Quantum Dynamic Imaging: Theoretical and Numerical Methods

CRM Series in Mathematical Physics Series Editorial Board: Joel S. Fieldman Department of Mathematics University of British Columbia Vancouver, British Columbia V6T 1Z2 Canada [email protected] Duong H. Phong Department of Mathematics Columbia University New York, NY 10027–0029 USA [email protected] Yvan Saint-Aubin D´epartement de Math´ematiques et de Statistique Universit´e de Montr´eal C.P. 6128, Succursale Centre-ville Montr´eal, Qu´ebec H3C 3J7 Canada [email protected] Luc Vinet D´epartement de Math´ematiques et de Statistique CRM, Universit´e de Montr´eal C.P. 6128, Succursale Centre-ville Montr´eal, Qu´ebec H3C 3J7 Canada [email protected]

For further volumes: http://www.springer.com/series/3872

André D. Bandrauk • Misha Ivanov Editors

Quantum Dynamic Imaging Theoretical and Numerical Methods

Editors André D. Bandrauk Département de chimie Université de Sherbrooke Sherbrooke, QC J1K 2R1 Canada [email protected]

Misha Ivanov Department of Physics Imperial College London London United Kingdom [email protected]

e-ISBN 978-1-4419-9491-2 ISBN 978-1-4419-9490-5 DOI 10.1007/978-1-4419-9491-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011930185 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

3UHIDFH

The “Quantum Dynamic Imaging” workshop, the first of its kind ever held in Canada, focused on the theoretical and mathematical problems associated with the imaging of quantum phenomena in matter from the femtosecond (1015 s) for nuclear motion to the attosecond (1018 s) for electron motion. Motion of a proton, one of the most important nuclei in chemistry and biology (e.g., DNA) has a natural time scale of 7 femtoseconds. Motion of electrons, responsible for chemical binding and electron transfer processes in natural phenomena, has a characteristic time scale of about 100 attoseconds (it takes an electron 152 attoseconds to go around the hydrogen atom). Both proton and electron motion can only be described by quantum mechanics, i.e., high dimension partial differential equations (HDPDEs). Furthermore such motions can only be monitored by ultrashort laser pulses. Thus the interaction of matter with such pulses can only be described by HDPDEs such as time-dependent Schr¨odinger and Dirac (for relativistic phenomena) equations coupled to the photons (Maxwell equations). The chapters of this book are based on lectures by invited speakers who are acknowledged experts in developing the necessary theories and numerical methods for treating photon-atom-molecule interactions in the nonlinear nonperturbative regime. In particular the generation of attosecond pulses is a spin-off of such theories of the nonlinear nonperturbative laser-matter interactions. The workshop was concerned with the mathematical problems and progress in developing and validating numerical methods used in the endeavor of imaging quantum phenomena with subfemtosecond temporal and sub-Angstrom spatial resolution. The experts addressed a very important problem of using quantum imaging methods discussed in the workshop and the massive amount of multidimensional information encoded in the time-dependent many-body wavefunctions for the eventual production of “molecular movies.” This is a new direction in molecular imaging with the aim of making quantum information available to researchers in the molecular sciences to visualize quantum phenomena on their natural, from the femto to attosecond, time scale. The scientific publisher, Springer has agreed to publish the lectures of the invited speakers in the CRM Series in Mathematical Physics , under the title of the workv

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Preface

shop. The co-organizers, who are co-editors of the book, wish to especially thank the CRM (Centre de recherches math´ematiques—Montr´eal) staff for its dedication and invaluable help in making this workshop a success and in preparing this book. Sherbrooke and London

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1RQSURGXFW 4XDGUDWXUH *ULGV 6ROYLQJ WKH 9LEUDWLRQDO 6FKUŽRGLQJHU (TXDWLRQ LQ G : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 Gustavo Avila and Tucker Carrington Jr. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Standard Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Calculations in Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Potential Matrix-Vector Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 Application of Smolyak to CH3 –CN . . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 0XOWL3KRWRQ 6LQJOH DQG 'RXEOH ,RQL]DWLRQ RI &RPSOH[ $WRPV E\ 8OWUDVKRUW ,QWHQVH /DVHU 3XOVHV : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : K. Bartschat, X. Guan, C.J. Noble, B.I. Schneider, and O. Zatsarinny 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Computational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Example Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Multiphoton Single-Ionization of Neon . . . . . . . . . . . . . . . 3.2 Multiphoton Single-Ionization of Argon . . . . . . . . . . . . . . 3.3 Two-Color Two-Photon Double Ionization of Helium . . . 4 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &RUUHODWHG (OHFWURQ1XFOHDU 0RWLRQ 9LVXDOL]HG 8VLQJ D :DYHOHW 7LPH)UHTXHQF\ $QDO\VLV : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Andr´e D. Bandrauk, Szczepan Chelkowski, and Huizhong Lu 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Computional Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Correlations in the Power Spectra of Electron and Protons . . . . . . . 4 Visualization of the Electron Returns Using the Time Profiles of Electron and Proton Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 16 16 16 18 20 21 23 23 24 27 29

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5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 $ 7KHRU\ RI '\QDPLF ,PDJLQJ RI &RKHUHQW 0ROHFXODU 5RWDWLRQV E\ +LJK +DUPRQLF *HQHUDWLRQ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : F.H.M. Faisal and A. Abdurrouf 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Wavefunction of the Interacting System . . . . . . . . . . . . . . . 2.2 HHG Amplitude and Dipole Expectation Value . . . . . . . . 2.3 Formulas of HHG Signals for N2 , O2 and CO2 . . . . . . . . . 3 Applications and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7KH RPDWUL[ &DOFXODWLRQV RI 2ULHQWDWLRQ DQG &RXORPE 3KDVH (IIHFWV LQ (OHFWURQ±0ROHFXOH 5H &ROOLVLRQV : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Alex G. Harvey and Jonathan Tennyson 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results for Molecular Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results for Carbon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9LVXDO $QDO\VLV RI 4XDQWXP 3K\VLFV 'DWD : : : : : : : : : : : : : : : : : : : : : : : : : : : : Hans-Christian Hege, Michael Koppitz, Falko Marquardt, Chris McDonald, and Christopher Mielack 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Image Sciences and Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Value of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Image Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Data Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Visualization Techniques for Quantum Data . . . . . . . . . . . . . . . . . . . 3.1 Standard Visualization Techniques . . . . . . . . . . . . . . . . . . . 3.2 Low-Dimensional Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Multidimensional Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Two Visualization Examples from Quantum Research . . . . . . . . . . . 4.1 Depiction of Coupled Electron and Nuclei Dynamics in H2C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Visualization of Data from a 3D Simulation of High-Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 39 39 41 43 45 53 53 55 55 56 59 64 68 69 71

71 72 72 73 74 76 76 76 79 81 81 82 85 86

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7KHRU\ RI '\QDPLF ,PDJLQJ RI 0ROHFXOHV ZLWK ,QWHQVH ,QIUDUHG /DVHU 3XOVHV : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 89 C.D. Lin, Anh-Thu Le, and Zhangjin Chen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2 Theoretical Tools for Studying Atoms and Molecules in Strong Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.1 Solution of the Time-Dependent Schr¨odinger Equation . . 91 2.2 Strong Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . 93 2.3 Laser-Free Elastic Scattering Theory . . . . . . . . . . . . . . . . . 93 2.4 Quantitative Rescattering Theory . . . . . . . . . . . . . . . . . . . . 95 2.5 The Rescattering Electron Wave Packet . . . . . . . . . . . . . . . 96 2.6 High-Order Harmonic Generation . . . . . . . . . . . . . . . . . . . . 98 2.7 Nonsequential Double Ionization Due to the Direct .e; 2e/ Collision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.8 Nonsequential Double Ionization Due to the Indirect Excitation-Tunneling Ionization Processes . . . . . . . . . . . . . 102 2.9 Laser-Induced Medium Energy Electron Diffraction of Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 $E ,QLWLR 0HWKRGV IRU )HZ DQG 0DQ\(OHFWURQ $WRPLF 6\VWHPV LQ ,QWHQVH 6KRUW3XOVH /DVHU /LJKW : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 107 M.A. Lysaght, L.R. Moore, L.A.A. Nikolopoulos, J.S. Parker, H.W. van der Hart, and K.T. Taylor 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 2 Time Propagation Using Taylor Series and Arnoldi Propagator Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3 Finite-Difference Grid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.1 Mixed Finite-Difference and Basis Set Techniques for Spatial Variables in Spherical Geometry with Application to Laser-Driven Helium . . . . . . . . . . . . . . . . . . 112 4 R-Matrix Basis Set Techniques for Spatial Variables in Spherical Geometry with Application to Multi-Electron Atoms . . . . . . . . . . . 116 4.1 The RMB Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5 Mixed Finite-Difference and R-Matrix Basis Set Technique for Spatial Variables in Spherical Geometry with Application to Multi-Electron Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.1 Outer Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.2 Inner Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

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6WURQJ)LHOG ,RQL]DWLRQ RI 0ROHFXOHV 6LPSOH $QDO\WLFDO ([SUHVVLRQV : : : : : 135 Ryan Murray, Serguei Patchkovskii, Olga Smirnova, and Misha Yu. Ivanov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2 General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3 Simple Analytical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5 Conclusion: Extension to Low-Frequency Laser Fields . . . . . . . . . . 144 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5HFHQW $GYDQFHV LQ &RPSXWDWLRQDO 0HWKRGV IRU WKH 6ROXWLRQ RI WKH 7LPH'HSHQGHQW 6FKUŽRGLQJHU (TXDWLRQ IRU WKH ,QWHUDFWLRQ RI 6KRUW ,QWHQVH 5DGLDWLRQ ZLWK 2QH DQG 7ZR (OHFWURQ 6\VWHPV $SSOLFDWLRQ WR +H DQG + : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 149 Barry I. Schneider, Johannes Feist, Stefan Nagele, Renate Pazourek, Suxing Hu, Lee A. Collins, and Joachim Burgd¨orfer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 2 An Introduction to the Finite Element Discrete Variable Representation (FEDVR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2.1 Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 2.2 Discrete Variable Representation . . . . . . . . . . . . . . . . . . . . . 152 2.3 FEDVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 3 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.1 Lanczos Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4 Remarks on the Numerics and Computer Implementation . . . . . . . . 164 5 He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.1 Remarks on the Calculation of Two Electron Integrals . . . 167 5.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.3 Ionization Probability Distributions . . . . . . . . . . . . . . . . . . 172 5.4 Total Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.5 Differential Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.6 Remarks on Two Photon Direct and Sequential Ionization of He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.7 Sequential and Nonsequential Regimes of TPDI . . . . . . . . 180 5.8 Results on Two Photon Direct Ionization of He . . . . . . . . . 182 5.9 Results on Two Photon Sequential Ionization of He . . . . . 187 6 HC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 2 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.3 Interference Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.1 Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.2 HC 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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,QIRUPDWLRQ RI (OHFWURQ '\QDPLFV (PEHGGHG LQ &RXSOHG (TXDWLRQV IRU )HPWRVHFRQG 1XFOHDU :DYHSDFNHWV : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 209 Kazuo Takatsuka 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 2 Photoelectron Velocity Map Imaging Reflecting Femtosecond Nuclear Wavepacket Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 2.1 Theory of Femtosecond Pump–Probe Photoelectron Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 2.2 Velocity Map Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 3 General Framework of Nonadiabatic Electronic Wavepacket Dynamics Driven by Nuclear Motion . . . . . . . . . . . . . . . . . . . . . . . . . 216 3.1 Branching Nuclear Paths Due to Nonadiabatic Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 3.2 Smoothly Branching Paths to Represent the Wavepacket Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 3.3 Quantization of the Non-Born–Oppenheimer Branching Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4 BO Representation and Electronic-Nuclear wavepacket Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 The Total Electronic and Nuclear wavefunction . . . . . . . . 224 4.1 For Continuum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 4.2 Separation of the Attosecond Oscillatory Factors . . . . . . . 226 4.3 Information of Electron Dynamics in the Nuclear Wavepackets . . . 227 5 Imaging and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 5.1 5.2 Reconstruction of the Electron-Wavepacket View from the Close-Coupling Equations . . . . . . . . . . . . . . . . . . . . . . . 228 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 233

/LVW RI &RQWULEXWRUV

A. Abdurrouf Department of Physics, Brawijaya University, Malang 65145, Indonesia, e-mail: BCEVSSPVG!CSBXJKBZBVBDJE Gustavo Avila Chemistry Department, Queen’s University, Kingston, ON K7L 3N6, Canada, e-mail: (VTUBWP@"WJMB!UFMFGPOJDBOFU Andr´e D. Bandrauk D´epartement de chimie, Universit´e de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada, e-mail: "OESF#BOESBVL!VTIFSCSPPLFDB Klaus Bartschat Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA, e-mail: LMBVTCBSUTDIBU!ESBLFFEV Joachim Burgd¨orfer Institut f¨ur Theoretische Physik, Technische Universit¨at Wien, Wiedner Hauptstraße 8, 1040 Wien, Austria, Austria, e-mail: CVSH!EPMMZXPPEJUQUVXJFOBDBU Tucker Carrington Jr. Chemistry Department, Queen’s University, Kingston, ON K7L 3N6, Canada, e-mail: 5VDLFS$BSSJOHUPO!DIFNRVFFOTVDB Szczepan Chelkowski D´epartement de chimie, Universit´e de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada, e-mail: 4$IFMLPXTLJ!VTIFSCSPPLFDB Zhangjin Chen Department of Physics, Cardwell Hall, Kansas State University, Manhattan, KS 66506, USA, e-mail: [KDIFO!QIZTLTVFEV

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List of Contributors

Lee A. Collins Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA, e-mail: MBD!MBOMHPW Farhad H.M. Faisal Fakult¨at f¨ur Physik, Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany, e-mail: GGBJTBM!QIZTJLVOJCJFMFGFMEEF Johannes Feist ITAMP, Harvard–Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA, e-mail: KGFJTU!DGBIBSWBSEFEV Xiaoxu Guan Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA, e-mail: YJBPYVHVBO!ESBLFFEV Alex G. Harvey Max-Born-Institute, Max-Born-Straße 2 A, 12489 Berlin, Germany, e-mail: "MFY)BSWFZ!NCJCFSMJOEF Hans-Christian Hege Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, e-mail: IFHF![JCEF Suxing Hu Laboratory for Laser Energetics, University of Rochester, 250 E. River Road, PO Box 278871, Rochester, NY 14623-1212 USA, e-mail: TIV!MMFSPDIFTUFSFEV Misha Yu. Ivanov Department of Physics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK, e-mail: NJWBOPW!JNQFSJBMBDVL Michael Koppitz Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, e-mail: LPQQJU[![JCEF Anh-Thu Le Department of Physics, Cardwell Hall, Kansas State University, Manhattan, KS 66506, USA, e-mail: BUMF!QIZTLTVFEV Chii-Dong Lin Department of Physics, Cardwell Hall, Kansas State University, Manhattan, KS 66506, USA, e-mail: DEMJO!QIZTLTVFEV Huizhong Lu D´epartement de chimie, Universit´e de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada, e-mail: )VJ[IPOH-V!VTIFSCSPPLFDB Micheal A. Lysaght School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: NMZTBHIU!RVCBDVL

List of Contributors

xv

Falko Marquardt Department of Mathematics, FU Berlin, Arnimallee 6, 14195 Berlin, Germany, and Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, e-mail: NBSRVBSEU![JCEF Chris McDonald University of Ottawa, 150 Louis Pasteur, Ottawa, Canada, e-mail: DNDEP!VPUUBXBDB Christopher Mielack Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, e-mail: NJFMBDL![JCEF Laura R. Moore School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: MNPPSF!RVCBDVL Ryan Murray Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada & Department of Physics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK, e-mail: SZBONVSSBZ!JNQFSJBMBDVL Stefan Nagele Institut f¨ur Theoretische Physik, Technische Universit¨at Wien, Wiedner Hauptstraße 8, 1040 Wien, Austria, e-mail: TUFGBOOBHFMF!UVXJFOBDBU Lampros A. A. Nikolopoulos School of Physical Sciences, Dublin City University, Dublin 9, Ireland, e-mail: MBNQSPTOJLPMPQPVMPT!EDVJF Cliff J. Noble Computational Science & Engineering Dept., Daresbury Laboratory, Warrington WA4 4AD, UK, e-mail: DKO!NBYOFUDPO[ Jonathan S. Parker School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: KQBSLFS!RVCBDVL Serguei Patchkovskii Max-Born-Institute, Max-Born-Straße 2 A, 12489 Berlin, Germany, e-mail: 4FSHVFJ1BUDILPWTLJJ!OSDDB Renate Pazourek Institut f¨ur Theoretische Physik, Technische Universit¨at Wien, Wiedner Hauptstraße 8, 1040 Wien, Austria, e-mail: SFOBUFQB[PVSFL!UVXJFOBDBU Barry I. Schneider Office of Cyberinfrastructure and Physics Division, National Science Foundation, 4201 Wilson Blvd., Arlington, VA 22230, USA, e-mail: CTDIOFJE!OTGHPW

xvi

List of Contributors

Kazuo Takatsuka Department of Basic Sciences, Graduate School of Arts and Sciences, The University of Tokyo, Komaba, 153-8902, Tokyo, Japan, e-mail: LB[UBL!NOT DVUPLZPBDKQ Kenneth T. Taylor School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: LUBZMPS!RVCBDVL Jonathan Tennyson Department of Physics and Astronomy, University College London, London WC1E 6BT, UK, e-mail: KUFOOZTPO!VDMBDVL Olga Smirnova Max-Born-Institute, Max-Born-Straße 2 A, 12489 Berlin, Germany, e-mail: TNJSOPWB!NCJCFSMJOEF Hugo W. van der Hart School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: IWBOEFSIBSU!RVCBDVL Oleg Zatsarinny Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA, e-mail: PMFH[BUTBSJOOZ!ESBLFFEV

1RQSURGXFW 4XDGUDWXUH *ULGV 6ROYLQJ WKH 9LEUDWLRQDO 6FKUŽRGLQJHU (TXDWLRQ LQ G Gustavo Avila and Tucker Carrington Jr.

$EVWUDFW The size of the quadrature grid required to compute potential matrix elements impedes solution of the vibrational Schr¨odinger equation if the potential does not have a simple form. This is due to the fact that potential matrix elements are typically computed with a direct product Gauss quadrature whose grid size scales as N D , where N is the number of points per coordinate and D is the number of dimensions. In this chapter we present new ideas for mitigating the quadrature-size problem. The grids we propose are nonproduct grids of the Smolyak type. They are applied using a Lanczos algorithm, to CH3 CN. All 12 coordinates are treated explicitly.

