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Attosecond Physics — Theory Lecture Notes 2013 (Work in Progress) Attosecond Physics — Theory Lecture notes 2013 (Work in progress) Armin Scrinzi November 19, 2013 DRAFT 2 DISCLAIMER Theses are not lecture notes, but rather scribbles. They need severe revision There are inconsistencies, errors, some false claims all things that were put there to be corrected at one point in their present status, the notes should not be used for actual computations without verification I reserve the right not to answer questions related to these notes DRAFT Contents 1 Introduction 7 2 Time scales, key experiments, units 9 2.1 What happens in an attosecond? ....................... 9 2.2 Photoionization ................................. 12 2.3 HHG — high harmonic generation ....................... 15 2.4 Units and scaling ................................ 17 3 Static field ionization 19 3.1 A quick estimate ................................ 19 3.2 A more accurate treatment ........................... 20 3.3 Numerical confirmations of the tunnel formula ................ 22 3.4 The Ammosov-Delone-Krainov (ADK) formula ................ 23 4 Above threshold ionization (ATI) 25 4.1 Initial velocity after tunneling ......................... 25 4.2 ATI - above threshold ionization ........................ 26 4.3 The 10 Up cutoff ................................ 29 5 Quantum mechanical description 31 5.1 Length- and velocity gauge ........................... 31 5.2 Volkov solutions ................................. 32 5.3 Quantum mechanics of laser-atom interaction ................ 33 5.4 A crash course in variational calculus ..................... 38 6 High harmonic generation 43 6.1 The classical model ............................... 43 6.2 A quantum calculation ............................. 44 6.3 The Lewenstein model ............................. 46 7 Pulse propagation 53 3 DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 4 CONTENTS 8 Photoionization in a laser pulse 57 8.1 Expansion into Bessel functions ........................ 57 8.2 Laser dressed photo-ionization ......................... 61 8.3 Tunneling vs.multi-photon regime ....................... 63 9 Attosecond measurements 69 9.1 The attosecond streak camera ......................... 69 9.2 Trains of attosecond pulses ........................... 73 9.3 HHG from CO2 ................................. 81 DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- Notation electric field E E0 ground state energy ωl laser fundamental (circular) frequency Tl laser period = laser optical cycle 5 DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 6 CONTENTS DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- Chapter 1 Introduction In the early years of 2000, extremely short, controlled light pulses could be produced for the first time. Typical duration of these pulses is 0.1 fs = 10−16 s = 100 10−18 s = 100 as with 1 as (attosecond) = 10−18 s. × Except for being short, the timing of these pulses can be controlled on a scale of about 10 as, 1/10 of their typical duration. These are the time scale on which electronic changes in ordinary matter happen. One or two decades earlier, the shortest pulses were in the & 10fs regime, the time scale on wich nuclei move during chemical reactions. This had given us the possibilty to directly observe what happens during chemical reactions. Atoms • Molecules • Solids • 7 DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 8 CHAPTER 1. INTRODUCTION DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- Chapter 2 Time scales, key experiments, units 2.1 What happens in an attosecond? 2.1.1 Electronic motion in an atom Velocity of an electron in the atom The quantum mechanical virial theorem for the Coulomb potential establishes a relation between the kinetic and the potential energy of an electron in an atom. It is strictly valid for any bound state of any Coulombic system: 1 T = V = E . (2.1) h kini −2h poti bind For any hydrogen-like ion one can relate the binding energy to the kinetic energy of a single electron. As Ebind is easily determined experimentally. The average velocity v0 of the electron in the ground state of the atom is m v2 T = e 0 = E = 13.6 eV (hydrogen). (2.2) kin 2 bind The Rydberg energy is defined as the binding energy of the hydrogen atom: 1Ry := 13.6058 eV. (2.3) Thus we find for the velocity 2Ry 27.2116 eV v = = = αc, (2.4) 0 m 0.511 MeV/c2 r e s α 1/137... fine structure constant, ≈ c... velocity of light. The typical speed of an electron in an atom is given by v0 = c/137 = αc. (2.5) 9 DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 10 CHAPTER 2. TIME SCALES, KEY EXPERIMENTS, UNITS It is easy to see that α is indeed exactly the fine-structure constant: expressing everything in terms of proton charge e, electron mass me and ~, e.g. table 2.1. For non-hydrogen like systems, we cannot associate Tkin with a single electron. Nev- ertheless, the