Several Molecular Layers Thick[END_REF][START_REF] Novoselov | Electric Field Effect in Atomically Thin Carbon Films[END_REF]H. Li et al., 2013) or chemical vapour deposition [START_REF] Elías | Controlled Synthesis and Transfer of Large-Area WS2 Sheets: From Single Layer to Few Layers[END_REF][START_REF] Van Der Zande | Grains and Grain Boundaries in Highly Crystalline Monolayer Molybdenum Disulphide[END_REF][START_REF] Chhowalla | The Chemistry of Two-Dimensional Layered Transition Metal Dichalcogenide Nanosheets[END_REF][START_REF] Lv | Transition Metal Dichalcogenides and Beyond: Synthesis, Properties, and Applications of Single-and Few-Layer Nanosheets[END_REF][START_REF] Manzeli | 2D Transition Metal Dichalcogenides[END_REF]. This family of 2D monolayer materials have been a topic of intensive research since the dawn of this millennium -kicking off with the discovery of graphene [START_REF] Novoselov | Electric Field Effect in Atomically Thin Carbon Films[END_REF]) -because they possess extraordinary physical properties compared to their bulk counterpart [START_REF] Castro Neto | The Electronic Properties of Graphene[END_REF][START_REF] Geim | Graphene: Status and Prospects[END_REF][START_REF] Splendiani | Emerging Photoluminescence in Monolayer MoS2[END_REF][START_REF] Lee | Atomically Thin p-n Junctions with van Der Waals Heterointerfaces[END_REF][START_REF] Eda | Photoluminescence from Chemically Exfoliated MoS2[END_REF][START_REF] Wang | Electronics and Optoelectronics of Two-Dimensional Transition Metal Dichalcogenides[END_REF][START_REF] Mak | Photonics and Optoelectronics of 2D Semiconductor Transition Metal Dichalcogenides[END_REF][START_REF] Berkelbach | Optical and Excitonic Properties of Atomically Thin Transition-Metal Dichalcogenides[END_REF]. The amazing aspect of these materials is that their properties can be extensively tuned using doping, strain, external fields and environmental effects, and they can be easily integrated into existing systems and devices [START_REF] Bonaccorso | Graphene Photonics and Optoelectronics[END_REF][START_REF] Grigorenko | Graphene Plasmonics[END_REF][START_REF] Xia | Two-Dimensional Material Nanophotonics[END_REF] to enhance their efficiency and performance.

Since the discovery of graphene new 2D materials have been added to the monolayer family, which now also includes hexagonal boron nitride, silicene [START_REF] Vogt | Silicene: Compelling Experimental Evidence for Graphenelike Two-Dimensional Silicon[END_REF], germanene [START_REF] Dávila | Germanene: A Novel Two-Dimensional Germanium Allotrope Akin to Graphene and Silicene[END_REF], black phosphorous [START_REF] Xia | Rediscovering Black Phosphorus as an Anisotropic Layered Material for Optoelectronics and Electronics[END_REF], borophene [START_REF] Mannix | Synthesis of borophenes: Anisotropic, two-dimensional boron polymorphs[END_REF] and of course transition metal dichalcogenides or TMDCs for short. The atomic unit of a monolayer TMDC is a group VI transition metal and two chalcogen atoms with the chemical formula 𝑀 𝑋 2 . They can be metallic or semiconducting depending on the metal or chalcogen atom. We are interested in the semiconducting group where the transition metal 𝑀 is either molybdenum (Mo) or tungsten (W) and the chalcogen 𝑋 is sulfur (S), selenium (Se), or tellurium (Te)1 . In contrast to the semimetallic character -no energy band gapof graphene, TMDCs display a direct energy band gap with absorption in the nearinfrared to the visible region of the electromagnetic spectrum. The presence of an energy band gap opens up new avenues for the application of TMDCs in photonics and optoelectronics.

Electronic Bandstructure

TMDCs are composed of three atomic planes consisting of a single layer of transition metal atoms sandwiched between two layers of chalcogen atoms in the trigonal prismatic structure [START_REF] Mattheiss | Band Structures of Transition-Metal-Dichalcogenide Layer Compounds[END_REF] (see Figure 4.1). They have out-of-plane mirror symmetry and broken in-plane inversion symmetry. Stacking monolayers in the The bulk TMDC is an indirect band gap semiconductor, but in the monolayer limit they alter to become a direct bandgap semiconductor (T. [START_REF] Li | Electronic Properties of MoS2 Nanoparticles[END_REF] (see Figure 4.2). This effect can be understood as a result of an increased indirectgap size due to the significant quantum confinement effect in the out-of-plane direction, whereas the direct gap at the K and the K' points remains mostly unaffected [START_REF] Berkelbach | Optical and Excitonic Properties of Atomically Thin Transition-Metal Dichalcogenides[END_REF]. This crossover from bulk to monolayer is supported by calculations using Density Functional Theory (DFT) [START_REF] Zhu | Giant Spin-Orbit-Induced Spin Splitting in Two-Dimensional Transition-Metal Dichalcogenide Semiconductors[END_REF] and GW approximation [START_REF] Lebègue | Electronic Structure of Two-Dimensional Crystals from Ab Initio Theory[END_REF][START_REF] Cheiwchanchamnangij | Quasiparticle Band Structure Calculation of Monolayer, Bilayer, and Bulk MoS 2[END_REF][START_REF] Ramasubramaniam | Large Excitonic Effects in Monolayers of Molybdenum and Tungsten Dichalcogenides[END_REF][START_REF] Shi | Quasiparticle Band Structures and Optical Properties of Strained Monolayer MoSe 2 and WS 2[END_REF][START_REF] Qiu | Optical Spectrum of MoS 2 : Many-Body Effects and Diversity of Exciton States[END_REF] and has been experimentally observed in 𝑀 𝑜𝑆 2 [START_REF] Jin | Photon Solid Phases in Driven Arrays of Nonlinearly Coupled Cavities[END_REF] and 𝑀 𝑜𝑆𝑒 2 (Y. [START_REF] Zhang | Direct Observation of the Transition from Indirect to Direct Bandgap in Atomically Thin Epitaxial MoSe2[END_REF].

The low energy bandstructure around the K and K' valleys can be modelled by an effective 2D massive Dirac Hamiltonian under the Tight Binding approximation with the spin and valley pseudospin degree of freedom (G.-B. [START_REF] Liu | Three-Band Tight-Binding Model for Monolayers of Group-VIB Transition Metal Dichalcogenides[END_REF][START_REF] Wang | Colloquium: Excitons in Atomically Thin Transition Metal Dichalcogenides[END_REF]. The spin-orbit interaction and the broken inversion symmetry breaks the spin degeneracy at each valley resulting in the splitting of the valence band (Xiao et al., 2012), but still the spin degeneracy is preserved at the conduction band2 . This splitting in the valence band manifests as two prominent transitions in the optical spectra, the so-called A and B excitons; A exciton is the more significant one -and the one of our interest -because it is at a lower energy compared to the B exciton. This results in the locking of the spin and valley degrees of freedom which allows us to selectively excite carries in one of the two valleys using circularly polarized light [START_REF] Zeng | Valley Polarization in MoS2 Monolayers by Optical Pumping[END_REF][START_REF] Mak | Control of Valley Polarization in Monolayer MoS2 by Optical Helicity[END_REF]. It means that the absorption of left (𝜎 + ) For further information on the history of 2D materials particularly TMDCs, their fabrication and synthesis, and further physical properties we refer the reader to the existing reviews on these topics (Q. H. [START_REF] Wang | Electronics and Optoelectronics of Two-Dimensional Transition Metal Dichalcogenides[END_REF][START_REF] Butler | Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene[END_REF][START_REF] Manzeli | 2D Transition Metal Dichalcogenides[END_REF][START_REF] Mak | Photonics and Optoelectronics of 2D Semiconductor Transition Metal Dichalcogenides[END_REF][START_REF] Berkelbach | Optical and Excitonic Properties of Atomically Thin Transition-Metal Dichalcogenides[END_REF] and the references therein. In the next section we'll mainly concentrate on their strong excitonic effects.

Excitonic effects

Strong excitonic effects in TMDCs (G. [START_REF] Wang | Colloquium: Excitons in Atomically Thin Transition Metal Dichalcogenides[END_REF] originates from the reduced dimensionality -which leads to quantum size effects -and the associated reduced dielectric screening of the Coulombic interaction between the electron and hole quasiparticles. It determines the strength of light-matter interactions, the radiative recombination lifetime of electron-hole (e-h) pairs and the material's optoelectronic response.

Screening of Coulomb interaction

Screening of the Coulomb interaction is an important ingredient in calculating the dielectric response of a material, which generally involves using the complete diagrammatic many theory [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF] within the Random Phase approximation (RPA) [START_REF] Hybertsen | Electron Correlation in Semiconductors and Insulators: Band Gaps and Quasiparticle Energies[END_REF]. On the other hand it is possible to drive an approximate screened interaction based on classical electrostatic arguments [START_REF] Schmitt-Rink | Linear and Nonlinear Optical Properties of Semiconductor Quantum Wells[END_REF].

The appropriate electrostatic problem is that of a dielectric slab of width 𝑑 and dielectric constant 𝜖, surrounded by uniform media with dielectric constants 𝜖 1 and 𝜖 2 above and below the slab (see Figure 4.3). It was shown in [START_REF] Rytova | The screened potential of a point charge in a thin film[END_REF][START_REF] Stern | Polarizability of a Two-Dimensional Electron Gas[END_REF][START_REF] Stern | Properties of Semiconductor Surface Inversion Layers in the Electric Quantum Limit[END_REF][START_REF] Keldysh | Coulomb Interaction in Thin Semiconductor and Semimetal Films[END_REF] during 1960s and 70s that when the dielectric mismatch between the environment and the material 𝜖 -𝜖 𝑛 (𝑛 = 1, 2) is large and 𝑘𝑑 ≪ 1 -where 𝑘 is the wavevector of the response laser beam -then the dielectric function and the associated real space potential 𝑊 (𝑟) between two charged particles 𝑞 1 and 𝑞 2 separated by a distance 𝑟 in the slab of the material can be written respectively as

𝜖(𝑘) ≈ 𝜖 1 + 𝜖 2 2 (1 + 𝑟 0 𝑘) (4.1) 𝑊 (𝑟) ≈ 𝜋𝑞 1 𝑞 2 𝑑𝜖 [ 𝐻 0 ( 𝑟 𝑟 0 ) -𝑌 0 ( 𝑟 𝑟 0 )] (4.2)
where 𝑟 0 is the 2D screening length equal to 𝜖𝑑/(𝜖 1 + 𝜖 2 ), 𝐻 0 is the Struve and 𝑌 0 the Bessel function of the second kind. For TMDCs, 𝜖 ≈ 14 and 𝑑 ≈ 6 Å [START_REF] Berkelbach | Theory of Neutral and Charged Excitons in Monolayer Transition Metal Dichalcogenides[END_REF]. If, for instance, the surrounding dielectric medium is vacuum then, the 2D screening length 𝑟 0 gives us a crossover scale between the 1/𝑟 form of Coulomb potential at large 𝑟 and a weaker log(𝑟) form of the potential 𝑊 (𝑟) at small inter-particle separation. Qualitatively the difference between the electrostatic potential at larger and smaller inter-particle separation can be understood as follows: The electric field lines between the particles, in general, pass through both the surrounding dielectrics and the thin 2D layer, but as the inter-particle interaction starts to increase the number Chapter 4. Background on TMDC monolayers of field lines passing through the 2D material decrease in number and eventually their number can be considered negligible compared to the amount of lines in the surroundings and if the surroundings on top and bottom are same we recover the Coulombic potential.

The validity of the above approximate approach to calculate the effective Coulomb interaction, when compared to the first principle calculations utilizing DFT and RPA, was carried out by Latini et al. [START_REF] Latini | Excitons in van Der Waals Heterostructures: The Important Role of Dielectric Screening[END_REF] who found deviations at length scales below 1 Å (𝑘 → ∞). Since the characteristic size of excitons in TMDCs is at least an order of magnitude larger than this length scale (see next subsection), this model based on classical electrodynamical arguments is highly justified.

Non-hydrogenic exciton states in TMDCs

Figure 4.4: The transition energies for the exciton s-states with as a function of the quantum number 𝑛. We can see that for the 1s and 2s exciton states the energies deviate from the hydrogenic Rydberg series. They match for higher excited levels with the hydrogenic model because their Bohr radius is larger compared to the 1s and 2s state; hence the electrostatic potential recovers the Coulombic form -which implies a hydrogenic Rydberg series on wards. The binding energies for the exciton states can be obtained as the difference between the band gap energy and their corresponding transition energy. Figure from [START_REF] Chernikov | Exciton Binding Energy and Nonhydrogenic Rydberg Series in Monolayer WS 2[END_REF] The reduced screening of the Coulomb interaction because of the two dimensional nature of TMDCs leads to a stronger attraction between the electrons and holes which ultimately leads to formation of various types exciton states which dominate its optical response spectrum. For TMDCs they have been numerically calculated using GW and Bethe-Salpeter approaches [START_REF] Qiu | Optical Spectrum of MoS 2 : Many-Body Effects and Diversity of Exciton States[END_REF] and semianalytically [START_REF] Wu | Exciton Band Structure of Monolayer ${\mathrm{MoS}}_{2}[END_REF] using the above form of Coulombic screening. As 4.2. Excitonic effects we saw in Chapter 1, that for the treatment of excitons in semiconductor quantum wells (QWs), we can write an effective mass Schrödinger equation (1.30) -which in essence is the 2D rescaled version of the Schrödinger equation for Hydrogen atomto get the Rydberg series of excitons and their binding energies. The same effective mass equation

( - 2 2𝜇 ∇ 2 r + 𝑊 (r) -𝐸 𝑔 ) Ψ 𝑛𝑙 (r) = 𝐸 𝑛𝑙 Ψ 𝑛𝑙 (r) , (4.3)
can also be used for the description of excitons in semiconducting 2D monolayers.