 ,QWURGXFWLRQ To compute ionization probabilities of atoms and to study the dissociation of diatomic molecules in intense fields one needs powerful numerical tools. One needs to solve either the time independent or the time dependent Schr¨odinger equation and this is typically done by choosing basis functions, computing Hamiltonian matrix elements, and using methods of numerical linear algebra. In this chapter we present new theoretical tools for solving the vibrational Schr¨odinger equation. Similar ideas can also be applied to electronic problems for which a nonperturbative approach is necessary. Algorithms for solving both the time dependent and the time independent Schr¨odinger equation require evaluating matrix-vector products. If it is not possible to obtain closed-form expressions for potential matrix elements it is necessary to Gustavo Avila Chemistry Department, Queen’s University, Kingston, ON K7L 3N6, Canada, e-mail: (VTUBWP@ "WJMB!UFMFGPOJDBOFU Tucker Carrington Jr. Chemistry Department, Queen’s University, Kingston, ON K7L 3N6, Canada, e-mail: 5VDLFS $BSSJOHUPO!DIFNRVFFOTVDB A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_1, © Springer Science+Business Media, LLC 2011

1

2

G. Avila and T. Carrington Jr.

use quadrature. We present new quadrature schemes for doing the integrals required to solve the vibrational Schr¨odinger equation. They are chosen to both minimize the number of points at which wavefunctions and the potential must be stored and to facilitate the computation of matrix-vector products. General basis set methods for solving the Schr¨odinger equation are limited by the size of the basis and size of the quadrature grid. For the purpose of studying the motion of nuclei on a potential surface, simple product basis sets are popular and useful [1–6]. For a problem with more than about 6 degrees of freedom product basis sets are very large, but if used in conjunction with an iterative algorithm for computing eigenvalues or propagating a wave packet they can, nonetheless, be useful. If n is a representative number of 1D functions the size of the product basis required to compute vibrational levels of a molecule with D coordinates is nD . As n for many problems is between 10 and 50, nD is huge, if D > 6. Unless the potential has a special, simple form, its matrix elements are evaluated with quadratures. When using a product basis it is common to use a product quadrature with N D points where N is a representative number of quadrature points for a single coordinate. Not only the basis size, but also the size of the quadrature grid is a significant problem. Important progress has been made in alleviating the basis-size problem. One common strategy uses basis functions that are products of functions of a single variable, but not DOO the functions of a direct product basis [1, 7–16]. Pruning a direct product basis according to some criterion can drastically reduce its size, nonetheless, unless something is done to reduce the size of the quadrature grid, calculations for molecules with more than 5 atoms are not possible. In this chapter we explore alternative new ideas for decreasing the size of the quadrature grid. If the basis functions are product functions, multi-d quadrature can be avoided when the potential is a sum of products of functions of a single variable (see, e.g., [17, 18]). Another way of obviating the huge multi-d grid problem, when using a product basis, is to use an order-of-coupling representation [19–26]. For a general potential one is forced to do do full-d quadrature. If one uses an iterative method and sequentially, coordinate by coordinate, transforms vectors from a basis representation to a grid, then it is necessary to keep large grid vectors in memory. This by itself makes some calculations impossible. To extend the realm of quantum vibrational calculations one needs to reduce ERWK basis and grid sizes.

 6WDQGDUG 4XDGUDWXUHV To solve a vibrational Schr¨odinger equation with D coordinates we expand the unknown wavefunctions in terms of basis functions, X iv .r1 ; : : : ; rD / D Cni n .r1 ; : : : ; rD / : (1) n

We use product basis functions, selected from a direct product basis, each of which has the form

Nonproduct Quadrature Grids: Solving the Vibrational Schr¨odinger Equation in 12d n .r1 ; : : : ; rD /

D n1 .r1 /    nD .rD / ;

3

(2)

where n is a composite index representing n1 ; n2 ; : : : ; nD . The solutions we seek must be in the space spanned by the basis. Standard 1D basis functions are of the form n1 .r1 / D 1 h1=2 Œ1 w.x1 /1=2 1 fn1 .x1 /, with n1 D 0; : : : ; n1max , where x1 n1 1 is a function of r1 , fn1 .x1 / is a classical orthogonal polynomial, 1 w.x1 / is the corresponding weight function, and 1 hn1 is a normalization factor. With these basis functions, it is common to use Gauss quadrature and n1max C 1, quadrature points to compute potential matrix elements. For notational simplicity we assume that N points are used for each of the D coordinates, in which case the direct product Gauss (PG) quadrature has N D points. A potential matrix element is, 0 .rD /jV .r1 ; : : : ; rD /jn .r1 /    n .rD /i hn01 .r1 /    nD 1 D Z 0 .rD /jV .r1 ; : : : ; rD /jn .r1 /    n .rD / D dx1 dx2    dxD n01 .r1 /    nD 1 D



N X N X i1

i2

N X  Tn10 ;i1Tn20 ;i2   TnD0 ;iDV .i1 r1 ; : : : ;iD rD /Tn11 ;i1Tn22 ;i2   TnDD ;iD ; (3) iD

1

2

D

where each T matrix is defined by Tn;i D

q

wi =w.x i /n .x i / ;

(4)

wi is a quadrature weight and x i is a quadrature point (and i r is the corresponding r value). If the potential can be represented by a multidimensional polynomial of low degree then potential matrix elements computed with a direct product Gauss quadrature are certain to be accurate if enough points and are used. The accuracy comes at a price: the D-dimensional quadrature grid is huge. The PG grid with N1 ; : : : ; ND points for coordinates xc c D 1; 2; : : : ; D correctly integrates the monomials dD x1d1    xD ;

with d1  2N1  1 ; : : : ; dD  2ND  1 :

(5)

P and therefore some monomials whose total degree is as large as D i D1 .2Ni  1/. A direct product basis set can be pruned by removing products whose zerothorder energies are larger than some threshold. For illustrative purposes, consider an example for which all coordinates have similar (1D) zeroth-order energies. In this case, removing multi-d basis functions whose zeroth-order energies are larger than some threshold is equivalent to retaining multi-d basis functions for which n1 C    C nD  b. It is well known that this truncated basis is large enough to accurately compute energy levels [1, 7, 27]. The maximum total degree of the functions in the truncated basis is b. If the potential can be represented with a polynomial of total degree p then the maximum total degree of the functions which must be integrated to compute the potential maP trix is 2b C p, which is much smaller than D i D1 .2Ni  1/. This means that the quadrature is better than necessary. We want to find a grid with which we can do all

4

G. Avila and T. Carrington Jr.

integrals of multivariate polynomials of total degree less than or equal to 2b C p, but not integrals of multivariate polynomials with total degree larger than 2b C p. When the coordinates do not have similar (1D) zeroth-order energies it is better to use the truncation condition ˛1 n1 C    C ˛D nD  b

(6)

where ˛i are integers chosen so that basis functions labelled by different sets of n1 ; : : : ; nD have approximately the same zeroth-order energy, or so that strongly coupled functions are included in the basis. There are so-called Smolyak quadrature schemes for integrating correctly only monomials with degree less than or equal to some maximum value [28–36]. The Smolyak quadrature equation for integrating a function F .x1 ; x2 ; : : : ; xD / can be written X S.D; H / D Ci1 ;:::;iD .˛1 ; : : : ; ˛D /Q i1 ˝    ˝ QiD ; (7) f .i /H

P where f .i / D  ˛ i and Qic .xc / .ic D 1; : : : ; K c / is a sequence of 1D quadrature rules for coordinate xc . Qi .x/ is an operator which when applied to a univariate function h.x/w.x/ gives Q0 .x/w.x/h.x/ D 0 ; i

Q .x/w.x/h.x/ D

Ni X

Z wi h.i x /

D1



b

w.x/h.x/dx ;

(8)

a

with i x  and wi being (respectively) the points and weights of the Ni -point Qi quadrature rule for the weight function w.x/ and the interval Œa; b. If ˛c D 1, P c D 1; : : : ; D , f .i / D  i D ji j and H D K C D  1, the coefficients in (7) are 1 1 1 X X X Ci1 ;:::;iD D  .1/1 C2 CCD ; (9) 1 D0 2 D0

D D0

with jj D H  ji j [35]. Applying S.D; H / to a multidimensional function F .x1 ; x2 ;    ; xD /w1 .x1 /w2 .x2 /    wD .xD / yields a quadrature approximation for the integral of F .x1 ; x2 ; : : : ; xD /w1 .x1 /w2 .x2 /    wD .xD /, where wc .xc / is a weight function for xc . Choosing f .i /; H , and the Qi .x/ determines which terms contribute to (7) and which multivariate polynomials will be integrated correctly. It has been proven [33, 36, 37] that if the 1D quadratures Qi .x/ are chosen so that Z

b m

w.x/x dx D a

for m D 0; : : : ; di D 2i  1 and if

Ni X D1

wi .i x /m ;

(10)

Nonproduct Quadrature Grids: Solving the Vibrational Schr¨odinger Equation in 12d

f .i / D

D X

i  H ;

5

(11)

D1

H D KCD1 then the Smolyak quadrature is exact for all integrals Z

Z

b1

b2

dx1 a1

Z dx2   

a2

bD aD

d

d

dxD x1 1    xDD w1 .x1 /    wD .xD /

(12)

with d1 C    C dD  2K  1. This choice of f .i / is appropriate if all the zerothorder energies are similar. If the zeroth-order energies are not all similar, one must choose D X f .i / D ˛i i   H ; (13) D1

where ˛i are the integers introduced in (6). For a given H and with the Qi chosen so that the i th quadrature correctly integrates polynomials of degree 2i 1, monomials d d x1 1    xDD are integrated correctly if dc < 2imax.c/  1, for c D 1; 2; : : : ; D, where imax .c/ is thePindex of the largest quadrature included when condition (13) is imposed ˛c .dc C 1/=2  H [38]. If ˛c ¤ 1, c D 1; : : : ; D, and P and if f .i / D c ˛c ic , Ci1 ;:::;iD D

1 1 X X 1 D0 2 D0



1 X

.1/1 C2 CCD ;

(14)

D D0

with the condition f ./  H  f .i /. If the 1D quadratures are nested one multivariate quadrature product grid Qj1 ˝    ˝ Qjd may share some of its points with other multivariate quadrature product grids Qi1 ˝    ˝ Qid , where for at least one coordinate ic ¤ jc . In the nested case the Smolyak grid is a subset of the points in the Qi1 DK ˝    ˝ Qid DK direct product grid.

 &DOFXODWLRQV LQ 1RUPDO &RRUGLQDWHV It is very common to compute vibrational spectra using a kinetic energy operator (KEO) written in normal coordinates and their conjugate momenta and basis functions that are products of functions of a single normal coordinate [4, 10, 11, 20–22]. As long as the transformation between the coordinates used for the KEO and the coordinates used for the potential is known, the potential can be a function of any coordinates. Watson discovered [39] that the normal coordiante KEO (in atomic units) can be written

6

G. Avila and T. Carrington Jr.

X 1 X @2 1X bW D 1 K .JO˛  O ˛ /˛ˇ .JOˇ  O ˇ /   ˛˛ ; 2 2 2 @Qk 8 ˛ ˛;ˇ

where

(15)

k

˛ˇ D .I 01 /˛ˇ ;

I 0˛ˇ D I ˛ˇ 

X

ˇ ˛ km lm Qk Ql ;

(16)

k;l;m ˛ and km are Coriolis parameters defined for example in [39]. I˛ˇ is the inertia tensor and the vibrational angular momentum, ˛ , are

˛ D i

X k;l

˛ kl Qk

@ : @Ql

(17)

It is common to neglect the  t  term which, for, J D 0, is the only nonseparable term. The  t  term is of the order of the rotation constants, and thus is generally small compared to other terms in the KEO. In our first application of the Smolyak quadrature scheme to calculations in normal coordinates we shall neglect  t . Owing to the fact that a quadratic approximation is fairly good, it is common to use Hermite basis functions. To use Hermite functions we need a Smolyak procedure for computing multidimensional integrals whose weight function is a product of Gaussian factors. The Smolyak procedure is most useful if one uses nested grids, i.e., the points of the Qj rule are be included in the the Qj C1 rule. The most obvious sequence of nested quadratures is a Kronrod extension of Gauss quadratures [40– 44]. Unfortunately, the number of points in Qj is approximately double the number of points in Qj 1 and this forces one to use huge grids. Even worse, in many cases the Kronrod points are complex or not in the required interval. Heiss and Winschelb (HW) has proposed a good set of nested grids for use with Hermite polynomials. [45]. HW use a program written by Patterson [41–44] that starts with n points and finds m new points and n C m new weights so that the quadrature approximation to integrals with maximum degree n C 2m  1 C , computed with the full set of n C m points and weights, is exact. Here  D 1 if weight function is even, n is odd and the starting points occur in plus/minus pairs; and  D 0 in other cases. HW then vary (by trial and error) the number of points added to the previous quadrature until they find a sequence of quadratures each of which has real points. and positive weights. Next they “delay,” an idea suggested by Petras [37]. Delay is motivated by the realization that the maximum 1D degree integrated by the Qi rule need only be 2i  1 [33, 36, 37]. If a sequence of grids obtained from the Patterson program is Q1 ; Q 2 ; Q3 ; : : : with degrees d1 ; d2 ; d3 ; : : : one may replace Qi with Qj where i > j and Qj is the smallest quadrature for which dj  2i  1. The final step is to replace quadratures Qi for which Ni > 2i  1 by a quadrature with di D 2i  1 and Ni D 2i  1 for which the points are a subset of the original Qi points and the weights are computed with Patterson’s program [41–43]. In this fashion they obtain quadratures with Ni points that do integrals of degree di where

Nonproduct Quadrature Grids: Solving the Vibrational Schr¨odinger Equation in 12d

7

Hermite–Heiss–Winschelb Ni D 1; 3; 3; 7; 9; 9; 9; 9; 17; 19; 19; 19; 19; 19; 19; 31; 33; 35; : : : ; di D 1; 5; 5; 7; 15; 15; 15; 15; 17; 29; 29; 29; 29; 29; 29; 31; 33; 35; : : : ; (18)

 3RWHQWLDO 0DWUL[9HFWRU 3URGXFWV Because the basis functions are orthonormal and the overlap matrix is exact we solve a standard (not generalized) eigenvalue problem. Energy levels were computed using the Lanczos algorithm [2, 46–49], which requires evaluating matrix-vector products. It is well known that if a full direct product basis and a full direct product grid are used it is possible to evaluate matrix-vector products at a cost that scales as N DC1 , where N is a representative number of 1d basis functions or grid points [2–4, 50–57]. Several years ago it was pointed out that if the basis set is truncated by applying a n1 C n2 C n3 C n4 C n5 C n6 C     b criterion, but a full direct product grid is used, it is also possible to retain the N DC1 scaling [27]. Despite the fact that Smolyak grid is a nondirect product grid, it is nonetheless still possible to compute matrix-vector products at a cost that scales as N DC1 . To demonstrate that this is possible, we begin by writing a Smolyak quadrature as a constrained sum over products of 1D grids. We explain the ideas for a problem with six degrees of freedom with similar zeroth-order frequencies. If we retained all of the points in a direct product grid we would sum, NK NK NK NK NK NK X X X X X X

w.k1 ; k2 ; k3 ; k4 ; k5 ; k6 /

k1 D1 k2 D1 k3 D1 k4 D1 k5 D1 k6 D1 k

k

k

k

k

k

 F .r1 1 ; r2 2 ; r3 3 ; r4 4 ; r5 5 ; r6 6 / ;

(19)

where NK is the number of points in the last (Kth) rule in the sequence of 1D quadratures and the weights are computed as explained in [38]. However, we do not retain all the points, but only those that are in at least one of the grids that satisfy max.K; D/  i1 C i2 C i3 C i4 C i5 C i6  D C K  1 (see (11) and (12)). How does one sum over the points that occur in (at least one of) the contributing grids? We use the HW grids and define the 1D quadratures so that (see (18)) Q1 D .r 1 / ; Q2 D .r 1 ; r 2 ; r 3 / ; Q3 D .r 1 ; r 2 ; r 3 / ; Q4 D .r 1 ; r 2 ; r 3 ; r 4 ; r 5 ; r 6 ; r 7 / ; Q5 D .r 1 ; r 2 ; r 3 ; r 4 ; r 5 ; r 6 ; r 7 ; r 8 ; r 9 / ; :: :

(20)

8

G. Avila and T. Carrington Jr.

The Smolyak quadrature is written, S.6; K/w1 .r1 /w2 .r2 /w3 .r3 /w4 .r4 /w5 .r5 /w6 .r6 /F .r1 ; r2 ; r3 ; r4 ; r5 ; r6 / D

N1 N2 N3 N4 N5 N6 X XXXXX k1

k2

k3

k4

k5

w.k1 ; k2 ; k3 ; k4 ; k5 ; k6 /

k6

 F .r1k1 ; r2k2 ; r3k3 ; r4k4 ; r5k5 ; r6k6 /

(21)

where Nj D N Œj  is the number of points in the j th quadrature, 

j 1

X

Nj D N .K C D  1/ 

 g.ki /  .D  j /

(22)

i D0

where g.k/ is an integer that is the index of the smallest quadrature rule Q i containing the point k. For the nested HW sequence g.0/ D 0; g.1/ D 1; g.2/ D 2; g.3/ D 2; g.4/ D 4; g.5/ D 4; g.6/ D 4; g.7/ D 4; g.8/ D 5; g.9/ D 5; g.10/ D 9; g.11/ D 9; g.12/ D 9; g.13/ D 9; g.14/ D 9; g.15/ D 9; g.16/ D 9; g.17/ D 9; g.18/ D 10; g.19/ D 10; : : : : (23) In (21) the sums from left to right are over k1 ; k2 ; : : : ; k6 but we could choose to do the sums in any order. Changing the order of the sums changes the upper limits. For example, the Smolyak sum can be done S.6; K/F .r1; r2 ; r3 ; r4 ; r5 ; r6 / D

N6 X N5 X N4 X N3 X N2 X N1 X k6

k5

k4

k3

k2

w.k6 ; k5 ; k4 ; k3 ; k2 ; k1 /

k1

 F .r1k1 ; r2k2 ; r3k3 ; r4k4 ; r5k5 ; r6k6 /

(24)

with N5 D NKCD1g.k6 /4 ; N4 D NKCD1g.k6 /g.k5 /3 ; N3 D NKCD1g.k6 /g.k5 /g.k4 /2 ; N2 D NKCD1g.k6 /g.k5 /g.k4 /g.k3 /1 ; N1 D NKCD1g.k6 /g.k5 /g.k4 /g.k3 /g.k2 / :

(25)

There is no need to compute potential matrix elements, all that is necessary is to evaluate potential matrix-vector products. [2–4, 50–57] Due to the fact that the quadrature sum can be written as a constrained sum over 1D point labels (21) it is possible, as was done in [27] with a direct product grid, to write (using (24)) the matrix-vector product as a sequence of sums,

Nonproduct Quadrature Grids: Solving the Vibrational Schr¨odinger Equation in 12d

9

u0i 0 ;i 0 ;i 0 ;i 0 ;i 0 ;i 0

1 2 3 4 5 6

D

N6 X

k

Pi60 .r6 6 /

N5 X

k6



k

Pi50 .r5 5 /

k5

N4 X

k

Pi40 .r4 4 /

k4

N3 X

k

Pi30 .r3 3 /

k3

N2 X

k

Pi20 .r2 2 /

k2

N1 X

k

Pi10 .r1 1 /

k1

k6 w.k1 ; k2 ; k3 ; k4 ; k5 ; k6 /V .r1k1 ; r2k2 ; r3k3 ; r4k4 ; r5k5 ; rD /

nmax 1

X

i1 D0

nmax 2

k Pi1 .r1 1 /

nmax nmax nmax nmax 3 4 5 6 X X X X k2 k3 k4 k5 k Pi2 .r2 / Pi3 .r3 / Pi4 .r4 / Pi5 .r5 / Pi6 .r6 6 / i2 D0 i3 D0 i4 D0 i5 D0 i6 D0

X

 ui1 ;i2 ;i3 ;i4 ;i5 ;i6 ;

