# final dissertation

Dissertation submitted to the Combined Faculties of the Natural Sciences and Mathematics of the Ruperto-Carola-University of Heidelberg, Germany for the degree of Doctor of Natural Sciences Put forward by Sven Ahrens, M.Sc. born in Erfurt Oral examination: 14.11.2012 Investigation of the Kapitza-Dirac effect in the relativistic regime Referees: Honorarprof. Dr. Christoph H. Keitel Priv.-Doz. Dr. Andrey Surzhykov Zusammenfassung Quantenmechanische Beugungseffekte sind von besonderem Interesse, da sie unserer Alltagserfahrung widersprechen. Diese theoretische Arbeit befasst sich mit der Beugung von Elektronen an stehenden Lichtwellen, dem sogenannten Kapitza-Dirac Effekt. Ein besonderer Fokus wird dabei auf eine spezielle Variante des Kapitza-Dirac Effektes gelegt, in welcher das Elektron mit drei Photonen wechselwirkt. Eine besondere Eigenschaft dieses 3-Photonen Kapitza-Dirac Effektes ist, dass in diesem Fall der Spin des Elektrons bei der Beugung am optischen Gitter gedreht wird. Die theoretischen Rechnungen in dieser Arbeit basieren auf verschiedenen relativistischen und nicht-relativistischen quantenmechanischen Wellengleichungen, die im Impulsraum formuliert werden. Einerseits wird die quantenmechanische Dynamik der gebeugten Elektronen numerisch im Impulsraum gelöst, um die Eigenschaften des 3-Photonen Kapitza-Dirac Effektes detailliert herauszuarbeiten. Andererseits werden die Gleichungen mit zeitabhängiger Störungstheorie gelöst und den numerischen Ergebnissen gegenüber gestellt. Im Gegensatz zu der von Kapitza und Dirac vorgeschlagenen Elektronenbeugung unter Beteiligung zweier Photonen, ist die Anzahl der vom Elektron absorbierten und emittierten Photonen beim 3-Photonen Kapitza-Dirac Effekt nicht gleich groß. Aus diesem Grund findet der in dieser Arbeit diskutierte Beugungsvorgang nur für Elektronen mit einem relativistischen Impuls in Laserpropagationsrichtung statt. Zudem sind sehr hohe Laserfeldstärken nötig, um den Übergang mit einer messbaren Übergangswahrscheinlichkeit zu treiben. Der Spin des Elektrons wird beim Beugungsvorgang um die Magnetfeldachse des Laserstrahls gedreht, mit einem Drehwinkel, der vom Elektronenimpuls in Laserpolaristationsrichtung abhängt. Die Wahrscheinlichkeit für das Umklappen des Elektronenspins lässt sich durch die Wahl des Elektronenimpulses in Laserpolaristationsrichtung gezielt einstellen. Eine experimentelle Untersuchung der Vorhersagen kann mit zukünftigen Röntgenlasern erreicht werden. Summary Quantum mechanical diffraction is of particular interest, because it contradicts our everyday life experience. This theoretical work considers the diffraction of electrons at standing waves of light, referred to as the Kapitza-Dirac effect. The work focuses on a special version of a Kapitza-Dirac effect in which the electron interacts with three photons. The particular property of this 3-photon Kapitza-Dirac effect is, that the electron spin is rotated. This work considers different relativistic and non-relativistic quantum mechanical wave equations which are described in momentum space. On one hand, the quantum dynamics of the diffracted electrons is solved numerically in momentum space and the properties of the 3-photon Kapitza-Dirac effect are investigated in detail. On the other hand, the quantum dynamics is solved via time-dependent perturbation theory and is compared with the numerical results. In contrast to the originally proposed Kapitza-Dirac effect with two interacting photons, the number of absorbed and emitted photons by the electron is not equal for the 3-photon Kapitza-Dirac effect. Therefore, the diffraction process only appears for relativistic electron momenta in laser propagation direction. Furthermore, a very high field strength of the laser beam is required for driving the KapitzaDirac effect with a measurable diffraction probability. The electron spin is rotated along the axis of the magnetic field of the laser beam, when it undergoes the diffraction process. The rotation angle of the spin rotation depends on the electron momentum component in laser polarization direction. Therefore, the probability for flipping the electron spin can be tuned by choosing the electron momentum in the direction of the laser polarization. An experimental investigation may by established by utilizing future X-ray laser facilities. In connection with this thesis, the following article has been published in a refereed journal: • Sven Ahrens, Heiko Bauke, Christoph H. Keitel, and Carsten Müller. Spin dynamics in the Kapitza-Dirac effect. Physical Review Letters, 109(4):043601, (2012) Contents 1 Introduction and Motivation 1.1 The Kapitza-Dirac effect . . . . . . . . . 1.2 Recent progress in laser technology . . . 1.3 Synopsis . . . . . . . . . . . . . . . . . . 1.3.1 State of knowledge . . . . . . . . 1.3.2 New aspects treated in my work 1.3.3 Applied methods . . . . . . . . . 1.3.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 12 15 15 15 16 16 2 Basic concepts related to the Kapitza-Dirac effect 2.1 Physical setup, geometry and notation . . . . . . . . . . . . . . . . . . 2.2 Conservation of energy and momentum . . . . . . . . . . . . . . . . . 2.2.1 Diffraction of photons . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Diffraction of electrons . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Energy-momentum conservation by graphical considerations 2.2.4 Analytic derivation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 The number of absorbed and emitted photons . . . . . . . . . 2.2.6 The limit of small laser frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 17 18 19 19 20 21 22 23 3 Theoretical framework: Quantum wave equations 3.1 Schrödinger equation . . . . . . . . . . . . . . . 3.2 Pauli equation . . . . . . . . . . . . . . . . . . . 3.3 Klein-Gordon equation . . . . . . . . . . . . . . 3.4 Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 25 26 26 28 4 Quantum wave equations in momentum space 4.1 Exemplification by the Schrödinger equation 4.2 Pauli equation . . . . . . . . . . . . . . . . . . 4.3 Klein-Gordon equation . . . . . . . . . . . . . 4.4 Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 34 36 39 5 Properties of the 2-photon Kapitza-Dirac effect 5.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Realistic pulse shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 41 42 43 6 Electron spin dynamics: Conceptual considerations 6.1 The propagator . . . . . . . . . . . . . . . . . . . 6.2 Spin dependence of the diffraction pattern . . . 6.3 Spin rotation in Pauli theory . . . . . . . . . . . . 6.4 Spin rotation in Dirac theory . . . . . . . . . . . . 47 47 48 49 50 . . . . . . . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 7 Electron spin dynamics: Analytical small-time behavior 7.1 General procedure . . . . . . . . . . . . . . . . . . . 7.2 Perturbation Theory for the Pauli equation . . . . . 7.2.1 Derivation . . . . . . . . . . . . . . . . . . . . 7.2.2 Interpretation . . . . . . . . . . . . . . . . . . 7.3 Perturbation Theory for the Dirac equation . . . . . 7.3.1 Derivation . . . . . . . . . . . . . . . . . . . . 7.3.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 53 54 54 59 61 61 66 8 Electron spin dynamics: Numerical results 8.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Spin properties of the 3-photon Kapitza-Dirac effect . . . 8.3 Variation of the spin rotation . . . . . . . . . . . . . . . . 8.4 The beam envelope in the 3-photon Kapitza-Dirac effect . 8.5 The resonance peak . . . . . . . . . . . . . . . . . . . . . . 8.6 Rabi frequency of the 3-photon Kapitza-Dirac effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 69 71 75 75 78 80 9 Conclusions and Outlook 9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 85 86 . . . . . . . . . . . . . . A Bi-scalar matrix relations 87 B Bi-spinor matrix relations B.1 Calculation of bi-spinor contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Verification of spinor properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 89 91 C Energy-momentum conservation C.1 Non-relativistic energy-momentum conservation . . . . . . . . . . . . . . . . . . . . . . . C.2 Relativistic energy-momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . 93 93 94 Chapter 1 Introduction and Motivation 1.1 The Kapitza-Dirac effect The diffraction of light has been known since the 17th century [1] and gives evidence for the wave nature of light. Light diffraction was also observed at a double slit by Young [2] in 1803 and at a grating by Gregory [3] in 1673. The nature of light was under heavy dispute and with the advent of quantum mechanics at the beginning of the 20th century it was realized that light ought to be described by a duality of particle and wave. In 1926 Davisson, Germer [4] and Thomson [5] showed that also electrons are subject to diffraction and therefore, subject to particle wave duality as well. The attribute, that matter has the same diffraction properties as light leads unavoidably to the question: May the roles of light and matter be interchanged in a diffraction experiment? Which means: Can electrons be diffracted by a grating of light as it is shown in figure 1.1? Since the superposition of two counter-propagating, monochromatic, coherent light waves of equal intensity forms a wave which has periodical nodes and anti-nodes in space, light may act as grating. The question, if electrons may be diffracted by a grating of light was first discussed by Pyotr Kapitza and Paul Dirac in 1933 [6]. The corresponding expression “Kapitza-Dirac effect” became a synonym for the diffraction of electrons, atoms and molecules at a standing wave of light [7]. Kapitza and Dirac considered electron diffraction by using the standing wave of light of an “ordinary mercury arc lamp” and concluded a tiny diffraction probability of 10−14 in their publication in 1933. It seems, that electron diffraction has not been tackled because of this small probability, until the availability of lasers in the 1960s. A few years after the invention of the laser, attempts for observing the Kapitza-Dirac effect have been made [8, 9, 10], but these early experiments were not able to prove the existence of electron diffraction by light. The enormous progress in laser technology in recent years permitted for light beams of very high intensities (see next section) and new experiments have been set up. The first confirmation of the Kapitza-Dirac effect of atoms was published in 1986 [11, 12]. The Kapitza-Dirac effect in the case of electrons was first observed by Bucksbaum in 1988 in the so-called diffraction regime [13]. The terms diffraction regime and also the complementary Bragg regime are introduced in chapter 5. The identification of single diffraction orders could be achieved the first time by Freimund, Aflatooni and Batelaan in 2001 [14] in a precise experimental setup based on improved technological devices. 11 12 CHAPTER 1. INTRODUCTION AND MOTIVATION Diffraction pattern Laser Laser Electrons Figure 1.1: This is a schematic sketch of the Kapitza-Dirac effect. The two counter-propagating laser beams form a standing light wave. An electron beam crosses the standing light wave and is diffracted at the light. A diffraction pattern can be observed by placing a screen behind the interaction point. 1.2 Recent progress in laser technology Since the invention of the laser in the 1960s [15], huge progress was made by increasing the laser intensity. Techniques like chirped pulse amplification [16] allow for laser facilities with intensities of 2 · 1022 W/cm2 at wavelengths of 800 nm (HERCULES laser [17]). Laser facilities with similar intensities exist, for example the “petawatt high energy laser for heavy ion experiments” (PHELIX) with a peak intensity of 1021 W/cm2 [18]. Table top laser systems reach intensities of 1018 W/cm2 . Even stronger laser facilities are planned for the future, as it is the case for the “extreme light infrastructure” (ELI) [19] which will provide an intensity of 1025 W/cm2 in the optical regime. Coherent light beams with short wavelengths are of particular interest in this thesis, because the Kapitza-Dirac effect is investigated at very short wavelengths. The wavelength of laser light can be shortened by nonlinear laser-matter interaction, yielding higher harmonics of the laser light. This technique of high harmonic generation (HHG) may be realized by a plasma mirror. An example of an experimental realization is given in [20]. The authors report extreme ultra violet radiation with wavelengths of 50 nm up to 100 nm at intensities of 1 · 1011 W/cm2 . Even higher intensities may be reached, as it is proposed in [21]. This publication claims attosecond pulses with a duration of 84 as and 1016 photons per pulse. The photon’s energy ranges from 20 eV to 70 eV. Assuming an average photon energy of 45 eV yields a pulse energy of 0.072 J and therefore, a pulse power of 8.6 · 1014 W. An intensity of 1.1 · 1023 W/cm2 would be accessible if one was able to focus this laser down to a beam spot diameter of 1 µm without any losses due to optical components. Another source of coherent X-ray light of high intensity are free electron lasers. For example, the linac coherent light source (LCLS) at the Stanford linear accelerator center (SLAC) provides 2 keV X-ray laser light at intensities of 1018 W/cm2 [22]. The European X-ray free-electron laser (European XFEL) [23] will provide a coherent X-ray beam of even higher intensity in the near future. The projected peak power of the European XFEL is 80 GW at a maximum photon energy of 17.5 keV. The laser will reach a peak intensity of 5.2 · 1018 W/cm2 at the intended beam spot diameter of 70 µm. The properties of the mentioned laser systems are listed in table 1.1 and illustrated in the wavelength- CHAPTER 1. INTRODUCTION AND MOTIVATION 13 intensity diagram of figure 1.2. Table 1.1: This table shows the properties of existing and prospective lasers. The intensities of the listed lasers may be even higher if one was able to narrow the beam focus. laser type intensity wavelength photon energy existing lasers optical laser (HERCULES) 2 · 1022 W/cm2 800.0 nm 1.5 eV optical laser (PHELIX) 1 · 1021 W/cm2 1064.0 nm 1.2 eV optical tabletop 1 · 1018 W/cm2 800.0 nm 1.5 eV plasma mirror HHG 1 · 1011 W/cm2 72.9 nm 17.0 eV 18 2 free electron laser (LCLS at SLAC) 1 · 10 W/cm 0.6 nm 2.0 keV proposed lasers optical laser (ELI) 1 · 1025 W/cm2 800.0 nm 1.5 eV 23 2 plasma mirror HHG 1 · 10 W/cm 27.6 nm 45.0 eV free electron laser (European XFEL) 5 · 1018 W/cm2 70.9 pm 17.5 keV observed 2-photon Kapitza-Dirac effect [13] optical laser [24] 1·1011 W/cm2 532.0 nm 2.3 eV proposed 3-photon Kapitza-Dirac effect (section 8) free electron laser [25] 2·1023 W/cm2 0.4 nm 3.1 keV 1014 1016 1018 tabletop LCLS SLAC 1020 European XFEL HERCULES PHELIX 1022 1024 ELI 1026 V0 > mc2 proposed plasma mirror proposed 3-photon Kapitza-Dirac effect in this thesis λ < h̄/mc 1028 1030 101 102 103 104 105 106 Eγ /eV I/(W/cm2 ) Figure 1.2: This figure shows the laser parameters of table 1.1 on a wavelength versus intensity chart. Laser wavelengths shorter than the reduced Compton wavelength (λ < h̄/mc) are located in the upper gray region. Laser intensities whose amplitude of the ponderomotive potential (see section 5) is larger than the electron restmass energy V0 > mc2 are located in the lower right gray region. Relativistic effects are assumed to be large in the gray regions. 1012 observed 2-photon Kapitza-Dirac effect existing plasma mirror 1010 10−6 10−7 10−8 10−9 10−10 10−11 10−12 λ/m CHAPTER 1. INTRODUCTION AND MOTIVATION 14 CHAPTER 1. INTRODUCTION AND MOTIVATION 1.3 1.3.1 15 Synopsis State of knowledge Most of the theory of the Kapitza-Dirac effect is based on non-relativistic quantum mechanics with a ponderomotive potential (introduced in chapter 2.1) [26, 27]. The resulting differential equation can be solved in the diffraction regime using Bessel functions. Furthermore, an adiabatic laser-electron interaction of the Kapitza-Dirac effect is considered by Fedorov [28]. The Kapitza-Dirac effect in the Bragg regime is solved in detail by Efremov and Fedorov by using second order time-dependent perturbation theory [29, 30]. Gush and Gush account for even higher orders of time-dependent perturbation theory [31]. A non-perturbative treatment of the Kapitza-Dirac effect, which employs Volkov states of the Schrödinger equation, is presented in [32, 33, 34]. The Kapitza-Dirac effect is also discussed in a relativistic treatment by employing the Klein-Gordon equation [35, 36]. Particle statistics (Bosons or Fermions) in the case of the Kapitza-Dirac effect is considered by Sanco [37]. Freimund and Batelaan consider a spin-dependent interaction in the Kapitza-Dirac effect [38] and treat the electron as a point-like, non-relativistic particle with a magnetic moment and investigate its trajectory in two counterpropagating plane waves of different wavelengths. A quantum mechanical, non-relativistic treatment of spin-flips in the Kapitza-Dirac effect is presented by Leonard Rosenberg in 2004 [39]. The author solves the quantum dynamics of a non-relativistic particle in a quantized external laser field by using time-dependent perturbation theory and an approximation in the diffraction regime consisting of Bessel functions. Both studies find negligible small spin-effects in the interaction regime considered. 1.3.2 New aspects treated in my work In my work, I investigate the Dirac equation, and I in particular exploit the electron spin, which is an intrinsic property of the Dirac equation. I discuss a general condition for absorption and emission of a certain number of photons by requiring energy and momentum conservation of classical particles. The condition’s analytical description is combined with geometric considerations, which – so far – cannot be found in such a detail in literature. Furthermore, the quantum dynamics of the electron-light interaction is solved numerically without applying approximations. The numerical results are exact in this sense. The quantum dynamics of the full time-dependent Pauli equation and the full time-dependent Dirac equation is also solved with the method of time-dependent perturbation theory. The comparison of both – the numerical and the analytical solution – features two advantages: First, even though reasoned approximations are assumed in time-dependent perturbation theory, the validity of the perturbative results can be checked by comparison with the numerical results. Second, one can easily provide scaling laws for the numerical results from perturbation theory. In view of the tremendous progress of available laser intensities and frequencies, the question arises which properties of the Kapitza-Dirac effect appear in these extreme fields. The methods appearing in this thesis (numerical simulations and perturbation theory) can compare the full and exact relativistic and non-relativistic properties of the Kapitza-Dirac effect. In particular, the newly introduced 3-photon Kapitza-Dirac effect is a quantum mechanical setup, which is a relativistic setup by its intrinsic properties (which has no non-relativistic limit) that demands for very high field strengths of the external X-ray laser field. Additionally, the spin-flip of the diffraction process is described by the electron wave-function propagator and facilitates the conclusion, that the spin of the electron is rotated. I point out, that the resonance condition from energy and momentum conservation allows to tune the quantum dynamics, such that the spin-dependent coupling terms, which are usually weak, can be amplified by a suitable choice of parameters. CHAPTER 1. INTRODUCTION AND MOTIVATION 1.3.3 16 Applied methods I derive a resonance condition from energy and momentum conservation, which can be utilized for determining laser and electron parameters, such that the electron will undergo a diffraction process, in which it absorbs and emits a certain number of photons in a classical interaction picture. I transform the Schrödinger-, Pauli-, Klein-Gordon- and Dirac equation into momentum space, such that each of them reduces into a system of coupled, ordinary differential equations. I implement these differential equations in a numeric code and investigate in this way the electron quantum dynamics. I derive the short-time quantum dynamics of the 3-photon Kapitza-Dirac effect by applying timedependent perturbation theory to the Pauli- and Dirac equation and identify characteristic properties of the diffraction process from this analytic solution. In particular, I identify an SU (2) representation of the propagator of the quantum dynamics and compare it with the numerical results. 1.3.4 Structure of the thesis The second chapter introduces the external laser field of the standing wave of light and considers classical energy and momentum conservation of the electron by graphical means. The third chapter introduces the quantum mechanical wave equations, namely the Schödinger equation, the Pauli equation, the Klein-Gordon equation and the Dirac equation. All four equations are transformed into momentum space in chapter 4. The resulting system of coupled differential equations is of relevance in this work, since the numerical and perturbative results are based on these equations. The fifth chapter discusses the original 2-photon Kapitza-Dirac effect and demonstrates, that the numerical implementation of the wave equations in chapter 4 reproduce the theoretically known and experimentally realized quantum dynamics of the 2-photon Kapitza-Dirac effect well. The sixth chapter infers general properties about the spin dependence of the diffraction pattern and the rotation of the electron spin from the propagator of the electron wave function. The seventh chapter calculates the perturbative short-time solutions of the Dirac equation and the Pauli equation with the method of time-dependent perturbation theory. Characteristic properties of the diffraction process, like the Rabi frequency and the spin-flip probability are derived. The eighth chapter applies the numerical implementation of the Dirac equation to the quantum dynamics of the 3-photon Kapitza-Dirac effect. The SU (2) property of the propagator and the properties from time-dependent perturbation theory are verified. The resonance peak of the transition is also discussed. The appendix contains the derivation of bi-scalar properties of the Klein-Gordon equation, bi-spinor properties of the mode expanded Dirac equation, and the constraint equations resulting from energy- and momentum conservation. Chapter 2 Basic concepts related to the Kapitza-Dirac effect The first part of this chapter introduces the vector potential of the external laser field, which is used throughout this thesis. The corresponding electric and magnetic fields of the laser beam are discussed as well as the effective ponderomotive potential. The second part of this chapter considers energy and momentum conservation of classical particles, in a graphical and intuitive picture. Even though these conservation laws are pure classical properties, they are a useful criterion for determining laser frequency and initial electron momenta, such that quantum dynamics undergoes an n-photon Kapitza-Dirac effect (see chapters 5, 7 and 8). The geometrical origin of the corresponding resonance condition is elaborately discussed. 2.1 Physical setup, geometry and notation In the Kapitza-Dirac effect, the electron moves in a standing wave of light (see figure 2.1), which can be described by an infinitely extended vector potential of the form ~ ~ ~ (~x, t) = − A0 sin(~k L · ~x − ωt) + A0 sin(~k L · ~x + ωt) A 2 2 ~ ~ = A0 cos(k L · ~x ) sin(ωt) , (2.1a) (2.1b) where ω is the angular frequency of the wave and ~k L is its wave vector. Note, that a small arrow is placed on top of each vector in this thesis. The vacuum Maxwell equations imply, that ω equals ck L , with k L = |~k L | and λ L = 2π/k L . In some parts of this thesis, the wave vector ~k L is considered to be parallel to the x1 axis, which is the case for the numerical chapters 5, 8, the sections, which discuss the resonance condition from energy and momentum conservation 2.2.4, 2.2 and the low laser frequency approximation of the perturbative calculations at the end of the subsections 7.2.1 and 7.3.1. A general ~k L is used everywhere else. The names ‘left’ and ‘right’ are used for the − x1 and x1 direction for convenience in section 2.2. The polarization direction and amplitude of the external vector potential of ~ 0 . The vacuum Maxwell equations also imply, that A ~ 0 and ~k L the laser beam is denoted by the vector A ~0 are always orthogonal to each other. Apart from this orthogonality constraint, the vectors ~k L and A can be chosen freely. Note, that line (2.1a) explicitly denotes the two counter-propagating laser beams with their vector potential amplitude, whereas line (2.1b) shows the combined potential, in which time and space dependence factorizes in a product of two trigonometric functions. 