Electron Positron Pair Production in Strong Electric Fields

Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.)

seit 1558

vorgelegt dem Rat der Physikalisch-Astronomischen Fakultät der Friedrich-Schiller-Universität Jena von M. Sc. Alexander Blinne geboren am 19.10.1987 in Essen Gutachter

1. Prof. Dr. Holger Gies Friedrich-Schiller-Universität Jena Theoretisch-Physikalisches-Institut

2. Prof. Dr. Burkhard Kämpfer Helmholtz-Zentrum Dresden-Rossendorf Institut für Strahlenphysik

3. Prof. Dr. Antony Ilderton Chalmers University of Technology Department of Physics

Datum der Disputation: 24. November 2016 Zusammenfassung

In dieser Arbeit wird Elektron-Positron Paarerzeugung in räumlich homogenen elektrischen (und magnetischen) Feldern behandelt. Verschiedene Feldkonfigurationen werden untersucht um verschiedene Phänomene wie Multiphoton-Paarproduktion, Sauter-Schwinger-Paarproduktion und dynamisch assistierte Paarproduktion zu stu- dieren. Der zentrale Fokus liegt auf gepulsten, rotierenden Feldern mit einer zentra- len Frequenzkomponente, die rotierende Sauter-Pulse genannt werden. Die Resultate werden mittels verschiedener numerischer Methoden gewonnen, die auf unterschiedlichen theoretischen Ansätzen basieren. Eine generische Methode, wel- cher eine modifizierte quantenkinetische Gleichung zu Grunde liegt, wird ausdem Dirac-Heisenberg-Wigner (DHW) Formalismus abgeleitet. Die numerische Lösung dieser Gleichung wird die Wigner-Methode genannt. Andere Arten von Gleichungen werden ebenfalls aus dem DHW-Formalismus abgeleitet und mit dem Ziel, magneti- sche Felder einzubeziehen, numerisch gelöst. Im Falle des rotierenden Sauter-Pulses wird zusätzlich eine komplett unabhängige numerische Methode entwickelt, welche auf einem semiklassischen Ansatz basiert und daher semiklassische Methode genannt wird. Eine Reihe von Parameterstudien wird durchgeführt, um Paarproduktion in rotierenden Sauter-Pulsen zu verstehen. In diesen Studien werden die Wigner-Methode und die semiklassische Methode ausschöpfend verglichen, wobei gefunden wird, dass sie sich ergänzen. Dadurch ist es möglich, den kompletten Parameterbereich des rotierenden Sauter-Pulses abzudecken, wodurch Paarproduktionsraten für Laser- basierte Experimente mit gegenläufig propagierenden zirkular polarisiertem Licht berechnet werden können. Die resultierenden Paarproduktionsspektren werden in- terpretiert. Durch die generische Natur der Wigner-Methode ist es möglich, darüber hinaus weitere Feldkonfigurationen zu untersuchen, beispielsweise Laser-Pulse mit elliptischer Polarisation, Pulse mit chirp oder Überlagerungen zweier rotierender Sauter-Pulse, so genannte bichromatische Pulse. Jede dieser Konfigurationen zeigt interessante Merkmale, unter anderem den dynamisch assistierten Schwinger-Effekt in bichro- matischen Pulsen, was die Planung von hochintensitäts Laser-Experimenten positiv beeinflussen kann.

Abstract

This work covers electron positron pair production in spatially homogeneous electric (and magnetic) fields. Different field configurations are looked at in order tostudy various phenomena including multiphoton pair production, Sauter-Schwinger pair production and dynamically assisted pair production. The main focus lies on pulsed, rotating fields with one main frequency component which are called rotating Sauter pulses. The results are obtained via different numerical methods, that rest on different theoretical approaches. A generic method is derived from the Dirac-Heisenberg- Wigner (DHW) formalism which entails a modified quantum kinetic equation. We call the numerical solution of this equation the Wigner method. Other types of equations are derived from the DHW formalism as well and numerically solved with the aim to include magnetic fields. In the case of rotating Sauter pulses a completely different numerical method is developed, which is based on a semiclassical approach and therefore called the semiclassical method. A number of parameter studies are conducted to understand pair production in these rotating Sauter pulses. In those studies the Wigner method and the semiclassical method are compared exhaustively and found to complement each other. This makes it possible to cover the complete range of parameters of the rotating Sauter pulse, which helps to calculate the pair production rates for experiments involving counter-propagating circularly polarized laser light. An interpretation of the resulting pair production spectra is given. Due to the general nature of the Wigner method it is possible to study more general field configurations which include pulses with elliptic polarization, chirped pulses or bichromatic rotating Sauter pulses. Each of these exhibit interesting features, including the dynamically assisted Schwinger effect in bichromatic pulses, which could be useful in planning high-intensity laser experiments.

iii

99 87 Conclusion 5. Fields Magnetic of Inclusion 4. 21 3 9 Fields Electric Homogeneous 3. Formalism Function Wigner The 2. Introduction 1. Contents of Table 97 . . . 89 . . . 92 ...... 74 . . 71 ...... 66 ...... Results . . . Preliminary ...... 4.3. . Implementation . . Numerical . . . . . Motion 4.2. of . . . Equation . . . . . 4.1...... 32 ...... Fields . . . Bichromatic . . 3.6. Pulses . . Chirped . . 3.5. . Polarization Generalized . 21 . 3.4...... 15 . 13 ...... 11 ...... 27 . . . . . Pulse . . Sauter . . . Rotating . . . . The ...... 3.3...... Fields . . Electric . . Two-Component . . for . . . Method . . . Semiclassical . . The ...... 3.2...... Method . Wigner . . . The . . . . 3.1...... States . Quantum . . . 2.3. . Observables . Motion 2.2. of Equation 2.1. 81 . . . 74 ...... 63 54 ...... 53 ...... Fields . . . . Linear . . Bichromatic . . . . . Fields 3.6.2. . Rotating . . Bichromatic ...... 3.6.1...... 37 . . . . 46 . . 28 ...... Moment . . . . 25 Magnetic . 31 ...... 3.3.6. . . Spectra . 33 . Typical ...... Methods . . . Yield . . Semiclassical 3.3.5. Particle and . . . . . Total Wigner . . . . . the Pulse 3.3.4. of . . . Sauter Comparison Pulse . Rotating . . Sauter the . . Rotating . 3.3.3. of the Phenomenology . . . for 23 . . . Method 3.3.2. Semiclassical . . . The ...... 3.3.1. Pairs . . . Produced . of . Spectrum . Equation . Momentum . Dirac . the . 3.2.2. of . Solution . . . . 3.2.1...... Calculations Equation Numerical Kinetic Quantum 3.1.2. Modified The 3.1.1. 1

0 π φ = 49 2 Table of Contents

A. AppendixI A.1. Matrices for the Equation of Motion Including Homogeneous Mag- netic Fields...... I A.2. Semiclassical Results for the Constant Rotating Pulse...... III A.3. Calculation of the Momentum Spectrum for the Sauter Pulse.....IV

B. BibliographyVII

C. List of Figures XIX

D. List of Symbols and Abbreviations XXI

E. Danksagungen XXV

F. Ehrenwörtliche Erklärung XXVII

2 tl pnqetos seily o ftepeitdefcso E aeyet to QED have of effects predicted the of lot a Especially, questions. fieldarethe open still areex- of theelectromagnetic positrons force. and electromagnetic the excitations the canbe electrons convey and which The field photons particles Dirac fields. that the quantized and of fields the citations account and of into particles excitation takes between as theory distinction viewed quantum no full is The there particles. that charged [6], classical Schwinger of S. J. motion for prize has Nobel 1965. shift in a Tomonaga Lamb S. to the and led [7], also theory Feynman P. and R. quantum precision This high electron explained. to the been of calculated moment perturbatively magnetic been anomalous The has developed. out,theexistence was 5]. (QED) [4, work dynamics later not ex- confirmed did experimentally possible this was a which as when conjectured, of but was thought first, positron were the at magnetic moment sea of [3] anomalous Dirac proton orthe proposed the shift this for in planation Lamb Holes was the electron. theory g., the the e. of but toexplain, atoms, indeed hydrogen-like was sufficient of it not understanding [2] theory the energy Dirac’s improve negative Using with to useful. eigenstates possible became the theory all the occupied, almost the be where When equa- should [1], Dirac state. ground the postulated stable was that no fact sea thus the and Dirac energy to negative due with nonrelativistic is eigenstates the has This because to tion consistent. problematic, analogy not in was is equation theory equation Dirac Dirac Schrödinger the of equation. interpretation Dirac Dirac one-particle called the a was this arela- involved time it the thespectrum, of At as developed. theory, was structure mechanics the exact quantum of toexplain kind tivistic sufficient not equation, Schrödinger’s very be charged to upon a out based negatively understand mechanics, turned and quantum to nucleus When urge charged inter- atom. the positively the theory ofelectromagnetic was electrons, a between force quantum state driving the bound The common 20th century developed. the been of has actions half first the In Introduction 1. etraieQDi eludrto,btbyn etrainter hr are there theory perturbation beyond but understood, well is QED Perturbative relativistic special with electrodynamics classical describes it limit classical the In electro- quantum of theory the when away went theory Dirac’s of problems These 3

1 π φ = 49 2 1. Introduction

a) e+ e− X′ b) e+ e−

t t ......

γ γ γ γ X

Figure 1.1.: Feynman diagrams for either a) Bethe-Heitler or b) Breit-Wheeler processes. be observed in experiment, e. g., light-light interaction via vacuum polarization [8, 9] or Sauter-Schwinger pair production [9–11]. Pair production by interaction of high-energy photons with laser light has been observed in the famous SLAC-144 experiment [12]. This work will concentrate on electron positron pair production in strong fields which is a phenomenon that has its roots in the fact that the spectrum of the Dirac equation is not bounded from below. If one thinks of slowly varying and very strong fields, pair production can be understood as a tunneling process. In the extreme, for homogeneous and constant fields, the pair production rate per unit spacetime volume is given by the Schwinger formula 2 2 (eE) −π m N = e eE , (1.1) 4π3 where m is the electron mass and e the (positive) elementary charge. This is called Sauter-Schwinger pair production, see Ref. [9–11]. The result has an essential singularity in the perturbative limit E → 0 and can not be found using perturbation theory. Because of this Sauter-Schwinger pair production is thought of as a nonperturbative effect. It was shown recently that the essential singularity is aconse- quence of the electric field being constant for all times. When the field is switched on adiabatically, it is possible to obtain analytic results which can be expanded into convergent power series. Furthermore the series coefficients can be reproduced by perturbation theory[13]. If the fields are quickly changing, it is possible to use perturbation theory. Of- ten the varying electromagnetic fields are then understood in terms of propagating photons. It is not possible to produce pairs from a single photon, as in the vacuum this would violate the conservation of energy and momentum. This can be reme- died in the presence of a nucleus X which can absorb excess momentum and this effect is called Bethe-Heitler pair production [14], an example for a Feynman graph

4 eprldpnece ftefed aet eacutdfor. beaccounted spatio- to hastobedetermined. as trivial have being fields from the far of is production pair this dependencies far, temporal so from becomputed,but Schwinger this performed hasto data calculations To quantify cascade the theinitial For cascade paircreation. the QED of all, only to about not first lead information observables, that carry final field to for electric expected difference the the be distribution, of can momentum directionality pairs initial the arising created isotropic electrons electron Schwinger an of an have of ensemble to by ensemble the likely seeded Whereas are cascade impurities production? QED from pair cascade a QED Schwinger a can from fields, by distinguished laser be seeded question: vacuum) strong imperfect crucial an in of impurities a production by intensities.(sourced pair to Schwinger critical rise of near gives discovery of possible this [35–44], generation a occur of the to view inhibit expected In fundamentally are even photons pair hard may a from accelerated swamp which of production entirely radiation particle or successive of and partly cascades charges could QED particular that In in signal. set production may processes physical possible intensity, critical the approach befound can fields such Schwinger in assisted [29–34]. production Refs. dynamically pair in into called research is Previous photons production. apromising energy pair production also pair medium is some for Schwinger adding potentially fields threshold by is different tunneling effective production of the pair superposition Lowering As the approach. process, [25–28]. nonlinear fields combination highly Coulomb many the a strong including to and also lead light [16–24], has discovery laser lasers first of a high-intensity (1.1). for X-ray Eq. schemes or to for due optical suggestions rates of production development small rapid pairproduction exponentially The the by for resonant hampered is thethreshold duction, well below = Ω frequencies nucleus. with a fields of absence tric in production pair pair multiphoton to and methods them production The non- apply pair we called Schwinger and is production. both production process cover pair work the multiphoton two this than just in greater described or is process photons Breit-Wheeler of in number linear displayed the and If [15] production 1.1b). pair Fig. Breit-Wheeler called is which around, photons 1.1 a). Fig. in found be can process this describing h is xesoso atrsoiia okcnetae onone-component workconcentrated original ofSauter’s extensions first The indeed gradually may systems in such intensities field highest the while However, elec- macroscopic in understanding this verifying at attempt any Unfortunately, two least at are there if possible only is vacuum in production pair Perturbative m hc r lsrt cwne arpouto hnmlihtnpi pro- pair multiphoton than production pair Schwinger to closer are which , I cr . = E cr 2 . =  m e 2  2 ≃ 4 . 3 × 10 29 W/cm 2 further , 5

2 π φ = 49 2 1. Introduction electric fields. In addition to unidirectional fields depending on either space[45–49] or time [49–57], exact solutions for specific classes of fields can be found in light cone variables [58, 59]. In Ref. [60] a connection between these three special cases was found using interpolating coordinates and the worldline instanton method. Recently more involved fields have also been studied including electric fields that are not necessarily unidirectional and are spatially inhomogeneous in up to three dimensions [18, 61–63] or depend on time [32, 33, 35–44, 64–66]. Also unidirectional fields that depend on space as well as ontime[67, 68] were studied. A lot of studies concentrate on field configurations that could be found in counterpropagating laser light; this includes (nonlinear) Breit-Wheeler pair production [69, 70] and pair production in pure electric fields near the antinodes of the magnetic field[32, 33, 64–66]. Different methods were developed including those that are exact on themean- field level, e. g., the quantum kinetic theory (QKT)[71, 72] and the real-time Dirac- Heisenberg-Wigner (DHW) formalism [73, 74]. A numerical scheme based upon the DHW formalism has been developed [75], which we will call the Wigner function method or, in short, the Wigner method. While it is possible to obtain exact results using a scattering ansatz [76, 77], it is combined with some kind of a semiclassical approximation [45, 50, 52, 57, 78– 85] in most of its applications. This combination is sometimes referred to as the Wentzel-Kramers-Brillouin (WKB) approach [60, 77, 86, 87] or even as the WKB approximation [76, 86], while only the ansatz, but not the approximation, is taken from the original WKB method. We will thus not refer to the method discussed here as the WKB method, but as the semiclassical scattering method. Other semiclassical methods include the worldline instanton method [48, 49, 88, 89]. It was shown that in the case of linearly polarized, purely electric fields the QKT is equivalent to the DHW formalism [90] as well as to the scattering approach [76]. The worldline instanton method and the semiclassical scattering method have been shown to agree for one-component fields [48, 49, 91] and for two-component fields [86]. In this Thesis, I consider a number of the aforementioned topics. For the most part, the Wigner method is used, which is based on the DHW formalism. Some aspects of the DHW formalism are explained in Cha.2. Pair production in time-dependent, spatially homogeneous electric fields is the focus of Cha.3. At first the modified quantum kinetic equations are derived fromthe DHW formalism and the Wigner method is explained in Sec. 3.1. After introducing a complementary semiclassical method in Sec. 3.2, the rotating Sauter pulse is in- 6 ntecneto h H omls sa xaso fteWge ehdis method Wigner the of expansion an presented. as are formalism results DHW preliminary and the explained of context the inthe in carrier frequencies such aselliptically with different chapter. fields the of of into account sections remaining kinds taken superpositions or are other Additionally field fields electric pulse. polarized the Sauter of rotating to dependencies the subject time of are of context [92] the method semiclassical in the comparison and and performance a [75] The method Wigner theory calculations. the field cascade of quantum QED accuracy the to bringing closer help step may substantial which a 3.3, studies Sec. in studied and troduced a prahfrteicuino ptal ooeeu antcfields magnetic homogeneous spatially of inclusion the for approach an 4 Cha. In 7

3 π φ = 49 2

be treated as a classical external field and its equation of motion will be reduced to be will of motion its equation the will and for field field requirement electromagnetic external the The classical a used. as is treated convention be sum Einstein’s that Note follow. chapter. this in given be will The thesis 90]. this Ref.[74, in in presented be found work the as can to the context fields relevant electric present called details particular the for also solutions for is exact summaries as formalism well Comprehensive Dirac-Heisenberg-Wigner formalism. The and function Górnicki Wigner 1991. byBialynicki-Birula, electron in of introduced [73] context first Husimi the quantum Rafelski was the In it formulate g., e. 96]. production to this, [95, pair do method function to positron Glauber-Sudarshan used a Wigner the also as The are or functions [94] [93] 4. Other function in Cha. 1930s space. phase the fields the since in magnetic well theory as known include 3.1 to is Sec. it in itself explained enhance and function to 3 attempt Cha. the in used for basis be as will the which is method formalism Wigner the function for Wigner or formalism Dirac-Heisenberg-Wigner The Formalism Function Wigner The 2. rmwihteDrceuto o h spinor the for equation Dirac the which from spinor h inrfnto omls sbsdo h emta E Lagrangian QED hermitian the on based is formalism function Wigner The L ˆ ( Ψ ˆ , = Ψ ˆ ∂ ∂ Ψ ˆ t t Ψ( Ψ( , ˆ ˆ A #‰ ˆ Ψ ˆ #‰ #‰ t , x t , x = ) † γ 0 = ) = ) 2 1  div div Ψ ˆ − − γ γ  B E #‰ #‰ µ 0 ∇ [i E #‰ γ #‰ rot 0 = = ∂ #‰ x µ  Ψ( and ˆ ρ ∇ #‰ − #‰ x #‰ t , x eA B − #‰ µ + ) ie ooe awl’ equations Maxwell’s obey to ] Ψ ˆ A #‰ ˆ ie − ( Ψ( #‰ t , x ˆ  [i #‰ t , x ∂ )  µ Ψ( + ˆ ) Ψ ˆ A #‰ ˆ rot eA #‰ t , x ( n t don omfrteadjoint the for form adjoint its and #‰ t , x µ B E #‰ #‰ ) ] Ψ ˆ − ) = =   imγ · γ − E #‰ ˙ #‰ γ γ µ B #‰ Ψ + ˙ ˆ 0  0 Ψ( µ ˆ + − 0 . j #‰ t , x im m Ψ ˆ Ψ( ) ˆ Ψ ˆ #‰ t , x − 1 4 ) F γ 0 µν F µν 9

4 π φ = 49 2 2. The Wigner Function Formalism

Starting from the fermionic field operator Ψˆ and its conjugate Ψˆ †, there are in principle two basic objects similar to two-point propagators,

± #‰ #‰ ˆ #‰ ˆ † #‰ ˆ † #‰ ˆ #‰ Cab(t, x1, x2) := ⟨0| Ψa(t, x1)Ψb(t, x2) |0⟩ ± ⟨0| Ψb(t, x2)Ψa(t, x1) |0⟩ , which could be used to build the Wigner function. Since C+ is the expectation value of the anticommutator of the field operators, which is by definition the delta function #‰ #‰ − δab · δ(x2 − x1), only C contains non-trivial information. Thus the commutator of the field operators will be the starting point in the definition of the Wigner function. #‰ A center of mass coordinate x for the underlying two-point correlator is introduced #‰ while s denotes the separation vector

#‰ 1 #‰ #‰ x := (x + x ) 2 1 2 #‰ #‰ #‰ s := x2 − x1 .

The Wigner operator Wˆ will now be defined as the Fourier (Wigner) transform of the equal time density operator of two Dirac field operators in the Heisenberg picture w. r. t. the separation vector. As we are interested in pair production from the vacuum, the Wigner function is defined by taking the vacuum expectation value ˆ ⟨0| ♢ |0⟩ of the Wigner operator. Note that the Dirac conjugate field operator Ψ = Ψˆ †γ0 is used instead of the adjoint field operator such that the Wigner operator transforms homogeneously under Lorentz transformations. #‰ #‰ #‰ #‰ W(t, x , p ) := ⟨0| Wˆ (t, x , p ) |0⟩  #‰ #‰ #‰ #‰ 1 #‰ − i p · s #‰ #‰ Wˆ (t, x , p ) := − d s e ℏ Cˆ (t, x , s ) ab 2 ab  #‰ #‰  (2.1) ˆ #‰ #‰ ˆ #‰ #‰ ˆ #‰ s ˆ #‰ s Cab(t, x , s ) := Φ(t, x , s ) Ψa(t, x + /2), Ψb(t, x − /2) #‰ #‰ s #‰  x − /2 #‰′ #‰′ #‰ #‰ −ie #‰ #‰ Aˆ(t, x )·d x Φ(ˆ t, x , s ) := Pe x + s /2

#‰ #‰ The Wilson line Φ(ˆ t, x , s ) was introduced to achieve gauge invariance [73, 74], the path ordering will be dropped when the electromagnetic field is treated classically. #‰ Choosing a straight line as integration path ensures the proper interpretation of p as kinetic momentum [74]. Please note that this is not the definition of the covariant Wigner function which depends on a spacetime point and a four-momentum, but the equal time Wigner function where an energy average has been taken [74]. The Fourier transform can be understood as measuring the plane wave content of the

10 stelte stetda lsia akrudfed h yaia qainfor equation dynamical as The written field, be field. electromagnetic then background the can classical for function a level Wigner as zeroth the treated the is at latter and the function as Wigner the for level the by operators of product a of values value expectation expectation the the of replace product We [73]. so Rafelski method Wigner the in be will used can approximation which tower only approximation, this the far. mean-field calculations, indeed or numerical is Hartree-type practical below, This do explained to [74]. order Kirkwood truncated G. In be Green, H. Born, [74]. M. Yvon tower is J. Bogoliubov, This N. and after as well. hierarchy ofthe BBGKY field electromagnetic the the evolution the only called of not evolution the describe result but which The function, equations Wigner derived. the definition ofcoupled be in can tower function infinite Wigner an operator the is for field motion every of equations for the motion (2.1), Eq. of equations Heisenberg the Using Motion of Equation 2.1. 74]. [73, density particle a as with particles of densities probability quasi as momentum interpreted then are correlator this origin in the to respect with correlator two-point fayosral hc sgvni em ftefedoeaos foeo thevariables of one If operators. field the of terms in given is which observable any of #‰ x h niietwro qain fteBGYheacyi rnae atthefirst istruncated BBGKYhierarchy the of ofequations tower infinite The and Górnicki Bialynicki-Birula, by introduced already was approximation This osqetyteWge ucincnb sdt aclt h xetto values expectation the calculate to used be can function Wigner the Consequently or #‰ p sitgae u,termiigfnto spstv n a einterpreted be can and positive is function remaining the out, integrated is #‰ p D tposition at t W = − 2 1 ⟨ ⟨ D #‰ 0 0 #‰ x | | #‰ x B E . ˆ ˆ  γ | O | O ˆ ˆ 0 #‰ , γ 0 0 ⟨ → ⟩ ⟨ → ⟩ W  − 0 0 | | B E i ˆ ˆ m | | 0 0  #‰ γ x ⟨ · ⟩ ⟨ · ⟩ 0 h mltdso h ln waves plane the of amplitudes The . , 0 0 W | | | O | O  ˆ ˆ − 0 0 ⟩ ⟩ i P #‰ , . ..Euto fMotion of Equation 2.1.  γ 0 #‰ , γ W  (2.2) 11

5 π φ = 49 2 2. The Wigner Function Formalism with the pseudo-differential operators

1/2  #‰ #‰ #‰ #‰ #‰ #‰ Dt = ∂t + e dλ E(t, x + iλ∇p ) · ∇p ,

−1/2

1/2 #‰ #‰  #‰ #‰ #‰ #‰ #‰ #‰ #‰ #‰ Dx = ∇x + e dλ B(t, x + iλ∇p ) × ∇p , (2.3)

−1/2

1/2 #‰  #‰ #‰ #‰ #‰ #‰ #‰ #‰ P = p − ie dλ λB(t, x + iλ∇p ) × ∇p .

−1/2

Here, we use the conventions {γµ, γν} = +2ηµν = +2 diag(1, −1, −1, −1) and work #‰ #‰ in temporal gauge A0 = 0. The electric field E and magnetic field B are given by #‰ #‰ E = −∂t A #‰ #‰ #‰ (2.4) #‰ B = ∇x × A.