(26)

where the upper limits on the 1D sums over the basis function labels are nmax D Nt ; 1 nmax D N t  i1 ; 2 max n3 D N t  i1  i2 ; nmax D N t  i1  i2  i3 ; 4 max n5 D N t  i1  i2  i3  i4 ; nmax D N t  i1  i2  i3  i 4  i 5 : 6

(27)

In this equation N t is the original number of basis functions for a single coordinate. In (26), as is done with direct product grids, summation signs are moved to the right so that sums can be evaluated sequentially. [2, 3, 50, 51, 53, 58]

 $SSOLFDWLRQ RI 6PRO\DN WR &+ ±&1 It has previously been demonstrated that Smolyak based approaches used in conjunction with a product basis set, ˚n1 ;n2 ;n3 ;n4 ;n5 ;n6 ;::: .r1 ; r2 ; r3 ; r4 ; r5 ; r6 ; : : :/ D n1 .r1 / n2 .r2 / n3 .r3 / n4 .r4 / n5 .r5 / n6 .r6 /   

(28)

and a truncation criterion, n1 C n2 C n3 C n4 C n5 C n6 C     b ;

(29)

enable a significant reduction in grid size for a molecule with six vibrational degrees of freedom [38]. In this chapter we present preliminary results showing, for the first time, that it is also possible to do calculations in 12d. Vibrational energy levels of CH3 –CN were calculated with a basis obtained by imposing the constraint

10

G. Avila and T. Carrington Jr.

7DEOH  Calculated and observed vibrational transitions of CH3 CN. The Smolyak grid is about a factor of 105 smaller than the Gauss grid mode

CC/B3 cm1

8˙1 280 28˙2 4 7˙1 38˙1 38˙3 8˙1 C 4 8˙1 C 71 a

Exp cm1

361 723 724 901 1034 1087 1088 1260 1395

362 [60], 365 [61] 739 [62] 717 [60, 62] 916 [60], 920 [62] 1041 [60], 1042 [62] 1077a [62] 1122a [62] 1290 [63] 1402 [63]

Estimated experimental transitions

f .n/ D

X

nj ˛j  27 ;

(30)

j

a HW Smolyak quadrature with H D 36, the force field of [59], and ˛ values: ˛1 D 3, ˛2 D 4, ˛3 D 3, ˛4 D 3, ˛5 D 3, ˛6 D 3, ˛7 D 4, ˛8 D 4, ˛9 D 3, ˛10 D 3, ˛11 D 1 and ˛12 D 1. There are 743103 basis functions. In Table 1 we present Smolyak/HW energy levels The product Gauss grid required to evaluate exactly all of the monomials that can be integrated correctly with the Smolyak/HW grid has more than 31013 points, whereas the Smolayk/HW grid that does the same integrals exactly has fewer than 2 108 points. The first  40 levels are converged to within  0:2 cm1 . This is established by comparing calculations with different basis sizes. It is difficult to compare our levels to those reported previously in [59] because some of the smallest force constants, used in the [59] calculation, are not available.

 &RQFOXVLRQ Due largely to the use of iterative algorithms (e.g., the Lanczos algorithm) it is now possible to compute vibrational spectra for molecules with more than five atoms. Iterative algorithms make it possible to compute spectra with bases having millions of functions, however, they are advantageous only if matrix-vector products can be done efficiently. If one uses a direct product basis and a direct product quadrature grid, good methods for doing the required matrix-vector products have been known for more than a decade. Years ago, it was recognized that similar efficiency is achievable if the basis is not a full direct product but includes functions selected from a direct product basis, provided a direct product quadrature grid is used [27]. Pruning the basis in this fashion enables one to work with bases much smaller than

Nonproduct Quadrature Grids: Solving the Vibrational Schr¨odinger Equation in 12d

11

direct product bases. This is good, nevertheless, for general potentials, one must introduce a quadrature grid and a direct product quadrature grid makes calculations with a basis of selected product functions impossible. We have recently shown that the quadrature grid size can also be drastically reduced by using a nondirect product grid [38]. In this chapter we explain the ideas and demonstrate that they can be used to do dynamics in 12d. $FNQRZOHGJHPHQWV This work has been supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chairs Program. We thank Paul Ayers for very useful discussions.

5HIHUHQFHV 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

S. Carter, N.C. Handy, Comput. Phys. Rep. , 115 (1986) M.J. Bramley, T. Carrington, Jr., J. Chem. Phys. , 8519 (1993) J.C. Light, T. Carrington, Jr., Adv. Chem. Phys. , 263 (2000) J.M. Bowman, T. Carrington, Jr., H.D. Meyer, Mol. Phys. , 2145 (2008) H.G. Yu, J.T. Muckerman, J. Mol. Spectrosc. , 11 (2002) G. Czako, T. Furtenbacher, A.G. Csaszar, V. Szalay, Mol. Phys. , 2411 (2004) R.J. Whitehead, N.C. Handy, J. Mol. Spectrosc. , 459 (1976) L. Halonen, D.W. Noid, M. Child, J. Chem. Phys. , 2803 (1983) L. Halonen, M.S. Child, J. Chem. Phys. , 4355 (1983) J.M. Bowman, S. Carter, X. Huang, Int. Rev. Phys. Chem. , 533 (2003) D. B´egu´e, N. Gohaud, C. Pouchan, P. Cassam-Chenai, J. Li´evin, J. Chem. Phys. , 164115 (2007) A. Maynard, R.E. Wyatt, C. Iung, J. Chem. Phys. , 9483 (1997) B. Poirier, J. Theo. Comput. Chem. , 65 (2003) R. Dawes, T. Carrington, Jr., J. Chem. Phys. , 134101 (2006) R. Dawes, T. Carrington, Jr., J. Chem. Phys. , 054102 (2006) J. Cooper, T. Carrington, Jr., J. Chem. Phys. , 214110 (2009) H. Romanowski, J. Bowman, L. Harding, J. Chem. Phys. , 4155 (1985) M.H. Beck, A. J¨ackle, G.A. Worth, H.D. Meyer, Phys. Rep. , 1 (2000) S. Carter, S.J. Culik, J.M. Bowman, J. Chem. Phys. , 10458 (1997) S. Carter, J.M. Bowman, N.C. Handy, Theor. Chem. Acc. , 191 (1998) D. Benoit, J. Chem. Phys. , 562 (2004) G. Rauhut, J. Chem. Phys. , 9313 (2004) S. Manzhos, T. Carrington, Jr., J. Chem. Phys. , 224104 (2008) S. Manzhos, T. Carrington, Jr., J. Chem. Phys. , 014103 (2007) S. Manzhos, T. Carrington, Jr., J. Chem. Phys. , 084109 (2006) O. Vendrell, F. Gatti, D. Lauvergnat, H.D. Meyer, J. Chem. Phys. , 184302 (2007) X.G. Wang, T. Carrington, Jr., J. Phys. Chem. A , 2575 (2001) S.A. Smolyak, Sov. Math. Dokl. , 240 (1963) F.J. Delvos, W. Schempp, %RROHDQ 0HWKRGV LQ ,QWHUSRODWLRQ DQG $SSUR[LPDWLRQ (Longman, Harlow, 1989) K. Frank, S. Heinrich, S. Pereverzev, J. Complexity , 17 (1996) A.C. Genz, SIAM J. Numer. Anal. , 1273 (1986) M. Griebel, M. Schneider, C. Zenger, in ,WHUDWLYH 0HWKRGV LQ /LQHDU $OJHEUD, ed. by R. Beauwens, P. de Groen (North-Holland, Amsterdam, 1992), p. 263 E. Novak, K. Ritter, Numer. Math. , 79 (1996) V.N. Temlyakov, $SSUR[LPDWLRQ RI 3HULRGLF )XQFWLRQV (Nova Science, Hauppauge, NY, 1994)

12

G. Avila and T. Carrington Jr.

35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

G.W. Wasilkowski, H. Wo´zniakowski, J. Complexity , 1 (1995) H.J. Bungartz, M. Griebel, Acta Numer. , 147 (2004) K. Petras, Numer. Math. , 729 (2003) G. Avila, T. Carrington, Jr., J. Chem. Phys. , 174103 (2009) J.K.G. Watson, Mol. Phys. , 479 (1968) A.S. Kronrod, Dokl. Akad. Nauk SSSR , 283 (1964) T.N.L. Patterson, ACM Trans. Math. Softw. , 137 (1989) T.N.L. Patterson, Numer. Math. , 511 (1993) T.N.L. Patterson, Math. Comput. , 847 (1969) T.N.L. Patterson, Math. Comput. , 892 (1969) F. Heiss, V. Winschelb, J. Econometrics , 62 (2008) J.K. Cullum, R.A. Willoughby, /DQF]RV $OJRULWKPV IRU /DUJH 6\PPHWULF (LJHQYDOXH &RPSX WDWLRQV 9ROV  DQG  (Birkh¨auser, Boston, MA, 1985) C. Iung, C. Leforestier, J. Chem. Phys. , 3198 (1989) F. Le Quere, C. Leforestier, J. Chem. Phys. , 247 (1990) A. McNichols, T. Carrington, Jr., Chem. Phys. Lett. , 464 (1993) T. Carrington, Jr., in (QF\FORSHGLD RI &RPSXWDWLRQDO &KHPLVWU\ 9RO  (Wiley, New York, NY, 1998), p. 3157 C. Leforestier, J. Chem. Phys. , 7357 (1994) C. Leforestier, L.B. Braly, K. Liu, M.J. Elrod, R.J. Saykally, J. Chem. Phys. , 8527 (1997) R. Chen, G. Ma, H. Guo, J. Chem. Phys. , 4763 (2001) X.G. Wang, T. Carrington, Jr., J. Tang, A.R.W. Mckellar, J. Chem. Phys. , 034301 (2005) P. Sakar, N. Poulin, T. Carrington, Jr., J. Chem. Phys. , 10269 (1999) T. Carrington, Jr., Can. J. Chem. , 900 (2003) X.G. Wang, T. Carrington, Jr., J. Chem. Phys. , 9781 (2001). Erratum, J. Chem. Phys. , 12682 (2003) M.J. Bramley, J.W. Tromp, T. Carrington, Jr., G.C. Corey, J. Chem. Phys. , 6175 (1994) D. B´egu´e, P. Carbonni`ere, C. Pouchan, J. Phys. Chem. A , 4611 (2005) I. Nagawa, T. Shimanouchi, Spectrochim. Acta , 513 (1962) M. Koivusaari, V.M. Horneman, R. Antilla, J. Mol. Spectrosc. , 377 (1992) A.M. Tolonen, M. Koivusaari, R. Paso, J. Schroderus, S. Alanko, R. Antilla, J. Mol. Spectrosc. , 554 (1993) R. Paso, R. Antilla, M. Koivusaari, J. Mol. Spectrosc. , 470 (1994)

47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63.

0XOWL3KRWRQ 6LQJOH DQG 'RXEOH ,RQL]DWLRQ RI &RPSOH[ $WRPV E\ 8OWUDVKRUW ,QWHQVH /DVHU 3XOVHV K. Bartschat, X. Guan, C.J. Noble, B.I. Schneider, and O. Zatsarinny

$EVWUDFW We present an DE LQLWLR and nonperturbative time-dependent approach to describe the response of a general atom to intense few-cycle laser pulses. After using a highly flexible B-Spline R-matrix method to generate field-free Hamiltonian and electric dipole matrices, the initial state is propagated in time using an efficient Arnoldi–Lanczos scheme. The method is illustrated with results for excitation and single ionization of Ne and Ar, as well as double ionization of He in a two-color pump-probe arrangement.

 ,QWURGXFWLRQ As noted in the cover article for a recent special issue of -RXUQDO RI 3K\VLFV % [1], “we are [currently] witnessing a revolution in photon science, driven by the vision to time-resolve ultra-fast electronic motion in atoms, molecules, and solids. . . ” InK. Bartschat Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA, e-mail: LMBVTCBSUTDIBU!ESBLFFEV X. Guan Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA , e-mail: YJBPYVHVBO!ESBLFFEV C.J. Noble Computational Science & Engineering Dept., Daresbury Laboratory, Warrington WA4 4AD, UK, e-mail: DKO!NBYOFUDPO[ B.I. Schneider Physics Division, National Science Foundation, Arlington, VA 22230, USA, e-mail: CTDIOFJE! OTGHPW O. Zatsarinny Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA, e-mail: PMFH[BUTBSJOOZ!ESBLFFEV A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_2, © Springer Science+Business Media, LLC 2011

13

14

K. Bartschat et al.

deed, the ongoing development of ultra-short and ultra-intense light sources based on high-harmonic generation and free-electron lasers is providing new ways to generate optical pulses capable of probing dynamical processes that occur on attosecond time scales. These capabilities promise a revolution in our microscopic knowledge and understanding of matter [2]. Over the past few years, our group has been working on the development of a general DE LQLWLR theoretical approach to describe short-pulse intense laser interactions with atoms, which is applicable to complex targets beyond (quasi) two-electron systems. In recent papers [3–5], we outlined how field-free Hamiltonian and electric dipole matrices generated with the highly flexible B-Spline R-matrix (BSR) [6] suite of codes may be combined with an efficient Arnoldi–Lanczos time propagation scheme to calculate multiphoton excitation as well as well as single and even double ionization, albeit the latter so far has been restricted to the helium target. A computer code is also publicly available [7]. In this contribution, we briefly summarize the computational method and then present a number of examples to demonstrate the feasibility of the approach. In addition to showing results for excitation and single ionization of Ne and Ar atoms by individual laser pulses of variable length, intensity, and central photon energy, we investigate pump-probe processes in He involving two XUV laser pulses whose time delay is varied. This allows us to visualize the competition between direct and sequential double ionization as a function of the time delay. Unless specified otherwise, atomic units (a.u.) are used in this manuscript.

 &RPSXWDWLRQDO $SSURDFK We start with the time-dependent Schr¨odinger equation @  .r 1 ; : : : ; r N I t/ @t D ŒH0 .r 1 ; : : : ; r N / C V .r 1 ; : : : ; r N I t/ .r 1 ; : : : ; r N I t/ i

(1)

for the N -electron wavefunction  .r 1 ; : : : ; r N I t/, where H0 .r 1 ; : : : ; r N / is the field-free Hamiltonian containing the kinetic energy of the N electrons, their potential energy in the field of the nucleus, and their mutual Coulomb repulsion, while V .r 1 ; : : : ; r N I t/ D

N X

E .t/  r i

(2)

i D1

represents the interaction of the electrons with the laser field E .t/ in the dipole length form. This gauge is generally preferable for complex atoms, for which highly accurate wavefunctions are very difficult to obtain and the region near the nucleus would need to be very well described for the velocity or acceleration gauges to be applicable. For an extended discussion of using different gauges for the helium

Ionization of Complex Atoms by Intense Laser Pulses

15

target, we refer to the recent paper by Hutchinson et al. [8], as well as the references therein. The tasks to be carried out in order to computationally solve this equation and to extract the physical information of interest are: 1. Generate a representation of the field-free Hamiltonian and its eigenstates; these include the initial bound state, other bound states, autoionizing states, as well as single-continuum and double-continuum states to represent electron scattering from the residual ion. 2. Generate the electric dipole matrices to represent the coupling to the laser field. 3. Propagate the initial bound state until some time after the laser field is turned off. 4. Extract the physically relevant information from the final state. The solution of the TDSE requires an accurate and efficient generation of the Hamiltonian and electronfield interaction matrix elements. In order to achieve this goal, we approximate the time-dependent wavefunction as X  .r 1 ; : : : ; r N I t/  Cq .t/˚q .r 1 ; : : : ; r N /: (3) q

The ˚q .r 1 ; : : : ; r N / are a set of time-independent N -electron states formed from appropriately symmetrized products of atomic orbitals. They are expanded as ˚q .r 1 ; : : : ; r N / X DA aijcq c .x1 ; : : : ; xN 2 I rO N 1 N 1 I rO N N /Ri .rN 1 /Rj .rN / : (4) c;i;j

Here A is the antisymmetrization operator, c .x1 ; : : : ; xN 2 I rO N 1 N 1 I rO N N / are channel functions involving the space and spin coordinates (xi ) of N  2 core electrons coupled to the angular (r) O and spin () coordinates of the two outer electrons, Ri .r/ is a radial basis function, and the aijcq are expansion coefficients. Although resembling a close-coupling ansatz with two continuum electrons, the expansion (4) contains bound states and singly ionized states as well. In general, the atomic orbitals, Ri .r/, are not orthogonal to one another or to the orbitals used to describe the atomic core. If orthogonality constraints are imposed on these functions, additional terms would need to be added to the expansion to relax the constraints. This possibility still exists as an option in our computer code. When the expansion (3) is inserted into the Schr¨odinger equation, we obtain iS

@ C .t/ D ŒH 0 C E.t/DC .t/ ; @t

(5)

where S is the overlap matrix of the basis functions, H 0 and D are representations of the field-free Hamiltonian and the dipole coupling matrices, and C .t/ is the timedependent coefficient vector in (3). The price to pay for the flexibility in the BSR approach, at least initially, is the representation of the field-free Hamiltonian and the dipole matrices in a nonorthog-

16

K. Bartschat et al.

onal basis. To solve this problem, we use a transformation into the eigenbasis by solving the field-free generalized eigenvalue problem first. Details can be found in [4]. In addition to simplifying the definition of the initial state and the extraction of the physically interesting information, this transformation makes it possible to cut unphysically high eigenvalues and the corresponding eigenvectors from the time propagation scheme, thereby improving its numerical stability and allowing us to use the standard Arnoldi–Lanczos method [8–10] to propagate the initial state in time until the observables of interest can be extracted from the wavefunction. These observables include the survival probability of the ground state, as well as the probabilities for single-electron excitation and ionization, ionization-excitation, and double ionization. Some of these probabilities are easy to extract via the coefficients of the time-propagated wavefunction in the eigenbasis of the problem, while others require projections to the one-electron or even two-electron continuum. Details can be found in [3–5]. In some cases, it is possible to derive a “generalized cross section” from these probabilities, but care has to be taken in the definition of the “effective interaction time” [11, 12].

 ([DPSOH 5HVXOWV  0XOWLSKRWRQ 6LQJOH,RQL]DWLRQ RI 1HRQ As our first example, Fig. 1 shows the response of the Ne atom to pulses with central photon energies of 11.6 and 7:3 eV, respectively. In these cases, at least two or three photons, respectively, need to be absorbed in order to ionize the system. Since this was our “proof-of-principle attempt” at such a problem, we only used the ground state .1s2 2s2 2p5 /2 P of NeC as the target state for the “half collision” of the ejected electron with the residual ion. In order to ensure converged results for the above cases, we actually coupled LS symmetries up to a total orbital angular momentum Lmax D 6 for the electron–ion collision system. Note that excitation rather than ionization appears as the dominating reaction process for ! D 0:27 a:u: and the laser parameters chosen here.

 0XOWLSKRWRQ 6LQJOH,RQL]DWLRQ RI $UJRQ For the argon target, we performed a more sophisticated calculation [4], in which we coupled three states of the ArC ion. In this project, we studied the effects of the finite pulse length and the intensity on various intermediate resonance states. Figure 2 depicts the excitation probability at different photon energies. So-called Rabi oscillations occur when the photon energy matches the energy gap between the ground state and, in this case, the .3p5 4s/1 P excited state. This matching leads

Survival probability

0.06 0.98

0.05

ω = 0.425 a.u.

