Christian Kohlfürst Electron-positron pair production in inhomogeneous electromagnetic fields Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften an der Naturwissenschaftlichen Fakultät der Karl-Franzens-Universität Graz Advisor: Univ-Prof. Dr. rer. nat. Reinhard Alkofer Institut für Physik Fachbereich Theoretische Physik 2015 2 Electron-positron pair production in inhomogeneous electromagnetic fields by Christian Kohlfürst Abstract The process of electron-positron pair production is investigated within the phase-space Wigner formalism. The similarities between atomic ionization and pair production for ho- mogeneous, but time-dependent linearly polarized electric fields are examined mainly in the regime of multiphoton absorption (field-dependent threshold, above-threshold pair produc- tion). Characteristic signatures in the particle spectra are identified (effective mass, channel closing). The non-monotonic dependence of the particle yield on the carrier frequency is discussed as well. The investigations are then extended to spatially inhomogeneous electric fields. New effects arising due to the spatial dependence of the effective mass are discussedin terms of a semi-classical interpretation. An increase in the normalized particle yield is found for various field configurations. Pair production in inhomogeneous electric and magnetic fields is also studied. The influence of a time-dependent spatially inhomogeneous magnetic field on the momentum spectrum and the particle yield is investigated. The Lorentz invari- ants are identified to be crucial in order to understand pair production by strong electric fields in the presence of strong magnetic fields. Zusammenfassung Elektron-Positron Paarerzeugung wird mittels des Wigner-Formalismus im Phasenraum untersucht. Die Gemeinsamkeiten von Atomionisation und Paarerzeugung werden dabei für homogene, zeitabhängige, linear polarisierte elektrische Felder im Bereich der Mehr- photonenabsorption untersucht (feldabhängige Produktionsschwelle etc.). Dabei wurden charakteristische Signaturen im Teilchenspektrum gefunden (effektive Masse, Channel Clos- ing). Außerdem wird der nicht-monotone Zusammenhang zwischen Produktionsrate und Feldfrequenz behandelt. Die Untersuchungen werden dann auf räumlich inhomogene Felder erweitert. Neue Effekte, die im Zusammenhang mit einer raumabhängigen effektiven Masse stehen, werden mithilfe semi-klassischer Methoden diskutiert. Ein Anstieg der räumlich normierten Produktionsrate für spezielle Feldkonfigurationen wurde gefunden. Die Unter- suchungen werden um räumlich und zeitlich inhomogene magnetische Felder erweitert, und deren Auswirkungen auf das Teilchenspektrum wird untersucht. Die Lorentzinvarianten werden als ausschlaggebend für die Produktionsrate durch starke elektrische Felder in An- wesenheit von starken magnetischen Feldern identifiziert. 3 4 Acknowledgments I would like to share my deepest gratitude for the help I received over the last years. This thesis would not have been possible without the permanent support of my colleagues, my friends and my parents. Nevertheless, I want to mention some of the “contributors” by name. At first I want to thank my advisor Reinhard Alkofer for ongoing support andgiving me free rein to follow my own ideas. I would also like to thank Holger Gies from the FSU Jena. He had a significant impact on my research and consulting him was always valuable. Additionally, I want to thank Tom Heinzl and colleagues at Plymouth University for their generous hospitality. I take this opportunity to express gratitude to all of the Department faculty members for their help and support. Furthermore, I want to thank the principal investigators of the Doktoratskolleg “Hadrons in Vacuum, Nuclei and Stars” for giving me the opportunity to participate in the graduate program. I also want to acknowledge the Austrian Science Fund, FWF, for financial support (FWF DK W1203-N16). Additionally, I want to mention the University of Graz’ research core are “Models and Simulation”. My office mates in Graz and Jena played a big role in the last three years, becausewe shared a lot of time together. Working on my PhD was often stressful and demanding, but with your help it was an enjoyable part of my life. Hence, my deepest gratitude to Richard Haider, Valentina Verduci, Milan Vujinovic, Mario Giuliani, Alexander Blinne and Nico Seegert. Besides my office mates, I want to thank Julia Borchardt, Ana Juricic, Ydalia Delgado Mercado, Matthias Blatnik, Hans-Peter Schadler, Pascal Törek, Markus Pak and Alexander Goritschnig. A special thank goes to our football team. It was a pleasure to participate. A note to Anita Ulz, conversations with you are and have always been refreshing and entertaining. A special thanks to Matthias Blatnik for careful proofreading my drafts. Ganz besonderen Dank gebührt meinen Eltern, Hans und Roswitha, für fast auf den Tag genau 26 Jahre vollste Unterstützung. Ein Dankeschön auch an meinen Bruder David, es macht immer wieder Freude mit dir etwas zu unternehmen. Ganz zum Schluss noch einen besonderen Dank an meine Oma Maria, die diese Arbeit leider nicht mehr miterleben konnte. 