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Quantum Theory of Light in Linear Media: Applications to Quantum Optics and Quantum Plasmonics Quantum theory of light in linear media : applications to quantum optics and quantum plasmonics Vincent Dorier To cite this version: Vincent Dorier. Quantum theory of light in linear media : applications to quantum optics and quan- tum plasmonics. Physics [physics]. Université Bourgogne Franche-Comté, 2020. English. NNT : 2020UBFCK006. tel-02954369 HAL Id: tel-02954369 https://tel.archives-ouvertes.fr/tel-02954369 Submitted on 1 Oct 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE DE DOCTORAT DE L’ETABLISSEMENT UNIVERSITE BOURGOGNE FRANCHE-COMTE PREPAREE AU LABORATOIRE INTERDISCIPLINAIRE CARNOT DE BOURGOGNE Ecole doctorale n°553 Carnot Pasteur Doctorat de Physique Par M. Vincent Dorier Quantum theory of light in linear media: Applications to quantum optics and quantum plasmonics Thèse présentée et soutenue à Dijon, le 21/02/2020 Composition du Jury : M. Gérard COLAS DES FRANCS Professeur, Laboratoire ICB, UMR 6303 CNRS Président M. Michael FLEISCHHAUER Professeur, Universitat Keiserslautern Rapporteur M. Stephan DE BIEVRE Professeur, Laboratoire Paul Painlevé, UMR 8524 CNRS Rapporteur Mme Almut BEIGE Professeur associé, School of Physics and Astronomy, University of Leeds Examinatrice M. Stéphane GUERIN Professeur, Laboratoire ICB, UMR 6303 CNRS Examinateur M. Hans-Rudolf JAUSLIN Professeur, Laboratoire ICB, UMR 6303 CNRS Directeur de thèse Titre : Théorie quantique de la lumière dans les milieux linéaires : applications à l’optique quantique et la plasmonique quantique Mots clés : Quantification, Plasmon, Optique Résumé : Ce travail de thèse présente une stratégie de quantification de la lumière dans les milieux linéaire. Basée sur la construction de modèles hamiltoniens compatibles avec les équations de Maxwell, elle utilise la diagonalisation de l’hamiltonien par une transformation canonique pour préserver la structure du modèle, et un principe de correspondance pour transformer les variables canoniques en opérateurs agissant sur un espace de Fock quantique. Nous montrons que cette stratégie générale peut être appliquée au cas de photons dans des milieux passifs (i.e., sans dissipation), ainsi que pour les plasmons dans des milieux de type Kramers-Kronig. Nous utilisons ce formalisme pour décrire l’effet Hong-Ou-Mandel résolu en temps, ainsi que l’émission spontanée de plasmons. Title : Quantum theory of light in linear media : applications to quantum optics and quantum plasmonics Keywords : Quantization, Plasmon, Optics Abstract : This PhD contains a strategy of quantization of light in linear media. Based on the construction of hamiltonian models compatible with Maxwell’s equations, it uses the diagonalization of the Hamiltonian via a canonical transformation to preserved the structure of the model, and a principle of correspondence to transform the canonical variables into operators acting on a quantum Fock space. We show that this general strategy can be applied to the case of photons in passive media (i.e., without dissipation), as well as plasmons in Kramers-Kronig media. We use this formalism to describe the time-resolved Hong-Ou-Mandel effect, and the spontaneous emission of plasmons. Université Bourgogne Franche-Comté 32, avenue de l’Observatoire 25000 Besançon Contents Contents 1 Acknowledgement4 Introduction 5 1 Classical electromagnetism 13 1.1 Circuits of light . 14 1.2 Maxwell’s equations . 15 1.2.1 Microscopic and macroscopic formulations . 15 1.2.2 Vector and scalar potentials . 17 1.2.3 Polarization density and electric susceptibility . 17 1.2.4 Fourier domain and dielectric coefficient . 