0.96

0.04

survival ionization excitation

0.03 0.02

0.94

0.01 0.92

0

1

2

3 4 5 6 7 8 Time (optical cycles)

9

0.00 10

Survival probability

1.00

0.05 0.04

0.98 ω = 0.27 a.u.

0.96

0.03

survival ionization excitation

0.02

0.94 0.92

0.01 0

1

2

3 4 5 6 7 8 Time (optical cycles)

9

0.00 10

Excitation/ionization probability

0.07

1.00

17 Excitation/ionization probability

Ionization of Complex Atoms by Intense Laser Pulses

)LJ  Ground-state survival (left scale) and total excitation and ionization probabilities (right scale) for Ne exposed to 10-cycle laser pulses of peak intensity 3:51014 W=cm2 with a Gaussian envelope. The central laser frequencies are 0:425 a:u: (11:6 eV, top panel) and 0:27 a:u: (7:3 eV, bottom panel). 1.0 Excitation probability

)LJ  Excitation probability of argon for a 30-cycle laser pulse with peak intensity of 2  1013 W=cm2 and photon energies of 10, 11, 12, and 13 eV. Note the different scales for the individual photon energies.

ω = 10 eV (×50) ω = 11 eV (×10) ω = 12 eV 0.8 ω = 13 eV

0.6 0.4 0.2 0.0

0

5

10 15 20 Time (optical cycles)

25

30

to oscillations in the excitation probability with a large amplitude and a long period. There are still oscillations when the photon energy is tuned away from the energy gap, but they have much smaller amplitudes and shorter periods. As a result of coupling several ionic states, Rydberg-type resonances converging to different thresholds can be observed. However, both the finite length of the pulse and its intensity have an effect on the details of the observed structures. Figure 3 shows the effect of the laser intensity on the generalized two-photon cross sec-

)LJ  Effect of the laser peak intensity on the generalized two-photon cross section for a 30-cycle laser pulse interacting with argon. The Floquetresults are from McKenna and van der Hart [13].

K. Bartschat et al. Generalized cross section (cm4 s)

18

10−48

R-matrix Floquet I0 = 1012 W/cm2 I0 = 1013 W/cm2

(3s3p6 3d) 1 D /(5s) 1S

(3p5 4s) 1 P

(3s3p6 4s) 1 S o

10−49

10−50 8

9

10 11 12 13 Photon energy (eV)

14

15

tion. While both peak intensities, 1012 and 1013 W=cm2 , still lie in the perturbative regime, the height of the first resonance peak is significantly diminished for the more intense laser field. Similar results have been obtained by another time-dependent Rmatrix approach, which is independently being developed by the Belfast group [14]. Figure 4 exhibits two examples for the single ionization rates in argon [4] obtained in few-cycle laser pulses of different lengths, intensities, and central wavelengths. There is clearly a nontrivial dependence on the various laser parameters, and once again resonances appear when the photon wavelength is varied. These predictions are currently awaiting experimental tests, but they show the richness of effects that can be expected in complex targets where inter-shell correlation effects play an important role.

 7ZR&RORU 7ZR3KRWRQ 'RXEOH ,RQL]DWLRQ RI +HOLXP The two-photon double ionization (DI) of the helium atom induced by intense short XUV laser pulses has received considerable attention from both theorists and experimentalists alike. Instead of listing a large number of references here, we note that much of the recent work was quoted in [11, 12], but several additional papers appeared since then or are currently in press. Interestingly, even the calculation of the total cross section for this process is under heavy dispute, with ongoing debates about the need, or lack thereof, to account for the electron-electron interaction in the final state and the role of the direct vs. the sequential process [15]. For more details, we refer to the above references and several papers in the special journal issue headed by [1]. Here we consider the process of double photoionization by absorption of two photons at different central photon energies. In other words, the target helium atom is exposed to the irradiation by two laser pulses, of potentially different frequencies and with a controllable time delay. Specifically, the following two-color laser parameters were used for the results shown in Fig. 5: Pulse 1 has a central photon

Ionization of Complex Atoms by Intense Laser Pulses

19

Ionization yield

10−1 10−2 10−3 τ = 30 o.c. τ = 10 o.c. I0 = 1013 W/cm2

10−4 10−5 5

6

7

8 9 10 11 12 13 14 15 Photon energy (eV)

100

Ionization yield

10−1 10−2 10−3 τ = 10 o.c. I0 = 1013 W/cm2 I0 = 1014 W/cm2

10−4 10−5 5

6

7

8 9 10 11 12 13 14 15 Photon energy (eV)

)LJ  Ionization yield as a function of photon energy in 10-cyle and 30-cycle laser pulses of peak intensity 1013 W=cm2 (top) and for 10-cycle laser pulses with peak intensities of 1013 W=cm2 and 1014 W=cm2 (bottom).

energy of 35:3 eV and a peak intensity of 1014 W=cm2 , while the corresponding parameters for pulse 2 are 57:1 eV and 1013 W=cm2 , respectively. The length of each pulse is 10 optical cycles with a Gaussian envelope, thus corresponding to pulse lengths of about 1.2 and 0.7 femtoseconds, respectively. We are interested in the mechanism for the ejection of two electrons when the time delay between the two pulses is varied. We define this delay as the time distance between the peak intensities. Consequently, there is no overlap at all between the two pulses for a delay of about 0.95 femtoseconds, corresponding to a little less than 40 atomic units of time (1 a:u:  24 attoseconds). Figure 5 shows the results for a variety of delays, ranging from about 120 attoseconds (i.e., the second photon with the higher energy comes first) to 600 attoseconds. Not surprisingly, the probability for double ionization is small in the first case, since the only chance for this process to happen is the two photons working together on the two electrons. Even when the two pulses come simultaneously, the probability for double ejection remains relatively small, and the peak for equal-energy sharing is a clear indication of the direct process. With increasing time delay, the peaks expected for the sequential process – one at 10:7 eV and the other at 2:7 eV, corre-

20

K. Bartschat et al.

)LJ  Energy distributions of the two escaping electrons in two-color laser pulses of 10 optical cycles each. The laser parameters are: !1 D 35:3 eV at a peak intensity of 1014 W=cm2 and !2 D 57:1 eV at a peak intensity of 1013 W=cm2 . The delay between the two pulses is varied between 121 as and C605 as. See text for details.

sponding to the 35:1 eV photon ionizing the neutral helium atom with an ionization potential of 24:6 eV and the other one ionizing the HeC .1s/ ion with an ionization potential of 54:4 eV – start to grow, but the two processes still have about equal weight for a time delay as large as 400 attoseconds. From this example, it is clear that the time delay plays a decisive role in determining how the two electrons are ejected by two-color XUV laser pulses. Depending on the details of the time delay, the electrons can be ejected in ways either similar to the sequential or the nonsequential process. Our findings qualitatively agree with those of Foumouo et al. [16] for the two-color problem and Feist et al. [17] in the singlecolor problem. They serve as an independent confirmation of their predictions, and also give us confidence in our computer code.

 6XPPDU\ DQG 2XWORRN We have presented a JHQHUDO method to calculate short-pulse intense laser interactions with complex atoms using a B-spline R-matrix approach in connection with an efficient Arnoldi–Lanczos time propagation scheme. Test calculations for several systems revealed good agreement with previous benchmark results obtained with different and entirely independent methods. Our application to two-photon double

Ionization of Complex Atoms by Intense Laser Pulses

21

ionization of helium using two time-delayed XUV laser pulses confirmed that attosecond spectroscopy will provide a “microscope” to examine and also control the way electrons interact in atomic and molecular targets. We are currently in the process of generating and then transforming the corresponding matrices for the two-photon double ionization problem of neon atoms. This will allow us to make a direct comparison with the recent experiments of Moshammer et al. [18] carried out at the FLASH facility in Hamburg. While the additional complication of a residual core with nonzero angular momentum is substantial, the current method has been formulated in such a way that these calculations are effectively limited by the available hardware (i.e., supercomputer facilities) rather than special-purpose software. Nevertheless, in order to move to systems like Ne and Ar, we need more work on parallelizing the code and substantial resources to handle the matrices that can quickly reach ranks of 50,000–100,000. To use the eigenbasis, we will need to solve a generalized eigenvalue problem RQFH for each partial-wave symmetry, and these matrices are not sparse. However, in light of currently available computational resources, we are confident that we will be able to generate results for complex targets, including simple molecules, in the near future. $FNQRZOHGJHPHQWV The work presented here was supported by the United States National Science Foundation under grants PHY-0757755 (KB and XG) and PHY-0901838 (KB,OZ,CJN), and supercomputer resources through the Teragrid allocation TG-PHY090031.

5HIHUHQFHV 1. R. Moshammer, J. Ullrich, J. Phys. B (134001) (2009) 2. M. Uiberacker, T. Uphues, M. Schultze, A.J. Verhoef, V. Yakovlev, M.F. Kling, J. Rauschenberger, N.M. Kabachnik, H. Schr¨oder, M. Lezius, K.L. Kompa, H.G. Muller, M.J.J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, F. Krausz, Nature , 627 (2007) 3. X. Guan, O. Zatsarinny, K. Bartschat, B.I. Schneider, J. Feist, C.J. Noble, Phys. Rev. A , 053411 (2007) 4. X. Guan, C.J. Noble, O. Zatsarinny, K. Bartschat, B.I. Schneider, Phys. Rev. A , 053402 (2008) 5. X. Guan, O. Zatsarinny, C.J. Noble, K. Bartschat, B.I. Schneider, J. Phys. B , 134015 (2009) 6. O. Zatsarinny, Comput. Phys. Commun. , 273 (2006) 7. X. Guan, C.J. Noble, O. Zatsarinny, K. Bartschat, B.I. Schneider, Comput. Phys. Commun. , 2401 (2009) 8. S. Hutchinson, M.A. Lysaght, H.W. van der Hart, J. Phys. B , 096603 (2010) 9. T.J. Park, J.C. Light, J. Chem. Phys. , 5870 (1986) 10. E.S. Smyth, J.S. Parker, K.T. Taylor, Comput. Phys. Commun. , 1 (1998) 11. X. Guan, K. Bartschat, B.I. Schneider, J. Phys. A , 043421 (2008) 12. J. Feist, S. Nagele, R. Pazourek, E. Persson, B.I. Schneider, L.A. Collins, J. Burgd¨orfer, Phys. Rev. A , 043420 (2008) 13. C. McKenna, H.W. van der Hart, J. Phys. B , 457 (2004) 14. M.A. Lysaght, H.W. van der Hart, P.G. Burke, Phys. Rev. Lett. , 25301 (2008) 15. P. Lambropoulos, L.A.A. Nikopoulos, M.G. Makris, A. Mihelic, Phys. Rev. A , 055402 (2008)

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16. E. Foumouo, P. Antoine, H. Bachau, B. Pireaux, New J. Phys. , 025017 (2008) 17. J. Feist, S. Nagele, R. Pazourek, E. Persson, B.I. Schneider, L.A. Collins, J. Burgd¨orfer, Phys. Rev. Lett. , 063002 (2009) 18. R. Moshammer, Y.H. Jiang, L. Foucar, A. Rudenko, T. Ergler, C.D. Schr¨oter, S. L¨udemann, K. Zrost, D. Fischer, J. Titze, T. Jahnke, M. Sch¨offler, T. Weber, R. D¨orner, T.J.M. Zouros, A. Dorn, T. Ferger, K.U. K¨uhnel, S. D¨usterer, R. Treusch, P. Radcliffe, E. Pl¨onjes, J. Ullrich, Phys. Rev. Lett. , 203001 (2007)

&RUUHODWHG (OHFWURQ1XFOHDU 0RWLRQ 9LVXDOL]HG 8VLQJ D :DYHOHW 7LPH)UHTXHQF\ $QDO\VLV Andr´e D. Bandrauk, Szczepan Chelkowski, and Huizhong Lu

$EVWUDFW We have solved numerically the time-dependent Schr¨odinger equation (TDSE) describing dissociative-ionization of a H2 molecule exposed to intense short-pulse laser light in one dimension. From the time dependent wave function we calculated the total average acceleration of the two electrons and the relative proton acceleration. We find that the general shape of the power spectra of electrons and protons is very similar except that the for the electrons the peaks occur at odd harmonics whereas for protons the peaks occur at even harmonics. The wavelet time-frequency analysis shows that, surprisingly, time profiles of electron and proton accelerations are nearly identical for high harmonics. The wavelet time profiles confirm predictions of the three-step quasi-classical model of harmonic generation by identifying several (up to three) electron return times with high precision.

 ,QWURGXFWLRQ High order harmonic generation, HOHG, is one of the most studied effects of the nonlinear nonperturbative response of atoms [1, 2] and molecules [3, 4] to short intense laser pulses with the results that the coherent emitted light can be harnessed to produce new coherent trains of extremely short ,“attosecond” (1 asec D 1018 s) pulses [3, 5]. The duration of these pulses is comparable to the time scale of elecAndr´e D. Bandrauk D´epartement de chimie, Universit´e de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada,e-mail: "OESF#BOESBVL!VTIFSCSPPLFDB Szczepan Chelkowski D´epartement de chimie, Universit´e de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada, e-mail: 4$IFMLPXTLJ!VTIFSCSPPLFDB Huizhong Lu D´epartement de chimie, Universit´e de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada, e-mail: )VJ[IPOH-V!VTIFSCSPPLFDB A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_3, © Springer Science+Business Media, LLC 2011

23

A.D. Bandrauk et al.

24

tron motion which is of the order of the period of the classical electron orbiting on the lowest Bohr orbit equal to 152 asecs. These new pulses will become new tools for visualizing such ultrafast motion in CEWs, coherent electron wavepackets prepared by such pulses [6]. A simple semiclassical (three-step) model, based on the idea of tunneling ionization of an electron in an atom followed by the laser induced recollision of the ionized electron with the parent ion [7–9], allows for interpreting the harmonic generation process in terms of recolliding electron trajectories. This model leads to a universal cutoff law for the harmonic spectrum, i.e, it predicts the maximum photon energy in harmonic generation is described by a simple formula: 2 „!max D 3:17Up C Ip , where Up D Emax =4me !L2 is the ponderomotive energy in an electric field of amplitude Emax and laser frequency !L , Ip is the ionization potential. Molecules introduce an additional complexity in harmonic generation due to possible recollison with neighboring ions which leads to larger maximum photon energies [10, 11]. Furthermore nuclear motion and charge resonance enhanced ionization, CREI, at distances larger than the equilibrium separation [12–14] influences significantly the harmonic generation process in molecules. Further complexity arises due to distortion of the molecular structures by a phenomenon called bond softening [15], as well as, by the influence of the charge transfer states like HC H in enhanced ionization [16] of the H2 molecule. Recently molecular high order harmonic generation, MHOHG [4], has been proposed as a novel method for orbital tomography [17] and also as a tool for probing the nuclear dynamics on the subfemtosecond time scale [18, 19]. The latter is based on the fact that in light molecules the nuclei move during the time interval in which an electron ionizes via tunneling and next recollides with the core. The semiclassical recollision model predicts that this time is approximately 23 of the optical cycle [7–9] in the cutoff (maximum photon energy) region of the harmonic spectrum. In the present work we investigate MHOHG using an additional tool which is the complete wavelet analysis of the electron and proton acceleration. Such a tool has becomes a general method for classifying multiscale dynamics in complex systems [20]. We show that this new tool allows to confirm predictions of the semiclassical three-step model with unprecedented precision: we show that the wavelet time-profile allows us to see even the third recollision of the electron. We also show that the time profile of the nuclear acceleration is nearly identical as the time profile of the electron motion thus illustrating the strong coupling of electron-proton dynamics in intense fields.

 &RPSXWLRQDO 'HWDLOV To study in detail the complete 1D 4-body dynamics of two-electron and two protons of a H2 molecule in intense (linearly polarized) laser pulse we use the complete 1D TDSE [21] written as (in a.u. e D „ D me D 1): i

@ .t/ D ŒH0 C V .z1 ; z2 ; R/ C E.t/.z1 C z2 / .t/; @t

(1)

Correlated Electron-Nuclear Motion Visualized

25

where H0 D

2 X i D1



1 @2 1 @2  ; 2 2 @zi 2 @R2

X 1 C Œ.z1  z2 /2 C d 1=2 C VC .zi / C V .zi / ; R 2

V .z1 ; z2 ; R/ D

i D1

2

V˙ .zi / D Œ.zi ˙ R=2/ C a E.t/ D ".t/ cos.!L t C / :

1=2

;

 D mp =2 is the reduced mass of the two nuclei each having mass mp and ".t/ is the the laser pulse envelope and is the CEP (carrier envelope phase) which we set to =2. The shape of the laser pulse E(t) is shown in Fig. 1. The Coulomb softening parameters a and d are chosen to reproduce faithfully the first three electronic potentials of H2 (a D 0:7, d D 1:2375) [22]. We assume that at t D 0 the molecule is in its lowest vibrational state of the ground electronic surface ˙g and is aligned parallel to the laser polarization. The TDSE (1), is solved using a split operator spectral method [23] for laser pulses of frequencies ! D 0:057 a:u: or ! D 0:038 a:u: which correspond to the wavelengths D 800 nm or D 1200 nm. The pulse envelope ".t/ rises linearly during the first two cycles, next is constant and decreases during the last two cycles yielding the electric field shown in Fig. 1. We used an electron grid of size jzk j < 512 a:u: containing 4048 grid points for each electron and the nuclear grid of size R < 12 a:u: containing 160 grid points. For very short pulses considered here, the electron wave function is still well contained within the grid since the maximum excursion of an ionized electron is ˛ D E=! 2 D 69 a:u: at I D 3:5  1014 W=cm2 , here E D 0:1 a:u: and ! D 0:038 a:u: We have verified that the absorber which was introduced in the electron grid led to only a few percent loss of the norm near the end of the pulse. The complete numerical two-electron nuclear wave function is then used to obtain the laser induced total dipole moment D.t/ and the nuclear displacement R.t/: D.t/ D zel .t/ D h .z1 ; z2 ; R; t/j.z1 C z2 /j .z1 ; z2 ; R; t/i ;

(2)

R.t/ D h .z1 ; z2 ; R; t/jRj .z1 ; z2 ; R; t/i :

(3)

Note that although in (2) only electron coordinates z1 ; z2 appear this expression describes the total dipole of the molecule since because of the molecular symmetry (both nuclei have the same mass) the nuclear relative distance R is absent in the total dipole moment. The proton average separation R.t/ and its temporal second derivative does not give any contribution to dipole electromagnetic high frequency radiation although it may contribute via quadripole radiation terms. In the above definitions the integration is performed over all three variables z1 ; z2 and R. The Fourier transform dF .!/ of the total dipole D.t/ gives the MHOHG power spectrum S.!/ D jdF .!/j2 ! 4 of the emitted dipole radiation. The power spectra can also be calculated directly from the dipole acceleration (i.e., its second temporal

A.D. Bandrauk et al. 0.1

E(t) (a.u.)

E(t) (a.u.)

