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The Photon Wave Function in Principle and in Practice Vincent Debierre To cite this version: Vincent Debierre. The Photon Wave Function in Principle and in Practice. Optics [physics.optics]. Ecole Centrale Marseille, 2015. English. NNT : 2015ECDM0004. tel-01406401 HAL Id: tel-01406401 https://tel.archives-ouvertes.fr/tel-01406401 Submitted on 1 Dec 2016 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. Thèse présentée à l’École Centrale de Marseille pour obtenir le grade de Docteur École Doctorale : Physique et Sciences de la Matière Spécialité : Optique, Photonique et Traitement d’Image LA FONCTION D’ONDE DU PHOTON EN PRINCIPE ET EN PRATIQUE Vincent DEBIERRE Institut Fresnel (Équipes CLARTÉ et ATHÉNA)—CNRS Soutenue publiquement le 25 Septembre 2015 devant un jury composé de Pascal DEGIOVANNI Rapporteur DR CNRS—Labo. Physique ÉNS Lyon Margaret HAWTON Rapporteur Professor Emeritus—Lakehead University Paolo FACCHI Examinateur Prof. Associato—Univ. Bari Evgueni KARPOV Examinateur Senior Researcher—Univ. Libre de Bruxelles Alexandre MATZKIN Examinateur CR CNRS—Labo. Phys. Théor. Modélisation Frédéric ZOLLA Examinateur PR—Aix-Marseille Université Thomas DURT Directeur de Thèse PR—École Centrale de Marseille André NICOLET Co-directeur de Thèse PR—Aix-Marseille Université This page intentionally left blank TABLE OF CONTENTS Introduction 1 I Photon Wave Function & Localisability 5 Introduction to Part I 7 1 Canonical Quantisation of the Electromagnetic Field 11 1.1 The Poincaré group . 13 1.2 Wigner-Bargmann equation for the Maxwell field . 16 1.3 Canonical quantisation of the Maxwell field: some preliminaries . 21 1.4 Coulomb gauge quantisation . 24 1.A Representations of the Poincaré group and Poincaré-invariant quantisation . 35 2 The Photon Position Operator 59 2.1 A brief history of the photon position operator . 60 2.2 The Hawton position operator . 61 2.A Commutation of the components of the photon position operator . 65 3 Photon Wave Functions 69 3.1 Photon wave function in momentum space . 70 3.2 Wave equation . 71 3.3 Energy and number densities—Nonlocality issues . 71 3.4 Photon wave functions through Glauber’s extraction rule . 73 3.5 Advanced topics . 75 3.6 Photon trajectories in the Bohmian picture of wave mechanics . 79 3.7 Outlook on the photon wave function . 83 3.A Kernel computation in spherical coordinates . 84 4 An Overview of Nonclassicality in Quantum Optics 91 4.1 Elements of photodetection theory . 93 4.2 Single-mode quantum optics: physical modes . 95 4.3 Single-mode quantum optics: quantum optics . 99 4.4 Interlude: n-photon wave functions . 106 4.5 Single mode quantum optics: effective one-point wave function . 110 4.6 Quantum optics and classical electrodynamics: an informal overview . 112 4.A Completeness relation for coherent states . 113 i II Quantum Electrodynamics & Light Emission 117 Introduction to Part II 119 5 Simple Interacting Systems 123 5.1 Nuclear decay and Gamow states . 125 5.2 On the two-level Friedrichs model . 130 5.3 A toy-model for the description of losses in cavity quantum electrodynamics . 134 5.A Quantum scattering: complex poles and their eigenfunctions . 141 6 Transient Regime in the Atom-Field Interaction 145 6.1 Interaction of light with nonrelativistic matter . 147 6.2 The dipole approximation and its shortcomings . 154 6.3 Improving on the dipole approximation: exact atom-field coupling . 158 6.4 Bonus: the ideal cutoff frequency for the dipole approximation . 163 6.5 Discussion . 164 6.A Some useful results of distribution theory . 166 7 Causality in Spontaneous Emission and Hegerfeldt’s Theorem 171 7.1 Hegerfeldt’s theorem as a Paley-Wiener theorem . 172 7.2 Photon propagation in spontaneous emission . 175 7.3 Discussion . 184 7.A The Paley-Wiener theorem . 