Here 𝜇 is the effective mass of the exciton, 𝑊 (r) is the screened Coulomb interaction, 𝐸 𝑔 is the energy bandgap and Ψ 𝑛𝑙 is the exciton wavefunction with principal quantum number 𝑛 and azimuthal quantum number 𝑙3 like TMDCs [START_REF] Cudazzo | Dielectric Screening in Two-Dimensional Insulators: Implications for Excitonic and Impurity States in Graphane[END_REF][START_REF] Berkelbach | Theory of Neutral and Charged Excitons in Monolayer Transition Metal Dichalcogenides[END_REF] with the added caveat that the screened Coulomb interaction term is not like in QWs -where its is -𝑒 2 /𝜖𝑟 -but the one of Eq. (4.2). The Rydberg series thus obtained is non-hydrogenic [START_REF] Chernikov | Exciton Binding Energy and Nonhydrogenic Rydberg Series in Monolayer WS 2[END_REF], as a consequence of which the binding energies [START_REF] He | Tightly Bound Excitons in Monolayer WSe 2[END_REF][START_REF] Ugeda | Giant Bandgap Renormalization and Excitonic Effects in a Monolayer Transition Metal Dichalcogenide Semiconductor[END_REF] of the first few terms in the series strongly deviate from the hydrogenic form that we obtained in Chapter 1 Eq. (1.30), as can be seen in Figure 4.4 for the case of 𝑊 𝑆 2 . [START_REF] Chernikov | Exciton Binding Energy and Nonhydrogenic Rydberg Series in Monolayer WS 2[END_REF].

Strong light-matter coupling

Liu et al (X. [START_REF] Liu | Strong Light-Matter Coupling in Two-Dimensional Atomic Crystals[END_REF] and Dufferwiel et al [START_REF] Dufferwiel | Exciton Polaritons in van Der Waals Heterostructures Embedded in Tunable Microcavities[END_REF] were the first to achieve a strong light-matter coupling between the excitonic states in a TMDC and photons in a microcavity. The characteristic anti-crossing signifying the formation of exciton-polaritons observed in X. [START_REF] Liu | Strong Light-Matter Coupling in Two-Dimensional Atomic Crystals[END_REF] is shown in Figure 4.5. They observed an order of magnitude larger Rabi splitting compared to the one observed in usually employed quantum wells of inorganic semiconductors, which opens a new possibility to achieve polariton condensates at room temperatures using the excitonic states present in TMDCs.

Polariton Interactions in MoSe 2

In this chapter, I'll present the work that we performed in experimental collaboration with the group of Dr. Maxime Richard at Néel Institute in Grenoble. We saw in the last chapter that the excitonic states in TMDCs are non-hydrogenic, because of which excitonmediated optical nonlinearities are predicted to be enhanced as compared to their hydrogenic counterpart. To quantify this enhancement, our experimental collaborators performed an optical transmission spectroscopy of a MoSe 2 monolayer placed in the strong coupling regime with the mode of an optical microcavity, and we analyzed the results quantitatively with a nonlinear input-output theory. In the subsequent sections, I'll present the data provided by the experiment and the numerical estimation of the nonlinear constants. Finally, I'll provide avenues for future research work.

Polariton resonances from Experiment

Our collaborators at Néel Institute in a team led by Dr. Maxime Richard provided us with nonlinear transmission measurements showing the two polariton resonances, which they observed by putting a MoSe 2 monolayer in the strong coupling regime with the resonance of the microcavity (see Figure 5.1 for a structural overview of the system and the characteristic anti-crossing of the polariton modes). They used a pulsed Ti:sapphire laser that delivered ∼ 200 fs pulses with a spectrum 𝐼 las (𝜔) of tunable mean energy 𝜔 las = [1640, 1660] meV and a bandwidth 𝛾 las ∼ 10 meV. Its purpose was the ultrafast creation of a dense polariton population resonantly, without overheating the sample, and to perform a broadband transmission spectrum measurement which was able to capture both the lower and upper polaritons. The beam was prepared into a Gaussian mode which was focused on the microcavity surface into a 𝜎 = 5.8 𝜇m waist size spot. They used a quarter-wave plate to tune the laser polarization among the states |𝜃⟩, where |𝜃⟩ = sin(2𝜃)/ √ 2|𝑥⟩ + [𝑖 + cos(2𝜃)] / √ 2|𝑦⟩, |𝑥, 𝑦⟩ is the linear polarization basis oriented, and 𝜃 is the wave plate rotation angle with respect to 𝑦. To get the first set of measurements, they used the 𝑦-polarized light (𝜃 = 0). The time-integrated transmitted light intensity 𝐼 𝑇 was collected with a microscope objective, and imaged at the entrance focal plane of a grating spectrometer. By doing so, they obtained the space and frequency-resolved transmission spectra 𝑇 (𝜔, 𝑦) =

Polariton resonances from Experiment

𝐼 𝑇 (𝜔, 𝑦)/𝐼 las (𝜔, 𝑦). Since our aim is to provide a quantitative estimate of the interactions, they also provided us with transmission spectrum measurements for pulses with different electromagnetic energies 𝑊 , which they measured from 𝑊 0 = 0.1 pJ, which is well below the onset of the nonlinear (interacting) regime, up to several hundreds of pJ, which is well above. One such set of experimental data, at temperatures 𝑇 = 127 K and 𝑇 = 105 K are shown in Figure 5.2. The excitonic fraction of the polariton field is |𝑋| 2 = 0.48 and |𝑋| 2 = 0.33, respectively. The plotted transmission spectra are normalized to their maximum 𝑇 𝑚 for clearer representation. To change the excitonic fractions they exploited the fact that the excitonic transition energy is temperature-dependent to control the detuning ∆(𝑇 ) = 𝜔 𝑐,0 -𝜔 𝑥 (𝑇 ) between the bare cavity (frequency 𝜔 𝑐,0 ) and the excitonic level (frequency 𝜔 𝑥 ), and hence the excitonic fraction |𝑋| 2 (|𝐶| 2 = 1 -|𝑋| 2 ) of the lower (upper) polariton states [START_REF] Boldt | Many-Body Theory for the Dense Exciton Gas of Direct Semiconductors II. Calculation of Exciton Level Shift and Damping in Dependence on Exciton Density[END_REF]. The effective polariton-polariton interaction constant is thus varied as it depends on |𝑋| 2 (see Chapter 1). The laser pulse spectral overlap with the polariton modes was also different in the two sets. We take advantage of these variations to test the robustness of our quantitative estimate of 𝑔 𝑠 and 𝑔 𝑥 , as they should not depend on these parameters.

Interpretation of experimental data

In the linear regime (lowest laser pump energy 𝑊 = 0.11 pJ), Figure 5.2 shows both the upper and lower polariton resonances, with a mostly equal weight for 127 K and with a dominant lower polariton peak at 𝑇 = 105 𝐾. This is consistent with their respective photonic fraction. Two smaller peaks are also visible in these spectra at 𝜔 = 1625.6 meV and 𝜔 = 1660.4 meV that our collaborators traced back, by real space analysis, to bare cavity resonances situated within the small gap separating the MoSe 2 monolayer from the bilayer. For this reason, we didn't include them in our theoretical analysis (see next section). Upon increasing the pump energy 𝑊 , the polaritonic resonances exhibit a clear and consistent trend: at moderate 𝑊 , the lower polariton peak blueshifts, while the upper polariton essentially does not. This difference in behaviour is the key to distinguishing between the contributions of 𝑔 𝑠 and 𝑔 𝑥 to the nonlinearity. Indeed, while Coulomb interaction contributes to blueshift both lower and upper polaritons, the saturation causes a reduction of the effective Rabi spitting, and thus shifts the lower and upper polaritons in opposite directions. The trend thus observed indicates that the saturation contributes significantly to the nonlinearity, which is consistent with recent reports [START_REF] Gu | Enhanced Nonlinear Interaction of Polaritons via Excitonic Rydberg States in Monolayer WSe 2[END_REF].

In the next section we'll explain these observed spectra by a numerical simulation of the polariton field ultrafast evolution, including the shape of the laser pulse in time and space. Specifically, we'll derive a mean-field input output theory in the exciton-photon basis -including exciton-exciton interaction and saturation at firstorder in the interaction strength -and using this theory we'll optimize the values of nonlinear constants 𝑔 𝑠 and 𝑔 𝑥 that fit best -in the sense of least square error fitting -with the observed experimental data. The optimised values of the nonlinear constants provided us with a quantitative estimate of 𝑔 𝑠 and 𝑔 𝑥 .

5.2 Numerical determination of g𝑥 (0) and g𝑠 (0)

Mean field Input Output Theory

We describe the dynamics of the system by the strongly-coupled Exciton-Photon Hamiltonian, including the spin anisotropic two-body contact interactions between excitons 𝑔 𝑥 and the saturation interaction due to the finite exciton oscillator strength 𝑔 𝑠 . The total Hamiltonian reads Ĥ = Ĥlin + Ĥint .

(5.1) 5.2. Numerical determination of g𝑥 (0) and g𝑠 (0)

Here Ĥlin is the linear Hopfield part which is written as

Ĥlin = ∫ 𝑑r ∑ 𝜎 [ ψ † x,𝜎 (r) ( 𝜔 x ) ψx,𝜎 (r) + ψ † c,𝜎 (r) ( 𝜔 c,0 - 2 ∇ 2 2𝑚 c ) ψc,𝜎 (r) + Ω 2 ( ψ † x,𝜎 (r) ψc,𝜎 (r) + h.c. ) ] , (5.2)
and Ĥint is the non-linear interaction part which is given as

Ĥint = ∫ 𝑑r ∑ 𝜎 [ ( 𝑔 𝑥,‖ 2
) ψ † x,𝜎 (r) ψ † x,𝜎 (r) ψx,𝜎 (r) ψx,𝜎 (r) - ( 𝑔 𝑠 2 ) ( ψ † c,𝜎 (r) ψ † x,𝜎 (r) ψx,𝜎 (r) ψx,𝜎 (r) + h.c. ) ] + 𝑔 𝑥,⊥ ψ † x,+ (r) ψ † x,-(r) ψx,-(r) ψx,+ (r).
(5.3)

Here ψ𝛼,𝜎 are the exciton (𝛼 = x) and cavity photon (𝛼 = c) field operators with circular polarization 𝜎 = {+, -} respectively, satisfying bosonic commutation relations, 𝜔 c,0 is the cavity photon frequency at vanishing in-plane wavevector 𝑘 ‖ = 0 and 𝑚 c its effective mass. 𝜔 x is the excitonic transition frequency, of which we neglect the kinetic contribution within the light cone. The excitonic level and cavity resonance are taken as polarization-isotropic. Ω is the Rabi splitting, 𝑔 𝑠 is the saturation interaction constant, 𝑔 𝑥,‖ and 𝑔 𝑥,⊥ are the Coulomb interaction constants between exciton of parallel and opposite spin respectively1 . Next, we derive the Heisenberg equations of motion as 𝑖 𝜕 𝑡 ψ𝛼,𝜎 = [ ψ𝛼,𝜎 , Ĥ], and take the mean field approximation ⟨ ψ𝛼,𝜎 ⟩ = 𝜓 𝛼,𝜎 . Using the input-output theory [START_REF] Verger | Polariton Quantum Blockade in a Photonic Dot[END_REF]Carusotto and Ciuti, 2013) to include the pump and losses, the equations of motion read

𝑖𝜕 𝑡 𝜓 c,+ = ( 𝜔 c,0 -2𝑚 ∇ 2 -𝑖 𝛾 𝑐 2 ) 𝜓 c,+ + ( Ω 2 - 𝑔 𝑠 2 |𝜓 x,+ | 2 ) 𝜓 x,+ + √ 2𝛾 in 𝐴 in,+ (5.4
)

𝑖𝜕 𝑡 𝜓 x,+ = ( 𝜔 x -𝑖 𝛾 𝑥 2 + 𝑔 𝑥,‖ |𝜓 x,+ | 2 + 𝑔 𝑥,⊥ |𝜓 x,-| 2 ) 𝜓 x,+ + ( Ω 2 -𝑔 𝑠 |𝜓 x,+ | 2 ) 𝜓 c,+ - 𝑔 𝑠 2 𝜓 2 x,+ 𝜓 * c,+
(5.5)

𝑖𝜕 𝑡 𝜓 c,-= ( 𝜔 c,0 -2𝑚 ∇ 2 -𝑖 𝛾 𝑐 2 ) 𝜓 c,-+ ( Ω 2 - 𝑔 𝑠 2 |𝜓 x,-| 2 ) 𝜓 x,-+ √ 2𝛾 in 𝐴 in,- (5.6
)

𝑖𝜕 𝑡 𝜓 x,-= ( 𝜔 x -𝑖 𝛾 𝑥 2 + 𝑔 𝑥,‖ |𝜓 x,-| 2 + 𝑔 𝑥,⊥ |𝜓 x,+ | 2 ) 𝜓 x,-+ ( Ω 2 -𝑔 𝑠 |𝜓 x,-| 2 ) 𝜓 c,- Chapter 5. Polariton Interactions in MoSe 2 - 𝑔 𝑠 2 𝜓 2 x,-𝜓 * c,-, (5.7)
where 𝐴 in,± (r, 𝑡) describes the incident laser pulse field density of total energy integrated over the pulse 𝑊 = ∫∫∫ d𝑡 d𝑥 d𝑦 |𝐴 in |2 , 𝛾 𝑐 = 𝛾 in + 𝛾 out is the cavity radiative decay rate, and 𝛾 in (𝛾 out ) are the coupling rate of the laser into the cavity (of the cavity into the outside detection channel) and finally 𝛾 𝑥 provides the excitonic dissipation.

In order to describe the experimental conditions, we rewrite these equations in the |𝜃, θ⟩ basis where 𝜃 is the rotation angle of the quarter waveplate with respect the linear polarization |𝑦⟩ which is that of the laser output. Algebraic manipulation leads to the following basis transformation

|𝜃⟩ = 1 2 (𝑖 + 𝑒 𝑖2𝜃 )|+⟩ - 1 2 (𝑖 + 𝑒 -𝑖2𝜃 )|-⟩ (5.8) | θ⟩ = 1 2 (-𝑖 + 𝑒 𝑖2𝜃 )|+⟩ + 1 2 (-𝑖 + 𝑒 -𝑖2𝜃 )|-⟩, (5.9)
which allows us rewriting the equations of motion, written above, in the (𝜃, θ) basis.