17 18 CHAPTER 2. BASIC CONCEPTS RELATED TO THE KAPITZA-DIRAC EFFECT x3 ~pout λL x2 ~0 A x1 left right ~pin Figure 2.1: This figure shows the coordinate system, which we use for the description of the Kapitza-Dirac effect. ~ 0 and wavelength λ L . The electron beam has the initial The laser beam has the vector potential amplitude A momentum ~pin and may be detected with the final momentum ~pout . The vector potential (2.1) results in the electric and magnetic fields ~E(~x, t) = ~E0 cos(~k L · ~x ) cos(ωt) ~B(~x, t) = ~B0 sin(~k L · ~x ) sin(ωt) , (2.2a) (2.2b) with electric and magnetic field amplitude vectors ~ 0kL ~E0 = − A ~0 . ~B0 = −~k L × A (2.3a) (2.3b) In a quantum mechanical description of the Kapitza-Dirac effect the vector potential (2.1) enters into the corresponding wave equation of the electron motion. The solution of this equation is, in general, demanding because of the separate space and time dependencies of the standing wave potential. For non-relativistic electron dynamics based on the Schrödinger equation, it has been shown that the effect of the vector potential can be well approximated by a static scalar potential [40, 41]. This so-called ponderomotive potential originates from a separation of fast and slow motion of a classical electron in the electro-magnetic fields (2.2) and a time average over the fast motion. The ponderomotive potential is given by V (~x, t) = V0 cos2 ~k L · ~x (2.4) with the amplitude ~2 e2 A 0 , (2.5) 4mc2 following the notion of [27]. The potential (2.4) varies periodically in laser propagation direction with spacial period λ L /2. This periodic structure allows to interpret the standing light wave naturally as an optical grating. V0 = 2.2 Conservation of energy and momentum The electron has the initial momentum ~pin and the final momentum ~pout . The vector ~k = ~pin /h̄ is used later in favor of a compact notation. This section considers the conservation of energy and momentum in the Kapitza-Dirac effect, by making the assumption that the electron with initial momentum ~pin absorbs an integer number of 19 CHAPTER 2. BASIC CONCEPTS RELATED TO THE KAPITZA-DIRAC EFFECT θn p1,out~e1 ~pin = p2,in~e2 g p2,in~e2 nλ θn sin θn = nλ g tan θn = (a) p1,out p2,in (b) Figure 2.2: The left subfigure shows the geometric condition for light of wavelength λ being diffracted at the angle θn at a grating with spacing period g. The right figure shows the the same diffraction process in momentum space. electrons and emits an integer number of photons, yielding the final electron momentum ~pout . It is worth to start with a general consideration on light diffraction first. 2.2.1 Diffraction of photons If light of wavelength λ impinges at a grating with spacial period g, it is diffracted at angles, which fulfill the condition (see figure 2.2(a)) sin θn = nλ/g , (2.6) for wavelengths λ, which are much smaller than the grating spacing g. The Compton effect [42] tells, that the in-falling photon has a momentum of p2,in = 2πh̄/λ . (2.7) If the light was detected at a small angle θn , when it passes the grating, it must have gathered the momentum p1,out = p2,in tan θn ≈ p2,in θn ≈ 2πnh̄/g (2.8) in the direction of the grating spacing (see figure 2.2(b)). Therefore, it stands to reason, that a grating with period g imposes multiples of momenta 2πh̄/g at in-falling photons. 2.2.2 Diffraction of electrons Figure 2.2 also holds for electrons and equation (2.7) is just de Broglie’s relation for the wave-particle duality of a massive particle [43]. Therefore, a standing light wave with period λ L is supposed to transfer multiples of momenta 2πh̄ = h̄k L (2.9) λL as well. In fact, Freimund and Batelaan observed a diffraction pattern at multiples of 2h̄k L in their experiment [14]. The same property shows up for the discrete momenta h̄k L in the mode expansion in chapter 4. Kapitza and Dirac also assumed a transfer of two photon momenta h̄k L in their proposal [6] of the Kapitza-Dirac effect (see figure 2.3). According to that publication, the electron should incline at the Bragg angle, which means that the incident electron should have a momentum of one h̄k L in laser propagation direction. The electron is reflected when it interacts with the laser, yielding an outgoing momentum of h̄k L in the opposite laser propagation direction. CHAPTER 2. BASIC CONCEPTS RELATED TO THE KAPITZA-DIRAC EFFECT E(~p) mc2 20 3 2 1 -2 0 -1 1 2 p1 mc Figure 2.3: This figure shows the Kapitza-Dirac effect in an electron energy over the electron momentum diagram, which is already conceived in [27]. In the case of the original publication of the Kapitza-Dirac effect [6], one photon is absorbed from the laser field and one photon is emitted to the laser field. The bended line shows the relativistic energy-momentum relation (2.11) of the electron. The interacting photons transfer an energy of ch̄k L and a momentum of h̄k L and therefore appear as diagonal lines in the figure. The total exchange of energy and momentum of the electron with the laser is represented by the dashed arrow. The electron interacts with two photons in the case of the originally published Kapitza-Dirac effect. Therefore, this originally published Kapitza-Dirac effect is referred to as 2-photon Kapitza-Dirac effect in this work. The question occurs, if other numbers of interacting photons are allowed by energy and momentum conservation. And, if this generalized version of the Kapitza-Dirac effect was possible: For what initial electron momenta and what laser frequencies does it occur? What are the final momenta of the electrons after the Kapitza-Dirac effect? 2.2.3 Energy-momentum conservation by graphical considerations These questions can be answered by a simple geometric argument. For simplicity, this geometric argument is discussed with an electron moving in x1 direction first. This means p2 and p3 are assumed to be zero. The general case, which includes non-vanishing momenta in x2 and x3 direction requires a minor modification of the geometric argument, which is discussed at the end of this subsection. Assume, the electron absorbed n a photons from the left laser beam and emits ne photons into the right laser beam, with n a , ne ∈ Z. Negative n a corresponds to photon emission to the left and negative ne corresponds to photon absorption from the right. Since a photon has the energy ch̄k L and the momentum h̄k L , the total transferred energy is ∆E = ch̄k L (n a − ne ) and the total transferred momentum is ∆p = h̄k L (n a + ne ). In case of figure 2.3, n a and ne are 1. Therefore the dashed arrow, which illustrates the totally transferred energy and momentum in figure 2.3 is horizontal. In fact, the slope of the dashed vector only depends on n a and ne by s= ∆E n a − ne =c . ∆p n a + ne (2.10) The relativistic energy momentum relation is taken as basis in the following considerations. It relates the kinetic energy E of an electron with restmass m and its momentum p1 by the relation q E( p1 ) = m2 c4 + p21 c2 . (2.11) The geometric argument works as follows: 1. Draw the relativistic energy momentum (2.11) relation in the electrons energy over time diagram, as it is done in figure 2.3. 21 CHAPTER 2. BASIC CONCEPTS RELATED TO THE KAPITZA-DIRAC EFFECT E(~p) mc2 3 2 1 p1,lim -2 -1 0 1 2 p1 mc Figure 2.4: This figure shows the same energy over momentum diagram as in figure 2.3. The difference to figure 2.3 is, that n a is 2 instead of 1. Therefore the dashed line of the total energy and momentum transfer has the slope s = c/3. The tangent line with slope c/3 at the dispersion relation consists of only one touching point with the initial and final electron momentum p1,lim . 2. Draw a line with slope s, which should intersect the relativistic energy momentum relation. 3. The intersection points of this secant are the initial and final momenta of the diffracted electron. The angular laser frequency can be obtained by dividing the transferred momentum by the number of photons ∆pc ω= . (2.12) h̄(n a + ne ) Figure 2.3 shows the geometric argument, in the case of the 2-photon Kapitza-Dirac effect, in which n a = 1, ne = 1 and s = 0. Another example of the geometric argument is shown in figure 2.4, in which the electron absorbs two photons n a = 2 and emits one photon ne = 1. Therefore, the slope is c/3 in this example. It remains to tell the modification of the geometric argument from above in the case of non-vanishing electron momenta perpendicular to the laser propagation direction. This corresponds to p2 6= 0 or p3 6= 0 or both. In this case one may formally replace the electron rest mass m with the increased mass q m0 = c−2 m2 c4 + c2 p22 + c2 p23 (2.13) in equation (2.11). The modified energy momentum relation q E( p1 ) = m02 c4 + c2 p21 (2.14) is reparameterized but identical to that of figures 2.3 and 2.4, if one replaces E/(mc2 ) by E/(m0 c2 ) and p1 /(mc) by p1 /(m0 c). Therefore, the geometric argument can be traced back to the situation with vanishing p2 and p3 . Note, that equation (2.14) is the same relativistic energy momentum relation as equation (3.22). This work will refer to the ladder equation from now on. 2.2.4 Analytic derivation The condition, of the initial electron momentum ~p1,in and the laser frequency ω, for an n-photon Kapitza-Dirac effect need not to be determined graphically. One may also derive a formula for these parameters from energy and momentum conservation, which is done in the appendix C. Energy conservation implies, that the final electron energy has to be the initial electron energy plus the energy of CHAPTER 2. BASIC CONCEPTS RELATED TO THE KAPITZA-DIRAC EFFECT 22 the absorbed photons, minus the energy of the emitted photons, yielding the equation E(~pout ) = E(~pin ) + n a h̄ω − ne h̄ω . (2.15) Similarly, momentum conservation implies, that the final electron momentum is the initial electron momentum plus the momentum of the absorbed photons from the left laser beam plus the momentum of photons, which are emitted into the right laser beam. c~pout = c~pin + n a h̄ω~e1 + ne h̄ω~e1 (2.16) If one inserts equation (2.16) into equation (2.15), one obtains an equation, which puts exactly one constraint on the four parameters h̄k L , p1,in , p2,in and p3,in of classical particles. The property of this resonance condition is, that one may freely choose three of these parameters. The fourth parameter however, must fulfill the combined equations (2.15), (2.16), otherwise no n-photon Kaptiza-Dirac diffraction will occur. One may, for example, solve both equations for the initial electron momentum in laser propagation direction and obtains s p1,in |n a − ne | h̄2 ω 2 n a + ne h̄ω 1 ± =− . (2.17) + 0 0 2 0 2 4 mc 2 2 n a ne mc m c One may analogously solve for the laser frequency, resulting in the dimensionless energy q c2 p21,in + m02 c4 p h̄ω 1 . = −(n a + ne ) 1,in ± |n a − ne | 0 0 2 0 2 mc 2n a ne mc mc (2.18) The computation for equation (2.17) and (2.18) is performed in appendix C.2. It should be mentioned, that the two solutions (2.17) and (2.18) are resulting from the relativistic energy momentum relation (3.22). In the case of the non-relativistic energy momentum relation (3.4), the corresponding solution of the system of equations (2.15) and (2.16) results in and p1,in n a − ne n a + ne h̄ω + =− , mc 2 n a + ne mc2 (2.19) h i p1,in h̄ω 2 = −( n + n ) + n − n ( ) a e a e mc mc2 ( n a + n e )2 (2.20) as it is also computed in appendix C.1. Note, that the equations (2.17) and (2.19) as well as the equations (2.18) and (2.20) differ from each other for n a 6= ne , because they are based on a different energy-momentum relation E(~k). The relativistic resonance condition (2.17) or (2.18) is applied for relativistic quantum wave equations, whereas the non-relativistic resonance condition (2.19) or (2.20) is applied for non-relativistic quantum wave equations. The important consequence, which is drawn out of this property is, that the simulation parameters for the relativistic and the non-relativistic quantum wave equations can never be exactly the same. Either the laser frequency or the initial electron momentum must be slightly different, if one switches between relativistic and non-relativistic quantum wave equations, for Kaptiza-Dirac scattering with n a 6= ne . 2.2.5 The number of absorbed and emitted photons The slope of the relativistic energy momentum dispersion relation is always in the interval ] − c, c[ , or in other words ∂E(~p) −c < <c ∀ p1 ∈ R . (2.21) ∂p1 CHAPTER 2. BASIC CONCEPTS RELATED TO THE KAPITZA-DIRAC EFFECT 23 Table 2.1: This table shows, for what values of n a and ne condition (2.22) is fulfilled. ‘and’ is a logical conjunction. All possible combinations of n a and ne are covered in this table. na na na na na na >0 <0 >0 <0 =0 ∈Z and and and and and and ne ne ne ne ne ne >0 <0 <0 >0 ∈Z =0 ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ condition (2.22) is fulfilled condition (2.22) is fulfilled condition (2.22) is not fulfilled condition (2.22) is not fulfilled condition (2.22) is not fulfilled condition (2.22) is not fulfilled Since the relativistic energy momentum dispersion relation is convex, only secants and tangents with a slope in this interval can be drawn at this dispersion relation. And since the slope is related to equation (2.10), n a and ne have to fulfill the condition n a − ne < 1. n a + ne −1 < (2.22) Table 2.1 shows for what values of n a and ne this condition is fulfilled and for what values it is not fulfilled. Note, that the cases with n a = 0 and ne 6= 0 or n a 6= 0 and ne = 0 contain the well known property, that an electron never scatters only at one photon, because of conservation of energy and momentum. One can further conclude that the Kapitza-Dirac effect only takes place, if photons are absorbed and emitted during the interaction process with the laser, which means, that both n a and ne must be positive or that both must be negative. Therefore, pure photon emission and pure photon absorption not allowed for Kapitza-Dirac scattering. Note, that the two allowed cases imply, that all photon momenta are transferred in one direction only. Either the electron only gathers photon momenta to the left (photon absorption from the left beam and photon emission into the right beam) or it only gathers photon momenta to the right (photon absorption from right beam and photon emission into the left beam). This is the reason, why the laser angular frequency in equation (2.12) can be obtained by dividing the transferred momentum by n a + ne . 2.2.6 The limit of small laser frequencies Figure 2.4 shows one unique point with momentum p1,lim , which is the tangent point at the dispersion relation with slope s. Parallel translation of the secant changes the size (but not the angles) of the triangle, formed by the dashed and solid arrows in figure 2.3 and 2.4. This means, that parallel translation of the secant towards the tangent line of the tangent point at p1,lim decreases the length of the dashed arrow. Decreasing the length of the dashed arrow implies in turn, that the momentum transfer ∆p and the angular frequency ω of the external laser field decrease too. The touch point of the tangent with momentum p1,lim at the energy-momentum relation marks therefore an unphysical transition, in which the laser frequency would be zero and the initial and final momentum would coincide. One may derive p1,lim by solving the condition ! s= ∂E( p1 ) ∂p1 (2.23) for p1 . The reason is, that the derivative of E( p1 ) with respect to p1 is the slope of the bended line at the momentum p1 in figures 2.3 and 2.4. The condition (2.23) is therefore the analytical formulation of the question “At what momentum p1 has the bended line in figures 2.3 and 2.4 the slope s?” The solution of (2.23) with respect to p1 is n a − ne 0 p1,lim = √ m c, 4n a ne (2.24) CHAPTER 2. BASIC CONCEPTS RELATED TO THE KAPITZA-DIRAC EFFECT 24 where p1 is denoted as p1,lim . In the case of the two-photon Kapitza-Dirac effect [6], ne and n a are 1, √ yielding p1,lim = 0. The case n a = 2, ne = 1 of figure 2.4 yields p1,lim = m0 c/ 8. If the secant in figures 2.3 and 2.4 is parallel translated towards the tangent, initial and final electron momentum gets closer to p1,lim and the laser frequency gets smaller. In the limiting case of an infinite small laser frequency, initial and final electron momenta will be infinitely close to p1,lim . This means, that the electron is nearly at rest, in the case of the 2-photon Kapitza-Dirac effect of figure 2.3. Chapter 3 Theoretical framework: Quantum wave equations Since the Kapitza-Dirac effect is a diffraction process, it requires a quantum mechanical description. Therefore, four different quantum wave equations are introduced in this chapter, which are the Schrödinger equation (section 3.1), the Pauli equation (section 3.2), the Klein-Gordon equation (section 3.3) and the Dirac equation (section 3.4). In the next chapter, all these quantum wave equations are transformed into momentum space. In the subsequent chapters, these transformed equations are applied for studying the quantum dynamics of the Kapitza-Dirac effect. All quantum wave equations have in common, that they can be written in the form ih̄ ∂ ψ(~x, t) = Ĥψ(~x, t) , ∂t (3.1) where ψ(~x, t) is the quantum mechanical wave function, whose time evolution is determined by the Hamiltonian Ĥ. Note, that all symbols, which are set in bold in this thesis have the structure of a n × n matrix, where n (n ∈ N) is the dimension of the wave function ψ(~x, t) in equation (3.1). The hat over a symbol means, that it contains spacial derivatives. Since the different quantum wave equations are determined by their Hamiltonians, it is sufficient to discuss these characteristic Hamiltonians in the following. 3.1 Schrödinger equation The Hamiltonian of the Schrödinger equation is [44] 1 ˆ e ~ 2 ~p − A + V . Ĥ = 2m c (3.2) The wave function of the Schrödinger equation has only one component. Therefore, the Hamiltonian ~ or the ponderomotive Ĥ it is not set in bold font. One may insert the vector potential (2.1) as A potential (2.4) as external potential V of the standing light wave in the Schrödinger equation. A plane wave with initial momentum ~pin = h̄~k has the wave function ~ ψ~k (~x ) = eik·~x . (3.3) The eigenvalue of the non-relativistic energy-momentum relation Enr (~k) = h̄2~k2 2m results, if the wave function is applied to the Hamiltonian. 25 (3.4) CHAPTER 3. THEORETICAL FRAMEWORK: QUANTUM WAVE EQUATIONS 3.2 26 Pauli equation The Hamiltonian of the Pauli equation is [44] Ĥ = 1 ˆ e ~ 2 eh̄ ~σ · ~B , ~p − A 1 + V1 − 2m c 2mc where ~σ is the vector of the Pauli matrices 0 1 0 σ1 = , σ2 = 1 0 i −i 0 , σ3 = 1 0 (3.5) 0 −1 , (3.6) and 1 is the identity matrix in two dimensions. Therefore, the wave function of the Pauli equation has two components, which are coupled only by the Pauli term ~σ · ~B at the right hand side of equation (3.5). The other part, which is proportional to the identity 1 is the same as the Schrödinger equation (3.2). Spinors are used to encode the two components of the Pauli equation. Spinors consist of two components 1 0 uP,1 = and uP,2 = . (3.7) 0 1 The plane wave function of the Pauli equation is similar to the wave function of the Schrödinger equation except the additional spin component uP,σ . ~ ψ~P,σ (~x ) = uP,σ eik·~x , σ ∈ {1, 2} . k (3.8) These two wave functions are degenerate with respect to the free Pauli equation, which means, that both have the eigen energy h̄2~k2 . (3.9) Enr (~k) = 2m In contrast to the eigen energy, the spinors (3.7) have different spin eigenvalues ( +h̄/2 , if σ = 1 Sσ = (3.10) −h̄/2 , if σ = 2 with respect to the third component of the Pauli spin operator h̄ P ~ S = ~σ . 2 (3.11) This thesis uses the additional index assignment • σ = 1 corresponds to spin up ↑ • σ = 2 corresponds to spin down ↓ . 3.3 Klein-Gordon equation The Schrödinger and Pauli equations are non-relativistic quantum wave equations, which are invariant under Galilei transformations. Since this thesis focuses on relativistic phenomena in the Kapitza-Dirac effect, relativistic quantum wave equations which are invariant under Lorentz transformations are required. The simplest, manifest covariant object for a relativistic quantum wave equation is the KleinGordon equation in covariant form [45]. " # 2 ∂ e ~ 2 2 ˆ 2 4 ih̄ − eΦ − c ~p − A − m c Υ = 0 (3.12) ∂t c CHAPTER 3. THEORETICAL FRAMEWORK: QUANTUM WAVE EQUATIONS 27 One may rewrite this equation, in order to fit it into the form of equation (3.1). According to [45], one introduces two variables φ and χ with the requirement ∂ Υ = φ+χ and ih̄ − eΦ Υ = mc2 (φ − χ) . (3.13) ∂t From these two requirements it follows, that φ and χ are related to the function Υ by ∂ 1 2 mc + ih̄ − eΦ Υ, φ= ∂t 2mc2 1 ∂ 2 χ= mc − ih̄ + eΦ Υ. ∂t 2mc2 (3.14) (3.15) Equation (3.12) can be rewritten by making use of the equations (3.13), (3.14) and (3.15). By defining the two component wave function φ ψ= , (3.16) χ the Hamiltonian of equation (3.1) transforms to Ĥ = σ 3 + iσ 2 ˆ e ~ 2 ~p − A + σ 3 mc2 + V1 , 2m c (3.17) where eΦ has been replaced by V. This equation is referred to as the Klein-Gordon equation of Hamiltonian form. Note, that the Hamiltonian (3.17) is non-Hermitian, because of the iσ 2 term. The Hamiltonian is split into a free part Ĥ 0 = (σ 3 + iσ 2 ) ~pˆ 2 + σ 3 mc2 (3.18) and an interaction part V̂ = (σ 3 + iσ 2 ) ~ · ~pˆ ~2 eA e2 A − + mc 2mc2 ! (3.19) + 1V for later convenience. The eigenfunctions of the free Klein-Gordon equation are ~ ψ~KG,σ (~x ) = uKG,σ (~k )eik·~x , σ ∈ {1, 2} , (3.20) k with the bi-scalars [45] u KG,1 (~k) = mc2 + E(~k ) mc2 − E(~k) ! , u KG,2 (~k) = mc2 − E(~k ) mc2 + E(~k ) ! , where E(~k) is the relativistic energy-momentum relation q E(~k) = m2 c4 + c2 h̄2~k2 . (3.21) (3.22) Since bi-scalars are no longer degenerate with respect to their eigen energy, they are functions of the wave vector ~k. Their different energy eigen values are ( + E(~k) , if σ = 1 Eσ (~k) = (3.23) − E(~k) , if σ = 2 , which is calculated in appendix A. This thesis uses the additional index assignment • σ = 1 corresponds to positive eigen energy + CHAPTER 3. THEORETICAL FRAMEWORK: QUANTUM WAVE EQUATIONS 28 • σ = 2 corresponds to negative eigen energy − . The bi-scalars may be written in a compact matrix notation, by employing the coefficients ~ dKG + (k) = −1 q mc2 + E(~k) 2 E(~k)mc2 and ~ dKG − (k) = q −1 2 E(~k)mc2 mc2 − E(~k) . (3.24) With these coefficients, the bi-scalars of the eigenfunctions of the free Klein-Gordon equation are the columns of the matrix KG ~ KG ~ ~ uKG (~k)† = dKG (k) . (3.25) + ( k )1 + d − ( k ) σ 1 = u It should be mentioned, that bi-scalars are not orthogonal to each other. One may need the pseudo orthogonal properties of bi-scalars [45] for later calculations. 0 uKG,σ (~k)σ 3 uKG,σ (~k ) = sign(σ)δσ,σ0 (3.26) The sign function returns the sign of its index: sign(1) = + , sign(2) = − . In particular, the scalar product (4.5), which is introduced in section 4.1 needs to be exchanged by hψa | ψb i = Z π/k L −π/k L ψa (~x )† σ 3 ψb (~x )dxk . (3.27) in the case of the Klein-Gordon equation. 