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Springer Series in Chemical Physics 116

Kaoru Yamanouchi
Wendell T. Hill III
Gerhard G. Paulus Editors

Progress in
Ultrafast Intense
Laser Science XIII
Springer Series in Chemical Physics

Volume 116

Series editors
A.W. Castleman Jr., University Park, USA
J.P. Toennies, Göttingen, Germany
K. Yamanouchi, Tokyo, Japan
W. Zinth, München, Germany
The Springer Series in Chemical Physics consists of research monographs in basic
and applied chemical physics and related analytical methods. The volumes of this
series are written by leading researchers of their fields and communicate in a
comprehensive way both the basics and cutting-edge new developments. This series
aims to serve all research scientists, engineers and graduate students who seek
up-to-date reference books.

More information about this series at http://www.springer.com/series/11752

Kaoru Yamanouchi ⋅ Wendell T. Hill III
Gerhard G. Paulus
Editors

Progress in Ultrafast Intense

Laser Science XIII

123
Editors
Kaoru Yamanouchi Gerhard G. Paulus
Department of Chemistry Institute of Optics and Quantum Electronics
The University of Tokyo Friedrich Schiller University Jena
Tokyo Jena, Thüringen
Japan Germany

Wendell T. Hill III

University of Maryland, College Park
College Park, MD
USA

ISSN 0172-6218
Springer Series in Chemical Physics
Progress in Ultrafast Intense Laser Science
ISBN 978-3-319-64839-2 ISBN 978-3-319-64840-8 (eBook)
https://doi.org/10.1007/978-3-319-64840-8
Library of Congress Control Number: 2017954485

© Springer International Publishing AG 2017

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
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Preface

We are pleased to present the thirteenth volume of Progress in Ultrafast Intense

Laser Science. As the frontiers of ultrafast intense laser science rapidly expand ever
outward, there continues to be a growing demand for an introduction to this
interdisciplinary research field that is at once widely accessible and capable of
delivering cutting-edge developments. Our series aims to respond to this call by
providing a compilation of concise review-style articles written by researchers at
the forefront of this research field, so that researchers with different backgrounds as
well as graduate students can easily grasp the essential aspects.
As in previous volumes of PUILS, each chapter of this book begins with an
introductory part, in which a clear and concise overview of the topic and its sig-
nificance is given, and moves onto a description of the authors' most recent research
results. All chapters are peer-reviewed. The articles of this thirteenth volume cover
a diverse range of the interdisciplinary research field, and the topics may be grouped
into four categories: atoms, molecules, and clusters interacting in an intense laser
field (Chaps. 1–5); high-order harmonics generation and its applications (Chaps. 6
and 7); photoemission at metal tips (Chap. 8); and advanced laser facilities (Chaps.
9 and 10).
From the third volume, the PUILS series has been edited in liaison with the
activities of the Center for Ultrafast Intense Laser Science at the University of
Tokyo, which has also been responsible for sponsoring the series and making the
regular publication of its volumes possible. From the fifth volume, the Consortium
on Education and Research on Advanced Laser Science, the University of Tokyo,
has joined this publication activity as one of the sponsoring programs. The series,
designed to stimulate interdisciplinary discussion at the forefront of ultrafast intense
laser science, has also collaborated since its inception with the annual symposium
series of ISUILS (http://www.isuils.jp/), sponsored by JILS (Japan Intense Light
Field Science Society).
We would like to take this opportunity to thank all of the authors who have kindly
contributed to the PUILS series by describing their most recent work at the frontiers
of ultrafast intense laser science. We also thank the reviewers who have read the
submitted manuscripts carefully. One of the coeditors (KY) thanks Ms. Mihoshi Abe

v
vi Preface

for her help with the editing processes. Last but not least, our gratitude goes out to
Dr. Claus Ascheron, Physics Editor of Springer-Verlag at Heidelberg, for his kind
support.
We hope this volume will convey the excitement of ultrafast intense laser sci-
ence to the readers and stimulate interdisciplinary interactions among researchers,
thus paving the way to explorations of new frontiers.

Tokyo, Japan Kaoru Yamanouchi

College Park, USA Wendell T. Hill III
Jena, Germany Gerhard G. Paulus
January 2017
Contents

1 Strong-Field S-Matrix Series with Coulomb Wave Final State .... 1

F.H.M. Faisal
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Three-Interaction Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Coulomb–Volkov Hamiltonian and Propagator . . . . . . . . . . . . . . 6
1.4 Coulomb-Volkov S-Matrix Series . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Strong-Field S-Matrix for Short-Range Potentials . . . . . . . . . . . . 11
1.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Multiconfiguration Methods for Time-Dependent Many-Electron
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15
Erik Lötstedt, Tsuyoshi Kato and Kaoru Yamanouchi
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15
2.2 Basics of Time-Dependent Multiconfiguration Methods . . . . . .. 17
2.3 Time-Dependent Multiconfiguration Methods
with Time-Independent Orbitals . . . . . . . . . . . . . . . . . . . . . . . .. 21
2.3.1 Time-Dependent Configuration Interaction
with Single Excitations . . . . . . . . . . . . . . . . . . . . . . . . .. 22
2.3.2 Time-Dependent Restricted-Active-Space
Configuration-Interaction . . . . . . . . . . . . . . . . . . . . . . .. 23
2.3.3 Time-Dependent R-Matrix Theory . . . . . . . . . . . . . . . .. 25
2.4 Time-Dependent Multiconfiguration Methods
with Time-Dependent Orbitals . . . . . . . . . . . . . . . . . . . . . . . . .. 26
2.4.1 Multiconfiguration Time-Dependent Hartree-Fock . . . . .. 28
2.4.2 Time-Dependent Complete Active-Space
Self-Consistent Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Factorized CI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

vii
viii Contents

3 Controlling Coherent Quantum Nuclear Dynamics in LiH by

Ultra Short IR Atto Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41
Astrid Nikodem, R.D. Levine and F. Remacle
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
3.2 Electronic Structure of LiH and Quantum Dynamics . . . . . . . . .. 43
3.3 Control of the Fragmentation Yields in the Σ Manifold by the
CEP of Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
3.4 Effect of the Non Adiabatic Coupling in the Σ Manifold . . . . . .. 51
3.5 Probing the Dynamics for a Superposition of Σ and Π States by
Transient Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61
4 Probing Multiple Molecular Orbitals in an Orthogonally
Polarized Two-Color Laser Field . . . . . . . . . . . . . . . . . . . . . . . . .. 67
Hyeok Yun, Hyung Taek Kim, Kyung Taec Kim
and Chang Hee Nam
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Two-Dimensional High-Harmonic Spectroscopy of Molecules . . . 69
4.2.1 HHG in an Orthogonally Polarized Two-Color Field . . . . 69
4.2.2 HHG from Linear Molecules . . . . . . . . . . . . . . . . . . . . . 72
4.3 Resolving High-Harmonics from Multiple Orbitals . . . . . . . . . . . 74
4.3.1 Qualitative Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Theoretical Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.3 Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Tracing Nonlinear Cluster Dynamics Induced by Intense XUV,
NIR and MIR Laser Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85
Bernd Schütte
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Ionization Dynamics of Clusters . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.1 XUV Multistep Ionization of Clusters . . . . . . . . . . . . . . . 87
5.2.2 Controlled Ignition of NIR Avalanching in Clusters . . . . . 89
5.2.3 MIR Strong-Field Ionization of Clusters Using
Two-Cycle Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 Expansion and Recombination Dynamics of Clusters . . . . . . . . . 94
5.3.1 Cluster Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.2 Frustrated Recombination . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.3 Reionization of Excited Atoms from Recombination . . . . 97
5.4 Autoionization and Correlated Electronic Decay . . . . . . . . . . . . . 101
5.4.1 Autoionization in Expanding Clusters . . . . . . . . . . . . . . . 102
5.4.2 Correlated Electronic Decay . . . . . . . . . . . . . . . . . . . . . . 104
Contents ix

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6 Molecules in Bichromatic Circularly Polarized Laser Pulses:
Electron Recollision and Harmonic Generation . . . . . . . . . . . . . . . 111
André D. Bandrauk, François Mauger and Kai-Jun Yuan
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Bicircular Recollision Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 Polarization of Molecular HHG . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7 High Harmonic Phase Spectroscopy Using Long Wavelengths . . . . 129
Antoine Camper, Stephen B. Schoun, Pierre Agostini
and Louis F. DiMauro
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.2 Reconstruction of the Attosecond Beating by Interference of
Two-Photon Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.3 High Harmonic Spectroscopy of Argon Cooper Minimum . . . . . 133
7.4 High Harmonic Spectroscopy of Aligned Nitrogen . . . . . . . . . . . 137
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8 Strong-Field-Assisted Measurement of Near-Fields and Coherent
Control of Photoemission at Nanometric Metal Tips . . . . . . . . . . . . 143
M. Förster, T. Paschen, S. Thomas, M. Krüger and P. Hommelhoff
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.3 Measurement of the Field Enhancement Factor at the Tip Apex
by Rescattering Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.4 Coherent Control of Photoemission . . . . . . . . . . . . . . . . . . . . . . 149
8.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9 Advanced Laser Facilities and Scientific Applications . . . . . . . . . . . 157
Luis Roso
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9.2 Different Approaches for a PW . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.3 Bottlenecks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
9.4 Applications of PW Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.5 Hard Laser Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.7 Appendix: The VEGA Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
x Contents

10 The Extreme Light Infrastructure—Attosecond Light Pulse

Source (ELI-ALPS) Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Dimitris Charalambidis, Viktor Chikán, Eric Cormier, Péter Dombi,
József András Fülöp, Csaba Janáky, Subhendu Kahaly,
Mikhail Kalashnikov, Christos Kamperidis, Sergei Kühn,
Franck Lepine, Anne L’Huillier, Rodrigo Lopez-Martens,
Sudipta Mondal, Károly Osvay, László Óvári, Piotr Rudawski,
Giuseppe Sansone, Paris Tzallas, Zoltán Várallyay and Katalin Varjú
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.2 The Mission and Structure of ELI-ALPS . . . . . . . . . . . . . . . . . . 184
10.3 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
10.3.1 The High Repetition Rate (HR) Laser System . . . . . . . . . 186
10.3.2 The Single-Cycle Laser System (SYLOS) . . . . . . . . . . . . 187
10.3.3 The High-Field (HF) Laser System . . . . . . . . . . . . . . . . . 189
10.3.4 The MIR System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.4 Secondary Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.4.1 The GHHG Beamlines . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.4.2 The Surface High Harmonic Generation (SHHG)
Development Beamlines . . . . . . . . . . . . . . . . . . . . . . . . . 202
10.4.3 The THz Beamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.4.4 The Electron Acceleration Beamlines . . . . . . . . . . . . . . . 209
10.5 Research Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
10.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Contributors