In this transformation, we use the fact that in the experiment, the source term 𝐴 𝑖𝑛, θ=0. Moreover, each nonlinear term involves products of three fields of the form 𝜓 𝜃, θ𝜓 * 𝜃, θ𝜓 𝜃, θ, such that none of them can serve as auxiliary source terms for the θ components of the field, as long as its initial amplitude is zero. As a result, only the nonlinear terms of the form 𝜓 𝜃 𝜓 * 𝜃 𝜓 𝜃 are nonzero, and we can drop the two equations of motions describing the θ-polarized field. We thus obtain the equations of motion for the 𝜃-polarized fields as 2

𝑖𝜕 𝑡 𝜓 c,𝜃 = ( 𝜔 c,0 -2𝑚 ∇ 2 -𝑖 𝛾 𝑐 2 ) 𝜓 c,𝜃 + √ 2𝛾 in 𝐴 in,𝜃 + ( Ω 2 - g𝑠 (𝜃) 2 |𝜓 x,𝜃 | 2
) 𝜓 x,𝜃 (5.10) 5.11) where

𝑖𝜕 𝑡 𝜓 x,𝜃 = ( 𝜔 x -𝑖 𝛾 𝑥 2 + g𝑥 (𝜃)|𝜓 x,𝜃 | 2 ) 𝜓 x,𝜃 + ( Ω 2 -g𝑠 (𝜃)|𝜓 x,𝜃 | 2 ) 𝜓 c,𝜃 - g𝑠 (𝜃) 2 𝜓 2 𝑥,𝜃 𝜓 * 𝑐,𝜃 (
g𝑥 (𝜃) = 𝑔 𝑥,‖ + 𝑔 𝑥,⊥ 2 + 𝑔 𝑥,‖ -𝑔 𝑥,⊥ 2 sin 2 (2𝜃) (5.12) g𝑠 (𝜃) = 𝑔 𝑠 2 [ (1 + sin 2 (2𝜃) ] , (5.13)
We finally include in the model the spatial fluctuations -disorder -of the cavity resonance and of the excitonic transition to include the inhomogenities of the TMDC material. To do so, we add the complex-valued potentials 𝑉 𝑐 (r) and 𝑉 𝑥 (r) 5.2. Numerical determination of g𝑥 (0) and g𝑠 (0) to equations (5.10) and (5.11) obtaining

𝑖𝜕 𝑡 𝜓 c,𝜃 = ( 𝜔 c,0 -2𝑚 ∇ 2 -𝑖 𝛾 𝑐 2 + 𝑉 𝑐 (r) ) 𝜓 c,𝜃 + ( Ω 2 - g𝑠 (𝜃) 2 |𝜓 x,𝜃 | 2 ) 𝜓 x,𝜃 + √ 2𝛾 in 𝐴 in,𝜃
(5.14)

𝑖𝜕 𝑡 𝜓 x,𝜃 = ( 𝜔 x -𝑖 𝛾 𝑥 2 + 𝑉 𝑥 (r) + g𝑥 (𝜃)|𝜓 x,𝜃 | 2 ) 𝜓 x,𝜃 + ( Ω 2 -g𝑠 (𝜃)|𝜓 x,𝜃 | 2 ) 𝜓 c,𝜃 - g𝑠 (𝜃) 2 𝜓 2 𝑥,𝜃 𝜓 * 𝑐,𝜃
(5.15)

Since we didn't have a functional form to include the disorder potentials, 𝑉 𝑥 (r) and 𝑉 𝑐 (r) are manually determined from the measured spatially-resolved spectra obtained at T=127K (see Figure 5.3) for each spatial point at very low excitation energy 𝑊 (for which the nonlinear terms are negligible). Note that as long as both the upper and lower polariton peaks are well distinguished in the spectra, 𝑉 𝑥 (r) and 𝑉 𝑐 (r) can be fully determined and distinguished from each other. When only the lower polariton peak is visible, since 𝑉 𝑥 exhibits much lower fluctuation amplitude than 𝑉 𝑐 , we fix Re[𝑉 𝑥 ] to a realistic value obtained in the other cases and then determine Re[𝑉 𝑐 ] from the experimental data.

Results of the Optimization procedure

We numerically solved the model derived above fully accounting for the time profile of the excitation pulse and of the Gaussian shape of the spot in real space. The experimental parameters entering the model are the microcavity and laser characteristics, which are known accurately. The interaction constants g𝑥 (𝜃) and g𝑠 (𝜃), are thus the only free parameters. We first apply this model to the spectra shown in Figure 5.2 (in the 𝜃 = 0 polarization state). g𝑥 (0) = ( 𝑔 𝑥,‖ + 𝑔 𝑥,⊥ ) /2 and g𝑠 (0) = 𝑔 𝑠 /2 are thus derived with their uncertainty by numerical optimization of the fit between the model and the measurements.

In order to determine the best fit of the model with the data, we use an optimization algorithm that minimizes the residues 𝑅 2 between the experimental data and the theoretical predictions of the model, where

𝑅 2 (g 𝑥 , g𝑠 ) = 1 𝑁 -2 𝑁 ∑ 𝑗=1 [𝑇 exp,j -𝑇 th (g 𝑥 , g𝑠 , 𝜔 𝑗 )] 2 ,
(5.16)

𝑇 exp,j is the measured transmission at the frequency 𝜔 𝑗 /2𝜋, and 𝑇 exp,j is the calculated transmission at this frequency. The function 𝑅 2 (g 𝑥 , g𝑠 ) is systematically swept within a reasonable range, and the best fit (g 𝑥,fit , g𝑠,fit ) is determined as the point for which 𝑅 2 (g 𝑥 , g𝑠 ) reaches its minimum 𝑅 2 min . The 1𝜎 uncertainty on the best fit is estimated as

𝛿g 2 = (𝑁 -2)𝑅 2 min 𝑁 ∑ 𝑗=1 [ 𝜕 𝜕g 𝑇 th (g 𝑥,fit , g𝑠,fit , 𝜔 𝑗 ) ] 2 .
(5.17) . This analysis yield g𝑥 (0) = 2.2 ± 1.6 𝜇eV.𝜇m 2 and g𝑠 (0) = 2.16 ± 0.5 𝜇eV.𝜇m 2 for the experiment at 𝑇 = 127 K. The experiment at 𝑇 = 105 K consistently yields g𝑥 (0) = 4.3 𝜇eV.𝜇m 2 , and g𝑠 (0) = 1.6 𝜇eV.𝜇m 2 , albeit with a much larger uncertainty due to the fact that the upper polariton contribution to the spectra is small, and hence prevents determining accurately the relative contribution of g𝑥 (0) and g𝑠 (0). We also derive the excitonic densities (half-width-at-half-maximum in time and space) that increases from 5 × 10 8 cm -2 (𝑊 = 0.11 pJ) to 9 × 10 11 cm -2 (𝑊 = 460 pJ). Note that at high 𝑊 , the saturation effect is large and our model is expected to overestimate it in this regime [START_REF] Rochat | Excitonic Bloch Equations for a Two-Dimensional System of Interacting Excitons[END_REF][START_REF] Emmanuele | Highly Nonlinear Trion-Polaritons in a Monolayer Semiconductor[END_REF] 2020; [START_REF] Combescot | The Many-Body Physics of Composite Bosons[END_REF][START_REF] Kyriienko | Nonlinear Quantum Optics with Trion Polaritons in 2D Monolayers: Conventional and Unconventional Photon Blockade[END_REF]. This is indeed the trend that we observe in the last four spectra in Figure 5.2, in which the theory predicts a slightly smaller Rabi splitting than in the experiment. Yet, except for this feature, the spectral shape and peak energies evolution for increasing 𝑊 are in very good agreement with the experiment.

Spatially-dependent transmission spectrum

We cross-checked this quantitative analysis by looking at another footprint of the nonlinearity: the nontrivial spatially-dependent transmission spectrum 𝑇 (𝑦, 𝜔)/𝑇 𝑚 that results from the interplay between the Gaussian shape of the spot and the nonlinearity. Figure 5.4 shows the lower polariton transmission peak energy 𝐸 lp (𝑦), plotted versus 𝑦, where 𝑦 is the position along a diameter of the laser spot provided by the experimental data. The lowest spectrum (black) is obtained in the linear regime (𝑊 = 0.11 pJ) and thus shows the lower polariton potential 𝑉 (𝑦), from which we derive 𝑉 𝑐 (𝑦). For increasing 𝑊 the nonlinearity changes this shape as the blueshift depends on the local density and excitonic fraction. We can reproduce this behaviour quantitatively with our model, and a good agreement is obtained for g𝑥 (0) = 4.3 +30 -4 𝜇eV.𝜇m 2 , and g𝑠 (0) = 3.2 ± 0.8, 𝜇eV.𝜇m 2 . The large uncertainty reflects the fact that the upper polariton contribution is weak in the dataset, and the relative contributions of 𝑔 𝑥 (0) and 𝑔 𝑠 (0) are hard to distinguish. Yet, the result Chapter 5. Polariton Interactions in MoSe 2 is consistent with the spectral analysis shown above.

Comparison of nonlinear constants in TMDCs

and QW materials As we saw in Chapter 1, the excitonic Bohr radius, in the Hydrogenic exciton (HE) picture depends on the reduced mass 𝜇 and dielectric constant 𝜖 𝑟 as 𝑎 𝐵 = (𝜖 𝑟 /𝜇) × (2𝜋𝜖 0 2 /𝑒 2 ) in SI units, where 𝜖 0 is the vacuum permittivity, and 𝑒 is the electron charge. 𝑔 𝑠 can thus be rewritten as [START_REF] Rochat | Excitonic Bloch Equations for a Two-Dimensional System of Interacting Excitons[END_REF]) (5.18) where 𝐶 0 = 𝜖 2 0 16𝜋 2 4 /(7𝑒 4 ). Using 𝑔 𝑥 = 3 2 /𝜇, 𝑔 𝑠 is thus connected to 𝑔 𝑥 as (5.19) where 𝐶 1 = 𝜖 2 0 16𝜋 2 /(63𝑒 4 ). We can draw two considerations from there. First, while 𝑔 𝑥 depends only on the exciton reduced mass 𝜇, Equation (5.18) shows that 𝑔 𝑠 depends on the dielectric constant as 𝜖 2 𝑟 . 𝑔 𝑠 is thus much more sensitive than 𝑔 𝑥 5.4. Dependence on polarization angle to any change in 𝜖 𝑟 , and second, a correction of 𝑔 𝑥 by a factor 𝛼 0 thus results in a correction of 𝑔 𝑠 by a factor 𝛼 2 0 , as is shown by Eq. (5.19). In Figure 5.5, we plotted the theoretical interaction constants stemming from the hydrogenic exciton picture as g𝑥 (0) ≃ 3𝛼 0 2 /2𝜇 (in which we assumed that |𝑔 ⊥ | ≪ 𝑔 ‖ ) versus 𝜇, and g𝑠 (0) = 𝛼 2 0 (2𝜋/7) Ω𝑎 2 𝐵 versus 𝑎 𝐵 (dashed lines). We introduce 𝛼 0 = 3.3 ± 0.8 in order for the theory to agree quantitatively with the measurement in [START_REF] Estrecho | Direct Measurement of Polariton-Polariton Interaction Strength in the Thomas-Fermi Regime of Exciton-Polariton Condensation[END_REF], where they found that 𝑔 𝑥,‖ = 13±3.4 𝜇eV.𝜇m 2 for a planar microcavity with GaAs quantum wells. This deviation might arise from the strict 2D approximation of the excitonic wavefunction in the theory, which is likely inaccurate in realistic quantum wells. Using excitonic reduced masses from the literature, a few materials are highlighted (squares) -see Table 5.1 for the material values -along these theoretical curve. In the left panel of Figure 5.5, the bulk exciton binding energies are also indicated for each material3 on the top axis as reference.

𝑔 𝑠 = 𝐶 0 Ω𝜖 2 𝑟 𝜇 2 ,
𝑔 𝑠 = 𝐶 1 Ω𝜖 2 𝑟 𝑔 2 𝑥 ,
The measurements obtained from the analysis of Figure 5.2 are shown as a red circle here. Our measured g𝑥 (0) is found to moderately exceed hydrogenic theory, and is fully compatible with the 30% enhancement (hollow square in left panel of Figure 5.5) predicted in [START_REF] Shahnazaryan | Exciton-Exciton Interaction in Transition-Metal Dichalcogenide Monolayers[END_REF], while g𝑠 (0) exceeds the hydrogenic theory by a large factor of 7±2. A possible origin of this larger deviation is visible in the hydrogenic picture, in which 𝑔 𝑠 depends directly on the square of dielectric constant (via 𝑎 𝐵 ), while 𝑔 𝑥 essentially does not depend on it.

Dependence on polarization angle

Our experimental collaborators also characterized the spin anisotropy of the nonlinearity at 𝑇 = 127 K, during the same experimental run as that shown in left panel of Figure 5.2, by measuring the transmission spectrum versus 𝜃 (see Figure 5.6). It was observed that upon increasing 𝜃 from 0 (linear polarization) to 𝜋/4 (circular polarization) at a fixed 𝑊 = 451 pJ, the spectrum exhibited a global redshift of 2.7 meV. Using our model and Equations (5.12) and (5.13), this behaviour implies that 𝑔 𝑥,⊥ is about twice larger than 𝑔 𝑥,‖ , and positive. In TMDC monolayers [START_REF] Shahnazaryan | Exciton-Exciton Interaction in Transition-Metal Dichalcogenide Monolayers[END_REF], like in conventional materials, the Coulomb interaction between polaritons is in principle dominated by exchange interaction, for which 𝑔 𝑥,⊥ is expected to be negative and small as compared to 𝑔 𝑥,‖ [START_REF] Vladimirova | Polariton-Polariton Interaction Constants in Microcavities[END_REF]. Our result differs from this picture, and is thus highly non-trivial. [START_REF] Nakwaski | Effective Masses of Electrons and Heavy Holes in GaAs, InAs, A1As and Their Ternary Compounds[END_REF][START_REF] Simmonds | Exciton Binding Energies in Shallow GaAs-Al 𝑦 Ga 1-𝑦 As Quantum Wells[END_REF][START_REF] Christen | Localization Induced Electron-hole Transition Rate Enhancement in GaAs Quantum Wells[END_REF] CdTe 0.1 0.55 0.085 10 7.3 [START_REF] Dang | Optical Detection of Cyclotron Resonance of Electron and Holes in CdTe[END_REF][START_REF] André | Spectroscopy of Polaritons in CdTe-Based Microcavities[END_REF][START_REF] Duan | One-pot synthesis of highly luminescent CdTe quantum dots by microwave irradiation reduction and their Hg 2+-sensitive properties[END_REF] [START_REF] Mita | Exciton Spatial Dispersion Determined through the Two-Photon Raman Scattering via Excitonic Molecule State at Large Wave Vectors in CuCl[END_REF][START_REF] Tang | Optical Selection Rule and Oscillator Strength of Confined Exciton System in CuCl Thin Films[END_REF][START_REF] Tang | Quantization of Excitons in CuCl Epitaxial Thin Films: Behavior between a Two-Dimensional Quantum Well and the Bulk[END_REF] MoSe 2 0.8 0.5 0.3 550 1.1 [START_REF] Kormányos | 𝐾 • 𝑝 Theory for Two-Dimensional Transition Metal Dichalcogenide Semiconductors[END_REF][START_REF] Ugeda | Giant Bandgap Renormalization and Excitonic Effects in a Monolayer Transition Metal Dichalcogenide Semiconductor[END_REF][START_REF] Goryca | Revealing Exciton Masses and Dielectric Properties of Monolayer Semiconductors with High Magnetic Fields[END_REF] Table 5.1: Material parameters mentioned in this work: electron (𝑚 * 𝑒 ) and heavyhole (𝑚 * ℎℎ ) effective mass, excitonic reduced mass 𝜇 𝑋 , and binding energies 𝐸 𝑏 . Bulk material is considered for GaAs, CdTe, ZnSe, CuBr, and CuCl. MoSe 2 cited effective masses are for a single monolayer. The cited binding energy is for a monolayer grown surrounded by a graphene bilayer on one side and vacuum on the other.