3.4 Dirac equation The Dirac equation is defined by using Dirac matrices, which have to fulfill the algebra α j αk + αk α j = 2δjk 1 , (3.28a) α j β + βα j = 0 , (3.28b) ββ = 1 . (3.28c) One realization of this algebra is the standard representation 0 0 0 1 0 0 0 1 0 0 0 σ 0 σ1 2 α2 = α1 = = = 0 1 0 0 , 0 σ1 0 σ2 0 1 0 0 0 i 1 0 0 1 0 0 0 0 −1 0 0 σ3 1 0 α3 = = β= = 1 0 0 0 , 0 σ3 0 0 −1 0 −1 0 0 0 0 0 0 i −i 0 0 0 0 1 0 0 −i 0 , 0 0 0 0 0 0 , −1 0 0 −1 which is employed in this thesis. In terms of these matrices, the Dirac Hamiltonian is [46] e ~ ·~α + V1 + mc2 β . Ĥ = c ~pˆ − A c (3.29) (3.30) The Hamiltonian is split into a free part Ĥ 0 = c~pˆ ·~α + mc2 β (3.31) ~ ·~α + V1 V = −e A (3.32) and an interaction part 29 CHAPTER 3. THEORETICAL FRAMEWORK: QUANTUM WAVE EQUATIONS for later convenience. The eigenfunctions of the free Dirac equation can be denoted by ~ ψ~σ (~x ) = uσ (~k)eik·~x , σ ∈ {1, 2, 3, 4} , (3.33) k where the uσ (~k) are bi-spinors. This thesis follows the convention of [46] for the introduction of bispinors. In a first step, the coefficients 1 d+ (~k) = √ 2 mc2 1+ E(~k ) !1 2 and h̄c d− (~k) = √ 2 1 E(~k)( E(~k) + mc2 ) !1 2 (3.34) are defined, where E(~k) is the relativistic energy-momentum relation (3.22). After that, these coefficients enter in the definition of the matrix u(~k ) := d+ (~k )1 + d− (~k ) β~α ·~k d+ 0 0 d + = (−k3 )d− (−k1 + ik2 )d− (−k1 − ik2 )d− (+k3 )d− (+k3 )d− (+k1 + ik2 )d− d+ 0 (+k1 − ik2 )d− (−k3 )d− 0 d+ (3.35) (−k1 + ik2 )d− (+k3 )d− . 0 d+ (3.36) and its adjoint matrix u(~k)† = d+ (~k )1 + d− (~k)~k ·~αβ = d+ (~k )1 − d− (~k) β~α ·~k d+ 0 (−k3 )d− 0 d (− k1 − ik2 )d− + = (+k3 )d− (+k1 − ik2 )d− d+ (+k1 + ik2 )d− (−k3 )d− 0 The bi-spinors uσ (~k ) are defined as the columns of this adjoint matrix. u(~k)† = u1 (~k), u2 (~k), u3 (~k), u4 (~k) (3.37) The energy eigen values of bi-spinors are ~ + E(k) + E(~k) Eσ (~k) = − E(~k) − E(~k) , if σ , if σ , if σ , if σ =1 =2 =3 = 4, (3.38) which is calculated in appendix (B.2). The spin eigen values of the bi-spinors are +h̄/2 −h̄/2 Sσ = +h̄/2 −h̄/2 , if σ , if σ , if σ , if σ =1 =2 =3 = 4, (3.39) with respect to the third component of the spin operator, which has been suggested by Foldy and Wouthuysen [47, 48, 49] h̄ ~ S FW = u(~pˆ )† ~Σ u(~pˆ ) , (3.40) 2 CHAPTER 3. THEORETICAL FRAMEWORK: QUANTUM WAVE EQUATIONS 30 where, for dimensional reasons the factor h̄ must be taken out of the relativistic energy-momentum relation (3.22) and the definition of the d− coefficient (3.34) for the operator version u(~pˆ ). The definition of this relativistic spin operator (3.40) makes use of the non-relativistic spin operator ~Σ −iα2 α3 ~ σ 0 ~Σ = = −iα3 α1 . (3.41) 0 ~σ −iα1 α2 This thesis uses the additional index assignment • σ = 1 corresponds to positive eigen energy with spin up + ↑ • σ = 2 corresponds to positive eigen energy with spin down + ↓ • σ = 3 corresponds to negative eigen energy with spin up − ↑ • σ = 4 corresponds to negative eigen energy with spin down − ↓ . It should be mentioned that reference [46] uses the coefficient 1 d˜− (~k ) = √ 2 mc2 1− E(~k) !1 2 (3.42) instead of d− (~k) of equation (3.34). One may rewrite d˜− (~k) by multiplying nominator and denominator with ( E(~k) + mc2 )1/2 , yielding 1 d˜− (~k) = √ 2 ( E(~k))2 − m2 c4 E(~k)( E(~k ) + mc2 ) !1 2 = |~k|d− (~k) . (3.43) From this transformation one may conclude, that the matrix ~k ũ(~k) = d+ (~k)1 + d˜− (~k) β~α · , |~k| (3.44) defined in [46] is equivalent to the matrix u(~k) of equation (3.35). The usage of the d− (~k) coefficients avoids the division by the factor |~k| in the calculation of the matrix (3.44). Since the avoidance of singularities at ~k = 0 appears to be more stable for numerical applications, the coefficients d− (~k) and u(~k ) are favored over d˜− (~k) and ũ(~k ). Chapter 4 Quantum wave equations in momentum space Even though the Kapitza-Dirac effect may be discussed by solving the quantum dynamics from first principles, i.e. by numerically implementing the equations of motion presented in section 3, one may solve these equations of motion with less effort, by rewriting them into a system of coupled ordinary differential equations in momentum space. The transformation of the quantum wave equations in chapter 3 from position into momentum space is performed in this chapter. Note, that the notion in momentum space is commonly used in literature of the Kapitza-Dirac effect. The transformation into momentum space is performed in the same order, as in chapter 3, which means, that this chapter starts with the Schrödinger equation 4.1, then discusses the Pauli equation 4.2, then the Klein-Gordon equation 4.3 and finally treats the Dirac equation 4.4. The first section in this chapter also contains an introductory explanation, of how the quantum wave equations are transformed into momentum space. The resulting, transformed equations are used for numerically replicating the 2-photon KapitzaDirac effect in chapter 5, for solving the Kapitza-Dirac effect with time-dependent perturbation theory in chapter 7 and for a numerical treatment of the 3-photon Kapitza-Dirac effect in chapter 8. 4.1 Exemplification by the Schrödinger equation All quantum wave equations can be transformed into momentum space by inserting the plane wave expansion (Fourier transform) of the wave-function in the quantum wave equation and projecting it with solutions of the free Hamiltonian of the quantum wave equation. In order to perform the transformation into momentum space, one might start out with the general ansatz for the wave-function ψ(~x ) = Z ∞ ~ d3 k ψ̃(~k)eik·~x (4.1) d3 x ψa (~x )† ψb (~x ) , (4.2) −∞ and use the general scalar product hψa | ψb i = Z ∞ −∞ for the computation of the projection. However, since the external vector potential (2.1) as well as ~ the external ponderomotive potential (2.4) contains multiples of plane waves eink L ·~x in every quantum wave equation, the generalized scalar product will result in delta spikes δ(~k,~k + n~k L ), n ∈ Z. The momentum integral of equation (4.1) will turn these delta spikes into a system of coupled differential 31 32 CHAPTER 4. QUANTUM WAVE EQUATIONS IN MOMENTUM SPACE equations of the subset ψ̃(~k + n~k L ), n ∈ Z. Therefore, the basis functions r k L i(~k+n~k L )·~x ψn (~x ) = e , 2π (4.3) with the plane wave ansatz ψ(~x ) = ∑ cn ψn (~x) = r kL 2π ~ (4.4) dxk ψa (~x )† ψb (~x ) (4.5) n ~ ∑ cn ei(k+nk L )·~x n and the scalar product hψa | ψb i = Z π/k L −π/k L are sufficient for rewriting the quantum wave equations into momentum space. The integration element dxk in equation (4.5) denotes, that the integration is performed in the ~k L direction. In the case of the Schrödinger equation (3.2) with the ponderomotive potential (2.4) ih̄ψ̇ = ~pˆ 2 ψ + V0 cos2 ~k L · ~x ψ , 2m (4.6) the projection with basis element ψn from the left would read + * ~pˆ 2 D E hψn | ih̄ | ψ̇i = ψn ψ + ψn V0 cos2 ~k L · ~x ψ . 2m (4.7) Each of the three terms can be computed separately. The time derivative term at the left-hand side results in hψn | ih̄ | ψ̇i = ih̄ Z π/k L k L −i(~k+n~k L )·~x e −π/k L 2π ~ ~ k L 2π ∑ ċa ei(k+ak L )·~x dxk = ih̄ 2π k L ∑ δn,a ċa = ih̄ċn . a a (4.8) The kinetic term at the right-hand side of equation (4.7) is rewritten into + Z * ~pˆ 2 π/k L k pˆ 2 ~ ~ L −i(~k +n~k L )·~x ~ c a ei(k+ ak L )·~x dxk ψn e ψ = ∑ 2m 2π 2m −π/k L a =∑ a h̄2 (~k + a~k L )2 h̄2 (~k + n~k L )2 δn,a c a = cn . 2m 2m (4.9) The ponderomotive coupling term at the right-hand side of equation (4.7) transforms into D k L −i(~k+n~k L )·~x V0 i2~k L ·~x ~ ~ ~ e e + 2 + e−i2k L ·~x ∑ c a ei(k+ak L )·~x dxk 2π 4 −π/k L a V0 V0 = (4.10) (δn,a+2 + 2δn,a + δn,a−2 ) c a = (cn−2 + 2cn + cn+2 ) . 4 ∑ 4 a E Z ψn V0 cos2 ~k L · ~x ψ = π/k L Plugging back the projections (4.8), (4.9) and (4.10) into equation (4.7) yields The Schrödinger equation with ponderomotive potential in momentum space ih̄ċn = V0 V0 h̄2 (~k + n~k L )2 cn + cn + ( c n −2 + c n +2 ) , 2m 2 4 (4.11) 33 CHAPTER 4. QUANTUM WAVE EQUATIONS IN MOMENTUM SPACE which is very similar to what Batelaan found in 2000 [27]. The numerical implementation of equation (4.11) requires a matrix notation of the Hamiltonian of equation (4.11), which is Ha,b = h̄2 (~k + a~k L )2 V0 V0 δa,b + δa,b + (δ + δa,b−2 ) . 2m 2 4 a,b+2 This can be checked by calculating ∑ Ha,b cb . (4.12) (4.13) b The result is identical with the right-hand side of equation (4.11). Tedious arrays of equations are resulting, if one expands the projection in equation (4.7) in every detail. Those arrays of equations have always the same structure and provide almost no insight about important physics to the reader. Therefore, the always reoccurring calculation steps are discussed here and are not shown later. Assume an operator diagonal in position space which, by the above considerations, is of the form ~ V (~x ) = ∑ a Va eiak L ·~x . Since the scalar product is linear, one may write it as the sum of all plane wave terms E D ~ (4.14) hψn | V (~x ) | ψi = ∑ ψn Va eiak L ·~x ψ . a The calculation of each term can be carried out separately. The expanded scalar product reads D E Z ~ ψn Va eiak L ·~x ψ = π/k L ~ ψn (~x )† Va eiak L ·~x ψ(~x )dxk r Z π/k L r k L −i(~k+n~k L )·~x kL ia~k L ·~x e Va e = 2π 2π −π/k L (4.15a) −π/k L ~ ~ ∑ cb ei(k+bk L )·~x dxk . (4.15b) b This can be rearranged and constants can be pulled out of the integral, where the integral reduces to a delta function. Z π/k L E D k ~ ~ ψn Va eiak L ·~x ψ = Va ∑ cb L ei(a+b−n)k L ·~x dxk = Va ∑ cb δb,n− a (4.16) 2π −π/k L b b The sum over b results in D E ~ ψn Va eiak L ·~x ψ = Va cn− a . (4.17) Resubstituting this into the expansion (4.14) yields hψn | V (~x ) | ψi = ∑ Va cn−a . a (4.18) In summary, one may perform the projection (4.14) by expanding the operator diagonal in position ~ space into plane waves and replace each plane wave eiak L ·~x by cn− a . On the other hand, operators occur which are diagonal in momentum space. Assume a func~ . This momentum operator acts on the plane wave tion V̂ (~pˆ ) of the momentum operator ~pˆ = −ih̄∇ ~k+b~k L )·~x i ( e , which by derivation turns into the number V (h̄(~k + b~k L )) in equation (4.15b). r E Z π/k L r k D kL ~ ~ L −i(~k +n~k L )·~x ˆ ~ ~ e cb ei(k+bk L )·~x dxk (4.19) ψn V̂ (~p) ψ = V (h̄(k + bk L )) ∑ 2π 2π b −π/k L The integral along dxk yields a delta function again, which in turn fixes the summation index b of the sum. E D ψn V̂ (~pˆ ) ψ = ∑ V (h̄(~k + b~k L ))cb δb,n = V (h̄(~k + n~k L ))cn (4.20) b CHAPTER 4. QUANTUM WAVE EQUATIONS IN MOMENTUM SPACE 34 In summary, one may compute the projection with an operator diagonal in momentum space, by ~ of the momentum operator with the wave vector i(~k + n~k L ), replacing the vector of derivatives ∇ where n and ~k are properties of the projecting basis element ψn . An expansion coefficient cn has to be multiplied at the converted operator. Operators, which are neither diagonal in position space, nor diagonal in momentum space may also occur. In all cases of this work those operators will appear as a product of position space operators and momentum space operators. Furthermore, all operators and basis elements in this work occur in a configuration, which allows to treat position and momentum space operators independently of each other according to the rules (4.18) and (4.20). For example, the scalar product of the operator ~ 0 (ei~k L ·~x + e−i~k L ·~x ) · ~pˆ turns into A E D ~ ~ i~k L ·~x ~ 0 (cn−1 + cn+1 ) · h̄(~k + n~k L ) = A ~ 0 · h̄~k(cn−1 + cn+1 ) . (4.21) + e−ik L ·~x ) · ~pˆ ψ = A ψn A 0 (e 4.2 Pauli equation The spinor eigenfunctions (3.8) are employed as basis elements, for rewriting the Pauli equation with Hamiltonian (3.5) into momentum space. Since the Pauli equation is a partial differential equation with two components, the basis elements consist of two components r k L P,σ i(~k+n~k L )·~x P,σ ψn (~x ) = u e (4.22) 2π with the two spinors uP,σ of equation (3.7). The expansion (4.4) of the wave function is extended by including an additional spinor index to the expansion coefficients. r kL ~ ~ σ P,σ cσn uP,σ ei(k+nk L )·~x (4.23) ψ(~x ) = ∑ cn ψn (~x ) = ∑ 2π n,σ n,σ The transformation of the Pauli equation into momentum space makes use of the linear property of the scalar product (4.5). This means, that each term of the Hamiltonian (3.5) and the wave equation (3.1) may be contracted separately with the projecting basis elements (4.22) and the wave function (4.23). The first term of interest is the time derivative of the wave function ∂ 0 0 0 ∂ ∂ (4.24) ψnP,σ ih̄ ψ = ih̄ ∑ uP,σ † uP,σ cσn = ih̄ ∑ δσ,σ0 ċσn = ih̄ċσn , ∂t ∂t ∂t σ0 σ0 σ0 where the spinor u originates from the wave function |ψi of equation (4.23) and the adjungated spinor uσ † originates from the projecting basis element hψnP,σ | of equation (4.22). The next term is the ~ )2 1/(2m) of the Pauli Hamiltonian (3.5), which can be expanded gauge invariant derivative (~pˆ − e A/c into ! ~ ~ · ~pˆ ~2 ~pˆ 2 1 ˆ e ~ 2 e~pˆ · A eA e2 A ~p − A 1 = − − + 1 2m c 2m 2mc 2mc 2mc2 ! ~ ) eA ~ · ~pˆ ~2 ~pˆ 2 e(~pˆ · A e2 A − − + 1. (4.25) = 2m 2mc mc 2mc2 ~ vanishes, because the divergence of the vector potential (2.1b) is zero. The term proportional to ~pˆ · A The mode expansion of the kinetic term proportional to ~pˆ 2 results in + * ~pˆ 2 0 0 0 0 h̄2 (~k + n~k L )2 h̄2 (~k + n~k L )2 h̄2 (~k + n~k L )2 σ ψnP,σ 1ψ = uP,σ † 1uP,σ cσn = uP,σ † uP,σ cσn = cn . ∑ ∑ 2m 2m 2m 2m σ0 σ0 (4.26) 35 CHAPTER 4. QUANTUM WAVE EQUATIONS IN MOMENTUM SPACE ~ · ~pˆ is similar to the example in equation (4.21), The mode expansion of the term proportional to A because the plane wave expansion of the vector potential (2.1b) is ~ ~ (~x, t) = A0 ei~k L ·~x + ei~k L ·~x sin(ωt) . A 2 (4.27) Therefore, one may write in analogy to equation (4.21) ! + * ~ 0 · h̄~k sin(ωt) σ ~ · ~pˆ eA eA P,σ 1ψ = − cn−1 + cσn+1 . ψn − mc 2mc (4.28) The last term of equation (4.25) contains the squared vector potential, which by expansion into plane waves reads as ~2 ~ (~x, t)2 = A0 ei2~k L ·~x + 2 + e−i2~k L ·~x sin2 (ωt) . (4.29) A 4 The mode expansion of this term yields + * 2~2 ~ 2 sin2 (ωt) e2 A P,σ e A 0 (4.30) 1ψ = cσn−2 + 2cσn + cσn+2 . ψn 2 2 2mc 8mc Since the scalar potential φ(~x, t) is zero, the external potential V = eφ is zero too. It remains the Pauli term, which is proportional to ~σ · ~B and is the last term in the Pauli Hamiltonian (3.5). The plane wave expansion of the magnetic field (2.2b) is ~ 0 ei~k L ·~x − e−i~k L ·~x sin(ωt) . ~B(~x, t) = i ~k L × A 2 (4.31) Therefore, the free state mode expansion of the Pauli term reads h i 0 eh̄ sin(ωt) eh̄ P,σ σ0 ~k L × A ~ 0 · uP,σ †~σuP,σ0 cσn− ~ ~σ · B ψ = −i ψn − ∑ 1 − c n +1 . 2mc 4mc σ0 (4.32) 0 Since the spinors uP,σ are the canonical unit vectors, the contraction uP,σ †~σuP,σ of the Pauli matrices with the spinors is the σth row and the σ0 th column of the Pauli matrices ~σ. Together with the sum over σ0 , one may write this as a matrix product of ~σ with the vector ! c↑n cn = . (4.33) c↓n This means the equality ∑0 σ h i 0 0 uP,σ †~σuP,σ cσn = ∑0 [~σ ]σ,σ 0 0 cσn = [~σcn ]σ (4.34) σ holds. Note, that the right-hand side of equation (4.34) makes use of the component vector (4.33), which is multiplied at ~σ. With this identity, equation (4.32) can be written as eh̄ sin(ωt) ~ eh̄ P,σ ~ 0 ·~σ (cn−1 − cn+1 ) . ~ ~σ · B ψ = −i kL × A (4.35) ψn − 2mc 4mc Adding up all terms of the Pauli equation (4.24), (4.26), (4.28), (4.30) and (4.35) yields CHAPTER 4. QUANTUM WAVE EQUATIONS IN MOMENTUM SPACE 36 The Pauli equation in momentum space # ~2 ~2 e2 A e2 A h̄2 (~k + n~k L )2 2 0 0 ih̄ċn = sin ( ωt ) c + sin2 (ωt) (cn−2 + cn+2 ) + n 2m 4mc2 8mc2 i eh̄ sin(ωt) h ~ ~ ~ 0 · (i~σ ) cn−1 + 2 A ~ 0 ·~k 1 −~k L × A ~ 0 · (i~σ ) cn+1 , − 2 A0 · k 1 +~k L × A 4mc " (4.36) which is expressed completely in terms of the two component vector (4.33) of the expansion coefficients cσn . The corresponding Hamiltonian matrix of this system of differential equations is H 0;a,b = h̄2 (~k + n~k L )2 1δa,b 2m (4.37) for the free Hamiltonian and V a,b = ~2 ~2 e2 A e2 A 2 0 0 sin ( ωt ) 1δ + sin2 (ωt)1 (δa,b+2 + δa,b−2 ) a,b 4mc2 8mc2 i eh̄ sin(ωt) h ~ ~ ~ 0 · (i~σ ) δa,b+1 + 2 A ~ 0 ·~k 1 −~k L × A ~ 0 · (i~σ ) δa,b−1 − 2 A0 · k 1 +~k L × A 4mc (4.38) for the interaction Hamiltonian. 4.3 Klein-Gordon equation Since the Klein-Gordon equation consists of two coupled partial differential equations, its basis elements include the two component bi-scalars (3.21), analogously to the Pauli equation. r k L KG,σ ~ ~ ~ KG,σ ψn (~x ) = u (k + n~k L )ei(k+nk L )·~x . (4.39) 2π However, the difference between bi-scalars and spinors is that bi-scalars are not canonical basis vectors and that bi-scalars depend on the wave vector ~k + n~k L . The expansion of the wave function includes bi-scalars and a corresponding bi-scalar index of the expansion coefficients. r kL ~ ~ σ KG,σ ψ(~x ) = ∑ cn ψn (~x ) = (4.40) ∑ cσn uKG,σ (~k + n~k L )ei(k+nk L )·~x 2π n,σ n,σ The ~k + n~k L dependence of the basis elements (4.39) must be considered by performing the free state ~ mode expansion. If one starts out with the mode expanded operator V a eiak L ·~x of equation (4.15), which has a matrix structure in the case of the Klein-Gordon equation and follows the steps of calculation in section 4.1, one recognizes that the step (4.16) is not affected by the bi-scalar components. However, in equation (4.17) the sum over the index b fixes the index of the wave vector argument ~k + b~k L of the bi-scalar. E D h i 0 ~ ψnKG,σ V a eiak L ·~x ψ = ∑ uKG,σ (~k + n~k L )V a uKG,σ (~k + b~k L ) cb δb,n− a b,σ0 h i 0 = ∑ uKG,σ (~k + n~k L )† V a uKG,σ (~k + (n − a)~k L ) cn−a (4.41) σ0 0 Since bi-scalars uKG,σ and uKG,σ † are the σ0 th column and the σth row of the uKG † and uKG matrices of equation (3.25), the contraction of the matrix V a with the two bi-spinors corresponds to the matrix 37 CHAPTER 4. QUANTUM WAVE EQUATIONS IN MOMENTUM SPACE element at the σ0 th column and the σth row of the matrix product uKG V a uKG † . This means, that the equality h iσ,σ0 0 (4.42) uKG,σ (~k )† V a uKG,σ (~k0 ) = uKG (~k)V a uKG (~k0 )† 0 holds, where [ M ]σ,σ is the σ0 th column and the σth row of a matrix M. The following results of the transformation into momentum space are expressed in terms of matrix entries of the matrix uKG V a uKG † . For a compact, analytic expression one may define the functions KG ~ KG ~ 0 ~ KG ~ 0 tKG (~k,~k0 ) = dKG + (k )d+ (k ) + d− (k )d− (k ) , (4.43a) KG ~ KG ~ 0 ~ KG ~ 0 sKG (~k,~k0 ) = dKG + (k )d− (k ) + d− (k )d+ (k ) , (4.43b) KG KG ~ KG ~ 0 ~ KG ~ 0 f (~k,~k0 ) = dKG + (k )d+ (k ) − d− (k )d− (k ) , KG ~ KG ~ 0 ~ KG ~ 0 rKG (~k,~k0 ) = dKG + (k )d− (k ) − d− (k )d+ (k ) . (4.43c) (4.43d) Matrix products, involving 1, σ 1 and σ 3 can be written as bi-scalar contractions uKG (~k)uKG (~k0 )† = tKG (~k,~k0 )1 + sKG (~k,~k0 )σ 1 , u KG u KG KG ~0 † KG ~0 KG (4.44a) ~0 (~k)σ 1 u (k ) = s (~k, k )1 + t (~k, k )σ 1 , (~k)σ 3 uKG (~k0 )† = f KG (~k,~k0 )σ 3 + rKG (~k,~k0 )iσ 2 (4.44b) (4.44c) and are derived in appendix A. The property, that bi-scalars are pseudo orthonormal (see the end of section 3.3) must be accounted for. The pseudo scalar product (3.27), of the basis elements (4.39) results in D E h iσ,σ0 0 ψaKG,σ σ 3 ψbKG,σ = uKG (~k + a~k L )σ 3 uKG (~k + b~k L )† δa,b h iσ,σ0 0 = uKG (~k + a~k L )σ 3 uKG (~k + a~k L )† δa,b = [σ 3 ]σ,σ δa,b . (4.45) The last equality makes use of equation (A.5). Therefore, each projection of the Klein-Gordon equation with the basis elements (4.39) must include a σ 3 at the left-hand side, which turns the object hψnσ | σ 3 | into the dual basis element of |ψnσ i. The computation of the mode expansion can be divided in projections of each term of equation (3.1) and (3.17), similar to the procedure of the Pauli equation. The time-derivative term in equation (3.1) can be transformed into h iσ,σ0 0 0 0 ∂ KG,σ KG ~ KG ~ † ~ ~ ψn σ ih̄ ψ = ih̄ u ( k + n k ) σ u ( k + n k ) ċσn = ih̄ ∑ [σ 3 ]σσ ċσn . (4.46) 3 3 L L ∑ ∂t σ0 σ0 The next term is the free Hamiltonian (3.18) of the Klein-Gordon Hamiltonian (3.17), which turns into the relativistic energy momentum relation. E D h iσ,σ0 0 ψnKG,σ σ 3 Ĥ 0 ψ = ∑ uKG (~k + n~k L )σ 3 H 0 (~k + n~k L )uKG (~k + n~k L )† cσn = E(~k + n~k L )cσn (4.47) σ0 The last equality made use of equation (A.9). The relativistic energy momentum relation E(~k + n~k L ) may be abbreviated by En = E(~k + n~k L ) . (4.48) The mode expansion of the interaction Hamiltonian (3.19) is similar to the mode expansion of the Pauli equation and results in D E ψnKG,σ V̂ ψ = · ∑ b,σ0 h uKG (~k + n~k L ) (1 + σ 1 ) uKG (~k + b~k L )† iσ,σ0 ~ 2 sin2 (ωt) ~ · h̄~k sin(ωt) e2 A eA 0 − 0 (δb,n−1 + δb,n+1 ) + (δb,n−2 + 2δb,n + δb,n+2 ) 2mc 8mc ! 0 cσb , (4.49) CHAPTER 4. QUANTUM WAVE EQUATIONS IN MOMENTUM SPACE 38 where a vanishing scalar potential φ is assumed and the mode expansion of the vector potential (4.27) and the squared vector potential (4.29) are used. The matrix uKG (~k + n~k L ) (1 + σ 1 ) uKG (~k + b~k L )† may be simplified to (4.50) uKG (~k + n~k L ) (1 + σ 1 ) uKG (~k + b~k L )† = t(~k,~k0 ) + s(~k,~k0 ) (1 + σ 1 ) . The sum t(~k,~k0 ) + s(~k,~k0 ) simplifies to mc2 t(~k,~k0 ) + s(~k,~k0 ) = d+ (~k) + d− (~k) d+ (~k0 ) + d− (~k0 ) = q . E(~k) E(~k0 ) (4.51) Therefore, one may write mc2 uKG (~k + n~k L ) (1 + σ 1 ) uKG (~k + b~k L )† = √ [1 + σ 1 ] . En Eb (4.52) The transformation of the Klein-Gordon equation into momentum space is the sum of the equations (4.46), (4.47) and (4.49). But the time-derivative of the expansion coefficients with a negative index c− n will be negative compared to the time-derivative of the expansion coefficients with positive index c+ n. It makes sense to demand, that the time-derivatives all have the same sign. Therefore, the equations (4.46), (4.47) and (4.49) are multiplied by σ 3 with respect to the index σ. The resulting system of coupled ordinary differential equations is The Klein-Gordon equation in momentum space ~ 0 · ch̄~k sin(ωt) [iσ 2 + σ 3 ]cn−1 [iσ 2 + σ 3 ]cn+1 eA √ + √ ih̄ċn = En σ 3 cn − 2 En En−1 En En+1 2 2 2 ~ e A0 sin (ωt) [iσ 2 + σ 3 ]cn−2 2[iσ 2 + σ 3 ]cn [iσ 2 + σ 3 ]cn+2 √ + + + √ , 8 En En En−2 En En+2 if the two component vector cn = c+ n c− n (4.53) (4.54) of the expansion coefficients is used. The matrix structure of the [iσ 2 + σ 3 ] matrix is [iσ 2 + σ 3 ] σ,σ0 = σ ↓ σ0 → 1 1 −1 −1 ! . (4.55) The matrix entries of the Hamiltonian of the Klein-Gordon equation (4.53) are H a,b ~ 0 · ch̄~k sin(ωt) [iσ 2 + σ 3 ] eA [iσ 2 + σ 3 ] √ = Ea σ 3 δa,b − δa,b+1 + √ δa,b−1 2 Ea Ea −1 Ea Ea +1 ~ 2 sin2 (ωt) [iσ 2 + σ 3 ] e2 A 2[iσ 2 + σ 3 ] [iσ 2 + σ 3 ] 0 √ δa,b+2 + δa,b + √ δa,b−2 . + 8 Ea Ea Ea −2 Ea Ea +2 (4.56) 39 CHAPTER 4. QUANTUM WAVE EQUATIONS IN MOMENTUM SPACE 4.4 Dirac equation The free state mode expansion of the Dirac equation is analogous to that of the Klein-Gordon equation. The functions (3.33) serve as basis elements r kL σ ~ ~ ~ σ u (k + n~k L )ei(k+nk L )·~x (4.57) ψn (~x ) = 2π with the four bi-spinors (3.37). The advantage of bi-spinors in contrast to bi-scalars is, that bi-spinors are orthonormal and not pseudo orthonormal. The expansion of the wave function with respect to these basis elements is r kL ~ ~ σ σ ψ(~x ) = ∑ cn ψn (~x ) = (4.58) ∑ cσn uσ (~k + n~k L )ei(k+nk L )·~x . 