Pierre Agostini Department of Physics, The Ohio State University, Columbus,

OH, USA
André D. Bandrauk Computational Chemistry & Molecular Photonics, Labora-
toire de Chimie Théorique, Faculté des Sciences, Université de Sherbrooke,
Sherbrooke, Québec, Canada
Antoine Camper Department of Physics, The Ohio State University, Columbus,
OH, USA
Dimitris Charalambidis FORTH-IESL, Heraklion, Greece; ELI-ALPS, ELI-Hu
Kft, Szeged, Hungary
Viktor Chikán Kansas State University, Manhattan, USA; ELI-ALPS, ELI-Hu
Kft, Szeged, Hungary
Eric Cormier University of Bordeaux, CEA, CNRS, CELIA, UMR 5107,
Talence, France; ELI-ALPS, ELI-Hu Kft, Szeged, Hungary
Louis F. DiMauro Department of Physics, The Ohio State University, Columbus,
OH, USA
Péter Dombi Wigner RCP, Budapest, Hungary; ELI-ALPS, ELI-Hu Kft, Szeged,
Hungary
F.H.M. Faisal Fakultaet Fuer Physik, Universitaet Bielefeld, Bielefeld, Germany;
Optical Sciences Center, University of Arizona, Tucson, AZ, USA
M. Förster Lehrstuhl für Laserphysik, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Erlangen, Germany
József András Fülöp University of Pécs, Pécs, Hungary; ELI-ALPS, ELI-Hu Kft,
Szeged, Hungary
P. Hommelhoff Lehrstuhl für Laserphysik, Friedrich-Alexander-Universität
Erlangen-Nürnberg, Erlangen, Germany

xi
xii Contributors

Csaba Janáky University of Szeged, Szeged, Hungary; ELI-ALPS, ELI-Hu Kft,

Szeged, Hungary
Subhendu Kahaly ELI-ALPS, ELI-Hu Kft, Szeged, Hungary
Mikhail Kalashnikov MBI, Berlin, Germany; ELI-ALPS, ELI-Hu Kft, Szeged,
Hungary
Christos Kamperidis ELI-ALPS, ELI-Hu Kft, Szeged, Hungary
Tsuyoshi Kato Department of Chemistry, School of Science, The University of
Tokyo, Tokyo, Japan
Hyung Taek Kim Center for Relativistic Laser Science, Institute for Basic Sci-
ence, Gwangju, Korea; Advanced Photonics Research Institute, Gwangju Institute
of Science and Technology, Gwangju, Korea
Kyung Taec Kim Center for Relativistic Laser Science, Institute for Basic Sci-
ence, Gwangju, Korea; Department of Physics and Photon Science, Gwangju
Institute of Science and Technology, Gwangju, Korea
M. Krüger Weizmann Institute of Science, Rehovot, Israel
Sergei Kühn ELI-ALPS, ELI-Hu Kft, Szeged, Hungary
Franck Lepine UMR 5306 CNRS Univ. Lyon 1, Villeurbanne Cedex, France;
ELI-ALPS, ELI-Hu Kft, Szeged, Hungary
R.D. Levine The Fritz Haber Center for Molecular Dynamics and Institute of
Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel; Crump Institute
for Molecular Imaging and Department of Molecular and Medical Pharmacology,
David Geffen School of Medicine and Department of Chemistry and Biochemistry,
University of California, Los Angeles, CA, USA
Anne L’Huillier Lund University, Lund, Sweden
Rodrigo Lopez-Martens LOA, UMR 7639, Palaiseau, France; ELI-ALPS,
ELI-Hu Kft, Szeged, Hungary
Erik Lötstedt Department of Chemistry, School of Science, The University of
Tokyo, Tokyo, Japan
François Mauger Department of Physics and Astronomy, Louisiana State
University, Baton Rouge, LA, USA
Sudipta Mondal ELI-ALPS, ELI-Hu Kft, Szeged, Hungary
Chang Hee Nam Center for Relativistic Laser Science, Institute for Basic Science,
Gwangju, Korea; Department of Physics and Photon Science, Gwangju Institute of
Science and Technology, Gwangju, Korea
Astrid Nikodem Département de Chimie, B6c, Université de Liège, Liège,
Belgium
Contributors xiii

Károly Osvay University of Szeged, Szeged, Hungary; ELI-ALPS, ELI-Hu Kft,

Szeged, Hungary
László Óvári University of Szeged, Szeged, Hungary; ELI-ALPS, ELI-Hu Kft,
Szeged, Hungary
T. Paschen Lehrstuhl für Laserphysik, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Erlangen, Germany
F. Remacle Département de Chimie, B6c, Université de Liège, Liège, Belgium;
The Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The
Hebrew University of Jerusalem, Jerusalem, Israel
Luis Roso Centro de Láseres Pulsados (CLPU), Villamayor, Salamanca, Spain
Piotr Rudawski Lund University, Lund, Sweden
Giuseppe Sansone Politecnico di Milano, Milan, Italy; ELI-ALPS, ELI-Hu Kft,
Szeged, Hungary
Stephen B. Schoun Department of Physics, The Ohio State University, Columbus,
OH, USA
Bernd Schütte Max-Born-Institut, Berlin, Germany
S. Thomas Lehrstuhl für Laserphysik, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Erlangen, Germany
Paris Tzallas ELI-ALPS, ELI-Hu Kft, Szeged, Hungary
Katalin Varjú University of Szeged, Dóm tér 9, 6720 Szeged, Hungary;
ELI-ALPS, ELI-Hu Kft, Szeged, Hungary
Zoltán Várallyay ELI-ALPS, ELI-Hu Kft, Szeged, Hungary
Kaoru Yamanouchi Department of Chemistry, School of Science, The University
of Tokyo, Tokyo, Japan
Kai-Jun Yuan Computational Chemistry & Molecular Photonics, Laboratoire de
Chimie Théorique, Faculté des Sciences, Université de Sherbrooke, Sherbrooke,
Québec, Canada
Hyeok Yun Center for Relativistic Laser Science, Institute for Basic Science,
Gwangju, Korea
Chapter 1
Strong-Field S-Matrix Series with Coulomb
Wave Final State

F.H.M. Faisal

Abstract Despite its long-standing usefulness for the analysis of various processes
in intense laser fields, it is well-known that the KFR or strong-field approximation
(SFA) does not account for the final-state Coulomb interaction for ionisation. Var-
ious ad hoc attempts have been made in the past to face this problem within the
SFA, however, till now no systematic S-matrix expansion accounting for it has been
found. To overcome this long standing limitation of SFA we present here a systematic
series expansion of the strong-field S-matrix that could accounts for the final-state
Coulomb interaction in all orders.

1.1 Introduction

Over the past several decades the well-known strong-field approximation in the form
of the so-called KFR or SFA ansatz [1–3] has provided much fruitful insights into
the highly non-perturbative processes in intense laser fields. However, it is also well-
known that SFA does not account for the Coulomb interaction in the final state that is
specially significant for the ubiquitous ionisation process in strong fields. Due to this
problem, many authors in the past decades have made various heuristic corrections to
the SFA. Thus, for example, attempts to account for the Coulomb effect appear within
early ionisation models [4–6]. Other approaches include WKB-like approximations
[7, 8], semi-clssical and/or “quantum trajectory” approach [9–11], semi-analytic R-
matrix approach [12], and more recently an approach employing ansätze with phase
correction plus inhomogeneous differential equation [13]. Until now, however, no
systematic strong-field S-matrix theory could be found that unlike the usual plane-
wave SFA would be able to account for the laser plus Coulomb interaction in the
final state to all orders.

F.H.M. Faisal (✉)

Fakultaet Fuer Physik, Universitaet Bielefeld, 33501 Bielefeld, Germany
e-mail: ffaisal@physik.uni-bielefeld.de
F.H.M. Faisal
Optical Sciences Center, University of Arizona, Tucson, AZ 85721, USA

© Springer International Publishing AG 2017 1

K. Yamanouchi et al. (eds.), Progress in Ultrafast Intense Laser Science XIII,
Springer Series in Chemical Physics 116, https://doi.org/10.1007/978-3-319-64840-8_1
2 F.H.M. Faisal

Here we present a strong-field S-matrix theory that overcomes this long standing
problem and derive a systematic all order S-matrix series explicitly incorporating
the laser plus Coulomb interaction in the final state. To this end we shall use be-
low a three-interaction formalism developed earlier in connection with the so-called
intense-field S-matrix theory or IMST (see, e.g. review [14] or, original references
cited therein).