In the language of Feynman diagrams, the mean-field approximation corresponds to neglecting radiative corrections, which is justified by the smallness of the fine- structure constant α. The Wigner function can be decomposed in terms of a com- 1 5 µ µ 5 µν i µ ν plete basis of the Dirac bilinears, ( , γ , γ , γ γ , σ := 2 [γ , γ ]), 1 W = (1s + iγ p + γµ v + γµγ a + σµν t ) (2.5) 4 5 µ 5 µ µν with correspondingly transformed coefficient functions, s, p, vµ, aµ, tµν. By inserting Eq. (2.5) into Eq. (2.2) the latter can be decomposed, yielding #‰ #‰ D s = +2P · t 1 t #‰ #‰ 2 0 Dt p = −2P · t −2m a #‰ #‰ D v0 = −D#‰ v t #‰x 0 #‰ #‰ Dt a = −Dx a +2m p #‰ #‰ #‰ #‰ #‰ (2.6) D v = −D#‰ v0 −2P × a −2m t 1 t #‰x #‰ #‰ #‰ 0 #‰ Dt a = −Dx a −2P × v #‰ #‰ #‰ #‰ #‰ D t 1 = −D#‰ × t 2 −2P s +2m v t #‰ #‰x #‰ #‰ 2 #‰ 1 Dt t = +Dx × t +2P p

12 r ie ytevcu ouinwt o-aihn components non-vanishing with solution vacuum the by given are For momentum energy of QED,thecon- and thesymmetries function charge the Wigner quantities of served definition the Using Observables (2.8) Eq. 2.2. condition initial the with combined problem. value (2.6) initial Eq. the defines motion of equation The where vectors two with h aumslto a lob rte nmti omas form matrix in written be also can solution vacuum The E #‰ and B #‰ Q P #‰ E htvns taypoial al times, early asymptotically at vanish that = = = #‰  t  e s #‰ t W vac  1 dΓ Γ( dΓ / 1 2 vac . dΓ =2 := otiigtecmoet fteatsmerctensor antisymmetric the of components the containing = #‰ p  = #‰ p v ω − v t 0 · ( 4 1 i 0 ( 2 0 v #‰ #‰ p ( t, ( m e #‰ t, 1s ( i ) #‰ , x t, , #‰ , x , vac #‰ , x #‰ p + ω #‰ p + ) ϵ 2 #‰ p + ) 1 2 ( , t, γ := + )   #‰ , x µ v  #‰ p d #‰ p vac m ) 3 d 2 x µ 3 s +  x = ) ( t,    m E #‰ E #‰ #‰ , x 2 ( ( − . t, t, #‰ t 2 v #‰ #‰ m p #‰ x #‰ x ω 2 vac )  ) ) := ) 1    × 2 . + = + t B t #‰ ij #‰ γ −∞ →    ( ω ϵ − B #‰ 2 t, ijk ( · ω 2 ( #‰ p #‰ x #‰ t, p e #‰ p #‰ ) k ) #‰ x . , . , nta conditions initial , )    ..Observables 2.2. 2  , (2.11) (2.10) (2.7) (2.8) (2.9) t µν 13

6 π φ = 49 2 2. The Wigner Function Formalism angular momentum

#‰   #‰ #‰ #‰ #‰ 1 #‰ #‰ #‰  S = dΓ x × p v0(t, x , p ) − a(t, x , p ) 2 (2.12)  #‰  #‰ #‰ #‰ #‰  + d3x x × E(t, x ) × B(t, x ) and Lorentz boost operator

#‰ #‰  #‰ #‰ #‰ #‰ #‰ #‰ #‰  K = tP − dΓ x p · v(t, x , p ) + m s(t, x , p ) 1   #‰ #‰ 2  #‰ #‰ 2 − d3x E(t, x ) + B(t, x ) 2    

#‰ 3 #‰ can be derived [73]. The phase space measure dΓ is given by dΓ = d3 x d p . The (2π)3 quantity #‰ #‰ #‰ #‰ #‰ #‰ #‰ #‰ ϵ(t, x , p ) = p · v(t, x , p ) + m s(t, x , p ) (2.13) in the integrand in Eq. (2.11) is of special interest as it can be interpreted as a (phase space) energy density of the fermionic fields. From it the particle density is #‰ #‰ calculated by subtracting the vacuum solution ϵvac = msvac + p · v vac = −2ω and normalizing the result to the energy of a particle pair

1 1 f := (ϵ − ϵ ) = ϵ + 1 . (2.14) 2ω vac 2ω In spatially homogeneous, unidirectional, purely electric fields this distribution function f was shown to be identical to the definition of the distribution function from QKT [90]. Observe that Eq. (2.13) can be written in a more generic way like

#‰ #‰ ϵ[W] = tr[W(m1 + p · γ )] . (2.15)

As a consequence, Eq. (2.14) can be written in terms of a projection of the Wigner function

1 #‰ #‰ f[W − W ] = tr [(W − W )(m1 + p · γ )] . (2.16) vac. 2ω vac. #‰ #‰ It is important to stress that the particle interpretation is only valid when E = B = 0. For the cases discussed in this work this is true at asymptotically large times. For instance, the total number of particles (= number of anti-particles = number of

14 o n matrix any for (2.19) It Eq. (2.19)? in Eq. described fulfills property that the result, regain a applied tothe A to get possible be to general order in theprojector in is should spinor how adjoint But the way. and/or spinor asymmetric in operators field )adteProjector the and Eq.(2.1)) (see operators field two of odfor hold projector a When not the has in function yield distribution asymptotic all the for calculated calculations been numerical the of some In by given is vacuum the of out produced pairs) ucini hti eae ne ojgto ieaDrcmatrix Wigner Dirac a the like of conjugation property under important behaves An pairs it produced that function. states the is Wigner quantum of function the the chirality various from and spin extracted hence about be and information can field the Dirac such gives As the function of electrons. Wigner of space the spinor function, the distribution to one-particle access full the to addition In States Quantum 2.3. pta arpouto est hc ilb aldtepril yield particle the tofindthe this called momentum as be over the space, will over which density integrate tointegrate production to pair sufficient sense spatial isthen make It not does in does infinity. function it yield arediscussed distribution would and the times position assumed, on is depend homogeneity at intermediate not spatial When 98]. numbers [97, particle Ref. for definitions Possible yrpaigi with it replacing by P W p hti eas h inrfnto sbidfo h commutator the from build is function Wigner the because is That . x - p y paei called is -plane P sapidt h inrfnto,ti rprymgtnot might property this function, Wigner the to applied is N #‰ p u nyfrasiewith slice a for only but , xy =lim := N N t B →∞ =lim := lim = N = t  W t →∞ xy →∞ 1 2 † d 2  n eie as defined and  p = π A  x dΓ γ +  (2 d 0 W γ 3 π d f 2 #‰ 0 p ) p π ( A γ 3 y t, f 0 † f γ ( . #‰ , x t, ( 0 p t,  z #‰ p #‰ p . #‰ p 0 = ) ) ) . . | p ntoecssteparticle the cases those In . z P =0 osntapyt both to apply not does . ..QatmStates Quantum 2.3. (2.19) (2.17) (2.18) 15

7 π φ = 49 2 2. The Wigner Function Formalism

Now Eq. (2.19) holds for B

1   γ0Bγ0 = γ0Aγ0 + A† = B† . 2 Applying this procedure to P W for any projector P results in

1   P @W := P W + γ0(P W)†γ0 (2.20) 2 1   = P W + Wγ0P †γ0 . 2 The operator @ is now defined as a shorthand for applying an operator to amatrix  1  in the way defined in Eq. (2.20). From i Pi = follows i Pi@W = W for any W. It should however be noted, that applying the operation P @ to W is not idempotent, because in general P @(P @W) ̸= P @W. What do P @W and specifically the two terms P W and Wγ0P †γ0 mean? Re- member that the Wigner function is basically build from a commutator

ˆ #‰ ˆ #‰ Wab ∼ [Ψa( x +, t), Ψb( x −, t)] .

A term P W results, when the spinor in the first argument of the commutator Ψˆ is replaced by the same spinor after the projection P was applied,

Ψˆ → P Ψˆ .

Because the entries of the projection matrix are just numbers and can be removed from the commutator we can deduce

ˆ ˆ Wab ∼ [Ψa, Ψb] ˆ ˆ → [PacΨc, Ψb] ˆ ˆ = Pac[Ψc, Ψb]

∼ PacWcb = (P W)ab .

Analogously a term Wγ0P †γ0 results from applying the projection operator to the adjoint spinor and replacing

Ψˆ → P Ψˆ .

16 rtr r h olwn [100] following the are results relevant erators the but in Ref.[99], those note from Please unchanged. different are thereof. are combinations definitions or these eigenstates that chiral eigenstates, charge states, sflprojectors Useful state any generic Diracfield the in the apply now function can density We the energy answered. is for function definitions Wigner the in operators spinor right the from projector adjoint the applying to identical is This h pnpoetrrequires projector spin The ewl o oka oepsil rjcin.Sm elkonpoeto op- projection known well Some projections. possible some at look now will We yteerslsteqeto fhwtepoetrsol eapidt oho the of both to applied be should projector the how of question the results these By hrlprojection Chiral pnprojection Spin projection Charge f s otepoetdWge function Wigner projected the to (2.16) and (2.15) Eqs. from n aclt h atcedsrbto o pcfcqatmstates. quantum specific for distribution particle the calculate and P ol orsodt aiu pnregntts .g,si eigen- spin g., e. eigenstates, spinor various to correspond could ϵ f s s ( [ W W a W 2 =tr[( := ) := ] + ab P b ∼ [ = → [ = ∼ ( 2 a,b,c 2 1 + ω  [ P [ P Ψ Ψ Ψ ˆ ˆ ˆ W Ψ ˆ p ϵ ) P r c , a a a / s / ± a 2 s e ( , , , l γ , @ W − W Ψ  Ψ 1 = ˆ = = = = ˆ ( 0 Ψ P ˆ W P d b ] ] † 1 2 1 2 1 2 1 2 Ψ) † ˆ  P )( γ ( ( ( ( oensure to γ 1 1 1 1 0 † m b 0  γ ] P ± ± ± ± vac ab 0 1  † ( (i Q γ + γ b . . aσ ) ] 5 ϵ aγ 0 = ) ) .  #‰ p rteoepril distribution one-particle the or 23 db 2 · γ P + 2 1 3 #‰ γ ( 2 a,b,c ( i + )] bσ 1 ∓ ) 31 bγ , ± γ ..QatmStates Quantum 2.3. + 3 = 0 γ ) cσ 1 P i + 12 ( a,b,c ) cγ . ) 1 , ± γ P . 2 )) s @ W (2.21) for 17

8 π φ = 49 2 2. The Wigner Function Formalism

Chiral Projection

The results for the energy density for the chiral states r/l are

m 1 #‰ #‰ #‰ ϵ (W) = s + ( p · (v ± a)) r/l 2 2 1 1 #‰ #‰ = ϵ(W) ± p · a . 2 2 For the distribution function f this turns into

1 f = ϵ (W − W ) r/l 2ω r/l vac. 1   = ϵ (W) − ϵ (W ) 2ω r/l r/l vac.   1 1 1 #‰ #‰ 1  =  ϵ(W) ± p · a − ϵ(Wvac.) 2ω 2    2 2    2ω(f−1) −2ω 1 = (f ± δf ) 2 c with the chiral asymmetry of the particle density

1 #‰ #‰ δf := p · a . (2.22) c 2ω

Charge Projection

When the charge projections Pp/e are applied the energy density and one-particle distribution function are

m   1 #‰ #‰ ϵ = s ∓ v0 + ( p · v) p/e 2 2 and 1 f = (f ∓ δf ) p/e 2 Q with the charge asymmetry of the particle density

1 δf = mv0 . (2.23) Q 2ω

This result is no surprise given that we already interpret v0 as the charge density due to Eq. (2.10).

18 ihtesi smer ftepril density particle the of asymmetry spin the with and h atcedniyi ie by given is density particle The obtain now We as givenby groups two into and charge spin specific combining with by moment states magnetic 4 to the correspond that projectors construct now can We let simplicity For by projection. (2.12), charge Eq. a to with correspond do projection that spin observables a construct combining to possible however is It Moment Magnetic the On (2.12). operators (2.5). Eq. spin Eq. the to function Wigner because correspond understandable, to perfectly seem is not for result does this result hand this other hand one the On direction any for result the projections spin the of case In Projection Spin i ̸= j eetthe select f ϵ δf , n , n #‰ #‰ ± ± n #‰ ϵ = = = µ #‰ t z ± 2 1 1 2 2 2 P P 1 := ω ( ( opnn,a seietfo h eopsto fthe of decomposition the from evident is as component, µ µ f f m z − z +  µ 1 2 − ± s z ± = = ( + #‰ m p = δf P P ( · p e #‰ p 1 2 n #‰ s P P P #‰ n ) u · ∓ ( u u f / a v #‰ + + d a 0 ± ) := z P + P ± + ) δf p e m P P P 2 1 µ d d (0 z #‰ p  #‰ n ) = = , 0 − , · · 1) 2 2 1 1 #‰ v #‰ p #‰ , t ± ( ( 2 · 1 1 ± .  #‰ n − + ( . a #‰ p γ γ 0 5 5 × + γ γ 3 3 #‰ t ) m ) . #‰ n ) z #‰ n ) with . ..QatmStates Quantum 2.3. · #‰ t 2  #‰ n 2 1 = is (2.24) σ 19 ij

9 π φ = 49 2 2. The Wigner Function Formalism with its asymmetry

1  #‰ #‰  δf = −ma + ( p × t ) . (2.25) µz 2ω z z The analogue results for magnetic moment in the x and y direction are

1  #‰ #‰  δf = −ma + ( p × t ) , µx 2ω x x 1  #‰ #‰  δf = −ma + ( p × t ) . µy 2ω y y #‰ These results contain the spin density a identified in Eq. (2.12).

20 nasailyhmgnosstp( setup homogeneous spatially a In Method Wigner The pulses 3.1. field electric different of number Towards a . 3.3.6 3.6) to to con- 3.4 3.3.4 studies (Secs. Secs. explored. the are chapter in of this presented results of are the end pulse Afterwards the Sauter rotating available 3.3.3. the the a Sec. compare on as in to ducted 3.3.2 example other an Sec. each of as in with phenomenology used explained methods is the is it Afterwards production before will pair consideration, 3.3.1. and regarding preliminary Sec. pulse pulse in Sauter Sauter case rotating rotating the semi- special the The to this only thesis. for this applied discussed of been part be far large so a up has makes method semiclassical which Sauter classical 3.3 , a rotating Sec. The Afterwards in 3.2. introduced derived. Sec. is is is introduced pulse method fields Wigner electric two-component the for and method case special this to homoge- time-dependent with in fields electric production neous pair positron electron cover will chapter This Fields Electric Homogeneous 3. notoidpnetst f1 n qain ah nytesto 0equations 10 of set the Only each. equations 6 functions and decouples the ofmotion 10 for of equations sets differential independent homogeneous two 16 into of system the result a As to simplify Eq.(2.3) from operators differential pseudo w =  D #‰ s , #‰ x 0 = v #‰ B #‰ , 0 = , #‰ a P #‰ , teWge ucinfraimi applied is formalism function Wigner the 3.1 Sec. In . = #‰ t #‰ p  ∇ #‰ ⊺ and #‰ x , 0 = D t ihapr lcrcfed( field electric pure a with ) = ∂ t + e  E #‰ #‰ t (  t ) i · := ∇ #‰ #‰ p t 0 . i − t i 0 B #‰ 0 = the ) (3.1) 21

10 π φ = 49 2 3. Homogeneous Electric Fields has a non-vanishing initial condition, while the remaining functions are zero for all times #‰ #‰ p = a0 = v0 = 0 , t 2 = 0 . (3.2)

What remains of the equations of motion can be written as

   #‰    s 0 0 0 2 p ⊺ s  #‰  #‰   #‰  #‰ #‰  v   0 0 −2 p × −2m v  #‰       ∂t + eE(t) · ∇p  #‰ =  #‰  ·  #‰  a   0 −2 p × 0 0   a   #‰  #‰   #‰ t −2 p 2m 0 0 t  #‰ #‰  #‰ ⇔ ∂t + eE(t) · ∇p w = M· w . (3.3)

In the numerical calculations a slightly different representation of the Wigner Matrix is used. The Wigner function is written as

 #‰ #‰  #‰ m p · v 0 #‰ #‰ p 4W = 2(f − 1) − 1 + iγ5p + γ v0 − γ · v + 2(f − 1) ω m ω (3.4) µ µν + γ γ5aµ + σ tµν .

By comparing the two representations we see #‰ #‰ #‰ 2m p · v #‰ #‰ 2 p s = (f − 1) − v = v + (f − 1) . (3.5) ω m ω Obviously this substitution does not change the degrees of freedom, as it is a linear #‰ ∂(s,v ) 2ω system of equations with determinant det #‰ = > 0 in each point in time. ∂(f−1,v n) m Additionally we need to make sure that the variable f as introduced by Eq. (3.4) is indeed the same object as the one-particle distribution function introduced in Eq. (2.14). To this end we will start with Eq. (2.13) and reproduce Eq. (2.14).

#‰ #‰ ϵ = ms + p · v #‰ #‰ #‰ 2m p · v  #‰  #‰ 2 p  = m (f − 1) − + p · v + (f − 1) ω m ω #‰ m2 #‰ #‰ #‰ #‰ p 2 = 2 (f − 1) − p · v + p · v + 2 (f − 1) ω ω #‰ m2 + p 2 = 2 (f − 1) ω = 2ω(f − 1)

22 oiidbsdsuigadfeetsmo o h Wigner components forthe symbol adifferent using numerical besides the modified for asymmetries these method calculate Wigner to the formulae for at as results. substitution arrive same we the (3.5), using Eq. purely and homogeneous, (3.2)) spatially (Eq. the Wigner case in remaining know electric we the what and Applying function distribution components. one-particle function the of terms in written by given are smere fteoepril itiuinfunction various distribution the for one-particle formulae the The of results. asymmetries numerical the from accessible readily is function hc sasse f1 nooeeu ata ifrnileutosfor equations differential partial inhomogeneous 10 quantities of auxiliary system a is which olw h ouino h lsia qaino oinfrapril ihcharge with particle a for motion momentum of kinetic equation the classical that the method requirement of the the solution apply on we the ODEs based follows into is PDEs This of (3.6) characteristics. Eq. of system the transform to Equation order In Kinetic Quantum Modified The 3.1.1. due vanish 2.23) ( and (2.24) Eqs. (3.2). from Eq. asymmetries to charge and spin the contrast In r not are (2.25) and (2.22) Eqs. asymmetries, moment magnetic and chiral The ,teoepril distribution one-particle the (3.5), Eq. in substitution the of nature the to Due eut in results (3.3) Eq. motion of equations the to substitution this Applying     ∂ ∂ ∂ ∂ t t t t + + + + e e e e E E E #‰ #‰ #‰ E #‰ · · · #‰ , v · ∇ ∇ ∇ #‰ #‰ #‰ ∇ #‰ #‰ #‰ #‰ p p p #‰ a #‰ p    δf  = δf #‰ #‰ #‰ a v t f µ #‰ a ( ( ( ( z c f t, t, t, t, , = = 0 = #‰ #‰ #‰ #‰ #‰ p p p p t 2 2 = ) ( 2 = ) = ) = ) 1 1 = ω ω , #‰  p #‰ #‰ v t − − 2 2 1 h nta odtosat conditions initial The . · ω ω 1 = ma #‰ p #‰ , a #‰ v 3 − e E × #‰  #‰ a + z ω #‰ p 1 ( + · = #‰ , v 2 #‰ ( p #‰ , v e #‰ p ( #‰ E t #‰ #‰ ( p #‰ p e = · E × #‰ · #‰ p #‰ v #‰ 0 · #‰ ) t )) #‰ f v − . ) z ) aealready are 2.3 Sec. in given ,  ω − . 2 ..TeWge Method Wigner The 3.1. e #‰ p E #‰ ×  ( #‰ a f t − − −∞ → 1) 2 #‰ , t o all for f n 9 and (3.6) (3.8) (3.7) 23 #‰ #‰ p p e 11 π φ = 49 2 3. Homogeneous Electric Fields

#‰ in the external field with canonical momentum q #‰ #‰#‰ #‰ p q (t) = q − eA(t) . (3.9) #‰ This allows the partial derivatives w. r. t. p to be absorbed into the total temporal derivative according to

d #‰   #‰  g(t, p #‰(t)) = ∂ + (∂ p )∂ g(t, p ) q t t i pi #‰ #‰#‰ dt p = p q (t)  #‰ #‰  #‰  = ∂ + eE(t)∇#‰ g(t, p ) . t p #‰ #‰#‰ p = p q (t)

#‰#‰ #‰ Additionally any function g(t, p q (t)) can now be reinterpreted as a function g˜(t, q ), #‰ where the canonical momentum q is now merely a parameter enumerating different trajectories. Along any of those trajectories the functions can be calculated by solving a set of ordinary differential equations without interchanging information with other trajectories. Since results are always being taken for t → ∞, it is convenient #‰ #‰#‰ #‰ to gauge the vector potential A in such a way that p q (t) → q for t → ∞ . The result of this procedure is a modified quantum kinetic equation [75], which can be solved numerically to directly calculate the one-particle distribution function #‰ #‰#‰ f(t, p ) at t → ∞. The kinetic momentum along the classical trajectory p q (t) will #‰ be denoted p in the context of the Wigner method. The modified quantum kinetic equations read

1 #‰ #‰ f˙ = eE · v , 2ω #‰ 1  #‰ #‰ #‰ #‰ v˙ = p (eE · p ) − ω2eE (f − 1) 2ω3 1 #‰ #‰ #‰ #‰ #‰ #‰ (3.10) − p (eE · v ) − p × a − 2 t , ω2 #‰ #‰ #‰ a˙ = − p × v , #‰ #‰ #‰ #‰ #‰ t˙ = 2 ( v + p ( p · v )) , where the dot above any function denotes the total temporal derivative, e. g., f˙ = d dt f. Combined with the initial conditions #‰ #‰ #‰ #‰ f = 0, v = a = t = 0 (3.11) at t → −∞ , the initial value problem is well defined. The modified quantum kinetic #‰ equations turn into a homogeneous ODE when E(t) is set to 0 . From this we can

24 eyltl rmtevcu ouin hc ed ols npeiinfo h dif- the from precision in loss wouldonlydiffer to leads solution which the solution, rates, vacuum the production pair from small little for very that is, problem calculations first one-particle These the calculate to potential. function necessity vector been the density to the have due for calculations problems solution numerical these showed analytic approaches, quickly an later differ- using all ordinary out obtain to to carried contrast applied In characteristics function of equations. theWigner method ential for the with equations (3.3) differential Eq. homogeneous from of set the with starting nu- calculate available needed to of all made terms for For was in precision, anyway. if decision numerically written used.Additionally, the calculated be be reasons not is must these can potential field vector or the re- found the electric functions, analytical be merical these can for change solution to configuration analytic have new no to a impractical time be each would trajectory.sults it different each code of along numerical lot over a a and In for over principle, motion in of is equation It classical same the solve to aihsfo h ueia acltoswe l lcrcfed r ie inunits charge elementary given asin are The fields strength electric mass. field all critical electron when the the of calculations is numerical momentum the from and vanishes energy of while unit calculations, the numerical the in used unit 1 base the is mass electron The Calculations Numerical 3.1.2. E utb nw.Ete n sal to able is one Either known. be potential must vector (3.9) the Eq. calculate momentum kinetic the merically, such that small is sufficiently field electric the time region a this pulse, for at field Sauter ODE electric the the the solving consider of start to suppression exponential useful the is to it Due test, benchmark a As time finite a at true also is (3.11) Eq. condition initial the that deduce hsmasta ieadlnt r esrdi nes lcrnmse,while masses, electron inverse in measured are length and time that means This . ( u is ueia ouin aebe on using found been have solutions numerical first Our nu- (3.10) equations kinetic quantum modified the solve to able be to order In t 0 = ) for t < t f 0 . .The (2.14). and Eqs. (2.13) using integration the finishing after t nea h einn fteprogram. the of beginning the at once , e E #‰ ( t A #‰ = ) ( t t 0 ) E e = orsodn oteiptfield input the to corresponding − cr E #‰ . 10 E ( #‰ E t τ ) ( cr osbet calculate to possible , t n tpa time a at stop and . ) = m 2 A #‰ E #‰ E ( ( cr t t ) ofa Mathematica Wolfram . ) ueial ihsufficient with numerically . ..TeWge Method Wigner The 3.1. f | ˙ t ≈ ≫ | 0 t 1 . τ E 10 = ti ufcetto sufficient is it , A #‰ ( t E ( #‰ = ) t ) ( τ t analytically. usd of Outside . ) E roehas one or 0 t ℏ sech 0 = [101], when c 2 25 τ = t . 12 π φ = 49 2 3. Homogeneous Electric Fields ference in Eq. (2.14). On the other hand the execution of Eq. (2.13) also includes large cancellations. This realization led to introducing the substitutions Eq. (3.5), which were then incorporated into any later numerical implementation. Because the integration processes for every single data point of the pair production spectrum are independent, parallelization of the calculation is trivial. Various technical approaches to split up the work and collect the results later have been tested.

