Florian Hebenstreit

Schwinger effect in inhomogeneous electric fields

Dissertation

zur Erlangung des Doktorgrades der Naturwissenschaften

verfasst am Institut für Physik, Fachbereich Theoretische Physik an der Karl-Franzens-Universität Graz

Betreuer: Univ.-Prof. Dr. rer. nat. Reinhard Alkofer

Graz, 2011

Abstract

The vacuum of is unstable against the formation of many- body states in the presence of an external electric field, manifesting itself as the creation of -positron pairs (Schwinger effect). This effect has been a long- standing but still unobserved prediction as the generation of the required field strengths has not been feasible so far. However, due to the advent of a new gener- ation of high-intensity laser systems such as the European XFEL or the Extreme Light Infrastructure (ELI), this effect might eventually become observable within the next decades.

Previous investigations of the Schwinger effect led to a good understanding of the general mechanisms behind the pair creation process, however, realistic electric fields as they might be present in upcoming high-intensity laser experiments have not been fully considered yet. Actually, it was only recently that it became possible to study the Schwinger effect in realistic electric fields showing both temporal and spatial variations owing to the theoretical progress as well as the rapid development of computer technology.

Based on the equal-time Wigner formalism, various aspects of the Schwinger ef- fect in such inhomogeneous electric fields are investigated in this thesis. Regarding the Schwinger effect in time-dependent electric fields, analytic expressions for the equal-time Wigner function in the presence of a static as well as a pulsed electric field are derived. Moreover, the pair creation process in the presence of a pulsed electric field with sub-cycle structure, which acts as a model for a realistic laser pulse, is examined. Finally, an ab initio simulation of the Schwinger effect in a simple space- and time-dependent electric field is performed for the first time, allowing for the calculation of the time evolution of various observables like the charge density, the particle number density or the number of created particles. Contents

1 Introduction 5

2 Overview and objectives 9 2.1 Schwinger effect ...... 9 2.2 Quantum kinetic methods ...... 13 2.2.1 Quantum Vlasov equation ...... 13 2.2.2 Wigner function formalism ...... 16 2.3 Current research topics ...... 19 2.3.1 Momentum distribution in realistic laser pulses ...... 19 2.3.2 Enhancement of the Schwinger effect ...... 20 2.3.3 Depletion of the laser field ...... 20 2.3.4 Schwinger effect in space- and time-dependent electric fields . 21 2.4 Objectives ...... 22

3 Schwinger effect in phase space 23 3.1 Covariant Wigner formalism ...... 23 3.1.1 Covariant Wigner operator ...... 24 3.1.2 Equation of motion ...... 25 3.1.3 Spinor decomposition ...... 26 3.2 Equal-time Wigner formalism ...... 27 3.2.1 Energy average ...... 28 3.2.2 Equation of motion ...... 29 3.2.3 Observables and conservation laws ...... 31

4 Schwinger effect for E(t) 35 4.1 Quantum Vlasov equation ...... 35 4.2 Equal-time Wigner formalism ...... 42 4.3 Analytically solvable electric fields ...... 46 4.3.1 Static electric field ...... 47 4.3.2 Pulsed electric field ...... 52 4.4 Pulsed electric field with sub-cycle structure ...... 58 4.4.1 Momentum distribution ...... 60 4.4.2 Quantum statistics effect ...... 67 Contents 4

5 Schwinger effect for E(x, t ) 69

5.1 Equal-time Wigner formalism in QED 1+1 ...... 69 5.1.1 QED in 1 + 1 dimensions ...... 71 5.1.2 Covariant Wigner formalism ...... 71 5.1.3 Equal-time Wigner formalism ...... 73 5.1.4 Observables and marginal distributions ...... 74 5.2 Solution strategies ...... 76 5.2.1 Derivative expansion in p-space ...... 77 5.2.2 Fu