1.2 Three-Interaction Formalism

In this section we briefly indicate the three-interaction technique suitable for the
problem at hand. The Schroedinger equation of the interacting atom+ laser field is

𝜕
(iℏ − H(t))|𝛹 (t)⟩ = 0 (1.1)
𝜕t

where H(t) is the total Hamlltonian of the system,

H(t) = Ha + Vi (t) (1.2)

For example, for an effective one electron atomic system interacting with a laser
field, we may take

𝐩op 2 Ze2
Ha = ( − + Vs.r. (𝐫)) (1.3)
2m r

where Z is the core charge and Vs.r. (𝐫) is a short-range potential that goes to zero for
asymptotically large r faster than the Coulomb potential.
The laser-atom interaction is assumed here in the minimal coupling gauge (in
“dipole approximation”)

e e2 A2 (t)
Vi (t) = (− 𝐀(t) ⋅ 𝐩op + ) (1.4)
mc 2mc2

where 𝐀(t) is the vector potential of the laser field, and 𝐩op ≡ −iℏ∇.
Since all information of the interacting system is contained in the full wavefunc-
tion 𝛹 (t) and in general this is not known explicitly, we shall consider a more useful
formal expression of the full wavefunction in terms of the appropriate partial in-
teractions among the sub-systems and, the associated sub-propagators (or Green’s
functions). The latter objects may be already known, or could be found, to expand
the total wavefunction in terms of them.
Thus, first, we may formally define the full propagator, G(t, t′ ), associated with
the total Hamiltonian H(t), by the inhomogeneous equation
1 Strong-Field S-Matrix Series with Coulomb Wave Final State 3

𝜕
(iℏ − H(t))G(t, t′ ) = 𝛿(t − t′ ). (1.5)
𝜕t
The solution of the Schroedinger equation (1.1) can then be expressed as

|𝛹 (t)⟩ = |𝜙i (t)⟩ + G(t, t1 )Vi (t1 )|𝜙i (t1 )⟩dt1 (1.6)
∫

where |𝜙i (t)⟩ is a given initial state. We may note here already that due to the implicit
presence of the Heaviside theta-function in all the propagators (see, for example, the
Volkov propagator given in the sequel) the time integration limits are always from
a given initial time ti to a given final time tf since the limits of the intermediate
time-integrations are automatically controlled by the propagators at the appropriate
positions by themselves. Usually the interaction time interval tf − ti is taken to be
long, e.g., from −∞ to +∞. Note, however, that there is no difficulty in using the
theory for interactions with any finite laser pulse, for during the rest of the time, from
and to the long-time limits, the pulse could be assumed to be vanishingly small.
In general, as for the full wavefunction, we do not have explicit knowledge of
the full propagator G(t, t′ ). Therefore, we intend to express it in terms of certain
most relevant sub-propagators. Clearly, the two most relevant states in any quantum
mechanical transition process are the initial state, in which the system is prepared,
and the final state, in which the system is detected. Since in any ionisation process
the final state interaction is governed by the long-range Coulomb interaction of the
outgoing electron and the residual ion-core, it is highly desirable that the final state
incorporates the long-range Coulomb interaction from the beginning.
Let us define a final reference Hamiltonian Hf (t) that incorporates the final-state
Coulomb interaction in the presence of the laser field. Hf , and the corresponding
final-state interaction Vf (t) are related to each other by the total Hamiltonian, H(t),

H(t) = Hf (t) + Vf (t) (1.7)

Formally, the final state propagator is then defined as usual by

𝜕
(iℏ − Hf (t))Gf (t, t′ )(t) = 𝛿(t − t′ ) (1.8)
𝜕t

Assuming for a moment that a suitable Hf (t) and Gf (t, t′ ) for the present purpose
could be found, the total G(t, t′ ) can be re-expressed in terms of Gf (t, t′ ) as

G(t, t′ ) = Gf (t, t′ ) + Gf (t, t1 )Vf (t1 )G(t1 , t′ )dt1 (1.9)

∫

Substituting this in |𝛹 (t)⟩ above we get a closed form expression of the full state
vector in the form
4 F.H.M. Faisal

|𝛹 (t)⟩ = |𝜙i (t)⟩ + dt1 Gf (t, t1 )Vi (t1 )|𝜙i (t1 )⟩

∫

+ dt2 dt1 Gf (t, t2 )Vf (t2 )G(t2 , t1 )Vi (t1 )|𝜙i (t1 )⟩ (1.10)
∫

This formally closed form of the wavefunction of the interacting system has been
originally derived and discussed in connection with non-sequential double ionization
processes (see, review [14]). Here we shall make use of it for the problem at hand.
In fact, the transition amplitude (or the S-Matrix element Sfi ) from an initial state,
|𝜙i (t)⟩, to a final state |𝜓f (t)⟩ of the system is given, by definition, by the projection
of the final state on to the total wavefunction evolving from the initial state. Thus,
using the above form of |𝛹 (t)⟩, we get

Sfi = ⟨𝜓f (t)|𝛹 (t)⟩

= ⟨𝜓f (t)|𝜙i (t)⟩ + dt1 ⟨𝜓f (t1 )|Vi (t1 )⟩|𝜙i (t1 )⟩ +
∫

+ dt2 dt1 ⟨𝜓f (t2 )|Vf (t2 )G(t2 , t1 )Vi (t1 )|𝜙i (t1 )⟩ + ... (1.11)
∫

This is a specially convenient general form of a transition amplitude from which

to generate the desired expansion of the ionisation amplitude. Now, G(t, t′ ) may be
expanded in terms of any suitable intermediate sub-propagator and the correspond-
ing intermediate interaction (without affecting the choice of the initial and the final
states and the respective rest-interactions). This generates a series expansion of the
strong-field S-matrix element of interest.
Before proceeding to derive the strong-field S-matrix series of present interest,
we may pause here briefly to make a few observations on the general character of
such series. Generally speaking, the strong-field S-matrix series are not perturbation
series based on a “small parameter”. Thus, for example, the usual plane wave SFA
expansion involves both the laser-atom interaction and the atomic potential. Indeed,
most of its useful applications using the first and the second order terms have been
for cases in which the laser field strength F had been weaker than the strength of
(eFa0)
the atomic potential or (Ze 2 ∕a0)
< 1, Z = nuclear charge; this is contrary to the view
sometimes held that SFA is a perturbative expansion where the “small parameter”
corresponds to the strength of the atomic potential (in comparison with the laser field
strength).
More appropriately viewed, strong-field S-matrix series are iterative series, where
each successive order of iteration corresponds to an additional intermediate
interaction-event or “collision” (involving the active electron and, either the atomic
potential or the laser-field, or both). With each increasing iteration order, the number
of intermediate “collisions” to occur also increases and hence the probability of its
significance for a given final event tends to decrease. Independent of this general but
qualitative expectation, the final results can be tested for quantitative accuracy only
a posteriori e.g. by comparison with accurate simulations (when feasible) and/or
1 Strong-Field S-Matrix Series with Coulomb Wave Final State 5

with experimental data (when available). Another physically significant usefulness

of such series is that they allow a systematic exploration of hypothesised mechanisms
behind a strong-field phenomenon. This is possible due to the ability of the S-matrix
series to systematically generate Feynman-like diagrams that can help to visualise
the underlying mechanism(s)suggested by the diagrams, as well as to estimate their
relative significance (see, e.g. [14]).
To continue with the problem at hand, we choose the strong-field Volkov propa-
gator GVol (t, t′ ) to expand the full G appearing in (1.11). The Volkov Hamiltonian is
given by the interaction of the free-electron with the laser field only, or

𝐩2op e e2 A2
HVol (t) = ( − 𝐀(t) ⋅ 𝐩 + ) (1.12)
2m mc 2mc2
The solutions of the corresponding Schroedinger equation are easily found
2
t pt′′
i
dt′′
𝜓𝐩 (𝐫, t) = ⟨𝐫|𝐩⟩e− ℏ ∫t′ 2m (1.13)

i
where 𝐩t ≡ (𝐩 − ec 𝐀(t)) and ⟨𝐫|𝐩⟩ = e ℏ 𝐩⋅𝐫 is a plane wave of momentum 𝐩.
The Volkov propagator GVol (t, t′ ) is the solution of the inhomogeneous equation

𝜕 𝐩2op e e2 A2
(iℏ − ( − 𝐀(t) ⋅ 𝐩 + ))GVol (t, t′ ) = 𝛿(t − t′ ). (1.14)
𝜕t 2m mc 2mc2
which is therefore given explicitly by:

i ∑ i t p2′′
dt′′
GVol (t, t′ ) = − 𝜃(t − t′ ) |𝐩⟩e− ℏ ∫t′
t
2m ⟨𝐩| (1.15)
ℏ 𝐩

Using the Volkov propagator we can expand

G(t, t′ ) = GVol (t, t′ ) + GVol (t, t1 )V0 (t1 )GVol (t1 , t′ )dt1 + ⋯ . (1.16)
∫

The intermediate interaction operator V0 (t) is accordingly defined by

V0 (t) = H(t) − HVol (t)

Ze2
= (− + Vs.r. (𝐫)) (1.17)
r
(which is time independent in the present case).
Since the initial state belongs to the atomic Hamiltonian Ha , therefore, the initial
rest-interaction Vi (t) is, as indicated earlier, simply
6 F.H.M. Faisal

Vi (t) = H(t) − Ha
e e2 A2
= (− 𝐀(t) ⋅ 𝐩op + ) (1.18)
mc 2mc2
For the final state, we intend to take account of the long-range Coulomb interac-
tion explicitly. One such state is the so-called “Coulomb-Volkov” state. It has been
introduced a long time ago [15, 16] by taking the usual stationary Coulomb-wave
and augmenting it heuristically by the time-dependent Volkov-phase:
2
t pt ′
−ℏ ∫i
dt′
𝛷𝐩 (𝐫, t) = 𝜙(−)
𝐩 (𝐫) × e
2m (1.19)

The stationary Coulomb waves, 𝜙(−)

𝐩 (𝐫), belong to the asymptotic atomic (or hydro-
genic) Hamiltonian HCou :

HCou = Ha − Vs.r. (𝐫)

𝐩2op Ze2
=( − ) (1.20)
2m r
They are given by [17]

1 𝜋 i i
𝜙(−)
𝐩 (𝐫) = e 2 𝜂p 𝛤 (1 + i𝜂p )e ℏ 𝐩⋅𝐫 1 F1 (−i𝜂p , 1, − (pr + 𝐩 ⋅ 𝐫)) (1.21)
L
3
2
ℏ

We have assumed them to be normalised in a large volume L3 with the understand-

∑ L 3
ing that, limit L → ∞, 𝐬 (⋯) ≡ ( 2𝜋 ) ∫ d3 s(⋯); 𝐩op ≡ −iℏ∇, and 𝜂p ≡ aZℏp is the
0
ℏ 2
so-called Sommerfeld parameter; a0 = Bohr radius = me 2
. Note that the “minus”
Coulomb wave is chosen above, which is appropriate for the ionisation final state. We
note in passing that for the laser assisted scattering problems the “plus” wave is also
of interest; they are related to each other by the transformation 𝜙(+) (−)∗
𝐩 (𝐫) = 𝜙−𝐩 (𝐫).
To determine the associated final-state interaction we need to know the appropri-
ate time-dependent Coulomb-Volkov Hamiltonian (call it HCV (t)). If it exists, HCV (t)
should be such, that the above defined Coulomb-Volkov state (1.19) should be a
member of the complete set of linearly independent fundamental solutions of the
associated Schroedinger equation.