A possible explanation that we could think of is the involvement of an intermediate state, like spin-2 dark excitons [START_REF] Glazov | Polariton-Polariton Scattering in Microcavities: A Microscopic Theory[END_REF] or biexcitons [START_REF] Takemura | Spin Anisotropic Interactions of Lower Polaritons in the Vicinity of Polaritonic Feshbach Resonance[END_REF][START_REF] Carusotto | Feshbach Blockade: Single-Photon Nonlinear Optics Using Resonantly Enhanced Cavity Polariton Scattering from Biexciton States[END_REF]. In such a mechanism, 𝑔 𝑥,⊥ is enhanced and takes a positive sign when the two-polaritons state is close, and on the high energy side of the intermediate state. In a MoSe 2 monolayer, the dark exciton state is a few meV above the bright one [START_REF] Robert | Measurement of the Spin-Forbidden Dark Excitons in MoS2 and MoSe2 Monolayers[END_REF], such that the upper polariton state, nominally 12.2 meV above the bright exciton, could benefit from this resonance at the peak intensity, when the saturation brings it closer. A resonance with the biexciton state is expected 10 meV [START_REF] Hao | Neutral and Charged Inter-Valley Biexcitons in Monolayer MoSe 2[END_REF][START_REF] Bleu | Polariton Interactions in Microcavities with Atomically Thin Semiconductor Layers[END_REF] below the bright exciton, which is 3 meV above the nominal energy of the lower polariton, and thus also favourable at the peak intensity. Finally, at such large 𝑊 , higher order many-body correlations and the composite nature of excitons might start to contribute, such that our mean field model might break down. The precise interpretation of the interaction constants then requires a fully dedicated investigation which could be a topic of future interest. The circle symbols show the lower polariton peak energy; the solid black line is a theoretical fit 𝐵 0 +𝐵 𝜃 sin 2 2𝜃, following Equations eq5.12 and (5.13), where 𝐵 0 = 7.9 meV and 𝐵 𝜃 = -2.7 meV.

Conclusions and Outlook

In this part we looked at the background of semiconducting transition metal dichalcogenides (TMDC) monolayers which belong to the growing family of two dimensional materials. Then, in collaboration with our experimental counterparts we took one member of this family -MoSe 2 monolayer -and showed that this monolayer in the strong light-matter coupling regime displays an enhanced excitonmediated optical nonlinearity than what would be expected if TMDC excitons were treated using the hydrogenic theory of excitons. In particular, the stark deviation from the hydrogenic theory is much more prominent in the nonlinearity caused due to the excitonic saturation mechanism.

To get a quantitative estimate of the nonlinear constants, 𝑔 𝑥 and 𝑔 𝑠 , we developed a mean field input output theory in the exciton photon basis complete both with spin anisotropy and including the effects of both the nonlinear constants. We numerically solved the input-output model and optimized its solution for the unknown parameters, 𝑔 𝑥 and 𝑔 𝑠 , to match with the experimental data.

Our collaborators also observed a non-trivial spin anisotropy of the interaction which deserves future investigation. Our results demonstrate that non-hydrogenic exciton in MoSe 2 , and in other TMDC materials where the nonlinearity enhancement could be potentially even larger, offer new perspectives for the engineering of excitonmediated optical nonlinearities which can lead to applications in quantum simulation of strongly interacting many-body phenomena far from equilibrium and application in quatum information devices based on photons.

Part III Polariton Interactions in Artificial

Honeycomb Lattice 99

Single Particle Dispersion in a Honeycomb lattice

In this chapter I will review the general features of a quantum particle in a honeycomb lattice, with specific attention to the features relevant for polariton experiments. In particular, I'll introduce the single particle dispersion using the tight binding approximation considering up to second nearest neighbour hopping. Then I'll introduce the effect of spin mixing on the dispersion particularly via spin orbit coupling or TE-TM splitting and birefringence in the context of an artificial honeycomb lattice engineered in a microcavity polariton system which we term as polaritonic graphene.

u 1 = 𝑎 ( √ 3 2 , 1 2 ) , u 2 = 𝑎 ( - √ 3 2 , 1 2
) , u 3 = 𝑎 (0, -1) (6.4) while the six second nearest neighbours (moving from a point in the same sublattice) are given by w 13/31 = ±a 1 , w 23/32 = ±a 2 , w 12/21 = ±(a 2a 1 ) (6.5)

Single particle dispersion: Tight-binding approximation

The single particle spectrum or bandstructure of graphene is well described under the tight binding approximation as the confinement energy on each lattice site is assumed to be much larger than the tunneling or hopping energy. The Hamiltonian under the tight-binding approximation including the nearest and the next neighbour hopping reads

ℋ 0 = -𝑡 ∑ ⟨𝑖,𝑗⟩ (𝑎 † 𝑖 𝑏 𝑗 + 𝑏 † 𝑗 𝑎 𝑖 ) -𝑡 ′ ∑ ⟨⟨𝑖,𝑗⟩⟩ (𝑎 † 𝑖 𝑎 𝑗 + 𝑏 † 𝑖 𝑏 𝑗 + h.c.). (6.6)
Here 𝑎 𝑖 (𝑏 𝑖 ) and 𝑎 † 𝑖 (𝑏 † 𝑖 ) are the particle annihilation and creation operators respectively at site A 𝑖 (B 𝑖 ) on sublattice 𝐴 (𝐵); 𝑡 is a phenomenological constant describing the nearest-neighbour hopping energy and similarly 𝑡 ′ is the next-nearest neighbour hopping energy. Here we are neglecting any particle spin related effects. This tight-binding Hamiltonian ℋ 0 can be diagonalized in wavevector space. Hence, Fourier expanding the creation and annihilation operators,

𝑎 𝑖 = 1 √ 𝑁 ∑ k𝑒 -𝑖k • A 𝑖 𝑎 k (6.7) 𝑏 𝑖 = 1 √ 𝑁 ∑ k 𝑒 -𝑖k • B 𝑖 𝑏 k , (6.8)
(where 𝑁 is the number of unit cells), we can write the tight-binding Hamiltonian in wavevector space in matrix form as

ℋ 0 = - ∑ k [ 𝑎 † k 𝑏 † k ] ⎡ ⎣ 𝑡 ′ 𝑓 (k) 𝑡𝜑(k) 𝑡𝜑 * (k) 𝑡 ′ 𝑓 (k) ⎤ ⎦ ⎡ ⎣ 𝑎 k 𝑏 k , ⎤ ⎦ (6.9) where 𝜑(k) = ∑ 3 𝑗=1 exp{(𝑖k • u 𝑗 )} = √ 3 + 𝑓 (k) and 𝑓 (k) = 2cos ( √ 3𝑘 𝑦 𝑎 ) + 4cos ( √ 3 2 𝑘 𝑦 𝑎 ) cos ( 3 2 𝑘 𝑥 𝑎 ) . (6.10)
The eigenvalues of this Hamiltonian provides the following dispersion relation with the two energy bands given by [START_REF] Wallace | The Band Theory of Graphite[END_REF])

𝐸(k) = ± 𝑡 √ 3 + 𝑓 (k) -𝑡 ′ 𝑓 (k). (6.11)
Here the plus sign refers to the 𝜋 * or upper band and minus refers to the 𝜋 or lower band. We can see from Eq. (6.11) that the spectrum is symmetric around 6.1. Honeycomb lattice zero energy if there is no next nearest neighbour hopping. For finite values of 𝑡 ′ , the bands are asymmetric as can be seen in Fig. 6.2.

Polaritonic graphene

The graphene-like dispersion relation -in Eq. ( 6.11) -can be simulated using model systems such as ultracold atoms in optical lattices [START_REF] Tarruell | Creating, Moving and Merging Dirac Points with a Fermi Gas in a Tunable Honeycomb Lattice[END_REF], arrays of photonic waveguides [START_REF] Polini | Artificial Honeycomb Lattices for Electrons, Atoms and Photons[END_REF] or polaritons by patterned engineering of the honeycomb lattice geometry [START_REF] Schneider | Exciton-Polariton Trapping and Potential Landscape Engineering[END_REF]. This quantum simulation of graphene's dispersion relation was recently observed for exciton-polaritons placed in an artificial honeycomb lattice [START_REF] Jacqmin | Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons[END_REF][START_REF] Milićević | Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene[END_REF], which is fabricated by etching hundreds of micropillars -quantum wells -in a planar semiconducting microcavity, as shown in Figure 6.3. From now on the honeycomb lattice structure for polaritons will be referred to as polaritonic graphene. In the last few years, a flurry of experimental activity in the field of topological photonics [START_REF] Ozawa | Topological Photonics[END_REF] utilized polaritonic graphene to measure topological invariants [START_REF] St-Jean | Measuring Topological Invariants in a Polaritonic Analog of Graphene[END_REF], photonic edge states [START_REF] Milićević | Edge States in Polariton Honeycomb Lattices[END_REF][START_REF] Milićević | Orbital Edge States in a Photonic Honeycomb Lattice[END_REF], quantum geometric tensor [START_REF] Gianfrate | Measurement of the Quantum Geometric Tensor and of the Anomalous Hall Drift[END_REF] and to realize an exciton polariton topological insulator [START_REF] Klembt | Exciton-Polariton Topological Insulator[END_REF]. Also, an anisotropic transport of polaritons in such structures was observed by creating a semi-Dirac cone [START_REF] Real | Semi-Dirac Transport and Anisotropic Localization in Polariton Honeycomb Lattices[END_REF]. These works didn't take into account interactions among the polariton particles. In the next chapter we will look into how the introduction of spin or polarization dependent polariton interactions changes the band structure of polaritonic graphene. Moreover, it gives us an opportunity to study particle interaction effects in a graphene simulator under a non-equilibrium driven-dissipative setting.

6.2 Effect of Spin-orbit coupling and birefringence

TE-TM splitting

Before delving into a study of interaction effects we have to see how spin related effects change the single particle spectrum of graphene. In particular, we'll look into two effects that create spin or polarization mixing in microcavity polaritons: spin-orbit coupling and optical birefringence.

Spin (or pseudospin to be precise) is an important property of polaritons which is linked with the polarization of photons absorbed or emitted by the microcavity, and the dipole polarization of QW excitons2 . It is possible to induce spin-orbit coupling (SOC) i.e. coupling between spin and motion of a particle in polaritonic graphene by utilizing the polarization dependent TE-TM energy splitting of polariton modes. TE-TM splitting leads to a change in the hopping or tunneling energy between two micropillars of the artificial lattice [START_REF] Sala | Spin-Orbit Coupling for Photons and Polaritons in Microstructures[END_REF]. The change in hopping energy is dependent on whether the polarization of light is longitudinal or transverse to the link connecting the two micropillars, as shown in Figure 6.4

To include the SOC effect caused due TE-TM splitting, we have to expand the tight-binding Hamiltonian in Eq. (6.6). We introduce two unit vectors êT𝑖 and êL𝑖 which point to a direction either transverse or longitudinal to the link connecting the pillar at lattice point 𝑖 to the pillar at point 𝑗 respectively. The Hamiltonian including the TE-TM splitting is written as

ℋ TB = - ∑ ⟨𝑖,𝑗⟩ { 𝑡 T (a † 𝑖 • êT𝑖 )(ê † T𝑖 • b 𝑗 ) + 𝑡 L (a † 𝑖 • êL𝑖 )(ê † L𝑖 • b 𝑗 ) + h.c.) } - ∑ ⟨⟨𝑖,𝑗⟩⟩ { 𝑡 ′ T (a † 𝑖 • êT𝑖 )(ê † T𝑖 • a 𝑗 ) + 𝑡 ′ L (a † 𝑖 • êL𝑖 )(ê † L𝑖 • a 𝑗 )
+ 𝑡 ′ T (b † 𝑖 • êT𝑖 )(ê † T𝑖 • b 𝑗 ) + 𝑡 ′ L (b † 𝑖 • êL𝑖 )(ê † L𝑖 • b 𝑗 )) } . (6.12) Here a † 𝑖 = [𝑎 † 𝑖,𝑥
𝑎 † 𝑖,𝑦 ] is the vector creation operators for polaritons where 𝑎 † 𝑖,𝑥(𝑦) creates a polariton at site 𝑖 in sublattice 𝐴 with 𝑥(𝑦) polarization, a similar definition holds for operators on sublattice B, 𝑡 T and 𝑡 L are the nearest neighbour transverse and longitudinal hopping energies respectively while 𝑡 ′ T and 𝑡 ′ L are the next nearest neighbour transverse and longitudinal hopping energies. The three nearest neighbour link unit vectors are:

Longitudinal:

( √ 3 2 , 1 2 ) , ( - √ 3 2 , 1 2 ) , (0, -1) Transverse: ( - √ 1 2 , - √ 3 2 ) , ( -1 2 , √ 3 2
) , (1, 0).