5 Contents List of Figures 11 List of Tables 12 Notation 13 I Considerations 17 1 Introduction 19 2 Overview 23 2.1 Historical remarks . 23 2.1.1 Electrodynamics . 23 2.1.2 Advances in Laser technology . 25 2.2 Theoretical considerations . 26 2.3 Laser beams . 28 2.4 Mechanisms for pair production . 30 2.5 Quantum kinetic methods . 33 2.6 Particle dynamics . 38 2.7 Objectives . 39 3 Dirac-Heisenberg-Wigner formalism 41 3.1 Wigner operator . 41 3.2 Equations of motion . 42 3.3 Pair production in arbitrary dimensions . 43 3.3.1 Pair production in the plane . 45 3.3.2 Pair production along a line . 51 3.4 Spatially homogeneous fields . 53 3.5 Cylindrically symmetric fields . 55 3.6 Observables . 59 3.7 Classical limit . 63 6 4 Solution strategies and models for the background field 67 4.1 Pseudo-spectral methods . 68 4.2 Pseudo-differential operators . 69 4.2.1 Operator expansion . 70 4.2.2 Full solution . 73 4.2.3 Filtering . 74 4.3 Model for the background field . 75 4.3.1 Parameter scales . 75 4.3.2 Model for the time-dependence . 76 4.3.3 Spatially inhomogeneous background fields . 77 4.3.4 Model for the magnetic field . 78 4.4 Final momentum distribution . 78 4.5 Semi-classical interpretation . 80 II Results 85 5 Multiphoton pair production 87 5.1 Effective mass signatures . 88 5.1.1 Quantum kinetic theory and background field . 88 5.1.2 Concept of an effective mass . 89 5.1.3 Particle yield . 91 5.1.4 Particle momentum distribution . 96 5.2 Pair production at extreme parameter regions . 98 6 Spatially inhomogeneous electric fields 101 6.1 Spatial focusing, full-operator DHW approach and background fields . 101 6.2 Ponderomotive forces for pair production in 3 + 1 dimensions . 102 6.3 Schwinger pair production in strongly localized fields . 110 7 Pair production in electric and magnetic fields 113 7.1 Pair production in the plane: Operator-expanded DHW approach and model for the fields . 113 7.2 Effective field amplitude . .116 7.2.1 Particle distribution . 117 7.2.2 Particle yield . 123 7 8 Summary and Outlook 127 8.1 Summary . 127 8.2 Outlook . 130 III Supplement 131 Appendix 132 Bibliography 199 8 Appendix A Detailed calculations 133 A.1 Equations of motion . 133 A.2 Energy averaging . 136 A.3 Transport equations for QED3+1 . 137 A.4 Transport equations for QED2+1 . 140 A.4.1 Four-Spinor formulation . 140 A.4.2 Two-Spinor formulation . 145 A.5 Transport equations for QED1+1 . 147 A.6 Alternative formulations . 149 A.7 Vacuum initial conditions . 154 A.8 Carrier Envelope Phase in the multiphoton regime . 156 A.9 Relativistic Lorentz force . 157 B Additional tables and figures 159 B.1 Figures . 159 B.1.1 Particle yield for homogeneous fields . 159 B.1.2 Carrier Envelope Phase regime . 160 B.1.3 Short pulsed, time-symmetric magnetic field . 160 B.1.4 Long pulsed, time-symmetric magnetic field . 164 B.1.5 Time-antisymmetric magnetic field . 167 B.2 Tables . 172 C Matlab solver 191 C.1 Code . 191 C.1.1 main.m . 191 C.1.2 Dgl.m . 194 C.2 Example calculations . 196 C.3 Parameter tables . 198 9 List of Figures Figure 2-1 Schematic picture displaying pair production via light-light interaction. 29 Figure 2-2 Schematic view of a transverse electromagnetic mode. 30 Figure 2-3 Pictorial description of the Schwinger effect. 31 Figure 2-4 Pictorial description of (multi)photon absorption. 32 Figure 2-5 Pictorial description of above-threshold pair production and the dy- namically assisted Schwinger effect. 32 Figure 2-6 Illustration of the number of non-trivial Wigner components depending on the dimension of the problem. 37 Figure 2-7 Illustration of the difference in the number of non-trivial Wigner com- ponents for ordinary and cylindrically symmetric problems. 37 Figure 4-1 Demonstration of particle-, charge-, positron- and electron distribution. 79 Figure 4-2 Final particle distribution in case of a single-peaked, homogeneous electric field. 80 Figure 4-3 2D picture of the distribution function for a single-peaked, inhomoge- neous electric field. 81 Figure 4-4 Modified particle distribution function demonstrating particle self- bunching. 82 Figure 4-5 Final particle distribution in case of a double-peaked, homogeneous background field. 83 Figure 4-6 2D picture of the distribution function for a double-peaked, inhomo- geneous electric field. 83 Figure 5-1 Particle yield as a function of the field frequency. 88 Figure 5-2 Momentum spectrum for Schwinger and multiphoton pair production. 91 Figure 5-3 Particle yield as a function of the field frequency in the multiphoton regime. 92 Figure 5-4.