1.3 Coulomb–Volkov Hamiltonian and Propagator

To determine the Hamiltonian HCV (t) to which the Coulomb-Volkov state belongs,
we introduce a vector operator defined by
1 Strong-Field S-Matrix Series with Coulomb Wave Final State 7
∑
𝜋c ≡ |𝜙𝐬 ⟩𝐬⟨𝜙𝐬 | (1.22)
𝐬

where |𝜙𝐬 ⟩ stands for the Coulomb wave (“+” or “-”) with momentum 𝐬, cf. (1.21).
Consider next the exponential operator

T(𝜋c ) = ei𝛼(t)⋅𝜋c (1.23)

t
where 𝛼(t) = mce
∫ 𝐀(t′ )dt′ . By expanding the exponential as a power series and
using the projection operator nature of the individual terms, it can be reduced to the
simple form
∑
T(𝜋c ) = 1 − |𝜙𝐬 ⟩(1 − ei𝛼(t)⋅𝐬 )⟨𝜙𝐬 | (1.24)
𝐬

We can write the Coulomb-Volkov Hamiltonian HCV (t) with the help of the op-
erator 𝜋c ,

𝐩2op Ze2 e2 A2 (t) e

HCV (t) = − + − 𝐀(t) ⋅ 𝜋c (1.25)
2m r 2mc2 mc

The corresponding Schroedinger equation is

𝜕 𝐩2op Ze2 e2 A2 (t) e

iℏ 𝛷j (t) = ( − + − 𝐀(t) ⋅ 𝜋c )𝛷j (t) (1.26)
𝜕t 2m r 2mc2 mc

The complete set of linearly independent solutions of (1.26) is

t 2 A2 (t′ )
i
(Ej + e )dt′ + ℏi 𝛼(t)⋅𝜋c
|𝛷j (t)⟩ = e− ℏ ∫ 2mc2 |𝜙j ⟩ (1.27)

where j ≡ 𝐩, stands for the momentum 𝐩 of the Coulomb wave state |𝜙(−) 𝐩 ⟩ and j ≡ D
stands for the discrete indices of the bound states |𝜙D ⟩ of the attractive Coulomb
potential. To establish that (1.27) indeed satisfies (1.26), let us first consider the case
{j ≡ 𝐩} and use (1.24) to calculate,
i
e ℏ 𝛼(t)⋅𝜋c |𝜙𝐩 ⟩ = T(𝜋c )|𝜙𝐩 ⟩
∑ i
= |𝜙𝐩 ⟩ − |𝜙𝐬 ⟩(1 − e ℏ 𝛼(t)⋅𝐬 )⟨𝜙𝐬 |𝜙𝐩 ⟩
𝐬
i
= |𝜙𝐩 ⟩ − |𝜙𝐩 ⟩(1 − e ℏ 𝛼(t)⋅𝐩 )
i
= e ℏ 𝛼(t)⋅𝐩 |𝜙𝐩 ⟩ (1.28)

Also we have
8 F.H.M. Faisal

e e
− 𝐀(t) ⋅ 𝜋c |𝜙𝐩 ⟩ = − 𝐀(t) ⋅ 𝐩|𝜙𝐩 ⟩ (1.29)
mc mc
Thus, substituting (1.27) in (1.26) for the continuum case, we get on the left hand
side
i t
(Ep + e
2 A2 (t′ )
)dt′ −𝛼(t)⋅𝐩) e2 A2 (t)
l.h.s. = e− ℏ (∫ 2mc2 (Ep + − 𝛼(t)
̇ ⋅ 𝐩)|𝜙𝐩 ⟩ (1.30)
2mc2
and on the right hand side,

i t
(Ep + e
2 A2 (t′ )
)dt′ −𝛼(t)⋅𝐩)
𝐩op 2 Ze2 e2 A2 (t′ ) e
r.h.s. = e− ℏ (∫ 2mc2 (( − )+ − 𝐀(t) ⋅ 𝐩)|𝜙𝐩 ⟩
2m r 2mc 2 mc
(1.31)

p 2
e
Noting that 𝛼(t)
̇ = mc 𝐀(t) and HCou |𝜙𝐩 ⟩ = Ep |𝜙𝐩 ⟩, where, Ep = 2m , one easily finds
from above that the l.h.s = r.h.s and hence the given solution is exactly fulfilled.
In a similar way it is easily seen that
∑ i
T(𝜋c )|𝜙D ⟩ = |𝜙D ⟩ − |𝜙𝐬 ⟩(1 − e ℏ 𝛼(t)⋅𝐬 )⟨𝜙𝐬 |𝜙D ⟩
𝐬
= |𝜙D ⟩ + 0
(1.32)

since, the overlap integral between the discrete and the continuum eigenstates of the
Coulomb Hamiltonian vanish by orthogonality, ⟨𝜙𝐬 |𝜙D ⟩ = 0. Hence, on substituting
(1.27) in (1.26) in the discrete case we get

i t
(ED + e
2 A2 (t′ )
dt′ +0) e2 A2 (t)
l.h.s. = e− ℏ ∫ 2mc2 (ED + + 0)|𝜙D ⟩ (1.33)
2mc2
and

i t
(ED + e
2 A2 (t′ )
)dt′ +0)
𝐩op 2 Ze2 e2 A2 (t′ )
r.h.s. = e− ℏ (∫ 2mc2 (( − )+ + 0)|𝜙D ⟩ (1.34)
2m r 2mc2
𝐩2 2
Moreover, ( 2m
op
− Zer )|𝜙D ⟩ = ED |𝜙D ⟩ and, therefore, the l.h.s = r.h.s and the verifi-
cation is complete.
To summarise, the complete set of solutions of the CV-Schroedinger equation
defined by (1.26) is given explicitly for the continuum case by
t p2
−ℏ ∫ i
( 2m + 12 A(t′ )2 − ec 𝐀(t′ )⋅𝐩)dt′
𝛷𝐩(−) (𝐫, t) = 𝜙(−)
𝐩 (𝐫)e
2c (1.35)

and for the discrete case by

1 Strong-Field S-Matrix Series with Coulomb Wave Final State 9

t p2
i
( 2m + 12 A(t′ )2 )dt′
𝛷D (𝐫, t) = 𝜙D (𝐫)e− ℏ ∫ 2c (1.36)

where we may recall that [17]

1 𝜋𝜂p i i
⟨𝐫|𝜙(−)
𝐩 ⟩ = e 2 𝛤 (1 + i𝜂p )e ℏ 𝐩⋅𝐫 1 F1 (−i𝜂p , 1, − (pr + 𝐩 ⋅ 𝐫)) (1.37)
L3∕2 ℏ
Zℏ
with 𝜂p = pa0
, and the well known hydrogenic bound states,

𝜙D≡(n,l,m) (𝐫) = Nnl Rnl (r)Ylm (𝜃, 𝜙)

Rnl (r) = (2𝜅n r)l e−𝜅n r F1 (−n + l + 1, 2l + 2, 2𝜅n r)
√
(2𝜅n )3∕2 𝛤 (n + l + 1)
Nnl = (1.38)
𝛤 (2l + 2) 2n𝛤 (n − l)
√
−2mED
with 𝜅n ≡ naZ = ℏ2
.
0
Having thus found the explicit form of both HCV (t), (1.25), and the complete set of
solutions [(1.27) or, (1.35) and (1.36)] of the Coulomb-Volkov Schroedinger equa-
tion, (1.26), we may now write down the associated Coulomb-Volkov propagator
GCV (t, t′ ) explicitly,

i
GCV (t, t′ ) = − 𝜃(t − t′ )
ℏ
∑ i t (𝐩− ec 𝐀(t′′ )2
dt′′
× { |𝜙𝐩 ⟩e− ℏ ∫t′ 2m ⟨𝜙𝐩 |
𝐩
∑ i t e2 A2 (t′′ )
)dt′′
+ |𝜙nlm ⟩e− ℏ ∫t′ (Enl + 2mc2 𝜙nlm |} (1.39)
nlm

1.4 Coulomb-Volkov S-Matrix Series

We are now in a position to obtain the desired S-matrix amplitude. From the knowl-
edge of HCV (t) obtained above the rest-interaction in the final-state turns out to be,

VCV (t) = H(t) − HCV (t)

e
= (− 𝐀(t) ⋅ (𝐩𝐨𝐩 − 𝜋𝐜 ) + Vs.r. (𝐫)) (1.40)
mc
Therefore, substitutions of the initial and the final rest-interactions as well as the
expansion of G(t, t′ ) in terms of the Volkov propagator and the intermediate rest-
2
interaction V0 (𝐫) = (− Zer + Vs.r (𝐫)) into the amplitude expression (1.11), immedi-
ately yield:
10 F.H.M. Faisal

Sfi =⟨𝛷𝐩 (t)|𝜙i (t)⟩

i
− dt ⟨𝛷 (t )|V (t )|𝜙 (t )⟩
ℏ∫ 1 𝐩 1 i 1 i 1
i e
− dt dt ⟨𝛷 (t )|(− 𝐀(t2 ) ⋅ (𝐩op − 𝜋c ) + Vs.r. (𝐫2 ))×
ℏ∫ 2 1 𝐩 2 mc
×GVol (𝐫2 , t2 ; 𝐫1 , t1 )Vi (t1 )|𝜙i (t1 )⟩
⋯ (1.41)

Thus, finally, we have arrived at the desired systematic S-matrix series for the strong-
field ionisation amplitude, which systematically accounts for the final state long
range Coulomb interaction through the Coulomb-Volkov state in all orders. We quote
the first three terms more explicitly and, give the rule of construction for all the higher
order terms of the series:
∞
∑
Sfi = Sfi(n) (1.42)
n=0

where,
Sfi(0) = ⟨𝛷𝐩 (𝐫, t)|𝜙i (𝐫, t)⟩ (1.43)

i e e2 A2 (t1 )
Sfi(1) = − dt1 ⟨𝛷𝐩 |(𝐫1 , t1 )(− 𝐀(t1 ) ⋅ 𝐩op + )|𝜙i (𝐫1 , t1 )⟩ (1.44)
ℏ∫ mc 2mc2

i e
Sfi(2) = − dt dt ⟨𝛷 (𝐫 , t )|(− 𝐀(t2 ) ⋅ (𝐩op − 𝜋c ) + Vs.r. (𝐫2 ))GVol (𝐫2 , t2 ; 𝐫1 , t1 )
ℏ∫ 2 1 𝐩 2 2 mc
e e2 A2 (t1 )
× (− 𝐀(t1 ) ⋅ 𝐩op + )|𝜙i (𝐫1 , t1 )⟩ (1.45)
c 2mc2

i e
Sfi(3) = − dt dt dt ⟨𝛷 (𝐫 , t )|(− 𝐀(t3 ) ⋅ (𝐩op − 𝜋c ) + Vs.r. (𝐫3 ))
ℏ∫ 3 2 1 𝐩 3 3 mc
Ze2
×GVol (𝐫3 , t3 ; 𝐫2 , t2 )(− + Vs.r. (𝐫2 ))GVol (𝐫2 , t2 ; 𝐫1 , t1 )
r2
e e2 A2 (t1 )
×(− 𝐀(t1 ) ⋅ 𝐩op + )|𝜙i (𝐫1 , t1 )⟩
mc 2mc2
… (1.46)

where the angle brackets stand for the integration with respect to the space coordi-
nates and “⋯” stands for the higher orders terms. The higher order terms can be
written down easily, if required, for they follow the same pattern as the third order
term but are to simply extended by an extra intermediate factor GVol V0 and an extra
time integration for each successive order, to all orders.
1 Strong-Field S-Matrix Series with Coulomb Wave Final State 11