While the six nearest neighbour link unit vectors are:

Longitudinal: ± (-1, 0), ± ( -1 2 , √ 3 2 , ) , ± ( 1 2 , √ 3 2
)

Transverse:

± (0, -1), ± ( - √ 3 2 , -1 2 ) , ± ( - √ 3 2 , 1 2 ) .
Similar to Eq. (6.9) using the above link unit vectors, we can diagonalize Eq. 6.12 in wavevector space to get the SOC dispersion relation and can rewrite it in matrix form as The matrix elements in A are explicitly given as

ℋ TB = ∑ k Ψ † k H(k) Ψ k = ∑ k [ a † k b † k ] ⎡ ⎣ A B B † A ⎤ ⎦ ⎡ ⎣ a k b k ⎤ ⎦ . ( 6
𝑀 11 (k) = -2𝑡 ′ L cos (𝑎k • w 12 ) - ( 3𝑡 ′ T + 𝑡 ′ L 2 ) [cos (𝑎k • w 13 ) + cos (𝑎k • w 23 )] (6.15a) 𝑀 22 (k) = -2𝑡 ′ T cos (𝑎k • w 12 ) - ( 𝑡 ′ T + 3𝑡 ′ L 2 ) [cos (𝑎k • w 13 ) + cos (𝑎k • w 23 )] (6.15b) 𝑀 12 (k) = - √ 3(𝑡 ′ T + 𝑡 ′ L ) 2 [cos (𝑎k • w 13 ) -cos (𝑎k • w 23 )] (6.15c) 𝑀 21 (k) = 𝑀 12 . (6.15d)
Whereas, the matrix elements for B are given as

𝑀 13 (k) = - ( 𝑡 T + 3𝑡 L 4 ) ( 𝑒 𝑎k • u 1 + 𝑒 𝑎k • u 2 ) -𝑡 T 𝑒 𝑎k • u 3 (6.16a) 𝑀 24 (k) = - ( 3𝑡 T + 𝑡 L 4 ) ( 𝑒 𝑎k • u 1 + 𝑒 𝑎k • u 2 ) -𝑡 L 𝑒 𝑎k • u 3 (6.16b) 𝑀 14 (k) = - √ 3(𝑡 T -𝑡 L ) 4 ( 𝑒 𝑎k • u 1 -𝑒 𝑎k • u 2 ) (6.16c) 𝑀 23 (k) = 𝑀 14 (6.16d)
The eigenvalues of H(k) provide the single particle dispersion with the SOC effect. The TE-TM splitting of the lower and higher bands as a result of the effective spin-orbit interaction can be seen in Figure 6.5a-b where we consider the splitting ∆𝑡 = 𝑡 T -𝑡 L = 0.2 𝑡 for nearest neighbour interaction and ∆𝑡 ′ = 𝑡 ′ T -𝑡 ′ L = -0.04 𝑡 for the next nearest neighbour and the energy scale parameter 𝑡 is equal to by

(𝑡 T + 𝑡 L )/2.

XY splitting

Similar to the SOC effect, the birefringence leads to a splitting between the 𝑥polarized and 𝑦-polarized energy bands, as can be seen in Figure 6.5c-d. The splitting essentially creates an energy shift in two opposite directions of the energy axis for the two polarizations. This energy shift -which we call as XY-splitting -parametrized 6.2. Effect of Spin-orbit coupling and birefringence

-3 -2 -1 0 1 2 3 4 K Γ K M K E/t -1.0 -0.5 0.0 0.5 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 E/t k y /(2π/a) -3 -2 -1 0 1 2 3 4 K Γ K M K E/t -1.0 -0.5 0.0 0.5 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 E/t k y /(2π/a) -3 -2 -1 0 1 2 3 4 K Γ K M K E/t -1.0 -0.5 0.0 0.5 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 a) b) c) d) e) f) E/t k y /(2π/a)
A = ⎡ ⎣ 𝑀 11 (k) 𝑀 12 (k) 𝑀 21 (k) 𝑀 22 (k) ⎤ ⎦ + ⎡ ⎣ 𝛿/2 0(k) 0 -𝛿/2 ⎤ ⎦
(6.17)

In Figure 6.5c-d we can see the effect of birefringence in absence of SOC, in which case it shifts the eigenvalues in Eq. (6.11) by an amount 𝛿/2 for 𝑥-polarized light and -𝛿/2 for y-polarized light. Further in Figure 6.5e-f we can see the combined effect of both the TE-TM and XY energy splittings.

Circular basis transformation

It will be helpful for us in the next chapter to transform the single particle Hamiltonian (6.13) to circular basis in order to include interactions which have a simpler form of the Hamiltonian in this basis. In order to transform to circular basis we use the basis transformation matrix Z which is written as

Z = ⎡ ⎣ T 𝑐 0 0 T 𝑐 ⎤ ⎦ and T 𝑐 = ⎡ ⎣ 1 𝑖 1 -𝑖 ⎤ ⎦ (6.18)
Utilizing this matrix we can transform our Hamiltonian as

ℋ TB = ∑ k [ 𝑎 † k+ 𝑎 † k-𝑏 † k+ 𝑏 † k- ] H 𝑐 (k) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑎 k+ 𝑎 k- 𝑏 k+ 𝑏 k- ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (6.19)
where

H 𝑐 (k) = Z H(k) Z † (6.20)
With this, we have completed the analysis of the single particle spectrum of a honeycomb lattice and we are ready to tackle the effects of interactions, drive and dissipation in polaritonic graphene. 110

Polariton-Polariton interactions in honeycomb lattice

Interactions modify the single particle dispersion of polaritonic graphene. In order to account for them, we'll obtain the Bogoliubov spectrum of elementary excitations over a steady state solution. First we'll introduce the interaction Hamiltonian, using which we will write a generalized Gross-Pitaevskii equation (GPE) in the mean field approximation. Finally, we'll calculate the Bogoliubov dispersion relation around the steady state solution of the generalized GPE and compare our predictions with the experimental data provided for coherent pumping at the Dirac point by our collaborators at Néel Institute.

In order to take into account the spin-dependent interaction effects, we add to the single particle Hamiltonian an interaction term which in the circular basis is given by

ℋ int = 1 2 { ∑ 𝑖∈𝐴 ∑ 𝜎 [ 𝑔 t 𝑎 † 𝑖,𝜎 𝑎 † 𝑖,𝜎 𝑎 𝑖,𝜎 𝑎 𝑖,𝜎 + 2𝑔 s 𝑎 † 𝑖,+ 𝑎 † 𝑖,-𝑎 𝑖,-𝑎 𝑖,+
] (7.1)

+ ∑ 𝑖∈𝐵 ∑ 𝜎 [ 𝑔 t 𝑏 † 𝑖,𝜎 𝑏 † 𝑖,𝜎 𝑏 𝑖,𝜎 𝑏 𝑖,𝜎 + 2𝑔 s 𝑏 † 𝑖,+ 𝑏 † 𝑖,-𝑏 𝑖,-𝑏 𝑖,+ ] } (7.2)
where 𝜎 = (+, -) is the circular polarization basis 𝑔 t and 𝑔 s are the singlet and triplet interaction constants as were introduced in Chapter 1.

Steady State with coherent pumping at 𝒦-point

In order to quantify the effect of polariton-polariton interactions on the dispersion relation of polaritonic graphene, first we calculate the steady state -parameterized by the pump strength 𝐹 0 . This is achieved by numerically solving the discrete driven-dissipative Gross-Pitaevskii equation, which is a non-linear system of four equations, succinctly written as

( H 𝑐 (k 𝑝 ) -(∆ + 𝑖Γ)1 + [ Ψ ss , ℋ int ] ) Ψ ss = -F pump (7.3)
where 1 is the identity matrix, ∆ = 𝜔 𝑝 -𝜔 LP (0) is the detuning energy, Γ is the dissipation constant, and

F pump = 𝐹 0 ( 𝜎 + , 𝜎 -, 𝑒 𝑖k • 𝛿 𝜎 + , 𝑒

Bogoliubov Spectrum

We linearize over the steady state solution using an ansatz (analogous to how we did in Chapter 1), i.e we take

Ψ(r, 𝑡) = [ Ψ ss + ∑ k 𝛿Ψ k (𝑡) 𝑒 𝑖k.r + ∑ k 𝛿Ψ * k (𝑡) 𝑒 -𝑖k.r
] 𝑒 𝑖(k𝑝.r-𝜔𝑝𝑡) (7.4) in Eq. ( 7.3), where 𝛿Ψ k = (𝛿𝑎 k+ , 𝛿𝑎 k,-, 𝛿𝑏 k,+ , 𝛿𝑏 k,-) 𝑇 is a vector describing the small plane wave fluctuations in k th wavevector mode. This yields a system of eight dynamical linear equations for each fluctuation mode and its conjugate, which are written as

𝑖 𝜕 𝜕 𝑡 ⎛ ⎝ 𝛿Ψ k (𝑡) 𝛿Ψ * -k (𝑡) ⎞ ⎠ = ℒ Bog (k) ⎛ ⎝ 𝛿Ψ k (𝑡) 𝛿Ψ * -k (𝑡) ⎞ ⎠ . (7.5)
Here, ℒ Bog (k) is the 8 × 8 Bogoliubov matrix which is written as

ℒ Bog = ⎡ ⎣ 𝒥 (k 𝑝 + k) 𝒦 -𝒦 * -𝒥 * (k 𝑝 -k) ⎤ ⎦ (7.6)
where

𝒥 (k 𝑝 + k) = H 𝑐 (k 𝑝 + k) -(Δ + 𝑖Γ)1 + ⎡ ⎢ ⎣ G 𝑎 0 0 G 𝑏 ⎤ ⎥ ⎦ (7.7) and 𝒦 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑔 t (𝑎 ss + ) 2 𝑔 s (𝑎 ss + ) (𝑎 ss -) 0 0 𝑔 s (𝑎 ss + ) (𝑎 ss -) 𝑔 t (𝑎 ss -) 2 0 0 0 0 𝑔 t (𝑏 ss + ) 2 𝑔 s (𝑏 ss + ) (𝑏 ss -) 0 0 𝑔 s (𝑏 ss + ) (𝑏 ss -) 𝑔 t (𝑏 ss -) 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (7.8)
The submatrices G 𝑎 and G 𝑏 are given as

G 𝑎 = ⎡ ⎢ ⎣ 2 𝑔 t |𝑎 ss + | 2 + 𝑔 s |𝑎 ss -| 2 𝑔 s 𝑎 ss + (𝑎 ss -) * 𝑔 s (𝑎 ss + ) * (𝑎 ss -) 2 𝑔 t |𝑎 ss -| 2 + 𝑔 s |𝑎 ss + | 2 ⎤ ⎥ ⎦ , (7.9) G 𝑏 = ⎡ ⎢ ⎣ 2 𝑔 t |𝑏 ss + | 2 + 𝑔 s |𝑏 ss -| 2 𝑔 s 𝑏 ss + (𝑏 ss -) * 𝑔 s (𝑏 ss + ) * (𝑏 ss -) 2 𝑔 t |𝑏 ss -| 2 + 𝑔 s |𝑏 ss + | 2 ⎤ ⎥ ⎦ .
(7.10)

Comparison with the experiment

We obtain the Bogoliubov spectrum by numerical diagonalization of the Bogoliubov matrix ℒ Bog . In Figure 7.1 we show the Bogoliubov spectrum for pumping at the 𝒦 point with right circularly polarized light for increasing values of pump strength 𝐹 0 . The normal branch spectrum is shown by the darker shade and the ghost branch by the lighter shade. We see that the normal and ghost branches are shifted in 𝑘-space which can be accounted by the fact that the normal Bogoliubov modes at wavector k have the corresponding ghost modes at wavevector 2k 𝑝k (Ciuti and Carusotto, 2005). In Figure 7.2 we show the reciprocal lattice for the normal and ghost modes with the pump at 𝒦 point marked with red circle. We also see that the part of the normal spectrum which is copolarized with the pump polarization has higher interaction energy for increasing pump strength and hence gets blueshifted more compared to the part of the spectrum which is cross-polarized with the laser pump. This can be linked to the difference in magnitude of the triplet and singlet interaction constants. For a microcavity polariton system with inorganic semiconductor QWs this ratio is approximately 𝑔 s ≈ -0.1 𝑔 t .

Comparison with the experiment

In order to match our theoretical predictions with the experimental observations, we perform a linear-response analysis for the fluctuation amplitude 𝛿Ψ = (𝛿Ψ k , 𝛿Ψ * -k ) 𝑇 where it is treated as a response to a steady state stochastic drive 𝛿F = (𝛿f k (𝜔), 𝛿f -k (-𝜔)) 𝑇 under the relation (7.11) We assume that the main mechanism for creation of fluctuations in the system is the coupling to the acoustic phonons [START_REF] Stepanov | Dispersion Relation of the Collective Excitations in a Resonantly Driven Polariton Fluid[END_REF]. Within such an assumption we can write 𝛿f k (𝜔) as 𝛿f k (𝜔) = 𝒫 k (𝜔) ( 𝑎 ss + , 𝑎 ss -, 𝑏 ss + , 𝑏 ss -) , (7.12)

𝛿Ψ(𝜔) = [𝜔1 -ℒ Bog (k)] -1 𝛿F
where 𝒫 k (𝜔) is a stochastic field created by the phonons. The experimental observable is the intensity of the fluctuations which can be obtained after averaging over random realizations of the phonon field. For instance, the intensity co-polarized with a right circularly polarized (+) pump is given by ⟨ |𝛿𝑎 k+ (𝜔) + 𝛿𝑏 k+ (𝜔)| 2 ⟩ 𝑝ℎ . We define the matrix elements of the susceptibility, 𝜒 k (𝜔) as In all of the above figures, we have also employed the interaction of the polaritons with an excitonic reservoir as another cause for the blueshift of the data for increasing pump powers [START_REF] Stepanov | Dispersion Relation of the Collective Excitations in a Resonantly Driven Polariton Fluid[END_REF]. The reservoir induced blueshift doesn't affect the shape of the Bogoliubov spectrum and just leads to an overall shift in the energy scale. It can be included in our model as an additional interaction energy equal to 𝑔 𝑅 (𝑛 𝑡 ) where 𝑛 𝑡 is the total density of the polaritons in the steady state.