2π n,σ n,σ Similar to equations (4.41) and (4.42) the transformation into momentum space of the Dirac equation can be written in terms of matrix products, for which the following functions are defined. t(~k,~k0 ) = d+ (~k )d+ (~k0 ) +~k ·~k0 d− (~k)d− (~k0 ) , l ~0 s (~k, k ) = r l (~k,~k0 ) = lq w (~k,~k0 ) = glq (~k,~k0 ) = (4.59a) ~0 k l d− (~k)d+ (k ) + k0l d+ (~k)d− (~k0 ) , k l d− (~k)d+ (~k0 ) − k0l d+ (~k)d− (~k0 ) , k l k0q d− (~k )d− (~k0 ) + k q k0l d− (~k)d− (~k0 ) , k l k0q d− (~k )d− (~k0 ) − k q k0l d− (~k)d− (~k0 ) , (4.59b) (4.59c) (4.59d) (4.59e) hl (~k,~k0 ) = ~el · ~k ×~k0 d− (~k)d− (~k0 ) (4.59f) These functions enter in the bi-spinor contractions u(~k)u(~k0 )† = t(~k,~k0 )1 + ∑ r l (~k,~k0 ) βαl + l ∑ 1≤ l < q l < q ≤3 glq (~k,~k0 )αl αq , (4.60a) ch̄k l mc2 β− αl , ~ E(k) E(~k) u(~k)αl u(~k0 )† = t(~k,~k0 )αl − ∑ wlq (~k,~k0 )αq + sl (~k,~k0 ) β u(~k) βu(~k)† = (4.60b) q q + ∑ r (~k, k ) βαq αl + hl (~k,~k0 )α1 α2 α3 , ~0 (4.60c) q6=l which are derived in the appendix B.1. The time projection turns into h iσ,σ0 0 ∂ σ ψn ih̄ ψ = ih̄ ∑ u(~k + n~k L )u(~k + n~k L )† ċσn = ih̄ċσn , ∂t σ0 (4.61) by using the orthonormality relation (B.16). The free Hamiltonian of the Dirac equation (3.31) results in the relativistic energy momentum relation ψnσ Ĥ 0 ψ = ∑0 h u(~k + n~k L ) H 0 (~k + n~k L )u(~k + n~k L )† σ iσ,σ0 0 0 0 cσn = E(~k + n~k L ) ∑ [ β]σ,σ cσn , (4.62) σ0 according to equation (B.18). The interaction Hamiltonian (3.32) with the vector potential (2.1b), in the form (4.27) and vanishing scalar potential Φ results in hψnσ | V | ψi = − e sin(ωt) 2 ∑0 b,σ h iσ,σ0 0 ~ 0 ·~α u(~k + b~k L ) u(~k + n~k L )† A (δn,b−1 + δn,b+1 ) cσb . (4.63) CHAPTER 4. QUANTUM WAVE EQUATIONS IN MOMENTUM SPACE 40 Adding up (4.61), (4.62) and (4.63) yields The Dirac equation in momentum space e sin(ωt) ~ ~ 0 ·~α u(~k + (n − 1)~k L )cn−1 u(k + n~k L )† A 2 e sin(ωt) ~ ~ 0 ·~α u(~k + (n + 1)~k L )cn+1 , − u(k + n~k L )† A 2 ih̄ċn = En βcn − (4.64) with the four component vector of expansion coefficients +↓ −↑ −↓ T cn = c+↑ . n , cn , cn , cn (4.65) The corresponding Hamiltonian matrix of this system of differential equations is H 0;a,b = Ea βδa,b (4.66) for the free Hamiltonian and e sin(ωt) ~ ~ 0 ·~α u(~k + b~k L )δa,b+1 u(k + a~k L )† A 2 e sin(ωt) ~ ~ 0 ·~α u(~k + b~k L )δa,b−1 − u(k + a~k L )† A 2 V a,b = − for the interaction Hamiltonian. (4.67) Chapter 5 Properties of the 2-photon Kapitza-Dirac effect In this chapter, the quantum dynamics of the 2-photon Kapitza-Dirac effect is solved by numerical application of the quantum wave equations in chapter 4. The chapters 6, 7 and 8 introduce and consider properties of the 3-photon Kapitza-Dirac effect and are based on the contents of this chapter. There are two simulation scenarios in this chapter: One scenario applies the Pauli equation 4.2 and reproduces the experiment, which has been carried out by Batelaan [50]. In the second scenario, the laser frequency and the laser intensity are substantially higher than in [50]. As a result, the quantum dynamics is faster in the second scenario and can be simulated by numerically solving the KleinGordon equation and the Dirac equation. The last section of this chapter discusses the influence of long turn on and long turn off times of the external laser field on the diffraction process. 5.1 Setup The simplest application of the derived wave equations in momentum space of chapter 4 is the 2photon Kapitza-Dirac effect for the parameters of its first experimental verification in the Bragg regime [50]. The optical wavelength of 532 nm translates in the wave number k L = 4.6 · 10−6 mc/h̄ for the laser and the kinetic energy of 380 eV translates in the momentum of about 0.039 mc for the electron. From the intensity of 3 · 108 W/cm2 one computes a ponderomotive amplitude (see equation (2.5)) of 7.5 · 10−12 mc2 . Since the electron inclines almost perpendicularly at the laser beam, one can use a geometry, in which the wave vector of the laser is ~k L = 4.6 · 10−6 mc/h̄ ~e1 and the electron momentum component perpendicular to the laser beam is h̄~k = 0.039 mc ~e3 . It remains to determine the small electron momentum in laser propagation direction from energy and momentum conservation, as it is discussed in section 2.2. According to the considerations in this section, the 2-photon Kapitza-Dirac effect should occur, if one photon is absorbed from the left laser beam and one photon is emitted into the right laser beam. Equation (2.19) tells, that the initial electron momentum must be minus one photon momentum. Therefore, the initial momentum vector of the incoming electron is ~pin = h̄(~k −~k L ) and the final momentum of the outgoing electron is ~pout = h̄(~k +~k L ). The initial momentum corresponds to the initial quantum state, in which the expansion coefficient c↑−1 (0) is 1.0 and all other expansion coefficients are zero at time 0. The choice of setting the c↑−1 (0) coefficient to one and the c↓−1 (0) coefficient to zero implies that the electron spin points in the x3 -direction, initially. 41 42 CHAPTER 5. PROPERTIES OF THE 2-PHOTON KAPITZA-DIRAC EFFECT diffraction probability 1.0 0.8 |c↑−1 |2 0.6 cos2 (Ω R T/2) |c1↑ |2 0.4 sin2 (Ω R T/2) 0.2 0.0 0.0 0.5 1.0 1.5 interaction time T/s 2.0 2.5 ×10−9 Figure 5.1: This figure shows the quantum mechanical time-evolution of the Kapitza-Dirac effect by integrating the Pauli equation (4.36). The result is directly compared with the analytical result (5.1) from Batelaan [50]. The simulation parameters are consistent with the first experimental demonstration of the Kapitza-Dirac effect by Freimund and Batelaan [50]. The parameters in [50] are a laser intensity of 3.0 · 108 W/cm2 , a laser wave length of 532 nm, an electron momentum of 19.7 keV/c and an interaction time in the range of nanoseconds, whereas the amplitude of the external laser field is turned on and turned off in ten laser cycles in the simulation. 5.2 Rabi oscillations The time evolution of the discussed initial quantum state is shown in figure 5.1, where the absolute square of the coefficients c↑−1 and c1↑ is plotted. Note, that the amplitude of the external vector potential is turned on and turned off by a sine shaped envelope of ten laser cycles for each data point in figure 5.1, according to equation (5.4). The turn on and turn off of the external field is discussed in detail in section 5.3. The absolute squares of all other coefficients than c↑−1 and c1↑ are negligibly small. This is consistent with the property, that the sum of |c↑−1 |2 and |c1↑ |2 is one, because the time-evolution is unitary. Figure 5.1 also shows, that the diffraction probability oscillates in Rabi cycles according to ↑ 2 2 ΩR T |c−1 | = cos (5.1a) 2 ΩR T . (5.1b) |c1↑ |2 = sin2 2 This property has been clearly pointed out in 1971 by Gush and Gush [31]. The Rabi frequency ΩR = V0 2h̄ (5.2) is proportional to the ponderomotive amplitude, according to Batelaan [27]. Equation (5.2) yields the frequency Ω R = 3.75 · 10−12 mc2 /h̄ for the ponderomotive amplitude V0 = 7.50 · 10−12 mc2 . The corresponding Rabi period of 2.16 ns is consistent with the simulation in figure 5.1. The experiment [50] also agrees with the numerical solution of the Pauli equation in figure 5.1. Note, that the negligibly small spin-flip probability |c1↓ |2 implies, that the quantum dynamics of the 2-photon Kapitza-Dirac effect does not affect the electron spin. CHAPTER 5. PROPERTIES OF THE 2-PHOTON KAPITZA-DIRAC EFFECT 43 Even though, the interaction time of one nanosecond appears short compared to the speed of human response, it is a long time compared to one laser period and an even longer time compared to the oscillation period of the electron phase in the relativistic quantum wave equations. In the example of figure 5.1, one Rabi cycle consists of 1.2 · 106 laser cycles and 2.7 · 1011 periods of the electron phase in the complex plane. A quantum dynamical simulation should resolve these oscillations appropriately, which means, that a simulation of the Pauli equation (4.36) requires at least one million time steps and a simulation with the Klein-Gordon or Dirac equation (4.53),(4.64) requires at least one trillion time steps. It seems that the Klein-Gordon and Dirac equation in the form (4.53),(4.64) are not suited for simulation parameters with optical light and low intensities. Therefore, the main part of this work will consider standing light waves of high frequencies in the X-ray regime with much higher intensities than used in the experiment [50]. The resonance conditions (2.17) and (2.19) of energy and momentum conservation imply, that the electron is always diffracted from momentum −k L (or k L ) to k L (or −k L respectively) in laser propagation direction in the case of the 2-photon Kapitza-Dirac effect. This means, the 2-photon KapitzaDirac effect would occur for every laser photon momentum h̄k L , if the initial electron momentum was ~pin = h̄(~k0 ± ~k L ). This means in turn, that the electron must approach the laser beam at the Bragg angle. The laser photon momentum k L = 0.05 mc/h̄ with a corresponding photon energy of 25.55 keV seems to fit well for a high but still non-relativistic photon energy. The ponderomotive amplitude of the external potential needs to be adjusted to the laser frequency. The amplitude should be as high as possible, such that one Rabi period is short and the number of time steps for simulations with relativistic quantum wave equations is short. On the other hand, the uncertainty in transition energy of the 2-photon Kapitza-Dirac effect should be larger than the energy spacing of different energy eigen values of the free Hamiltonian. This requirement led Batelaan to the condition h̄ (5.3) ∆E T 2 for the so-called “Bragg regime” [27], where ∆E is the recoil shift h̄2 k2L /(2m) and T is one Rabi period. The opposite case would be the so-called “Diffraction regime”. If one chooses a ponderomotive amplitude of V0 = 2.0 · 10−5 mc2 , the inequality of condition (5.3) turns into 157.1 1 and the quantum dynamics corresponds to the Bragg regime. The duration of one Rabi cycle reduces to 0.77 fs due to the higher amplitude of the ponderomotive potential. Figure 5.2 shows a simulation of the 2-photon Kapitza-Dirac effect with the new parameters. Like in figure 5.1, the data points from the simulation with the Pauli equation fit to the analytical solution (5.1). The same holds for the simulation data of the Klein-Gordon and Dirac equation. The spin-flip probability is negligibly small again. 5.3 Realistic pulse shape The turn on and turn off time of the external vector potential is only 10 laser cycles for the quantum dynamics in the sections 5.1 and 5.2. This is a very short time compared to the full interaction time T. The advantage of this short turn on and short turn off time is, that the time evolution is numerically easier to compute and that the quantum dynamics evolves in a more systematic behavior, which makes it easier to investigate it. One may ask, whether the Kapitza-Dirac effect takes place for a longer turn on, a longer turn off and a shorter plateau phase of the external potential, given by the envelope function t , if 0 ≤ t ≤ ∆T sin2 π2 ∆T 1 , if ∆T < t < T − ∆T (5.4) A3 (t) = A3,max · ( T −t) π 2 sin 2 ∆T , if T − ∆T ≤ t ≤ T 0 else. The turn on and turn off duration ∆T = f T/2 is the fraction f of the full interaction time T. The fraction f may vary between 0 and 1, where f = 0 corresponds to an instantaneous turn on and turn off and f = 1 corresponds to a vanishing plateau phase. 44 CHAPTER 5. PROPERTIES OF THE 2-PHOTON KAPITZA-DIRAC EFFECT cos2 (Ω R T/2) diffraction probability 1.0 sin2 (Ω R T/2) 0.8 |c↑−1 |2 Dirac 0.6 |c1↑ |2 Dirac 0.4 |c↑−1 |2 Pauli |c1↑ |2 Pauli 0.2 0.0 |c↑−1 |2 Klein-Gordon 0 1000 2000 3000 4000 5000 T/Tcycle (number of laser cycles) 6000 |c1↑ |2 Klein-Gordon Figure 5.2: This figure shows the same time evolution, as in figure 5.1, but with other laser parameters and for a different initial electron momentum. The result of simulations with relativistic quantum wave equations (KleinGordon and Dirac equation equation) is shown in addition to the results of the Pauli equation. The laser intensity of this simulation corresponds to 2.32 · 1022 W/cm2 , with a laser wave length of 48.5 nm. The electron momentum perpendicular to the laser propagation direction is 0.05mc = 25.55 keV/c. If the external potential is turned on and off more slowly, the Rabi cycle will be delayed, which has to be accounted for in the analytic solution (5.1) of the Kapitza-Dirac effect. The solution (5.1) originates from the truncated Schrödinger equation with a ponderomotive potential (4.11) V0 (t) c 4 1 V0 (t) ih̄ċ1 = c , 4 −1 ih̄ċ−1 = (5.5a) (5.5b) which in this case has been shifted in energy by −h̄2 (~k + n~k L )2 /2m − V0 /2 , resulting in a timedependent, dispensable change of the global phase of the solution. A solution of equation (5.5), whose time-dependent ponderomotive coupling is related via equation (2.5) to the amplitude (5.4) is given by c −1 ( t ) = cos(t0 ) 0 c1 (t) = −i sin(t ) , with the warped time parameter t0 (t) = Z t V0 (τ ) 0 4h̄ dτ . (5.6a) (5.6b) (5.7) ! If one performs the integral (5.7) and requires, that t0 ( T ) = π/2 and solves for T, one obtains T= π 8 π 16mc2 h̄ 8 = , 2 e2 A23,max 8 − 5 f ΩR 8 − 5 f (5.8) where the ponderomotive amplitude (2.5) is reidentified in equation (5.8) and also the Rabi frequency (5.2) is resubstituted. Equation (5.8) tells, that one half Rabi cycle π/Ω R needs to be extended by the 45 CHAPTER 5. PROPERTIES OF THE 2-PHOTON KAPITZA-DIRAC EFFECT 1.0 probability 0.8 0.6 2 |c+↑ −1 | |c1+↑ |2 0.4 0.2 0.0 0 1000 2000 3000 4000 5000 t/Tcycle (number of laser cycles) 6000 Figure 5.3: This figure shows the same Kapitza-Dirac effect as in figure 5.2, simulated by using the Dirac equation. In contrast to figure 5.2, the turn on and turn off time ∆T is the half of the interaction time T, which corresponds to f = 1. Note, that in contrast to the figures 5.1 and 5.2, the in-field quantum dynamics is shown, in which the external field is not smoothly turned off. If, according to (5.8) the full interaction time T is stretched by the factor 8/3, a full quantum transition from mode −1 to mode 1 appears for an interaction time T, which is larger than in figure 5.2. factor 8/(8 − 5 f ), if the fraction f T of the interaction time elapses for the turn on and turn off of the external laser field. The extension by this factor compensates the turn on and turn off phase of the interaction such, that the occupation probability fully evolves from c−1 to c1 after the interaction. Figure 5.3 shows the stretched quantum dynamics by an explicit example, in which f equals 1. The property, that the electron beam is always diffracted by 100%, if one accounts for the envelope form (5.4) and stretches the interaction time according to equation (5.8), is demonstrated in figure 5.4. 46 CHAPTER 5. PROPERTIES OF THE 2-PHOTON KAPITZA-DIRAC EFFECT 1.0 probability 0.8 2 |c+↑ −1 | 0.6 |c1+↑ |2 0.4 0.2 0.0 0.0 0.2 0.4 0.6 f = ( Ton + Toff )/T 0.8 1.0 Figure 5.4: This figure shows the final diffraction probability of the Kapitza-Dirac effect, for a variation of the fraction f in the time-dependent envelope function (5.4) of the potential amplitude of the external laser field. For each f , the interaction time T has been chosen according to (5.8). The simulations are identical to the simulations in figure 5.3, except the different potential envelope. One can see, that the electron is always diffracted by 100%, if one accounts for the extension factor 8/(8 − 5 f ) for the interaction time T. The figures 5.2 and 5.3 show the time-evolution of the extreme cases, in which f almost vanishes or equals 1, respectively. Chapter 6 Electron spin dynamics: Conceptual considerations No spin effects appeared in the quantum dynamics of the 2-photon Kapitza-Dirac effect in chapter 5. This is expected from the Schrödinger equation (4.11). Even in the case of the Pauli equation (4.36), it seems, that the spin-dependent coupling term ~σ · ~B plays a minor role in the quantum dynamics of the 2-photon Kapitza-Dirac effect. This is different for the case of the 3-photon Kapitza-Dirac effect, which is discussed analytically in chapter 7 and numerically in chapter 8. Whereas the 2-photon Kapitza-Dirac effect is not relying on a distinction of the spin-up and spindown components of the wave function, the 3-photon Kapitza-Dirac effect connects both components with each other. The general diffraction properties and an interpretation of the quantum dynamics of the 3-photon Kapitza-Dirac effect are discussed in this chapter. In order to do so, the propagator of the wave function is introduced. The propagator contains not only information about the time-evolution of one quantum state but it contains information about the time-evolution of any quantum state. Therefore, the information, which can be extracted from properties of the propagator is comprehensive. A subsequent consideration discusses, whether the configuration of the initial electron spin affects the diffraction pattern. By anticipation of the results from perturbation theory in chapter 7 and the numerical investigation in chapter 8 it is concluded, that the diffraction probability does not depend on the initial electron spin. This implies, that the spin-dependent part of the propagator can be parameterized by a SU (2) representation. A further analysis of the properties of the SU (2) representation illustrates, that the electron spin is rotated, when it is diffracted. 6.1 The propagator The quantum state of the wave functions’ expansion coefficients cσa may be mapped from the initial 0 time t0 to the final time t by the propagator U σ,σ a,b ( t, t0 ) according to cσa (t) = 0 0 σ ∑0 U σ,σ a,b ( t, t0 ) cb ( t0 ) . (6.1) b,σ The propagation from time t0 to t0 is the quantum state itself. Therefore, the propagator has the property 0 U σ,σ (6.2) a,b ( t0 , t0 ) = δa,b δσ,σ0 . This means, that the propagator is related to a solution of the fundamental system of the differential equation ih̄ċσa = σ,σ0 σ0 cb ∑0 Ĥ a,b b,σ 47 , (6.3) CHAPTER 6. ELECTRON SPIN DYNAMICS: CONCEPTUAL CONSIDERATIONS 48 which is the general quantum wave equation (3.1) in momentum space. The propagator entries with 0 an identical mode index U σ,σ a,a ( t, t0 ) correspond to the probability, that the electron does not change 0 its momentum after Kapitza-Dirac scattering. All other propagator entries U σ,σ a,b ( t, t0 ) with a 6 = b correspond to a transition matrix element, which describes the change of the electron momentum from h̄(~k + b~k L ) to momentum h̄(~k + a~k L ) after Kapitza-Dirac scattering. In the case of the 3-photon Kapitza-Dirac effect, the electron starts in mode 0, with momentum h̄~k and is diffracted to mode 3 with momentum h̄(~k + 3~k L ). Therefore, the propagator subentry U 3,0 (t, t0 ) is of interest in the following. In the case of the Pauli equation, the propagator U 3,0 (t, t0 ) is a 2 × 2 matrix. However, in the case of the Dirac equation, U 3,0 (t, t0 ) is a 4 × 4 matrix. But the propagator entries, which relate the negative energy eigenstates to the positive energy eigenstates are negligibly small and are not of interest here. The propagator entries of interest for the Kapitza-Dirac effect, are those, which relate the positive energy eigenstates σ0 ∈ {+ ↑, + ↓} to the positive energy eigenstates σ ∈ {+ ↑, + ↓}. The corresponding subentry of U 3,0 (t, t0 ) of the propagator of the Dirac equation is a 2 × 2 matrix, too. The following considerations only refer to these 2 × 2 subentries of the propagator in the case of the Dirac equation. The propagator is denoted by q U 3,0 (t, t0 ) = P(t, t0 )eiφ(t,t0 ) S(t, t0 ) , (6.4) p where P(t, t0 ) is some amplitude, eiφ(t,t0 ) is some phase and S(t, t0 ) is a spin-dependent part. S(t, t0 ) is denoted by α(t, t0 ) α(t, t0 ) ~n(t, t0 ) ·~σ , S(t, t0 ) = cos 1 − i sin (6.5) 2 2 with some angle α and some vector ~n. If one requires, that P, φ, α ∈ R, ~n ∈ C3 and |~n|2 = 1, the representation (6.4) has the 8 degrees of freedom, corresponding to the 8 degrees of freedom of a complex 2 × 2 matrix. 6.2 Spin dependence of the diffraction pattern With the two component vector (4.33) of the Pauli equation or the corresponding two component vector of positive eigen energies of the Dirac equation ! c+↑ n + (6.6) cn = c+↓ n the quantum state propagation (6.1) from mode 0 to mode 3 can be noted by and c3 (t) = U 3,0 (t, t0 )c0 (t0 ) for the Pauli equation (6.7a) c3+ (t) (6.7b) = U 3,0 (t, t0 )c0+ (t0 ) for the Dirac equation. The vector of positive eigen energy coefficients (6.6) of the Dirac equation should not be confused with the positive eigen energy coefficient of the Klein-Gordon equation in section 4.3. The diffraction probability to mode 3 may therefore be expressed in terms of the initial quantum state of mode 0 by |c3↑ (t)|2 + |c3↓ (t)|2 = ||c3 (t)||2 = c3 (t)† c3 (t) = c0 (t0 )† U 3,0 (t, t0 )† U 3,0 (t, t0 )c0 (t0 ) . (6.8) In the case of the Dirac equation an index + needs to be added at each expansion coefficient to denote the positive eigen energy expansion coefficients only. The product of the adjoint propagator with itself in equation (6.8) can be expanded to h α α α α i † U 3,0 U 3,0 = PS† S = P cos2 + |~n|2 sin2 1 + 2 cos sin Im(~n) ·~σ = (6.9a) 2 2 2 2 = P [1 + sin (α) Im(~n) ·~σ ] , (6.9b) CHAPTER 6. ELECTRON SPIN DYNAMICS: CONCEPTUAL CONSIDERATIONS 49 where the time-dependence (t, t0 ) is omitted in this notion. The identity term in (6.9b), would yield P3,0 (t, t0 )c0 (t0 )† c0 (t0 ) = P3,0 (t, t0 )||c0 (t0 )||2 = P3,0 (t, t0 ) |c0↑ (t0 )|2 + |c0↓ (t0 )|2 (6.10) in equation (6.8). Since probability |c3↑ (t)|2 + |c3↓ (t)|2 and the probability |c0↑ (t0 )|2 + |c0↓ (t0 )|2 are completely spin independent, the identity term in equation (6.9b) does not induce any spin dependence in the diffraction pattern. In contrast, the term, which is proportional to the imaginary part of ~n in equation (6.9b) results in (6.11) P3,0 (t, t0 ) sin (α3,0 (t, t0 )) Im(~n3,0 (t, t0 )) · c0 (t0 )†~σc0 (t0 ) . This diffraction probability is spin dependent. If, for example, the parameters in equation (6.10) and √ (6.11) were P = 1/ 2, φ = 0, α = π/2 and ~n = i~e3 , then the initial quantum state c0↑ = 1, c0↓ = 0 is diffracted with probability 1, if one sums up the equations (6.10) and (6.11). On the other hand, if the initial quantum state was a spin down state c0↑ = 0, c0↓ = 1, the sum of the diffraction probabilities (6.10) and (6.11) yields 0. The results from perturbation theory (7.39) and (7.83) contain no imaginary part of the vector ~n. And also the numerical results in chapter 8 fit to a propagator, in which the imaginary part of ~n is negligible. Since the imaginary part of ~n is vanishingly small, the diffraction probability in the case of the 3-photon Kapitza-Dirac effect is independent of the incoming electron spin. A spin independence of the diffraction pattern still allows for a rotation of the electron spin, which is described in the next section. Due to the vanishing imaginary part of ~n the electron cannot be sorted out by its initial spin configuration, as it has been suggested by Batelaan [38]. 6.3 Spin rotation in Pauli theory Since the imaginary part of ~n vanishes, the spin-dependent part (6.5) of the propagator (6.4) looses 3 of its 6 degrees of freedom and therewith fulfills the properties of an SU (2) representation of rotations. In fact, the expectation value of the spin operator (3.11) is rotated by the SU (2) representation of the propagator. In order to explain this property, the initial quantum state of the electron is written in terms of the Bloch state θ θ ↓ ↑ φ0 iϕ φ0 , cb = e e sin . (6.12) c0 = e cos 2 2 The spin expectation value of the operator (3.11) with respect to this quantum state results in the vector h̄~n0s /2 † c0 σ 1 c0 sin(θ ) cos( ϕ) D P E h̄ h̄ c0 ~ S c0 = c0† σ 2 c0 = sin(θ ) sin( ϕ) = ~n0s , (6.13) 2 2 cos(θ ) c0† σ 3 c0 whose direction is parameterized by the angles θ and φ of spherical coordinates. The spin expectation value of the quantum state c3 in equation (6.7) evaluates to D E h̄ h̄ h̄ h̄ s † s c3 SiP c3 = P3,0 c0† S3,0 σ i S3,0 c0 = P3,0 ∑ Rij c0† σ j c0 = P3,0 ∑ Rij n0,j = P3,0 n3,i , (6.14) 2 2 j 2 j 2 where the left lower index of the vector ~ns denotes the mode index and the right lower index denotes the three spacial components of the vector. The matrix Rij is the SO(3) rotation matrix R = ~ ) with the generating Lie algebra D1 = δ3,2 − δ2,3 , D2 = δ1,3 − δ3,1 and D3 = δ2,1 − δ1,2 . exp(−α3,0~n3,0 D The matrix Rij acts as right-handed rotation around the axis ~n3,0 with the rotation angle α on vectors in R3 . In particular, the direction ~n0S is right-handed rotated by the angle α3,0 around axis ~n3,0 to the angle ~n3S , if the SU (2) representation S3,0 acts at the quantum state c0 in mode 0. This is illustrated in figure 6.1. CHAPTER 6. ELECTRON SPIN DYNAMICS: CONCEPTUAL CONSIDERATIONS 50 ~e3 ~n3,0 θ ~n0s ~n3s ~e2 ϕ ~e1 α3,0 Figure 6.1: The direction of the spin ~n0s of the quantum state c0 is rotated by the SU (2) representation S3,0 to the direction ~n3s of the quantum state c3 around the axis ~n3,0 by the angle α3,0 . In this illustrative sketch, the rotation axis coincides with the 3-axis, such that the azimuthal angle θ of the spherical coordinates is not changed, but the polar angle ϕ of the polar coordinates is increased by α a,b modulo 2π. This implies, that the spin is conserved in ~n3,0 direction but changes in all directions perpendicular to ~n3,0 . The statement of this figure also applies for Dirac theory, if the electron spin of the incoming and outgoing electron is considered in its rest frame of reference. 