1.5 Strong-Field S-Matrix for Short-Range Potentials

Before ending this report it is interesting to consider the S-matrix expansion of the
strong-field amplitude in the presence of an asymptotically short range potential.
This can be obtained simply by taking the limit Z = 0 in the result derived above. In
i
this limit the Coulomb waves 𝜙𝐩 (𝐫) reduce to the plane waves e ℏ 𝐩⋅𝐫 and the Coulomb-
Volkov state 𝛷𝐩 (𝐫, t) (1.19) reduces to the Volkov state (1.13). This implies that
the factor with the final state interaction in all terms, beginning with the second
order term, reduces to the short range potential Vs.r. (𝐫) only, due to the following
simplification
e e
⟨𝐩|(− 𝐀(t) ⋅ (𝐩op − 𝜋c ) + Vs.r. (𝐫)) = (− 𝐀(t) ⋅ (𝐩 − 𝐩)⟨𝐩| + ⟨𝐩|Vs.r. (𝐫))
mc mc
= ⟨𝐩|Vs.r. (𝐫) (1.47)

Also, the intermediate interaction V0 in all terms (from the second order term on-
wards) for Z = 0 reduces to the short-range potential Vr.s. (𝐫) only. Hence, in general,
for Z ≡ 0, the Coulomb-Volkov S-matrix series, (1.41), goes over to the simpler se-
ries

Sfi (Z = 0) = ⟨𝜓𝐩 (t)|𝜙i (t)⟩

i
− dt ⟨𝜓 (t )|V (t)|𝜙i (t)⟩
ℏ∫ 1 𝐩 1 i
i
− dt dt ⟨𝜓 (t )|V (𝐫 )G (𝐫 , t ; 𝐫 , t )V (t )|𝜙 (t)⟩
ℏ ∫ 2 1 𝐩 2 s.r. 2 Vol 2 2 1 1 i 1 i
i
− dt dt dt ⟨𝜓 (t )|V (𝐫 )G (𝐫 , t ; 𝐫 , t )V (𝐫 )
ℏ ∫ 3 2 1 𝐩 3 s.r. 3 Vol 3 3 2 2 s.r. 2
× GVol (𝐫2 , t2 ; 𝐫1 , t1 )Vi (t1 )|𝜙i (t1 )⟩
+ …. (1.48)

Equation (1.48) provides a strong-field S-matrix expansion when there is no long-

range Coulomb interaction preset asymptotically. This occurs with all asymptotically
neutral systems with effective core charge Z = 0 (as seen by the ejected electron) e.g.
for the case of electron-detachment from negative ions. Equation (1.48) is apparently
analogous in structure to the usual SFA with the plane-wave Volkov final state. Note,
however, that in (1.48) the plane-wave Volkov final state and the short-range potential
Vs.r. , appear self-consistently together.

1.6 Concluding Remarks

We may end this report with a few short remarks.

12 F.H.M. Faisal

(a) For the sake of concreteness we have presented the result starting with the
Schroedinger equation of the interacting system in the minimal coupling gauge (so-
called velocity gauge). A similar result can be derived in the same way (or by a
gauge transformation) starting from the Schroedinger equation in the so-called length
gauge. This and the issue of gauge invariance of the theory will be presented and
discussed elsewhere.
(b) It is expected that the present theory would be helpful in clarifying a number
of issues of much current interest in strong-field physics involving (i) the shape of the
so-called “low energy structure” (LES) [18], (ii) the number of peaks associated with
the “very low energy structures” (VLES) [19, 20], (iii) origin of the “zero energy
structure” (ZES) [21], and (iv) possible existence of an as yet unknown “threshold
law” for the energy dependence of the strong-field ionization probability. Most or
all of these issues possibly depend crucially on the role of the long-range final state
Coulomb interaction specially in the low energy regime (e.g. c.f. [22]).
(c) The explicit expression of the Coulomb-Volkov propagator (or Green’s func-
tion) GCV (t, t′ ) given here suggests that the theory also would be useful for strong-
field processes involving excitation of the discrete states, either as a final state, or as
intermediate mediating states, or both, for example, in connection with the so-called
“frustrated ionization” (e.g. [23]) in strong fields.
(d) We may point out for that the terms of the S-matrix series (1.42), for example,
the amplitudes Sif(1) and Sif(2) , could be evaluated by a combination of stationary phase
method and numerical evaluation, provided the coordinates dependent Coulomb in-
tegrals can be evaluated analytically, for example by Norsieck’s method [24]. The
Coulomb integral of the first order amplitude (and of the first factor of the second
order amplitude) are of the form

(1) e
M𝐩,i = 𝜙(−)∗
𝐩 (𝐫)(− 𝐀(t) ⋅ 𝐩op )e−𝜅r d3 r (1.49)
∫ mc
Zℏ
where 𝜂(p) ≡ pa0
. The second Coulomb integral of the 2nd order amplitude is of the
form

(2) e
M𝐩,𝐤 = 𝜙(−)∗
𝐩 (𝐫)(− 𝐀(t) ⋅ (𝐩op − 𝜋c ))ei𝐤⋅𝐫 d3 r
∫ mc
e
= (− 𝐀(t) ⋅ (𝐤 − 𝐩)) 𝜙(−)∗ (𝐫)ei𝐤⋅𝐫 d3 r
mc ∫ 𝐩
(1.50)

They have the same form as of the following two prototypical integrals which we
give explicitly below:

I1 = e−i𝐬⋅𝐫 1 F1 (i𝜂s , 1, i(sr + 𝐬 ⋅ 𝐫))(𝜀 ⋅ 𝐩op )e−𝜅r d3 r

∫
= 8𝜋ℏ𝜅(1 + i𝜂s )(𝜀 ⋅ 𝐬)∕((𝜅 + is)(2−i𝜂s ) (𝜅 − is)(2+i𝜂s ) ) (1.51)
1 Strong-Field S-Matrix Series with Coulomb Wave Final State 13

I2 = (𝜀 ⋅ (𝐤 − 𝐬)) × Lim.𝜆→0 e−i𝐬⋅𝐫 1 F1 (i𝜂s , 1, i(sr + 𝐬 ⋅ 𝐫)ei𝐤⋅𝐫 e−𝜆r d3 r

∫
8𝜋s𝜂s q2
= (𝜀 ⋅ 𝐪) × ( 2 )i𝜂s (1.52)
q2 (q2+ 2𝐪 ⋅ 𝐬) q + 2𝐪 ⋅ 𝐬
Z
where, 𝐪 ≡ 𝐤 − 𝐬, 𝜂s = sa0 , and 𝜀 stands for an unit vector. The additional integra-
tion over the intermediate momentum 𝐤 can be performed e.g. by the stationary
phase method (or otherwise), and the first time-integration can be done either ana-
lytically or by the stationary phase method, while the second time-integration can be
done e.g. numerically. (Application of the theory to the observed low energy struc-
tures/phenomena [18–21] with more details of the calculations and discussions of
the results will be presented elsewhere).

References

1. L.V. Keldysh, Sov. Phys. JETP 20, 1307 (1965)

2. F.H.M. Faisal, J. Phys. B: At. Mol. Opt. Phys. 6, L89 (1973)
3. H.R. Reiss, Phys. Rev. A 22, 1786 (1980)
4. A.I. Nikishov, V.I. Ritus, Sov. Phys. JETP 23, 168 (1966)
5. A.M. Perelomov et al., Sov. Phys. JETP 50, 1393 (1966)
6. V.S. Popov, Phys. Usp. 42, 733 (1999)
7. M. Klaiber et al., Phys. Rev. A 87, 023417 (2013)
8. V.P. Krainov, J. Opt. Soc. Am. B 14, 425 (1997)
9. S.V. Popruzhenko et al., Phys. Rev. A 77, 053409 (2008)
10. T.-M. Yan et al., Phys. Rev. Lett. 105, 253002 (2010)
11. X.-Y. Lai et al., Phys. Rev. A 92, 043407 (2015)
12. L. Torlina, O. Smirnova, Phys. Rev. A 86, 043408 (2012)
13. A. Galstyan et al., Phys. Rev. A 93, 023422 (2016)
14. A. Becker, F.H.M. Faisal, J. Phys. B: At. Mol. Opt. Phys. 38, R1 (2005)
15. M. Jain, N. Tzoar, Phys. Rev. A 18, 538–45 (1978)
16. C. Leone et al., Nuouo Cimento D 9, 609 (1987)
17. L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Pergamon Press, Oxford, 1965)
18. C.I. Blaga et al., Nature Phys. 5, 335 (2008)
19. W. Quan et al., Phys. Rev. Lett. 103, 093001 (2009)
20. C. Wu et al., Phys. Rev. Lett. 109, 043001 (2012)
21. J. Dura et al., Nature Sci. Rep. 3, 2675 (2013)
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24. A. Nordsieck, Phys. Rev. 93, 785 (1954)
Chapter 2
Multiconfiguration Methods
for Time-Dependent
Many-Electron Dynamics

Erik Lötstedt, Tsuyoshi Kato and Kaoru Yamanouchi

Abstract A concise overview of time-dependent multiconfiguration methods for the

approximate solution of the time-dependent Schrödinger equation for many-electron
systems in intense laser fields is presented. In all the methods introduced, the total
wave function of the system is written as a linear combination of Slater determinants.
The methods can be divided into two classes, one class in which the orbitals used to
construct the Slater determinants are time-independent, and the other class in which
the orbitals are time-dependent. The key ideas of these two classes are reviewed,
focusing on the scheme used for reducing the number of Slater determinants in the
expansion of the wave function. Also described is our recent proposal, in which
the number of Slater determinants is not reduced, but the matrix of configuration-
interaction coefficients is approximated by a product of three smaller matrices.