26

0.0 -0.1 2

4

6

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0

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0

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WLPHᤡODVHUF\FOHVᤢ

)LJ  First column shows the shape of the electric field, the induced total dipole D.t/ D hz1 C z2 i, and the dipole acceleration ael .t/. In the second column we plot again the electric field, the average proton separation hRi and the relative proton acceleration aR .t/. The laser pulse wavelength is  D 800 nm and the laser intensity is I D 4  1014 W=cm2 . One laser cycle D 2:67 fs

derivative) therefore we have also computed the electron and proton accelerations using the equations: d2 D.t/ @V @V D h .z1 ; z2 ; R; t/j C j .z1 ; z2 ; R; t/i ; 2 dt @z1 @z2 d2 R.t/ @V aR .t/ D  D h .z1 ; z2 ; R; t/j j .z1 ; z2 ; R; t/i : dt @R ael .t/ D

(4) (5)

The molecular dipole D.t/ D hz1 C z2 i, R.t/ and accelerations ael .t/, aR .t/ are shown in Fig. 1. Note, that the electron acceleration ael is nearly proportional to the electric field whereas the amplitude of electron dipole oscillation grows as function of time. Also note that proton acceleration reveals a clean second harmonic component not seen in the plot of hRi as function of time. We calculate the power spectra directly from the accelerations given in the above equations since it was shown in the past [24] that it is numerically more advantageous and accurate to use the electron acceleration instead of the electronic dipole. Our goal is to investigate how the system emits harmonics in time and look for the possible correlation between the harmonic emission determined by the Fourier transform of ael .t/ and the nuclear motion characterized by the power spectrum of aR .t/. The appropriate mathematical tools for such an investigation is the Gabor or the Morlet-wavelet transform [20,25]. In our previous paper [21] we used the Gabor transform whereas in this work we find more convenient to use the Morlet-wavelet

Correlated Electron-Nuclear Motion Visualized

27

W Œf  D w.u; s/ of a function f .t/ defined via [20, (2)] via: W Œf  D w.u; s/     Z 1 1  .t 0  u/2 0 0 D p f .t / exp i.t  u/ exp  dt 0 s 2s 2  2 s 1=2 1

(6)

where s is the time-scale and  s is the width of the Gaussian time-window. Since we are interested in a wavelet for a chosen value of a harmonic ! D nH !L D  D

 ; s

where nH is the harmonic order, we replace the variable s in (6) by s D =! and we set u D t (the symbols ; s; u;  and  are the same as in [20]). Thus we get W Œf  D w.!; t/   Z 1 r ! ! 2 .t 0  t/2 0 0 D f .t / expŒi!.t  t/ exp  dt 0 : (7) 2 2 2  1=2  1 Thus the wavelet, as used for our application, does not depend on all three mathematical parameters , s and  but it depends only on the ratio =s and product  which have the following physical meaning. ! D =s is the angular frequency and the product  D 2=T is proportional to  which is the time width of the Gaussian filter expŒ.t 0  t/2 =2 2  used in (7) where T D 2=! is the period corresponding to the angular frequency of a given harmonic !. Thus =.2/ is the width of the Gaussian time-window expressed in periods of a given harmonic. We tried two values for this width:  D 2 and  D 12:0 and found that in most cases better temporal resolution was achieved for the first, smaller value, except the near the harmonic cutof case where better temporal resolution is achieved for a larger value  D 12:0. The time profile analysis [25] based on the wavelet or Gabor transform provides the recollision time c [8, 9] of the electron trajectories involved in the harmonic generation process, and informs us about the depopulation in time of the state to which the electron returns. We used the definition (7) to calculate the wavelet transforms of electron and proton accelerations, i.e., we substituted f .t/ D ael .t/ or f .t/ D aR .t/.

 &RUUHODWLRQV LQ WKH 3RZHU 6SHFWUD RI (OHFWURQ DQG 3URWRQV In this section, we calculate the power spectra of the electron acceleration ael .t/ and proton aR .t/ which are determined via their Fourier transform expression by: ˇ Z ˇ 1 Sel .!/ D ˇˇ T tot

Ttot 0

ˇ2 ˇ dt ael .t/H.t/ expŒi!tˇˇ ;

(8)

A.D. Bandrauk et al.

28

and

ˇ ˇ2 Z Ttot ˇ 1 ˇ SR .!/ D ˇˇ dt aR .t/H.t/ expŒi!tˇˇ ; Ttot 0

(9)

  1 t H.t/ D 1  cos 2 ; 2 Ttot

where

is the Hanning filter [20], Ttot is the total duration of the pulse shown at the top of Fig. 1, Ttot D 10 optical cycles, T D 2:67 fs for the 800 nm laser. The Hanning filter reduces the background introduced by the presence of a non decaying component in accelerations which is artificially cut in (8), (9) by performing the Fourier transform over the finite time in (8) and (9), i.e., over 10 optical cycles only. We show the acceleration power spectra of electrons and protons in Figs. 2 and 3. The first shows the plateau and cutof part of the spectrum whereas the second show the lower order harmonics. Clearly, the general shape of harmonics present in electron and proton motion is very similar with the exception the fact that the electrons oscillate with the superposition of odd order frequencies whereas the relative proton motions contains only even order harmonics. This difference can be readily explained by the fact that when we calculate the average accelerations using (4) and (5) we use the operator @V =@z1 C @V =@z2 for the electron power spectra, and we use the operator @V =@R for the proton spectra. Note that the latter is an even function with respect to the inversion of the electron coordinates: zi ! zi (i D 1; 2) whereas the former is an odd function. Thus if one expands the electron wave function in the continuum as a

-7

H 2 -electron spectra from a el(t)

(a)

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H 2 -electron spectra from ael(t)

-4

(c)

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log10 power spectrum

-6 -9

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H 2 - proton spectra from μ x a R (t)

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H 2 - proton spectra from μ x a R(t) (d)

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harmonic order

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harmonic order

)LJ  (a) and (c) show the electron power spectra Sel . Figures (b) and (d) show he corresponding proton spectra SR . (a) and (b) show the plateau and cutof part whereas (c) and (d) show lower harmonics. In Figs. 1–6 the laser pulse wavelength is  D 800 nm and laser intensity is I D 4  1014 W=cm2

Correlated Electron-Nuclear Motion Visualized

log10 power spectrum

-2

29

-2

electron spectrum from ael(t)

(a) -4

proton spectra from μ x aR(t)

(b)

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2

| F.T.[E(t)] |

2

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| F.T. [E (t)] |

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)LJ  (a) shows the electron power spectra Sel .!/ and (b) the proton power spectra for harmonics 0-2. They are compared with the Fourier transforms of the electric field E.t / and E 2 .t/

series of electronic eigenstates, the ground state will have nonzero matrix elements with even or odd states, respectively. The similarity of the shapes of both spectra suggests that the oscillatory motion of protons and electrons is highly correlated. Note that only the power spectrum Sel of acceleration of electrons ael represents the real electromagnetic dipole radiation since ael .t/ is the second time derivative of the total dipole D.t/ of the H2 molecule. Although there is no real dipole electromagnetic radiation related to the proton power spectrum SR .!/, the function SR .!/ does inform us about the presence of the oscillatory motion R.t/ which is very clearly correlated to the electron motion. This nuclear acceleration originates from the DV\PPHWU\ of the electron cloud induced by the laser fields oscillating as shown in Fig. 1. This asymmetry leads to the acceleration with positive sign at each laser half-cycle whereas the electron acceleration changes sign at each half-cycle. Consequently, since the periodicity of this electronic cloud displacement is a half-laser period the principal peak in the proton power spectra occurs at frequency ! D 2!L , followed by the peaks separated by 2!L . In Fig. 3, which shows the lowest frequency part, we notice a series of weak peaks separated by by 0:1!L or 0:2!L . These are not related to the the molecular vibrations but they are the replicas of the shape of the Fourier transform of the laser electric field E.t/ and its square E 2 .t/ as illustrated in Fig. 3. Note that these side peaks are 3 orders of magnitude weaker than the principal peaks.

 9LVXDOL]DWLRQ RI WKH (OHFWURQ 5HWXUQV 8VLQJ WKH 7LPH 3URILOHV RI (OHFWURQ DQG 3URWRQ +DUPRQLFV The goal of our investigation in this section is two-fold: first, we wish to reveal a possible influence of the laser induced electron oscillations related to the harmonic generation in time on nuclear oscillations, and second, we want to visualize with the help of the wavelets predictions of the three-step model. This model emphasizes the classical aspects of the harmonic generation process in which the electron is

A.D. Bandrauk et al.

30

tunneling at a specific time t0 (t0 is slightly larger than the electric field maximum or minimum), next accelerates in the laser electric field and the radiation is emitted during consecutive recollisions of the electron with the core [7–9] at times tf when the electric field changes sign. The basic assumption of the model is that when the electron at recollision time tf has kinetic energy Ef D v.tf /2 =2 a photon is emitted with energy: ! D nH !L D Ef C Ip where Ip is the ionization potential of an atom or of a molecule. The initial velocity of the electron at the tunneling time t D t0 is assumed to be zero. The solution of Newton’s equation with the condition that at t D tf the electron returns to the core and emits a photon imposes two equations on parameters t0 , tf and Ef . Thus when we select a value of a harmonic ! D nH !L we fix the value of the return energy and the return time tf . Consequently the tunneling time t0 is also fixed via solutions of the Newton equation. Therefore, for each harmonic order nH , we expect to see in the wavelet time profiles the peaks at return times tf predicted by the three-step model. More specifically (we use the notation of our previous paper [9]) this model gives us the specific relation between the electron return energy Ef (which is known for a fixed harmonic nH and for a given Ip ) and the return time tf . This relation can be derived by solving the classical Newton equations of motion of an electron in a laser field E.t/ with an initial velocity v0 at initial time t0 (in atomic units, me D e D „ D 1): d2 z dv D D E0 cos.!L t/ ; dt 2 dt

v.t0 / D v0 ;

z.t0 / D z0 :

(10)

Its general solution, at a final time tf (starting at the initial time t0 ), is: vf D v.tf / D v0 C X sin. 0 /  X sin. f / ; z.t0 / D z0 : v0 C X sin. 0 / X z.tf / D . f  0 / C Œcos. f /  cos. 0 / C z0 ; !L !L

(11) (12)

where X D E0 =!L , 0 D !L t0 , f D !L tf . We set in above equation v0 = 0 (this is the basic assumption of the three-step model) and use the condition of the electron return: z.tf / D z0 . Thus, using (12), we obtain 0 as function of  D f  0 (note that =!L D tf  t0 is the time it takes to move the electron to recollide after it had tunneled at t D t0 ) :   1  cos  0 D 0 ./ D arctan : (13)   sin  Thus, using (11), we get the final electron velocity vf at the recollision time as the following simple function of : vf D vf ./ D X Œsin 0 ./  sin. 0 ./ C / :

(14)

This relation allows us to plot the harmonic order nH D nH ./ D

vf2 =2 C Ip !L

(15)

Correlated Electron-Nuclear Motion Visualized

31

as function of . Since for each value of  we can calculate the recollision time tf D tf ./ D

0 ./ C  !L

(16)

we thus can also plot the harmonic order nH as function of the return time tf . We plot this relation between the recollision time tf and the harmonic order nH in Figs. 4(c), 5(d), 7(c) for the trajectories initialized at t0 ’s slightly greater than the field maximum occurring at t D 4:25 optical periods T . From these graphs we can deduce the values of subsequent (we show up to three returns for short and long trajectories) recollision times tf for a given harmonic order nH for a fixed laser frequency and for fixed ionization potential Ip and compare these predictions with the maxima seen in the wavelet graphs displayed in Figs. 4(a)–8(a). We show the time frequency profiles for two wavelengths, in Figs. 4–6 for D 800 nm, I D 4  1014 W=cm2 and in Figs. 7 and 8 for D 1200 nm, I D 2  1014 W=cm2 . The numbers seen near the peaks of the wavelet transforms are deduced from the corresponding plots of nH .tf / Figs. 4(c), 5(c) and Fig. 7(c). We conclude that the return times predicted by the three-step model (i.e., by (15), (14)) agree with the peaks position of time-profiles (obtained from TDSE) with a surprisingly high precision. The nuclear and electron motions are highly correlated: their time-frequency profiles are nearly identical. However, we note that in most cases the relative intensities of contributions from short and long trajectories is different for the electron time profiles as compared to the proton time profiles: for the electron case the long trajectories contribution is stronger than for short trajectories whereas for the proton case inverse relation occurs. We do not have an explanation for this, we believe that 3D calculations are necessary for investigating this effect since in 3D the transverse spreading of the electron wave packet reduces the intensity of radiation originating from a long trajectory [5].

 &RQFOXVLRQ We have solved numerically the time-dependent Schr¨odinger equation describing the (four-body) dynamics of a H2 molecule exposed to intense short-pulse laser light in one dimension. From the wave function (which depends on the two electron coordinates and the relative proton coordinate) we calculated the total average acceleration of the two electrons ael .t/ and the relative proton acceleration aR .t/ as functions of time. Next, we calculated the power spectra of these accelerations and their wavelet time profiles. We find the striking similarity in the overall shapes of the electron and nuclear power spectra and in the cutoff position except that the electron spectra show odd harmonics whereas the nuclear spectra show even harmonics. This difference is related to the opposite parities of the derivatives of the potential: the derivative with respect to the electronic variable is an odd function of the electronic variable whereas the derivative with respect to the internuclear separation R is an even function of electronic variable in the acceleration matrix elements.

A.D. Bandrauk et al. 39-th harmonic 30

60

ση=12

(c)

ση=2π

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0 4.6

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tf for long tr.

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time profile of aR (10 a.u.)

classical return times

ᤡ Dᤢ

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(Ef + Ip)/ωlaser

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time profile of ael (10 a.u.)

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classical return times. t0 is near 4.25c, t0>4.25c 4.0

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time prof. of aR(t) (10 a.u.)

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35-th harmonic

25

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time prof. of ael(t) (10 a.u.)

electric field

)LJ  (a) time profile of the electron acceleration ael , and (b) time profile of the proton acceleration aR for the harmonic order nH D 39. In (c) we show the harmonic nH order as function of the return times tf obtained from the Newton’s equations (10), using (16).

40

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)LJ  (a) the electric field E.t/, (b) time profile of the electron acceleration ael , and (c) time profile of the proton acceleration aR for the harmonic order nH D 35. In (d) we show the harmonic nH order as function of the return times tf obtained from the Newton’s equations (10), using (16). The dots denote six possible return times tf for the 35th harmonic predicted by the semiclassical three-step model

time profile of ael (10-4 a.u.) field E(t) (a.u.)

Correlated Electron-Nuclear Motion Visualized 0.1

0.1

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time profile of aR (10-4a.u.)

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W IRUVKRUWWU

4.0 60

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57-th harmonic W IRUORQJWU

50

6.0 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 69-th harmonic ση=12

(b)

40

60 tf = 4.95 for a single return.

40

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0 4.0 40

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6.0 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 69-th harmonic

(c)

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30 20

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time (cycles)

time (cycles)

)LJ  First column: (a) the electric field E.t /, (b) time profile of the electron acceleration ael , and (c) time profile of the proton acceleration aR for the harmonic order nH D 57. The second column shows the same as the first column but for a higher harmonic nH D 69 (near the cutof).

110

time profile of ael (10-4a.u.)

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5.2

lo.(1) (a) 5.114

1.5

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sh.(3) 4.862

lo.(2) 5.0365

0.0 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2

(b) 2.0 1.5

I=2x1014, λ=1200 nm

100

sh.(1) 4.772 sh.(2)

(Ef + Ip)/ωlaser

field E(t)

λ=1200 nm, H2 , I=2x1014 W/cm2 0.1

(c)

90 80 70 60 50

1.0

40

0.5 0.0 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2

time (cycles)

5.0

5.5

6.0

tf (cycles)

)LJ  First column: upper panel: the electric field E.t /, (a) time profile of the electron acceleration ael , and (b) time profile of the proton acceleration aR , both for the harmonic order nH D 57. The laser pulse wavelength is  D 1200 nm and its intensity I D 2  1014 W=cm2 in this and in the next figure. 5 return time are identified. Note that the third long return overlaps with the second long return denoted by the symbol “lo(2).”

A.D. Bandrauk et al.

)LJ  Same as in the first column of Fig. 7 but for the harmonic order nH D 71. Up to three returns are seen, the third is a single possible return denoted by the abbreviation ”sing(3)”. The second return denoted by “lo(2)” cannot be identified since it happens at the same time as the “lo(1)” return.

time profile of aR (10-4 a.u.) time profile of ael (10-4 a.u.) field E(t) (a.u.)

34

14 2 λ=1200 nm , I=2x10 W/cm 71-th harmonic

ᤡDᤢ

0.1 0.0

-0.1 4.4 2.5

4.6

4.8

5.0

71-th harmonic sh.(1)

lo.(1) 5.081

4.810

2.0 1.5

5.2

ᤡEᤢ sing.(3) 4.989

sh.(2) 4.902

1.0

lo.(2) 5.065

0.5 0.0

4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2

ᤡFᤢ

2.0 1.5 1.0 0.5 0.0

4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2

time (cycles)

We note that the electronic power spectra describe the dipole electromagnetic radiation emitted by the molecule since they are derived from the total dipole of electrons and protons; because of symmetry of H2 the nuclear component is absent in the total molecular dipole D.t/. Since the protons have same charge (with the same sign, unlike lectron proton pair) there is no dipole related to the nuclear average separation R and its second time derivative, and consequently no dipole emission related to the nuclear power spectra SR .!/ is expected. Nevertheless, our analysis reveals the strict correlation of the high frequency components in the electron and nuclear motions. In particular, we find that the wavelet time profiles for electron and proton motion are nearly the same. Particularly interesting is the conclusion of our analysis of positions of peaks of time-frequency profiles for a fixed harmonic frequency ! D nH !L . We have identified for each peak the corresponding classical return times tf which agree with precision down to 0.05 of the laser period and have identified the second and third electron return. This is an important quantitative confirmation of the validity of the semiclassical two-step model [7–9]. Note, that we obtained the values of peak positions of time profiles using exact (1D) quantum dynamics based on the TDSE the two-electron and two-proton motion without using any semiclassical approximations.

Correlated Electron-Nuclear Motion Visualized

35

$FNQRZOHGJHPHQWV We are very grateful to Turgay Uzer (Georgia Tech, Atlanta) for stimulating discussions regarding the applications of wavelets in complex systems.