6.4 Spin rotation in Dirac theory The objects of interest in Dirac theory are the spin expectation value of the Foldy-Wouthuysen spin operator (3.40) with the basis functions (3.33). Since u(~k ) of equation (3.35) is a unitary matrix and the bi-spinors uσ (~k) are columns of u† (~k), the uσ (~k ) are mapped at the unit vectors ~eσ of R4 , by u(~k). Therefore, the matrix entries of the spin operator (3.40) with respect to the basis elements (4.57) at mode n 0 0 0 h̄ h̄ h̄ (σ,σ0 ) hψnσ |S FW |ψnσ i = uσ (~k + n~k L )† u(~k + n~k L )† ~Σ u(~k + n~k L )uσ (~k + n~k L ) = ~eσ† ~Σ~eσ = ~Σ (6.15) 2 2 2 is equivalent to the matrix entries of the non-relativistic spin operator (3.41). Similar to section 6.3, the positive eigen energy quantum state of mode 0 can be expressed in terms of the Bloch state θ θ +↑ +↓ φ0 φ0 iϕ (6.16) c0 = e cos , c0 = e e sin , c0−↑ = 0 , c0−↓ = 0 . 2 2 The occupation probability of all other modes is 0. Therefore, the expectation value of the wave function (4.58) reduces to h̄ h̄ h̄ hψ|S FW |ψi = c0† ~Σc0 = c0+† ~σc0+ = ~n0s . (6.17) 2 2 2 The quantum state of the diffracted mode is given by equation (6.7b). If one assumes a 100% diffraction probability P3,0 = 1, the occupation probability of mode 0 is 0. Hence, the spin expectation value after the diffraction results in h̄ † h̄ h̄ s s hψout |S FW,i |ψout i = c0+† U 3,0 σ i U 3,0 c0+ = ∑ Rij n0,j = n3,i . 2 2 j 2 (6.18) Therefore, the spin rotation described by Pauli theory also applies to quantum dynamics with the Dirac equation, if the spin was measured with the Foldy-Wouthuysen spin operator. Since the bispinors uσ (~k) at rest are just the unit vectors of R4 , the Foldy-Wouthuysen spin operator measures CHAPTER 6. ELECTRON SPIN DYNAMICS: CONCEPTUAL CONSIDERATIONS 51 the spin in the rest-frame of the electron. Therefore, the statement in the case of the Pauli equation, that electron spin ~n0s is right-handed rotated around the axis ~n3,0 with the angle α3,0 also applies to Dirac theory. The difference to Pauli theory is, that this statement applies to the electron spin in the rest-frame of the electron, in the case of Dirac theory. Chapter 7 Electron spin dynamics: Analytical small-time behavior This chapter solves the quantum dynamics of the 3-photon Kapitza-Dirac effect, by utilizing timedependent perturbation theory with the quantum wave equations in momentum space of chapter 4. The result of the calculation is identified with the SU (2)-representation (6.4) in chapter 6. Diffraction properties, like the Rabi frequency and the rotation angle, with which the spin is rotated are concluded from the perturbative result. These results are checked numerically in chapter 8. This chapter starts with a general summary of time-dependent perturbation theory in section 7.1. The lowest order contributions of time-dependent perturbation theory are computed for the 3-photon Kaptiza-Dirac effect for the case of the Pauli equation (section 7.2) and for the case of the Dirac equation (section 7.3). The resonance condition resulting from considerations of energy and momentum conservation of chapter 2.2 is explicitly used in the calculations and plays an important role in the derivation of the perturbative result. Only terms, with a divergent time dependence are dominant over all other terms and are accounted in the calculation. 7.1 General procedure This summary of time-dependent perturbation theory is based on the lecture notes of Christof Wetterich from the year 2009 [51]. Time dependent perturbation theory relies on the identity U (t, t0 ) = U 0 (t, t0 ) + which satisfies 1 ih̄ Z t t0 dt1 U 0 (t, t1 )V̂ (t1 )U (t1 , t0 ) , ∂ 1 U (t, t0 ) = ( Ĥ 0 + V̂ (t))U (t, t0 ) ∂t ih̄ with the free propagator i U 0 (t, t0 ) = e− h̄ (t−t0 ) Ĥ 0 . (7.1) (7.2) (7.3) and the interaction Hamiltonian V̂ . Since equation (7.2) is equivalent to equation (3.1), the solution (7.1) contains the time evolution of the wave function. Inserting eq. (7.1) recursively into itself yields a series, which is assumed to converge on the interval [t0 , t]. The first four terms of this series are U (t, t0 ) = U 0 (t, t0 ) + U st (t, t0 ) + U nd (t, t0 ) + U rd (t, t0 ) + higher terms , (7.4) where U 0 (t, t0 ) is the interaction-less time propagation, U st (t, t0 ) = 1 ih̄ Z t t0 dt1 U 0 (t, t1 )V̂ (t1 )U 0 (t1 , t0 ) 53 (7.5) CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 54 is the first order perturbation, Z t 1 U nd (t, t0 ) = (ih̄)2 t0 dt2 Z t2 t0 dt1 U 0 (t, t2 )V̂ (t2 )U 0 (t2 , t1 )V̂ (t1 )U 0 (t1 , t0 ) (7.6) is the second order perturbation and 1 U rd (t, t0 ) = (ih̄)3 Z t t0 dt3 Z t3 t0 dt2 Z t2 t0 dt1 U 0 (t, t3 )V̂ (t3 )U 0 (t3 , t2 )V̂ (t2 )U 0 (t2 , t1 )V̂ (t1 )U 0 (t1 , t0 ) (7.7) is the third order perturbation in time-dependent perturbation theory. 7.2 Perturbation Theory for the Pauli equation This section considers time-dependent perturbation theory of the Pauli equation. The time-dependent perturbation theory of the Dirac equation of section 7.3 is based on the concepts of this section. 7.2.1 Derivation Since the Pauli equation contains coupling terms to its neighboring and second next neighboring modes, the lowest order non-vanishing contribution of time-dependent perturbation theory of the 3-photon Kapitza-Dirac effect is of second order. The second order term (7.6) of time-dependent perturbation theory can be written as (7.8) U nd;3,0 (t, t0 ) = 1 ( ih̄ )2 n ,i ∈{1,2,3,4} ∑ Z t i t0 dt2 Z t2 t0 dt1 U 0;3,n1 (t, t2 )V n1 ,n2 (t2 )U 0;n2 ,n3 (t2 , t1 )V n3 ,n4 (t1 )U 0;n4 ,0 (t1 , t0 ) , with the free propagator i nr ( t − t U 0;a,b (t, t0 ) = e− h̄ Ea 0) δa,b 1 (7.9) from equation (4.37) and the abbreviation of the energy Ennr = (~k + n~k L )2 . 2m (7.10) Since the quantum mechanical operators U 0 and V̂ change into the matrices (7.9) and (4.38), the second order perturbation contribution (7.6) is converted in a matrix product of five matrices in equation (7.8). The matrices (7.9) and (4.38) are indexed with the mode index a and b, whereas the spin-dependent part (1 and ~σ) is still kept as 2 × 2 matrix and is not indexed. Accordingly the matrix products in equation (7.8) are sums over the mode indices n1 , n2 , n3 and n4 and matrix products in 2 × 2 spinor space. The 2 × 2 matrix products in spinor space are denoted by the bold symbols U 0;a,b and V a,b . The result of equation (7.8) is a matrix with mode indices a and b and a matrix structure in 2 × 2 spinor space, denoted by the bold symbol U nd;3,0 . The free propagator (7.9) does not change the mode index, because it only contains a Kronecker delta δa,b . This property results in the conditions 3 = n1 , n2 = n3 and n4 = 0 in the sum of equation (7.8). The interaction Hamiltonian (4.38) contains the Kronecker deltas δa,b−2 , δa,b−1 , δa,b , δa,b+1 and δa,b+2 . Therefore, there are two combinations of interaction Hamiltonian terms, which contribute in the propagator for the desired initial mode index 0 and final mode index 3, which are the terms − ~2 e2 A e sin(ωt1 ) ~ 2 0 ~k 1 + h̄~k L × A ~ 0 · (i~σ ) sin ( ωt ) 1 2 A · h̄ 0 2 4mc 8mc2 (7.11) CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR Energy E3 E2 U 0;3,3 (t, t2 ) U 0;1,1 (t2 , t1 ) V 1,0 (t1 ) V 3,1 (t2 ) V 3,2 (t2 ) U 0;2,2 (t2 , t1 ) U 0;3,3 (t, t2 ) Mode E1 3 2 V 2,0 (t1 ) 1 55 E0 U 0;0,0 (t1 , t0 ) U 0;0,0 (t1 , t0 ) 0 Figure 7.1: This picture illustrates the two contributions (7.13) (dashed arrows) and (7.14) (dotted arrows) of second order time dependent perturbation theory of the Pauli equation for the 3-photon Kapitza-Dirac effect. In the dashed arrows, the electron starts in mode 0, is scattered to mode 1 at time t1 and is again scattered to mode 3 at time t2 . The roles of 2-photon and 1-photon scattering are interchanged in the case of the dotted line. In the dotted arrows, the electron starts in mode 0, is scattered to mode 2 at time t1 and is again scattered to mode 3 at time t2 . Note, that the 2-photon scattering is caused by the ponderomotive term of the Pauli equation, which does not influence the electron spin. In contrast, to the 1-photon scattering term of the Pauli equation influences the electron spin. for the product V 3,1 (t2 )V 1,0 (t1 ) and − 2~2 e sin(ωt2 ) ~ ~ 0 · (i~σ ) e A0 sin2 (ωt1 )1 2 A0 · h̄~k 1 + h̄~k L × A 4mc 8mc2 (7.12) for the product V 3,2 (t2 )V 2,0 (t1 ). Consequently, second order time-dependent perturbation theory (7.8) consists of the two contributions ~2 e 1 e2 A 0 ~ 0 · h̄~k 1 + h̄~k L × A ~ 0 · (i~σ ) 2 A (7.13) (ih̄)2 8mc2 4mc Z t Z t2 i · dt2 dt1 exp − ( E3nr (t − t2 ) + E1nr (t2 − t1 ) + E0nr (t1 − t0 )) sin2 (ωt2 ) sin(ωt1 ) h̄ t0 t0 U nd;3,0 (t, t0 ) = − and 2~2 e ~ 1 ~k 1 + h̄~k L × A ~ 0 · (i~σ ) e A0 (7.14) 2 A · h̄ 0 (ih̄)2 4mc 8mc2 Z t Z t2 i · dt2 dt1 exp − ( E3nr (t − t2 ) + E2nr (t2 − t1 ) + E0nr (t1 − t0 )) sin(ωt2 ) sin2 (ωt1 ) . h̄ t0 t0 U nd;3,0 (t, t0 ) = − These two contributions are sketched in figure (7.1). All four sums in equation (7.8) are collapsed according to the considerations above. The 2 × 2 identity of the ponderomotive 2-photon coupling term of the interaction Hamiltonian (4.38) is multiplied at the other 1-photon interaction term and vanishes thereby. It remains to compute the time integrals over t1 and t2 of equation (7.13) and (7.14). The integration can be performed by expanding the sine functions into exponentials. If the phase is constant with respect to the integration variables t1 and t2 , the integrals will diverge for infinite long times. This means, that parts of the integral with a constant phase will dominate over all other terms for long times t − t0 . The following considerations focus on the identification of these divergent, constant phase terms, in order to neglect all other terms with a fast oscillating phase. The resonance condition from energy and momentum conservation (C.3) may cause a constant phase in the time integral of (7.13) and (7.14). In the case of the 3-photon Kapitza-Dirac effect, with n a = 2 and ne = 1, equation (C.3) reduces to E3nr = E0nr + h̄ω . (7.15) CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 56 If one solves this equation for h̄ω, one will know, that the energy difference has the property ! nr ∆Ea,b = Eanr − Ebnr = h̄ω , (7.16) if either condition (2.19) or condition (2.20) was fulfilled. The factor h̄ω times an integration variable t1 or t2 in the exponent appears from the expansion of the sine functions in the time integral of (7.13) and nr or ∆Enr times the time integration (7.14). Therefore, one has to search for the energy difference ∆E3,0 0,3 variable t1 or t2 in the exponential of the time integral of (7.13) and (7.14), because this energy difference plus h̄ω yields a constant phase, which in turn yields a diverging and dominating integral. The exponent resulting from the free electron propagation (7.9) − i i nr nr nr t2 + ∆E0,1 t1 ( E3 (t − t2 ) + E1nr (t2 − t1 ) + E0nr (t1 − t0 )) = − E3nr t − E0nr t0 + ∆E1,3 h̄ h̄ (7.17) of equation (7.13) or − i i nr nr nr t2 + ∆E0,2 t1 ( E (t − t2 ) + E2nr (t2 − t1 ) + E0nr (t1 − t0 )) = − E3nr t − E0nr t0 + ∆E2,3 h̄ 3 h̄ (7.18) nr times t or t , nor ∆Enr times t or t . of equation (7.14) neither contains an energy difference ∆E3,0 2 2 1 1 0,3 However, the upper limit of the integral over t1 in (7.13) and (7.14) changes t1 into t2 , resulting in − i i nr nr nr Enr t − E0nr t0 + ∆E1,3 t2 + ∆E0,1 t1 → − E3nr t − E0nr t0 + ∆E0,3 t2 t1 → t2 h̄ 3 h̄ (7.19) for the term (7.17) and − i i nr nr nr E3nr t − E0nr t0 + ∆E2,3 t2 + ∆E0,2 t2 t1 → − E3nr t − E0nr t0 + ∆E0,3 t → t h̄ h̄ 2 1 (7.20) nr t /h̄ should be compenfor the term (7.18). According to the considerations above, the term −i∆E0,3 2 sated by a term −iωt2 from the expansion of the sine functions in equations (7.13) and (7.14), which has to be seeked for. The sin2 (ωt2 ) sin(ωt1 ) of equation (7.13) results in sin2 (ωt2 ) sin(ωt1 ) = The terms i 2iωt2 +iωt1 e − e2iωt2 −iωt1 − 2 eiωt1 − e−iωt1 + e−2iωt2 +iωt1 − e−2iωt2 −iωt1 . 8 (7.21) i −2iωt2 +iωt1 i i e = exp − (2h̄ωt2 − h̄ωt1 ) 8 8 h̄ and i −iωt1 i i e = exp − (h̄ωt1 ) 4 4 h̄ (7.22) in equation (7.21) will change into an exponential of −iωt2 , by focusing again only on the upper limit of the first integral over t1 . Similarly, the sin(ωt2 ) sin2 (ωt1 ) term in equation (7.14) may be expanded into i 2iωt1 +iωt2 sin(ωt2 ) sin2 (ωt1 ) = e − e2iωt1 −iωt2 − 2 eiωt2 − e−iωt2 + e−2iωt1 +iωt2 − e−2iωt1 −iωt2 . 8 (7.23) One identifies the terms i −2iωt1 +iωt2 i i i −iωt2 i i e = exp − (2h̄ωt1 − h̄ωt2 ) and e = exp − (h̄ωt2 ) , (7.24) 8 8 h̄ 4 4 h̄ which contain an exponential of −iωt2 after the integration over t1 by taking only the upper limit. One concludes, that the integrals of (7.17) times the terms (7.22) of equation (7.13) and the integrals of (7.18) times the terms (7.24) of equation (7.14) are the only diverging contributions and are therefore computed in the following. As discussed above, only the upper limit of the integral over t1 contributes CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 57 to the diverging terms. Therefore, the lower limit of the integral over t1 is neglected in the following calculations. The integral over t1 of the exponential of (7.17) times the exponential of the left term of (7.22) of equation (7.13) results in i 8 Z t t0 dt2 Z t2 i nr nr nr nr dt1 exp − E3 t − E0 t0 + (∆E1,3 + 2h̄ω )t2 + (∆E0,1 − h̄ω )t1 h̄ Z 1 i h̄ t nr nr nr = − nr dt2 exp − E3 t − E0 t0 + (∆E0,3 + h̄ω )t2 . ∆E0,1 − h̄ω 8 t0 h̄ (7.25) The integral over t1 of the exponential of (7.17) times the exponential of the right term of (7.22) of equation (7.13) results in i 4 Z t t0 dt2 Z t2 i nr nr dt1 exp − E3nr t − E0nr t0 + ∆E1,3 t2 + (∆E0,1 + h̄ω )t1 h̄ Z h̄ t i 1 nr dt2 exp − E3nr t − E0nr t0 + (∆E0,3 + h̄ω )t2 . = − nr ∆E0,1 + h̄ω 4 t0 h̄ (7.26) The integral over t1 of the exponential of (7.18) times the exponential of the left term of (7.24) of equation (7.14) results in i 8 Z t t0 dt2 Z t2 i nr nr nr nr dt1 exp − E3 t − E0 t0 + (∆E2,3 − h̄ω )t2 + (∆E0,2 + 2h̄ω )t1 h̄ Z h̄ t i 1 nr nr nr dt2 exp − E3 t − E0 t0 + (∆E0,3 + h̄ω )t2 . = − nr ∆E0,2 + 2h̄ω 8 t0 h̄ (7.27) The integral over t1 of the exponential of (7.18) times the exponential of the right term of (7.24) of equation (7.14) results in i 4 Z t t0 dt2 Z t2 i nr nr nr nr dt1 exp − E3 t − E0 t0 + (∆E2,3 + h̄ω )t2 + ∆E0,2 t1 h̄ Z i 1 h̄ t nr nr nr dt2 exp − E3 t − E0 t0 + (∆E0,3 + h̄ω )t2 . = − nr ∆E0,2 4 t0 h̄ (7.28) The time integration over t2 is identical for all four terms (7.25 - 7.28) and may therefore be discussed once by ignoring the prefactors of the integration of t1 . The integral over t2 evaluates to Z t 1 nr nr i nr nr nr e−i( E3 t−E0 t0 )/h̄ , (7.29) dt2 exp − E3 t − E0 t0 + (∆E0,3 + h̄ω )t2 = i e−i∆ωt − e−i∆ωt0 h̄ ∆ω t0 with the laser detuning nr ∆ω = ω − ∆E3,0 /h̄ . (7.30) This integral is undefined for zero detuning ∆ω = 0, which is the case, if the resonance condition (2.15) is fulfilled. If one considers the first order Taylor expansion of i e−i∆ωt − e−i∆ωt0 = ∆ω (t − t0 ) + O(∆ω 2 ) (7.31) with respect to ∆ω, one recognizes, that the nominator vanishes as fast as the denominator in equation (7.29) such that the integral is well-defined. Furthermore, one may compute the integral (7.29) for the case of zero detuning ∆ω = 0, resulting in Z t nr nr i nr nr dt2 exp − ( E3 t − E0 t0 ) = (t − t0 )e−i(E3 t− E0 t0 )/h̄ . (7.32) h̄ t0 CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 58 One would obtain the same result, if one substituted the first order Taylor expansion (7.31) in the result of the integral (7.29). Summing up the double integral of all divergent contributions of the second order perturbation theory (7.8) yields U nd;3,0 (t, t0 ) = − 2~2 1 e ~ ~ 0 · (i~σ ) e A0 PF 2 A0 · h̄~k 1 + h̄~k L × A 64h̄ mc 4mc2 ( nr nr t − t0 · e−i(E3 t−E0 t0 )/h̄ i −i∆ωt − e−i∆ωt0 ∆ω e with the prefactor PF = on resonance off resonance , 1 2 1 2 + + + nr nr nr − h̄ω ∆E0,1 + h̄ω ∆E0,2 + 2h̄ω ∆E0,2 (7.33) (7.34) nr ∆E0,1 of the integrals (7.25 - 7.28). The propagator can either be noted with an off-resonant term, from the time integral (7.29), or with a resonant term, which originates from the resonant integral (7.32). It is difficult to read of a frequency dependent scaling law from equation (7.33), because of the prefactor (7.34). A simpler value of this prefactor in the resonant case ∆ω = 0 and small laser frequencies h̄ck L mc2 would be useful and is derived in the following. First, h̄ω in the prefactor PF may be nr as it is implied by the resonance condition (7.16), yielding replaced with ∆E3,0 PF = = 2 1 2 1 nr − ∆Enr + ∆Enr + ∆Enr + ∆Enr + 2∆Enr + ∆Enr ∆E0,1 3,0 3,0 0,2 3,0 0,2 0,1 2 2 1 1 nr + ∆Enr + ∆Enr + ∆Enr + ∆Enr + ∆Enr . ∆E0,2 0,3 3,2 3,0 3,1 0,1 (7.35) If one inserts the definition of the energy difference (7.16) and the energy (7.10) into equation (7.35), one obtains PF = − 2h̄2~k2L m h̄2~k2L m − . h̄4 (~k L ·~k +~k2L )(~k L ·~k + 2~k2L ) h̄4 (4~k L ·~k + 5~k2L )(4~k L ·~k + 7~k2L ) (7.36) This term may be simplified by imposing the limit h̄ck L mc2 of small laser frequencies. In the case of the 3-photon Kapitza-Dirac effect, the initial electron momentum in laser propagation direction (2.19) changes into n a − ne 1 lim h̄k1 (k L ) = mc = mc (7.37) n + n 3 k L →0 a e for small laser frequencies. Note, that this limit corresponds to the momentum p1,lim of subsection 2.2.6, but for the case of the non-relativistic energy momentum relation (3.4). This implies k L k1 and therefore ~k2L ~k L ·~k. Using this property simplifies the prefactor in equation (7.36) to P̃F ≈ − h̄2~k2L m h̄2~k2L m 9m 81 − =− 2 2 =− , 4~ 4 8mc2 8h̄ k1 h̄ (k L ·~k)2 8h̄ (~k L ·~k)2 (7.38) where the momentum k1 for small k L in equation (7.37) is inserted in the last equality. The perturbation theory propagator (7.33) results in the simpler form U nd;3,0 (t, t0 ) = 2~2 e ~ 81 ~k 1 + h̄~k L × A ~ 0 · (i~σ ) e A0 2 A · h̄ 0 2 512h̄ m2 c3 ( 4mc t − t0 · e−i(E3 t−E0 t0 )/h̄ i −i∆ωt − e−i∆ωt0 ∆ω e in the case of the simplified prefactor (7.38). on resonance off resonance (7.39) CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 7.2.2 59 Interpretation Electron spin rotation Equation (7.39) has the same form as the propagator (6.4), with a vanishing imaginary part of the unit vector ~n. As it is discussed in section (6.2), the vanishing imaginary part of ~n implies, that the diffraction probability of the 3-photon Kapitza-Dirac effect does not depend on the initial electron spin and the spin part (6.5) of the propagator turns into a SU (2)-representation of a rotation of the electron spin. The parameters for the ansatz (6.2) in the resonant case of equation (7.39) assume the values 2 2 21 ~ 2 81 e e2 A 0 ~ 0 · h̄~k + h̄~k L × A ~0 2 A ( t − t0 ) , 512h̄ m2 c3 4mc2 E3 t − E0 t φ3,0 (t, t0 ) = −i , h̄ 2 2 − 21 α3,0 (t, t0 ) ~ ~ ~ ~ ~ ~ = 2 A0 · h̄k 2 A0 · h̄k + h̄k L × A0 , cos 2 2 2 − 21 α3,0 (t, t0 ) ~ ~ ~ ~ ~ ~ ~n3,0 (t, t0 ) sin = −h̄k L × A0 2 A0 · h̄k + h̄k L × A0 . 2 P3,0 (t, t0 ) = (7.40a) (7.40b) (7.40c) (7.40d) Note, that the four equations are unique. In particular, the division by the square root in the last two equations in (7.40) is necessary, because the unity of the determinant of the SU (2) representation (6.5) requires, that α3,0 (t, t0 ) α3,0 (t, t0 ) ! ~n23,0 (t, t0 ) = 1 . (7.41) + sin2 cos2 2 2 This unitarity constraint is only valid, if one substitutes the left-hand side of the equations (7.40c) and (7.40d) by their right-hand side in equation (7.41). Furthermore, the angle α3,0 (t, t0 ) is constant in time, because the right-hand side of equation (7.40c) is constant in time. For similar reasons ~n3,0 (t, t0 ) is constant in time, because α3,0 and the right-hand side of equation (7.40d) are constant in time. Since ~n3,0 is a unit vector in the SU (2) representation, (7.40d) also implies, that ~k × A ~0 ~B ~n3,0 = − L = 0 . (7.42) ~ 0| |~B0 | |~k L × A This means, that the rotation axis of the SU (2) representation points in the direction of the magnetic field ~B0 of the laser beam and therewith the electron spin is rotated around this axis. The rotation angle α3,0 can be resolved by multiplying equation (7.40d) with equation (7.42) and dividing it by equation (7.40c), resulting in tan α 3,0 2 = ~ 0| |~k L × A . ~ 0 ·~k 2A (7.43) The tangent needs to be inverted on the interval [−π/2, π/2] in order to solve for the rotation angle α3,0 . Rabi frequency Section 8 suggests, that the diffraction probability of the 3-photon Kapitza-Dirac effect depends on the interaction time T with the probability ΩR T , (7.44) P3,0 ( T, 0) = sin2 2 CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 60 if the turn on and turn off time of the interaction was infinitely short. If one uses this diffraction probability in the propagator ansatz (6.4) and accounts for the vanishing imaginary part of ~n one can compute ΩR T † 1. (7.45) U nd;3,0 ( T, 0)U nd;3,0 ( T, 0) = sin2 2 All parameters of the SU (2) representation and the complex phase φ3,0 ( T, 0) drop out in equation (7.45). The only quantity, which remains is the diffraction probability (7.44) times the 2 × 2 identity 1. A second order Taylor expansion with respect to T of the right-hand side of equation (7.45) results in † U nd;3,0 ( T, 0)U nd;3,0 ( T, 0) = ΩR T 2 2 1 + O( T 3 ) . (7.46) On the other hand, one may compute the same quantity for the derived, resonant propagator (7.39), resulting in † U nd;3,0 ( T, 0)U nd;3,0 ( T, 0) = 81 512h̄ 2 2 2 e 2 ~ ~ ~ ~ 2 A0 · h̄k + h̄k L × A0 m2 c3 ~2 e2 A 0 4mc2 !2 T 2 1 . (7.47) This short-time result is of the same structural form as the second order Taylor term in equation (7.46) and one may extract the Rabi frequency Ω R from it by requiring equality between the second order Taylor term and the right-hand side of equation (7.47), yielding r 2 2 ~2 81 2 e e2 A 0 ~ 0 · h̄~k + h̄~k L × A ~0 . 2A ΩR = (7.48) 2 3 2 256h̄ m c 4mc Note, that the Rabi frequency also appears in the diffraction probability (7.40a), such that one may write Ω P3,0 (t, t0 ) = R (t − t0 ) (7.49) 2 in equation (7.40a). The property, that second order time-dependent perturbation theory (7.39) is the first term of a power series of a sine function suggests, that higher order perturbation theory results in higher order terms of the sine function. In fact, a publication of Gush and Gush in 1971 [31] computes all higher order terms of time-dependent perturbation theory of the Schödinger equation with an external vector potential for the 2-photon Kapitza-Dirac effect. The authors show, that the sum over all terms, up to infinite order perturbation theory results in a sine time-dependence of the diffraction probability. In the case of the discussed 3-photon Kapitza-Dirac effect, which is solved by using time-dependent perturbation theory of the Dirac equation, the next order term of the sine series is expected to originate from 9th order time-dependent perturbation theory1 . Spin-flip probability The time evolution of the expansion coefficients cσn is given by equation (6.1). If one assumes an initial configuration, in which only the 0 mode with spin up is occupied, one may compute ~2 ~ 0 · h̄~k e2 A 81 2e A 0 −i( E3nr t− E0nr t0 )/h̄ e (t − t0 )c0↑ (t0 ) , 2 3 512h̄ m c 4mc2 ~2 ~ 0 e2 A 81 eh̄~k L × A 0 −i( E3nr t− E0nr t0 )/h̄ e c3↓ (t) = (U nd;3,0 )↓,↑ (t, t0 )c0↑ ( T ) = (t − t0 )c0↓ (t0 ) . 2 3 512h̄ m c 4mc2 c3↑ (t) = (U nd;3,0 )↑,↑ (t, t0 )c0↑ (t0 ) = 1 The 9th (7.50a) (7.50b) order perturbation theory corresponds to the coupling from mode 0 to mode 3, back to mode 0 and back to mode 3. CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 61 With the more specific initial condition c0+↑ (0) = 1 one obtains |c3↑ ( T )|2 |c3↓ ( T )|2 !2 81 512h̄ 2 ~ 0 · h̄~k 2e A m2 c3 81 512h̄ 2 ~0 eh̄~k L × A m2 c3 = = ~2 e2 A 0 4mc2 !2 !2 ~2 e2 A 0 4mc2 T2 !2 and (7.51a) T2 . (7.51b) after time T. One may define the non-spin-flip probability Pnoflip = |c3↑ ( T )|2 (7.52) |c3↑ ( T )|2 + |c3↓ ( T )|2 and the spin-flip probability Pflip = |c3↓ ( T )|2 (7.53) |c3↑ ( T )|2 + |c3↓ ( T )|2 ~ 0 ·~k/| A ~ 0| of the diffracted beam. If the wave vector component parallel to the laser polarization k k = A ~ ~ ~ is introduced and one considers, that k L × A0 = k L | A0 | for the vacuum Maxwell equations, one can compute Pnoflip = Pflip = 7.3 ~ 0 · h̄~k)2 (2k k )2 (2e A 1 = = 2 2 2 ~ 0 · h̄~k)2 + (eh̄~k L × A ~ 0 )2 (2k k ) + (k L ) 1 + k L /(4k2k ) (2e A and ~ 0 )2 (eh̄~k L × A ( k L )2 1 . = = 2 2 2 + ( k )2 2 2 ~ ~ ~ ~ ( 2k ) 4k k /k L + 1 L (2e A0 · h̄k) + (eh̄k L × A0 ) k (7.54a) (7.54b) Perturbation Theory for the Dirac equation The derivation and the results of time-dependent perturbation theory of the Dirac equation is often similar to the derivation and the results from time-dependent perturbation theory of the Pauli equation of section 7.2. Therefore, the concepts of section 7.2 are adopted and analogous explanations are referred to section 7.2. 