2.1 Introduction

A common theme in theoretical chemistry is to develop efficient procedures for

describing electron-electron interaction in atoms and molecules. Starting with the
Hartree-Fock approximation, which includes the electron-electron interaction only
on a mean-field level, researchers have proposed a variety of methods such as config-
uration interaction (CI) and coupled-cluster (CC) theory [1] to solve approximately
the Schrödinger equation (SE)
H𝛹 = E𝛹 , (2.1)

E. Lötstedt (✉) ⋅ T. Kato ⋅ K. Yamanouchi

Department of Chemistry, School of Science, The University
of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
e-mail: lotstedt@chem.s.u-tokyo.ac.jp
T. Kato
e-mail: tkato@chem.s.u-tokyo.ac.jp
K. Yamanouchi
e-mail: kaoru@chem.s.u-tokyo.ac.jp

© Springer International Publishing AG 2017 15

K. Yamanouchi et al. (eds.), Progress in Ultrafast Intense Laser Science XIII,
Springer Series in Chemical Physics 116, https://doi.org/10.1007/978-3-319-64840-8_2
16 E. Lötstedt et al.

where H is the Hamiltonian, E is the eigenenergy, and 𝛹 is the wave function. The
non-relativistic ground state energy levels E0 of many-electron molecules (where the
number of electrons N ≤ 15) can be obtained to about 0.01 eV accuracy using the CI
method [2] or the CC method [3]. For small systems like He [4, 5] and H2 [6], the
exact eigenenergies of the bound states can be calculated. It remains a challenge to
obtain reliable estimates of the energies of highly excited states [7–9].
Much less effort has been spent on developing methods for obtaining solutions to
the time-dependent Schrödinger equation (TDSE),