5HIHUHQFHV 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

T. Brabec, F. Krausz, Rev. Mod. Phys. , 545 (2000) P. Sali`ere, A. L’Huillier, P. Antoine, M. Lewenstein, Adv. At. Mol. Opt. Phys. , 83 (1999) P.B. Corkum, F. Krausz, Nature Phys. , 381 (2007) A.D. Bandrauk, S. Barmaki, S. Chelkowski, G.L. Kamta, in 3URJUHVV LQ 8OWUDIDVW ,QWHQVH /DVHU 6FLHQFH ,,,, 6SULQJHU 6HULHV LQ &KHPLFDO 3K\VLFV, vol. 89, ed. by K. Yamanouchi, S.L. Chin, P. Agostini, G. Ferrante (Springer, New York, 2008), chap. 9 P. Agostini, L.F. DiMauro, Rep. Prog. Phys. , 813 (2004) S. Chelkowski, G.L. Yudin, A.D. Bandrauk, J. Phys. B , S409 (2006) P.B. Corkum, Phys. Rev. Lett. , 1994 (1993) W. Becker, S. Long, J.K. Mclver, Phys.Rev.A , 1540 (1994) A.D. Bandrauk, S. Chelkowski, S. Goudreau, J. Mod. Opt. , 411 (2005) A.D. Bandrauk, S. Barmaki, G.L. Kamta, Phys. Rev. Lett. , 013001 (2007) X.B. Bian, A.D. Bandrauk, Phys. Rev. Lett. , 093903 (2010) T. Zuo, A.D. Bandrauk, Phys. Rev. A , 2511 (1995) T. Seideman, M.Y. Ivanov, P.B. Corkum, Phys. Rev. Lett. , 2819 (1995) S. Chelkowski, A.D. Bandrauk, J. Phys. B , L723 (1995) A. Zavriyev, P.H. Bucksbaum, in 0ROHFXOHV LQ /DVHU )LHOGV, ed. by A.D. Bandrauk (Dekker, New York, 1994), chap. 2 K. Harumiya, H. Kono, Y. Fujimura, I. Kawata, A.D. Bandrauk, Phys. Rev. A , 043403 (2002) J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. P´epin, J.C. Kieffer, P.B. Corkum, D.M. Villeneuve, Nature , 867 (2004) S. Baker, J.S. Robinson, C.A. Haworth, H. Teng, R.A. Smith, C.C. Chirilˇa, M. Lein, J.W.G. Tisch, J.P. Marangos, Science , 424 (2006) M. Lein, Phys. Rev. Lett. , 053004 (2005) C.C. Wiggins, U. T, Phys. D , 171 (2003) A.D. Bandrauk, S. Chelkowski, H. Lu, J. Phys. B , 075602 (2009) A.D. Bandrauk, H.Z. Lu, Phys. Rev. A , 023408 (2005) S. Chelkowski, C. Foisy, A.D. Bandrauk, Phys. Rev. A , 1176 (1998) K. Burnett, V.C. Reed, J. Cooper, P.L. Knight, Phys. Rev. A , 3347 (1992) P. Antoine, B. Piraux, A. Maquet, Phys. Rev. A , R1750 (1995)

$ 7KHRU\ RI '\QDPLF ,PDJLQJ RI &RKHUHQW 0ROHFXODU 5RWDWLRQV E\ +LJK +DUPRQLF *HQHUDWLRQ F.H.M. Faisal and A. Abdurrouf

$EVWUDFW A dynamic theory of mapping coherent molecular motions from high harmonic generation signals is presented. Application to mapping of coherent rotational motions of linear molecules is made. Results of concrete calculations for N2 , O2 and CO2 are analyzed both in time and frequency domains. A “magic angle” for the polarization geometry is predicted at which the HHG signals for all pump-probe delay times become equal for linear molecules of g orbital symmetry. In contrast only a “crossing neighborhood” near the magic angle is predicted for molecules with orbitals of  symmetry. They are expected to help in identifying the orbital symmetry in the inverse problem of orbital reconstruction from experimental data. Comparison with available experimental data show remarkable agreement with all the salient properties of dynamic HHG signals of linear diatomic molecules, N2 and O2 , and some simple organic molecules. Additional results for the triatomic molecule CO2 are discussed that might help to test the theory further.

 ,QWURGXFWLRQ With the advent of ultrashort DQG very intense laser pulses in the femtosecond to attosecond regime real time imaging of coherent molecular motions has become a topic of much interest. In recent years many research groups have succeeded in inducing and monitoring coherent rotational motion of the molecular frame of simple linear molecules using intense-field ultrashort pump-probe technique of high harmonic generation (or HHG), e.g., [1–10]. In these experiments the “pump” pulse F.H.M. Faisal Fakult¨at f¨ur Physik, Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany, e-mail: GGBJTBM!QIZTJLVOJCJFMFGFMEEF A. Abdurrouf Department of Physics, Brawijaya University, Malang 65145, Indonesia, e-mail: BCEVSSPVG! CSBXJKBZBVBDJE A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_4, © Springer Science+Business Media, LLC 2011

37

38

F.H.M. Faisal and A. Abdurrouf

sets a thermal ensemble of gas molecules in coherent rotational motion, and the “probe” pulse induces the rotating molecules to emit high harmonics of the probe radiation, at any multiple of its carrier frequency. The emitted HHG intensity is recorded as a function of the delay, td , between the two pulses, for a fixed relative angle, ˛, between their linear polarization directions. The recorded signal modulation as a function of the delay td provides a real time imaging of the underlying coherent nuclear motion. Additional information can be obtained if the HHG signal is recorded, at a given delay td , as a function of the relative polarization angle ˛. Recent advances in theory and analysis of molecular HHG process in intensefield ultrashort pump-probe experiments have provided much insight into the relationship between the HHG and the molecular orbitals and the underlying rotational dynamics (see, e.g., [11–18]). Here we present and discuss a recently developed analytic quantum theory [13,14,16,18] of intense-field pump-probe HHG experiments for imaging nuclear rotations.1 The theory is illustrated with results of concrete calculation with diatomic N2 and O2 as well as triatomic CO2 molecules. Before proceeding further, in Fig. 1 we show schematically the alignment of the molecular axis R with respect to the polarization direction (along z-axis) of the pump laser pulse and the polarization direction of the probe laser pulse (along z 0 axis). The angles ,  0 and correspond, respectively, to the angle between the molecular axis and the pump pulse, the probe pulse, and the azimuth angle measured from the common z-z 0 -x plane; the angle between the pump and the probe polarization directions (assumed to be linearly polarized) is denoted by ˛. It should be noted that the angles ;  0 and generally remain unobserved in the laboratory. In contrast the angle ˛ is perfectly measurable and indeed in a fully controlled manner in the laboratory. Clearly, the other control parameter in the laboratory is the time delay, td , between the pump and the probe pulse. The goal of the theory is to connect the rotational coherent motion (induced by the pump pulse) in real time with the observed high harmonic signal (induced by the probe pulse) as a function of the delay td , as well as with the variation of the relative polarization angle ˛. We shall give useful algebraic formulas that provide these connections, help to analyze

)LJ  Molecular axis R and relative alignment angles  ,  0 and ; pump polarization along z-axis, probe polarization along z 0 -axis; relative angle between the pump and probe polarization, ˛ 1

A number of alternative models appear to be of limited use for the present purpose (for an assessment, see Sects. V A (last para.) and VII D in [16])

Theory of Dynamic Imaging by HHG

39

and interpret the characteristic experimental observations, and also to predict further characteristics that might be observed in the future.

 7KHRUHWLFDO )RUPXODWLRQ In this section we give a brief outline of the mathematical formulation of dynamic HHG signals from molecular targets in intense-field ultrashort pump probe experiments.

 :DYHIXQFWLRQ RI WKH ,QWHUDFWLQJ 6\VWHP The total Hamiltonian of the molecular system interacting with a pump pulse L1 at a time t , and a probe pulse L2 applied after a delay-time td can be written ( [14, 16]) within the Born–Oppenheimer approximation as (we use Hartree atomic units: „ D e D m D ˛c D 1, unless given explicitly otherwise): .0/

Htot .t/ D HN C VNL1 .t/ C He.0/ C VeL2 .t  td /

(1)

where the subscripts N and e stand for the nuclear and the electronic subsystems, respectively. An intense ultrafast pump pulse is assumed to interact with the molecular polarizability, via VNL1 .t/, and sets it into coherent free rotation described quantum mechanically by the nuclear wavepacket states created by the pump pulse (e.g., [16]): X .J M / j˚J0 M0 .t/i D aJM0 0 .t/ exp.iEJM t/jJM i : (2) JM

Each of these states characterizes a mode of quantum coherent rotational motion of the nuclear frame of the molecule that is initially in the rotational eigenstate jJ0 M0 i exp.iEJ0 M0 t/. One may ask if the induced coherent motion lasts only as long as the pump pulse acts on the molecule, or it can survive even after the driving pulse is over. In Fig. 2 we show the evolution of the component-coefficients of the coherent wavepacket state (2) that is calculated for an intense femtosecond pump pulse of 800 nm wavelength interacting with a N2 molecule for times before, during and after the pump pulse is applied (the time profile of the pulse is shown by the dotted line). It is clear from the figure that after the pulse is over the coefficients of the wavepacket does not die out but in fact attain generally nonvanishing constant values for all future times. Clearly, therefore, the coherent wavepacket motion once excited by an ultrafast pump pulse can persist forever (if left undisturbed). Thus at any time, while the molecule keeps executing the coherent rotation, an ultrafast probe pulse can be applied with a delay td (with respect to the application of the pump pulse) to induce the high harmonic generation (HHG) signals. Such signals

40

F.H.M. Faisal and A. Abdurrouf

)LJ  Survival of rotational wave packet states after the passage of an ultrashort pump pulse (dotted line). Modulo square of the component-coefficients, jaJM j2 , of a coherent wavepacket state evolving from a given initial rotational eigenstate J0 M0 (indicated by ). Note the nonzero constant values of the coefficients DIWHU the passage of the pulse; pulse profile shown as dashed curve, intensity D 0:5  1014 W=cm2 , FWHM 40 fs, 800 nm

measured as a function of the delay td can provide a real time image of the coherent quantum rotation of the nuclear frame. We note that there are as may linearly indendent wavepacket states (2) as there are initially occupied eigenstates jJ0 M0 i of the molecule, that are populated with a Boltzmann distribution %.J0 M0 / D .1=ZP / exp.EJ0 M0 =kT /, where ZP is the partition function and T is the rotational temperature. Thus, after the pump pulse is applied, the state of the molecule is characterized by the corresponding ensemble of product states, j i .t/i, with i fe; J0 ; M0 g, composed of the ground electronic state j e.0/ .t/i and the complete set of coherent wavepacket states j˚J0 M0 .t/i: j i .t/i D j e.0/ .t/ij˚J0 M0 .t/i :

(3)

To be able to theoretically investigate the HHG signal we need first to construct the appropriate wavefunction of the total interacting system. To this end we generalize the strong-field KFR ansatz (e.g., [19]) for the dynamics of laser molecule interaction and express the system wavefunction, evolving from each of the ensemble of the initial states (3) as ( [16] and/or [18]):

Theory of Dynamic Imaging by HHG

41

Z ji .t/i D j i .t/i C

dt 0 G0 .t; t 0 /VeL2 .t 0  td /j i .t 0 /i

(4)

where, the Green’s function G0 .t; t 0 / of the system is given by X .C/   G0 .t; t 0 / D i.t  t 0 / j j ij p .t  td /ij˚JM .t/i exp iEjC .t  t 0 / j pJM

 h˚JM .t 0 /jh p .t 0  td /jh j

.C/

j (5)

j p .t/i are Volkov states (e.g., [19]). We have also introduced the j th “Dyson or.D/ .C/ bital” j j .t/i defined by the overlap of the states j j i: .D/

j j

.C/

.t/i h j

.1; 2; : : : ; Ne  1I t/j 0 .1; 2; : : : ; Ne I t/i

D

.C/ h j .1; 2; : : : ; Ne



j j.D/ i exp.i"j t/

 1/j 0 .1; 2; : : : ; Ne /i exp.i"j t/ (6)

.C/

where, "j jEj  E .0/ j, j D 0; 1; 2; : : : are the successive ionization potentials of the molecule. .D/ We may add parenthetically that the Dyson orbitals, j j i, and the corresponding energies, "j , implicitly include effects of electron-electron correlation that the usual Hartree–Fock orbitals can not. Note that the present theory incorporates the Dyson orbitals in a natural way, that can be used more appropriately in place of the Hartree–Fock MOs and the associated HOMO, HOMO-1, etc. if needed. We have also found it useful, e.g., in the context of molecular orbital recovery problem (in order to distinguish them from the corresponding Hartree–Fock orbitals) to refer to .D/ the sequence of Dyson orbitals j j i; j D 0; 1; 2; : : :, as HODO (highest occupied Dyson orbital), HODO-1, etc. [18]. The above wavefunction explicitly shows how the present theory can account for the appearance of the HODO as well as HODO1, etc. on the same footing. We note, however, that for the numerical calculations presented in this work the usual HOMO of the molecule concerned is used.

 ++* $PSOLWXGH DQG 'LSROH ([SHFWDWLRQ 9DOXH The quantum transition amplitude T .n/ .td ; ˛/ for the coherent emission of the nth harmonic photon of energy „˝ D n„! by a transition from the initial state (3) into the interacting state (4) and recombining back into the VDPH initial state (3) is related to the nth Fourier transform (FT) of the expectation value of the dipole operator of the system D.t/ as ( [16]): p T .n/ .td ; ˛/ D 2.n!/"n  D .n/ .td ; ˛/ (7) where "n is the polarization vector of the emitted (nth) harmonic and

42

F.H.M. Faisal and A. Abdurrouf

D .n/ .td ; ˛/ D FTfExp: Val:ŒD.t/g.n!/ D FTfh i .t/jdji .t/i C c:c:g.n!/ D hJ0 M0 .; ; td /jD .n/ e .; I ˛/jJ0 M0 .; ; td /i

(8)

.n/

where D e .; I ˛/ is the nth FT of the electronic part of the expectation value above (for a given alignment of the molecular axis, .; /, and for a given relative polarization angle ˛ between the pump and the probe polarization). This can be calculated explicitly once the molecular orbitals of interest are given, by first transforming them from the body frame to the space fixed frame of reference, obtaining the electronic part of the dipole expectation value and then taking the expectation value with respect to the rotational wavepacket states jJ0 M0 .; ; td /i. The probability per unit time of emission of the nth harmonic photon is then obtained from the effective “golden rule,” and the final expression for the dynamic HHG signal for the nth harmonic, S .n/ .td ; ˛/, is obtained by taking the statistical average of the LQGHSHQGHQW emission probabilities,2;3 associated with each of the thermally occupied initial states (3), and multiplied by the density of the final wave-number states. Thus, for the nth HHG signal we get: S .n/ .td ; ˛/ D

1 .n!/3 X %.J0 M0 / 2 c 3 J0 M0

2 jh˚J0 M0 .; ; td /j"n  D .n/ e .; ; ˛/j˚J0 M0 .; ; td /ij :

(9)

The active orbital (in the body fixed frame) of a linear molecule is conveniently expressed in a single-center expansion in spherical harmonics in the form: X .r/ D Clj mj Rj .r/Ylj mj .; / (10) j

where C ’s are linear combination coefficients and Rj .r/ are conveniently taken in j 1 j r the generalized Slater-like functions with asymptotic form R e , pj .r/ D r where j is given by the orbital ionization energy Ij , j D 2Ij (a.u.), and j D Z= j , Z is the asymptotic nuclear charge. The HHG operator for linear molecules is given more explicitly as X 0 "n  D .n/ .; ; ˛/ D aQ .n/ .l; l 0 ; L; m/PL .cos  0 / (11) l;l 0 ;L

where, 2

We may point out explicitly that the rotational coherence is not destroyed by the thermal averaging of the emission probabilities since the former coherence is encoded in the individual rotational wavepacket states themselves from which the individual emission probabilities of interest are obtained. For an extension of the theory to ro-vibrational coherence, see [18]. 3 We would like to take this opportunity to correct the following: (i) in (29) of [18] the term “Cc:c:” should appear inside the curly brackets (ii) a missing factor 1=.2/3 should appear in the density of wavenumber states in Sect. IV and in the constant “C” in Sect. V of [16].

Theory of Dynamic Imaging by HHG

43

0

.n/ aQ .n/ .l; l 0 ; L; m/ D .1/m hl; l 0 ; m; mjL; 0ihl; l 0; 0; 0i  aQ zz .l; l 0 ; m/; (12)

cos  0 D cos ˛ cos  C sin ˛ sin  cos ;

(13)

.n/

and the coefficients aQ zz .l; l 0 ; m/ are given by the nth FT of the matrix elements between the lth and the l 0 th orbital components, for a given conserved value of m, that arise in calculating the electronic dipole expectation value.

 )RUPXODV RI ++* 6LJQDOV IRU 12  22 DQG &22 Below we give the dominant terms of the dynamic signals4 for the nth harmonic, S .n/ .td ; ˛/, as a function of the delay td , and the relative polarization angle ˛. Signal for N2 (orbital symmetry g , m D 0; l.l 0 / D 0; 2; 4): ˛ .n/ .n/ ˝ S .n/ .td ; ˛/ D c00 C c01 hcos2  0 i.td / C    .n/ .n/ D .c00 C 12 c01 sin2 ˛/ .n/

C c01 .cos2 ˛ 

1 2

˝ ˛ sin2 ˛/ hcos2  i.td / C    :

(14)

Signal for O2 , (molecular orbital symmetry g , m D 1; l.l 0 / D 2; 4): ˛2 .n/ ˝ S.td ; ˛/ D c11 hsin2  0 cos2  0 i.td / C    D

.n/ c11 hf.3  30 cos2 ˛ C 35 cos4 ˛/hsin2  cos2 i.td / 64  .1  6 cos2 ˛ C 5 cos4 ˛/hcos2  i.td /

C .4 sin2 ˛  3 sin4 ˛/g2 i C    :

(15)

Signal for CO2 , (orbital symmetry , m D 1; l.l 0 / D 2; 4; 6): S.td ; ˛/ D c11 hhsin2  0 cos2  0 i2 .td /i C    .n/

D

.n/ c11 hf.3  30 cos2 ˛ C 35 cos4 ˛/hsin2  cos2 i.td / 64  .1  6 cos2 ˛ C 5 cos4 ˛/hcos2  i.td /

C .4 sin2 ˛  3 sin4 ˛/g2 i C    :

(16)

.n/

In the above formulas the coefficients cij ’s are given by simple combinations of .n/

the parameters aQ zz .l; l 0 ; m/ for each case [16]. The inner angle brackets stand for the expectation value of the trigonometric operators with respect to the wavepacket states, and the outer angule brackets, for the thermal average with respect to the For the details of the derivation LQFOXGLQJ the higher order terms (not shown here) we refer the interested reader to Sects. IV and V in [16] 4

44

F.H.M. Faisal and A. Abdurrouf

Boltzmann distribution of the initially occupied rotational states at a temperature T and where Z.T / is the rotational partition function, ˝ ˛ h  i.td / X 1 D exp.EJ0 M0 =kT /h˚J0 M0 .; ; td /j  j˚J0 M0 .; ; td /i : (17) Z.T / J 0 M0

It should be noted that the signals for O2 and CO2 have analogous mathematical form due to the same orbital symmetry (, m D 1) of the two molecules. The quantitative differences arise from the difference in the nth order FT coefficients .n/ aQ zz .l; l 0 ; m/ for different number of l.l 0 / and the radial basis functions associated with them, for the two molecules. We may note in passing that for the special case of parallel orientation of the pump and the probe pulse, ˛ D0, (14) and (15) reduce properly to the special limiting cases discussed earlier in [13].