order of magnitude estimate for the kinetich energyi of any valence electron by its ionization potential is valid. The motion of the electrons in an atom is sub-relativistic on the scale αZ∗, where Z∗ is some effective screened charge. For the valence electrons of a neutral Z∗ 1. For larger atoms only outer electrons can be treated non-relativistically. ∼ Distance of the electron to the nucleus In order to estimate the radius of the electron orbit, we again use the virial theorem for the Coulomb potential: e2 Vpot = 2Ry = , (2.6) h i − −4πǫ0a0 where a0 is the searched for radius. ǫ0 is the dielectric constant in the vacuum, ǫ0 = 8.85 10−12As/V m. Hence we get · 2 e −10 a0 = 0.529 10 m 0.05 nm. (2.7) 8πǫ0 Ry ≈ × ≈ a0 is called Bohr’s atomic radius. The classical orbit time of a valence electron Figure 2.1: An electron as it travels around a nucleus... If the atom moves with velocity v0 on a circular orbit with radius a0 around the nucleus, the orbit time is 2πa 0.529 10−10 τ = 0 =2π × 137s 2π 24.188 10−18s 150 as. (2.8) orbit v 3 108 × ≈ · × ≈ 0 × as stands for attosecond = 10−18s. DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 2.1. WHAT HAPPENS IN AN ATTOSECOND? 11 2.1.2 Transition energies — time scales Electron density of a superposition state Starting from the Schr¨odinger equation (with ~ = 1, atomic units) ~2∆ ∂ + V (r) Ψ(r, t)= i Ψ(r, t), (2.9) − 2m ∂t we can make an ansatz for quasi-stationary solutions Ψ(r, t) := e−iEtφ(r), (2.10) with φ(r) fulfilling the time independent Schr¨odinger equation ~2∆ + V (r) φ(t)= Eφ(t). (2.11) − 2m Quasi-stationary means that the electron density is time independent, ρ(r)= Ψ(r, t) 2 = φ(r) 2. (2.12) | | | | If we have a superposition of two such solutions of different energy, −iE1t −iE2t Ψ(r, t)= e φ1(r)+ e φ2(r), (2.13) we find for the electron density ρ(r, t)= Ψ(r, t) 2 = φ 2 + φ 2 + φ∗φ e−i(E2−E1)t + h.c. (2.14) | | | 1| | 2| 1 2 Notation: h.c. ...“hermitian conjugate” (which is here just the complex conjugate). Notice that the electron density of the superposition of two quasi-stationary states is time dependent. The time dependent part is periodic with a period 2π τ = . (2.15) E E | 2 − 1| Characteristic times of quantum systems 2π τ (~ =1) (2.16) ∼ ∆E ∆E...characteristic energy differences ∆E τ vibrational motion of nuclei in molecules 100 meV 20 fs valence electrons in atoms/molecules∼ 13 eV∼ 150 as inner shell electrons 1 keV 2 as nuclear fusion d + t He++ + n∼ 17 MeV ∼10−7 as. → ∼ DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 12 CHAPTER 2. TIME SCALES, KEY EXPERIMENTS, UNITS Other time scales Solids: thermalization relaxation of defects Clusters: ionization, electron detachment Coulomb explosion of the positively charged cluster 100 fs ∼ Attosecond physics is the physics of the dynamics of valence electrons 2.2 Photoionization Multi-photon ionization Wave length 800 nm, laser photon energy 1.5 eV Ionization potential (hydrogen) 13.6 eV ∼ An electron needs to absorb at least 9 photons to leave the atom and reach the continuum. Above threshold ionization The electron absorbs many more photons than needed for escaping (and thus escapes with considerably larger velocity)! “Above threshold ionization — ATI” , Figure 2.2: Multiphoton-ionization and above threshold ionization DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 2.2. PHOTOIONIZATION 13 Figure 2.3: Early experimental results on multi-photon ionization and ATI electrons. Photoelectrons from the lower ~ω and higher intensity appear at higher energies - “above threshold”. From Agostini et al., (1997) [1] DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 14 CHAPTER 2. TIME SCALES, KEY EXPERIMENTS, UNITS Figure 2.4: Measured at MPQ, Paulus et al. 1993 [13]. Depending on intensity, cuttoffs and plateaus are visible. Why and how? to be explained later → Photo-ionization at extreme intensities Some phenomenology: The 2 Up and the 10 Up cutoff. Note: 2 Up is the maximum kinetic energy of a free electron in a laser field (see below) Where do the 10 Up come from? The ponderomotive potential Up We want to determine the mean energy of a (classical) electron in a plane wave electro- magnetic field: Continuous wave (cw) laser field The electric field of a continuous, monochromatic wave with frequency ω is given by (t)= cos ωt. (2.17) E E0 Here dipole approximation is used, i.e. the field does not depend on space coordinates (which is a reasonable assumption as long as the wave length of the laser field is large compared to the atom). DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 2.3. HHG — HIGH HARMONIC GENERATION 15 Kinetic energy With ve(0) = 0 we find for the velocity of an electron in such a field t e (t) e F =v ˙(t)m = e (t) v(t)= E dt = E0 sin ωt. (2.18) e − E ⇒ − m −m ω Z0 e e With e := 1, me := 1 the kinetic energy of the electron at a time t is given by v (t) 2 2 T (t)= | e | = E0 sin2 ωt.
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