𝜕𝛹 (t)
iℏ = H(t)𝛹 (t), (2.2)
𝜕t

where the Hamiltonian H(t) depends on time. The time-dependence of the Hamil-
tonian usually arises from the coupling with a laser field. Equation (2.2) describes
an initial-value problem, where we provide the initial wave function 𝛹 (t = 0) = 𝛹0 ,
which is the lowest energy solution to (2.1), and we seek an approximation to the
wave function 𝛹 (t) for 0 ≤ t ≤ T for some finite time T. To compute 𝛹 (t) could be a
significantly more difficult problem than solving the time-independent SE (2.1). The
reason is that we require a method that describes well the electron correlation not
only in the initial state 𝛹 (0) but also at later time t. Particularly in the case of intense
laser-molecule and laser-atom interactions, the wave function is strongly perturbed
by the laser field so that 𝛹 (t) becomes in general very different compared to the ini-
tial state 𝛹 (0). After the interaction with the laser field, the wave function becomes
a superposition of the initial state and more than a few excited states, which may
include singly excited states, doubly excited states, Rydberg states, and continuum
states representing ionization. Ideally, a theoretical method we develop can describe
all of these components equally well.
On the experimental side, a variety of phenomena have been observed, the mech-
anisms of which need to be explored theoretically. The best studied example is the
correlated motion of the two electrons in helium, leading to non-sequential double
ionization [10, 11] and the creation of wave packets consisting of doubly-excited
states [12]. Signatures of correlated electron dynamics can also be seen in high-
harmonic spectra of molecules [13, 14] and atoms [15], and in molecular dissocia-
tion [16, 17].
In the first part of this chapter, Sect. 2.2, we introduce the basics of multicon-
figuration-based wave function approaches to the approximate solution of the TDSE
(2.2). A multiconfiguration wave function is a wave function which consists of
several or many Slater determinants. The simplest wave function of this kind, in
which only one Slater determinant is included, is called the Hartree-Fock (HF)
wave function. Adding more Slater determinants gives a better description of the
dynamics of the system. Since the inclusion of all possible Slater determinants
is computationally unfeasible for any system having more than two electrons, the
essential point of a multiconfiguration method is to find a way of effectively reduc-
ing the number of Slater determinants. In Sect. 2.3, we review three approaches for
reducing the number of Slater determinants: the time-dependent configuration inter-
Another random document with
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 Powerful arms were about her, she was being supported. The
fumes of raw spirit were in her nostrils, a glass was pressed against
her lips. She fought again to get free, only feebly now, for this was
but a last reaction of a dying will. Yet the final word of all was
nature’s. When mind itself had ceased to count, the life-force
grasped wildly at the proffered means of life.
“Thank God!” she heard a thick voice mutter. “I felt sure you were
a goner.”
A livid face, whose eyes seemed to blind her own, materialized
suddenly before her. “Drink it up, damn you!” said the voice hoarsely.
“And then get out—you——!”
It was insult for the sake of insult, and therefore the full measure
of her victory. But it meant less than nothing to June now. She
scarcely heard, or hearing did not comprehend. Beyond pain and
suffering, beyond good and evil her torn spirit only craved release.
As soon as the fire in the glass had kindled her veins this desire
was met, less, however, by the operation of her own will than by the
will of Keller. As if she had been a noisome reptile whom his flesh
abhorred, and yet had a superstitious fear of killing, he dragged her
out of the room, along the short passage as far as the door of the
flat. Slipping back the catch, he flung her out on to the landing.
As she fetched up against the iron railing opposite the door,
which guarded the well of the staircase, she heard a low hiss: “Take
yourself off as soon as you like, you——, or you’ll find the police on
your track.”
 XLII
June had no idea of the time that she lay in a huddle against the
railing. But it may not have been so long in fact as it was in
experience. Shattered she might be, yet unknown to herself, there
was still a reserve of fighting power to draw upon.
Cold iron, moreover, and raw air had a magic of their own. Clear
of that mephitic room and the foul presence of Keller, a fine human
machine began slowly to renew itself. Except for a faint gleam from
the room out of which she had just come, stealing through the
fanlight of the door out of which she had been flung, there was not a
sign of light upon the staircase. The entire building appeared to be
deserted. Its stone-flagged steps were full of echoes as soon as she
ventured to move upon them; and when clinging to the railing for
support she had painfully descended two she entered a region of
total darkness.
It was like going down into a pit. Could she have only been sure
that death awaited her below, she might have been tempted to fling
herself into it headlong. But she knew that the ground was not far off.
Three or four steps more brought her to the vestibule. At the end
of it was a door, open to the street. Outside this door shone a faint
lamp, round which weird shadows circled in a ghostly witch-dance.
The night beyond was a wall of horrors, which she had lost the will to
face.
Met by this pitiless alternative, she recoiled against the wall of the
vestibule, huddling in its darkest corner, behind the stairs. Crouching
here, like a hunted thing at bay, she fought for the courage to go out
and face her destiny.
She fought in vain. Half collapsed as she now was, a spur was
needed to drive her into the grim wilderness of the open street. One
glance at the crypt outside sufficed to tell her that with no point to
make for, it would be best to stay where she was and hope soon to
die.
Why had she not had the sense to throw herself down the stairs
and kill herself? A means would have to be found before the night
was out. She could bear no more. A terrible reaction was upon her. It
was as if a private door in her mind had suddenly given way and a
school of awful phantoms had rushed in and flooded it.
She was living in a nightmare that was too bad to be true. But it
was true and there lay its terror. Adrift in the dark canyons of that
vast city, penniless and alone, with the marks of thieves and
murderers upon her bruised body, and her treasure stolen, there was
only one thing to look for now.
Death, however, would not be easy to come by. As she huddled
in cold darkness in the recess behind the stairs she felt that her will
was going. To enter the night and make an end would need courage;
but a miserable clapping together of the jaws was sign enough that
the last hope of all was slipping away from her.
 XLIII
Cowering in body and spirit in that dark corner, time, for June,
became of no account. Perhaps, after all, she might be allowed to
die where she was. As a kind of inertia crept upon her she was able
to draw something of comfort from the thought. It would be better
than the river or being run over in the street.
She grew very cold; yet a lowering of the body’s temperature
induced a heightened consciousness. Aches and pains sprang into
life; the forces of her mind began to reassert themselves; the
phantoms about her took on new powers of menace. Gradually it
became clear to June, under the goad of this new and sharper phase
of suffering, that mere passivity could not induce the death she
longed for.
No, it was not in that way the end would come. She would have
to go into the shadow-land beyond the lamp, and seek some positive
means of destroying herself. For that reason she must hold on to the
fragment of will that now remained to her. It alone could release her
from the awful pit in which she was now engulfed.
She gathered herself for an effort to move towards the fog-
encircled light at the entrance to the street. But the effort, when
made, amounted to nothing. Her limbs were paper, all power of
volition was gone.
The October raw struck to her blood. She began to whimper
miserably. To pain of mind was added pain of body, but the delicate
apparatus from whose harmony sprang the fuse of action was out of
gear. Something must be done; yet no matter how definite the task,
any form of doing was beyond her now.
At this dire moment, however, help came. It came, moreover, in
an unlooked-for way. She heard a door slam over head. There was
the sound of a match being struck, and then came a gingerly shuffle
of feet on the stone stairs.
Someone was coming down. June cowered still lower into the
dark recess at the back of the stairway. A man was approaching.
And by the flicker of the match which he threw away as he reached
the floor of the vestibule she saw that the man was Keller.
Faint and but momentary as was the glimpse afforded, June, with
every sense strung again to the point of intensity, saw that under
Keller’s arm was a brown paper parcel. The sight of it was like a
charm. Some fabulous djinnee might have lurked in that neat
package, who commanded a miraculous power of reaction upon the
human will.
Keller struck a second match and peered into the shadows. June
knew that he was looking to see if she had lingered there, but the
light could not pierce to the corner in which she crouched; and it
burnt itself out, leaving him none the wiser.
Without striking another match Keller moved away from her
towards the doorway, and as he did so June felt a swift release of
heart and brain. A thrill of new energy ran through her. No sooner
had Keller passed out of the vestibule, beyond the lamp into the fog,
than without conscious impulse or design she began to follow him.
It may have been the reasoned act of a lucid being, but at first it
did not appear to be so. Once, however, her limbs were moving, all
her faculties, now intensely awake, seemed as if by magic to bear
them company. As soon as she reached the open street, with Keller
a clear ten yards ahead, the keen air on her face had an effect of
strong wine. Her nerves felt again the sense of motion; the impulse
of the natural fighter unfurled strong pinions within her. All the virile
sense and the indomitable will of a sound inheritance rallied to her
need.
Growing sensibly stronger at every yard, she followed Keller
round the corner into Manning Square. The mist was thick, the lamps
poor and few, but as well as she could she kept on his track. Lurking
pantherlike in the deep shadows of the house-walls, she had
approached within five yards of him by the time he had turned the
corner into a bye-street. He went a few yards along this, and then
zigzagged into a squalid ill-smelling thoroughfare whose dismal
length seemed unending.
June had no difficulty in keeping up with these twists and winds,
for Keller, impeded by the fog, moved slowly. For her, however, the
fog had its own special problem, since there was a danger of losing
him if he was allowed to get too far ahead; and yet if his steps were
dogged too closely there was always the fear that he might turn
round suddenly and see her.
At last the interminable street seemed to be nearing its end. For
June, whose every faculty was now strung up to an unnatural
acuteness, saw but a short distance in front the brightly lit awning of
the Underground looming through the fog.
In a flash she realized the nature of the peril. Only too surely was
this the bourn for which Keller was making. Once within its precincts
and her last remaining hope would be gone.
It must be now or never. The spur of occasion drove deep in her
heart. She knew but too well that the hope was tragically small, but
wholly desperate as she was, with the penalty of failure simply not to
be met, she would put all to the touch.
Closer and closer she crept up behind the quarry. But the
entrance to the Tube loomed now so near that it began to seem
certain that she must lose him before she could attempt what she
had to do. Abruptly, however, within ten yards or so of his goal,
Keller stopped. He began to search the pockets of his overcoat for a
box of matches to relight his pipe which had gone out. While so
doing, and in the preoccupation of the moment, he took the parcel
from under his right arm, and set it rather carelessly beneath his left.
Providence had given June her chance. Like a falcon, she
swooped forward. Aim and timing incredibly true, at the instant Keller
struck a match and bent over his pipe, her fingers closed on the Van
Roon, and whisked it out of his unguarded grasp.
 XLIV
AsherJune turned and ran she heard a wild and startled oath. Before
was the eternal fog-laden darkness of the narrow street. But
now it struck her with a thrill of pure terror that the mist was not thick
enough to conceal her flight. The swift surprise of the onset had
gained for her a start of a few yards, but instantly she knew that it
would not suffice.
She ran, all the same, as if her heart would burst. But her legs
seemed to wear the shackles that afflict one in a dream. Her most
frantic efforts did not urge them on, and yet, in spite of that, they
bore her better than she knew. Not a soul was in sight. She could
hear Keller’s boots echo on the damp pavement as they pounded
behind her. It could only be a matter of seconds before his fingers
were again on her throat. But this time, before robbing her of the Van
Roon and getting clear, he would have to kill her.
The vow had hardly been made, when at the other side of the
street she saw a thread of light. It came from a house whose door
was open. Instinctively she turned and made one final dash for it.
This was the last wild hope there was.
A man, it seemed, was in the act of leaving the house. Wearing
overcoat and hat, he stood just within the doorway peering into the
murk before venturing out. June flung herself literally upon him.
“Save me! Save me!” she was able to gasp. “A man! A man is
after me!”
The house was of the poverty-stricken kind whose living room
opens on to the street. June had a confused vision of a glowing
lamp, a bright fire, a dingy tablecloth and several people seated
around it. Her wild impact upon the man who was about to put off
from its threshold drove him backwards several paces into the room.
At the same instant a female voice, loud and imperious, rose from
the table.
“Shut the door, Elbert, can’t yer? The fog’s comin’ in that thick it’ll
put out the perishin’ fire.”
The bewildered Elbert, raked fore and aft by fierce women,
automatically obeyed the truculent voice at his back, even while he
gave ground in a collision which seemed to rob him of any wit that he
might possess. With a deft turn of the heel, he dealt the door a kick
which effectually closed it in the murderous face of the halting and
hesitating Keller.
June, shuddering in every vein, clung to her protector.
“Gawd-love-us-all!” Cries and commotion arose from the table,
yet almost at once the imperious voice soared above the din. “Set
her down, can’t yer, Elbert? Didn’t yer see that bloke?”
“Ah—I did,” said Elbert, stolidly pressing his queer armful into a
chair near the fire.
“Better git after him lively,” said the voice at the table. “He’s the
one as did in Kitty Lewis last week.”
Elbert, a young man six feet tall and proportionately broad of
shoulder, was not however a squire of dames. With a scared look on
a face that even in circumstances entirely favourable could hardly
rank as a thing of beauty, he moved to the door and slipped a bolt
across. “Not goin’ near the——” he said, sullenly. “Not goin’ to be
mixed up wiv it—not me.”
The voice at the table, whose owner was addressed as Maw,
proceeded “to tell off” Elbert. He was a skunk, he was no man, he
was a mean swine. In the sight of Maw, who ran to words as well as
flesh, Elbert was all this and more. She rose majestically, threatening
to “dot him” if he didn’t “’op in,” and she came to June with an
enormous bosom striving to burst from its anchorage, an apron that
had once been white, and with her entire person exuding an odour
peculiar to those of her sex who drink gin out of a teacup.
 Three other people were at the table, and they were engaged
upon a meal of toasted cheese, raw onions and beer. Of these, two
were girls about sixteen, scared, slatternly and anæmic; the third
was a toothless hag who looked ninety; and as the whole family,
headed by Maw, suddenly crowded round June, the terrified fugitive,
shuddering in the chair by the fire, hardly knew which of her
deliverers was the most repulsive.
June fought with every bit of her strength against the threat of
total collapse that assailed her now. In the desperate hope of
warding off disaster, she gathered the last broken fragment of will.
But nature had been driven too hard. For the second time within the
space of one terrible hour, she lost the sense of where she was.