)LJ  Calculated image of rotational coherent motion of N2 as obtained from the 19th harmonic dynamic signal in 3D representation as a simultaneous function of the delay td and the relative polarization angle˛; shown on the right are sections of the signal at a sequence of ˛ at an interval of 10ı . Pump intensity I D 0:8  1014 W=cm2 , probe intensity, I D 1:7  1014 W=cm2 , pulse durations, FWHM 40 fs, carrier wavelength 800 nm

Theory of Dynamic Imaging by HHG

45

 $SSOLFDWLRQV DQG 'LVFXVVLRQV We shall illustrate the application of the present theory to the linear molecules N2 , O2 , and CO2 . In Fig. 3 we show the calculated dynamical image of the coherent rotational wavepacket motion of the linear molecules N2 as imaged by the HHG signal at the 19th harmonic, as a function of the delay td , and also of relative polarization angle ˛. Note the modulation of the signal with several revivals as a function of the delay td . Also shown on the right are sections through ˛ at an interval of 10ı , near the first revival time td between 5.3 and 6:3 ps. Note that near ˛  55ı the signal appears to be virtually constant for all values of the delay td implying a steady p output of the harmonic emission at this “magic” angle ˛ D arctan 2, the origin of which will be clarified in the discussion below. In Fig. 4 and Fig. 5 we present the corresponding calculated images for the coherent rotational motion of the diatomic O2 and triatomic CO2 molecules. To gain further insights into the variation of the signals both with td and with ˛, in these 3D images, we consider in Fig. 6 the results of computations using (14) for N2 as a function of the delay time td only, at three selected polarization angles, ˛ D 0ı , 45ı and 90ı . The results show a full revival with a period Trev 1=.2Bc/ D 8:4 ps (B is the usual rotational constant in cm1 and c is the velocity of light) and two fractional 12 and 14 revivals, for all three ˛ values. ˝ ˛ The hcos2  i .td / term is known to govern the 12 and 14 revivals and the associated Raman allowed spectral lines (e.g., [13]). Interestingly, the signals for ˛ D 0ı and ˛ D 90ı are found to be in RSSRVLWH phase, while that for ˛ D 0ı and ˛ D 45ı are in the same phase. Exactly the same phase relation between the ˛-dependence of the td -signal from N2 has been observed in recent experiments (e.g., [2, 3, 10]). To analyze the origin of the phase reversal and the occurrence of the “magic angle” with respect to ˛, we consider the second line of (14) for the signal for N2 more explicitly. For ˛ D 0ı (parallel polarization), we have, S.td I 0ı /  ˛ .n/ .n/ ˝ c00 C c01 hcos2  i .td / and for the perpendicular polarization we get, S.td I 90ı /   .n/ 1 .n/  1 .n/ ˝ ˛ 2 ˝ c00 2C 2˛c00  2 c01 hcos  i .td /. Clearly due to the opposite sign of the moment hcos i .td /, they vary in RSSRVLWH phase to each other from their respective bases.  .n/ ˛ .n/  .n/ ˝ In contrast, the signal at ˛ D 45ı , S.td ; 45ı /  c00 C 14 c01 C 14 c01 hcos2 i .td /, ˝ ˛ has the same sign of the hcos2 i term as for ˛ D0, which makes them to vary LQ phase. This behavior is what can be seen in the full calculations in Fig. 3, which agree with the experimental observations (e.g., [1–3, 10]). The simple formula predicts further that the extrema of the signal should occur for sin ˛ cos ˛ D 0, with a maximum at ˛ D 0ı and a minimum at ˛ D 90ı . This is also what has been seen experimentally [1, 3,p 10]. Finally, the same formula (14) predicts a universal “magic angle” ˛c D arctan 2  54:7ı , given by the condition .cos2 ˛c  12 sin2 ˛c / D 0 (or P2 .cos ˛c / D 0) at which the HHG signals essentially become LQGHSHQGHQW of the delay td between the pulses. Exactly such a “magic” crossing angle for N2 signals has also been experimentally confirmed [10]. Note that the sequence of signals shown on the right-hand side of Fig. 3 confirms the approach of the steady character of the HHG signal at the magic angle  54:7ı as well as the reversal of the phase of the signal from one side of this angle to the other. Clearly, this geometry can be

46

F.H.M. Faisal and A. Abdurrouf

used in femtosecond pulse-probe experiments to generate a steady HHG signal from freely rotating N2 . We may note further that the appearance of the magic angle in the HHG signal provides a definite signature of the g orbital symmetry of N2 , and thus can be useful for the problem of orbital reconstruction from experimental data (e.g., [12]) and/or help to identify an unknown orbital symmetry in the HHG data. In Fig. 7 we present the results of calculations for the 19th harmonic signal for O2 , using (15) for three geometries, ˛ D 0ı , 45ı and 90ı . Similarly, in Fig. 8 we show the 9th harmonic signals for CO2 at ˛ D 0ı , 45ı and 90ı . Because of the common -symmetry the main revival structures in the two cases are similar. The differences in details are due to the different rotational periods and parity of their rotational states. The signals for O2 are seen to be characterized by a full revival at Trev D 1=.2Bc/ D 11:6 ps and also by the fractional 12 and 14 revivals, like in N2 . But, in contrast to N2 , there is an DGGLWLRQDO 18 -revival (and correspondingly 38 , 58 and 78 revivals), for all the three polarization geometries; the same characteristics have been observed experimentally for O2 (e.g., [1, 2, 10]). The existence of the 18 -revival is due to the presence of the higher cosine-moments than ˝ ˛ hcos2 i .td / in the analytic signal, that couple the Raman-forbidden (J D ˙4) and the “anomalous” transitions (jJ j > 4) between the rotational states [13]. An examination of the second line of (15) or (16) shows that for ˛ D 0ı and 90ı the signals at the full, 12 and 14 revivals should be in RSSRVLWH phase, and that at the 18

)LJ  Calculated image of rotational coherent motion of O2 as obtained from the 19th harmonic dynamic signal in 3D representation as a simultaneous function of the delay td and the relative polarization angle ˛; shown on the right are sections of the signal for a sequence of ˛ at an interval of 10ı . Pump intensity I D 0:8  1014 W=cm2 , probe intensity, I D 1:7  1014 W=cm2 , pulse durations FWHM 40 fs, carrier wavelength 800 nm

Theory of Dynamic Imaging by HHG

47

)LJ  Calculated image of rotational coherent motion of CO2 as obtained from the 19th harmonic dynamic signal in 3D representation, as a simultaneous function of the delay td and the relative polarization angle˛; shown on the right are sections of the signal for a sequence of ˛ at an interval of 10ı . Pump intensity I D 0:8  1014 W=cm2 , probe intensity, I D 1:7  1014 W=cm2 , pulse durations, FWHM 40 fs, carrier wavelength 800 nm

)LJ  Calculated 19th harmonic dynamic signal for N2 ; ˛ D 0ı , 45ı and 90ı ; pump intensity I D 0:8  1014 W=cm2 , probe intensity, I D 1:7  1014 W=cm2 , duration 40 fs, and wavelength 800 nm, T D 200 ı K

48

F.H.M. Faisal and A. Abdurrouf

)LJ  Calculated 19th harmonic dynamical signal for O2 , for ˛ D 0ı , 45ı and 90ı ; pump I D 0:5  1014 W=cm2 , probe I D 1:2  1014 W=cm2 , pulse durations FWHM D 40 fs, wavelength 800 nm, T D 200 ı K

)LJ  Calculated 9th harmonic dynamical signal for CO2 , at ˛ D 0ı , 45ı and 90ı ; pump I D 0:56  1014 W=cm2 , probe I D 1:3  1014 W=cm2 , durations FWHM D 40 fs, wavelength 800 nm, T D 300 ı K

Theory of Dynamic Imaging by HHG

49

revival be LQ phase. A direct comparison of the calculations in Figs. 7 and 8 and the full calculations5 in Figs. 4, and 5 are in close agreement. The analytic signal for O2 add CO2 show that the coefficient of the 4th cosine-moment hcos4  i.td / would vanish for .35 cos4 ˛  30 cos2 ˛ C 3/ D 0, i.e., at ˛C D 70:12ı and ˛ D 30:55ı . Thus at these two polarization angles the 18 -revival seen in these figures would disappear and exhibit only the reduced number of fractional revivals as in the case of N2 . Also, the modulation of the 18 -revival would be a maximum for .@=@˛/.35 cos4 ˛  30 cos2 ˛ C 3/ D 0, i.e., for ˛ D 49:11ı . Thus the following behavior of the 18 -revival of O2 and CO2 (and generically for molecules with active MO of -symmetry) is predicted to hold: the fractional 18 -revival would reduce with increasing ˛  30:55ı , would increase between 30:55ı  ˛  49:11ı , reduce again between 49:11ı  ˛  70:12ı and finally increase for ˛  70:12ı. Next, we compare the relation between the orbital symmetry and the presence or absence of the magic angle in Fig. 9 for the three molecules N2 , O2 and CO2 by plotting the ˛-dependence of the signals at three different values of td in the vicinity of the 12 -revival time: at the “top” revival time or maximum alignment (solid curve), at the “base” or average alignment (dotted curve), and at the “antitop” revival time or perpendicular alignment (dashed-dot curve). In the case of g -symmetry (N2 ) the three signals cross at a common point that occurs as expected at the magic angle (54:7ı ) discussed above. For the case of -symmetry (O2 and CO2 ) the three

)LJ  Dynamic HHG signal vs. ˛, near td D 12 -revival for N2 (orbital symmetry g ), O2 (orbital symmetry g ), and CO2 (orbital symmetry g ); top alignment (solid line), average alignment (dashed line), antitop alignment (dashdotted line); pulse parameters are the same as in Fig. 6 5

As indicated above, the numerical calculations are performed using the usual HOMO. The calculations using the leading DQG the higher order terms for the signals are referred to as “full calculation.” It is useful to note that the leading term formulas (14), (15) and (16) are sufficient to provide excellent approximation for the signals – they are hardy distinguishable graphically in the scale of the figures presented from the results of the full calculation.

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F.H.M. Faisal and A. Abdurrouf

signals do not cross exactly at a common point, rather the pair-wise crossings of the three curves at the points “a,” “b,” and “c” occur in a region surrounding the magic angle. This is because of the presence of the additional but relatively weaker contribution of the 4th cosine-moment (compared to the contribution of the 2nd cosinemoment) in the analytic signal for -symmetry, that hinders the common crossing at the magic angle. This near common crossing characteristic in the -symmetric case is, for the ease of reference, dubbed [14] a crossover or crossing “neighborhood” (a-b-c) near the magic angle. We may note explicitly that the symmetry induced effects should tend to occur independently of the harmonic order. This is illustrated by the 9th and 19th harmonic signals (H9 and H19) for CO2 in Fig. 9. It also shows that there exists a phase reversal between the H9 and H19 signals in the crossing neighborhood, i.e., the point “a” lies above/below that of the points “b” and “c” in the two cases, respectively). This prediction of the phase reversal between the 9th and the 19th harmonics might also be tested by future HHG experiments with CO2 . It is interesting to note that without making detailed calculations but merely from the knowledge of linearity of a more complex organic molecule, one may predict the symmetry of the active orbital from the presence of the magic angle at 54:7ı (generic g symmetry), or the “crossing neighborhood” (generic  symmetry) or its absence (e.g., nonlinear) in the ˛ dependent HHG signal. To this end, we show a set of recent experimental data in Fig. 10 (after [8]), of dynamic HHG signals for the organic molecules, acetylene, ethylene and allene for three different delays. It is seen

)LJ  Dynamic HHG signal in rel. units (vertical axis) vs. relative polarization angle ˛ (horizontal axis), for the organic molecules (a) acetylene, (b) ethylene and (c) allene [8]. Note the presence of the “crossing neighborhood” near the “magic angle .54:5ı /, for the two linear molecules with orbital symmetry : acetylene (panel a) and allene (panel c), and its absence for ethylene (panel b), consistent with the present theory, after [8]

Theory of Dynamic Imaging by HHG

51

that a “crossing neighborhood” near the magic angle (54:7ı ) appears for acetylene (panel a) and allene (panel c); this is consistent with the known linear structure with active  orbitals for these two molecules. In contrast, the data for ethylene (panel b) shows neither a common crossing at the magic angle or a “crossing neighborhood” near it, indicating that ethylene is unlikely to be a linear molecule (probable structure, asymmetric top). The dynamic signals could also be analyzed with advantage in the frequency domain, i.e., the domain of FT of the signal vs. td . In Fig. 11 we show the FT for the dynamic signal of CO2 at the 19th harmonic. We note the presence of the following discrete series: (a) a most prominent series .22; 30; 38; 46; : : :/Bc, (b) a second prominent series .52; 60; 68; 76; : : :/Bc, (c) extending to the series .180; 196; 212; 228; : : :/Bc and (d) a minor series .8; 16; 24; 32; : : :/Bc. The appearance of these Fourier series can provide a more stringent tests for the theories of dynamical HHG signals of various molecules, since the discrete nature of the Fourier spectra are more amenable to accurate comparison than the dynamic signals themselves. We conclude by investigating a variation of the ultrashort pump pulse for inducing coherent rotational motions. This is the so-called sudden switch-off method in which a long pump pulse is initially allowed to evolve slowly (in ps scale) until its peak value is reached when it is switched-off suddenly (in fs scale). In this way one may expect that the modulation of the dynamical HHG signal could be HQKDQFHG as much as the maximum strength of the adiabatic signal at the peak of the pulse. In Fig. 12 we show the calculated dynamical signal (for N2 ) using a suddenly switched-off pump pulse (that evolves slowly to its maximum near 10 ps, is

)LJ  Fourier spectrum of calculated dynamic HHG signal for CO2 at 19th harmonic (H19); pump I D 0:531014 W=cm2 , probe I D 1:51014 W=cm2 , wavelength 800 nm, pulse durations FWHM D 40 fs, T D 300 ı K. Predicted spectral series: (a) series .22; 30; 38; 46; : : :/Bc, (b) series .52; 60; 68; 76; : : :/Bc, (c) series .180; 196; 212; 228; : : :/Bc and (d) series .8; 16; 24; 32; : : :/Bc

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F.H.M. Faisal and A. Abdurrouf

)LJ  Dynamic HHG in N2 under sudden switch-off of an adiabatic pump pulse; panel (a) signal vs. delay, (b) signal at three given delays vs. relative polarization angle ˛, (c) Fourier spectrum of the dynamic signal in (a). Note the “top signal” occurs at full revival with the same amplitude as when the pulse is suddenly switched-off

then switched off within 24 fs.). The results are shown for three different relative polarization angles ˛ D 0ı , 45ı and 90ı . Panel (a) shows the dynamic HHG signal. It adiabatically follows the pulse envelope to its maximum value at 10 ps, then exhibits the characteristic revival structure typical of N2 , i.e., a fractional 14 -revival is followed by a 12 -revival, a 34 -revival and the full revival (“top signal”). The full revival at “top-signal” (for ˛ D 0ı ) equals the maximum of the adiabatic signal at the pulse maximum (near 10 ps). We note that the “modulation depth,” measured by the difference between the “top-signal” and the “antitop signal” (near above 14 ps), also becomes the largest. In panel (b) we show the ˛ dependence of the “top,” “antitop,” and the “average” signals which behave as for the case of an ultrashort pulse (cf. Fig. 3). In panel (c) is shown the Fourier spectrum of the dynamic signal of panel (a). It is seen to be characterized by two prominent series .6; 14; 22; 30; : : :/Bc and .10; 18; 26; 34; : : :/Bc, a weak series .20; 28; 36; 44; : : :/Bc, and the remnant of an “anomalous series” .: : : ; 8; : : : ; 16; : : :/Bc; these series for N2 we had identified and fully analyzed elsewhere (cf. Fig. 1 in [13]). Finally, we note that all the salient results of the present theory have been compared directly with recent experiments for N2 and O2 , both in the time domain [10], and also in the frequency domain [13,16], that are in excellent agreement; the additional results including that for CO2 discussed above provide extra opportunities for further tests of the theory.

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 6XPPDU\ A general theory of dynamic HHG signal for real time imaging of coherently rotational motions of linear molecules is given. Simple analytic formulas for the simultaneous dependence of the signal on the relative polarization angle, ˛, and the time delay, td , between a pump and a probe pulse, are presented. The result of HHG signals of the molecules N2 , O2 and CO2 are analyzed in time and frequency domains. A number of critical values of ˛ are predicted that are correlated with the orbital symmetry of the molecule, and hence could be used effectively in identifying the latter, in problems of molecular orbital reconstruction from HHG data. The salient theoretical results show excellent agreement with the available experimental results and provide a unified interpretation of the same. Additional results discussed above would allow further tests of the theory.

5HIHUHQFHV 1. J. Itatani, D. Zeidler, J. Levesque, M. Spanner, D.M. Villeneuve, P.B. Corkum, Phys. Rev. Lett. , 123902 (2005) 2. T. Kanai, S. Minemoto, H. Sakai, Nature , 03577 (2005) 3. K. Miyazaki, M. Kaku, G. Miyaji, A. Abdurrouf, F.H.M. Faisal, Phys. Rev. Lett. , 243903 (2005) 4. J. Levesque, J. Itatani, D. Zeidler, H. P´epin, J.C. Kieffer, P.B. Corkum, D.M. Villeneuve, J. Mod. Opt. , 185 (2006) 5. S. Patchkovskii, Z. Zhao, T. Brabec, D.M. Villeneuve, Phys. Rev. Lett. , 123003 (2006) 6. S. Baker, J.S. Robinson, C.H. Haworth, H. Teng, R.A. Smith, C.C. Chirila, M. Lein, J.W.G. Tisch, J.P. Marangos, Science , 424 (2006) 7. N.L. Wagner, A. Wuest, I.P. Christov, T. Popmintchev, X. Zhao, M.M. Murname, H.C. Kapteyn, Proc. Natl. Acad. Sci. U.S.A. , 13279 (2006) 8. N. Kajumba, R. Torres, S. Baker, J.S. Robinson, J.W.G. Tisch, J.P. Marangos, J.G. Underwood, R. de Nalda, C. Altucci, R. Velotta, W.A. Bryan, I.C.E. Turcu, in &HQWUDO /DVHU )DFLOLW\ $QQXDO 5HSRUW  (Rutherford Appleton Lab., Chilton, 2006), p. 80 9. R. Torres, N. Kajumba, J.G. Underwood, J.S. Robinson, S. Baker, J.W.G. Tisch, R. de Nalda, W.A. Bryan, R. Velotta, C. Altucci, I.C.E. Turcu, J.P. Marangos, Phys. Rev. Lett. , 203007 (2007) 10. K. Yoshii, G. Miyaji, K. Miyazaki, Phys. Rev. Lett. , 183902 (2008) 11. H. Stapelfeldt, T. Seideman, Rev. Mod. Phys. , 543 (2003) 12. M. Lein, J. Phys. B , R135 (2007) 13. F.H.M. Faisal, A. Abdurrouf, K. Miyazaki, G. Miyaji, Phys. Rev. Lett. , 143001 (2007) 14. F.H.M. Faisal, A. Abdurrouf, Phys. Rev. Lett. , 123005 (2008) 15. A.D. Bandrauk, S. Chelkowski, S. Kawai, H. Lu, Phys. Rev. Lett. , 153901 (2008) 16. A. Abdurrouf, F.H.M. Faisal, Phys. Rev. A , 023405 (2009) 17. A.D. Bandrauk, S. Chelkowski, H. Lu, J. Phys. B , 075602 (2009) 18. F.H.M. Faisal, Theor. Chem. Acc. , 175 (2010) 19. A. Becker, F.H.M. Faisal, J. Phys. B , 1 (2005)

7KH RPDWUL[ &DOFXODWLRQV RI 2ULHQWDWLRQ DQG &RXORPE 3KDVH (IIHFWV LQ (OHFWURQ±0ROHFXOH 5H &ROOLVLRQV Alex G. Harvey and Jonathan Tennyson

$EVWUDFW Electron recollision in strong laser fields is usually studied with oriented molecules. This introduces orientation effects into the recollision problem which are generally not present in usual treatments of electron–molecule collisions. In addition this collision occurs with a molecular ion which means that the dominant electron– molecule interaction is the long-range Coulomb potential, which competes asymptotically with the strong laser field. Different workers have performed treatments varying between the complete inclusion of all asymptotic Coulomb effects to their complete neglect. Three possible treatments of the Coulomb problem are explored using H2 and CO2 as prototypical systems. Calculations based on R-matrix studies of the (re-)collision, which neglect the effects of the laser field, show that inclusion of the complete Coulomb interaction leads not only to the well-known singularity problems for forward scattering but also leads to the washing out of much of the detailed, angular structure in the differential cross section of the oriented molecules.