𝜒 k (𝜔) 𝑖𝑗 = ( [𝜔1 -ℒ Bog (k)] -1 ) 𝑖𝑗 . ( 7
The agreement between the theory and the experiment is excellent, taking into account the simplicity of our model1 . In particular, we see that not all the Bogoliubov modes are observed in the theoretical linear response spectrum or the experimental emission spectrum. This feature can be attributed to Brillouin zone selection effects in a honeycomb lattice [START_REF] Shirley | Brillouinzone-selection effects in graphite photoelectron angular distributions[END_REF], which is a generic feature of the lattice geometry, wherein the wavefunctions from the two lattice sites in each polaritonic graphene unit cell destructively interfere along certain high symmetry directions to produce no amplitude for detection.

Notice also that specific features emerge in the spectra at increasing pump strength; for instance, a clear distortion with respect to the non-interacting single particle spectra, obtained at very low pump powers, and the emergence of flat parts in the Bogoliubov dispersion which are the precursors of instabilities in the system. In addition, the intensity of ghost branch is negligible compared to the normal branch because we are at low interaction energies and Bogoliubov mixing between the two branches is very less.

Conclusions and outlook

In this part we studied the effect of interactions on the dispersion relation of polaritonic graphene. To this aim, we developed a Bogoliubov theory of elementary excitations on the honeycomb lattice and used it for comparison with the experimental observations after performing a linear response analysis of the fluctuation fields.

The work presented in this chapter is part of an ongoing research work and in the future it would be interesting to to study the behaviour of the Bogoliubov dispersion when we pump at other high symmetry points such as the Γ point where it was suggested by [START_REF] Bleu | Photonic versus Electronic Quantum Anomalous Hall Effect[END_REF] that interactions can lead to an opening of a topological gap. Another direction could be to use this theory to study interaction effects in different lattice geometries such as the Kagomé, Lieb or Mielke lattices. The study of polariton excitations in exotic lattice geometries can open pathway for quantum simulations of interaction effects in relativistic systems [START_REF] Fagnocchi | Relativistic Bose-Einstein Condensates: A New System for Analogue Models of Gravity[END_REF].

Conclusions and perspectives

In this thesis we looked at three different cases of particle interaction effects in exciton polaritons:

F irst, we looked at the interaction of an impurity with a bath of exciton polaritons under the context of Bose polaron problem in the Fröhlich regime. In particular, we studied the motion of the impurity in the polariton fluid under a coherent drive and dissipation, assuming a weak coupling between the impurity and the fluid, thereby extending the description of Bose polarons to the non-equilibrium case of a bosonic driven-dissipative quantum fluid.

We determined the polaron effective mass and showed that it is possible to tune it to values both smaller and larger than the bare one by adjusting the detuning between the laser pump frequency and the frequency at the minima of the lower polariton branch. We also determined the drag force acting on the impurity, as well as the polaron trajectories at a semiclassical level and found a counter-intuitive regime of acceleration of the impurity against the flow of the fluid. This work showed that the impurity dynamics can be used as a test particle to probe the different regimes of non-equilibrium quantum flow in quantum fluids of light.

From an experimental point of view, a free moving obstacle (or impurity) of finite mass has not been realized so far and yet, realistic strategies can be envisaged. For example, a long-lifetime dark exciton droplet can be created by a second laser beam in the middle of the polariton flow. Such excitons can be dark (i.e. not in the strong coupling regime) due to the symmetry of their wavefunction in multiplequantum-well microcavities, or can have momenta outside the light cone, as a result of thermalization. Another possible impurity, much lighter in mass, can be made up of a cross-polarized polariton droplet created by a second laser. While such polaritons have lower interaction with cross-polarized polaritons, it is still nonzero and could be enhanced by utilizing Feshbach resonance with the biexciton.

In the future it would be interesting to explore corrections to the Markovian approximation and to look at the effects of non-Markovianity in the system. Another possible avenue to look could be to go beyond the semiclassical description of the polaron to study strong interaction between the impurity and the bath of polaritons.

S econd, we looked at the polariton-polariton interactions in a monolayer of Chapter 7. Interaction effects in Polaritonic Graphene semiconducting Transition metal dichalcogenides (TMDC) monolayer. This work was performed in collaboration with our experimental counterparts who placed one member of this family -MoSe 2 -inside a microcavity in the strong coupling regime with the cavity photons and provided us with transmission spectrum at increasing intensities of the pump laser. We statistically analyzed the transmission spectrum using a mean field input-output theory of the coupled the TMDC exciton and the cavity photon fields -in the spirit of generalized Gross-Pitaevskii equation -to get a quantitative estimate of the nonlinear constants, 𝑔 𝑥 and 𝑔 𝑠 .

We showed that this TMDC monolayer displays an enhanced exciton-mediated optical nonlinearity in particular via the excitonic saturation mechanism. This enhanced nonlinearity was linked with the non-hydrogenic form of the exciton wavefunction in TMDC monolayers. It proved that by utilizing semiconducting monolayers of TMDCs we can achieve higher optical nonlinearities at room temperatures. Our collaborators also observed a non-trivial spin anisotropy of the interaction which deserves a future investigation. Early results point to the fact that we may need a description for exciton-polaritons beyond the mean field regime to understand this discrepancy. Hence, it would be interesting to also go beyond the mean-field approximation and consider the composite nature of excitons for very high pump intensities.

In the future we can also look at the option of further increasing the optical nonlinearities using excitons polaritons in bilayer TMDC Moiré lattices, which are predicted to have strong nonlinearites due to exciton blockade (L. [START_REF] Zhang | Van Der Waals Heterostructure Polaritons with Moiré-Induced Nonlinearity[END_REF].

T hird, we are looking at the effects of polariton-polariton interaction on the Bogoliubov dispersion in a honeycomb lattice, which is as of writing this thesis is under progress. This work is also in conjunction with experiment where our collaborators have provided us with the transmission spectrum generated by coherently pumping an artificial honeycomb lattice, placed inside a microcavity, with a frequency and wavevector close to the Dirac point. To match the experimentally observed spectra with theory we utilized the Bogoliubov theory on a honeycomb lattice and are looking for signatures of non-trivial dispersion relations due to the driven-dissipative nature of the system.

As a next step, it would be interesting to use polaritonic graphene as a quantum simulator for the study of interaction effects in relativistic Hamiltonians [START_REF] Fagnocchi | Relativistic Bose-Einstein Condensates: A New System for Analogue Models of Gravity[END_REF] at far from equilibrium conditions. Further, it opens up the pathway for the study of interaction effects in other exotic lattice geometries like the Kagomé, Lieb and Mielke.

B Generalized Bogoliubov Transformation

In the case of positive detuning, the quadratic form of lower polariton Hamiltonian in (3.5) after undergoing Bogoliubov approximation is non-positive definite for a finite range of wavevector k values. This precludes its diagonalization using the standard Bogoliubov operators bk and b † k (3.6a, 3.6b) in this range of k values [START_REF] Rossignoli | Complex modes in unstable quadratic bosonic forms[END_REF] These operators satisfy the bosonic commutation relation with the condition 𝑢 2 k -𝑣 2 k = 1 but in this case bk𝑝+k ̸ = b † k𝑝+k which results, in general, complex Bogoliubov energy and onset of dynamical instability. However, we stay in the regime where the imaginary part of the Bogoliubov energy is less than the dissipation rate, Γ which stabilizes the system against the instability caused due to complex Bogoliubov energies. Proceeding forward with these non-standard Bogoliubov operators all the subsequent results remain the same as derived in the main paper, albeit replacing b † k𝑝+k with bk𝑝+k . The use of non-standard Bogoliubov operators when detuning ∆ is positive also results in a small imaginary contribution to the excitation momentum Π when we solve the self-consistency relation in (3.26). We have taken it into account by adding Interactions entre polaritons excitoniques : des polarons aux matériaux 2D, et aux réseaux en nid d'abeille Interactions in exciton polaritons: from polarons to 2D materials and honeycomb lattices Résumé Dans cette thèse, nous avons examiné théoriquement l'effet des interactions à deux corps dans différents systèmes de polariton-excitoniques en microcavité, en situation d'excitation laser résonante. Dans le premier cas, nous avons étudié la dynamique d'une impureté mobile en interaction avec un bain de polaritons, qui constitue un problème de type "polaron Bosonique", dans un cadre hors-équilibre. L'impureté est ainsi soumise à une force d'entrainement cohérente exercée par le fluide de polaritons, dont ce dernier a la particularité d'etre dans une situation stationnaire gain/dissipation. Cette situation résulte en des caractéristiques uniques du spectre d'excitation du fluide de polariton, qui conduisent à différents régimes dynamiques pour le mouvement du polaron, dont en particulier un régime non-trivial d' "accélération à contre-courant". Nous avons ensuite étudié les interactions excitoniques dans une monocouche de dichalcogénures de métaux de transition (TMDC) inséré dans une microcavité. Dans ces matériau d'épaisseur atomique, le potentiel d'interaction de Coulomb entre électron et trou est fortement influencée par la forte discontinuité de la fonction diélectrique selon l'axe perpendiculaire à la couche. Cette caractéristique donne lieu à des états excitoniques non-hydrogéniques, offrant une plus forte interaction à deux corps que dans le cas des états hydrogéniques conventionnelles. Pour étudier cette amélioration, noos avons analysé les résultats d'une expérience de spectroscopie de transmission optique d'une monocouche de MoSe2 en régime de couplage fort avec le mode d'une microcavité optique de manière quantitative, à l'aide de notre théorie entrée-sortie non-linéaire. Nous avons pu ainsi determiner que l'experience montre une amélioration de l'interaction excitonexciton et de la saturation fermionique excitonique par rapport aux valeurs mesurées dans des systèmes où l'exciton est hydrogénique. Dans une troisième partie, nous avons étudié l'effet des interactions polariton-polariton lorsque ces derniers sont confinés dans un réseau artificiel en nid d'abeille, de même géométrie que celui du graphène. Nous avons analysé les excitations élémentaires d'un condensat de polaritons situé au point K de ce système en nous basant sur le théorie de Bogoliubov généralisée, et nous avons pu ainsi retrouver plusieurs caractéristiques observés dans les expériences.

Mots-clés :

Polaritons excitoniques, systèmes entraînés-dissipatifs, équation de Gross-

Pitaevskii

Abstract

In this thesis we studied theoretically the effect of particle interactions in three different cases of a resonantly driven microcavity exciton polariton system. First, we studied the motion of an impurity interacting with a bath of polaritons or the Bose polaron problem in a nonequilibrium setting. We considered the impurity to be embedded in a quantum fluid of light realised by microcavity polaritons, subject to a coherent drive and dissipation on account of pump and cavity losses. We find different dynamical regimes for the motion of polaron, originating from the unique features of the excitation spectrum of driven-dissipative polariton fluids, in particular a non-trivial regime of acceleration against the flow. Second, we looked at the excitonic interactions in a monolayer of transition metal dichalcogenides (TMDC). In these layers of atomic scale thickness, the electron-hole Coulomb interaction potential is strongly influenced by the sharp discontinuity of the dielectric function across the layer plane. This feature results in peculiar non-hydrogenic exciton states, in which exciton-mediated optical nonlinearities are predicted to be enhanced as compared to their hydrogenic counterpart. To demonstrate this enhancement, we analysed the results of an optical transmission spectroscopy experiment of a monolayer -MoSe2 -placed in the strong coupling regime with the mode of an optical microcavity quantitatively with our nonlinear input-output theory. We find an enhancement of both the exciton-exciton interaction and of the excitonic fermionic saturation with respect to realistic values expected in the hydrogenic picture. Third, we studied the effect of polaritonpolariton interactions on the dispersion relation of polaritonic graphene which is an artificial honeycomb lattice etched on a semiconductor and placed inside a microcavity. We analysed the condensate elementary excitations using the Bogoliubov theory for resonantly driven polaritons on a lattice and compared with experimental observations.

Introduction1

  Introduction to Exciton Polaritons 1.1 Concept of a polariton . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Exciton Polaritons in bulk semiconductors . . . . . . . . . . . . . . . 1.3 Exciton Polaritons in a semiconductor microcavity . . . . . . . . . . . 1.4 Generalized Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . 1.5 Bogoliubov dispersion of elementary excitations . . . . . . . . . . . . I Mobile Impurity in a polariton fluid: Bose Polaron Polariton 2 Background on Polarons 2.1 What are Polarons? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fröhlich Polaron in an atomic BEC . . . . . . . . . . . . . . . . . . . 3 Bose Polaron in a quantum fluid of light 3.1 Physical system and model Hamiltonian . . . . . . . . . . . . . . . . 3.2 Quantum dynamical equations and observables . . . . . . . . . . . . . 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . II Polariton-Polariton Interactions in Monolayer TMDC Microcavities 4 Background on TMDC monolayers 4.1 What are TMDC monolayers? . . . . . . . . . . . . . . . . . . . . . . 4.2 Excitonic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Strong light-matter coupling . . . . . . . . . . . . . . . . . . . . . . . i Introduction L' interaction entre les particules dans un fluide quantique conduit souvent à

Figure 1 . 1 :

 11 Figure 1.1: Various flavours of polaritons in layers of 2D materials. Conduction electrons in graphene and topological insulators (surface plasmon polaritons), infrared-active phonons in boron nitride (phonon polaritons), excitons in dichalcogenide materials (exciton polaritons), superfluidity in FeSe and Cu based superconductors with high critical temperature Tc (Cooper-pair polaritons), and magnetic resonances (magnon polaritons).Figure taken from[START_REF] Basov | Polaritons in van Der Waals Materials[END_REF]

  Figure 1.1: Various flavours of polaritons in layers of 2D materials. Conduction electrons in graphene and topological insulators (surface plasmon polaritons), infrared-active phonons in boron nitride (phonon polaritons), excitons in dichalcogenide materials (exciton polaritons), superfluidity in FeSe and Cu based superconductors with high critical temperature Tc (Cooper-pair polaritons), and magnetic resonances (magnon polaritons).Figure taken from[START_REF] Basov | Polaritons in van Der Waals Materials[END_REF]

Chapter 1 .

 1 Figure 1.2: Schematic illustration of exciton formation in a semiconductor with the absorption of a photon of energy close to the band gap.