7.3.1 Derivation Since the interaction Hamiltonian (4.67) of the Dirac equation contains only next neighbor coupling terms, the lowest non-vanishing contribution in time-dependent perturbation theory is of third order. The term (7.7) of third order time-dependent perturbation theory can be written as U rd;3,0 (t, t0 ) = ∑ ni ,i ∈{1,2,3,4,5,6} 1 (ih̄)3 Z t t0 dt3 Z t3 t0 dt2 Z t2 t0 dt1 U 0;3,n1 (t, t3 )V n1 ,n2 (t3 ) · U 0;n2 ,n3 (t3 , t2 )V n3 ,n4 (t2 )U 0;n4 ,n5 (t2 , t1 )V n5 ,n6 (t1 )U 0;n6 ,0 (t1 , t0 ) . (7.55) with the free propagator i U 0;a,b (t, t0 ) = e− h̄ Ea β(t−t0 ) δa,b = exp (−iEa (t − t0 )/h̄) 1 0 0 δ exp (iEa (t − t0 )/h̄) 1 a,b from equation (4.66) and the abbreviation for the energy q En = m2 c4 + c2 h̄2 (~k + nk L )2 . (7.56) (7.57) CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 62 The matrix V a,b is the interaction Hamiltonian (4.67). Note, that the free propagator (7.56) has a 4 × 4 bi-spinor matrix form, which is denoted by four 2 × 2 matrices on the right-hand side of equation (7.56). The free propagator (7.9) does not change the mode index, because it only contains a Kronecker delta δa,b . This property results in the conditions 3 = n1 , n2 = n3 , n4 = n5 and n6 = 0 in the sum in (7.55). Since the interaction Hamiltonian (4.67) contains the Kronecker deltas δa,b−1 and δa,b+1 , only one combination of interaction Hamiltonian terms contribute to the propagator. Therefore equation (7.55) reduces to U rd;3,0 (t, t0 ) = 1 (ih̄)3 Z t t0 dt3 Z t3 t0 dt2 Z t2 t0 dt1 U 0;3,3 (t, t3 )V 3,2 (t3 ) · U 0;2,2 (t3 , t2 )V 2,1 (t2 )U 0;1,1 (t2 , t1 )V 1,0 (t1 )U 0;0,0 (t1 , t0 ) Z t Z t3 Z t2 1 dt dt dt1 U 0;3,3 (t, t3 ) = 3 2 (ih̄)3 t0 t0 t0 e sin(ωt3 ) ~ ~ 0 ·~α u(~k + 2~k L )U 0;2,2 (t3 , t2 ) ·− u(k + 3~k L )† A 2 e sin(ωt2 ) ~ ~ 0 ·~α u(~k + 1~k L )U 0;1,1 (t2 , t1 ) ·− u(k + 2~k L )† A 2 e sin(ωt1 ) ~ ~ 0 ·~α u(~k + 0~k L )U 0;0,0 (t1 , t0 ) . ·− u(k + 1~k L )† A 2 (7.58) In contrast to the computation of time-dependent perturbation theory with the Pauli equation, the free propagators U 0;a,b do not commute with the interaction Hamiltonian, because of the β in the exponential of (7.56). It is necessary to expand the matrix product further, because the time-dependence of all four propagators enters in the time integration over the variables t1 , t2 and t3 . Therefore the matrix part of the interaction Hamiltonian is broken up into four 2 × 2 matrices ~0 + A ~ 0 ·~α uσ0 (~k + (n − 1)~k L ) = t A ~ 0 ·~α − A ~ 0T w~α + A ~ 0 ·~s β + iβ~Σ · ~r × A ~ 0 ·~h α1 α2 α3 uσ (~k + n~k L )† A uu ~ 0 ·~σ ~ 0 ·~s 1 + i ~r × A ~ 0 ·~σ − A ~ T w ~σ + i A ~ 0 · ~h 1 A tA M ud 0 = M = , (7.59) ~ 0 ·~σ − A ~ T w ~σ + i A ~ 0 · ~h 1 − A ~ 0 ·~s 1 + i ~r × A ~ 0 ·~σ M du M dd tA 0 where ~Σ is the non-relativistic spin operator (3.41). Note, that the variables t, s, r, w and h are functions with the two parameters ~k + n~k L and ~k + (n − 1)~k L . Equation (7.59) rewrites the bi-spinor contractions (4.60c) by converting (7.60) ∑ A0l rq (~k,~k0 ) βαq αl = ∑ A0l rq (~k,~k0 )ε mql iβΣm = ∑ ~r × A~ 0 iβΣm = iβ~Σ · ~r × A~ 0 q6=l m l,m,q m and employs the notion of the the tensor contraction ~ 0T w~α = − ∑ A0l wlq αq . −A (7.61) l,q In summary, equation (7.59) introduced the 2 × 2 matrices dd ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ M uu = − M = A ·~ s ( k + n k , k + m k ) 1 + i r ( k + n k , k + m k ) × A ·~σ 0 0 L L L L m,n m,n M ud m,n = (7.62) ~ ~ ~ ~ ~ σ−A ~ 0T w(~k + n~k L ,~k + m~k L )~σ + i A ~ 0 · ~h(~k + n~k L ,~k + m~k L ) 1 . M du m,n = t ( k + n k L , k + m k L ) A0 ·~ (7.63) The matrix product in equation (7.58) can be performed with the help of the matrices (7.62) and (7.63). CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 63 Only the diffraction of quantum states with positive eigen energy into states of positive eigen energy is of relevance in the case of the Kapitza-Dirac effect, which corresponds to the upper left 2 × 2 matrix subentry of equation (7.58), which results in Z t3 Z t2 e 3 Z t dt3 dt2 dt1 sin(ωt3 ) sin(ωt2 ) sin(ωt1 ) U rd,ul;3,0 (t, t3 ) = − i2h̄ t0 t0 t0 uu uu uu ud du uu · phase1 M 3,2 M 2,1 M 1,0 + phase2 M 3,2 M 2,1 M 1,0 uu ud du + phase3 M 3,2 M 2,1 M 1,0 ud dd du + phase4 M 3,2 M 2,1 M 1,0 (7.64) ! . The additional “ul” index in U rd,ul;3,0 denotes, that only the upper left 2 × 2 matrix subentry of equation (7.58) is considered. The propagator (7.64) contains the four different phases i phase1 = exp − ( E3 (t − t3 ) + E2 (t3 − t2 ) + E1 (t2 − t1 ) + E0 (t1 − t0 )) h̄ i uu uu uu = exp − E3 t − E0 t0 + ∆E2,3 t3 + ∆E1,2 t2 + ∆E0,1 t1 , h̄ i phase2 = exp − ( E3 (t − t3 ) − E2 (t3 − t2 ) + E1 (t2 − t1 ) + E0 (t1 − t0 )) h̄ i du ud uu E3 t − E0 t0 + ∆E2,3 t3 + ∆E1,2 t2 + ∆E0,1 t1 = exp − , h̄ (7.65) i phase3 = exp − ( E3 (t − t3 ) + E2 (t3 − t2 ) − E1 (t2 − t1 ) + E0 (t1 − t0 )) h̄ i uu du ud E3 t − E0 t0 + ∆E2,3 t3 + ∆E1,2 t2 + ∆E0,1 t1 = exp − , h̄ (7.66) i phase4 = exp − ( E3 (t − t3 ) − E2 (t3 − t2 ) − E1 (t2 − t1 ) + E0 (t1 − t0 )) h̄ i du dd ud = exp − E3 t − E0 t0 + ∆E2,3 t3 + ∆E1,2 t2 + ∆E0,1 t1 , h̄ (7.67) (7.68) with the abbreviations uu ∆En,m = En − Em , ud ∆En,m = En + Em , du ∆En,m = − En − Em and dd ∆En,m = − En + Em . (7.69) Note, that the up (u) and down (d) indices of the matrices M coincide with the up and down indices of the energy differences En,m of the corresponding phases. One may assign some physical interpretation to the product of the 2 × 2 matrices times the phase. The phase starts out with the positive oscillation uu du energy E0 times t1 − t0 . At time t1 either the matrix M 1,0 or the matrix M 1,0 is applied. In the case, uu in which M 1,0 is applied, the phase has again a positive oscillation energy E1 times t2 − t1 . In the du case of the application of M 1,0 however, the phase has a negative oscillation energy E1 times t2 − t1 . In general, the sign of the energy En−1 of the phase before time tn is related to the upper right index of the matrix M ?? n,n−1 and the sign of energy En after the time tn is related to the upper left index. Therefore, ?? ?? ?? the products of the matrix product M 3,2 M 2,1 M 1,0 times the preceding phase can be interpreted as a 64 CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR Energy E3 Time E2 t3 E1 E0 t2 t1 U 0;3,3 (t, t3 ) V 3,2 (t3 ) U 0;2,2 (t3 , t2 ) V 2,1 (t2 ) U 0;1,1 (t2 , t1 ) V 1,0 (t1 ) U 0;0,0 (t1 , t0 ) Positive eigen energy uu M3,2 +mc2 −mc2 uu M2,1 ud M2,1 ud M3,2 du M1,0 Negative eigen energy Mode uu M1,0 du M2,1 3 2 dd M2,1 1 0 Figure 7.2: This figure illustrates the third order time dependent perturbation theory of equation (7.64) in a similar way as it is presented in figure 7.1. The solid arrows correspond to the contribution, in which the electron propagates in positive energy eigenstates in the whole diffraction process. The dotted arrows correspond to the contribution, in which the electron is scattered to a negative energy eigenstate at time t1 and is scattered back into a positive energy eigenstate at time t2 . The dashed arrows correspond to the contribution, in which the electron is scattered to a negative energy eigenstate at time t2 and is scattered back into a positive energy eigenstate at time t3 and the dash-dotted line corresponds to a contribution, in which the electron is scattered to a negative Energy eigenstate at time t1 stays a negative energy eigen at t2 and is scattered back into a positive energy eigenstate at time t3 . quantum mechanical pathway of an electron which may be scattered into positive or negative energy eigenstates, depending on the upper indices of the matrices M ?? n,n−1 . Equation (7.64) contains all four possible combinations of matrix products, which are sketched in figure 7.2. The upper left index of the left matrix and the upper right index of the right matrix is an u, because positive incoming and outgoing electron eigen states are required by considering the upper left 2 × 2 subentry of equation (7.58). Similar to section 7.2 a time integration over the vanishing term uu ∆E0,3 + h̄ω (7.70) is demanded, because this term will diverge in time and dominate over all other contributions. And similarly to section 7.2, this term occurs in the last integral over t3 , where the former integration times t1 and t2 are substituted into t3 by the upper integration limits. Therefore, the time-dependent phase exp(−iωt3 ) has to be searched in the expansion of sin(ωt1 ) sin(ωt2 ) sin(ωt3 ), in which t1 and t2 have to be substituted into t3 . From the expansion of the sine product " i sin(ωt1 ) sin(ωt2 ) sin(ωt3 ) = − eiω (−t1 −t2 −t3 ) + eiω (+t1 −t2 −t3 ) + eiω (−t1 +t2 −t3 ) − eiω (+t1 +t2 −t3 ) 8 # + eiω (−t1 −t2 +t3 ) − eiω (+t1 −t2 +t3 ) + eiω (−t1 +t2 +t3 ) + eiω (+t1 +t2 +t3 ) (7.71) only the terms i iω (+t1 −t2 −t3 ) i iω (−t1 +t2 −t3 ) i iω (−t1 −t2 +t3 ) e , e , e (7.72) 8 8 8 contribute after the substitution of t1 and t2 . The time integration over t1 and t2 for all three terms times phase1 will be calculated in the following. The same integration of all three terms times phase2 , CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 65 ?? are phase3 and phase4 is very similar, except that the u and d indices of the energy differences ∆En,m different. But Different u and d indices do not affect the calculation. The integral of the first term of (7.72) times phase1 results in Z Z t3 Z t2 i i t uu uu uu dt3 dt2 dt1 exp − E3 t − E0 t0 + (∆E2,3 + h̄ω )t3 + (∆E1,2 + h̄ω )t2 + (∆E0,1 − h̄ω )t1 8 t0 h̄ Z t Z t3 i 1 h̄ uu uu dt3 dt2 exp − E3 t − E0 t0 + (∆E2,3 + h̄ω )t3 + ∆E0,2 t2 = − uu (7.73) ∆E0,1 − h̄ω 8 t0 h̄ Z i 1 ih̄2 t 1 uu dt3 exp − E3 t − E0 t0 + (∆E0,3 + h̄ω )t3 . = − uu uu 8 ∆E0,1 − h̄ω ∆E0,2 h̄ t0 The integral of the second term of (7.72) times phase1 results in Z Z t3 Z t2 i i t uu uu uu dt3 dt2 dt1 exp − E3 t − E0 t0 + (∆E2,3 + h̄ω )t3 + (∆E1,2 − h̄ω )t2 + (∆E0,1 + h̄ω )t1 8 t0 h̄ Z Z t3 1 i h̄ t uu uu = − uu dt3 dt2 exp − E3 t − E0 t0 + (∆E2,3 + h̄ω )t3 + ∆E0,2 t2 (7.74) ∆E0,1 + h̄ω 8 t0 h̄ Z i 1 ih̄2 t 1 uu dt exp − E t − E t + ( ∆E + h̄ω ) t = − uu . 3 3 0 0 3 0,3 uu 8 ∆E0,1 + h̄ω ∆E0,2 h̄ t0 The integral of the third term of (7.72) times phase1 results in Z Z t3 Z t2 i t i uu uu uu dt3 dt2 dt1 exp − E3 t − E0 t0 + (∆E2,3 − h̄ω )t3 + (∆E1,2 + h̄ω )t2 + (∆E0,1 + h̄ω )t1 8 t0 h̄ Z t Z t3 h̄ 1 i uu uu dt3 dt2 exp − E3 t − E0 t0 + (∆E2,3 − h̄ω )t3 + (∆E0,2 + 2ωt)t2 = − uu ∆E0,1 + h̄ω 8 t0 h̄ Z ih̄2 t 1 1 i uu = − uu dt exp − E t − E t + ( ∆E + h̄ω ) t . (7.75) 3 3 0 0 3 0,3 uu + 2ωt 8 ∆E0,1 + h̄ω ∆E0,2 h̄ t0 Summing up the three integrals (7.73), (7.74) and (7.75) yields PF1 = 1 1 1 1 1 1 uu − h̄ω ∆Euu + ∆Euu + h̄ω ∆Euu + ∆Euu + h̄ω ∆Euu + 2ωt ∆E0,1 0,2 0,2 0,2 0,1 0,1 times Z t t0 i uu dt3 exp − E3 t − E0 t0 + (∆E0,3 + h̄ω )t3 h̄ (7.76) (7.77) times −ih̄2 /8. The integration over the phases (7.66), (7.67) and (7.68) results in the prefactors PF2 = PF3 = PF4 = 1 1 1 1 1 1 + + , uu − h̄ω uu + h̄ω uu + h̄ω ud ud ud + 2ωt ∆E0,1 ∆E0,1 ∆E0,1 ∆E0,2 ∆E0,2 ∆E0,2 (7.78) 1 1 1 1 1 1 uu + uu + uu + 2ωt , ud ud ∆E ∆E ∆E − h̄ω ∆E0,1 + h̄ω ∆E0,1 + h̄ω 0,2 0,2 0,2 (7.79) ud ∆E0,1 1 1 1 1 1 1 + + ud ud ud ud ud − h̄ω ∆E0,2 ∆E0,1 + h̄ω ∆E0,2 ∆E0,1 + h̄ω ∆E0,2 + 2ωt ud ∆E0,1 (7.80) times equation (7.77) times −ih̄2 /8. The last integral over t3 of equation (7.77) is identical to (7.29), if one accounts for the different energy-momentum relation Ennr → En in the Dirac case. Taking together CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 66 all calculation results, one obtains uu uu uu ud du uu uu ud du ud dd du U rd,ul;3,0 (t, t0 ) = PF1 M 3,2 M 2,1 M 1,0 + PF2 M 3,2 M 2,1 M 1,0 + PF3 M 3,2 M 2,1 M 1,0 + PF4 M 3,2 M 2,1 M 1,0 ( on resonance e3 −i( E3 t− E0 t0 )/h̄ t − t0 · − 6 e (7.81) i −i∆ωt − e−i∆ωt0 2 h̄ e off resonance , ∆ω for equation (7.58). A first order Taylor expansion with respect to k L and k3 for vanishing A2 and vanishing k2 yields e 1 U rd,ul;3,0 (t, t0 ) = 12h̄ m2 c3 5 √ A3 h̄k3 1 − h̄k L A3 (iσ 2 ) 2 e2 A23 −i(E3 t− E0 t0 )/h̄ e ( t − t0 ) , 4mc2 (7.82) where the vector~k L is pointing in the x1 -direction. The vanishing A2 , k2 and the direction of k L has been chosen, such that a Taylor expansion is feasible with Mathematica. In order to recover an equation, which is independent of this specific geometry, one may perform the following replacements in (7.82). ~ 0 /| A ~ 0 | of the vector ~k, parallel to the The product A3 k3 is nothing, but the component k k = ~k · A ~ 0 |, vector potential direction of the external laser beam times the amplitude of the vector potential | A 2 2 ~ ~ ~ ~ resulting in A3 k3 = | A0 |k k = k · A0 . Similarly, A can be written as A . The term, containing the σ 2 3 0 ~ 0 . Inserting all can be recovered as scalar product of the vector ~σ times the cross product of ~k L with A replacements in (7.82) yields U rd,ul;3,0 (t, t0 ) = 1 e 12h̄ m2 c3 5 ~ ~ ~ ~ √ A σ) 0 · h̄ k1 + h̄ k L × A0 · (i~ 2 ~2 e2 A 0 −i( E3 t− E0 t0 )/h̄ e ( t − t0 ) , 4mc2 (7.83) which is of the same form as the propagator of perturbation theory with the Pauli equation (7.39), except that the prefactors of the 2 × 2 matrices are different. 7.3.2 Interpretation Comparison with Pauli theory There is no non-relativistic limit, in which the propagator of the Dirac equation (7.83) and the propagator of the Pauli equation (7.39) coincide. The reason for this counter-intuitive property is, that there is no configuration for which the laser frequency and initial electron momentum are non-relativistic for the 3-photon Kapitza-Dirac effect. This property is already discussed in section 2.2.4. Therefore, the 3-photon Kapitza-Dirac effect always occurs in a parameter regime, in which the validity of Pauli theory breaks down. Thus, even though the results of Pauli and Dirac theory are similar, it is clear, that they cannot coincide excactly. Electron spin rotation Similar to subsection 7.2.2, equation (7.83) has the same form as the proposed SU (2)-ansatz (6.4) for the propagator matrix entry, with vanishing imaginary part of ~n. The parameters for this ansatz assume 67 CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR the values " #1 2 2 2 ~2 5 1 e e2 A 0 ~ 0 · h̄~k + h̄~k L × A ~0 √ A P3,0 (t, t0 ) = ( t − t0 ) , 12h̄ m2 c3 4mc2 2 E3 t − E0 t , h̄ " # 1 2 2 − 2 α3,0 (t, t0 ) 5 5 ~ ~ 0 · h̄~k + h̄~k L × A ~0 √ A cos = √ A0 · h̄~k , 2 2 2 # 1 " 2 2 − 2 α3,0 (t, t0 ) 5 ~ ~ ~ ~ ~ ~ √ A0 · h̄k + h̄k L × A0 ~n3,0 (t, t0 ) sin = −h̄k L × A0 . 2 2 (7.84a) (7.84b) φ3,0 (t, t0 ) = −i (7.84c) (7.84d) Like in subsection 7.2.2, the four equations (7.84) are unique, because of the unitarity of the SU (2) representation. One analogously concludes as in subsection 7.2.2, that the rotation angle α3,0 (t, t0 ) and the rotation axis ~n3,0 (t, t0 ) are constant in time. The unit vector ~n3,0 = − ~k L × A ~0 ~B = 0 ~ 0| |~B0 | |~k L × A (7.85) of the rotation axis of the electron spin points in the direction of the magnetic field ~B0 of the laser beam, too. The rotation angle α3,0 is again computed by multiplying equation (7.84d) with equation (7.85) and dividing it by equation (7.84c), resulting in √ α ~ 0| 2 |~k L × A 3,0 tan = . (7.86) ~ 0 ·~k 2 5 A Rabi frequency The quantity † U nd,ul;3,0 ( T, 0)U nd,ul;3,0 ( T, 0) = 1 12h̄ 2 # " 2 2 5 ~ e 2 ~ ~ ~ √ A0 · h̄k + h̄k L × A0 m2 c3 2 ~2 e2 A 0 4mc2 !2 T2 1 , (7.87) originating from propagator subentry (7.83) is of the same form as the Taylor expansion (7.46). Therefore one may conclude the Rabi frequency s 2 2 ~2 5 1 e e2 A 0 ~ ~k L × A ~ ~0 . √ ΩR = A · h̄ k + h̄ (7.88) 0 6h̄ m2 c3 4mc2 2 of the 3-photon Kapitza-Dirac effect, similarly to subsection 7.2.2. Spin-flip probability Similarly to subsection 7.2.2, the time evolution of the expansion coefficients cσn can be computed by equation (6.1). If one assumes an initial configuration, in which only the 0 mode with spin up is occupied, one may compute ~2 ~ · h̄~k e2 A 1 5e A 0 −i( E3nr t− E0nr t0 )/h̄ √ 0 e (t − t0 )c0+↑ (t0 ) , (7.89a) 12h̄ 2m2 c3 4mc2 ~2 ~ 0 e2 A 1 eh̄~k L × A 0 −i( E3nr t− E0nr t0 )/h̄ c3+↓ (t) = (U nd;a,b )(+↓,+↑) (t, t0 )c0+↑ ( T ) = e (t − t0 )c0+↓ (t0 ) . (7.89b) 2 3 12h̄ m c 4mc2 c3+↑ (t) = (U nd;a,b )(+↑,+↑) (t, t0 )c0+↑ (t0 ) = CHAPTER 7. ELECTRON SPIN DYNAMICS: ANALYTICAL SMALL-TIME BEHAVIOR 68 With the more specific initial condition c0+↑ (0) = 1 one obtains |c3+↑ ( T )|2 |c3+↓ ( T )|2 1 12h̄ 2 ~ · h̄~k 5e A √ 0 2m2 c3 1 12h̄ 2 ~0 eh̄~k L × A 2 3 m c = = !2 !2 ~2 e2 A 0 4mc2 !2 ~2 e2 A 0 4mc2 T2 !2 and T2 . (7.90a) (7.90b) after time T. Analogous to subsection 7.2.2 one can compute the non-spin-flip and spin-flip probabilities √ √ ~ 0 · h̄~k/ 2)2 (5k k / 2)2 (5e A 1 √ Pnoflip = = = and (7.91a) √ 2 2 2 ~ 0 · h̄~k/ 2)2 + (eh̄~k L × A ~ 0 )2 1 + 2k L /(25k2k ) (5k k / 2) + (k L ) (5e A Pflip = ~ 0 )2 ( k )2 1 (eh̄~k L × A √ L = = . √ 2 / (2k2 ) + 1 2 2 2 2 ~ ~ ~ ~ 25k (5k k / 2) + (k L ) (5e A0 · h̄k/ 2) + (eh̄k L × A0 ) L k (7.91b) Chapter 8 Electron spin dynamics: Numerical results In this chapter, the quantum dynamics of the 3-photon Kapitza-Dirac effect is solved by numerical application of the quantum wave equations in chapter 4, in analogy to the 2-photon Kapitza-Dirac effect of chapter 5. A focus is on the spin properties, which clearly shows up in the 3-photon KapitzaDirac effect, in contrast to the 2-photon Kapitza-Dirac effect. The considerations on the general electron spin properties of chapter 6, together with the perturbative short-time solution of chapter 7 are verified with the numerical simulations in this chapter. The chapter starts with the detailed presentation of the 3-photon Kapitza-Dirac effect of reference [25]. After that, the spin-flip probability and the spin rotation properties of the 3-photon KapitzaDirac effect are discussed in section 8.2. The properties of a slower turn on and turn off of the laser field amplitude are considered similarly to section 5.2. At the end, the resonance peak structure of the diffraction probability is discussed (section 8.5) and the Rabi frequencies from the perturbative solution 7 are verified (section 8.6). 8.1 Setup One may use the resonance condition (2.17) from energy and momentum conservation for the determination of the initial momentum ~p1,in of the 3-photon Kapitza-Dirac effect, similarly to chapter 5. The number of absorbed photons is 2 and the number of emitted photons is 1. The photon energy of 3.1 keV = 6.1 · 10−3 mc2 of the laser beam is adopted from [25], as well as the initial electron momentum in laser polarization direction of 2h̄k L /5. Equation (2.17) yields a value of 176 keV/c = 0.347mc for the initial momentum of the electron in laser propagation direction. The ponderomotive amplitude is chosen to be 1.16 · 10−2 mc2 , as it is used in [25]. The resulting quantum dynamics is shown in figure 8.1. The initial condition of the simulation is +↑ c0 (0) = 1 and all other expansion coefficients are 0. The amplitude of the vector potential is turned on for 10 laser cycles and off for 10 laser cycles, according to the envelope function (5.4). One can identify Rabi oscillations in the quantum dynamics and may use the Rabi frequencies and spin-flip probabilities from perturbation theory of section 7.3 for expressing the curves with sine and cosine functions as it is done for the 2-photon Kapitza-Dirac effect in section 5. For the chosen parameters, the spin-flip probability (7.91b) evaluates to 1/3 and the probability of no spin-flip (7.91a) evaluates to 69 70 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 1.0 |c0+↑ |2 probability 0.8 |c3+↑ |2 0.6 |c3+↓ |2 P3 0.4 P3↑ 0.2 0.0 P3↓ 0 500 1000 1500 T/TL time in laser cycles 2000 2500 Figure 8.1: This figure shows a similar time evolution of the Dirac equation (4.64), as in figure 5.2, but this time for the case of the 3-photon Kapitza-Dirac effect. The corresponding parameters of the laser and the electron are introduced in section 8.1. The laser intensity of this simulation corresponds to 2.0 · 1023 W/cm2 , with a laser wave length of 0.4 nm. The electron momentum is 176 keV/c in laser propagation direction and 1.2 keV/c in laser polarization direction. The solid lines are the analytical estimate (8.1) and the crosses are data points of the numerical simulation [25]. The Rabi transition evolves between mode 0 and 3, implying 3 transferred photon momenta, in contrast to the two photon momenta of the 2-photon Kapitza-Dirac effect in section 5. Furthermore, +↓ the diffraction probability of the electron beam consists of a spin-flipped part |c3 |2 and a not spin-flipped part |c3+↑ |2 . 2/3. Therefore, one may assume ΩR ΩR P0 ( T ) = cos T− 13.75 TL , 2 2 2 ΩR ↑ 2 ΩR T− 13.75 TL , P3 ( T ) = sin 3 2 2 ΩR 1 ↓ 2 ΩR T− 13.75 TL , P3 ( T ) = sin 3 2 2 2 (8.1a) (8.1b) (8.1c) where P0 ( T ), P3↑ ( T ) and P3↓ ( T ) should approximate the probabilities |c0+↑ ( T )|2 , |c3+↑ ( T )|2 and |c3+↓ ( T )|2 in figure 8.1. The frequency Ω R is the Rabi frequency (7.88), derived from perturbation theory of the Dirac equation and TL is the time of one laser period TL = 2π/ck L . The number of 13.75 laser periods accounts for the turn on and turn off time of the external laser field and is determined according to the considerations of section 8.4. Figure 8.1 may mislead the reader to assume, that the quantum dynamics of the 3-photon KapitzaDirac effect only takes place in the 0th and the 3rd mode and all other modes are not occupied. However, figures 8.2, 8.3 and 8.4 show, that the opposite is the case. If the external laser field is turned on, the occupation probability distributes over many neighboring modes and rejoins back in mode 3 after the interaction of 700 laser periods. This is a remarkable property, because there is a priori no reason that quantum dynamics oscillates according to the sine shaped Rabi oscillations (8.1) and that parameters like the Rabi frequency and the spin-flip probability agree with the perturbatively derived values. In particular, the spread of the occupation probability over many neighboring modes implies, that higher order terms of time-dependent perturbation theory need to be accounted for, in order to 71 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS | c −4 | 2 1.0 | c −2 | 2 | c −1 | 2 probability 0.8 | c0 |2 0.6 | c1 |2 | c2 |2 0.4 | c3 |2 0.2 0.0 | c4 |2 0 5 10 15 20 25 t/Tcycle (number of laser cycles) 30 | c5 |2 | c7 |2 Figure 8.2: This figure shows the same time evolution as in figure 8.1 for the duration of the first 35 laser periods. The difference is that the occupation probability ∑σ |cσn |2 for each mode n is plotted while the external field amplitude is not turned off. In contrast to that, a turn off of 10 laser cycles is propagated for each data point in figure 8.1. One can see, that the occupation probability of the 0 mode decreases and the probability is distributed over the neighboring modes during the turn on phase of the interaction. assure validity of perturbation theory. However, the results in this, the next section and in section 8.6 indicate, that the predictions of third order time-dependent perturbation theory are applicable even in a non-perturbative parameter space. It should be pointed out that the numerical results in figure 8.2, 8.3 and 8.4 demonstrate impressively, that the numerical solution of the quantum wave equation is a powerful tool to prove the analytical estimations of chapter 7. Even though, the perturbative solutions of chapter 7 provide useful equations for describing the 3-photon Kapitza-Dirac effect, only a numerical solution can answer, how quantum dynamics evolves without approximations. 