185 Conclusion 189 ii WORD OF THANKS “After all, science is essentially international, and it is only through lack of the historical sense that national qualities have been attributed to it.” Marie Curie First of all, of course, I wish to thank my thesis advisors. A PhD project is always a collaborative affair, and I am glad to have worked under the supervision of Profs. Thomas Durt and André Nicolet. I have truly enjoyed academic freedom without being left on my own. Both Thomas and André were always interested in what I had to say, or at least, that is the impression I had! I found that Thomas’s style of investigation, radically different from mine, often made us a very complementary pair. Meanwhile, along with “third man” Frédéric Zolla, André provided very helpful suggestions along the way. I am of course also indebted to the members of the defense committee and especially to the two reviewers of this work who kindly accepted the time-consuming task of reading the present manuscript. Warm thanks also go to other collaborators outside and inside the lab, especially Édouard Brainis, Brian Stout and Guillaume Demésy and to past and present officemates Brice Rolly, Victor Grigoriev and Mahmoud Elsawy—not least for coping up with my frequent mumbling to myself. During my stay in Institut Fresnel I am lucky to have met many fellow students, some of them scientific collaborat- ors, and all of them friendly co-workers. A list of all the people I’m thinking about while writing these lines would be too long—and I do mean that—but I want to mention Yoann Brûlé and Isabelle Goessens who were—I feel—the closest to me. I embarked on this PhD adventure at the same time and on the same campus as friends Arnaud Demion, Laurent Chôné and Pierre Manas. Thanks for the shared adventure and the shared moments, guys. A PhD project lives and dies by intellectual curiosity, and I owe mine to my parents, who always encouraged me. Thanks for all the help, and for having so many books at home. Maybe in spite of the epigraph with which I open these acknowledgments, I wish to express my gratitude to the French taxpayer who, whether they liked it or not, provided for me during these three years. As I see it, there’s no better arrangement than being paid to perform calculations. And also a mention to the people at CUMSJ badminton, with whom I shared most of my weeknights. Especially to Vladyslav Atavin who very generously took two full days to sit down with me and patiently translate a paper by Shirokov from Russian. iii AN INCOMPLETE GUIDE TO NOTATIONS General conventions Throughout this work we endeavoured to keep the following conventions: The prime notation x′ is used to denote the image of a quantity x under a Poincaré transformation. The boldface notation x is used to denote three-vectors. The hat notation xˆ is used to denote quantum operators on Hilbert spaces. The overbar notation x¯ is used to denote the Fourier transform of a quantity x. The star notation x∗ is used to denote complex conjugation. The upper plus/minus notation x± is used to denote the positive or negative frequency part of a quantity x. The longitudinal part Fk (x) of a vector field is the inverse Fourier transform of (6.1.6a). The transverse part F⊥ (x) of a vector field is the inverse Fourier transform of (6.1.6b). The brace notation x,y is used to denote the Poisson bracket of two quantities x and y. { } The square bracket notation [ˆx, yˆ] is used to denote the commutator of two operators xˆ and yˆ. Latin indices such as i run over 1,2,3. Greek indices such as µ run over 0,1,2,3. Tensor algebra and the Poincaré group The following notations are especially relevant to chapter 1: The twice covariant metric tensor ηµν is defined at (1.1.1a). The twice contravariant metric tensor ηµν is defined at (1.1.1d). µ The mixed metric tensor η ν is defined at (1.1.2). The covariant Levi-Civita symbol ǫµνρσ is the totally antisymmetric tensor. The Poincaré generator P µ generates spacetime translations. The Poincaré generator J µν generates Lorentz transf. (i.e., space rot. and Lorentz boosts). The Pauli-Lubanski´ four-pseudovector Wµ is defined at (1.1.12). The invariant volume element on the lightcone d˜k is defined at (1.4.23). iv Electrodynamics The four-vector potential Aµ is formed by the scalar pot. A0 and the vector pot. A. The Faraday tensor F µν is defined at (1.2.20) in terms of the electric E and magnetic B fields. The charge density-current four-vector J µ is formed by the charge density ρ and the current density j. The polarisation vectors ǫ(λ) (k) in the Coulomb gauge are defined at (1.4.19). µ The polarisation four-vectors ǫ(λ) (k) in the Lorenz gauge are defined at (1.A.51), (1.A.54) and (1.A.55). The connection α (k) on the lightcone is defined at (1.4.28). The generators Sˆi of the rotation group SO (3) are defined at (2.1.2). (+1/2) The photon wave function ψ(λ) is the Riemann-Silberstein wave function (see sects. 3.3 and 3.4). (0) The photon wave function ψ(λ) is the Landau-Peierls wave function (see sect.
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