Figure 1

 1 Figure 1.3: Left: Absorption spectrum of a sample of 𝐺𝑎𝐴𝑠 crystal at 1.2 K. The 𝑛 = 1, 2 and 3 peaks are shown along with the extrapolated band gap characterized by 𝑛 = ∞.The fine structure of the excitons is too narrow to be resolved here. The peak 𝐷 0 -𝑋 corresponds to an exciton 𝑋 bound to a neutral donor atom 𝐷 0 -an impurity in the sample -which is not considered in our hydrogenic model of excitons as we treat our semiconducting crystal as an ideal crystal devoid of any impurities. Figure fromWeisbuch, Benisty, et al., 2000. Right: Absorption spectrum of a sample of bulk 𝐶𝑢 2 𝑂 at 1.2 K showing exciton resonances for principal quantum number, 𝑛, up to 25. Figure from[START_REF] Kazimierczuk | Giant Rydberg Excitons in the Copper Oxide Cu 2 O[END_REF]

  -Mott excitons only (for the rest of the text exciton implies Wannier-Mott exciton). An illustration of a Wannier-Mott and a Frenkel exciton is shown in Figure 1.4. For a detailed difference between the physical properties of the two types of excitons I'd refer the reader to Klingshirn, 2007 or Haug and S. W. Koch, 1994 which provide a comprehensive account of optical properties of semiconductors.

Figure 1

 1 Figure 1.4: A schematic illustration of a Wannier-Mott (a) and a Frenkel exciton (b).

Figure 1 . 5 :

 15 Figure 1.5: Bulk exciton polariton dispersion showing the upper and lower polariton branches (in solid purple) and their characteristic anticrossing behaviour. The blue dotted line represents the dispersionless exciton whereas the red dotted line represents the linear dispersion of a free photon. Units are = 𝑐 = 1, 𝐸 0 = 𝐸 exc and 𝑘 0 = 𝐸 exc .

Figure 1

 1 Figure 1.6: a) Schematic illustration of a quantum well structure b) Energy structure along the growth or 𝑧 direction showing the formation of discrete subbands due to quantum confinement of the electron and hole quasiparticles.

Figure 1 . 8 :

 18 Figure 1.8: Three cases of exciton polariton dispersion (right column) and their respective Hopfield coefficients (left column) for 𝛿 = 𝜔 0 cav -𝐸 exc : -0.05 𝐸 0 (a-b), 0 (c-d), and 0.05 𝐸 0 (e-f ). The dashed lines are the bare exciton (blue) and cavity photon (red) dispersion. Solid purple curves represent exciton-polaritons with upper polariton (UP) being the higher energy branch and the lower polariton (LP) being the lower energy one. In the right column blue represents the photonic fraction |𝐶 k || | 2 and orange the excitonic fraction |𝑋 k || | 2 . Here we took Ω = 0.05 𝐸 0 . 𝐸 0 can be any relevant energy scale.

Figure 1

 1 Figure 1.9: Observation of microcavity polaritons in the strong-coupling regime. a) Calculated (dotted line) and measured emission (full line) from a planar microcavity. b) polariton dispersion curve deduced from peaks in photoluminescence spectrum. b) Photoluminescence spectra for various various values of angles. Figure from Houdréet al., 1994.

Figure 1 .

 1 Figure 1.10: Upper panel: polariton density as a function of the pump intensity with the left column showing the optical delimiter regime and the right column showing the optical bistable regime for k p = 0 . The dashed line marks the region of instabilities in the system. Lower panel: real and imaginary parts of the Bogoliubov spectrum corresponding to the points marked by letters in the upper panel. The blue palette spectrum are in the stable regime and the red palette one are in the unstable regime. In this figure we took 𝜔 𝑝 -𝐸 0 𝑙 = -0.5𝛾 for the delimiter regime and 𝜔 𝑝 -𝐸 0 𝑙 = 2𝛾 for the bistable regime. Further, 𝑔 LP = 0.1𝛾𝜉 -2 0 where 𝜉 0 = √ 2 /2𝑚 LP 𝑔 LP 𝑛 LP .

Figure 1

 1 Figure 1.11: Upper panel: polariton density as a function of the pump intensity with the left column showing the optical delimiter regime and the right column showing the optical bistable regime for k p = 0 .1 𝜉 -1 0 . Lower panel: Tilting of the Bogoliubov spectra for a polariton fluid in motion as a consequence of Doppler shift. The simulation parameters are the same as in Figure 1.10.

Contents 2 . 1

 21 What are Polarons? . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.2 Fröhlich Polaron in an atomic BEC . . . . . . . . . . . . . . . . 54 2.2.1 Derivation of Bogoliubov-Fröhlich Hamiltonian . . . . . 55 51 Chapter 2. Background on Polarons

Figure 2 . 1 :

 21 Figure 2.1: Schematic illustration of an electron in an ionic lattice and formation of a polaron. Figure from Franchini et al., 2021.

Figure 2 . 2 :

 22 Figure 2.2: Qualitative phase diagram describing various approaches to solve the polaron problem. Figure from Grusdt and Demler, 2015

Figure 3 . 1 :

 31 Figure 3.1: Schematic of the system setup considered in the present work. Left panel: excitons (x) and impurities (i) are placed in a photonic cavity (c), pumped by laser light and subjected to losses. Central panel: as a result of the strong light-matter coupling, a polariton fluid is created and the impurity is dressed by its excitations, giving rise to a Bose polaron (BP). Right panel: illustration on how this set up could be experimentally realized in semiconductor microcavities. To achieve the quasi-onedimensional geometry considered in the present work, a specific etched structure can be realized. The figure shows also how to set into motion the polariton fluid by suitably choosing the angle of incidence of the laser beam.

Figure 3

 3 Figure 3.2: a) Ratio of effective mass to bare mass, 𝑀 eff /𝑀 of the impurity as a function of the detuning, ∆ -in the units of blueshift 𝑔𝑛. The shaded regions from left to right depicts the regions (I), (II) and (III) corresponding to different types of excitation spectrum of the fluid shown in Panel b) and the dashed lines indicate the point where effective mass coincides with bare mass of the impurity. Here the impurity-polariton mass ratio is 𝑀/𝑚 = 10. b) Imaginary (top) and real (bottom) parts of the Bogoliubov spectrum of polaritons for varying values of detuning ∆ leading to (I) gapless or gapped spectrum, (II) diffusive spectrum around k = 0 and (III) diffusive spectrum around finite k points. In all the figures we have used Γ = 1.5 𝑔𝑛 and k 𝑝 = 0.

Figure 3 . 3 :

 33 Figure 3.3: Drag Force, 𝐹 drag , experienced by polaron at rest as a function of the detuning ∆ -in units of interaction energy, 𝑔𝑛 -and the polariton pump wavevector k 𝑝 -in units of inverse healing length 𝜉 = / √ 𝑚𝑔𝑛 -which quantifies the fluid flow velocity, 𝑣 fl = k 𝑝 /𝑚. The red line depicts the Landau critical velocity, 𝑣 𝑐 as function of detuning ∆, with 𝑣 𝑐 = min k 𝐸 𝑏 (k)/|k|. Here we have taken 𝑀/𝑚 = 10 for the impurity-polariton mass ratio and Γ = 1.5 𝑔𝑛.

Figure 3

 3 Figure 3.4: a) Excitation momentum Π as a function of polaron momentum 𝑝, both in units of 𝑝 0 = √ 𝑚𝑔𝑛 for various values of detuning ∆ = -1.5, 0, 1.5, 2.5, 3.5 -in units of 𝑔𝑛 -going from darker to lighter shaded curves respectively, for a fluid at rest. The arrows depict the momentum flow towards the long time limit state, where the polaron attains a saturation or terminal momentum. The saturation momentum values are depicted by the black dots on 𝑝 axis (roots of excitation momentum Π(𝑝)) and are dependent on the initial value of the impurity momentum 𝑝. In the inset we show the roots of the excitation momentum, as a function of ∆ in units of 𝑔𝑛. The gray-shaded region in the inset marks the region of negative drag. b) and c) Heatmaps of impurity momentum, p/𝑝 0 as a function of time, 𝑡, in units of 𝑡 0 = /𝑔𝑛, and detuning ∆, in units of 𝑔𝑛, for different values of the initial momentum: c) 𝑝 = 6 𝑝 0 , d) 𝑝 = 12 𝑝 0 . In all panels we have taken 𝑀/𝑚 = 10, Γ = 1.5 𝑔𝑛 and the fluid is at rest.

Figure 3 . 5 :

 35 Figure 3.5: Polaron saturation momentum: absolute value of the polaron saturation momentum 𝑝 in units of 𝑝 0 = √ 𝑚𝑔𝑛 as a function of detuning, ∆, and dissipation, Γ -both in units of blueshift 𝑔𝑛, for a fluid at rest. The gray shaded area represents the region of dynamical instability of the Bogoliubov theory. The region separating zero and positive saturation momentum of the impurity corresponds to the negative drag regime.

Contents 4 . 1

 41 What are TMDC monolayers? . . . . . . . . . . . . . . . . . . . 78 4.1.1 Electronic Bandstructure . . . . . . . . . . . . . . . . . . 78 4.2 Excitonic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2.1 Screening of Coulomb interaction . . . . . . . . . . . . . 81 4.2.2 Non-hydrogenic exciton states in TMDCs . . . . . . . . . 82 4.3 Strong light-matter coupling . . . . . . . . . . . . . . . . . . . . 84

Figure 4

 4 Figure 4.1: a) The trigonal prismatic structure of TMDCs where the honeycomb lattice can be observed when viewed from the top and the first Brillouin zone with the points of high symmetry. Figure from Mak and Shan, 2016

Figure 4 . 2 :

 42 Figure 4.2: The band structure of 𝑀 𝑜𝑆 2 of (i) bulk and (ii) monolayer, calculated from first principles showing the transition from and indirect to a direct bandgap semiconductor. Figure from Berkelbach and Reichman, 2018

Figure 4 . 3 :

 43 Figure 4.3: Schematic of the dielectric model of a TMDC monolayer sandwiched between surrounding layers of a different dielectric material (for instance in vacuum 𝜖 1 = 𝜖 2 = 1).Figure from Berkelbach and Reichman, 2018

Figure 4

 4 Figure 4.5: Exciton-Polariton dispersion observed by X. Liu et al., 2015 using angleresolved reflection spectroscopy utilizing 𝑀 𝑜𝑆 2 monolayer in a microcavity. Left panel shows the reflectivity spectra showing the formation of the two polariton modes. Right panels shows the dispersion relation obtained using the spectral data showing the anti-crossing between the A exciton and cavity photon modes.

Contents 5 . 1

 51 Polariton resonances from Experiment . . . . . . . . . . . . . . . 86 5.1.1 Interpretation of experimental data . . . . . . . . . . . . 88 5.2 Numerical determination of g𝑥 (0) and g𝑠 (0) . . . . . . . . . . . . 88 5.2.1 Mean field Input Output Theory . . . . . . . . . . . . . 88 5.2.2 Results of the Optimization procedure . . . . . . . . . . 91 5.3 Comparison of nonlinear constants in TMDCs and QW materials 94 5.4 Dependence on polarization angle . . . . . . . . . . . . . . . . . 95 5.5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . 97

Figure 5

 5 Figure 5.1: (a) Outline of the microcavity structure: a MoSe2 monolayer (red) and bilayer (green) embedded in polymethyl methacrylate (PMMA) (pale blue) are sandwiched between two Bragg mirrors (DBR). (b) White light image of the microcavity; the MoSe2 layers have been color shaded and outlined. The experiments performed at 105 K and 127 K were realized in the area labeled (1) and (2)respectively. The bare cavity measurements have been done in the area labeled (3). (c) Polaritonic splitting of the excitonic and the cavity photon modes as seen the experiment. The upper and lower polariton modes, the bare excitonic transition, and the bare cavity mode are calculated in a coupled oscillators model (see next section), are shown in solid green and black, dashed red, and dashed black respectively.

Figure 5 . 2 :

 52 Figure 5.2: Normalized transmission spectra 𝑇 (𝜔)/Max(𝑇 ) at a) 𝑇 = 127 K and b) 𝑇 = 105 K, provided from experiment. The spectra are stacked from the lowest used pulse energy 𝑊 (bottom) to the highest (top). The pulse energy 𝑊 used for each spectrum is indicated on the right axis. The laser pulse spectrum is shown as a red dotted line. The dashed vertical black lines in highlight the polaritonic resonances in the linear (non-interacting) regime. The theoretical predictions are shown as solid gray lines.

Figure 5 . 3 :

 53 Figure 5.3: Spatial fluctuations of the cavity resonance and excitonic transition (a) Cavity resonance 𝜔 𝑐 (r) = 𝜔 𝑐,0 + Re[𝑉 𝑐 (r)] frequency (red) and excitonic transition frequency 𝜔 𝑥 (r) = 𝜔 𝑥,0 + Re[𝑉 𝑥 (r)] (blue) as a function of position across the laser spot. (b) Cavity resonance dissipation rate 𝛾 𝑐 = 𝛾 𝑐,0 + Im[𝑉 𝑐 (r)] (red) and excitonic dissipation rate 𝛾 𝑥 (blue) as a function of position across the laser spot. The latter is assumed constant across the excitation spot diameter. These data have been obtained from the spatially resolved spectra obtained at T=127K

Figure 5

 5 Figure 5.4: Spatially resolved lower-polariton transmission peak energy at 𝑇 = 127 K, across the excitation spot diameter, for increasing 𝑊 (same color code as in the spectra of Figure 5.2). The spatial laser intensity profile is shown as a red dotted line. The spectra in Figure 5.2 were measured at 𝑦 = 0 represented by the dashed vertical line. The solid circles with error bars represent the experimental data and solid lines represent the theoretical predictions.

Figure 5

 5 Figure 5.5: g𝑥 (0) dependence on 𝜇, and d) g𝑠 (0) dependence on 𝑎 𝐵 . The dashed line and grey area show the hydrogenic theory corrected by a factor 𝛼 0 = 3.3 ± 0.8 following (Estrecho et al., 2019). The black squares highlight the hydrogenic theory for CdTe, ZnSe, ZnO, CuBr and CuCl. The hollow square shows the 30% enhanced non-hydrogenic theory for MoSe 2 . The blue diamond is a measurement in a GaAs microcavity taken from. Our quantitative estimates for g𝑥 (0) and g𝑠 (0) are shown as red circles. The upper axis in left panel indicates the bulk exciton binding energy for each materials. Excitons that are stable at room temperature are shown on the right side of the vertical dashed line.

Figure 5

 5 Figure 5.6: Color-scaled normalized transmission spectra obtained at 𝑊 = 451 pJ versus the polarization state |𝜃⟩ (vertical axis).The circle symbols show the lower polariton peak energy; the solid black line is a theoretical fit 𝐵 0 +𝐵 𝜃 sin 2 2𝜃, following Equations eq5.12 and (5.13), where 𝐵 0 = 7.9 meV and 𝐵 𝜃 = -2.7 meV.