Ppsolve

When the high-level nature of Wolfram Mathematica [101] hampered the numerical performance, a new implementation was created written in the Python programming language using the Scipy [102, 103] package for scientific computation. Scipy contains an implementation of a generic ODE solver called LSODA, part of ODEPACK [104]. The advantage of the LSODA algorithm is that it switches between stiff and non-stiff methods adaptively. The implementation accepts twopa- rameters, rtol (relative tolerance) and atol (absolute tolerance), and then chooses the step size such that it ensures the approximate integration error ε to be smaller than ε = atol + |x| rtol , where |x| is the Euclidean norm of the solution vector. Previous experience [75] showed, that rtol should be set to 0, because of big intermediate function values, which would spoil the overall precision. As a result the only external parameter to the numerical calculations is the absolute error tolerance atol = 10−k. The evaluation of the right-hand sides of the equations was carried out using a compiled C++ component in order to improve performance. This component was built using a specialized part of Scipy, called weave. It used the Blitz++ library [105] to generate optimized code. For parallelization a work queue was implemented using MPI4Python such that it was possible to use the computer cluster of the university. This work queue would distribute all the single ODE solver runs across all the processors of all the nodes assigned to a job, while a single job could easily contain a multi-dimensional parameter scan. The workload distribution using this model turned out to be quite uniform, maximizing the computation load. The total cpu time for a parameter scan was however difficult to predict such that jobs were often cancelled when they exceeded their wall time limit.

26 qainfrterfeto ofiin n oslei ueial as wasdonefor solve itnumerically and to Riccati a construct to coefficient possible reflection is it the Indeed, for performed. equation is (3.26 ) Eq. in proximation coordinates prefac- interpolating in method. in as 60] [ instanton to method Ref. worldline in referred instanton and the (often method for worldline order scattering the next the for the [92] include using Ref. in to using the tor) and extended was 86] This studied [79, [108]. been Ref. Ref. has in fields rotating method constant scattering 3.3.1 . for inSec. production given is pair Schwinger pulse ofanumerical field specific example the a An of to choices method other this berequired. con- for of a application for but might rate [92], calculations production analytically numerical pair field becalculated , 81 The electric 80, ansatz. can , 78 scattering field 53, a gener- rotating , 52 on a stant [50, is based fields This are electric which 99]. 85] unidirectional method[92, 84, for methods byasemiclassical similar of becomputed elec- alization homogeneous also two-component can by fields production tric pair method, Wigner the Besides Two-Component for Method Semiclassical The 3.2. error absolute the for settings with motion as of small equations enabled as This the tolerances integrate calculations. point to floating code in registers the cpu of precision excess the from using method outsomeof Wigner totry the of wish implementation The new a the to interface. lead aunified implementations the these of through part algorithms different is of which lot the focus, implementations, into solver came ODE [106] available other into looking While Charwigner putations faster. The switch has the side-effect of changing side-effect the standard has switch The faster. putations switch to similar tolerances error abserr relative and absolute giving by controlled oeta h cteigast rsne ntefloigi xc ni h ap- the until exact is following the in presented ansatz scattering the that Note Sauter- of factor) exponential as to referred (often order semiclassical leading The oto h D ovr in solvers ODE the of Most C++ -ffast-math lcrcFields Electric and agae The language. to relerr -fexcess-precision=fast ..TeSmcasclMto o w-opnn lcrcFields Electric Two-Component for Method Semiclassical The 3.2. hc ialssrc EE74cmlac nodrt ocom- do to order in compliance 754 IEEE strict disables which , nti ae oal rc a h s fteg+compiler g++ the of use the was trick notable A case. this in 10 − Blitz++ 14 with boost.odeint double a otne ob used. be to continued was [105] library hc nbe h optto obenefit to computation the enables which , lo dpiese iecnrlta a be can that control size step adaptive allow rcso arithmetic. precision boost irr rjc.I fesa offers It project. library [107] -fexcess-precision= LSODA odeint called , library 27

13 π φ = 49 2 3. Homogeneous Electric Fields one-component fields in Ref. [76, 77]. However for one-component electric fields the Riccati approach has been shown to be equivalent to the QKT [76], which in turn is equivalent to the Wigner method [90]. While this is not necessarily true for the two-component case we still expect the Riccati approach to have a numerical behavior comparable to the one of the Wigner method.

3.2.1. Solution of the Dirac Equation

We start from the Dirac equation

µ ˆ #‰ ([i∂µ − eAµ(x)] γ − m) Ψ( x , t) = 0 and decompose the Dirac field as

3  d q #‰ #‰  †  ˆ i q x  #‰ #‰ ˜#‰ ˆ #‰ Ψ(x, t) = 3 e ψ q ,s(t)ˆa q ,s + ψ q ,s(t)b− q ,s , (2π) s=±1 where

˜#‰ #‰ ∗ 2 0 ψ q ,s(t) := Cψ q ,s(t) , C = iγ γ .

The Dirac field satisfies the canonical equal-time anticommutation relations

 ˆ #‰ ˆ † #‰  3 #‰ #‰ Ψa( x , t), Ψb( y , t) = δab · δ ( x − y ) , provided the mode operators obey the corresponding relations

 †  #‰ #‰ aˆ #‰ , aˆ #‰ = (2π)3δ3( k − q )δ , q ,s k ,r rs  †  #‰ #‰ ˆb #‰ , ˆb #‰ = (2π)3δ3( k − q )δ , q ,s k ,r rs  †  aˆ #‰ , ˆb #‰ = 0 q ,s k ,r and the modes satisfy the Wronskian condition

   #‰ #‰ † ˜#‰ ˜#‰ † ψ q ,s(t)ψ q ,s(t) + ψ q ,s(t)ψ q ,s(t) = 1. (3.12) s=±1

28 for osletesurdDrceuto.Deto derived Due be equation. Dirac can squared equation e., the i. solve Dirac to the [92] Ref. of in ( used solution is the that ansatz for the ansatz from This introduced. is ansatz the make can one where i.e. representation, Weyl the in work to choose we convenience For Inserting this ansatz into the Dirac equation leads to the defining equations for the equations defining the to leads functions equation mode Dirac unknown the into ansatz this Inserting h ouin ewl idblwfor below find will we solutions The hr ehv defined have we where via (3.14) Eqs. equation in Dirac present the in course coupling of minimal the is from potential vector the that note Please q z o w-opnn ilssll eedn ntime[ on depending solely fields two-component For + s = sϵ σ ± ⊥ j )( 1 r h al matrices. Pauli the are ihafxdnraiainconstant normalization fixed a with , q z − sϵ ..TeSmcasclMto o w-opnn lcrcFields Electric Two-Component for Method Semiclassical The 3.2. γ ⊥ j = ) = i i   ψ ψ − ˙ ˙ − ψ 2 1 s s m 0 ( ( σ ψ #‰ ,s q t t j 2 + ) + ) 1 s / ( eosrethat observe we 2 p t σ ψ = ) 0 x ϵ s ϵ s j ± #‰ ,s q   y ⊥ ⊥ ( γ , C ( t ψ ψ ϵ s := ) t ⊥ 2 s 1 1 ) s s = ( ( †        := t t #‰ p · ) ) − ± ψ p − − ( = s q 1 x q #‰ , q z ( 2 ( z hsrpeettoidpnetsolutions. independent two represent thus p s p s q t − + #‰ q ψ m s z + ) s ψ m + x x ( ± ψ − m sϵ t − + 0 = ) #‰ ,s q y y sϵ i 2 e ⊥ p ( ( 2 C s 1 s A #‰ t t ( y ) ⊥ ( ( ) ) s t ( t ψ t ) ( eetetases energy transverse the Here . ψ ψ ) t ) ) t ) ψ 2 s . 2 2 s s ) 0 and ( . ( ( h rnka odto in condition Wronskian The . 1 s t t t = ( ) 0 = ) 0 = ) A t )   µ        ψ ( 1 0 x #‰ , q (0 = ) , , − 1 0 s (   t ) , r independent, are A , x ( t ) A , y ( (3.13) (3.14) t ) , 0) 29 ] 14 π φ = 49 2 3. Homogeneous Electric Fields

Eq. (3.12) holds if

s 2 s 2 |ψ1(t)| + |ψ2(t)| = 1 (3.15) and 1 Cs =  . 2ϵ⊥(qz + ϵ⊥)

s The unknown mode functions ψ1/2 are now replaced with a WKB ansatz

  i i  cp (t) − 2 Ks(t) 2 Ks(t) s x−y  e e ψ1(t) =  cp∥(t) αs(t) + iβs(t)  , (3.16) 2ω(t) ω(t) + sϵ⊥ ω(t) − sϵ⊥

  i i  cp (t) − 2 Ks(t) 2 Ks(t) s x+y  e e ψ2(t) = s  cp∥(t) αs(t) − iβs(t)  , (3.17) 2ω(t) ω(t) − sϵ⊥ ω(t) + sϵ⊥ introducing the Bogoliubov coefficients αs(t) and βs(t). See Ref. [92] for a motiva- tion. The integrals are given by

Ks(t) := K0(t) − sKxy(t) , (3.18) t  ′ ′ K0(t) := 2 ω(t ) dt , (3.19) −∞ t  ′ ′ ′ ′ p˙x(t )py(t ) − p˙y(t )px(t ) ′ Kxy(t) := ϵ⊥ ′ ′ 2 dt (3.20) ω(t )p∥(t ) −∞ with

2 2 2 p∥(t) := px(t) + py(t) , #‰ #‰ ω(t)2 = ( q − eA(t))2 + m2 .

From this ansatz, using Eq. (3.14), the evolution equations for the Bogoliubov coefficients ω˙ (t) α˙ (t) = Gs (t)eiKs(t)β (t) , (3.21) s 2ω(t) + s ω˙ (t) β˙ (t) = Gs (t)e−iKs(t)α (t) (3.22) s 2ω(t) − s

30 efind we (3.15) Eq. condition normalization the in 3.17) ( and (3.16) Eqs. Using h otu eetattetrigpit o which for points turning the extract we contour the that fact the points turning use classical now the We around [ 109]. regions by Ref. dominated in are introduced integrals ideas the the following (3.22) for and description a multiple-integral find can one function a as pairs positron electron momentum produced canonical of the number of the as interpreted be can probability Pairs transmission The Produced of Spectrum Momentum 3.2.2. where found, are onsrpeetplso order of poles represent points the (2.9) Eq. to According unn on ftepair the of point turning following the in If ..TeSmcasclMto o w-opnn lcrcFields Electric Two-Component for Method Semiclassical The 3.2. G β t ± s p s ( ( sue ihu h superscript the without used is −∞ t ω ω i = ) ˙ ( ( t t ) ) 0 = ) t p ± t s ≈ p ± p #‰ q .Asmn htteturning the that Assuming (3.25). Eq. fulfills which ∥ ϵ dK r on ncmlxcnuaepis ydeforming By pairs. conjugate complex in found are | W sn prpit onaycniin [77] conditions boundary appropriate Using . ( ⊥ α ν α , t t s ) dt p s ( 0 109 ] [92, finds one , ( ℑ ± t ( #‰ ) q t ω [ | ) K 2 =lim := ) p p ˙ ˙ ( ν x x + t 0 t ( ( p ± p ( ν t t t | =0 := ) t ) 2 + ) β p p t p p →∞ )] s y x ( ( ( t < t K t ) ) ˙ + ) | | β 2 β 0 . 0 − ˙ ( s ± 1 = . t ( p ) p t ( ˙ ) t y y − | ) ( ( 1 2 . s t t yieaieyuigEs (3.21) Eqs. using iteratively by K ) ( ) p −∞ p ± 0 x y ( ( ( t t t twl lasrfrt the to refer always will it p ) ) 1 = ) ) p ω . ∥ ( ( t t ) ) , . (3.23) (3.26) (3.24) (3.25) 31

15 π φ = 49 2 3. Homogeneous Electric Fields

One can now approximate the preexponential factor in each integrand in the multiple-integral series by its behavior around the poles tp given by Eq. (3.26) to find

   πνtp β (∞) ≈ −2 e−iKs(tp) sin . (3.27) s 2(ν + 2) tp tp

This approximation is semiclassical in the sense that the exponential factor, which is not approximated, presents the leading semiclassical order. The approximation in Eq. (3.27) breaks down if the turning points get too close to each other in the complex plane, such that Eq. (3.26) is no longer valid, see Sec. 3.3.3. Since the examples covered in the present work have simple turning points, i.e.

νtp = 1, the semiclassical momentum spectrum of Eq. (3.23) takes the form

 2   s #‰  −iK (t ) W ( q ) =  e s p  . (3.28)    tp 

The total particle yield is then given by summing over the independent solutions

#‰ #‰ #‰ W ( q ) = W +( q ) + W −( q ) . (3.29)

3.3. The Rotating Sauter Pulse

Now that two methods are available for pair production in homogeneous electric fields, let us define an electric field which we will study and use asanexampleto compare the performance and accuracy of the methods. A spatially homogeneous, monochromatic rotating electric field under a cosh2-envelope in the absence of a magnetic field will be referred to as a rotating Sauter pulse. It isgivenby

cos(Ωt) #‰ E0   E(t) = 2 sin(Ωt) , (3.30) cosh (t/τ)   0 characterized by a maximum field strength E0, an angular rotation frequency Ω and a pulse duration τ. This field configuration can be viewed as a model for the fieldin an anti-node of a standing wave mode with appropriate circular polarization. This field has been at the center for a lot of the studies presented in this thesis.Later also superpositions of those fields will be taken into account, see Sec. 3.6.

32 fet si a etasomdt eob oaino h oriaesysteminthe coordinate ofthe rotation bya zero to betransformed ( Note can it [90]. as isoneofthe analytically calculated effect, phase which envelope be carrier can field, a function that Wigner Sauter-type the anon-rotating where to examples few collapses field rotating the where o a wyfo h eie ouin o eibenmrclsac.I,however, If, search. numerical are reliable points a parameter these for Unfortunately, strength solutions field point. desired the starting the a and from as away A.2 used The far appendix in be too solution. may desired discussed (A.3) the field Eq. to rotating in close constant given sense the some for in points is by turning that known done guess first is initial This rates, an found. needs production be which to pair need semiclassical potential given the solving the rotating numerically calculate of the points to for turning order solve classical becalculated the to In can required field are pulse. rotating calculations constant Sauter numerical a for but field rate electric [92], production two-component analytically pair generic The a 3.2. for Sec. Pulse discussed in Sauter was Rotating method semiclassical the The for Method Semiclassical The 3.3.1. time Compton the of units in mass electron the e., i. m scale, QED the given of units be in will dura- parameters pulse dimensionful the The within tion. cycles rotation full of ber nt fteciia il strength 1 field critical the of units where parameters dimensionless ,y x, . 3 o ntne h us uaini measured is duration pulse the instance, For . ntelimit the In 3.1. inFig. is illustrated field this of evolution time The o h icsin ti sflt nrdc the introduce to useful is it discussion, the For · ) 10 -plane. ε ℏ 18 esrstemxmmfedsrnt in strength field maximum the measures = V / c m 1 = ε and = . E E σ cr ω 0 . samauefrtenum- the for measure a is ( σ , t p 0 = ) Ω = ε ε → o complex for ϕ t srpae ytepleshape pulse the by replaced is (A.3) Eq. in c , , τ ihtereplacement the with E 1 = E E ( cr t cr j /m . ) . = = nunits in m cosh t (3.31) e p 2 ℏ c sn etnRpsnmto [110], method Newton-Raphson a using 3 2 ≈ ε  ℜ ( Tm eednyof dependency Time 3.1.: Figure τ t j E ..TeRttn atrPulse Sauter Rotating The 3.3. )  y ( Ω , t ) t → Ω pulse. Sauter rotating the t + ϕ ol aeno have would E E 0 = Ω ( cr E t j . ) x ( t t 33 ) , 16 π φ = 49 2 3. Homogeneous Electric Fields

Finding the classical turning points 7 Constant field 6.5 Guess for pulsed field 6 Result for pulsed field

] 5.5 c t 5 )[ p t 4.5

Im( 4 3.5 3 2.5 −30 −20 −10 0 10 20 30 Re(tp)[ tc]

Figure 3.2.: This figure shows how the turning points for the rotating Sauter pulse are found. The turning points of the constant rotating field are shifted away from the real axis by replacing the field strength parameter by the pulse shape (upward arrow). Afterwards the correct turning points can be found by a numerical search (smaller arrow). the result is a sufficient guess for the starting point (see Fig. 3.2 for a depiction of this behavior). In this way we also get a nomenclature for the turning points, by giving them the same name as the corresponding ones of the constant rotating field. For the computation the momentum grid is divided into several parts for parallelization. For each of these parts the number of used pairs of turning points is chosen adaptively. To this end, the turning point t0 is considered first. Afterwards, for increasing integer j, tj and t−j are added in pairs until their contribution to #‰ Ws( q ) is less than 0.1%. The semiclassical method relies heavily on integrals in the complex plane. These are expressed in terms of multiple real integrals by parameterization of the integration paths. Afterwards the GNU Scientific Library [111] is used to carry out the real-valued integrals, specifically with the use of adaptive Gauss-Kronrod [112] and Clenshaw-Curtis [113] rules. The adaptive algorithms are also tuned by an absolute and a relative error tolerance. Still, it is necessary to evaluate the vector potential #‰ #‰ #‰ d A with E = − dt A for complex times. The indefinite integral of the field given by

34 icniute.Teesnuaiisaefudo h mgnr xsat axis imaginary the on found are singularities These discontinuities. of solutions at larities where oeta ueial nagi nodrt s pieinterpolation. spline vector use thecomplex to integral order in line precomputes grid a method a performing Another requested on by numerically each field. potential scratch, computes electric from testing, complex potential for the vector over used complex mainly the methods, implemen- of these the value in of available One are potential vector tation. complex the evaluating for methods the and named code the a of to evaluation left the is arguments Finally AEAE complex axis. with leaving imaginary function axis hypergeometric the imaginary Gaussian on the of strictly side discontinuities right-hand the the all towards mirrored carefully of be thesymmetries can Byexploiting else. field electric everywhere the is continuous positive towards it axis real sin- the while the infinity, to at parallel real continue start and that axis lines imaginary straight the on on discontinuities gularities has (3.32) Eq. in given form The with as given be can (3.30) Eq. o ag ausof values large For 1 2 hc sdsrbdi e.[115]. Ref. in described is which F AEAE .Det h singu- the to Due [114]. function hypergeometric Gaussian the denotes  E #‰ irr al ocmueisvle eiby nteecssdifferent cases these In reliably. values its compute to fails library E ( t x d ) / y H H ( σ t t 2 1 ) = cosh h ruet otehpremti ucinbcm large become function hypergeometric the to arguments the = = ± → εE 1 2 1 2 F F 2 2 cr E (   . t / τ x 1 1 τ / , , 0 = ) y · ( t               − k 1 2 i = t 2 − ℜ ) − ℜ h otnosrgo ihngtv elpart real negative with region continuous the , h etrptnilms aebac cut branch have must potential vector the ,  2 2 i i ie  2 τ τ e k − Ω Ω − 2 i 1 + i t , , t Ω Ω H 2cos( +2 H 2sin( +2 2 1 τ. πτ 1 1 − − e + − 2 2 i i e τ τ 0 ..TeRttn atrPulse Sauter Rotating The 3.3. τ t τ Ω Ω t t t (2 )tanh( Ω) (2 )tanh( Ω) , , − − i i τ τ − − Ω) Ω) e e − 2+i 2 2 τ τ t t 2+i τ τ   Ω τ τ t Ω τ t , , ) Ω τ ) Ω H H 2  2                (3.32) 35

17 π φ = 49 2 3. Homogeneous Electric Fields

The Locally Constant Rotating Field Approximation

It is possible to approximate the momentum spectrum of the rotating Sauter pulse using the result for the rotating rectangular pulse field

cos(Ωt)   #‰ t   E = εEcr. Rect sin(Ωt) , τ   0 with the rectangular box function

Rect(x) = Θ(x) − Θ(x − 1) , where Θ(x) is the Heaviside step function. Fields of this form can be treated analytically as shown in Ref. [92], in a way similar to the constant rotating field

cos(Ωt) #‰   E(t) = εEcr. sin(Ωt) . (3.33)   0

The idea is to replace the field by a sum of rectangular pulses with pulse length τ0 and a different constant field strength given by the form of the pulse E(t), i.e. replace E(t) by

∞   1    t  1    t  E(t) ≈ E − j τ0 Rect + j + E + j τ0 Rect − j . j=0 2 τ0 2 τ0

Now one can compute the momentum spectrum of the pair creation rate using the analytic result for the pair creation rate for each of these pulses. The shorter the length τ0 the better becomes this approximation which we call the locally constant rotating field approximation, or LCRFA. Using that for the rectangle pulse theonly turning points which contribute are those whose real part lies within the pulse range, it is possible to perform the limit τ0 → 0 which leads to

 2  ∞ #‰ #‰  s #‰   K ( q ,E(ℜ[t ])) K ( q ,E(ℜ[t ])) W ( q ) =  e s j + e s −j  . (3.34) approx   j=0  #‰ Here Ks( q , E) is the integral from Eq. (3.18) which is given by Eqs. (A.4) and (A.5) and tj, represent the turning points given in Eq. (A.3).

36 h o-oaigSue us,tecaatrsi rqec srpae yteinverse the by replaced yielding is duration frequency pulse characteristic duration the pulse, pulse Sauter because the non-rotating enough, the parameter: good not important is another parameter disregards this it For here described used. pulse was Sauter in Eq.(3.31) rotating the defined parameter field-strength dimensionless the where as QED field to according electron bound inasinu- the field of ionization the of in atomic parameter frequency adiabaticity characteristic example the Keldysh the for There known is field. quickly varying electric in soidal ofregimes production pair slowly duality in This the multi-photon production fields. pair and byat Sauter-Schwinger fields strong the is provided very processes, varying, fields distinct strong quite in two production pair least introduction, the Pulse in Sauter mentioned As Rotating the of Phenomenology 3.3.2. pulse rotation the the of of scale scale time time the the which pulsearene- than in the approxima- the smaller pulses of is enough Accordingly long shape intact. for the rotation reasonable the by of is tion effects caused the leaving time variation while the glected, from effects Therefore h usinbisdw oakn htfeunycnb huh fa charac- scales. a time as independent of two thought has that be pulse, can laser frequency a what in asking frequency to teristic down boils question The strength field the given h CF prxmtstefedb osatrttn il teverytime. at field rotating aconstant by field the approximates LCRFA The n osblt ol eacmiaino ohsae fteform the of scales both of combination a be would possibility One γ := E Ω 0 Ω T n idn energy binding and , = Ω ˜  γ γ τ Ω  := a := Ω τ 1  τεm ihteitisctneigfrequency tunneling intrinsic the with τ εm 2 Ω 1 i.e. , (Ω) + E , . b Ω σ hshsbe dpe nstrong- in adopted been has This . T 2 =Ω := ..TeRttn atrPulse Sauter Rotating The 3.3. . = √ τ 2 eE ≫ mE 0 1 b . γ , τ sue ocompare to used is hnanalyzing When . (3.35) (3.36) 1 Ω / 37 Ω T

18 π φ = 49 2 3. Homogeneous Electric Fields

A somewhat less general idea of a combined Keldysh parameter γ∗ has already been introduced in Ref. [75], which used a = 1. When a much larger set of data was analyzed, it turned out that the parameter a should be varied. We propose a new π definition with a = 2

 1 π 1 2 γ∗ := + Ω2 (3.37) εm 2 τ  1 π 2 = + σ2 τεm 2 which far better fits the numerical data, as can be seen inFig. 3.11. Obviously the limits

∗ σ→∞ γ −−−→ γΩ (3.38) π γ∗ −−−→σ→0 γ 2 τ hold. The combined Keldysh parameter γ∗ can also be written in terms of the

Keldysh parameters γΩ and γτ as

 π 2 γ∗ = γ + γ2 . (3.39) 2 τ Ω

While the combined parameter γ∗ might be well suited for a collective quantitative description, when used to separate parameter regimes its meaning is just

∗ γ ≪ 1 ⇔ (γτ ≪ 1) ∧ (γΩ ≪ 1) (3.40) ∗ γ ≫ 1 ⇔ (γτ ≫ 1) ∨ (γΩ ≫ 1) . (3.41)

In different parameter regimes, different expectations arise for the resulting spectra from both Schwinger and multiphoton pair production. In order to be able to discuss the results in the later section, let us first talk about these processes.