19 1.3 Models for matter . 20 1.3.1 Passive media . 21 Definition of a passive medium . 21 Maxwell’s equations in a passive medium . 22 Generalized Coulomb gauge . 22 1.3.2 Drude model . 23 1.3.3 Lorentz model . 26 1.3.4 Drude-Lorentz model . 26 1.4 Summary on classical electromagnetism . 27 I Photons 29 2 Canonical optics 31 2.1 Hamiltonian structure . 32 2.1.1 Starting equations . 32 2.1.2 Hamiltonian function . 32 2.2 Electromagnetic configurations . 34 2.2.1 Phase space and complex representation . 34 2.2.2 Modes and eigenconfigurations . 36 2.2.3 Normal mode decomposition . 38 2.2.4 Spectral structure (in vacuum) . 39 2.3 Summary on the classical Hamiltonian system . 41 1 CONTENTS 3 Quantum optics 43 3.1 Quantization procedure in a reduced model . 44 3.1.1 Principle of correspondence . 44 3.1.2 General configurations . 45 3.1.3 Ground states . 46 3.1.4 Ladder space . 46 3.2 Photons . 47 3.2.1 Bosonic Fock space . 47 3.2.2 Creation-annihilation operators . 48 3.2.3 Number operator . 49 3.2.4 Hilbert spaces of classical electromagnetic configurations . 49 3.2.5 Electromagnetic field operators and Hamiltonian . 51 3.2.6 The quantum field operator . 52 3.3 Summary on the spaces . 53 3.4 The key to the full model . 54 3.4.1 The infinite Fock space . 54 3.4.2 The different roles of operators in quantum optics . 55 3.4.3 Remarks on the physical status of the reduced model . 56 3.5 Summary on the quantum model . 56 4 Photon dynamics 59 4.1 Time evolution of photon states . 60 4.2 One photon through a beam splitter . 61 4.2.1 Classical description . 61 4.2.2 Quantum dynamics of the photon . 63 4.3 Hong-Ou-Mandel effect . 63 4.3.1 General setup . 64 4.3.2 Usual HOM effect . 65 4.3.3 Discussion on classical dynamics . 67 4.4 Hong-Ou-Mandel with a fast detector . 67 4.4.1 Time-resolved HOM effect . 68 4.4.2 Time-resolved HOM with phase flip . 69 4.4.3 Multi-pulses HOM with phase flips . 71 4.5 Summary on the dynamics of photons . 72 II Plasmons 75 5 Canonical plasmonics 77 5.1 Microscopic model . 78 5.1.1 Light . 79 5.1.2 Matter . 79 5.1.3 Interaction . 80 5.1.4 Equivalence with Maxwell’s macroscopic equations . 81 5.1.5 Hamiltonian structure . 83 5.1.6 Coupling function in standard models . 84 Drudemodel .................................. 85 Lorentz model . 85 5.2 Canonical diagonalization of the model . 86 5.2.1 Pre-diagonalization of the electromagnetic part . 86 5.2.2 Integral operators, matrix form and canonical transformations . 87 5.2.3 Finding the harmonic-like Hamiltonian . 89 2 CONTENTS 5.2.4 Møller wave operator . 90 5.2.5 Lippmann-Schwinger equation . 92 5.2.6 Degeneracy and block structure . 94 5.2.7 Phase space of plasmonic configurations . 96 5.2.8 Orthonormalization and completeness of the eigenconfigurations . 97 5.2.9 Diagonal Hamiltonian . 98 5.3 Summary on the classical Hamiltonian system . 99 6 Quantum plasmonics 101 6.1 Quantization . 102 6.1.1 Bosonic operators and Hamiltonian . 102 6.1.2 Fock space and quantum plasmons . 103 6.1.3 Electric field operator . 104 6.1.4 Other fields and commutation relations . 105 6.1.5 No-coupling limit . 106 6.2 Solving the Lippmann-Schwinger equation . 107 6.2.1 Perturbative series . 107 6.2.2 Small coupling regime . 108 Zeroth order . 108 First order . 109 6.2.3 Small volume regime . 111 Partial exact diagonalization . 111 Zeroth order . 114 First order . 114 Second order . 115 6.3 Comments on the literature . 115 6.3.1 Summary of the previous approaches in literature . 116 6.3.2 The problem of the no-coupling limit . 118 For the bosonic operators . 118 For the Hamiltonian . 119 For the electric field operator . 119 Remarks on an infinite bulk.
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