# Elements of Photoionization Quantum Dynamics Methods

- Lampros A A Nikolopoulos

- March 2019

### Description

*Elements of Photoionization Quantum Dynamics Methods* focuses on the RMT approach to multiphoton quantum dynamics, which is employed to tackle the problem of laser-induced atomic dynamics. This recently developed formulation has shown the potential to become a mainstream *ab initio* theoretical approach, which is capable of describing the quantum dynamics of multielectron quantum systems exposed in ultrashort intense laser pulses lasting at the femtosecond timescale (∼10^{−18} sec). The method itself builds on the division-of-space concept and has evolved as an extension of the well-established atomic *R*-matrix method to incorporate time in its formulation. It is this way that its name was inherited, *R*-matrix with time incorporation (RMT), although no *R*-matrix appears anywhere in the formulation.

### About Editors

Lampros A A Nikolopoulos is a lecturer at the School of Physical Sciences at Dublin City University (DCU). He earned his BSc (Hons) in physics from the physics department of the University of Athens, and his MSc and PhD in theoretical atomic physics from the University of Crete, Greece. His research interests include ultrashort laser-matter quantum dynamics and the development of high-performance computational methods for their*ab initio*description. He has (co)authored more than 80 journal articles, two book chapters, and has co-edited a special issue on short-wavelength free electron laser.

### Table of Contents

I Basics on ab-initio quantum dynamics

1 Introduction

1.1 The theoretical minimum

1.2 Laser-atom quantum dynamics methods

2 Hilbert-space dynamics

2.1 Time-development

2.2 TDSE on the position representation

2.3 TDSE in the energy representation

3 Hilbert space rotations

3.1 Introduction

3.2 Central Hamiltonian

3.3 Angular momentum representation

3.3.1 Laplace operator

3.3.2 Dipole operators

3.3.3 Molecular diatomic potential

II One-electron atomic systems in laser ﬁelds

4 Radial SE of the atomic & molecular hydrogen

4.1 Atomic hydrogen radial SE

4.2 Molecular diatomic

4.2.1 Dipole matrix elements

5 SE on a ﬁnite basis

5.1 The TD wavefunction in a ﬁnite region

5.2 Finite basis expansion

5.2.1 Fixed boundary conditions

5.2.2 Free boundary conditions

5.3 Molecular hydrogen ion

5.4 Appendix: Normalization of the continuum states

6 Laser pulses

6.1 Radiation gauge

6.2 Laser beams in the paraxial approximation

6.2.1 Fourier-limited pulses

7 TDSE

7.1 Hamiltonian

7.2 TDSE of atomic hydrogen

7.2.1 Energy-angular momentum basis

7.2.2 TDSE on the angular momentum basis

7.3 TDSE of the molecular hydrogen ion

7.3.1 Energy-angular momentum basis

7.3.2 Angular momentum basis

7.4 TDSEs matrix notation

7.4.1 Initial state calculation-diﬀusion equation

8 TDSE post-processing

8.1 Preliminary

8.2 Bound state populations and ionization yield

8.2.1 Photoelectron Angular distribution (PAD)

8.2.2 High harmonic generation (HHG)

III The RMT division-of-space method

9 Division-of-space method

9.1 Introduction

9.2 RMT in 1-dimensional square well

9.3 TDSE in the inner region

9.3.1 Fixed boundary conditions (Green's theorem)

9.3.2 Free-boundary conditions (Bloch-operator)

9.4 Outer-region, ﬁnite-diﬀerence approach

10 RMT in hydrogenic systems

10.1 Introduction

10.2 Inner and outer TDSE

10.3 The atomic case

10.4 Population and ionization yields

10.4.1 Bound state populations and ionization yield

Appendices

A Spherical harmonics and Glebsch-Gordan coeﬃcients

A.1 Basic properties

A.2 Expansions on spherical harmonics

B Numerical approximations of operators

B.1 Spatial operators

B.1.1 Finite-diﬀerences

B.1.2 B-splines

B.2 Time-evolution operators

B.2.1 The Truncated Taylor Series Propagator

B.2.2 Runge-Kutta methods### Bibliographic

Paperback ISBN: 9780750329743

Ebook ISBN: 9781681747156

DOI: 10.1088/2053-2571/aaf10d

Publisher: Morgan & Claypool Publishers