Multiphoton Pair Production and Effective Mass

Multiphoton pair production rests on the idea that the electric field can be described in terms of photons which correspond to Fourier modes. If the field contains photons with high enough frequencies, some of them can combine to produce an electron-

38 ti ekda h us frequency pulse the at peaked is It h ttccmoet oprdt h ekseta opnn,i ie by given is component, spectral peak the to compared component, static The en htte ilol eoepr n lal eaae rmShigrpair Schwinger from separated clearly and higher pure for forsmaller become effects production only not expected be will does they This that not production. means should pair effects multiphoton multiphoton clear that expect mean can we and spectrum enough the large for that means That function peaked symmetric, a with a under wave plane complex a and neglected is polarization simplicity For pair. positron ocet neeto-oirnpi.Teeeto n oirnne ob on-shell, be to need positron and electron The means that pair. electron-positron an create to large for suppressed exponentially is which htde utpoo arpouto en h dai that is idea The mean? production pair multiphoton does What cosh #‰ p 2 2 neoei osdrd t ore rnfr sgvnby given is transform Fourier Its considered. is envelope = ω E 2 ˜ h − ( ω ( x m := ) := ) 2 2 = = ihtettlenergy total the with −∞ εE  σ E ∞ sinh εE ˜ E . max h cr d ( cr . ( t t πτ π 2 σ = ) . .  E e h τ x E σ ˜ ) ˜ = i Ω π 2 ωt 2 max h pcrldniypa at peak density spectral the , (0) −−−→ x σ ( E E E ε ( n a aiu value maximum a has and →∞  ω ˜ τ . ( Ω 2 = (Ω) − = (Ω t = cr ) . Ω) h h π 2 − cosh σ ( ( σ − because , σ sinh ω e e x − ) εE )) − ) iΩ  , π 2 h , τ t t cr ω  σ  . π hsttleeg oe from comes energy total This . 2 . ..TeRttn atrPulse Sauter Rotating The 3.3. . τ 2 τ 0 1 = (0) 1 ( ω − Ω) .  n ω htn interact photons σ Ω = tal tonly it all, at dominates (3.42) 39

19 π φ = 49 2 3. Homogeneous Electric Fields n photons of frequency Ω, but only half of the total energy goes into either the positron or the electron. The total momentum for any of the particles is thus given by

  2 #‰  nΩ | p | =  − m2 2 1 = (nΩ)2 − (2m)2 . (3.43) 2 Charged particles in oscillating fields do behave as if their mass is slightly higher due to the kinetic energy of their periodic movement (ponderomotive potential). This has been shown to be the case also for pair production in pulsed, oscillating fields [116]. For linearly polarized fields a good approximation for the effective mass is given by  e2 ε2 m = m 1 + . ∗ m2 2Ω2 For the rotating field, which has the same x-component and an additional y-component, the analogue result is

 e2 ε2 mrot. = m 1 + . ∗ m2 Ω2

Exchanging the mass in Eq. (3.43) for the effective mass we expect pairs tobe created by multiphoton pair production to have momenta of

#‰ 1 | p | = (nΩ)2 − (2m∗)2 (3.44) 2 for n > nmin., where nmin. is given by the requirement of a positive radicand

2m∗  n = , (3.45) min. Ω where ⌈x⌉ is the smallest positive integer k with k ≥ x. For short pulses the spectrum of the pulse has a high bandwidth which offers the possibility of having multiphoton pair production with different momenta, because then not only photons with the characteristic frequency are in the pulse. The components of the spectrum where 2m∗ = kω for some integer k will produce pairs at rest. Depending on the interplay between the time scales of multiphoton pair production and the pulse, the produced pairs are subject to acceleration by the electric #‰ #‰ field and end up with p = A(0).

40 n icsigispoete.Tefunction The σ properties. its discussing and function discussed  with in resulting case, this in at strength vectorpotential by the time. isdictated creation their which peak at distribution production final the pair in a momentum be kinetic should strength there field dominant, the is when production pair Schwinger If Production Pair Schwinger eednisfo h parameters the from dependencies for result the 0 1 ∞ → hl h eutfor result the While field the of maximum global the have always will pulse Sauter rotating The d x H 1 1 − n − x ie by given x n en h nltccniuto fteHroi Numbers Harmonic the of continuation analytic the being . t 0 = A y h etrptnilat potential vector The . snta tagtowr oudrtn.W a eaaethe separate can We understand. to straightforward as not is h A #‰ ( σ 0 = (0) g ) ( A σ )by (3.42)) Eq. (see | A = = E x #‰ = ) g y ( | sqiesml n a ewitnwt h previously the with written be can and simple quite is  E  0 0 0 = (0) σ ∞ ∞ a lcl aiu.Toeprilswudhv a have would particles Those maximum. (local) a has := ) cr n  E cosh #‰ ∞ =1 . A τ ε ( E t x ( d ) E cr σ 4 0 = (0) − 2   τ . ( cr  4) ε t t σ / 4 . and H τ g τ ε n  ) i H (4 σ 4   E i σ − n cr cos(Ω sin(Ω σ ( 4 σ σ . − ydfnn afunction defining by π − H 2 2 τh ε ) n σ 1) − / − g t H sinh ( 1 2 1 t t 0 = σ ζ − ) ( +i ) σ   (1 ) 1 2 ( σ 4 ) +i π 2 a ipesre xaso at expansion series simple a has d − , ..TeRttn atrPulse Sauter Rotating The 3.3. c + σ σ 4 t ) a eaayial calculated analytically be can 2 c + n . c ) .  . c .    , g ( H σ ) n codn to according =  k n =1 (3.46) k 1 41 =

20 π φ = 49 2 3. Homogeneous Electric Fields

σ   g(σ) = Hi σ − H− 1 +i σ + c.c. 0.7 4 4 2 4 g(σ) (1.51, 0.57) 0.6 g∞(σ) 0.5 g0(σ) 1/σ 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 σ

Figure 3.3.: This figure shows the asymptotics of the function g(σ) which is used to calculate the y-component of the vector potential at t = 0. with the Riemann ζ function. The first few terms are

1 2 16 272  1  g(σ) = + + + + O . σ σ3 σ5 σ7 σ9 At σ = 0 another series expansion is given by

∞ n n  (−1) (4 − 1)ζ(2n + 1) 2n+1 g(σ) = ln(2) σ + n σ . n=1 16

A useful approximation of g(σ) for σ > 3 or σ < 1 is given by taking these expansions to the orders σ−3 and σ5, respectively

1 2 g∞(σ) := + σ σ3 3 15 g0(σ) := ln(2) σ − ζ(3)σ3 + ζ(5)σ5 16 256 ≈ 0.693σ − 0.225σ3 + 0.0608σ5 , which is displayed in Fig. 3.3. The maximum of g(σ) is located at approximately

σmax. ≈ 1.506 with g(σmax.) ≈ 0.571295.

42 y limits the taking by Lc ftevco oeta nuisof units in potential vector the of Loci 3.4.: Figure ntrso h rqec ae eds parameter Keldysh based frequency the of terms in cmoettersl ntelaigodrin order leading the in result the -component oeitrsigrltost h eds daaiiyprmtr a efound be can parameters adiabaticity Keldysh the to relations interesting Some

Ay(t)[τ ε Ecr.] Ay(t)[τ ε Ecr.] − − − 0 0 0 0 0 0 0 0 . . 0 05 05 ...... aia il teghocr.Tedse iemrsti on forall point this marks line Thedashed at σ potential occurs. the vector strength mark field maximal Crosses ranges. axes different h uvsfrhge ausof values higher for curves The A 2 0 2 4 6 8 1 0 1 . y (0) − 0 . −−−→ 0 0 1 σ →∞ σ A 0 0 → x ( E t σ σ [ ) 0 cr . 20 = 10 = E ε τ and τ ε Ω σ cr 1 τ . . ] ∞ → 1 1 5 = E . 1 A cr .Frthe For (3.46). Eq. in given result the for . x ( Ω ε t σ [ ) − − − − = 0 0 0 0 0 0 0 0 E ε τ r hw nsprt lt,nt the note plots, separate in shown are σ ...... E 04 03 02 01 01 02 03 04 0 ∞ → cr . cr ..TeRttn atrPulse Sauter Rotating The 3.3. E ε τ mc γ . ] ℏ − Ω . 2 0 is . γ cr 2 0 0 025 1 Ω . A . 2 5 o ifrn ausof values different for = x σ ( t σ σ σ σ σ E 0 = σ σ [ ) t cr 0 = 5 = 3 = 2 = 1 = 0 = 50 = 30 = E ε τ . . t 1 c γ 1 cr Ω . ] . t 025 0 = where (3.47) 43 σ . 21 π φ = 49 2 3. Homogeneous Electric Fields

In the limit of σ → 0, taking only the linear term into account, the result is

σ→0 σ Ay(0) −−→ ln(2) Ecr. ε τ σ = ln(2) Ecr. tc γτ with the time scale based Keldysh parameter γτ . The limits of the x-component are given by

σ→∞ − π σ Ax(0) −−−→ Ecr. ε τ π σ e 2 → 0 (3.48)  2 2  σ→0 π σ 1 Ax(0) −−−→ Ecr. ε τ 1 − = Ecr. tc . 24 γτ

The complete result for the vector potential at t = 0 is visualized as the dashed line in Fig. 3.4. For σ → 0 the vector potential at t = 0 is dominated by Ax, for large σ the Ay-component is dominating.

Consequences for Pair Production Spectra

As is evident from QKT and the DHW formalism, the pair production spectrum in unidirectional electric fields have cylindrical symmetry with respect to the direction of the field. In the language of the rotating Sauter pulse that means thatfor σ = 0, where the field is in fact a unidirectional oscillating field as investigated inRef.[116],  2 2 the pair production probability depends on pq = px and p⊥ = py + pz. In the rotating field for large σ, another symmetry arises. In the case of a constant rotating  2 2 field (see Eq. (3.33)), the pair production probability depends on pq = px + py and p⊥ = pz. This is because a translation of the constant rotating field in time can be canceled by a rotation around the pz-axis. As the pair production probability should not depend on time it must also be the same for every angle of rotation around the pz-axis. As will be shown through a number of examples in Sec. 3.3.5, this is true also in the pulsed case for σ ≳ 20, independent of which pair production process is dominant. In both of these cases the subscripts q and ⊥ refer to the momentum direction being parallel or perpendicular to the electric field, however with linear fields the py and pz direction are both perpendicular to the field while in rotating fields only the pz direction is perpendicular to the field.

44    to according symmetry this ,ti orsod to corresponds this (3.47), Eq. to pair favor according Schwinger would sigma, which large varying, For and slowly production. is strong field the that implies p with etoe odto o utpoo arpouto.I ocuinoecnsay can previously one the conclusion of In inverse the production. to pair that exactly multiphoton for correspond condition conditions mentioned the (3.40), Eq. to to implies and pro- at pair multiphoton potential for vector allow the to If exist should duction. energies enough large with photons from away moving when quickly decay point in the momenta at small the peak in always a is with potential potential, vector the to of Due locus gaps. the have of not may vicinity or may that spheres, have to but the completed fields, When be oscillating interference. should linear of rings with result present the are are rings that These gaps (3.45). Eq. with by (3.44) given Eq. by given radii with rings of values non-vanishing A #‰ 1 a ecluae yexploiting by calculated be can Eq. (2.17) in defined as yield particle total The ftecaatrsi rqec cl slreenough, large is scale frequency characteristic the If the in lie should pairs produced the dominant, is production pair Schwinger If the within expect, we dominant, is production pair multiphoton If (0) = γ ∗ γ p    p = ∗ q 1 ≪ ≪ E = = γ z 1 τ √ 1 . 0 = p ≪ p γ and ⊥ x τ + 1 = ≪ o l ie eas have also we times all for n ntr implies turn in and ,    p z A #‰ √ y 1 direction, (0) ,ti corresponds this (3.48), Eq. to according sigma, small For . p and N y    + σ = = = ≫ p p h ultredmninlitga a ob calculated. be to has integral three-dimensional full the p 2  z (2 (2 1 z = and π π 1 1 0 = r hnmnlgclyeuvln,a are as equivalent, phenomenologically are | ) ) p p 2 3 (2 z ⊥ d −∞ −∞ |   3 π ln.Hnepissol epoue nywith only produced be should pairs Hence plane. p ∞ ∞ < = #‰ p 2 ) 3 d d 1 t p = f p p z m 0 = ( 2 2 γ p #‰ o oaigple with pulses rotating for p   0 0 p n h arpouto rbblt should probability production pair the and , ∞ ∞ Ω z n q ) d d 0 = ≪ = = slre(.g rae than greater g. (e. large is p p A 1 1 n z .  p 0 p 1 2 min 1 π x nbt fteecss according cases, these of both In . 0 = d f for . p φ ( n , p p ..TeRttn atrPulse Sauter Rotating The 3.3. z 1 min o l ie,hnetevector the hence times, all for 1 p , σ xsi ae noacut the account, into taken is axis f 2 . ( ) 0 = 1 + p 1 γ p , ∗ n , 2 ≫ (linear fields) or with fields) (linear ) min σ 1 . 2 + ≳ seE.(3.41)), Eq. (see . . . , 20 o small, For . p E γ #‰ x p ∗ cr with - p . ≫ = γ t y c Ω ,this ), plane, (3.49) 1 A #‰ ≪ n and (0) min 45 1 , . . 22 π φ = 49 2 3. Homogeneous Electric Fields

3.3.3. Comparison of the Wigner and Semiclassical Methods

As we have three different methods for calculating pair production spectra atour disposal (the Wigner method, the full semiclassical method and the LCRFA), we should start with a comparison of these methods. We do so for the example of the rotating Sauter pulse in Eq. (3.30). The limit to the non-rotating pulse Ω → 0 can be treated analytically with both the Wigner method and the scattering approach (see Appendix A.3 for more details). As for the constant rotating field discussed in Ref. [92] we find that there is an infinite number of turning points for the rotating Sauter pulse. But incontrast to the constant field case the turning points in the general case have different real and imaginary parts (see Fig. 3.2 for a plot of the turning points). This would in principle require a separate treatment of all of them. However, the closer a pair of turning points is to the real axis, the bigger is its influence on the pair creation rate [77], such that it is sufficient to study a finite number of turning points inorder to have a good approximation for the pair creation rate (see Appendix 3.3.1 for details). Note that this holds true also within the LCRFA where it is sufficient to evaluate the sum in Eq. (3.34) up to a finite |j|. In order to compare the momentum spectra calculated by all three methods, let us choose an illustrative example. Let τ = 10/m and σ = 6. Fig. 3.5 shows the spectrum as computed by all the available methods. In the semiclassical result and the result of the LCRFA the sum over both solutions according to Eq. (3.29) is taken. It turns out that the semiclassical method overestimates the pair production probability by roughly 12 percent as compared to the result of the Wigner method, which should be considered as exact within the chosen numerical precision. The result in LCRFA also has the same order of magnitude as the other results but underestimates certain features of the momentum spectrum. This is due to the small number of field cycles as the approximation gets better for bigger σ, see Fig. 3.6. We can compute the total particle yield per volume from the momentum spectrum by integrating over momentum space

 d3q #‰ N := W ( q ) . sc (2π)3 s

Comparing the results we find that the methods agree for an intermediate rangeof pulse length τ (see Fig. 3.6).

46 Mmnu pcrmo h atrplefor pulse Sauter the of spectrum Momentum 3.5.: Figure eutdvddb .2 ae ) h euto h CF iie by divided LCRFA the of result The semiclassical The c): Panel b): Panel 0.83. 1.12. result. by Wigner The divided result a): Panel box. color the ϵ 0 = py [m] py [m] py [m] − − − . 1 − − − 0 0 0 0 0 0 h eeso h otu ie r niae ytemrsof marks the by indicated are lines contour the of levels The ...... 1 5 0 5 1 1 5 0 5 1 1 5 0 5 1 LCRFA method Semiclassical method Wigner 15- 0500511.5 1 0.5 0 -0.5 -1 -1.5 c) b) a) p x [ m ] ..TeRttn atrPulse Sauter Rotating The 3.3. W W / / τ 1 0 . . 12 83 f 10 = [10 [10 [10 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 − − − /m 6 6 6 ] ] ] , σ 6 = and 47

23 π φ = 49 2 3. Homogeneous Electric Fields

N (ε, σ, τ), ε = 0.1 10−2 σ = 50 10−4 σ = 30 10−6 σ = 20 σ = 10 ] −8 −14 3 10 10 σ = 6

m −10 [ 10 σ = 0

N −12 10 10−15 10−14 10−16 10−18 500 750 1000 1 10 100 1000 10−2 σ = 30 10−4 σ = 20 10−6 σ = 10

] −8 3 10

m −10 [ 10

N 10−12 10−14 10−16 10−18 1 10 100 1000 τ [ tc]

Figure 3.6.: Comparison of the total particle number per Compton volume of the rotating Sauter pulse for ε = 0.1 as a function of the pulse length τ. Top panel: Solid lines show particle yield as calculated using the Wigner method, dashed lines show particle yield as calculated using the semiclassical method. In the cases σ ∈ {6, 10} a noise suppression method has been used when integrating over the spectra of the Wigner method to obtain the three dimensional totals. Bottom panel: Solid lines show the particle yield as calculated using the numerical semiclassical method, dashed lines show particle yield as calculated using the LCRFA. One finds that for long enough pulses the approximation agrees with the numerical results.

48 Cmaio ftettlpril ubrprCmtnvlm ftero- the of volume Compton per number particle total the of Comparison 3.8.: Figure of Value 3.7.: Figure

N [m3] y − − 10 10 10 10 − 1 0 0 1 10 10 10 10 . . . . 5 1 5 0 5 1 5 − − − − − − − − h te ad eoe ueial hae o ogrpulses. longer for on cheaper method, numerically semiclassical becomes length, The hand, pulse other method unfeasible. increasing the Wigner numerically with The becomes time it computational spectrum. until in per time increases yield, quickly processor showparticle lines show Solid lines dashed precision. regarding settings different for for pulse Sauter tating not is point turning every for holds (3.26) anymore. Eq. satisfied that assumption the and q 16 14 12 10 0 0 8 6 4 2 z 0 = 0101000 100 10 1 eseta o small for that see We . | ω ( t σ ) | 20 = . for 1 1 5 σ t eilsia,lwprecision low semiclassical, , | 20 = ω = N ( ϵ Re ℜ ( 0 = inr o precision low Wigner, , ,σ τ σ, ε, [ ( t t p τ ε σ p i + ] . i + ) τ 1 20 = σ , τ ε [ [ ) t t , c yτ c σ ] yτ ] ε 20 = h unn ons(e ie e closer get line) (red points turning the semiclassical , 20 = 0 = ) eedn on depending | [ m . safnto fteplelength pulse the of function a as 1 Wigner , ..TeRttn atrPulse Sauter Rotating The 3.3. ] . 2 5 τ ε for q x 3 = 1 10 100 1000 ,q m, 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

tcomp. [hrs] y 49 = τ

24 π φ = 49 2 3. Homogeneous Electric Fields

We also find that the semiclassical method is not stable for short pulses. This can be explained by looking at the non-rotating Sauter pulse which is studied in more detail in Appendix A.3. Taking the turning points given by Eq. (A.6) into consideration, we find that, for decreasing eE0τ/m, the turning points get closer in the complex plane (see Fig. 3.7 for a plot of |ω(t)| around the turning points). The approximation performed in Sec. 3.2 assumes that Eq. (3.26) holds for every turning point. This is not the case if the different turning points get too close to each other in the complex plane. For longer pulses the numerics of the Wigner method become challenging. This is due to the fact that the integration from t = −10τ to t = 10τ, which is performed analytically in the semiclassical method, needs more steps the longer the pulse becomes. For pulses that are too long the precision of the result is limited by computational errors which leads to an overestimation of the total particle yield due to summing up numerical noise. We find that for σ = 20 both numerical methods have a comparable computation time for pulse durations τ ∈ (40 tc, 100 tc), see Fig. 3.8. For shorter pulses the Wigner method is computationally faster, while for longer ones the semiclassical method should be preferred. The computation time when using the LCRFA is in comparison to the numerical semiclassical method negligible. We find that the approximation gets better fora longer pulse length τ and a higher σ (see Fig. 3.6, observe, that within the LCRFA calculations, the number of considered turning points has been fixed to nine, in contrast to the adaptive method used for the numerical method described in Ap- pendix 3.3.1). This can be explained by the fact that the approximation of the pulse being locally constant becomes better for longer pulses, and that a larger σ means more rotations per pulse length and hence larger rotation effects compared to pulse shape effects.

Interpretation of the Independent Solutions

In Sec. 3.2 we found two independent solutions of the Dirac equation which were interpreted as different spin components in Ref. [92]. To compare these with the

Wigner method we can construct a projector Ps, by requiring

#‰ #‰ #‰ Ps · ψ q ,s = ψ q ,s and Ps · ψ q ,−s = 0 for the two independent solutions of the Dirac equation defined in Eq. (3.13).

50 hyaeieptn n orthogonal and idempotent are they semiclassical the for spectra of Comparison 3.9.: Figure h eutn rjcosread projectors resulting The pz [m] pz [m] 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 0 0 0 0 . . . . 0 0 1 1 0 2 4 6 8 1 . . . . . 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 eilsia (-) Semiclassical inrmto (-) method Wigner xetfra nefrnepteni h otmrgtpo around plot right bottom the the in by pattern indicated are are interference parameters p lines an pulse contour The for the method box. Except of Wigner color levels the the of from The marks spectra (3.51)). corresponding Eq. as (see well as (3.28)) Eq. tprgtpo) h pcr ftetomtosarewt ahother. each with agree methods two the of spectra the plot), right (top x ∼ P 0 s . P := 4 = s hc sntfudi h orsodn eilsia result semiclassical corresponding the in found not is which , p · x 1 2 2 1 P [ m 1 1 s − − ] = s s P 2 1 2 1 s P , q  z ϵ γ q ⊥ z 5 W ( f + P ϵ − − ⊥ r γ m [10 [10 − 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 5 P − − γ 7 7 l + ) 3 ] ] . . . . . 1.4 1.2 1 0.8 0.6 0.4 0.2 0 . . . . . 1.4 1.2 1 0.8 0.6 0.4 0.2 0 eilsia (+) Semiclassical inrmto (+) method Wigner ϵ m ⊥  ..TeRttn atrPulse Sauter Rotating The 3.3. P s µ · z + P − − s + P p 0 = µ x z − and [ m  τ .  ] 46 = , − . ouin (see solutions 42 t W c f , + + σ (3.50) [10 [10 20 = 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 − − 51 9 9 ] ] . 25 π φ = 49 2 3. Homogeneous Electric Fields

N (ε, σ, τ), ε = 0.1, σ = 20 10−2 − −4 semiclassical, W 10 semiclassical, W + 10−6 Wigner, f − + −8 Wigner, f

] 10 3 −10 m

[ 10

N 10−12 10−14 10−16 10−18 1 10 100 1000 τ [1/m]

Figure 3.10.: Comparison of the particle yield for the semiclassical W + and W − solutions (see Eq. (3.28)) with the corresponding Wigner function projections f + and f − (see Eq. (3.51)). The Wigner method data suffer from a lack of precision for τ > 100, which results in some artifacts in the blue and yellow lines. The data points on top of the lines have been calculated using higher precision. We find that the results agree with each other.

They are also complete, i. e.,

Ps + P−s = 1 .

Accordingly they project onto the two parts of the spectrum which correspond to these solutions. In Eq. (3.50) it is evident that the two solutions from the scattering method correspond to a linear combination of chirality and magnetic momentum. While in the context of the Wigner function, which contains the full spinor information, δfc and δfµz , given in Eqs. (3.7) and (3.8) respectively, are the physically meaningful observables, we will construct f s to show the connection to the solutions of the semiclassical method. The projected one-particle function, as defined in Eq. (2.21), using this projection is given by

s f = f [Ps (W − Wvac.)] 1 = (f − s δf ) (3.51) 2 sc with the corresponding asymmetry δfsc. The latter can be related to the chiral and

52 ie ihteivretm cl ftepledrto ofn obndKeldysh acombined find to duration pulse the of scale time inverse the with bined appropriate. where used were method spectra. the of shape the of of amplitude complexities observable fixed the interesting a disregarding first For the yield, happening, particle is total what the of picture is general a get to order In Yield Particle Total the 3.3.4. of Eq. (3.50). solutions the in independent of combination two specified methodagree linear projection the moment the Wigner that magnetic of and and shows eigenstates chirality This spinor thesemiclassical represent for 3.10). method data semiclassical and the 3.9 Figs. that (see find we this Using asymmetries momentum magnetic the of data The pulse. Sauter rotating the of yield particle Total 3.11.: Figure and hti h oainlfrequency rotational the if that [75] previously shown been has It

τ N 4 ∈ τ [m ] 10 10 10 10 10 10 [1 10 10 10 10 t − − − − − − c − − − − 20 18 16 14 12 10 , 8 6 4 2 atceyedt h us uainaduigtecmie Keldysh combined the using and duration pulse parameter. the the to normalizing yield by aligned particle closely are scan parameter two-dimensional 1000 . 10 01 t c ] a cne.Bt h inrmto n h semiclassical the and method Wigner the Both scanned. was ε σ σ σ σ σ σ σ 0 = 50 = 20 = 6 = 3 = 2 = 1 = 0 = . δf 1 1 . 1 sc h w-iesoa aaee pc of space parameter two-dimensional the = δf arpouto rate production Pair ϵ q c γ ⊥ z ∗ and δf = c τε + δf 1  µ ϵ m ⊥ z  δf π epcieyas respectively 2  µ 2 z ..TeRttn atrPulse Sauter Rotating The 3.3. + . σ 2 02 1000 20 10 γ ∗ | τ Ω=2 ≫ 1 = Ω m t c σ σ τ ∈ scom- is [0 , 50] 53

26 π φ = 49 2 3. Homogeneous Electric Fields

 2 ∗ 1  1  2 parameter γ = εm τ + Ω , a lot of the particle yield data fall on a common line. Further parameter scans demonstrated, that the alignment of the data can be improved if the particle yield is normalized by the pulse duration τ and if the combined Keldysh parameter is modified to the definition in Eq.(3.37). Fig. 3.11 shows that the particle yield is mainly a function of the combined Keldysh parameter modified with effects of multiphoton resonances which are visible inthe interval γ∗ ∈ [4, 30] and are more pronounced for larger σ. This is because, for large ∗ σ, γ = γΩ (see Eq. (3.38)) and γτ → 0 (see Eq. (3.39)), which indicates clearly separated regimes and clear domination of multiphoton pair production.