8.2 Spin properties of the 3-photon Kapitza-Dirac effect Figure 8.1 contains the probability that an initial electron with spin up is diffracted to a final electron with spin down, where the spin quantization axis points in x3 -direction. In general, the initial electron spin could point in any direction ~n, with polar coordinates θ, ϕ and one may detect, if the final spin points in the ~n0 direction, with polar coordinates θ 0 , ϕ0 . If the initial quantum state is parameterized by the Bloch state c0 (θ, ϕ) of equation (6.16) and the final quantum state, on which the wave function is projected to, is parameterized by the similar Bloch state c3 (θ 0 , ϕ0 ) with different angles θ 0 and ϕ0 , then the transition probability from c0 (θ, ϕ) to c3 (θ 0 , ϕ0 ) is given by the matrix element c3 (θ 0 , ϕ0 )† U 3,0 (t, t0 )c0 (θ, ϕ) . (8.2) This matrix element can be reasoned with the scalar product of c3+ (θ 0 , ϕ0 ) with c3+ of equation (6.7b), which is the spin-dependent transition probability. The scalar product of c3+ (θ 0 , ϕ0 ) with the righthand side of equation (6.7b) yields the matrix element (8.2). Therefore, the knowledge of the matrix U 3,0 (t, t0 ) can answer the above question of a spin dependent diffraction probability for a general incoming spin direction and a general outgoing spin direction. Chapter 6 assumes, that this propagator subentry U 3,0 (t, t0 ) should be of the form (6.4), with the SU (2) representation (6.5) and vanishing 72 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS | c −4 | 2 1.0 | c −2 | 2 | c −1 | 2 probability 0.8 | c0 |2 0.6 | c1 |2 | c2 |2 0.4 | c3 |2 0.2 0.0 665 | c4 |2 670 675 680 685 690 t/Tcycle (number of laser cycles) 695 700 | c5 |2 | c7 |2 Figure 8.3: This figure shows the same time evolution as in figure 8.1 for the duration of the last 35 laser periods of the interaction time of 700 laser periods. Similarly to figure 8.2, the occupation probability ∑σ |cσn |2 for each mode n is plotted while the external field amplitude is not turned off. When the external laser field is smoothly turned off all the distributed occupation probability joins back into mode 3 at time t = 700TL . Figure 8.4: This figure shows exactly the same data as in the figures 8.2 and 8.3, but for the whole interaction time of 700 laser periods and in a rainbow colored density plot. The color coding is the logarithm to base 10 of the occupation density ∑σ |cσn |2 of each mode at time t. Occupation probabilities lower than 10−2.5 are represented in dark blue. One can see, that the probability density distributes in momentum space during the interaction. The modes −4, −2, 0, 2 and 4 exchange their occupation probability with the modes −1, 1, 3, 5 and 7 in the interacting process. In the turn off phase of the last 10 laser cycles, the occupation probability accumulates to mode 3, resulting in the data point at interaction time T = 700TL in figure 8.1. 73 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 10−3 10−4 10−5 || M ||Fr 10−6 10−7 10−8 a=0 a=3 10−9 10−10 10−11 10−12 10−13 10−14 0 500 1000 1500 T/Tcycle (number of laser cycles) 2000 2500 Figure 8.5: This figure shows the Frobenius norm (8.3) of the propagator subentry U a,b (t, t0 ), subtracted by its approximation (6.4) for the data of the simulation, which is described in section 8.1. The data with a = 0 originates from the propagator U 0,0 (t, t0 ), which describes the quantum dynamics of the undiffracted beam, whereas a = 3 corresponds to the propagator U 3,0 (t, t0 ) of the diffracted beam. The propagator entry of the undiffracted beam is approximated well. In the case of the diffracted beam, the value between 10−4 till 10−5 means, that the parameters P, Φ, α and ~n are approximated well up to 4 till 5 decimal places, in the case of a full diffraction probability P ≈ 1. imaginary part of ~n3,0 . This property would be quite useful, because the diffraction properties, which are encoded in U 3,0 (t, t0 ) are reduced to the degrees of freedom of a diffraction probability, a complex phase, a rotation angle and a rotation direction, where the rotation refers to the rotation of the spin expectation value of the Foldy-Wouthuysen spin operator (3.40). In order to check, whether the approximation (6.4) can be applied to the simulated propagator subentry, the parameters P3,0 , φ3,0 , α3,0 and ~n3,0 are determined numerically and the Frobenius norm || M ||Fr = ∑ | M i,j | 2 !1 2 (8.3) i,j of the matrix M = U a,b (t, t0 ) − q iφa,b (t,t0 ) Pa,b (t, t0 )e cos α a,b (t, t0 ) 2 1 − i sin α a,b (t, t0 ) ~n a,b (t, t0 ) ·~σ 2 (8.4) is computed in figure 8.5. The Frobenius norm is used, because it is a matrix norm, which is compatible to the euclidean scalar product || · ||2 and it is easy to compute. Figure 8.5 tells, that the propagator subentry U 3,0 can be approximated with a precision of 4 to 5 decimal places and U 0,0 can be approximated even better. The successful approximation of the propagator by the parameters P, φ, α and ~n, with vanishing imaginary part ~n implies already, that the electron spin is rotated by the angle α around the direction ~n. The question arises, how these parameters evolve in time in the case of Kapitza-Dirac scattering. The figures 8.6 and 8.7 present the relevant data over the interaction time T. According to figure 8.6(a), the spin of the undiffracted beam remains unchanged. The figures 8.6(b) and 8.7 tell, that the electron spin is rotated by an angle of 70.6 degrees around the direction of the magnetic field of the counterpropagating laser beam. As stated above, the spin-flip probability of the diffracted beam can be deduced from the matrix element of the SU (2) representation of rotations (6.5). If the initial electron points in the x3 -direction 74 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 10−9 1.2 1.0 10−10 |~n0,0 sin(α0,0 /2)| 0.8 10−11 n1;3,0 0.6 n2;3,0 n3;3,0 0.4 10−12 0.2 10−13 10−14 0.0 0 500 1000 1500 2000 2500 T/Tcycle (number of laser cycles) −0.2 0 500 1000 1500 2000 2500 T/Tcycle (number of laser cycles) Figure 8.6: (a) Left figure. The value |~n0,0 sin(α0,0 /2)| is plotted for the data of the simulation, which is described in section 8.1. Since this value is below 10−9 , the angle α0,0 is also below 10−9 , which means, that the change of the electron spin of the undiffracted beam is negligible. (b) Right figure. The three components of the unit vector ~n3,0 are plotted for the data of the simulation, which is described in section 8.1. The vector points precisely in the x2 -direction for all interaction times T, which is the direction of the magnetic field of the external vector potential (2.1b). 90 α3,0 / degree 75 60 45 30 15 0 0 500 1000 1500 T/Tcycle (number of laser cycles) 2000 2500 Figure 8.7: The rotation angle α3,0 is plotted for the simulation data, shown in figure 8.1. The value of this angle is 70.6◦ independent of the interaction time T. CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 75 (θ = 0, φ = 0), as well as the final projecting state (θ 0 = 0, φ0 = 0) and the rotation axis was pointing in x2 -direction (~n3,0 = (0, 1, 0) T ), the matrix element (8.2) evaluates to c3 (0, 0)† S c0 (0, 0) = † 1 cos(α/2) 0 sin(α/2) − sin(α/2) cos(α/2) α 1 . = cos 0 2 (8.5) If the final projecting state was pointing in the − x3 -direction (θ 0 = π, φ0 = 0), the matrix element evaluates to † α 0 cos(α/2) − sin(α/2) 1 . (8.6) c3 (π, 0)† S c0 (0, 0) = = sin 1 sin(α/2) cos(α/2) 0 2 Note, that the two matrix elements (8.5) and (8.6) correspond to the observables |c3+↑ |2 and |c3+↓ |2 of figure 8.1 at the initial condition c0+↑ (0) = 1. The absolute square of these matrix elements with the angle α = 70.6 results in | cos(α/2)|2 = 0.666 and | sin(α/2)|2 = 0.334 , which is consistent with the spin-flip probabilities (7.91a) and (7.91b). This demonstrates, that the knowledge of the parameters P, φ, α and ~n of the propagator subentry U 3,0 (t, t0 ) is sufficient for computing the spin-flip probability in figure 8.1 and any other directed spin change probability. The rotation angle and the rotation axis of the propagator of the diffracted and the undiffracted electron beam do not depend on the interaction time. This means, that the electron spin of the undiffracted beam never changes its direction. Similarly, the electron spin of the diffracted beam is always rotated by the angle 70.6 degrees around the y direction for all times T. The only property, which evolves in time (and oscillates in Rabi cycles) is the diffraction probability of the diffracted and the undiffracted beam. 8.3 Variation of the spin rotation The spin-flip probability of the numerical results in figure 8.1 is deduced from perturbation theory by using the equations (7.91a) and (7.91b). Both probabilities depend on the parameters k L and k k . If these parameters are changed, the spin-flip probability changes, which implies, that also the angle of the electron spin rotation changes. This is tested by varying the electron momentum in laser polarization direction h̄k k in figure 8.8. The spin-flip probability changes indeed according to (7.91b). The parameters of the propagator subentries U 0,0 (t, t0 ) and U 3,0 (t, t0 ) are determined from the simulation data, similarly to section 8.2. Figure 8.9 shows, that the propagator is approximated well by the SU (2) ansatz (6.4). Therefore, one may proceed in interpreting the parameters of the propagator approximation. Figure 8.10(a) tells, that the spin of the undiffracted electron beam remains unchanged, analogously to section 8.2. One concludes from figure 8.10(b), that the vector ~n3,0 points in the x2 direction, which means that the spin of the diffracted electron is rotated around the magnetic field axis of the laser beam. Figure 8.11 tells, that the rotation angle of the of the electron spin rotation can be varied by varying k k , according to equation (7.86). 8.4 The beam envelope in the 3-photon Kapitza-Dirac effect Like in chapter 5, the amplitude of the external vector potential is turned on and turned off in a very short time, as compared to the full interaction time of the electron with the laser beam. And like in section 5.3, it is of interest to consider a more realistic time-dependent envelope of the external potential. This section considers the same envelope function (5.4), as in section 5.3. In order to treat the delay of the Rabi cycle, caused by the lower potential amplitude during the 76 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 1.0 analytical numerical 0.8 | sin(α3,0 /2)|2 Pflip 0.6 0.4 0.2 0.0 0.0 0.5 1.0 |k k | / k L 1.5 2.0 Figure 8.8: This figure shows the spin-flip probability of the diffracted beam for the data of the simulation, which is described in section 8.1 at the interaction time of one half Rabi period π/Ω R . The only parameter, which is varied, is the electron momentum in laser polarization direction h̄k k . The solid, analytical line originates from equation (7.91b), the red crosses represent the spin-flip probability from numerical simulation (see also reference [25]) and the blue pluses are the absolute value squared matrix elements (8.6). The angle alpha in (8.6) is the rotation angle of figure 8.11. All three methods for determining the electron spin-flip probability agree well with each other. 10−3 10−5 10−7 10−9 || M ||Fr 10−11 10−13 10−15 10−17 a=0 a=3 10−19 10−21 0.0 0.5 1.0 |k k | / k L 1.5 2.0 Figure 8.9: This figure shows the same Frobenius norm plot as in figure 8.5, but now for the data set presented in figure 8.8. Similarly to the conclusion of figure 8.5 the parameters P, Φ, α and ~n are approximated well up to 4 or 5 decimal places in the case of the diffracted beam and are determined even more accurately for the undiffracted beam. 77 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 10−8 1.0 0.5 10−10 0.0 10−11 −0.5 |~n0,0 sin(α0,0 /2)| 10−9 n1;3,0 n2;3,0 n3;3,0 −1.0 10−12 0.0 0.5 1.0 |k k | / k L 1.5 2.0 0.0 0.5 1.0 |k k | / k L 1.5 2.0 Figure 8.10: (a) Left figure This figure shows the same |~n0,0 sin(α0,0 /2)| as in figure 8.6, but for a variation with respect to k k . One similarly concludes, that the spin of the undiffracted beam remains unchanged. (b) Right figure This figure shows the same ~n3,0 as in figure 8.6(b) but also for a variation with respect to k k . Similarly, the electron spin is rotated around the x2 -direction of the magnetic field of the laser beam. 180 α3,0 / degree 150 120 analytical numerical 90 60 30 0 0.0 0.5 1.0 |k k | / k L 1.5 2.0 Figure 8.11: This figure shows the α3,0 parameter of the SU (2) ansatz (6.4), which has been determined for the numerical data of figure 8.8. The solid line corresponds to the analytical property (7.86), which has been derived by applying third order, time-dependent perturbation theory at the Dirac equation. The numerical data agrees well with the analytical derivation. 78 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS turn on and turn off, the Rabi transition is modeled by the differential equation Ω R (t) c3 2 Ω (t) ċ3 = i R c0 , 2 ċ0 = i (8.7a) (8.7b) similarly to the system of ordinary differential equations (5.5). The Rabi frequency (7.88) of the 3photon Kapitza-Dirac effect scales with the third power of the potential amplitude, such that one may write ( A3 )3 . (8.8) Ω R ( A3 ) = Ω R,max ( A3,max )3 The Rabi frequency Ω R,max is the frequency (7.88) with the maximal field amplitude A3,max . Analogously to section 5.3, the differential equation (8.7) is solved by the solution c0 (t) = cos(t0 ) (8.9a) 0 c3 (t) = i sin(t ) , with the warped time parameter t0 (t) = Z t Ω R (τ ) 0 2 (8.9b) dτ . (8.10) ! Like in section 5.3, one may compute this integral and solve the requirement t0 ( T ) = π/2 for T, resulting in 16 π . (8.11) T= Ω R,max 16 − 11 f Figure 8.12 tests, whether the quantum system undergoes a full Rabi cycle, if the fraction f is varied. For large f the diffraction probability decreases. This may be attributed to the small resonance peak, which is described in the next section. If one compares the Rabi frequencies of the 2-photon KapitzaDirac effect with the 3-photon Kapitza-Dirac effect, one obtains s 2 2 Ω R,3photon 1 e 5 ~ ~ ~0 . √ = + h̄~k L × A (8.12) A · h̄ k 0 2 3 Ω R,2photon 6h̄ m c 2 This means, that the Rabi frequency of the 3-photon Kapitza-Dirac effect is a factor 3.8 · 10−4 times smaller than the Rabi frequency of the 2-photon Kapitza-Dirac effect, for the chosen parameters of figure 8.12. A high amplitude of the external potential detunes the laser frequency of the resonance peak slightly. If the external potential is detuned during the turn on and turn off, the transition of the 3-photon Kapitza-Dirac effect may not take place to 100%, because there will always be a time, in which the Rabi transition is off-resonant. 8.5 The resonance peak Within this thesis, the resonance peak is referred to as the property, that the Kapitza-Dirac effect takes place only, if constraint of energy- and momentum conservation (see section 2.2) are fulfilled. If the laser frequency or the electron momentum are not fine tuned to each other no diffraction takes place. Of course, the parameter range for diffraction is not of measure zero, which means that there is a smooth transition between resonant parameters and off-resonant parameters and the diffraction probability smoothly decreases, if the parameters of the laser and the electron smoothly deviate from the resonance condition. It should be mentioned, that these peak properties have already been illustrated by Gush and Gush [31] for the case of the 2-photon Kapitza-Dirac effect. 79 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 1.0 probability 0.8 0.6 |c0+↑ |2 |c3+↑ |2 + |c3+↓ |2 0.4 0.2 0.0 0.0 0.2 0.4 0.6 f = ( Ton + Toff )/T 0.8 1.0 Figure 8.12: This figure shows the final transition probability of the 3-photon Kapitza-Dirac effect, if the fraction ∆T = f T of the turn on and turn off is varied, according to the envelope function (5.4). All simulation parameters are the same as in figure 8.1. The interaction time is extended by the factor 16/(16 − 11 f ) of equation (8.11), in order to compensate for the lower Rabi transition time. One can see, that the diffraction probability decreases for an increasing fraction of turn on and turn off duration. The resonance peak can be described by off-resonant Rabi cycles. Therefore, it is worth to exploit the theory of Rabi transitions of a two-level quantum system. The differential equation, which describes Rabi transitions is a more sophisticated version of the differential equation (8.7) and can be found for example in subsection 5.2.1 in the book of Scully and Zubairy [52]. Ω R −iφ+∆ωt e c3 2 Ω ċ3 = i R eiφ−i∆ωt c0 . 2 ċ0 = i (8.13a) (8.13b) This is a simple differential equation of a two level quantum system with detuning ∆ω, which evolves in Rabi cycles. The analytical solution of equation (8.13) can also be found in [52]. ∆ωt Ωt i∆ω Ωt Ω R −iφ Ωt c0 (t) = c0 (0) cos − sin + c3 (0)i e sin ei 2 (8.14a) 2 Ω 2 Ω 2 ∆ωt Ωt Ωt Ωt i∆ω Ω c3 (t) = c3 (0) cos + sin + c0 (0)i R eiφ sin e −i 2 (8.14b) 2 Ω 2 Ω 2 The frequency Ω= q Ω2R + ∆ω 2 (8.15) is the oscillation frequency of the off-resonant transition probability and is always larger or equal to the Rabi frequency Ω R . Similar to the initial condition of figure 8.1, the initial condition c0 (0) = 1 and c3 (0) = 0 may be inserted in the solution (8.14b), resulting in the time-dependent transition probability |c3 (t)|2 = Ω2R sin2 Ω2 Ωt 2 . (8.16) CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 80 This equation is not only to be seen as function of the time t, but also implicitly as function of the detuning ∆ω. The structure of equation (8.16) tells, that the transition probability would be peaked for zero detuning. Figure 8.13 shows a similar peak also for the case of the 3-photon Kapitza-Dirac effect, if the off-resonant frequency peak was adapted by a scaling of the detuning ∆ω. r ∆ω 2 (8.17) Ω = Ω2R + 2 b If one accounts for the effective peak-broadening parameter b = 45.7, which is obtained by a fit to the numerical data, equation (8.16) seems to approximate the resonance peak well. From the variation of the resonance condition (2.17), one also expects a detuning by changing the initial momentum of the electron, which should be related to the energy detuning by ∂p |n a + ne | 3 h̄∆ω = h̄∆ω . (8.18) |∆p1,in | = 1,in ∆ω ≈ ∂ω 2c 2c Therefore, one would expect the off-resonant frequency s 2c∆k1 2 Ω = Ω2R + 3b (8.19) in dependence of the momentum h̄k1 of the electron in laser propagation direction. Figure 8.14 verifies this property. It should be mentioned, that the resonance condition (2.17) yields the momentum p1 = 0.344468mc, which corresponds to 176.022 keV/c, but the position of zero detuning in figure 8.14 has the momentum p1 = 0.347017mc, corresponding to 177.325 keV/c. One may assume, that this detuning originates from the coupling of mode 0 and mode 3 to the neighboring modes. Since the classically expected and the numerically found momenta of the resonance peak differ by 1.303 keV/c, either the initial electron momentum or the laser frequency must be tuned. Otherwise, the diffraction probability would be negligibly small. 8.6 Rabi frequency of the 3-photon Kapitza-Dirac effect The Rabi frequency Ω R depends on various parameters of the laser and the electron, according to subsection (7.3.2). Figure 8.15 tests the dependency of Ω R on the momentum h̄k k in laser polarization direction of the electron. The Rabi frequency can be measured, by fitting the function ΩR T sin2 +φ (8.20) 2 at the time evolution of the numerical data |c3+↑ |2 + |c3+↓ |2 . One can see, that the numerical data agrees with the analytical model (7.88), of third order time-dependent perturbation theory. There is a small deviation, which can also be found in the plot of the time-dependent dynamics 8.1. This deviation can be attributed to the interactions of the modes 0 and 3 with the neighboring modes. ~ 0 | of the laser beam, which is One may also vary the frequency ck L or the potential amplitude | A shown in figure 8.16 and 8.17. The setups of both figures differ from the setup, which is described in section 8.1. The reason is, that if the interaction parameter and in particular the laser frequency are changed by orders of magnitudes, the quantum dynamics of the 3-photon Kapitza-Dirac effect may interfere with other quantum dynamical effects, for example the 2-photon Kapitza-Dirac effect. Therefore, even though there always exists an analytical short-time propagator from perturbation theory and therewith a Rabi-frequency, there is no guarantee, that a well-formed 3-photon Kapitza-Dirac transition, as it is shown in figure 8.1, takes place in the quantum system. In order to fit at least the short-time time evolution of the diffraction probability, it is useful to start with the mode of the higher 81 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 1.0 P3 ( T ) probability 0.8 P3 ( T ) fit P3,max 0.6 P3,max fit 0.4 0.2 0.0 −400 −300 −200 −100 0 h̄c∆k L /(eV) 100 ↑ 200 300 400 ↓ Figure 8.13: This figure shows the diffraction probability P3 = |c3 |2 + |c3 |2 , for the setup of section 8.1, but for a variation of the laser frequency ck L by c∆k L . The interaction time T is one half Rabi cycle π/Ω R . P3 is fitted at the function (8.16) with off-resonant frequency (8.17) and fit parameter b = 45.7 . The maximum diffraction probability, independent of interaction time T is also plotted, with the corresponding function (8.16), in which the sine is set to 1. 1.0 P3 ( T ) probability 0.8 P3 ( T ) fit P3,max 0.6 P3,max fit 0.4 0.2 0.0 −400 −300 −200 −100 0 100 h̄∆k1 /(eV/c) 200 300 ↑ ↓ 400 Figure 8.14: This figure shows the same variation of the diffraction probability P3 = |c3 |2 + |c3 |2 as figure 8.13, but this time with respect to the variation of the initial electron momentum h̄k1 in laser propagation direction. Equation (8.16) also applies for the parameter variation with respect to h̄k1 in the case of the detuning (8.19) and peak-broadening parameter b = 45.7 . 82 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS 8 7 analytical numerical Ω R /Ω R,0 6 5 4 3 2 1 0 0.0 0.5 1.0 |k k | / k L 1.5 2.0 Figure 8.15: This figure shows the Rabi frequency Ω R of the 3-photon Kapitza-Dirac effect, for the variation of the momentum h̄k k in laser polarization direction of the electron. The numerical points are fits of the function (8.20) at the numerical data, presented in figure 8.8. The analytical line originates from equation (7.88) of time-dependent perturbation theory of the Dirac equation. The numerical data is in good agreement with the analytical prediction. momentum (3-mode) and to evaluate the mode with lower momentum. The Rabi frequency from mode 3 to mode 0 should be the same as from mode 0 to mode 3. Since there might not occur a full Rabi cycle, it might not be possible, to find a resonance peak, as described in section 8.5 and the offresonant quantum dynamics, described by equation (8.16) is fitted. However, in the case of a very small detuning, the fit parameters Ω R and Ω are almost degenerate and equation (8.16) is no longer a good fitting function. Therefore, depending on the dynamics, the short time evolution of mode 0 may only be fitted at the fourth order Taylor expansion of equation (8.16) with respect to t Ω2R 2 1 Ω2R Ω2 4 t − t + O(t5 ) . 4 3 4 4 (8.21) The turn-on and turn-off time of the external potential is zero in the case of the figures 8.16 and 8.17. This is reasoned by the amplitude of the external potential, which is much lower that the one in figure 8.1. Therefore, the approximation of the quantum dynamics with an infinite short turn-on and turn-off does not affect the diffraction probability very much. Since there is no turn on and no turn off time, the φ-offset in the time argument is omitted in the fitting functions, which are used for the figures 8.16 and 8.17. 83 CHAPTER 8. ELECTRON SPIN DYNAMICS: NUMERICAL RESULTS Ω R / (mc2 /h̄) 10−5 analytical Taylor expansion numerical 10−6 10−7 10−8 10−2 10−1 k L / (mc/h̄) 100 Figure 8.16: This figure shows the Rabi frequency of the 3-photon Kapitza-Dirac effect for an electron, which moves collinear√to the laser propagation direction. The amplitude of the external potential is A2 = 0 and A3 = 0.5 mc2 / h̄c. The final electron momentum in laser propagation direction is given by equation (2.17). The analytical curve is from the evaluation of U 0,3 in equation (7.81), whereas the dashed line originates from the Taylor expansion of the propagator (7.82). The crosses of the fits of the numerical time-evolution data agree well with the analytical curve. 