 XLV
The faces, with one exception, had receded into the background,
when June returned slowly and painfully to a knowledge of what
was happening. Maw was bending over her, and holding a cracked
cup to her lips, and also “telling off” the others with a force and a
scope of language that added not a little to June’s fear.
Perhaps the smell of its contents had quite as much effect upon
the sufferer as the cup’s restorative powers. It was so distasteful to
one who had been taught to shun all forms of alcohol, that a sheer
disgust helped to bring her round.
At first, however, her mind was hardly more than a blank. But
when, at last, a few links of recognition floated up into it out of the
immediate past and hitched themselves to this strange present, a
shock of new terror nearly overwhelmed her again. Recollection was
like a knife stab. The Van Roon! The Van Roon! Where was it? Oh,
God—if she had not got it after all!
The thought was pain, pure and exquisite. But the case did not
really call for it. She was clutching the Van Roon convulsively to her
breast like a child holds a doll. As she wakened slowly to this fact her
brain wonderfully cleared.
The mind must be kept alive, if only to defend this talisman for
whose sake she had already suffered so outrageously. She did not
know where she was, and the evil presence holding the foul cup to
her lips, and those other evil presences filling the background
beyond gave her an intense apprehension.
Maw, however, in spite of a general air of obscenity, meant well. It
was not easy for this fact to declare itself through that loud voice and
ruthless mien; but gradually it began to percolate to June’s violated
nerves, and so gave her a fleck of courage to hold on to that sense
of identity which still threatened at the first moment again to desert
her.
“Where was you goin’, deery?”
Rude the tone, but when June’s ear disentangled the words, she
was able to appreciate that they were spoken in the way of kindness.
But if the knowledge brought a spark of comfort it was quickly
dowsed. Where was she going? To that grim question there was no
possible answer.
“Scared out of her life, poor lamb!” said Maw. With furtive
truculence she announced the fact to the rather awed spectators
who gathered once more about the sufferer.
“Where you come from?”
June’s only answer was a shiver. The frozen silence was so full
of the uncanny that Maw shook her own head dismally and tapped it
with a grimy finger.
In the view of Maw, for such a calamity there was only one
remedy. Once more the cup was pressed to June’s lips; once more it
was resisted, this time with a hint of fierceness reassuring to the
onlookers, inasmuch that it implied a return of life.
“Looks respectable,” said the cracked voice of the crone, who
was now at Maw’s elbow.
“Where was you goin’?” demanded Maw again.
June was beyond tears, or she would have shed them. Now that
the facts of the situation in all their hopelessness were streaming
back to her, a feeling of sheer impotence kept her dumb.
“Off her rocker,” said Elbert gloomily.
 XLVI
Amid the silence which followed Elbert’s remark, June fought hard
to cast her weakness off. She wanted no longer to die. The
recovery of the talisman inhibited, at least for the time being, that
desire. Acutely aware that the Van Roon was still miraculously hers,
she felt that come what might she must go on.
But her position was hopeless indeed. She dare not venture out
of doors, with a murderous thief waiting to spring upon her. And if
venture she did, there was nowhere she could go. Besides, had
there been any place of refuge for such a weary bundle of frightened
misery, without money and with a sorry ignorance of the fog-bound
maze of bricks and mortar in which she was now lost, there would
have been no means of getting to her destination.
At the same time, she had no wish to stay with these uncouth, ill-
looking, evil-smelling people one moment longer than was
necessary. In a curiously intimate way she was reminded of that grim
story Oliver Twist, which had so powerfully haunted her youth. To her
distorted mind, this squalid interior was a veritable thieves’ kitchen,
the crone a female Fagin, the angel of the cup, a counterpart of Bill
Sikes, and the gloomy, beetle-browed Elbert a kind of Artful Dodger
grown up. She and her treasure could never be safe in such a place,
yet at the other side of the door nameless horrors awaited her.
In June’s present state it was far beyond her power to cope with
so dire a problem. Keeping a stony silence as those faces, devoured
by curiosity, pressed ever closer upon her, she half surrendered to
her weakness again.
Amid the new waves of misery which threatened to submerge
her, she was wrenched fiercely back to sensibility. The Van Roon
was torn by a strong hand from her grasp. As if a spring had been
pressed in her heart she rose with a little cry. Maw was in the act of
handing the picture to Elbert. “There’s a label on it, ain’t there?” she
said.
Still half stupefied, June clung to the table for support, while
Elbert, who was evidently the family scholar, read out slowly the
name and address that was written upon the parcel: “Miss
Babraham, 39b Park Lane, W.”
June was hardly in a state just then to grasp the significance of
the words. Her mind was wholly given up to concern for the treasure
which had passed to alien hands. And yet the words had
significance, even for her, as the mind-process they induced soon
began to reveal.
A locked door of memory, of which she had lost the key, seemed
to glide back. Thoughts of William, of his friend, the tall, beautiful and
distinguished wearer of the blue crepe de chine, and of Sir Arthur,
her father, came crowding into her brain. And with them came a
perceptible easing of spirit, as if they had been sped by the kindly
hand of that Providence, of whom she had never been so much in
need.
The recognition of this acted upon her like a charm. Girt by the
knowledge that she was not alone in the world after all, and that
friends might be at hand if only she could reach out to them, her
mind began once more to function.
Even while Maw and Elbert were occupying themselves with the
parcel’s address and its specific importance, June was fain to inquire
of an awaking self how such magic words came to be there at such a
moment. Casting back to recent events, over which oblivion had
swept, she was able to recall certain strands in the subtle woof of
Fate. Days ago, years they seemed now, Miss Babraham had sent
to William a picture frame to be restored. The stout brown paper in
which it had been wrapped appealed to June’s thrifty soul, and she
had stowed it away in her box for use on a future occasion. Her
mind’s new, almost dangerous clarity, enabled her to remember that
upon the paper’s inner side was an old Sotheran, Bookseller,
Piccadilly label which bore the name and address of Miss Babraham.
 The piecing together of this slender chain gave June the thing
she needed most. At this signal manifestation of what Providence
could do, hope revived in her. If only she could get to Park Lane—
wherever Park Lane might be!—to Miss Babraham.
As if in answer to the half-formed wish, Maw’s dominant voice
took up the parable. “Elbert, you’d better see this lidy as fur as Park
Lane.”
 XLVII
Elbert did not welcome the prospect with open arms. Nature had
not designed him for such a task. All the same, Maw was imaged
clearly in his mind as one whose word was law.
At the best of times, Elbert’s obedience to that word was apt to
be grudging. And to-night, with murder lurking outside in the
darkness, he was full of a disgusted reluctance at having to face
such a prospect. Even in circumstances wholly favourable to it, the
countenance of Elbert was not attractive; to June at this moment it
was very much the reverse. She felt that its owner was not to be
trusted an inch.
Meanwhile her mind was growing very active. Miss Babraham’s
name and address, that magic omen, was like an elixir; it quickened
the blood, it strengthened the soul. If only she could bear her
treasure to Park Lane all might yet be well!
Urged by this spur, native wit sprang to her aid. The first thing to
be done was to get clear of present company. She was haunted still
by the likeness to Fagin’s kitchen; but also there was a recollection
of the fact that a Tube Station was only a few yards along the street.
That was the haven wherein salvation lay.
Pressing hard upon the hope, however, was the dismal
knowledge that only one penny remained in her pocket. This sum
could not take her to Park Lane, unless that Elysium was close at
hand. Alas, it was not at all likely. Her ignorance of London was so
great, moreover, that she would need help to find her way there; and
in the process of obtaining it in her present state of weakness she
might be caught by new perils. For it was only too likely that Keller
was lurking outside in the fog, waiting to spring upon her and tear the
Van Roon from her grasp at the first chance that arose.
 Beset by such problems, June felt that she was between the devil
and the deep sea. Perhaps the best thing she could do was to dash
along the street to the Tube, and then put herself in the hands of the
nearest policeman. But even to attempt such a feat was to run a
grave risk.
Elbert, in the meantime, scowling and disgruntled, was bracing
himself under further pressure from Maw to brave the perils of the
night. June felt, however, that it would be wise not to saddle herself
with this reluctant champion if it could be avoided. To this end, she
was now able to pluck up the spirit to ask what was the best means
of getting to Park Lane.
Maw did not know, but Elbert when appealed to said that she
could take the Tube to Marble Arch, or she might turn the corner at
the end of the street and pick up a bus in Tottenham Court Road.
How much was the fare? Twopence, Elbert thought. Alas, June
had only a penny. She was painfully shy about confessing this
difficulty, but there was no help for it.
“Don’t you worry, Miss. Elbert is goin’ to see you all the way.” And
Maw fixed a savage eye upon her son.
Much as June would have preferred to forego the services of this
paladin, Maw’s ferocious glance settled the matter finally.
“And you’ll carry the pawcel for the lidy,” said Maw, as Elbert,
scowling more darkly than ever turned up the collar of his overcoat.
 XLVIII
The Van Roon, at that moment, was in the hand of Maw. And
although June was on fire to get it back, her natural faculties had
the authority to tell her that undue eagerness would be most unwise.
She must be content to await her chance, yet there was no saying
when that chance would come; for Maw was careful to hand
personally the parcel to Elbert.
Before June set out on her journey one of the girls pressed a cup
of tea from the family brew upon her. It was lukewarm and thrice-
stewed, but June was able to drink a little and to feel the better for it.
She was in a high state of tension, all the same, when Elbert opened
the street door, her treasure under his arm, and she followed close
behind him into the darkness.
Surely Keller must be out there in the fog, waiting to attack them.
Her heart beat wildly as she marched side by side with Elbert along
the street towards the Tube. Distrust of her cavalier was great.
Should he guess the value of the thing he bore, as likely as not he
would play her a trick. But for the moment, at any rate, this fear was
merged in the sharper one of what was concealed by the fantastic
shadow shapes of that dark thoroughfare. Less than a hundred
yards away, however, was the Tube Station. And to June’s
unspeakable relief they gained its light and publicity without
misadventure. Here, moreover, was her chance. While Elbert
searched his pockets for fourpence to purchase two tickets for
Marble Arch, she insisted on relieving him of the parcel. Once
restored to her care, she clung to it so tenaciously that the puzzled
Elbert had reluctantly to give up the hope of getting it back again.
Going down in the lift to the trains, with the surge of fellow
passengers guaranteeing a measure of safety, June allowed herself
to conclude that Elbert, after all, might be less of a ruffian than he
looked. If he had no graces of mind or mansion, he was yet not
without a sort of rude care for her welfare. By no wish of his own was
he seeing a distressed damsel to her home, yet the process of doing
so, once he grew involved in it, seemed to minister in some degree
to a latent sense of chivalry. At all events he had a scowl for anyone
whose elbows came too near his charge.
Arriving at Marble Arch in due course, the heroic Elbert piloted
the fugitive out of the station and across the road into Park Lane.
Here, under a street lamp, they paused a moment to examine the
label on the parcel for the number of the house they sought. Thirty-
nine was the number, and it proved to be not the least imposing
home in that plutocratic thoroughfare.
Elbert accompanied June as far as its doorstep. Before ringing
the bell she said good-bye to her escort with all the gratitude she
could muster, begging him to give her his name and address, so that
she might at least restore to him the price of her fare. Yet the squire
of dames saw no necessity for this. His scowl was softened a little by
her thanks, but his only answer was to press the electric button and
then, without a word, to slink abruptly away into the fog.
 XLIX
June felt a wild excitement, as she stood waiting for the answer to
her ring. The stress of events had buoyed her up, but with Elbert
no longer at her side and the door of a strange house confronting
her, trolls were loose once more in her brain. A fresh wave of panic
surged through her, and again she feared that she was going to faint.
The prompt opening of the door by a gravely dignified
manservant acted as a strong restorative. June mustered the force
of will to ask if she could see Miss Babraham. Such a request, made
in a nervous and excited manner, gave pause to the footman, who at
first could not bring himself to invite her into the large dimly lighted
hall. Finally he did so; closed the door against the fog, and then
asked her name with an air of profound disapproval, which at any
other time must have proved highly embarrassing.
“I’m Miss Gedge,” said June. “From the second-hand shop in
New Cross Street. Miss Babraham’ll remember me.”
The servant slowly repeated the fragmentary words in a low voice
of cutting emphasis. “I’m afraid,” he said, while his eye descended to
June’s shoes and up again, “Miss Babraham will not be able to see
you to-night. However, I’ll inquire.”
Superciliously the footman crossed the hall, to discuss the matter
with an unseen presence in its farthest shadows. The conference
was brief but unsatisfactory, for a moment later the unseen presence
slowly materialized into the august shape of a butler, who seemed at
once to diminish the footman into a relative nothingness.
“Perhaps you’ll let me know your business,” said the butler, in a
tone which implied that she could have no business, at any rate with
Miss Babraham, at such an hour.
June, alas, could not explain the nature of her errand. These two
men were so imposing, so unsympathetic, so harsh, so frightening
that had life itself depended upon her answers, and in quite a special
degree she now felt that it did, she was yet unequal to the task of
making them effective.
“Miss Babraham cannot see you now,” said the slow-voiced
butler, with an air of terrible finality.
“But I must see her. I simply must,” wildly persisted June.
“It’s impossible to see her now,” said the butler.
The words caused June to stagger back against the wall. In
answer to her tragic eyes, the butler said reluctantly: “You had better
call again some time to-morrow, and I’ll send in your name.”
“I—I must see her now,” June gasped wildly.
The butler was adamant. “You can’t possibly see her to-night.”
“Why can’t I?” said June, desperately.
“She is going to a ball.”
The words were like a blow. A vista of the fog outside and of
herself wandering with her precious burden all night long in it
homeless, penniless, desolate, came upon her with unnerving force.
“But—please!—I must see her to-night,” she said, with a shudder of
misery.
Faced by the butler’s pitiless air, June felt her slender hope to be
ebbing away. She would be turned adrift in the night. And what
would happen to her then? She could not walk the streets till
daybreak with the Van Roon under her arm. Already she had
reached the limit of endurance. The dark haze before her eyes bore
witness to the fact that her strength was almost gone. No matter
what the attitude of the butler towards her she must not think of
quitting this place of refuge unless she was flung out bodily, for her
trials here were nought by comparison with those awaiting her
outside.
June’s defiance was very puzzling to the stern functionary who
quite plainly was at a loss how to deal with it. But in the midst of
these uncertainties the problem was unexpectedly solved for him. A