3.3.5. Typical Spectra

In Sec. 3.3.2 the different parameter regimes regarding multiphoton and Schwinger pair production have been discussed from a phenomenological viewpoint. In Fig. 3.12 the parameter space spanned by the pulse duration τ and the pulse frequency Ω is  #‰ ∗   displayed for fixed amplitude ε = 0.1. The theoretical boundaries γ = 1, A (0) =  #‰   ε Ecr. τ and A (0) = 2 ε Ecr. τ are plotted and additionally a collection of lines with constant σ = Ωτ. The latter are important, because for a fixed σ the locus of the vector potential always has the same shape and is only scaled by ε τ. In this section a number of example spectra out of this parameter space will be discussed according to the parameters and the shape of the spectrum. These example spectra are organized into 4 groups (Figs. 3.13 to 3.16) of 4 spectra (“a)” to “d)”) each, as can be told by the different kinds and colors of markings inFig. 3.12, referring to these figures. The first double page, showing Figs. 3.13 and 3.14, features short pulse durations, whereas the second double page with Figs. 3.15 and 3.16 features long pulse durations, both with increasing frequencies from left to right. It is not possible to classify every spectrum as either multiphoton pair production or Schwinger pair production. A large portion of the parameter space does not clearly fall into either category and the spectrum can only be explained by a mixture of both processes. There are a number of different possible interpretations which we assign to each of the spectra.

1. Clear multiphoton pair production

2. Multiphoton pair production, modified by influences of the vector potential

3. A complete mixture of both processes with no recognizable tendency

54 oainlsmer.Ti stecs o l h lt with plots the all for case the is This symmetry. rotational ) )add) eadeso h us uainbigqiesala in as small quite being duration 3.16c). pulse Fig. in the as of large regardless or 3.14 d) d)), Fig. and c) 3.16 b), Figs. markings. already the are to close classification numbers this the by regarding 3.12 spectra Fig. example in noted the of assignments The of Curves pulse. Sauter rotating of range parameter over Overview 3.12.: Figure ssae ale,frlreenough large for earlier, stated As CerShigrpi production pair Schwinger Clear 5. produc- pair ofmultiphoton influences by modified production, pair Schwinger 4.

tion τ [tc] 1000 100 10 1 0 . 0 0 001 iae bv h lt h ahdrdlnsgive lines red dashed The plot. the above dicated parameters constant 3.3.5. inSec. given classification spec- the example to respectively. of according interpretation 58f, tra indicate and markings 56f on pages to labels the on corresponds Numbered Markings above markings spectra green indicated example and . 3.6 as corresponding black Sec. 3.16 , the the of to in arrangement 3.13 relevance The Figs. of plot. in be spectra will example which indicate amplitude, lower a of amplitude an and line) dashed e otdln hc smarked is which line dotted red

1 4 5 σ 5 σ σ

= 1 = 0 = 0 ε

0 = . . 1 01

. 01 | A #‰ | A #‰ (0) ε γ 3.15 Fig. see 3.13 Fig. see (0) . τ 10 01 0 = |

= σ 2 = | | 5 A #‰ = 10 = π 2 . | 1 , E E hrzna e ahdln) hr relevant Where line). dashed red (horizontal sasmd ihteol xeto fthe of exception only the with assumed, is γ σ cr cr ∗ ti xetdta h pcr hwa show spectra the that expected is it , 2 2 2 2 . . t t [ Ω and c c 4 m ε σ ] 0 = r ie ndfeetclr asin- colors different in given are 3 . ..TeRttn atrPulse Sauter Rotating The 3.3. . 1 1 01 σ 4 h atrgives latter The .

= 100 3.16 Fig. see 3.14 Fig. see σ const = γ ∗ 1 2 3 σ γ 1 = ≥ Ω . 1 = 20 3.14d), (Fig. vria red (vertical γ ∗ 1 = for 55

27 π φ = 49 2 3. Homogeneous Electric Fields

τ Ω τ Ω = 4.0 , = 0.025 , σ = 0.1 = 4.0 , = 0.25 , σ = 1.0 tc m tc m ∗ ∗ γτ = 2.5 , γΩ = 0.25 , γ = 4.0 f [10−8] γτ = 2.5 , γΩ = 2.5 , γ = 4.7 f [10−7] 1.5 10.0 1.5 8.0 a) b) 1 8.0 1 6.0 0.5 0.5

] 6.0 m

[ 0 0 4.0 y

p 4.0 −0.5 −0.5 2.0 −1 2.0 −1 −1.5 0.0 −1.5 0.0 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5

τ Ω τ Ω = 14.7 , = 0.07 , σ = 1.0 = 14.7 , = 0.34 , σ = 5.0 tc m tc m ∗ ∗ γτ = 0.7 , γΩ = 0.7 , γ = 1.3 f [10−12] γτ = 0.7 , γΩ = 3.4 , γ = 3.6 f [10−8] 2 3.0 1.5 2.0 c) d) 1.5 2.5 1 1 1.5 2.0 0.5

] 0.5 m

[ 0 1.5 0 1.0 y p −0.5 1.0 −0.5 −1 0.5 −1.5 0.5 −1 −2 0.0 −1.5 0.0 −2−1.5−1−0.50 0.5 1 1.5 2 −1 0 1 px [m] px [m]

Figure 3.13.: A collection of pair production spectra for the rotating Sauter pulse as marked in Fig. 3.12 as black boxes ( ). The levels of the contour lines are indicated by the marks of the color box. The Parameters for the rotating Sauter pulse are given on#‰ top of the individual plots. The red cross marks the vector potential A at t = 0. The green circles, if present, show the locations of the expected multiphoton rings for appropriate values of n.

56 Acleto fpi rdcinsetafrterttn atrpulse Sauter rotating the for spectra production pair of collection A 3.14.: Figure γ γ t t τ py [m] τ py [m] c c τ τ − − − − 2 = 4 = 0 = 14 = − − 1 0 0 1 1 0 0 1 ...... 5 1 5 0 5 1 5 5 1 5 0 5 1 5 − − . . . 5 0 7 1 1 . 7 . . c) a) γ , , γ , 5 5 , m Ω − − Ω m Ω fpeet hwtelctoso h xetdmlihtnrnsfor rings multiphoton expected the of of locations values the appropriate show present, if ( crosses black as 3.12 Fig. in marked as h oaigSue us r ie ntpo h niiulpos The plots. potential individual vector for the Parameters the of The top marks on box. cross given color red are the pulse of Sauter marks rotating the the by indicated are lines Ω 1 1 7 = 0 = − − 6 = 0 = 0 0 . . . . 5 75 0 0 5 0 0 5 . . p 8 68 γ , x γ , σ , [ m σ , ∗ ] ∗ 8 = 3 = . . 6 = 10 = 1 1 5 1 1 5 . . 5 0 . 9 f f . . n 5 5 [10 [10 . 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 2.0 4.0 6.0 8.0 10.0 − − 5 5 ] ] γ γ t t τ τ c c τ τ − − − − 4 = 14 = 0 = 2 = − − 0 0 1 1 1 0 0 1 ...... 5 0 5 1 5 1 5 5 5 0 5 1 5 1 − . . . 0 7 5 1 . 7 A ..TeRttn atrPulse Sauter Rotating The 3.3. . #‰ b) d) , γ , γ , 5 , m Ω − − Ω m at Ω Ω .Telvl ftecontour the of levels The ). 1 1 0 1 1 = 12 = − 1 = 14 = t 0 0 = . . 26 0 0 5 . . p 36 6 x γ , σ , γ , h re circles, green The . [ m σ , ∗ ] ∗ 5 = 13 = . 20 = 14 = 1 1 5 . 2 f f . 5 [10 [10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.5 1.0 1.5 2.0 2.5 − − 57 3 3 ] ] 28 π φ = 49 2 3. Homogeneous Electric Fields

τ Ω τ Ω = 100 , = 0.001 , σ = 0.1 = 100 , = 0.01 , σ = 1 tc m tc m ∗ ∗ γτ = 0.1 , γΩ = 0.01 , γ = 0.16f [10−14] γτ = 0.1 , γΩ = 0.1 , γ = 0.19 f [10−14] 15 5.0 6.0 a) 4.5 10 b) 10 4.0 5.0 5 3.5 5 4.0

] 3.0 m

[ 0 2.5 0 3.0 y

p 2.0 −5 1.5 −5 2.0 −10 1.0 1.0 0.5 −10 −15 0.0 0.0 −15−10 −5 0 5 10 15 −10 −5 0 5 10

τ Ω 2.5 τ Ω 12.5 = 400 , = 3 , σ = 1 = 400 , = 3 , σ = 5 tc m 10 tc m 10 ∗ ∗ γτ = 0.03 , γΩ = 0.03 , γ = 0.05 f [10−14] γτ = 0.03 , γΩ = 0.12 , γ = 0.13f [10−14] 4.5 10 5.0 40 c) 4.0 d) 4.5 4.0 3.5 5 20 3.0 3.5 ] 2.5 3.0 m

[ 0 0 2.5

y 2.0 p 2.0 −20 1.5 1.5 −5 1.0 1.0 −40 0.5 0.5 0.0 −10 0.0 −40 −20 0 20 40 −10 −5 0 5 10 px [m] px [m]

Figure 3.15.: A collection of pair production spectra for the rotating Sauter pulse as marked in Fig. 3.12 as green boxes ( ). The levels of the contour lines are indicated by the marks of the color box. The Parameters for the rotating Sauter pulse are given on#‰ top of the individual plots. The red cross marks the vector potential A at t = 0.

58 Acleto fpi rdcinsetafrterttn atrpulse Sauter rotating the for spectra production pair of collection A 3.16.: Figure γ γ t t τ py [m] τ py [m] c c τ τ − − 100 = 0 = 400 = 0 = − − − 1 0 0 1 . . . . 2 1 0 1 2 5 1 5 0 5 1 5 . . − 03 1 c) a) γ , 1 , , − γ , . m 5 Ω m Ω Ω fpeet hwtelctoso h xetdmlihtnrnsfor rings multiphoton expected the of of locations values the appropriate show present, if ( crosses green as 3.12 Fig. in marked as h oaigSue us r ie ntpo h niiulpos The plots. potential individual vector for the Parameters the of The top marks on box. cross given color red are the pulse of Sauter marks rotating the the by indicated are lines 2 − Ω 1 = 0 = 1 0 = 0 = − − 2 1 0 1 0 . γ , . . . 1 p 0 0 5 5 05 x σ , γ , [ m σ , ∗ ] 1 = 10 = ∗ . 1 1 5 0 = 20 = . 5 . f f 5 n [10 [10 . − − 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 12 13 ] ] γ γ t t τ τ c c τ τ − − − − 100 = 0 = 398 = 0 = − − 1 0 0 1 0 0 1 1 ...... 5 5 0 5 1 5 5 0 5 1 5 1 1 5 − . . 03 1 1 A ..TeRttn atrPulse Sauter Rotating The 3.3. . #‰ b) d) γ , 5 , , γ , m − − Ω m at Ω Ω Ω .Telvl ftecontour the of levels The ). 1 0 1 1 0 = 5 = − 1 = 0 = t 0 0 = . . γ , 0 0 5 . . 5 p 3 126 x σ , γ , h re circles, green The . [ m ∗ σ , ] 50 = 5 = ∗ . 1 1 5 1 = 50 = . 3 f f . [10 5 [10 − 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 − 11 59 6 ] ] 29 π φ = 49 2 3. Homogeneous Electric Fields

z = 0 τ = 100 tc, Ω = 0.5 mp x = 0 fp 1.5 10−5 10−6 1 10−7 0.5 10−8

] 10−9 m

[ 0

y −10

p 10 −0.5 10−11 10−12 −1 10−13 −1.5 10−14 −1.5 −1 −0.5 0 0.5 1 1.5 0 0.5 1 1.5 px [m] |pz| [m]

Figure 3.17.: This plot shows two different slices of a three dimensional spectrum of pairs produced by a σ = 50 pulse. The pulse parameters are identical to Fig. 3.16b), the spectrum only seems different due to the logarithmic color scale. The vertical cyan line in the large plot shows the position of the slice that is shown in the smaller plot. The green circles show the expected position of pairs due to multiphoton pair production. The innermost green circle corresponds to the energy-momentum-relation with the energy of 5 photons.

Fig. 3.14c) and Fig. 3.16a) both have the same value of σ, but only the first has rotational symmetry, so σ = 10 is not large enough to guarantee rotational symmetry on its own. In Fig. 3.14c) γΩ is already large enough to allow multiphoton pair production, which improves the rotational symmetry, but only the outer ring lines up with the expected momentum, so the multiphoton interpretation is not complete. We can clearly see multiphoton pair production in Fig. 3.16b) and Fig. 3.14d). The produced pairs line up exactly with the prediction of Eq. (3.44), and the Keldysh parameters are large enough to support multiphoton pair production. This interpretation is backed up by the full three dimensional data from which a selection of slices is displayed in Fig. 3.17. The multiphoton rings that are visible in the pz = 0 slice are in fact part of multiphoton spheres with suppressed pair production around their pole regions at maximum and minimum pz. The 4-photon-process is suppressed, because 4 photons do not have enough total energy (4Ω = 2 m) to over- come the pair production threshold 2 m∗ > 2 m. However, due to the pulse length

60 h oio h etrptnilfrdfeetvle of values different for with potential decays vector pairs the produced of of loci number The the that visible clearly increasing is It the and displayed. potential are vector the at of and peaked locus the 3.14d) is along spectrum Fig. momenta have spectra pairs produced The the All 3.17. Fig. of interpretation to region assigned inner are the 3.16b) Fig. in with produced some photons pairs and to isbroadened than pulse larger field electric energies the of spectrum Fourier the spectrum dimensional athree of slices different three shows plot This 3.18.: Figure oprsn hs pcr r sindt interpretation to assigned are spectra These comparison. nepeain o loigShigrpi rdcini infcn quantity. significant a different in a production indicate pair rather Schwinger parameters allowing Keldysh not The interpretation, production. pair Schwinger interpretations intermediate the to assigned be will epooete odvlpabte nesadn ftepi rdcinregimes production pair the of the understanding boundaries. way. better their mathematical t. a and r. a develop w. in to stringent spectra them not production propose are We and pair 3.3.2 the Sec. interpreting in explained to phenomenology due are assignments these 3.15. Fig. of all in production pair Schwinger see clearly can we hand other the On l h eann pcr ipa etrso ohpi rdcinpoessand processes production pair both of features display spectra remaining the All o hr ussw e arpouto ekdnear peaked production pair see we pulses short For

pz [m] py [m] − 4 5 0 0 p . . . . z 4 5 5 5 6 5 0 5 hw h ou ftevco oeta safnto ftime. of function a the curve as show red potential plot The vector large plots. the the smaller of the in locus in by lines shown the indicated cyan are shows are that The slices lines the box. contour of color the position the of of levels marks The the b). 3.15 Fig. a in by shown produced pairs of u osntflo peodlsaea ntemlihtncase. multiphoton the in as shape spheroidal a follow not does but p z y 011 10 9 8 7 6 5 4 3 0 = 5 = 0 = Ω . m 19 A #‰ (0) m . 3.15b) Fig. for data dimensional three the 3.18 Fig. In . 5 m xs n nbete4poo-rcs,wihleads which 4-photon-process, the enable and exist p x [ τ σ m 100 = ] 1 = #1 . us.The pulse. t c , 0 = Ω ..TeRttn atrPulse Sauter Rotating The 3.3. #2 . 01 σ A p #‰ x p #5 to for 3.4 Fig. in shown are (0) mp z 6 = . #4 hc ol suggest would which , 0 = . 05 0.5 -0.50 76 laent that note Please . p z m lc a already was slice [ m f ] [10 − 0.0 2.0 4.0 14 ] 61

30 π φ = 49 2 3. Homogeneous Electric Fields

y = −0.372 τ = 17.1 tc, Ω = 0.35 mp 0.5 ] m [

z 0.25 p 0 f [10−8] pz = 0 px = 0 1 1.5

0.5

] 1.0 m [ y

p 0

0.5 −0.5

0.0 −1 −0.5 0 0.5 1 0 0.25 0.5 px [m] pz [m]

Figure 3.19.: This plot shows three different slices of a three dimensional spectrum of pairs produced by a σ = 6 pulse. The pulse parameters are not identical but similar to Fig. 3.13d). The levels of the contour lines are indicated by the marks of the color box. The cyan lines in the large plot show the position of the slices that are shown in the smaller plots. The green circles show the expected position of pairs due to multiphoton pair production. The red curve shows the locus of the vector potential. The yellow curve shows the value of py averaged over pairs with either py < 0 or py > 0 and px = 0 for all pz.

This apparent contradiction is resolved by noting that the broad bandwidth of the #‰ short rotating Sauter pulse enables multiphoton pair production at p = 0 at t = 0. Those pairs are subsequently accelerated by the electric field which explains the shape of the spectra. Conclusively, the spectra Figs. 3.13a), b), Figs. 3.14a), b) and c) will be assigned to interpretation #2. Another kind of explanation can be applied to Fig. 3.16a), c) and d). The Keldysh parameters do not strongly support multiphoton pair production. Pairs are produced close to the vector potential at t = 0, but as σ is large enough in the latter

62 ocertnec em ob vdn.Te r sindt interpretation to assigned are and They production evident. be pair to Schwinger seems and tendency production clear pair no multiphoton of features show process interpretation dominating to assigned the be as will production spectra pair these multiphoton absence and discount the rings and with parameters inner rings exactly Keldysh the inner The is of the missing. rings while simply the production, are between numbers ringsinthese pair photon distance multiphoton lower in the filigree expected that forthe be noted, Schwinger would be the what must by pairs account it produced However are those spectra. pairs between while interferences cycles, and of vec- effect number the that a argue takes could One potential symmetry. tor cylindrical observe and expect we cases two ar nedlnsu ihtecluae utpoo peewith sphere multiphoton calculated the with up lines indicated indeed as pairs circle), the (green of sphere part lower multiphoton the the with average up the line by to seem not does to assigned those to nepeain.Cermlihtnpi rdcinsest eur ag enough large a require to seems σ production pair t. absolute multiphoton r. w. constant Clear continuous of interpretations. at lines be potential the to vector and seems the parameters assignment of Keldysh value the the that that noted and is parameters the it 3.12, Fig. of see result space, a just line). also yellow is (lower line spheres. yellow expectation partial upper overlapping the two the with over of averaging up misalignment If anaverage line the that not suggests does This also order. result multi-photon the any spheres, fits radius of the radius of a of with regardless sphere visible, production also pair is partial weaker somewhat A obnto fsi n hre sacniaefrmaigu information. a meaningful moment, various for magnetic about candidate the a information that is out to charge, turned access and It gain spin of pairs. to combination produced possible the is of it states 2.3 quantum Sec. in discussed As Moment Magnetic 3.3.6. ≳ )add), and 3.13 c) Figs. examples, displayed all of spectra remaining two only The sostetredmninlsetu o us ihprmtr close parameters with pulse a for spectrum dimensional three the shows 3.19 Fig. hnteasgmnsaetknit con n apdoe h parameter the over mapped and account into taken are assignments the When p y 20 o ar with pairs for and Ω ≳ p y 0 . ftoepiswith pairs those of 3 #3 m p y p . x h arpouto eki h pe ato h spectrum the of part upper the in peak production pair The . < 0 = 0 scluae,wihaeae vrbt fteepartial these of both over averages which calculated, is lti hudb oe htaprino h produced the of portion a that noted be should it plot t 0 = r ufcetgie odsigihsm ofthe some todistinguish guides sufficient are p y > 0 ipae steuprylo ie In line. yellow upper the as displayed ..TeRttn atrPulse Sauter Rotating The 3.3. #4 . | | #‰ p #‰ p ≈ | ≈ | #3 0 0 . 66 . 37 . m 63 m . 31 π φ = 49 2 3. Homogeneous Electric Fields

∗ τ = 4.0 tc, Ω = 0.025 m, σ = 0.1, γτ = 2.5, γΩ = 0.25, γ = 4.0 −8 −8 f + [10 ] f − [10 ] µz µz 1.5 5.0 1.5 5.0 a) b) 1 4.0 1 4.0 0.5 0.5

] 3.0 3.0 m

[ 0 0 y

p 2.0 2.0 −0.5 −0.5 −1 1.0 −1 1.0 −1.5 0.0 −1.5 0.0 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5

∗ τ = 4.0 tc, Ω = 1.26 m, σ = 5, γτ = 2.5, γΩ = 12.6, γ = 13.2 −6 −3 f + [10 ] f − [10 ] µz µz 1.5 5.0 1.5 1.2 c) d) 1 4.0 1 1.0 0.5 0.5 0.8

] 3.0 m

[ 0 0 0.6 y

p 2.0 −0.5 −0.5 0.4 −1 1.0 −1 0.2 −1.5 0.0 −1.5 0.0 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 px [m] px [m]

Figure 3.20.: Panels a) and b): Pair production spectrum Fig. 3.13a) separated by magnetic moment. Panels c) and d): Pair production spectrum Fig. 3.14b) separated by magnetic moment. The levels of the contour lines are indicated by the marks of the color box. The red crosses indicate the vector potential at = 0. The green circles, if present, show the expected position of pairs due to multiphoton pair production.

64 )separated 3.15d) Fig. spectrum production Pair b): and a) Panels 3.21.: Figure τ τ 100 = p [m] p [m] 400 =

z − y − 0 0 0 0 1 1 10 10 ...... 0 2 4 6 8 1 2 4 5 0 5 − 0 0 . . 10 0 0 a) c) t t c c . , , 50 25 xetdpsto fpisdet utpoo arproduction. pair multiphoton to due pairs indicate spectrum of crosses position production red at expected The Pair potential moment. d): vector magnetic and the by c) separated Panels 3.16 b) Fig. moment. magnetic by 0 = Ω 0 = Ω − 10 5 0 5 . 0 5 p . . x 50 013 . 511 1 75 [ m ,σ m, ] ,σ m, 50 = . 5 = 51 25 f γ , . . 0 µ 5 f γ , z + τ µ z + [10 0 = τ 0.0 0.5 1.0 1.5 2.0 2.5 10 10 10 10 10 10 10 0 = 0 = − − − − − − − − . 14 11 10 9 8 7 6 5 1 h re ice,i rsn,so the show present, if circles, green The . γ , . ] 03 − Ω γ , − 0 0 0 0 1 1 10 10 5 = ...... 0 2 4 6 8 1 2 4 5 0 5 Ω − 0 0 0 = 10 . 0 b) d) γ , ..TeRttn atrPulse Sauter Rotating The 3.3. . . 12 50 25 ∗ − 5 = γ , 10 5 0 5 . ∗ 0 5 . p 0 = 0 x . 511 1 75 [ m . 1 ] . 51 25 f . µ f 5 z − µ z − [10 0.0 0.5 1.0 1.5 2.0 2.5 10 10 10 10 10 10 10 − − − − − − − − 14 65 11 10 9 8 7 6 5 ] 32 π φ = 49 2 3. Homogeneous Electric Fields

The spectra in Figs. 3.20 and 3.21 show the pair production spectra projected onto specific states of magnetic moment, according to Eq.(3.8) and

1 f ± = (f ± δfµ ) . µz 2 z Interestingly, in some cases there is nearly no asymmetry, as in Figs. 3.20a) and b) and Figs. 3.21a) and b), while in other cases the asymmetry is drastic. In Fig. 3.20c) + pair production of pairs in the state µz is suppressed by three orders of magnitude. The pair production spectra for both states also have a completely different shape in this case. The suppression in Fig. 3.21c) is weaker, but nevertheless apparent on a logarithmic scale. The relevance of spin effects to multiphoton pair production has been demonstrated in Ref. [117]. We conjecture that the susceptibility to spin effects is linked with the characteristic frequency of the pulse. In pulses of long duration as in Fig. 3.21, the Schwinger- dominated spectra are symmetric w. r. t. magnetic moment, while the multiphoton dominated spectra show an asymmetry. When the pulse duration is very short as in Fig. 3.20, the picture is not so clear. The broad spectrum of short pulses enables resonant pair production which peaks #‰ #‰ at p = A(0), this is the case in Figs. 3.20a), b) and d). In the first two cases the spectra are nearly symmetric w. r. t. magnetic moment, while in Figs. 3.20c) the resonant pair production process due to the broadened spectrum seems to be completely suppressed and a multiphoton ring remains which is slightly modified by the vector potential.