Figure 6

 6 Figure 6.2: a) The two band single-particle spectrum of graphene with a closeup near a Dirac point. b) Top view of the upper energy band where the first Brillouin zone in a hexagon shape can be easily seen. c) Dispersion relation with a cut along the triangular section marked in red in b). This cuts traverses all the high symmetry points: Γ at the center of the Brioullin zone, two equivalent Dirac points 𝒦 and 𝒦 ′ and the 𝑀 point at the center of the line connecting two adjacent Dirac points. Here energy is in units of nearest neighbour hopping energy, 𝑡 and 𝑡 ′ = -0.1 𝑡.

Figure 6

 6 Figure 6.3: (a) SEM image of the corner of the fist honeycomb lattice sample realized in the group of J. Bloch in Orsay. Blue circles design six pillars form-ing one hexagon of the lattice. (b) Top view of the corner hexagon (c) Top view of the lattice close to the edge. Potential minima are at the center of each pillar belonging to hexagonal lattice. Holes in between pillars, which represent potential maxima, form triangular lattice visible in the image. (d) zoomed view of the lattice edge from (c).Figure from Milicevic, 2018.

Figure 6 . 4 :

 64 Figure 6.4: Illustration of hopping energies due to TE-TM splitting. Figure from Nalitov et al., 2015

Figure 6 . 5 :

 65 Figure 6.5: In left column we can see the effect of SOC or TE-TM splitting (a), Birefringence or XY-splitting (c) and the combination of both effects (e) on the single particle dispersion relation in a honeycomb lattice. The right column shows SOC (b), birefringence (d) and both (f ) close to the Dirac point 𝒦 along the 𝑘 𝑦 axis, with 𝑘 𝑥 fixed at 2/3 √ 3 (2𝜋/𝑎), describing the position of the Dirac point in wavevector space.

  7.1. Polariton-Polariton interactions in honeycomb lattice

Figure 7 . 2 :

 72 Figure 7.2: Illustration showing the shift of reciprocal lattice (grey) for ghost Bogoliubov modes with respect to the reciprocal lattice (grey) of normal Bogoliubov modes when we pump at the 𝒦 point marked by red circle.

  (7.14) where 𝜌 k (𝜔) is the phonon density of state and is taken as a constant in this case. We can similarly express the other fluctuation amplitudes.In Figures 7.4, 7.5 and 7.6 we show the comparison between the experimental observations and our theoretical predictions for a detuning value equal to 0.7 meV using linear response theory for increasing pump powers (increasing with number of rows: 20mW, 183mW and 375mW) in the polarization basis co-polarized with the pump polarization along different cuts of the Brillouin zone as shown in Figure7.3: 𝑀 Γ𝑀 cut (red line in Figure7.4 ), 𝒦 cut (green line in Figure7.5) and 𝒦 ′ 𝒦 ′ cut (blue line in Figure7.6). Inside each figure the left panel is the experiment and the right one is the theoretical prediction. The white curves indicates the normal branch solutions of the Bogoliubov matrix.

Figure 7 . 3 :

 73 Figure 7.3: The reciprocal k space of a honeycomb lattice showing the different cuts which we use for comparison with experimental data.

Figure 7 . 4 :

 74 Figure 7.4: Comparison between the experimental observations and theoretical predictions for a detuning value equal to 0.7 meV along the 𝑀 Γ𝑀 cut. The experimental parameters used for the theoretical analysis are 𝑡 L = 0.317 meV, 𝑡 T = 0.338 meV, 𝑡 ′ L = -0.016 meV, 𝑡 ′ T = -0.017 meV, 𝛿 = 0.09 meV, Γ = 0.32 meV and the 𝒦 point energy is 1480.86 meV.

Figure 7 . 5 :

 75 Figure 7.5: Comparison between the experimental observations and theoretical predictions for a detuning value equal to 0.7 meV along the 𝒦 cut. The experimental parameters are the same as in Figure 7.4 .

Figure 7 . 6 :

 76 Figure 7.6: Comparison between the experimental observations and theoretical predictions for a detuning value equal to 0.7 meV along the 𝒦 ′ 𝒦 ′ cut. The experimental parameters are the same as in Figure 7.4.

Figure 7 . 7 :

 77 Figure 7.7: Comparison between the experimental observations and theoretical predictions for a detuning value equal to 0.34 meV along the Γ cut. The experimental parameters are the same as in Figure 7.4.

Figure 7 . 8 :

 78 Figure 7.8: Comparison between the experimental observations and theoretical predictions for a detuning value equal to 0.34 meV along the 𝒦 cut. The experimental parameters are the same as in Figure 7.4.

  . This happens because bk and b † k fail to satisfy the bosonic commutation relations upheld by the condition |𝑢 k | 2 -|𝑣 k | 2 = 1 which in this instance is zero. However, we may still diagonalize the Hamiltonian using non-standard Bogoliubov operators given by bk𝑝+k = 𝑢 k âk𝑝+k + 𝑣 -k â † k𝑝-k (B.1) bk𝑝-k = 𝑣 k âk𝑝+k + 𝑢 -k â † k𝑝-k . (B.2)

  Renard, J., Schneider, C. & Richard, M., Exciton-

	Exciton Interaction beyond the Hydrogenic Picture in a MoSe 2 Monolayer in
	the Strong Light-Matter Coupling Regime, Phys. Rev. Lett. 126, 167401
	(2021)
	• Vashisht, A., Stepanov, P., Harder, T., Victorin, N., Benatre, D., Schneider,
	C., Klembt, S., Minguzzi, A. & Richard, M. Excitation spectrum of a polariton

condensate sitting at the K point of a Honeycomb lattice, A soumettre

TABLE OF CONTENTS

 OF physics, a polaron state describes electrons dressed by lattice phonons

TABLE OF CONTENTS

 OF

TABLE OF CONTENTS

 OF six corners of the first Brioullin zone.

  Contents 6.1 Honeycomb lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 102

	6.1. Honeycomb lattice
	𝐴 to 𝐵) in real space are given by
	6.1.1 Single particle dispersion: Tight-binding approximation . 104
	6.1.2 Polaritonic graphene . . . . . . . . . . . . . . . . . . . . 105
	6.2 Effect of Spin-orbit coupling and birefringence . . . . . . . . . . 106
	6.2.1 TE-TM splitting . . . . . . . . . . . . . . . . . . . . . . 106
	6.2.2 XY splitting . . . . . . . . . . . . . . . . . . . . . . . . . 108
	6.2.3 Circular basis transformation . . . . . . . . . . . . . . . 110

  𝑖k

  • 𝛿 𝜎 - ) 𝑇 is a vector signifying the pumping strength and polarization of the pump. The steady state wavefunction is a bispinor of the form, Ψ ss = 𝑒 𝑖k𝑝 • a 𝑖 -𝑖𝜔𝑝𝑡 ( 𝑇 where k 𝑝 and 𝜔 𝑝 are the pump wavevector and frequency respectively.

	𝑎 ss + , 𝑎 ss -, 𝑏 ss + , 𝑏 ss

-

)

  .13) Correspondingly, we have |𝑎 k+ (𝜔)| 2 = |𝜒 11 𝑎 ss + + 𝜒 12 𝑎 ss -+ 𝜒 13 𝑏 ss + + 𝜒 14 𝑏 ss --𝜒 15 𝑎 ss +

* -𝜒 16 𝑎 ss -* -𝜒 17 𝑏 ss + * -𝜒 18 𝑏 ss -* | 2 𝜌 k (𝜔),

An exciton state will usually imply 1𝑠 state in the Rydberg series onwards

It is also possible to introduce interactions via a Kerr nonlinearity among the photonic fraction of polaritons by using a nonlinear optical medium[START_REF] Boyd | Nonlinear optics[END_REF]

We'll explicitly calculate this later in part I chapter 3 of the thesis.

Hysteretic behaviour with proper turning points ('S like' shape) and not just a signature of the behaviour is, however, only obtained when 𝜔 𝑝 > 𝐸 𝑙,k𝑝 + √ 3/2 𝛾 LP

Note that in the unstable regions where the maxima of the imaginary part of the Bogoliubov modes is greater than zero our linearization procedure is no longer valid as such a regime leads to a rapid depletion of the steady state condensate because of a diverging elementary excitation spectrum.

As opposed to the Holstein or small polaron. See(Franchini et al.,

2021) for the latest review.

The formalism is valid for and easily translated to lower dimensions

For the case of diffusive spectrum, Δ > 0, the effective mass was extracted using the real part of the impurity momentum Π, as its imaginary part adds to the dissipation of the polariton fluid hence does not contribute to the effective mass.

Moving forward the semiconductor and monolayer nature is implied when we say TMDCs.

There is splitting also in the conduction band but it is very small compared to the valence band one that for our purposes we can consider it to be effectively spin degenerate

We considered the ground state energy to be zero just by changing the energy scale and are in the center of mass frame.

Note that exciton-exciton induced broadening is a second-order contribution[START_REF] Boldt | Many-Body Theory for the Dense Exciton Gas of Direct Semiconductors II. Calculation of Exciton Level Shift and Damping in Dependence on Exciton Density[END_REF] which is typically ten times smaller than the first-order ones in TMDCs(Moody et al.,

2015)

Since in the experimental data polariton modes didn't exhibit any birefringence, we have neglected its effects here.

We consider that an excitonic state is stable at room temperature when its binding energy 𝐸 𝑏 = 0.5𝑘 𝑏 × 300 K where the factor 0.5 accounts for the fact that in a quantum well of thickness comparable with the Bohr radius, 𝐸 𝑏 is typically twice that in the bulk[START_REF] Mathieu | Simple Analytical Method for Calculating Exciton Binding Energies in Semiconductor Quantum Wells[END_REF].

There are three equivalent pairs of Dirac points in the first Brillouin zone. Here I have mentioned only such pair as the physics remains unchanged regardless of the pair chosen.

For a review of polarization dependent phenomenon in polaritons see[START_REF] Shelykh | Polariton Polarization-Sensitive Phenomena in Planar Semiconductor Microcavities[END_REF]

Even though we are considering a lot of parameters in the model the simplicity lies in the fact that it is still obtained using a mean field tight binding approach.

Honeycomb lattice

The honeycomb lattice structure is primarily responsible for the novel single particle properties of graphene [START_REF] Castro Neto | The Electronic Properties of Graphene[END_REF]. Its hexagonal structure is made up of two interpenetrating triangular sublattices 𝐴 and 𝐵 or it can be regarded as a triangular Bravais lattice with a two point basis, 𝑎(0, 0) and 𝑎(0, -1) (see Fig. 6.1a)). The lattice vectors are given as

where, 𝑎 is the lattice constant. It's reciprocal lattice is also hexagonal with the reciprocal lattice vectors,

The high symmetry points of the Brillouin zone are also shown in Fig. 6.1b), out of which the points 𝒦 and 𝒦 ′ -called Dirac points -at the corners of the hexagon of the Brillouin zone are particularly important. Their positions are given as

) (6.

3)

The three nearest-neighbours to a lattice point (moving from a point in sublattice

Interaction effects in Polaritonic Graphene

In this chapter I'll introduce the polariton-polariton interactions into the Hamiltonian of the honeycomb lattice system within the tight binding approximation. Next, I'll calculate the Bogoliubov dispersion for elementary excitations over a steady state of the system, considering the case of coherent pumping at 𝒦 point. I'll compare our predictions with the experimental observations by conducting a linear response analysis of the fluctuation amplitudes.

A Input Output model in Polariton basis

In order to better understand the 𝜃-dependence of the spectral shift that we observe in spin anisotropic observations in Chapter 5, we can further transform the equations of motion to write them into the polariton basis.

is the polaritons basis expressed in the exciton photon basis. |𝐿, 𝑈 ⟩ are the upper and lower polariton states, |𝑥, 𝑐⟩ are the exciton and cavity photon states, and (𝑋, 𝐶) are the usual excitonic and photonic Hopfield coefficients. The equation of motions in this basis read

where 𝜔 𝐿,𝑈,0 , 𝛾 𝐿,𝑈 , and 𝑚 𝐿,𝑈 are the lower and upper polaritons rest energy, linewidth and effective mass respectively. 𝐴 𝐿,𝑈,𝑖𝑛 is the laser amplitude at the lower and upper polaritons energy. ∆ 𝑈/𝐿 gather all the nonlinear terms in the upper (U) and lower (L) polariton field equations.

Appendix

• ∆ 𝐿,𝑟 contains two terms proportional to |𝜓 𝐿 | 2 𝜓 𝐿 |𝜓 𝑈 | 2 𝜓 𝐿 . The frequency of these two terms is resonant with the lower polaritons frequency such that ∆ 𝐿,𝑟 contributes directly to the field dynamics.

• ∆ 𝐿,𝑛𝑟 contains terms proportional to 𝜓 2 𝐿 𝜓 * 𝑈 and 𝜓 2 𝑈 𝜓 * 𝐿 , that oscillate at the frequencies 2𝜔 𝐿 -𝜔 𝑈 and 2𝜔 𝑈 -𝜔 𝐿 respectively. Neither frequencies is resonant either with the upper or the lower polariton frequencies. Upon integration over time, ∆ 𝐿,𝑛𝑟 is thus going to vanish and can be neglected.

• ∆ 𝐿,Ω contain terms proportional to |𝜓 𝐿 | 2 𝜓 𝑈 and |𝜓 𝑈 | 2 𝜓 𝑈 . These terms are not resonant with the lower polariton frequency, but with the upper. Since analogue terms are present in the upper polariton equation of motion, ∆ 𝐿,Ω is a nonlinear correction to the Rabi splitting. However, owing the the large frequency splitting between the upper and lower polaritons as compared to the interaction energy, one can easily show that as long as these two terms are much smaller than 𝜔 𝑈 -𝜔 𝐿 > Ω, ∆ 𝐿,Ω can be neglected as compared to ∆ 𝐿,𝑟 .

After discarding the terms that average to zero, ∆ 𝑈/𝐿 read These equation allow a very general understanding of the different contributions to the spectral shifts of the lower and upper polaritons, including when a comparable population occupies the lower and upper polariton states, like in our case.

Chapter B. Generalized Bogoliubov Transformation an imaginary contribution to the energy spectrum as Im{𝐸 b -𝑖Γ} +Im{Π}.