10−4 Ω R / (mc2 /h̄) 10−5 analytical numerical 10−6 10−7 10−8 10−9 −1 10 A3 / (mc2 / √ 100 h̄c) Figure 8.17: This figure shows the same, as figure 8.16, but this time for a variation of the potential amplitude A3 of the external laser field. The laser wave number is 0.1mc/h̄, which corresponds to a photon energy of 51.1 keV or a laser wave length of 24.3 pm. The solid line is the Rabi frequency Ω R of equation (7.88) and it agrees well with the crosses of the fits of the numerical time-evolution data. Chapter 9 Conclusions and Outlook 9.1 Conclusions The main result of this thesis is the discovery and characterization of a Kapitza-Dirac effect, in which a diffracted electron interacts with 3 laser photons. A special feature of this 3-photon Kapitza-Dirac effect is the influence of the diffraction process on the electron spin. The 3-photon Kapitza-Dirac effect only appears, if the initial electron momentum and the frequency of the standing light wave are tuned with respect to each other. This resonance condition originates from conservation of energy and momentum of the combined system of the electron and the laser photons it interacts with. Section 2.2 provides a graphical interpretation of this energy- and momentum conservation, which allows a more intuitive understanding of the scaling of the resonance condition. One can infer from this resonance condition, that there is no non-relativistic limit for the 3-photon Kapitza-Dirac effect, which demands for a relativistic description of the effect. Even the resonance condition from the relativistic energy-momentum relation itself differs from the resonance condition from the non-relativistic energy-momentum relation in the case of the 3-photon Kapitza-Dirac effect. The quantum dynamics in this thesis is described by the Schrödinger equation, the Pauli equation, the Klein-Gordon equation, and the Dirac equation which are introduced in chapter 3. All four quantum wave equations are transformed into momentum space in chapter 4. Chapter 4 thereby makes use of the bi-scalar and bi-spinor matrix contractions of the appendices A and B. The quantum dynamics of the Kapitza-Dirac effect is solved numerically in the chapters 5 and 8. The quantum dynamics of the 2-photon Kapitza-Dirac effect is verified in chapter 5 with the numerical solution of the quantum wave equations in momentum space. The quantum dynamics of the 3-photon Kapitza-Dirac effect is elaborately discussed in section 8. The numeric solution is able to prove that a full Rabi cycle of the diffraction probability takes place for the 3-photon Kapitza-Dirac effect, and shows that the quantum dynamics features an in-field interaction of the electron with many neighboring modes in momentum space. A full Rabi transition is possible, even for a sine-shaped time-dependent envelope of the laser amplitude, in the case of the 2-photon Kapitza-Dirac effect. In the case of the 3-photon Kapitza-Dirac effect, the diffraction probability decreases to 50% for a sine-shaped laser envelope. A short time solution of the 3-photon Kapitza-Dirac effect is also provided in chapter 7 by applying time-dependent perturbation theory to the full Pauli equation (section 7.2) and the full Dirac equation (section 7.3). The derivation of the perturbative solution of the Dirac equation thereby takes advantage of the matrix notation of the bi-spinor matrix contractions in appendix B. A low laser frequency solution is obtained by a Taylor expansion of the perturbative solution. The perturbative solutions of the Pauli equation and the Dirac equation differ from each other even for the low frequency limit. This discrepancy is reasoned by the inapplicability of Pauli theory in the case of the always relativistic parameters of the 3-photon Kapitza-Dirac effect. The solution of time-dependent perturbation theory allows for the analytical derivation of the Rabi frequency, the spin-flip probability, and also the parameters of the SU (2)-rotation of the electron spin. These analytical parameters of the 3-photon 85 CHAPTER 9. CONCLUSIONS AND OUTLOOK 86 Kapitza-Dirac effect are verified by the numerical solution of the Dirac equation in section 8. The Rabi frequency of the 3-photon Kapitza-Dirac effect is, in contrast to the 2-photon Kapitza-Dirac effect, low and demands for high laser field strengths, which in turn require a relativistic description of the effect. Even though high laser intensities and short wavelengths in the X-ray regime are required for an experimental realization, a detection of the 3-photon Kapitza-Dirac appears feasible in the near future. Reference [25] discusses the implementation of such an experiment at the European X-ray free electron laser facility. A special property of the 3-photon Kapitza-Dirac effect is the rotation of the electron spin, when the electron is diffracted. The rotation angle of the electron spin depends on the electron momentum in laser propagation direction. This property opens the possibility to tune the spin-flip probability of the diffracted electron. The spin properties of the electron and the diffraction pattern are inferred from the propagator of the electron wave function. It is important to mention, that only the diffraction probability of the electron varies in time, whereas the spin properties and, in particular, the spin-flip probability of the diffracted electron are time-independent. The theoretical framework and the concepts which are introduced in this thesis open a precise understanding of the 3-photon Kapitza-Dirac effect and its spin properties. The methods presented can be generalized to n-photon Kapitza-Dirac effects. 9.2 Outlook This work considers the standard field configuration of the Kapitza-Dirac effect with two counterpropagating linearly polarized laser beams of equal laser frequency. This standard scenario may be modified and extended towards more general geometries such as unequal laser frequencies and different laser polarizations. In this way the understanding of the effect could be deepened further. The whole setup of the 3-photon Kapitza-Dirac effect may be Lorentz transformed into a frame, in which the absolute value of the incoming electron momentum equals the outgoing one. In this frame the two counter-propagating laser fields have unequal frequencies. As a consequence, according to the graphical considerations in section 2.2, the initial and final electron momentum and the laser frequency can be reduced at the same time for this configuration. This means that the 3-photon Kapitza-Dirac effect in a two color laser field has a non-relativistic limit in this inertial frame. This opens the possibility to compare the quantum dynamics of the Pauli equation and the dynamics of the Dirac equation in the non-relativistic limit, which would be an important consistency check for the theory of Kapitza-Dirac scattering. The investigation of the Kapitza-Dirac effect is based on two counter-propagating laser beams with equal linear polarization in this thesis. One may study the quantum dynamics of the Kapitza-Dirac effect also for different light polarizations. It is possible that the quantum dynamics is completely different from that of linearly polarized light and therefore new effects are expected. The change of the electron spin is of essential interest because the 3-photon Kapitza-Dirac effect explicitly demonstrates a change of the electron spin in the laser polarization direction, while the laser photons only carry an angular momentum in their propagation direction. An intuitive description of the angular momentum transfer from the laser photons to the electrons would be highly desirable, in particular for various laser polarizations. Appendix A Bi-scalar matrix relations The contraction of the identity matrix with the bi-scalar matrices (3.25) yields KG ~ 0 KG ~ 0 KG ~ ~ d ( k ) 1 + d ( k ) σ uKG (~k)uKG (~k0 )† = dKG ( k ) 1 + d ( k ) σ 1 1 + − + − KG ~ KG ~ 0 KG ~ KG ~ 0 KG ~ KG ~ 0 ~ KG ~ 0 = d+ (k)d+ (k ) + d− (k)d− (k ) 1 + dKG + (k )d− (k ) + d− (k )d+ (k ) σ 1 = tKG (~k,~k0 )1 + sKG (~k,~k0 )σ 1 . (A.1) The contraction of the σ 1 matrix yields uKG (~k )σ 1 uKG (~k0 )† = uKG (~k )uKG (~k0 )† σ 1 = tKG (~k,~k0 )1 + sKG (~k,~k0 )σ 1 σ 1 = sKG (~k,~k0 )1 + tKG (~k,~k0 )σ 1 . (A.2) The contraction of the σ 3 matrix yields KG ~ KG ~ 0 KG ~ 0 ~ uKG (~k)σ 3 uKG (~k0 )† = σ 3 dKG ( k ) 1 − d ( k ) σ d ( k ) 1 + d ( k ) σ 1 1 + − + − KG ~ KG ~ 0 KG ~ KG ~ 0 KG ~ KG ~ 0 ~ KG ~ 0 = dKG + ( k ) d+ ( k ) − d− ( k ) d− ( k ) σ 3 + d+ ( k ) d− ( k ) − d− ( k ) d+ ( k ) iσ 2 = f KG (~k,~k0 )σ 3 + rKG (~k,~k0 )iσ 2 . (A.3) If ~k equals ~k0 , the functions tKG , sKG , f KG and rKG of equation (4.43) simplify to KG ~ 2 ~ 2 tKG (~k,~k ) = dKG + (k ) + d− (k ) = mc2 + E(~k) 2 2 + mc2 − E(~k) 4E(~k)mc2 mc2 + E(~k) mc2 − E(~k ) = (mc2 )2 + E(~k)2 2E(~k )mc2 (A.4a) (mc2 )2 − E(~k)2 c2~p2 =− (A.4b) 2E(~k)mc2 2E(~k)mc2 2E(~k)mc2 2 2 mc2 + E(~k) − mc2 − E(~k) 4mc2 E(~k KG ~ 2 ~ 2 f KG (~k,~k ) = dKG = =1 (A.4c) + (k ) − d− (k ) = 4E(~k)mc2 4E(~k)mc2 rKG (~k,~k ) = 0 . (A.4d) ~ KG ~ sKG (~k,~k ) = 2dKG + (k )d− (k ) = = Therefore, the contraction of σ 3 in equation (A.3) turns into the property of uKG,σ (~k) beeing pseudo orthonormal u(~k)σ 3 u(~k)† = σ 3 . (A.5) 87 88 APPENDIX A. BI-SCALAR MATRIX RELATIONS Furthermore, the pseudo-orthonormal contracted free Hamiltonian turns into the relativistic energymomentum relation. This can be seen by expanding the pseudo-orthonormal contracted free KleinGordon Hamiltonian (3.18) into ~p2 KG ~ † KG ~ KG ~ 2 ~ u (k)σ 3 H 0 (k)u (k) = u (k)σ 3 (σ 3 + iσ 2 ) + σ 3 mc uKG (~k)† 2m 2 ~p + 2m2 c2 ~p2 KG ~ + σ1 uKG (~k)† (A.6) = u (k) 1 2m 2m ~p2 + 2m2 c2 ~p2 = tKG (~k,~k) + sKG (~k,~k) 1 2m 2m ~p2 ~p2 + 2m2 c2 + tKG (~k,~k) σ1 . + sKG (~k,~k) 2m 2m The prefactors of the 1 matrix simplify into the relativistic energy-momentum relation 2 + 2m2 c2 2 (mc2 )2 + E(~k)2 ~p2 + 2m2 c2 + (mc2 )2 − E(~k)2 ~p2 ~ ~ p p tKG (~k,~k) + sKG (~k,~k) = 2m 2m 4E(~k)m2 c2 2(mc2 )2 2c2~p2 + 2(mc2 )2 + 2E(~k )2 2(mc2 )2~p2 + (mc2 )2 + E(~k)2 2(mc2 )2 = (A.7) = 4E(~k )(mc2 )2 4E(~k)(mc2 )2 4(mc2 )2 E(~k)2 = = E(~k) 4E(~k)(mc2 )2 and the prefactor of the σ 1 matrix vanishes. sKG (~k,~k) = ~p2 + 2m2 c2 2m + tKG (~k,~k) −~p2 c2~p2 + 2(mc2 ) 2 ~p2 2m = −c2~p2 ~p2 + 2m2 c2 + (mc2 )2 + E(~k)2 ~p2 + 2(mc2 )2 + c2~p 4E(~k)m2 c2 4E(~k)m2 c2 2 ~p2 =0 (A.8) Taking together both results (A.7) and (A.8) yields uKG (~k)σ 3 H 0 (~k)uKG (~k)† = E(~k)1 , for equation (A.6). (A.9) Appendix B Bi-spinor matrix relations B.1 Calculation of bi-spinor contractions The calculations in this subsection make use of the commutation relations (3.28) of the Dirac matrices. The product of two spinor matrices u(~k )u(~k0 )† in equation (4.60a) may be expanded to: u(~k)u(~k0 )† = ∑ d+ (~k )1 + k l d− (~k) βαl · d+ (~k0 )1 + k0q d− (~k0 )αq β l,q = d+ (~k)d+ (~k0 ) +~k ·~k0 d− (~k)d− (~k0 ) 1 + ∑ k l d− (~k)d+ (~k0 ) − k0l d+ (~k)d− (~k0 ) βαl l + ∑ k l k0q d− (~k)d− (~k0 )αl αq l 6=q = d+ (~k)d+ (~k0 ) +~k ·~k0 d− (~k)d− (~k0 ) 1 + ∑ k l d− (~k)d+ (~k0 ) − k0l d+ (~k)d− (~k0 ) βαl (B.1) l + ∑ 1≤ l < q l < q ≤3 k l k0q − k q k0l d− (~k)d− (~k0 )αl αq = t(~k,~k0 )1 + ∑ r l (~k,~k0 ) βαl + l ∑ 1≤ l < q l < q ≤3 glq (~k,~k0 )αl αq . The contraction of β, namly u(~k) βu(~k0 )† of equation (4.60b) can be split up into three terms. Note, that the parameters of the left and right spinor matrix have the same parameter ~k, because different parameters are not required. u(~k ) βu(~k)† = ∑ d+ (~k )1 + k l d− (~k) βαl β d+ (~k )1 + k q d− (~k )αq β l,q = d+ (~k)2 β (B.2a) + ∑ k l d+ (~k)d− (~k) βαl β + k q d+ (~k)d− (~k) βαq β (B.2b) + ∑ k l k q d− (~k)d− (~k) βαl βαq β (B.2c) l,q l,q The second term (B.2b) can be simplified by using βαl β = − ββαl = −αl and noticing that the sum over the indices q and l yields the same term twice. The third term (B.2c) is a doubled sum consisting of 89 90 APPENDIX B. BI-SPINOR MATRIX RELATIONS 3 × 3 = 9 terms. The three terms with equal index l = q can be simplified by βαq βαq β = − ββαq αq β = − β and summed up, yielding −~k2 d− (~k)2 β. The six terms with unequal indices l 6= q are a sum over anti-symmetric matrices βαl βαq β = − βαq βαl β multiplied with symmetric factors k l k q . Therefore, these six terms vanish and the sum of (B.2a) and (B.2c) yields mc2 d+ (~k)2 −~k2 d− (~k)2 β = β. E(~k) (B.3) Therefore, the matrix (B.2) can be rewritten into ch̄k l mc2 u(~k) βu(~k)† = β− αl . E(~k ) E(~k ) (B.4) The contraction of αl , namly u(~k)αl u(~k0 )† of equation (4.60c) splits up into three terms too. u(~k)αl u(~k0 )† = ∑ d+ (~k)1 + k q d− (~k ) βαq αl d+ (~k0 )1 + k0j d− (~k0 )α j β q,j = d+ (~k)d+ (~k0 )αl + ∑ k q d− (~k)d+ (~k0 ) βαq αl + k0 d+ (~k)d− (~k0 )αl α j β j q,j + ∑ k q k0j d− (~k)d− (~k0 ) βαq αl α j β (B.5a) (B.5b) (B.5c) q,j The line (B.5b) consists of a sum of six different terms. In the case of equal indices q = l and j = l of the α matrices, line (B.5b) simplifies to k q d− (~k)d+ (~k0 ) β + k0j d+ (~k)d− (~k0 ) β = sl (~k,~k0 ) β . (B.6) If the indices of the α matricies in the sum are not equal (q 6= l and j 6= l), they need to be commuted, resulting in another minus sign. ∑ kq d− (~k)d+ (~k0 ) βαq αl + k0j d+ (~k)d− (~k0 )αl α j β = ∑ kq d− (~k)d+ (~k0 ) βαq αl − k0q d+ (~k)d− (~k0 ) βαq αl (B.7) q6=l q6=l j6=l Therefore, line (B.5b) can be reformulated into ∑ kq d− (~k)d+ (~k0 ) βαq αl + k0j d+ (~k)d− (~k0 )αl α j β = sl (~k,~k0 ) β + ∑ rq (~k,~k0 ) βαq αl . q,j (B.8) q6=l Line (B.5c) consists of a doubled sum containing 3 × 3 = 9 terms. In contrast to (B.2) the calculation depends additionally one the index l of the αl matrix. Therefore the nine terms in (B.5b) can be divided in • one term consisting of three identical α matricies (q = l = j), • two terms for which the indices of ~k and ~k0 are the same but not equal to the contracted αl (q = j 6= l), • four terms for which ~k or ~k0 have the same index of the contracted αl , but not the same index as the other ~k0 or ~k (q 6= l = j or q = l 6= j), • and two terms in which all three α have different indices (q 6= l 6= j,q 6= j) . 91 APPENDIX B. BI-SPINOR MATRIX RELATIONS If all three α are the same (q = l = j), the matricies in line (B.5c) simplify to βαl αl αl β = βαl β = − ββαl = −αl , yielding − k l k0l d− (~k)d− (~k0 )αl . (B.9) In the case (q = j 6= l) there is another minus sign from the commutation of alphas βαq αl αq β = − ββαq αl αq = αq αq αl = αl , resulting in the expression ∑ kq k0q d− (~k)d− (~k0 )αl (B.10) q6=l for line (B.5c). If the indices are (q 6= l = j or q = l 6= j) the matrices turn into βαq αl αl β = − ββαq = −αq , which yields ∑ kq k0l d− (~k)d− (~k0 ) βαq αl αl β + ∑ kl k0j d− (~k)d− (~k0 ) βαl αl α j β q6=l j6=l = − ∑ k l k0q d− (~k)d− (~k0 ) + k q k0l d− (~k)d− (~k0 ) αq = − ∑ wlq (~k,~k0 )αq . (B.11) q6=l q6=l The terms which consist of three different α matricies can be reduced by βαq αl α j β = − ββαq αl α j = −αq αl α j . The remaining three α matricies anti-commute with each other and are totally anti-symmetric with respect to their indices. Since this is a property of the totally anti-symmetric Levi-Civita tensor, one may write αq αl α j = ε qlj α1 α2 α3 . Furthermore, ε qlj is zero if two indices are equal. Therefore, the sum over the terms in line (B.5c) with three different indices can be rewritten into ∑ (see text) k q k0j d− (~k )d− (~k0 ) βαq αl α j β = − ∑ k q k0j d− (~k)d− (~k0 )ε qlj α1 α2 α3 q,j = ~el · (~k ×~k0 )d− (~k)d− (~k0 )α1 α2 α3 = hl (~k,~k0 )α1 α2 α3 , (B.12) where only terms with three different indices are counted in the sum of the first line. The identity ~k × ~k0 = ε lqj~k q~k j~el was used in the last but one step. Note, that α1 α2 α3 equals iγ5 of the Lorentz-kovariant Clifford algebra representation γµ . The results (B.5a), (B.9), (B.10) and (B.11) can be combined to d+ (~k)d+ (~k0 )αl − k l k0l d− (~k)d− (~k0 )αl + ∑ kq k0q d− (~k)d− (~k0 )αl − ∑ wlq (~k,~k0 )αq q6=l q6=l = d+ (~k)d+ (~k0 )αl − 2k l k0l d− (~k)d− (~k0 )αl + ∑ k q k0q d− (~k)d− (~k0 )αl − ∑ wlq (~k,~k0 )αq q q6=l = t(~k,~k0 )αl − ∑ wlq (~k,~k0 )αq . (B.13) q The sum of the terms (B.8), (B.12) and (B.13) results in a transformed equation (B.5). u(~k )αl u(~k0 )† = t(~k,~k0 )αl − ∑ wlq (~k,~k0 )αq + sl (~k,~k0 ) β + q B.2 ∑ rq (~k,~k0 ) βαq αl + hl (~k,~k0 )α1 α2 α3 (B.14) q6=l Verification of spinor properties The spinors (3.37) are orthogonal and are eigen vectors of the free Dirac Hamiltonian H 0 (~k ). This property is shown in terms of matrix relations in this section. In order to do so, the matrix relations (4.60) are employed with ~k = ~k0 for the left and right spin matrix. Therefore, the functions (4.59) need 92 APPENDIX B. BI-SPINOR MATRIX RELATIONS to be evaluated for ~k = ~k0 , resulting in t(~k,~k) = d+ (~k)2 +~k2 d− (~k)2 = 1 , ch̄k l sl (~k,~k) = 2k l d+ (~k)d− (~k ) = , E(~k) (B.15a) (B.15b) r l (~k,~k) = 0 , lq (B.15c) ql w (~k,~k) = 2k l k q d− (~k )d− (~k) = w (~k,~k ) , glq (~k,~k ) = 0 , l h (~k,~k ) = 0 . (B.15d) (B.15e) (B.15f) Inserting these coefficients into (4.60a) yields the orthonormality property of spinors in matrix notation. u(~k)u(~k)† = 1 (B.16) The contraction of the free Dirac Hamiltonian requires the evaluation of the term ch̄k ∑ u(~k)αl u(~k)† kl = ∑ kl αl t(~k,~k) − kl 2kl kq d− (~k)2 αq + kl E(~k)l β l l,q = ∑ k l αl t(~k,~k) − 2k l~k2 d− (~k)2 αl + l ch̄~k2 β E(~k ) (B.17) ch̄~k2 mc2 ~ ch̄~k2 = d+ (~k)2 −~k2 d− (~k)2 ~k ·~α + β= k ·~α + β. E(~k ) E(~k) E(~k) With this and equation (4.60b), the contraction of the free Hamiltonian H 0 (~k ) results in the signed relativistic energy-momentum relation. u(~k) H 0 (~k)u(~k)† = u(~k) ch̄~k ·~α + βmc2 u(~k )† ! ! 2 mc2 ~ ch̄2~k2 mc c =c h̄k ·~α + β + mc2 β− h̄~k ·~α (B.18) E(~k) E(~k) E(~k) E(~k) ! c2 h̄2~k2 + m2 c4 =β = E(~k) β . E(~k ) Appendix C Energy-momentum conservation This appendix derives analytical formulas for the resonance condition from energy and momentum conservation, which is discussed geometrically in section 2.2. The conservation of energy of the incoming and outgoing electron, together with the absorbed and emitted photons can be written as E(~pout ) = E(~pin ) + n a h̄ω − ne h̄ω . (C.1) The corresponding conservation of momentum reads c~pout = c~pin + n a h̄ω~e1 + ne h̄ω~e1 . C.1 (C.2) Non-relativistic energy-momentum conservation If (C.2) is inserted in (C.1) and the non-relativistic energy-momentum relation (3.4) is used, one obtains # " 2 p2 + p22,in + p23,in 1 h̄ω (n a + ne ) + p1,in + p22,in + p23,in = 1,in + (n a − ne )h̄ω . (C.3) c 2m 2m This can be rearranged to h̄ω (n a + ne ) p1,in h̄2 ω 2 ( n a + n e )2 + = h̄ω (n a − ne ) , 2 mc 2mc (C.4) and devided by h̄ω (n a + ne ) p1,in h̄ω ( n a + n e )2 + = (n a − ne ) . 2 mc 2mc A further rearrangement yields h i p1,in h̄ω 2 = −( n + n ) + n − n ( ) a e a e 2 mc mc ( n a + n e )2 (C.5) (C.6) for the dimensionless energy or p1,in n a + ne h̄ω n a − ne =− + . 2 mc 2 n a + ne mc for the dimensionless initial electron momentum in laser propagation direction. 93 (C.7) 94 APPENDIX C. ENERGY-MOMENTUM CONSERVATION C.2 Relativistic energy-momentum conservation One may also use the relativistic energy-momentum relation (3.22) in the combination of equation (C.1) and (C.2), resulting in q q m2 c4 + ((n a + ne )h̄ω + cp1,in )2 + c2 p22 + c2 p23 = m2 c4 + c2 p21,in + c2 p22 + c2 p23 + n a h̄ω − ne h̄ω (C.8) The momentum in 2 and 3-direction is absorbed in the increased mass q (C.9) m0 c2 = m2 c4 + c2 p22 + c2 p23 , which is already introduced in section 2.2. The square of equation (C.8) is m02 c4 + ((n a + ne )h̄ω + cp1,in )2 = m02 c4 + c2 p21,in + (n a − ne )2 h̄2 ω 2 + 2 (n a − ne ) h̄ω q m02 c4 + c2 p21,in . (C.10) It is useful to perform the following side calculations for a further transformation of this equation. ((n a + ne )h̄ω + cp1,in )2 = n2a h̄2 ω 2 + n2e h̄2 ω 2 + c2 p21,in + 2n a ne h̄2 ω 2 + 2cp1,in (n a + ne )h̄ω c2 p21,in + (n a − ne )2 h̄2 ω 2 = c2 p21,in + n2a h̄2 ω 2 + n2e h̄2 ω 2 − 2n a ne h̄2 ω 2 (C.11) (C.12) Inserting these side calculations in equation (C.10) results in 4n a ne h̄2 ω 2 + 2cp1,in (n a + ne ) h̄ω = 2 (n a − ne ) h̄ω q m02 c4 + c2 p21,in . (C.13) Dividing this by 2h̄ω yields 2n a ne h̄ω + cp1,in (n a + ne ) = (n a − ne ) q m02 c4 + c2 p21,in . (C.14) This equation is squared again. 4n2a n2e h̄2 ω 2 + 4n a ne h̄ωcp1,in (n a + ne ) + c2 p21,in (n a + ne )2 = (n a − ne )2 m02 c4 + c2 p21,in (C.15) Rearranging and dividing by 4n2a n2e m02 c4 yields p21,in 1 h̄ω p1,in n a + ne h̄2 ω 2 ( n a − n e )2 = 0. + + − m 0 c2 m 0 c n a n e m 02 c2 n a n e 4n2a n2e m 02 c4 The dimensionless energy h̄ω/m0 c2 in this equation has the two solutions q c2 p21,in + m02 c4 p h̄ω 1 , ± |n a − ne | = −(n a + ne ) 1,in 0 0 2 0 2 mc 2n a ne mc mc where the following expression appears in the square-root and is simplified to c2 p21,in (n a + ne )2 − 4ne n a c2 p21,in + (n a − ne )2 m02 c4 /4n2a n2e m02 c4 = c2 p21,in (n a − ne )2 + (n a − ne )2 m02 c4 /4n2a n2e m02 c4 = (n a − ne )2 c2 p21,in + m02 c4 /4n2a n2e m02 c4 . (C.16) (C.17) (C.18) APPENDIX C. 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Danksagung Zu allererst möchte ich meinem Betreuer Prof. Christoph H. Keitel herzlich danken, dass er mir die Möglichkeit gegeben hat, in seiner Arbeitsgruppe zu promovieren. Dies umfasst auch die vielen Tagungen und Konferenzen, die Sommer- und Winterschulen und Graduiertenseminare, die mein Verständnis von der Welt erweitert haben. Ich danke Prof. Keitel auch für den offenen und einfachen Umgangston, den ich sehr schätze. Ich danke auch meinem Mitbetreuer Prof. Carsten Müller, der immer Zeit für mich hatte und mir mit ruhiger und besonnener Stimme in allen Lagen weiter geholfen und Mut zugesprochen hat. Ich danke ihm für das aufmerksame Zuhören, seine aufgeschlossene Art ohne die ich meine Arbeit wohl nur schwer hätte zu Ende führen können. Ich danke zudem auch meinem Mitbetreuer Heiko Bauke, der immer Rücksprache mit mir gehalten hat und mir viele Möglichkeiten aufgezeigt hat, meine Arbeit zu verbessern. Seine Ratschläge haben die Struktur meiner Arbeit und auch die Struktur meines eigenen Arbeitens verbessert. Herzlichen Dank möchte ich auch an Prof. Rainer Grobe und die Kollaboration mit ihm richten. Ich danke ihm für die freundlichen und aufgeschlossenen Diskussionen, die er mit mir geführt hat. Ich danke auch meinen Eltern Gotthelf Ahrens und Marion Ahrens, die zwar physisch weit weg, aber im Geiste immer bei mir waren. Danke, dass ihr immer für mich da wart. Ebenso danke auch meinem Bruder Marco Ahrens, seiner Freundin Lena und meiner kleinen, lieben Nichte Klara. Einen ganz besonderen Dank möchte ich auch meinem freundlichen Kollegen Matthias Ruf zusprechen, der mich auf seine erfrischende Art in schwierigen Phasen meiner Promotion immer wieder aufgepäppelt und einen lebensfrohen Menschen aus mir gemacht hat. Einen großen Dank möchte ich auch an meinen Bürokollegen Wen-Te Liao richten, der mit der Reise nach Taiwan und vielen anderen Ausflügen in mir die wohl wertvollste Erinnerung in meiner Promotionszeit hinterlassen hat. Ich danke auch meinen beiden anderen Bürokollegen Enderalp Yakaboylu and Felix Mackenroth für die schöne Arbeitsatmosphäre in meinem Büro. Ein besonderes Dankeschön möchte ich auch an die Computeradministratoren Dominik Hertel, Peter Brunner und Frank Köck richten. Zum Einen für die Hilfestellung bei Computerproblemen aller Art, deren Lösung sie jeder Zeit bereitwillig in Angriff genommen haben, zum Anderen für die interessanten und unterhaltsamen Gespräche, die wir geführt haben. Ich danke auch Markus Kohler für viele wichtige Tipps und Einsichten rund ums promovieren und die vielen Wettläufe. Zudem danke ich Benjamin Galow, Kilian Heeg, Jonas Gunst und Carmen Leimer für das Korrekturlesen meiner Arbeit. Ich möchte auch all meinen anderen Kollegen danken, insbesondere Sebastian Meuren, Martin Gärtner, Stefano Cavaletto, Sven Augustin, Stephan Helmrich, Norman Neitz, Anton Wöllert, Andreas Danksagung 102 Reichegger, Lida Zhang, Huayu Hu, Sarah Müller, Michael Klaiber, Matteo Tamburini und Jörg Evers. Außerdem danke ich meinen WG-Mitbewohnern insbesondere Philipp Paa, Kai Becker und Sebastian Richtarsky, deren Gespräche mich und meine Weltsicht während meiner Promotionszeit nachhaltig geprägt haben. Auch meine Freunde vom Tanzsport, vor allem Andre, Jürgen und Katja seien in diesem Zusammenhang dankend erwähnt. Zudem auch großen Dank an meine ehemaligen Kommilitonen von der TU-Darmstadt. Es war insbesondere schön, einige von meinem Jahrgang des Sommersemesters 2004 auch während meiner Promotionszeit wieder gesehen zu haben – Hinter der grünen Tür! Und natürlich danke ich auch meinen Rollenspielfreunden aus Wiesbaden, die seit meinem Studium einen wichtigen Ankerpunkt in meinem Leben bilden. Zu guter Letzt möchte ich auch Marius Schollmeier danken, ohne welchem die vergangen drei Jahre meiner Promotion gar nicht erst stattgefunden hätten. Danke Marius!

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