3.4. Generalized Polarization

In an experiment real laser light would never have perfect circular polarization, or even perfect linear polarization. It is as such necessary to look into the intermediate regime of elliptic polarization. While some studies have already been conducted [118–121], there is still a large parameter region unexplored. The parameter that interpolates between the linear and circular regimes is the phase shift between the orthogonal components of the electric field. If it is introduced to both, the x and y component in a symmetric way, we can rotate the resulting field such that it

66 loi h litccs.Tecliae uc tl xssi h ihrfrequency higher the in exists still bunch collimated The at the minimum case. case. for a elliptic peaks as the symmetric Two interpreted in other. be also the also in symmetric can peak two centered to have frequency linear one lower the difference and in a case which of one higher, enough in slightly is peaks is which 3.23a)) frequency and the 3.22 d) Only (Figs. f). case to 3.22 d) Figs. in those col- a resembles spectrum production bunch. pair electron the limated but appears symmetry cylindrical no parameter dimensionless the Also equivalent. but 120], σ [119, Refs. in used that from for (3.30) Eq. from factor pulse Sauter normalization the to Due for [116] in Ref. field pulsed oscillating linear the with coincides ) de- 3.22b)) Fig. moderate g. (e. for already spectra symmetry intermediate this the velop 3.22 c)), Fig. (see variety.spectra inthe symmetric of drically lot of the a show also offeatures can cases the variety intermediate on the corresponding the cases, for Depending extreme fields. spectra the rotating or between linear smoothly interpolate fields elliptical the Ω = )t )hv aaeesqiesmlrto similar quite parameters have c) to 3.23a) Figs. in spectra production pair The ftenme frotations of number for the spectra If The 3.23. and 3.22 Figs. in depicted are spectra example of set A τ stesm si h oaigSue pulse. Sauter rotating the in as same the is E #‰ ( t := ) = cosh √ E ε cos( 2 2 σ crit ( t slreadtesetu nterttn aei cylin- is case rotating the in spectrum the and large is E ε / . φ 2 τ φ ) cosh ) crit     = 1 / tan( ( . π √ / cos( 2 2 2 oeta hsdfnto ssihl different is slightly definition this that Note . cos(Ω ( φ 2 t / sin(Ω ) τ φ 0 φ 2 ) ) e), 3.22 Fig. example for cases, other In . R ) t ) hsas onie ihterotating the with coincides also this  − t ) π     4  .     ..GnrlzdPolarization Generalized 3.4. cos(Ω cos(Ω #‰ p 0 t t − + 0 = φ φ 2 2 φ ) )     0 = hc spresent is which eutn in resulting , 67

33 π φ = 49 2 3. Homogeneous Electric Fields

τ Ω 1 φ f [10−6] τ Ω 1 φ f [10−7] t = 61, m = 2 , π = 0 t = 10, m = 2 , π = 0 c 5.0 c 6.0 1 a) d) 4.0 5.0 0.5 4.0

] 3.0 m

[ 0 3.0 y

p 2.0 −0.5 2.0 1.0 1.0 −1 0.0 0.0 τ Ω 1 φ f [10−6] τ Ω 1 φ f [10−7] t = 61, m = 2 , π = 0.38 t = 10, m = 2 , π = 0.41 c 0.9 c 6.0 1 b) 0.8 e) 5.0 0.7 0.5 0.6 4.0 ] 0.5 m

[ 0 3.0

y 0.4 p −0.5 0.3 2.0 0.2 1.0 −1 0.1 0.0 0.0 τ Ω 1 φ f [10−6] τ Ω 1 φ f [10−7] t = 61, m = 2 , π = 0.5 t = 10, m = 2 , π = 0.5 c 3.5 c 8.0 1 c) f) 3.0 0.5 2.5 6.0

] 2.0 m

[ 0 4.0 y 1.5 p −0.5 1.0 2.0 0.5 −1 0.0 0.0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

px [m] px [m]

Figure 3.22.: Example spectra of pair production in linear, elliptical and circular polarization for different parameters. The levels of the contour lines are indicated by the marks of the color box. The amplitude is ε = 0.1 in all cases. The left hand set of spectra has σ = 30.5 and thus cylindrical symmetry is expected and present in the circular case. Already in the elliptical regime the symmetry can be observed. The right hand set of spectra has σ = 5 and is far away from cylindrical symmetry in the elliptical case.

68 Eapesetao arpouto nlna,elpia n circular and elliptical linear, in production pair of spectra Example 3.23.: Figure py [m] py [m] py [m] − − − − − − 0 0 0 0 0 0 ...... 1 5 0 5 1 1 5 0 5 1 1 5 0 5 1 ε ε ε c) b) a) 1-. . 1 0.5 0 -0.5 -1 0 = 0 = 0 = h pcr ntergthn ieueawae il ihahigher with field weaker a use side hand right of the the frequency cases on these spectra In The box. color has the spectra of marks the amplitude by indicated are with lines pulses short for polarization . . . 1 1 1 , , , m m m Ω Ω Ω 0 = 0 = 0 = p x [ . . . m ε 6 6 6 σ , , , ] σ n h frequency the and φ φ φ π π π 6 = 10 = 0 = 0 = 0 = 3.5a). Fig. in featured also is case circular the and . . . 41 5 f f f [10 [10 [10 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.5 1.0 1.5 2.0 2.5 − − − 6 6 6 ] ] ] Ω τ ε ε ε d) e) f) 1-. . 1 0.5 0 -0.5 -1 r aid h ethn e of set hand left The varied. are 0 = 0 = 0 = 10 = . . . 01 01 01 ..GnrlzdPolarization Generalized 3.4. t c , , , h eeso h contour the of levels The . m m m Ω Ω Ω p 1 = 1 = 1 = x [ m , , , ] φ φ φ π π π 0 = 0 = 0 = . . 5 37 f f f [10 [10 [10 0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 0.2 0.4 0.6 0.8 1.0 − − − 69 8 8 7 ] ] ] 34 π φ = 49 2 3. Homogeneous Electric Fields

Ω 1 ε = 0.1, m = 2 3 · 10−8 τ = 61 tc −8 2.5 · 10 τ = 10 tc

−8

] 2 · 10 2 m [ 1.5 · 10−8 xy

N 1 · 10−8 5 · 10−9 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 φ [π]

Figure 3.24.: Two dimensional particle yield in general polarization for spectra shown in Fig. 3.22.

τ t = 10 10−8 c ε = 0.1, Ω = 0.5 m ε = 0.01, Ω = 1 m 10−9 ] 3 m [

N 10−10

10−11 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 φ [π]

Figure 3.25.: Total particle yield in general polarization for spectra shown in Fig. 3.23.

Additional spectra corresponding to Figs. 3.23a) to c) with φ covering in the interval [0, π/2] are printed on the top right of the odd pages in this dissertation and can be viewed like a flip-book. In Figs. 3.23d) to f) the amplitude of ε = 0.01 is lower, completely suppressing Schwinger pair production. In combination with the sufficiently large Ω = 1 m those spectra clearly demonstrate multiphoton pair production.

70 The 3.25. Fig. the in in displayed minimum is the yield if total question the and calculated was dataset for dimensional spectra the of in duration yield total yield at The minimum particle ellipticity. a dimensional varying two for the (2.18), displays Eq. 3.24 Fig. and able rdcinsetafrtoeknso uss oe o hre asinpulse Gaussian chirped a for model A calculate pair to pulses. of itispossible is kinds inthose might for a method and spectra offers the Wigner production production pulse pair Using chirped for effects. A energies interesting troduce photon compression. pulseamplifica- available after of chirp chirped bandwidth residual called broader a a technique have may using (CPA) amplified tion were that pulses Laser Pulses Chirped 3.5. available the with answered be not can yield particle total the data. in also or yield ticle itrstesetu o n loicesstepril il shge photon of higher sign as the yield chirp When particle The the spectrum. 3.27 . flipped increases pulse and also the 3.26 and enter Figs. lot modes see a pulses, spectrum chirped the in distorts production pair of behavior ol w iesoa aawith data dimensional two only 3.22 Fig. in spectra the For sn hsmdlafweape aebe acltdadso ut complex quite show and calculated been have examples few a model this Using p x − → τ 10 = p φ x ≈ )adc). and 3.26 b) Fig. in seen be can as E #‰ t c 0 ( t tecmlt three complete the 3.23 Fig. in spectra the For monotonic. is . 3 = ) π hl h il ihapulse a with yield the while 3.24, Fig. in seen be can as E ε crit τ . e 61 = − ( τ t ) t 2 c        aeeit nyi h w iesoa par- dimensional two the in only exists case cos sin   σ σ   b b   0 τ τ t t   2 2 b + + sfipd h pcrmis spectrum the flipped, is τ τ t t          ..CipdPulses Chirped 3.5. . N p xy z sdfndin defined as , 0 = τ 61 = r avail- are t (3.52) c has 71

35 π φ = 49 2 3. Homogeneous Electric Fields

b = 0 f [10−6] b = 0.05 f [10−6] 3.0 4.0 1 a) d) 2.5 0.5 3.0 2.0 ] m

[ 0 1.5 2.0 y p 1.0 −0.5 1.0 0.5 −1 0.0 0.0 b = −0.2 f [10−6] b = 0.3 f [10−5] 8.0 1.5 1 b) e) 0.5 6.0 1.0 ] m

[ 0 4.0 y p 0.5 −0.5 2.0

−1 0.0 0.0 b = 0.2 f [10−6] b = 0.4 f [10−5] 8.0 4.0 1 c) f) 0.5 6.0 3.0 ] m

[ 0 4.0 2.0 y p −0.5 2.0 1.0

−1 0.0 0.0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

px [m] px [m]

Figure 3.26.: A set of example spectra of pair production in chirped pulses using the model in Eq. (3.52). The levels of the contour lines are indicated by the marks of the color box. The Pulse duration is τ = 10 tc with a frequency of ω = 0.6 m , and consequently σ = 6.

72 Asto xml pcr fpi rdcini hre ussuigthe using pulses chirped in production pair of spectra example of set A 3.27.: Figure py [m] py [m] py [m] − − − − − − 0 0 0 0 0 0 ...... 1 5 0 5 1 1 5 0 5 1 1 5 0 5 1 b b b 0 = 0 = 0 = c) b) a) 1-. . 1 0.5 0 -0.5 -1 rqec of frequency h ak fteclrbx h us uainis duration by Pulse indicated The are lines box. contour color the of the levels of The marks the (3.52). Eq. in model . . 2 1 p x [ m ] ω = 30 7 m ≈ f f f 0 [10 [10 [10 . 23 − − − 0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 10 10 10 m ] ] ] n consequently and , b b b 0 = 0 = 0 = d) e) f) 1-. . 1 0.5 0 -0.5 -1 . . . 3 5 4 p x [ m ..CipdPulses Chirped 3.5. σ ] τ 7 = 30 = . t f c f f [10 iha with [10 [10 − 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 − − 10 73 9 9 ] ] ] 36 π φ = 49 2 3. Homogeneous Electric Fields

3.6. Bichromatic Fields

A popular idea in the field of pair production in strong fields is dynamically assisted pair production. It is based on superposing a low amplitude, high frequency pulse and a high amplitude, low frequency pulse. The result would be a deliberate combination of multiphoton and Schwinger pair production. The tunneling process is helped by absorbing some photons, resulting in a lower tunneling barrier. If this is the case, then Schwinger pair production should still have an influence on where the particles end up in the spectrum and pair production should be strongest at those t where the field strength peaks.

3.6.1. Bichromatic Rotating Fields

Assume a superposition of two rotating Sauter pulses with amplitudes ε0,1, frequency ratio |n| > 1 and phase shift φ

  ε0 cos(Ωt) + ε1 cos(nΩt + φ) #‰ Ecr.   E(t) =  2 ε0 sin(Ωt) + ε1 sin(nΩt + φ) . (3.53) cosh t   τ 0

We will call the ε0 component the “fundamental” pulse and the ε1 component the “harmonic” pulse, having integer n in mind. The frequency ratio does not necessarily have to be integer, but for simplicity we will restrict ourselves to integer n most of the time. If n is negative, the two components are counter-rotating. Atomic ionization with counter- and co-rotating circularly polarized lasers has been experimentally studied in Ref. [122, 123]. Both components on their own could be classified according to the scheme explained in Sec. 3.3.5. Depending on the relation of those two classifications, the outcome in the superposed case could be quite different. To be close to the initial idea, superposing a low amplitude, high frequency pulse and a high amplitude, low frequency pulse, we should choose parameters such that pair production by the fundamental pulse would be dominated by the Schwinger effect. The harmonic pulse should at least fall into an intermediate regime, butdoes not have to allow for a lot of multiphoton pair production on its own. The dotted ∗ line in Fig. 3.12 shows, that for a small amplitude of ε1 = 0.01 the relation γ ≫ 1 of the combined Keldysh parameter holds for the majority of parameter choices. According to the Schwinger effect, pair production should occur where the field strength has a maximum. In the superposition of pulses given in Eq. (3.53), the absolute field strength has not only one global maximum, but a set of local maxima

74 n aigueof use making and eut ofr h rdcin aei h rvossection. in theprevious a numerical and made the production production, predictions pair pair the multiphoton confirm Schwinger by results by dominated is dominated that is harmonic weaker that fundamental Pulses in a Sauter case Rotating Choosing the in Production is Pair as Schwinger points Assisted those Dynamically of vicinity the in 123]. produced [122 , ionization be atomic to pairs expect We of expansion true series still via is (3.54) this Eq. case from pulsed approximation the In wave. fundamental the where of cycle each for maxima to corresponds This and disregarded is envelope the If well. as hnpisaepoue ttoetmsadgi oetmol yclassical by asin only momentum potential the vector gain by and are given times those momenta final at their produced acceleration, are pairs When at maxima local of number infinite an has scniee,i olw htteeeti il strength field electric the that follows it considered, is | t 0 k | ≲ τ nta aetelcto ftemxm a ecluae nagood a in calculated be can maxima the of location the case that In . | E #‰ 0 tanh( E #‰ ( #‰ p t t k k 0 ) ( | = = t = = ) x t  t k ) ∞ 0 E k ≈ e cr E    E #‰ . t cr x  1 0 ( . k t − ε hc eut in results which     d ) 0 2 = ε ε 1 + 0 0 t N ( − 2 cos(Ω sin(Ω n ε = πk 1 2 3 = − 2 + A #‰  − | 1)Ω ( t n t t 0 τ t k ε + ) + ) φ k − 0  ) ε 2 k , − 1 1 1 + ε ε | cos 0 1 1 A #‰ ∈ sin( cos( ( n ( 2(  ∞ − Z φ ε 1) n 0 . n = ) 2 + ( + Ω ε Ω ε 0 | 1 t E ε t #‰ ) 1 n 2 + A #‰ + σ | 2 − ( pt ierodrat order linear to up φ φ    ..BcrmtcFields Bichromatic 3.6. t k ) 1)Ω ) )     . . t  (3.56) (3.55) (3.54) t 75 0 k 37 π φ = 49 2 3. Homogeneous Electric Fields

−12 −11 ε1 = 0 f [10 ] n = 2, φ = 0 f [10 ] 1.0 1.6 1.5 a) 0.9 b) 1.4 0.8 1 1.2 0.7 0.5 ] 0.6 1.0 m

[ 0 0.5 0.8 y

p 0.4 −0.5 0.6 0.3 0.4 −1 0.2 0.2 −1.5 0.1 0.0 0.0 n = 3, φ = 0 f [10−11] n = 4, φ = 0 f [10−10] 4.0 1.4 1.5 c) d) 3.5 1.2 1 3.0 1.0 0.5

] 2.5 0.8 m

[ 0 2.0 y 0.6 p -0.5 1.5 0.4 -1 1.0 0.5 0.2 -1.5 0.0 0.0 n = 6, φ = 0 f [10−9] n = 6, φ = π f [10−9] 2.5 2.5 1.5 e) f) 1 2.0 2.0 0.5 ] 1.5 1.5 m

[ 0 y p -0.5 1.0 1.0 -1 0.5 0.5 -1.5 0.0 0.0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 px [m] px [m]

Figure 3.28.: Example spectra for dynamically assisted Schwinger pair production in co-rotating Sauter pulses with harmonic order n. The red lines display the locus of the vector potential. The green crosses give the predicted pair production peaks as given by Eq. (3.56). Amplitudes are ε0 = 0.1 and ε1 = 0.01 at a pulse duration of τ = 100 tc and fundamental frequency Ω = 0.1 m.

76 h pcrm oehrwt h ou ftevco oeta saandslydin n displayed again is potential vector the production. of pair 3.28a). locus Schwinger Fig. the mostly with exhibits together it spectrum, as The #4, interpretation assigned was it hc si codnewt h ueia aa hsi n oeidctrthat indicator more pulse. one fundamental the is in This process dominant the data. indeed numerical yield, is production the production pair pair with Schwinger in amplitude accordance enhancement higher in of slightly magnitude is this one which than (1.1) more Eq. in from results formula alone amplitude. Schwinger higher slightly the a with to pulse their According fundamental and the same but the nothing amplitude, is their to from superposition extend apart field are, bichromatic pulse the harmonic of and of fundamental yield 3.30 . enhancement particle inFig. the strong of is displayed The data effect The Schwinger second [122]. assisted dynamically its fashion atomic this and counter-rotating in in production beam and pair observed laser co- been polarized fields a circularly already counter-rotating in a have for harmonic of 3.29a) true superposition and also a using 3.28b) is ionization 3.28e) this Figs. Figs. that to in seen shows similar be 3.29 Spectra can as Fig. spectrum the f). in in and peaks phase is production relative pair spectrum the the the of Changing in positions peaks (3.55). production Eq. only pair with with of case accordance number the The was it pulse. like fundamental much the behave deviations the while momenta, predicted | frequency with own pulse its harmonic On regime. of multiphoton accuracy the into with pulses pulses harmonic harmonic the the by production pair frequency #‰ p 0 = Ω h amncodri aidfo o6 eutn nhroi rqece of frequencies harmonic in resulting 6, to 2 from varied is order harmonic The )t ) h rdcdpisaecoet the to close are pairs produced the f), to 3.28b) Figs. in observed be can As amplitude the has pulse fundamental The | = 1 2  . 2 ,..., . . . m, (4 0 = Ω 10 · 0 − . 6 14 . 0 m 1 enn oacranbepi rdcin nthe In production. pair ascertainable no meaning , . 6 ) m 2 m )and 3.16a) Fig. in shown already was spectrum resulting Its . − obndwt h oe mltd of amplitude lower the with Combined . (2 m n ∗ ≤ ) 2 5 0 = Ω aeterdsrbto function distribution their have harm . 663 . Ω=0 = 6Ω = m with ε 0 0 = m . 6 ∗ m . 1 = 1 h duration the , rdcsamlihtnrn at ring multiphoton a produces . 00014 φ ..BcrmtcFields Bichromatic 3.6. m eut nrttn the rotating in results . f n ε 1 eo h numeric the below 1 = τ 0 = 100 = n At . . 6 = 01 t n hsputs this c < n aethe case 1 = n the and 0 the . 77

38 π φ = 49 2 3. Homogeneous Electric Fields

n = −2 f [10−11] n = −5 f [10−10] 1.6 4.5 1.5 a) 1.4 b) 4.0 1 1.2 3.5 0.5 1.0 3.0 ] 2.5 m

[ 0 0.8 y 2.0 p -0.5 0.6 1.5 0.4 -1 1.0 0.2 0.5 -1.5 0.0 0.0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 px [m] px [m]

Figure 3.29.: Example spectra for dynamically assisted Schwinger pair production in counter-rotating Sauter pulses with harmonic order n. The red lines display the locus of the vector potential. The green crosses give the predicted pair production peaks as given by Eq. (3.56). Amplitudes are ε0 = 0.1 and ε1 = 0.01 at a pulse duration of τ = 100 tc and fundamental frequency Ω = 0.1 m. Subplot a) shows the pair production spectrum by the fundamental pulse only.

harmonic frequency nΩ[m] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10−8 fundamental, ε = 0 10−9 1 harmonic, ε0 = 0 10−10 bichromatic

] −11

2 10 m [ 10−12

xy −13

N 10 10−14 10−15 10−16 1 2 3 4 5 6 7 8 harmonic order n

Figure 3.30.: Pair production yield of dynamically assisted Schwinger pair production in rotating Sauter pulses. Amplitudes are ε0 = 0.1 and ε1 = 0.01 at a pulse duration of τ = 100 tc and fundamental frequency Ω = 0.1 m. Two dimensional integral over particle yield. Example spectra given in Fig. 3.28.

78 hti o iil at visible not is bunch. that electron collimated more a in resulting narrower, much gets peak production tepril il snal osati h is ae while case, first the in when constant lowered drastically nearly is is As it yield present. case particle clearly second the is the 3.31 ring in Fig. multiphoton in partial seen a be spectrum can latter the In 3.32f). Fig. whenthe example, first the In from changed is production. phase pair relative relative multiphoton by the influenced changing is by pulse introduced variation the pulses, phase Sauter rotating Pulses short Sauter In Rotating Bichromatic Short in Dependency Phase rela- on depending pulse, Sauter rotating bichromatic in yield Particle 3.31.: Figure ntescn xml,b hnigterltv hs,amlihtnsignature, multiphoton a phase, relative the changing by example, second the In

−9 2 φ Nxy [10 m ] smc agr u otesotpledrto led h fundamental the already duration pulse short the to Due larger. much is 10 20 30 40 50 60 70 80 90 0 ie:pril il yhroi us.Coss atceyedby yield particle Crosses: pulse. harmonic by pulse. yield bichromatic particle lines: phase tive 0 0 φ 0 = φ oi ie:pril il yfnaetlple Dashed pulse. fundamental by yield particle lines: Solid . ) a emd ietyvsbeat visible directly made be can 3.32d), Fig. in . φ 1 1 5 0 = )to b) 3.32 Fig. in 0 = Ω 0 = Ω φ / π . . 5 2 ,n m, ,n m, φ ≈ π 2 = 6 = φ . = π ..BcrmtcFields Bichromatic 3.6. . 2 5 )tepair the c) 3.32 Fig. in φ = π 79 in

39 π φ = 49 2 3. Homogeneous Electric Fields

Ω −10 Ω φ −6 = 0.2, ε1 = 0 f [10 ] = 0.5, n = 2, = 0 f [10 ] m 10.0 m π 6.0 1.5 a) d) 1 8.0 5.0 0.5 4.0 ] 6.0 m

[ 0 3.0 y

p 4.0 -0.5 2.0 -1 2.0 1.0 -1.5 0.0 0.0 −6 −6 Ω = 0.2, n = 6, φ = 0 f [10 ] Ω = 0.5, n = 2, φ = 1 f [10 ] m π 2.5 m π 2 4.0 1.5 b) e) 2.0 1 3.0 0.5 ] 1.5 m

[ 0 2.0 y p -0.5 1.0 1.0 -1 0.5 -1.5 0.0 0.0 −6 −6 Ω = 0.2, n = 6, φ = 1 f [10 ] Ω = 0.5, n = 2, φ = 1 f [10 ] m π 3.0 m π 2.0 1.5 c) f) 2.5 1 1.5 0.5 2.0 ] m

[ 0 1.5 1.0 y p -0.5 1.0 0.5 -1 0.5 -1.5 0.0 0.0 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 px [m] px [m]

Figure 3.32.: Spectra for a selection of bichromatic rotating Sauter pulses with varying relative phase φ. The levels of the contour lines are indicated by the marks of the color box. The red lines display the locus of the vector potential. The green circles show the expected position of pairs due to multiphoton pair production. Amplitudes are ε0 = 0.1 and ε1 = 0.01 at a pulse duration of τ = 10 tc. Subplots a) to c) have fundamental frequency Ω = 0.2 m and harmonic order n = 6 while subplots d) to f) have Ω = 0.5 m and n = 2.

80 h udmna us nteeeape is examples these in of pulse duration fundamental pulse the a at harmonic 5th )to b) 3.33 theamplitude Fig. when comparing 3.33 f). by Fig. seen stronger to be is also 3.33e) can Fig. which or effect 3.33 c) increased, this Fig. is hand pulse other harmonic the the 3.33c) Fig. On of or 3.33e) Fig. 3.33f). to pa- Fig. 3.33b) witha Keldysh Fig. to comparing bigger cases by a observed for be inthose accounts can which which is stronger rameter, pulse, toward fundamental effect spectrum the this for the amplitude hand shifts lower one harmonic the higher On the all behavior. of In multiphoton addition occur. the might expected, features as additional however cases, fundamental, is not are amplitude spectrum the tion ofrings when absence spectrum the the of change 3.33d). which explains qualitative Fig. in the changed potential, and the vector 3.33 a) of Fig. in influence the under γ areconsidered, pulse and the fundamental 3.33 Fig. of Pulses frequencies Sauter different Linear Two in Production Pair Schwinger Assisted Dynamically uss h lcrcfedfrteesuisi given by is studies these Sauter for oscillating field witha linear electric fields bichromatic, The pulsed of or examples pulses. analyze will fields 33 ]. we oscillating ofcases[29–31, envelope, infinitely number uniform on a in focused studies previous been studied While already have fields linear Bichromatic Fields Linear Bichromatic 3.6.2. hr o nyterltv phase relative the only not where 3.49) ( Eq. to according yield particle the calculate using can we [116], Ref. in studied pulses set will we so pulses, cycle phase Ω hntehroi aei de,tecagst h hp ftepi produc- pair the of shape the to changes the added, is wave harmonic the When stepi rdcinsetahv h aesmere steoclaigSauter oscillating the as symmetries same the have spectra production pair the As 2 = φ . 88 0 sicue.Hwvri sepce htti osol lyarl o few for role a play only does this that expected is it However included. is ) ecntu xetbgnigmlihtnpi production pair multiphoton beginning expect thus can We 3.33d). Fig. in 0 = Ω E x ( t = ) . 15 oh( cosh m N nbt ae h ussaepie ihtheir with paired are pulses the cases both In 3.34 . Fig. in E = φ cr t / . (2 0 = τ ) π 1 2 )  . 2 ε δφ −∞ 0  ∞ cos(Ω τ = d p 50 = φ x t  − 0 ∞ + d φ γ t φ p c Ω 0 h oetKlyhprmtrfor parameter Keldysh lowest The . y u loa vrl are envelope carrier overall an also but , 0 + ) 1 = p y f ε . ( 5 1 p y cos( ) h ihs is highest the 3.34a), Fig. in p , x n ) Ω . ..BcrmtcFields Bichromatic 3.6. t + φ )  , 0 = Ω . 23 (3.57) m 81 in

40 π φ = 49 2 3. Homogeneous Electric Fields

−8 −11 ε0 = 0.12, ε1 = 0 f [10 ] ε0 = 0.08, ε1 = 0 f [10 ] 1 1.2 4.5 a) d) 4.0 1.0 0.5 3.5 0.8 3.0 ] 2.5 m

[ 0 0.6 y 2.0 p 0.4 1.5 −0.5 1.0 0.2 0.5 −1 0.0 0.0 −6 −6 ε0 = 0.12, ε1 = 0.008 f [10 ] ε0 = 0.08, ε1 = 0.008 f [10 ] 1 3.0 0.5 b) e) 0.4 2.5 0.4 0.5 2.0 0.3

] 0.3 m

[ 0 1.5 0.2 y

p 0.2 1.0 0.2 -0.5 0.5 0.1 0.1 -1 0.0 0.0 −6 −7 ε0 = 0.12, ε1 = 0.012 f [10 ] ε0 = 0.08, ε1 = 0.012 f [10 ] 1 6.0 16.0 c) f) 5.0 14.0 0.5 12.0 4.0 10.0 ] m

[ 0 3.0 8.0 y p 2.0 6.0 -0.5 4.0 1.0 2.0 -1 0.0 0.0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 px [m] px [m]

Figure 3.33.: Bichromatic linear Sauter pulses with τ = 50 tc, Ω = 0.23 m and n = 5 and varying field strengths. Panels a) and d): pairs produced by fundamental pulse only. Remaining panels: pairs produced by bichromatic pulses. The green circles, if present, show the locations of the expected multiphoton rings for appropriate values of n.

82 Dnmclyasse cwne arpouto nlna atrpulses Sauter linear in production pair Schwinger assisted Dynamically 3.35.: Figure with pulse Sauter linear Bichromatic 3.34.: Figure py [m] -0.8 -0.6 -0.4 -0.2 3 0.2 0.4 0.6 0.8 ε

N [m ] 0 0 0 = 10 10 10 10 10 a) 10 10 10 − − − − − . 0500.5 0 -0.5 − − − ihfeunyratio frequency with oetetopoo eoac ftehroi aeat wave harmonic the of resonance two-photon the Note panel: Right only. pulse pulse. fundamental bichromatic by by produced produced pairs pairs panel: Left 1 14 13 12 11 10 9 8 7 ε , 0 0 . . 40 14 0 7 1 0 = p x [ m ] . . 60 16 0 8 fundamental, harmonic, bichromatic udmna frequency fundamental amncfrequency harmonic n f . . 80 18 5 = 1 1 9 [10 ε 0 − 0 = 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ε n amplitudes and 11 1 ] 0 = ε . 0 2 0 0 = b) τ . 0500.5 0 -0.5 50 = 1 Ω[ 5Ω ε , . [ Ω . 20 22 1 1 1 m t 0 = ε m c 0 , ] p ..BcrmtcFields Bichromatic 3.6. ] 0 = Ω x 0 = . 012 [ m . ] 12 . . 40 24 . 1 2 15 and m Ω=1 = 5Ω and ε 1 . f 0 = . 26 3 n [10 m 5 = . . 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 01 − 83 8 ] . . 41 π φ = 49 2 3. Homogeneous Electric Fields

r = N⊥/N 0.7 0.24 0.6 0.22 0.5 ]

m 0.2 0.4 Ω[ 0.18 0.3

0.16 0.2 0.1 0 0.5 1 1.5 2 φ [π]

Figure 3.36.: Portion of particles accelerated in transverse direction as defined by Eq. (3.58). Pluses mark the parameters for the examples in Fig. 3.37. Amplitudes are ε0 = 0.08 and ε1 = 0.01, harmonic order is n = 5. The low resolution of the plot and apparent discontinuity w. r. t. the frequency ω is due to the limited set of available data.

The total particle yield may be strongly enhanced by the added harmonic wave as shown in Fig. 3.35. The multiphoton resonance is visible in the particle yield of the harmonic pulse, but not in the respective data for the bichromatic pulse. This can be understood in the context of dynamically enhanced Schwinger pair production, because the necessary number of absorbed photons does not suddenly decrease. Instead a smaller but constant number of photons is absorbed and the increasing frequency continuously lowers the tunneling barrier.

Off-Axis Particle Acceleration

An additional phenomenon in bichromatic linear Sauter pulses is that there exist a number of cases, where a large portion of the created particles are accelerated perpendicularly to the electric field. This offers a possibility to maximize the portion of produced pairs that hit the detector in an experimental setup. In order to quantify this behavior, we will consider every created particle to be transversely accelerated, ◦ if its momentum encloses an angle of more than 60 with the longitudinal px axis.

84 Eapesetafo ihoai ierSue usswt trans- with pulses Sauter linear bichromatic from spectra Example 3.37.: Figure a eietetases atceyield particle transverse the define can

ihaHaiiefnto,we function, Heaviside a with (3.57) Eq. of integrand the multiplying by Thus, py [m] py [m] − -0.5 − 0.5 0 0 -1 . . 0 = Ω 0 = Ω 0 1 1 5 0 5 1 c) a) N 1-. . 1 0.5 0 -0.5 -1 ⊥ . . upelnsdslytebudr o rnvreaclrto sgiven as acceleration transverse for boundary by the is display duration lines Pulse Purple 3.36. are Fig. in amplitudes marked as acceleration verse 19 2 := ,φ m, = = ,φ m, arctan (2 (2 (2 π π π 1 1 1 0 = p ) ) ) 0 = x 2 2 2 [ −∞ −∞ −∞  m    | | ∞ ∞ ∞ p p ] x y | | d d d  p p p 60 = ε x x x 0 √   0 0 1 ∞ ∞ 3  0 = | ∞ d d p x ◦ p p | . y y d . f f p p 08 p y y [10 [10 y f f p ( ( and 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 y − − p p f x x 7 7 p , p , ( ] ] p ε x y y 0 = Ω 0 = Ω 1 p , )Θ( )Θ 0 = y d) b)  ) | 1-. . 1 0.5 0 -0.5 -1 p arctan . . y . 16 24 01 − | ,φ m, ,φ m, amncodris order harmonic , √  3 | p ..BcrmtcFields Bichromatic 3.6. | | p p p 0 = = x x x y [ | | | m π )  ] − 60 ◦  τ f f 50 = n [10 [10 5 = 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 − − t 85 8 6 c ] ] , . 42 π φ = 49 2 3. Homogeneous Electric Fields and the relative transverse particle yield

N r := ⊥ . (3.58) N As can be seen in Fig. 3.36, the transverse particles peak at a number of different parameter sets. Some example spectra are given in Fig. 3.37 and their parameters are marked with plus symbols in Fig. 3.36. There are at least two quite different types of spectra with transverse acceleration of particles. In some cases, collimated particle bunches with high velocity are accelerated in a transverse direction. Two examples of this are displayed in Fig. 3.37a) and b). In other cases, particles are accelerated in all directions with a slight tendency towards the p⊥-direction. Due to the cylinder symmetry this accounts for a large relative transverse particle yield.

86 eeu eutfrtepril il stesm stersl rmitgaigover integrating from result the as same the is yield an particle results at the correct wavelengths for to achieve Compton result geneous 50 important than is of smaller strength field scales field electric at magnetic inhomogeneities the spatial that for shown was it and tem- because fields, have to induction electric of possible homogeneous law not Faraday’s spatially technically the of and also reconsider fields is to magnetic it have changing hand would porally other one the and On formalism, levels the Wigner condition. Landau in initial atempo- introduce include would field this magnetic tosimilarly because homogeneous for thataccommodate spatially not straightforward and is states constant It ofbasis rally similar set field. equation different magnetic kinetic a constant a using the There derived been Ref.[125]. in has QKT studied to been has field magnetic stant into account executed. be be taken to to has have longer expansion operators derivative no differential a can pseudo or Eq.(3.1) full calcu- from the making Instead thesimplification is used. latter be case because The difficult, special discarded. more the much is only lations homogeneity not spatial case also this haveconsid- but In abandoned, DHWformalism 124]. the [68, in fields inhomogeneous fields ered magnetic including of attempts Previous Fields Magnetic of Inclusion 4. ai.Toerslswr oprdt aclto htatfcal fixed artificially that calculation a to compared were results Those valid. omls.Teeas antcfedwsitoue and a introduced InRef.[68] was field magnetic yield. a scale also local particle alength There should formalism. When with the integrated yield field particle electric yield. from total localized differ theparticle the spatially to decreased start tocalculate is point inhomogeneities some localized the at is of scale integrate field and length the rate the production where fieldis pair constant region the alocally calculate the by locally over to of able inhomogeneities the approximation be and spatial should we the enough correct, of is large thataremuch scale scales the field length If electric over the wavelength. change that Compton fields the for than larger amodel often are fields neous arpouto nuiietoa lcrcfed ntepeec facliercon- ofacollinear the presence in fields electric unidirectional in production Pair hnpi rdcini ooeeu lcrcfed sdsusd thehomoge- is discussed, fields electric homogeneous in production pair When ε 0 = . × ∇ 707 ttesm iei a hw htteinhomo- the that shown was it time same the At . E #‰ = − B #‰ ˙ . λ a tde sn h Wigner the using studied was × ∇ E #‰ = − B #‰ ˙ a kept was B #‰ B #‰ 0 = 0 = 87 is

43 π φ = 49 2 4. Inclusion of Magnetic Fields the particle yield for locally homogeneous fields at a variation scale of 50 Compton wavelengths or larger. When the magnetic field is changing, Faraday’s law requires us to also introduce a spatial inhomogeneity of the electric field of the same order. If we make sure that this spatial inhomogeneity of the electric field has a large enough scale we should be allowed to neglect it in the calculation. In other words we technically include the correct spatially inhomogeneous electric field to ensure the validity of Faraday’s law while at the same time approximating the inhomogeneous electric field by a homogeneous field for the calculations. We could thus start the time evolution at a point in time where there is no magnetic field and the initial conditions Eq.(2.8) are valid and slowly switch the magnetic field on afterwards. Before ending the time evolution, the magnetic field must be switched off in a similar manner. Switching the magnetic field onandoff is implemented by a function

 0 x < 0   Θ(˜ x) := 1 x > 1   6x5 − 15x4 + 10x3 otherwise, which is also known as the smootherstep function [126]. It has zero first and second order derivatives at x = 0 and x = 1. Its first derivative has a maximum at x = 1/2 ′ with Θ˜ (1/2) = 15/8 < 2. The maximum rate of change for the magnetic field when

 t  B(t) := B · Θ˜ (4.1) 0 ∆t

2B0 is gB = ∆t which should not exceed gmax. Requiring gB < gmax yields ∆t > 2000 tc · B 2 2 . Thus for B = 0.01B , with the critical field strength B = m c /e ≈ 4.4GT, Bcr. cr. cr. ℏ the time scale for switching the magnetic field should be ∆t > 20 tc.

88 h eodse ste ovn h eutn e fOE iha prpit solver. appropriate an with ODEs of set resulting the solving then is step second The ntecs fhmgnoseeti n antcfed h suodfeeta op- differential pseudo the to fields simplify magnetic (2.3) Eq. and from electric erators homogeneous of case the In Motion of Equation 4.1. eprldrvtvso h ucinvle ttegi onso ftecoefficients, the form of the the or into points calculate put grid to is the spatial possible PDE at the is the values if provide it function to those the able of Using derivatives be for functions. temporal coefficients should the discretized decomposition or the This grid of a functions. derivatives of basis vertices Those of the spatial of numbers. at set the values Thefirst set a represent finite function steps. a to the with be decomposition time two could in basis point numbers in each of at done form functions be the appropriate Solving of an can dependence case. use electric to by equation purely is equation the differential step in differential partial done evolving ordinary been time an has a as into characteristics equation itisnot of resulting method ( 2.6). the the Eq. various functions, from to the turn over start to apply to possible have operators but differential (3.3) different Eq. now use Because cannot we thus nonzero, be may nti aeteeuto fmto osntdcul n l h 6components 16 the all and decouple not does motion of equation the case this In D #‰ ∂ D P t #‰ #‰ x t W = = = = ∂ e #‰ . p B t #‰ F + ( ( t t, e ) E #‰ W × ( ∇ , t #‰ ) ∇W #‰ #‰ p · ∇ #‰ , #‰ p ) . , ..Euto fMotion of Equation 4.1. (4.3) (4.2) 89

44 π φ = 49 2 4. Inclusion of Magnetic Fields

The equations of motion Eq. (2.6) can be easily brought into the form Eq. (4.3) by inserting Eq. (4.2) and sorting the terms accordingly. This results in

 #‰ #‰  #‰ #‰ #‰ 1 ∂ts = −e E · ∇p s +2 p · t  #‰ #‰  #‰ #‰ #‰ 2 0 ∂tp = −e E · ∇p p −2 p · t −2m a  #‰ #‰   #‰ #‰  0 #‰ 0 #‰ #‰ ∂tv = −e E · ∇p v −e B(t) × ∇p · v  #‰ #‰   #‰ #‰  0 #‰ 0 #‰ #‰ ∂ta = −e E · ∇p a −e B(t) × ∇p · a +2m p  #‰ #‰   #‰ #‰  #‰ (4.4) #‰ #‰ #‰ #‰ 0 #‰ #‰ 1 ∂t v = −e E · ∇p v −e B(t) × ∇p · v −2 p × a −2m t  #‰ #‰   #‰ #‰  #‰ #‰ #‰ #‰ 0 #‰ #‰ ∂t a = −e E · ∇p a −e B(t) × ∇p · a −2 p × v #‰  #‰ #‰  #‰  #‰ #‰  #‰ 1 #‰ 1 #‰ 2 #‰ #‰ ∂t t = −e E · ∇p t −e B(t) × ∇p × t −2 p s +2m v #‰  #‰ #‰  #‰  #‰ #‰  #‰ 2 #‰ 2 #‰ 1 #‰ ∂t t = −e E · ∇p t +e B(t) × ∇p × t +2 p p .

If this system of equations would be used, the numerical difficulties in calculating the one-particle distribution function f as described in the homogeneous case in Sec. 3.1 would reappear. It is possible to do a similar substitution also in this case, but due to the more complex structure of Eq. (4.4), a lot more additional terms arise. The task of carrying out this substitution and turning the resulting equations into C++ code was much too tedious to do by hand. In order to automate the process, a Mathematica [101] notebook has been written, which created the C++ code automatically. For this to work, the system of equations was brought into the form

W W W ∂t = Ci (∂pi ) + M W W = Ci ,pi + M (4.5)  #‰ #‰ #‰ #‰ ⊺ with W = s p v0 v a0 a t 1 t 2 .

W W Here the short-hand notation ,pi is introduced for ∂pi . Determining the Matrices

Ci is a matter of carefully evaluating the dot products, double cross products and triple products in Eq. (4.4). The details can be found in App. A.1, where Ci and M are given in Eqs. (A.1) and (A.2), respectively. The substitution from Eq. (3.5) can be written in terms of the vacuum solution Eq. (2.8)

#‰ #‰ #‰ #‰ #‰ #‰ s( p , t) = (1 − f( p , t)) svac.( p ) − p · v ( p , t) , #‰ #‰ #‰ #‰ #‰ #‰ #‰ v( p , t) = (1 − f( p , t)) v vac.( p ) + v ( p , t) .

90 with form the takes it and on h ubro iesossol erdcdi re ohv manageable a have this to At order together. in evolved reduced and be stored should be dimensions complete to a of true, has number not space the is momentum point this in Eq. (4.7) points of equation differential grid partial the of that context means, the This space evolution. momentum their in during point every information trajectories exchange the not characteristics, do of and decouple method the to due space method, momentum dimensional three a transformation. Fourier in a by done is this executed, The be can code. code C++ this into before motion calculated of equations 16 the automatically write that realizing (4.5) Eq. into inserted is (4.6) Eq. for When solved used. and been has notation matrix block where of terms In l h ie qain pt hspitwr upsdt eciepi production pair describe to supposed were point this to up equations given the All ∂ t W = #‰ p M  ftefnlmmnu pcrmcnb acltdidpnety In independently. be calculated can spectrum momentum final the of ∂ A t W W W − W W 1 vac C vac h eutn qainreads equation resulting the ( ( A i . A #‰ #‰ t , p t , p . ( ( 0 =  #‰ #‰ p p W = ) = ) = ) = ) sn hsrsl n h nw arcsi spsil to possible is it matrices known the and result this Using . ,p i                    A   + s f − ( − vac A #‰ p v #‰ s − 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v p ) . vac vac 1 W ( 0 0 . . C ( 0 i #‰ t , p A 0 0 0 0 ,p #‰ p #‰ v + ) i v #‰ + hteovswt time with evolves that vac a − MA W 0 . 1 #‰ p vac 0 ⊺ #‰ a ) . W ( 0 0 0 0 0 0 0 0 #‰ #‰ p 0 #‰ t ) + 1 1 #‰ A 0 − #‰ t 1 0 0 ..Euto fMotion of Equation 4.1. 1 2 #‰ C 0  1  i 0 ⊺ ∂ ⊺                    p i W , t nteWigner the In . vac W ,p . , i edt be to need (4.6) (4.7) 91

45 π φ = 49 2 4. Inclusion of Magnetic Fields

memory usage. Setting Ez = Bx = By = 0 has the effect that all derivatives with respect to pz are dropped from the equations, because the matrix Cz, defined in Eq. (A.1), is identically zero. This enables two-dimensional calculations with a p -p grid, for any given value of p . x y #‰ #‰ z Setting B → 0 in the resulting equation of motion results in Eq. (3.6), which is equivalent to Eq. (3.10) plus another set of equations for the remaining functions, that completely decoupled from the one-particle distribution function.

4.2. Numerical Implementation

Following the previous section, there are now various possibilities to combine the available forms of the equation of motion with the numerical methods. They are listed together with their development name in Tab. 4.1 in chronological order.

Table 4.1.: Table listing the various numerical codes. #‰ differential eq. subst.?1 B name #‰ 2 ODE, Eq. (3.3) no = #‰0 no name, first attempts with Mathematica ODE, Eq. (3.10) yes = #‰0 Wigner method (ppsolve, charwigner) 3 PDE, Eq. (3.6) yes = #‰0 fftwignerh PDE, Eq. (4.7) yes ̸= #‰0 fftwignerb PDE, Eq. (4.4)4 no ̸= 0 fftwignerb-nosubst 1 Indicates whether the variables have been substituted to solve for f directly. Otherwise still a substitution is applied to remove the vacuum solution.2,4 2 The system of equation for this case results from Eq. (3.3) by substituting w w′ w w′ → + vac. and solving for ˙ . #‰ #‰ 3 Eq. (3.6) follows from Eq. (4.7) by inserting B = 0 . 4 The system of equation for this case results from Eq. (4.4) by substituting #‰ ′ #‰′ #‰ ′ #‰′ (s, v) → (s , v ) + (svac., v vac.) and solving for ∂t(s , v ).

The solution scheme is based upon setting the PDE Eq. (4.7) onto a grid in #‰ p -space and calculating W,p using a Fourier transformation. This removes the #‰ i derivatives w. r. t. p and turns the system of 16 continuous PDEs into a system of 16 · Nx · Ny · Nz ODEs which can be solved using, e. g., a Runge-Kutta method.

The momentum grid covers in each direction a range pi ∈ (−Pi,Pi). Usually two- dimensional calculations are done using Nx = Ny = N and Px = Py = P . One of the fastest available implementations of the fast Fourier transformation is the FFTW [127], which is a pure C library. As the library exists in multiple variants in order to support multiple floating point data types, most of the function names

92 py [m] Toeapesetacluae by calculated spectra example Two 4.1.: Figure hscd okdwl nuhteeuto fmto a hne oicuethe include to changed renamed Once was code motion algorithm. the and of numerical terms the equation and of the functions stability enough additional the well improve worked and code test called this was to code used This was decomposition. basis and Fourier a using (3.6) Eq. solves individual the accessing right threads, the parallel evaluating using when motion using access of functions equation memory the minimize of to side order hand in preferred using also array, was single a in basic full state transform the the independent implement Using multiple only array. out interface they one carrying within abstract so for allow an Doing not provide FFT. does the to the which try functionality, of takes also details which functions. they the correct struct, but hides the exist, that C++ to already references a do contains pureClibrary basically and wrappers parameter Such is the template automatically a wrapper names to as function This type correct data wrapper the time. selecting AC++ of compilation task at the type. has point which built floating was used the on depend ,acd a enwitnthat written been has code a ofEq.(4.7), full problem the towards step first a As -1.5 -0.5 0.5 1.5 -1 N 0 1 15- 0500511.5 1 0.5 0 -0.5 -1 -1.5 128 = a) inbek onatrhg ore oe r ouae exponentially. size populated Grid are 3.5. modes b): Fig. Fourier Panel in high after found down be breaks size can tion pa- Grid methods available with a): other Eq.(3.30) Panel the pulse by Sauter calculated as therotating by rameters is given field electric rmsest ge ihterfrnesetu,btatfcso high of artifacts but spectrum, visible. reference are the modes with Fourier agree to seems trum P , Blitz++ 5 = p x τ [ m 10 = [105]. ] t c FFTW , FFTW N N 0 = Ω 128 = 1024 = f sbiti aalls.Ti eoylayout memory This parallelism. built-in ’s [10 nefc ti osbet tr h complete the store to possible is it interface . -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 6 oetmextent momentum , − m oetmextent momentum , 6 ] eeec pcrmfrti pulse this for spectrum reference A . N 15- 0500511.5 1 0.5 0 -0.5 -1 -1.5 1024 = fftwignerh b) ..NmrclImplementation Numerical 4.2. fftwignerb P , 20 = p ihu apn.The damping. without x [ P m P . ] 5 = 20 = h calcula- The . h spec- The . fftwignerh f FFTW [10 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 − 93 6 ] 46 π φ = 49 2 4. Inclusion of Magnetic Fields

−6 −6 ds = 1.5, dp = 0 f [10 ] ds = 1.5, dp = 1.5 f [10 ] 3.5 3.0 4 a) 3.0 b) 2.5 2.5 2 2.0 2.0 ] 1.5 m

[ 0 1.5 y 1.0 p -2 0.5 1.0 0.0 0.5 -4 -0.5 -1.0 0.0 -4 -2 0 2 4 -4 -2 0 2 4 px [m] px [m]

Figure 4.2.: Two example spectra calculated by fftwignerh with damping. Grid size in both cases N = 128, momentum extent P = 5. The electric field is given by the rotating Sauter pulse Eq. (3.30) with parameters τ = 10 tc, Ω = 0.6 m. Common damping parameters are xp = xs = 0.7 and hp = hs = 20. Panel a): Dampening only in Fourier space. The central region of the spectrum agrees with the reference spectrum Fig. 3.5, but some artifacts remain at the boundaries. Panel b): Both damping terms enabled. The central region of the spectrum agrees with the reference spectrum, the only artifact is a smooth ring at the border of the damping term.

During the first tests with fftwignerh it became evident that it is necessary to get rid of the high Fourier modes which grow during time evolution. If the integration steps are too large, the high Fourier modes start growing exponentially due to numerical instability. Even if the step size is small enough, high Fourier modes will grow due to propagation terms in Fourier space. After some time the bounds of the reciprocal grid are reached and the evolution breaks down, an example can be seen in Fig. 4.1a). In principle it is possible to compensate by using a large enough grid, but this consumes lots of memory and computation time. Even though a number of example calculations were successful using this approach and an example is given in Fig. 4.1b), it was found to be computationally too expensive. Another idea was to add a linear damping term to the equation of motion

  #‰   −1 | s | #‰′ #‰ #‰ ∂tW → ∂tW − FT us · FT[W ( p )]( s ) ( p ) , smax resulting in an exponential decay of any population of the high modes. This term

94 aedmigtr a de oteeuto fmto nmmnu space momentum in motion of the equation seenin artifacts, the be those to can from added arising was anexample term problems damping numerical same spectra, further prevent final To the 4.2a). in Fig. remained still artifacts but results, and value that of u half is damping the where where 4.2b). an Fig. problems, in numerical produce seen to be likely can less example are that artifacts smoother much parameters have damping of set second a introducing as as evolve defined modes was lower prefactor the The that such dictates. modes, motion higher of the equation on the act only course of should s ( a ) sn hsdmigfrtehg ore oe mrvdtenumerical the improved modes Fourier high the for damping this Using . d s ol eietedmigsrnt o h highmodes, the for strength damping the define would u s ∂ ( a t W = ) → d 2 s ∂  t ah( tanh W − u h p s (  h a s p | − eie h tens fthefunction of steepness the defines max #‰ p d p x | , s  x )+1 + )) ..NmrclImplementation Numerical 4.2. p · W and  , h p h eutn spectra resulting The . x s eie h point the defines 95

47 π φ = 49 2 4. Inclusion of Magnetic Fields

−9 −8 t = −42 tc,B = 0 f [10 ] t = −42 tc,B = 0.001 Bcr. f [10 ] 1.5 1.5 8.0 a) b) 1 6.0 0.5 1.0 ] m

[ 0 4.0 y p -0.5 0.5 2.0 -1 -1.5 0.0 0.0 −6 −6 t = 42 tc,B = 0 f [10 ] t = 42 tc,B = 0.001 Bcr. f [10 ] 1.5 2.5 2.5 c) d) 1 2.0 2.0 0.5

] 1.5 1.5 m

[ 0 y p 1.0 1.0 -0.5 -1 0.5 0.5 -1.5 0.0 0.0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 px [m] px [m]

Figure 4.3.: Comparison of the time evolution with and without magnetic field. The electric field is given by the rotating Sauter pulse Eq.(3.30) with parameters τ = 10 tc, Ω = 0.6 m. The magnetic field is given by Eq.(4.1) with a switching parameter of ∆t = 100 tc. The levels of the contour lines are indicated by the marks of the color box. Panel a) and b): Intermediate stages of the time evolution before the peak of the electric field pulse without magnetic field or while the magnetic field is switched on. Panel c): An intermediate stage of the time evolution without magnetic field after the peak of the electric field pulse. The distribution has almost arrived at its final shape. Panel d): An intermediate stage of the time evolution while the magnetic field is being switched off after the peak of the electric field pulse. Some differences to Panel c) are visible that vanish completely once the magnetic field is completely switched off.

96 ene osrs htteeaevr al eut n ute eerhnest be to needs research But further and useful. results be early very not are might these for approach it done. that assumptions our that stress our that conclude to that and we need right be we so might not result, ques- conclusion are the open approximation possible change poses the One not approach does stable. our size numerically that grid is indicates the clearly Increasing and tions. possible be not should pcr oti eaievle for values negative contain spectra seilydrn h wthn ftemgei il,tesae r ifrn.Some aredifferent. 4.3. Fig. states in the seen field, times, be magnetic intermediate can the At examples of field. switching magnetic the without during as especially same the is spectrum production .Tevle for values The in Eq.(4.1). asdefined parameter added was field B magnetic the stage that At Results Preliminary 4.3. for results the of Comparison 4.4.: Figure ncs ftesrne antcfield magnetic stronger the of case In ncs ftewa antcfield magnetic weak the of case In p [m]

eecoe obe to chosen were y -3 -2 -1 N 0 1 2 3 3- 10123 2 1 0 -1 -2 -3 128 = a) ∆ ihparameters with Eq.(3.30) pulse Sauter 10 the rotating by given field t 100 = t P , c , 0 = Ω 5 = p t B c x . [ m . 0 = 6 ] m . h eutsest esal .r .tegi size. grid the t. r. w. stable be to seems result The . 001 B cr . n later and f f B This 4.4 . Fig. in seen be can example an , [10 0 = B -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 B − 6 0 = . ] 0 = 001 N B . 01 . 3- 10123 2 1 0 -1 -2 -3 B 01 256 = 0 = b) B cr B . cr ttrsotta h ia pair final the that out turns it cr . . 01 . h eutn arproduction pair resulting the P , ihdfeetgis Electric grids. different with B 5 = cr ..PeiiayResults Preliminary 4.3. . p ohwt switching a with both , x [ m ] f [10 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 τ − 97 = 6 ] 48 π φ = 49 2

yadtoa eerh h eut thn r led sflt ana overall an gain to useful already fields. are rotating in hand production at pair results of picture the research, additional by results. preliminary our discussed and setup of variant another discovered. case linear case is the rotating features In the useful in experimentally ionization. atomic spectra from resulting results the experimental compared to We pulses. Sauter rotating amplification. and pulse chirped using lasers shapes, high-power chaotic quite for chirp show relevant spectra a be resulting If could the bunches. pulse which Sauter electron rotating collimated pa- the of pulsesdisclosed to production added Sauter the is to polarized lead which inelliptically sets rameter production pair Studying moment. magnetic field. the regarding asymmetry rotating of the measure in some production has from pair pulse apart Multiphoton Sauter parameter, 3.11. Keldysh Fig. combined see a resonances, and on unit multiphoton only production per dependent yield pair particle is total Schwinger volume the spacetime that potential, demonstrated We vector production. the pair multiphoton of terms in phenomenology frequency rotation strength field easier substantially is durations pulse method. Compu- long semiclassical for pulse. the spectra Sauter using production rotating pair the the in of production tation and pair calculate Eq.(3.10) to a method equation developed classical also kinetic We fields. quantum solver. electric modified numerical rotating a the implemented in constructed we production end pair this positron To electron investigated have We Conclusion 5. hl lal oeo h eut rsne nti oksol ecomplemented be should work this in presented results the of some homogeneous aspatially clearly in While fields magnetic include to approach an on worked We linearoscillating superposed of form in the studied been have fields Bichromatic electric homogeneous aspatially for configurations of kinds other considered We sub-critical the with pulse Sauter rotating the in production pair studied We E 0 = Ω . on ow nepee h pcr ihrsetterrich their respect with spectra the interpreted we so Doing . 1 E cr . ndpnec fteprmtr us duration pulse parameters the of dependency in C++ oebsdo h semi- the on based code τ and 99

49 π φ = 49 2

A. Appendix

A.1. Matrices for the Equation of Motion Including Homogeneous Magnetic Fields

The equation of motion for the inclusion of homogeneous magnetic fields in Chap- ter4, Eq. (4.5), was given in a general form. The matrices are still to be defined, which will be done via block matrices. Let us first define some vectors and matrices used to carry out the various vector operations that occur in the equation of motion. #‰ #‰ Evaluating a cross product B × v can be expressed using a matrix b defined by

#‰ #‰ #‰ b v := B × v .

The matrix b is then given by

  0 −Bz By    #‰ #‰ #‰ b =  B 0 −B  =: b b b ,  z x x y z −By Bx 0 #‰ #‰ #‰ where the columns of b are designated b , b , b . The two types of terms containing #‰ #‰ x y z #‰ B × ∇p can then be expressed using these vectors as

 #‰ #‰  #‰ #‰  B × ∇p a = bi (∂pi a) i=x,y,z  #‰ #‰  #‰ #‰ #‰  #‰ B × ∇p · v = bi · (∂pi v ) . i=x,y,z

 #‰ #‰  #‰ #‰ Terms with a triple cross product B × ∇p × t need to be evaluated separately resulting in

 #‰ #‰  #‰  #‰ #‰  B × ∇p × t = Ti ∂pi t i=x,y,z

I A. Appendix

with matrices Ti defined by

      0 By Bz 0 −Bx 0 0 0 −Bx       Tx := −B 0 0  ,Ty := B 0 B  ,Tz :=  0 0 −B  .  y   x z  y −Bz 0 0 0 −Bz 0 Bx By 0

To define the matrices Ci and M for Eq. (4.5) block matrix notation is used. They are given by

  0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0   #‰  0 0 0 b ⊺ 0 0 0 0   i   #‰  0 0 b 0 0 0 0 0   i  1 Ci := e  #‰  − eEi 16 (A.1) 0 0 0 0 0 b ⊺ 0 0   #‰ i    0 0 0 0 b 0 0 0   i    0 0 0 0 0 0 0 Ti   0 0 0 0 0 0 Ti 0   −eEi 0 0 0 0 0 0 0      0 −eEi 0 0 0 0 0 0   #‰   0 0 −eE b ⊺ 0 0 0 0   i i   #‰   0 0 b −eE 1 0 0 0 0   i i 3  = e  #‰   0 0 0 0 −eE b ⊺ 0 0   #‰ i i     0 0 0 0 b −eE 1 0 0   i i 3     0 0 0 0 0 0 −eEi13 Ti    0 0 0 0 0 0 Ti −eEi13 and

 #‰  0 0 0 0 0 0 2 p ⊺ 0  #‰   ⊺  0 0 0 0 −2 m 0 0 −2 p     0 0 0 0 0 0 0 0       0 0 0 0 0 −2 P 2 · 1 0   3  M :=   (A.2)  0 2 m 0 0 0 0 0 0       0 0 0 −2 P 0 0 0 0     #‰  −2 p 0 0 2 · 13 0 0 0 0   #‰  0 2 p 0 0 0 0 0 0

II A.2. Semiclassical Results for the Constant Rotating Pulse

with

#‰ #‰ #‰ P v := p × v   0 −pz py   ⇒ P :=  p 0 −p  .  z x −py px 0

A.2. Semiclassical Results for the Constant Rotating Pulse

The integrals of Eqs. (3.19) and (3.20) at the turning points for the constant rotating field have been computed beforehand in[92]. Here we reproduce them for the sake of completeness. The turning points and the integrals can be found to be

 2     2 2   Ω  2 q q + ϵ⊥ + m Ωt± = arcsin x ± i arcosh  ∥ mε  + 2πk . (A.3) k   Ω   q∥ 2 mε q∥ m

The integrals K(tk) and Kxy(tk) are given by

     2 2 4ϵ⊥ 2 y− − y+ K(tk) = 1 − y+ 2k E  2  + Φ Ω 1 − y+      (A.4)   2  2 4ϵ⊥ 2 1 − y− 1 − y− + i 1 − y+ E  2  − K  2  , Ω 1 − y+ 1 − y+    2 2 2  2 2 g(y− − 1) y+ 1 y− − y+ y− − y+ Kxy(tk) = 2k  Π  ,   2 y y2 1 − y2 1 − y2 1 − y+ − − + +    2 2 g (1 − y−y+) y− − y+ + 2k  K   + Φxy 2 1 − y2 1 − y+ +    (A.5)  2 ig(y−y+) 2 1 − y− +  Π 1 − y−,   2 1 − y2 1 − y+ +    2 ig 1 − y− −  K   , 2 1 − y2 1 − y+ +

III A. Appendix with Ω q ± eεE ∥ cr.  2 2 y± := i , q∥ := qx + qy Ω ϵ⊥ and the elliptic integrals [114]

π/2  1 K(x) := dΘ √ 1 − x2 sin2 Θ 0 π/2   E(x) := dΘ 1 − x2 sin2 Θ 0 π/2  1 Π(n, x) := dΘ √ . (1 − n sin2 Θ) 1 − x2 sin2 Θ 0

The quantities Φ and Φxy are physically irrelevant global phases.

A.3. Calculation of the Momentum Spectrum for the Sauter Pulse

In this appendix we want to calculate the integral K0(t) for the non-rotating Sauter pulse which is given by Eqs. (3.30) for Ω = 0. We start from the potential

Aµ(x) = (0,A(t), 0, 0),   t  A(t) = eE τ 1 + tanh . 0 τ

The turning points, as defined in Eq. (3.24), are found to be

  ± qx ± i˜ϵ⊥ tj = τ artanh − 1 + iπjτ , (A.6) eE0τ for j ∈ N with

2 2 2 ϵ˜⊥ := ϵ⊥ + qy .

IV A.3. Calculation of the Momentum Spectrum for the Sauter Pulse

This means we find an infinite number of turning points which all have thesame real part

 2 2  ± 1 qx +ϵ ˜⊥ sj = ℜ(tj ) = log 2 2 . 4 (qx − 2eE0τ) +ϵ ˜⊥

The integral from Eq. (3.19) gives

2  γ  t  K (t) = −τ log (ω(t) + q ) + tanh 0 γ m x τ     t  − τ lql log 1 − l tanh l=±1 τ  γ   γ  t  − log q + l q + tanh (A.7) m x m x τ 2  γ  2  + qlω(t) +ϵ ˜  m2 ⊥ + Φ˜ , where Φ˜ is a physically irrelevant global phase and we introduced the Keldysh parameter for the pulse length τ which is defined as m γ := eE0τ and we also defined   2  ms q :=  q − (1 ± 1) +ϵ ˜2 . ± x γ ⊥

Using the explicit form of the turning points of Eq. (A.6) and assuming E0 > 0 we find that the imaginary part of the integral from Eq.(3.19) at the turning points is given by

π 1 ℑ[K (t±)] = ∓ τ (γq + γq − 2m) . 0 j 2 γ + −

According to the condition in Eq. (3.25) only the turning points for which the imaginary part is negative contribute. That still leaves an infinite number of turning points which will give the same contribution to the sum in Eq. (3.28). However Eq. (3.28) only holds if the turning points have a different real part. This is con- nected to how the contour is deformed to extract the contributions of the poles. We V A. Appendix

chose the contour such that it follows the real axis up to sp and then approaches the turning point in a line parallel to the imaginary axis. If turning points have the same real part it is sufficient to take one integral which encircles all of the turning points. Using Eq. (A.7) we find

± tj+1  ω(t′)dt′ = 0 .

± tj

+ This means that only the integral from sp to tp contributes for the Sauter pulse, since the contributions of the other ones vanish due to the periodic form of ω(t). Accordingly the semiclassical momentum spectrum defined in Eq. (3.28) takes the form

 π 1 γq γq  W SC(⃗q) = exp − + + − − 2 . s ϵ γ2 m m

This can be compared to the exact result, which for instance can be obtained in the real-time DHW formalism and is [74, 90]

 1 π 1  γq+ γq−   1 π 1  γq+ γq−  sinh 2 ϵ γ2 2 + m − m sinh 2 ϵ γ2 2 − m + m Ws(⃗q) = .  π 1 q+   π 1 q−  sinh ϵ γ m sinh ϵ γ m

1 1 τm Using the fact that sinh(x) ≈ 2 exp(x) for large x we find for ϵγ ∼ / ≪ 1

τm≫1 SC Ws(⃗q) ≈ Ws (⃗q) such that the semiclassical result is approximating the exact one well for long enough pulses. As described in Sec. 3.3.3 for shorter pulses the turning points get too close in the complex plane and the approximation in Eq. (3.26) breaks down.

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XVIII C. List of Figures

1.1. Feynman diagrams for either a) Bethe-Heitler or b) Breit-Wheeler processes...... 4

3.1. Time dependency of the rotating Sauter pulse...... 33 3.2. Positions of the turning points of the rotating Sauter pulse...... 34 3.3. Asymptotics of an auxiliary function used to calculate the vector potential of the rotating Sauter pulse...... 42 3.4. Loci of the vector potential of the rotating Sauter pulse...... 43 3.5. Momentum spectrum of a specific rotating Sauter pulse as given by three different methods...... 47 3.6. Comparison of the total particle number per Compton volume of rotating Sauter pulses as a function of the pulse length τ...... 48 3.7. Energy of particles with specific kinetic momentum in the constant rotating field with different parameters at complex times...... 49 3.8. Comparison of the total particle number per Compton volume of rotating Sauter pulses as a function of the pulse length τ for different methods...... 49 3.9. Spectra of the particle yield for the independent solutions of the semiclassical method compared to the corresponding projections of the Wigner function...... 51 3.10. Comparison of the particle yield for the independent solutions of the semiclassical method with the corresponding projections of the Wigner function...... 52 3.11. Total particle yield of the rotating Sauter pulse...... 53 3.12. Overview over parameter range of rotating Sauter pulse...... 55 3.13. Collection of pair production spectra for the rotating Sauter pulse (1/4) 56 3.14. Collection of pair production spectra for the rotating Sauter pulse (2/4) 57 3.15. Collection of pair production spectra for the rotating Sauter pulse (3/4) 58 3.16. Collection of pair production spectra for the rotating Sauter pulse (4/4) 59 3.17. Slices of a three dimensional spectrum (1/3)...... 60 3.18. Slices of a three dimensional spectrum (2/3)...... 61 3.19. Slices of a three dimensional spectrum (3/3)...... 62 3.20. Spectra of pairs with specific magnetic moment (1/2)...... 64 3.21. Spectra of pairs with specific magnetic moment (2/2)...... 65 3.22. Spectra of pair production in general polarization (1/2)...... 68

XIX C. List of Figures

3.23. Spectra of pair production in general polarization (2/2)...... 69 3.24. Particle yield in general polarization (1/2)...... 70 3.25. Particle yield in general polarization (2/2)...... 70 3.26. Spectra of pair production in chirped pulses (1/2)...... 72 3.27. Spectra of pair production in chirped pulses (2/2)...... 73 3.28. Example spectra for dynamically assisted Schwinger pair production in co-rotating Sauter pulses...... 76 3.29. Example spectra for dynamically assisted Schwinger pair production in counter-rotating Sauter pulses...... 78 3.30. Pair production yield of dynamically assisted Schwinger pair production in rotating Sauter pulses...... 78 3.31. Particle yield in bichromatic rotating Sauter pulses...... 79 3.32. Spectra for a selection of bichromatic rotating Sauter pulses with varying relative phase φ ...... 80 3.33. Spectra of pair production in bichromatic linear Sauter pulses (1/2). 82 3.34. Spectra of pair production in bichromatic linear Sauter pulses (2/2). 83 3.35. Dynamically assisted Schwinger pair production in linear Sauter pulses 83 3.36. Transverse portion of the particle yield in bichromatic linear Sauter pulses...... 84 3.37. Example spectra from bichromatic linear Sauter pulses with transverse acceleration...... 85

4.1. Example spectra of fftwignerh without damping...... 93 4.2. Example spectra of fftwignerh with damping...... 94 4.3. Comparison of the time evolution with and without magnetic field.. 96 4.4. Pair production spectra for B = 0.01 Bcr...... 97

XX D. List of Symbols and Abbreviations

Symbols

µ, ν, α, β Greek indices running from 0 to 3 over spatial & time indices

i, j, k, l Latin indices running from 1 to 3 over spatial indices

#‰ i #‰ #‰ v = v ei Cartesian unit vectors ei and Einstein’s sum convention

a, b spinor indices

Lˆ operators in a Hilbert space carry a hat

ˆ ˆ (P Ψ)a = PabΨb spinor indices are also summed if they are on the same level W W ,pi = ∂pi short-hand comma notation for derivatives

1 identity matrix

ϵijk, ϵµναβ Levi-Civita pseudo tensor

ei·π + 1 = 0 Euler’s constant e and the imaginary unit i

ηµν Minkowski metric, ηµν = diag(1, −1, −1, −1)

γµ Dirac matrices, {γµ, γν} = 2ηµν

σi Pauli matrices

#‰ 3 #‰ dΓ = d3 x d p phase space measure dΓ (2π)3 #‰ #‰ #‰ E, B, A electric field, magnetic field and vector potential (Eq.(2.4)) #‰ q , qx,y,z canonical momentum #‰ #‰ #‰ #‰ p , px,y,z kinetic momentum, p = q − eA #‰ #‰ #‰ Dt , Dx , P pseudo-differential operators (Eq. (2.3)) XXI List of symbols and abbreviations

W Wigner function, 4 by 4 matrix, transforms as a Dirac matrix #‰ 1,2 s, p, vµ, aµ, t Wigner function components (Eqs. (2.5) and (2.7)) #‰ #‰ #‰ t in the spatially homogeneous case t = t 1

w vector containing the 10 components of the spatially homogeneous Wigner function in the purely electric case  #‰ #‰ #‰⊺ w = s v a t

W vector containing the 16 components of the Wigner function  #‰ #‰ #‰ #‰ ⊺ W = s p v0 a0 v a t 1 t 2

ϵ phase space energy density of the Dirac field (Eq. (2.13))

f one-particle distribution function (Eq. (2.16))

N total particle yield (Eq. (2.17))

Nxy particle yield for z = 0 plane (Eq. (2.18))

c speed of light, here c ≡ 1

e elementary charge, e ≈ 1.6 · 10−19 C

ℏ Planck’s constant, here ℏ ≡ 1

m mass of the electron, unit of mass and due to c = ℏ = 1 unit of momentum in the numerical calculations

ℏ 1 −21 tc Compton time tc = c2m = m ≈ 1.29·10 s, unit of time and due to c = 1 unit of length in numerical calculations

2 3 m c 18 V Ecr. critical field strength Ecr. = ≈ 1.3 · 10 /m eℏ ε amplitude of electric field in units of the critical field strength

Ω frequency of electric field pulse #‰ ω total energy of a particle with momentum p according to ω2 = #‰ m2 + p 2

τ pulse duration

XXII Abbreviations

γΩ Keldysh adiabaticity parameter regarding oscillation/rotation fre- Ω Ω quency Ω, γΩ = mc eE = εm (Eq. (3.35))

1 γτ Keldysh adiabaticity parameter for Sauter pulse, γτ = mc τ eE = 1 τεm (Eq. (3.36))

γ∗ combined Keldysh adiabaticity parameter for rotating fields, γ∗ =  2 1  π 1  2 εm 2 τ + Ω (Eq. (3.37))

ℜ (·) , ℑ (·) taking the real or imaginary part of a complex number

+ c.c. plus complex conjugated, z + c.c. := z + z∗ = 2ℜ (z).

Abbreviations

DHW Dirac-Heisenberg-Wigner (formalism)

QED Quantum electrodynamics

QKT Quantum kinetic theory

WKB Wentzel-Kramers-Brillouin (approach)

ODE ordinary differential equation

PDE partial differential equation

XXIII

E. Danksagungen

An dieser Stelle möchte ich allen danken, die mich bei der Erstellung dieser Disser- tation unterstützt haben. Eine besondere Stellung nimmt hier mein Betreuer Holger Gies ein, dem ich die Möglichkeit verdanke, an diesem spannenden Thema zu arbeiten. Ebenfalls erwähnt werden muss ebenfalls die Finanzierung durch die DFG in den Bereichen SFB/TR-18 und GRK 1523. Ohne die inspirierenden Gespräche mit den Kollegen wären viele Ideen so sicher nicht entstanden. Daher möchte ich meinen Dank insbesondere Julia Borchardt, Tom Dörffel, Michael Kahlisch, Steven Krause, Christian Kohlfürst, Julian Leiber, Stefan Lippoldt, David Schinkel und Nico Seegert aussprechen. Nicht zuletzt möchte ich auch meiner Familie für ihre Unterstützung danken und meinen Freunden, die für meine wunderbare Zeit in Jena während des Studiums und der Promotion gesorgt haben. Danke Luja.

XXV

F. Ehrenwörtliche Erklärung

Ich erkläre hiermit ehrenwörtlich, dass ich die vorliegende Arbeit selbstständig, ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfs- mittel und Literatur angefertig habe. Die aus anderen Quellen direkt oder indirekt übernommenen Daten und Konzepte sind unter Angabe der Quelle gekennzeichnet. Bei der Auswahl und Auswertung des Materials haben mir die nachstehend aufge- führten Personen unentgeltlich geholfen:

1. Prof. Dr. Holger Gies 2. Dr. Christian Kohlfürst

Weitere Personen waren an der inhaltlich-materiellen Erstellung der vorliegenden Arbeit nicht beteiligt. Insbesondere habe ich hierfür nicht die entgeltliche Hilfe von Vermittlungs- bzw. Beratungsdiensten (Promotionsberater oder andere Personen) in Anspruch genommen. Niemand hat von mir unmittelbar oder mittelbar geld- werte Leistungen für Arbeiten erhalten, die im Zusammenhang mit dem Inhalt der vorgelegten Dissertation stehen. Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörte vorgelegt. Die geltende Promotionsordnung der Physikalisch-Astronomischen Fakultät ist mir bekannt. Ich versichere ehrenwörtlich, dass ich nach bestem Wissen die reine Wahrheit gesagt und nichts verschwiegen habe.

Ort, Datum Unterschrift d. Verfassers

XXVII