 ,QWURGXFWLRQ The detailed treatment of the dynamics of a molecule trapped in a strong laser field presents a number of theoretical challenges. This is particularly true when considering electron recollision as for this case it is necessary to treat effects due to the field and the physics of the recollision process itself. In the context of electron–molecule collisions, the recollision problem actually involves the ionized electron recollid-

Alex G. Harvey Max-Born-Institute, Max-Born-Strasse 2A, 12489 Berlin, Germany, e-mail: "MFY)BSWFZ! NCJCFSMJOEF Jonathan Tennyson Department of Physics and Astronomy, University College London, London WC1E 6BT, UK, email: KUFOOZTPO!VDMBDVL A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_5, © Springer Science+Business Media, LLC 2011

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A.G. Harvey and J. Tennyson

ing with the residual parent ion [1]. The theoretical treatment of field-free electron molecule collisions is itself a complex problem and an active area of research. The R-matrix method [2] has proved highly successful in treating field-free electron-molecule collisions and is particularly well-suited to treating electron collisions with molecular ions. The method has been used to study molecular photoionization problems in weak [3] and intermediate [4, 5] fields, but has so far not been adapted for the strong field problem. The treatment of recollisions with molecules in strong fields introduce a number of effects which are not present in standard scattering calculations. The one we consider here is the fact that it is usual in strong field experiments to work with oriented molecules and therefore recollision occurs against a target ion with a definite orientation. This introduces orientation effects which are routinely averaged out when considering collisions with field-free molecules which are assumed to be oriented completely randomly. In this chapter we consider how orientation effects alter the results of these calculations. This extends work reported by us previously [6]. The particular issue we discuss here is the role of the long-range Coulomb potential as felt by the (re-)scattered electron in these recollisions. Short-range potentials that decay asymptotically faster r 2 are easily dealt with in the context of rescattering. The situation is less straightforward for the Coulomb potential which depends on r 1 . In a field free collision the range of the Coulomb potential is infinite which leads to an infinite cross section in the forward direction in the asymptotic behavior of the scattering problem. Additionally the well-known Coulomb phase is introduced due to the use of Coulomb functions instead of Bessel functions to represent the radial part of the scattering wave function. This phase can have a profound influence on on the angular distribution of scattered electrons as it changes the relative phase between partial wave components of the scattered wave, leading to constructive or destructive interference. In a strong laser field the field dominates the asymptotic behavior of the scattering and the rescattering event is in any case not actually a full collision; under these circumstances the precise role of the infinity in the forward direction and the Coulomb phase needs to be explored. All calculation presented below are performed with the R-matrix method but in fact the reorientation issues considered below are common to all electron-molecule scattering procedures. For this reason we will not present a detailed consideration of the R-matrix method and will instead direct readers to a recent review of electronmolecule scattering using the R-matrix method by one us [7].

 *HQHUDO &RQVLGHUDWLRQV C We discuss electron (re-)collision with aligned HC 2 and CO2 for a range of energies and alignment angles. These are high symmetry systems whose neutral ground states are both of 1 ˙gC symmetry. It therefore worthwhile to start by considering some effects due to symmetry. In the special case where the orientation of the molecule is such that the molecular axis is perfectly aligned parallel to the laser field

The R-matrix Calculations of Orientation and Coulomb Phase Effects

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polarization, dipole selection rules restrict the overall symmetry of the scattering wave function. During ionization and subsequent acceleration in the laser field absorption of laser photons can only change the inversion symmetry of the overall wavefunction; it is clear that linearly polarized light cannot change the angular momentum of a wave function around an axis collinear with itself. This means for example that in the ionization from the CO2 1 ˙gC ground state leaving the ion in its 2 ˘g groundstate the continuum electron must have g or u symmetry and must be rotating in the opposite sense to the ion in order to preserve the component (along the molecular axis) of the angular momentum of the total wavefunction. Taking a group theory perspective we see that the dipole operator has u character for this alignment. For any other alignment however it also has a u component and so absorption of multiple photons can lead to many different symmetries. With this in in mind we restrict the symmetry of our overall scattering wavefunctions to 1 ˙uC and 1 C ˙g corresponding to absorption of an odd or even number of photons respectively during the ionization process. We do this even in the case of nonparallel alignment, with the desire to keep the model simple in the examination of Coulomb and orientation effects. Our techniques is however fully capable of calculating the other relevant symmetry components of the wavefunction. Finally it should be noted that in a real experiment perfect alignment is not achievable and further averaging over a range molecular orientations would be required. One technical issue is that the CO2 scattering calculations were performed using the UK R-matrix polyatomic code [8, 9], which only works in Abelian point groups. It was therefore necessary to transform our T -matrices from D2h to D1h symmetry [10]. To do the transformation it was necessary to rerun the scattering calculation, so that both degenerate target ground states were calculated (2 B2g and 2 B3g ). Furthermore, as discussed below, the lowering of the symmetry introduces a mixing of partial waves. This is particularly important for CO2 where ˙ and  symmetry scattering can be important. In the remainder of this section we give a brief description of our method for introducing orientation to the UK R-matrix code. We describe the more complex case of D1h molecules, for C1v molecules we simply need to suppress the summation over gerade and ungerade symmetries, denoted  below, and relax the parity restrictions on the initial and final electron angular momenta, denoted li and lj respectively below. We consider an incident electron with kinetic energy Ei D 12 ki2 and spin projection msi on the molecular axis from a neutral N electron target molecule of electronic state i (note: both ki and U are in the lab frame). The target electronic wave function in channel i is specified by the quantum numbers fi ; Si ; MSi ; i g. i is the projection of the angular momentum on the molecular axis, Si is the spin, MSi is the spin projection on the molecular axis, i is the gerade/ungerade nature of the state and msi is the spin projection of the incident electron. The Euler angles (˛; ˇ; ) specify the orientation of the molecule. f; S; MS ; g are the quantum numbers of the N C 1 system of the target plus the scattering electron and are invariant throughout the collision. The molecular frame T -matrix calculated by the UK R-matrix codes is related to the lab frame scattering

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amplitude in the following manner [10] C

Fij .; I ˛; ˇ;  / D fi ./ıi li jlj

p  C p i ki kj

X

N

N

ili lj

q

2lNi C 1e

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lNi  T{N S C  D0  .˛; ˇ;  / lNi |N lNj i i

j

(1)

where the Coulomb phase is given by li D arg  .li C 1 C ii / ZN i D  ki

(2) (3)

and Ci=j D hSMS jSi=j MSi=j 12 msi=j i are Clebsch–Gordan coefficients. The differential cross section is then ˇ d ˇˇ kj D jFij j2 ˇ d˝ ij ki    C .ad d /  kj .add/ 2 C 2 C  .add/ D jfi j ıi li jlj C jFij j C fi Fij C fi Fij ıi li jlj ki   h i  kj .add/ .ad d /  D jFij j2 C ıi li jlj jfiC j2 C 2 RE fiC Fij ; (4) ki where Fij.add/ is the second term on the right-hand side of (1). We note that Fij.add/ differs from the oriented scattering amplitude for a neutral molecule only in the the i. C / term e li lj involving the Coulomb phases. Several models for calculating aligned differential cross sections are examined. The first (Model I), which has been explored by us previously [6], considers neither the pure Coulomb scattering and interference term (that exist in the elastic channel), nor the Coulomb phase in the short-range term [6]. The second (Model II), following Sheehy et al. [11] includes the Coulomb phase in the short range term but not the Coulomb scattering and interference terms. In the third model (Model III) the effects of adding the entire Coulomb scattering and interference terms are considered in full. This approach, which was used recently by Chen et al. [12], has issues with infinite cross sections for forward scattering although there do appear to be regularization procedures for treating this issue [13]. Results are presented for each of these models in the illustrative calculations presented below.

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 5HVXOWV IRU 0ROHFXODU +\GURJHQ The T -matrix elements obtained previously [1] were used for all the calculations. Only the states of 1 ˙uC and 1 ˙gC symmetry are considered, which is sufficient for the case of parallel alignment. Figure 1 shows the total cross section for electron rescattering from parallel aligned H2 against incident electron energy using Model I. The cross sections are approximately 3–4 times greater in magnitude than the orientationally averaged case, this is what we might expect taking the simplistic view that the electron takes a longer path through the molecule and so has more opportunity to interact with it. Compared to previous, orientationally averaged calculations [1, 14], they show similar resonance structure up to the first electronic excitation threshold; however the series of Feshbach resonances converging on the second excited state are heavily suppressed. 0RGHO , Figure 2 shows parallel aligned differential cross sections for a range of nonresonant energies. With this alignment, due to the symmetry of the system, the differential cross section is a function of  alone. We see that 1 ˙gC symmetry differential cross sections are markedly different in shape than the orientationally averaged case. All the differential cross sections are symmetric about 90ı ; this is to be expected as elastic scattering of single total symmetry will have a continuum described entirely by a sum of either gerade or ungerade spherical harmonics depending on the gerade/ungerade symmetry of the target state; these sums are symmetric around 90ı . For energies below the first resonance at, 12:87 eV, we can see that the differential cross sections have strong peaks in the forward and backwards direction and slightly lower peaks at 90ı . Interestingly at 15:5 eV, in between the first and second resonances, we see a significant change in shape that appears to be an inversion, 20

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with minima in the forwards and backwards direction. This may be a phase effect due to passing through the first resonance. Above the first excited state we again get strong forwards and backwards scattering, however the peak at 90ı is significantly less pronounced. The 1 ˙uC symmetry shows strong forwards and backwards scattering, unlike 1 ˙gC symmetry, there is no sideways scattering and no significant shape difference between the aligned and orientationally averaged cases, or change of shape as a function of energy.

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0RGHO ,, Figure 3 shows the same plots as Fig.2 but now with the Coulomb phase added. We can see now that the addition make a marked qualitative difference to the differential cross section. For the 1 ˙gC symmetry, for energies less than 15:5 eV the phase (which has an energy dependence) tends to enhance the cross section in the forward and backwards directions and suppress scattering at 90ı . The effect is extreme in the orientationally averaged case which has been almost inverted when compared to its Model I counterpart, in the parallel aligned case the central peak (at 90ı ) is strongly suppressed. The 19:5 eV cross section, which appeared inverted in Model I is still inverted and the central peak has been enhanced. The differential cross section at

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21 eV now also appears to be inverted. This however must be due to the Coulomb phase and not from passing through a resonance as the Model I cross section is not inverted. So we can see that the addition of a Coulomb phase makes the shape of these differential cross sections strongly energy dependent, as might be expected due to the energy dependence of the Coulomb phase. The 1 ˙uC symmetry for the orientationally averaged case has undergone a less dramatic transformation, with forward and backwards scattering being slightly suppressed. For the parallel aligned cases the major change is again suppression of the forward and backwards direction and the addition of a slight maximum between 30–35ı, these shape appears to be stable over the energy range considered. 0RGHO ,,, Addition of the Coulomb scattering and interference term, Fig. 4, largely obscures the features due to short range forces, especially for small angle scattering. This is as one might expect on the consideration that the Coulomb scattering term is significantly higher than the short range forces term across the majority of the angular range. One way to extract information from these cross sections is to normalize them to the pure Coulomb differential cross section at the same energy. Figure 5 shows the differential cross sections normalized in this fashion, they show the proportional effect the short range force and interference terms have on the the pure Coulomb differential cross section. One common feature to all of the normalized cross sections is that the proportional deviation from Coulomb scattering is low at small angles and increases for larger angles. This is to be expected as the scattering due to short range forces does not vary greatly in magnitude over the angular range in comparison to Coulomb scattering which changes by many orders of magnitude. We also see that for large angle scattering, with  > 90ı , that the angular structure seen in Model II is to a certain extent preserved. In Fig. 6 we have plotted total cross section versus alignment angle ˇ. We see that both the 1 ˙gC and 1 ˙uC components are strongly peaked for parallel alignment,

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with 1 ˙gC having an additional smaller peak at perpendicular alignment while 1 ˙uC is zero. Due to the 2 ˙gC ground state of the cation we expect the contribution of higher symmetries for nonparallel alignments to be relatively minor. These observations justify the use of parallel alignment in rescattering experiments on H2 . It is apparent that the addition of a Coulomb phase makes no difference to the integral cross section.

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)LJ  Total cross section as function of alignment angle ˇ for the aligned electron–HC 2 collision problem at a range of energies:top figures Model I, bottom figures Model II, left figures 1 ˙gC total symmetry, right 1 ˙uC symmetry.

 5HVXOWV IRU &DUERQ 'LR[LGH The T -matrix elements obtained previously [1] were used for all the calculations. Figure 7 compares the orientationally averaged D1h total cross sections (calculated in Model I) for 1 ˙gC and 1 ˙uC symmetries with the corresponding results for 20

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1

Ag and 1 B1u from the original D2h calculations. These are the D2h symmetries which contain the 1 ˙ C contributions. We see that the 1 Ag and 1 B1u D2h symmetry cross sections are approximately three to four times larger than their respective D1h symmetry counterparts, 1 ˙gC and 1 ˙uC . This is because the D2h cross sections contain significant contributions from the 1  components. The 2 ˘g target and  continuum couple to give 1 ˙ and 1  total symmetry and both these symmetries are contained in the corresponding D2h T -matrix elements. It can be anticipated that cross sections of 1 ˘g total symmetry will be dominant at low collision energies as they couple to a g continuum. It should be noted that the resonance structure is unchanged, we still see infinite series of Feshbach resonances below each threshold [1], a higher energy resolution would resolve more of these. Figure 8 shows total cross section for 1 ˙ C symmetries as a function of the alignment angle ˇ. We immediately see that for both symmetries we get zero total cross section for the parallel aligned case and therefore zero differential cross sections; this is a direct consequence of the 2 ˘g symmetry of the COC 2 target. It is important to note that, for alignments other than parallel, it is necessary to sum over higher symmetries to give the correct differential cross sections, these are expected to be significant due to the 2 ˘g ground state of the cation. We see that for the 1 ˙gC component the cross sections have a maxima between 50–55ı and 140–145ı, depending on energy, and are zero at 90ı and 0ı . 4 3.00eV 6.00eV 9.00eV

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For 1 ˙uC there is a maximum at 90ı in contrast with 1 ˙gC , and there are smaller peaks between 35–40ı and 125–130ı. These observations agree with the recent study on HHG with CO2 by Mairesse et al. [15]. Again there is no difference between Model I and Model II integral cross sections for 1 ˙gC . However there is a slight depression of the central peak and raising of the side peaks for the 1 ˙uC . 0RGHO , Figure 9 shows the 1 ˙ C part of the differential cross section given in polar coordinates with r being the magnitude of the differential cross section and  measured from the z-axis. The differential cross sections are symmetrical around the molecular axis, this suggests that the zx plane, containing the molecule, is an appropriate plane in which to examine them. We see that for 1 ˙gC the differential cross section is zero parallel and perpendicular to the molecular axis. 1 ˙uC is also zero in the parallel direction but has a small peak perpendicular to the molecular axis and is approximately 4 times greater in magnitude than 1 ˙gC . 0RGHO ,, The addition of the Coulomb phase makes little difference to the 1 ˙gC differential cross sections across the energy range studied in contrast to the H2 case. For the 1 ˙uC symmetry we see a strong enhancement of scattering perpendicular to the molecular axis and a corresponding reduction in the maxima in either directions. The direction of the maxima and minima are stable across the energy range considered. Further work to extend the energy range by including more target states and using the molecular R-matrix with pseudo state method [16, 17] for energies near and above the ionisation threshold would allow us to further examine the effect of the Coulomb phase with increasing energy. 0RGHO ,,, Figure 10 shows differential cross sections normalized to the Coulomb cross section, for scattering angles in the backwards direction. For this model the polar plots of the last two models have been abandoned as they would show small deviation from a unit circle. The backward direction was chosen as the deviation from Coulomb scattering is small for small angle scattering. Examining the figure we see that zero scattering due to short range forces in the direction of the molecular axis is preserved (corresponding to a value of unity) in all cases, as one would expect. For the 1 ˙gC the position of maxima is shifted slightly to larger scattering angles. The Model II maxima at 150ı is less than unity in Model III, this is due to the relative phase of the Coulomb scattering amplitude and the amplitude due to short range forces which controls whether the interference term is additive or subtractive. Comparing

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Model II to Model III for 1 ˙uC we see that the maxima are again preserved as in the 1 C ˙g case, however maxima separated by nonzero local minima become harder to resolve as the minima become much less pronounced.

 &RQFOXVLRQV Integral and differential cross sections for rescattering from parallel aligned H2 and CO2 are presented which consider three different models for treating the effects of the long-range Coulomb potential. In H2 for all models, the results show that for the 1 ˙gC symmetry both integral and differential cross sections are greater in magnitude and that differential cross sections also differ in shape in comparison to the nonaligned situation. We see strong forward and backwards scattering in contrast to the minima at these direction in the unaligned case. We also see that the shape of the differential cross section can change significantly before and after a resonance. The addition of a Coulomb phase has a marked effect on the cross sections, with a tendency to enhance or suppress forwards, backwards and sideways scattering dependent on the energy, due to constructive or destructive interference between partial waves. Inclusion of the Coulomb scattering and interference terms obscures much of the angular structure seen in the differential cross sections, however through normalization to pure Coulomb scattering some of the structure can be recovered giving qualitatively similar shapes for scattering angles greater than 90ı For CO2 we find that parallel alignment gives zero cross sections due to short range forces indicating a need to perform this type of experiment at nonparallel alignments. We also see that perpendicular alignment gives a zero cross section for 1 C ˙g and a maximum for 1 ˙uC , however further work to include higher symmetries is required to gain an accurate picture of nonparallel alignments. Again examining the differential cross sections we see that the addition of the Coulomb phase has an effect of suppressing and accentuating the various maxima. In this case the strongest effect was seen for the 1 ˙uC symmetry. The 1 ˙gC symmetry showed little change with the addition of the Coulomb phase. Inclusion of the Coulomb scattering and interference terms has a greater effect in obscuring the structural information present in scattering from short range forces than in H2 . We note that the latest models of recollision find it important to explicitly allow for electron impact electronic excitation of the target ion [18]; this process is automatically included in our scattering treatment. This study ignores the undeniably important effect of the external laser field on the scattering electron. It is straightforward to add a weak field using our current techniques [3] but adding a strong field requires a time dependent treatment. There are two possible approaches to overcoming the omission of the laser field. One is to incorporate the laser effects within the R-matrix approach. We note that the formalism for this was given sometime ago [19] but has only recently been implemented for the atomic case [20]. Generalization of this to molecular systems would give a

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consistent treatment of the problem but remains both technically and computationally challenging. Work in this direction is just commencing. The other possibility is to incorporate our field free scattering calculation in a strong field model. The Coulomb singularity in the forward direction is a consequence of the breakdown of the validity (in the forward direction) of the asymptotic division of the scattering wavefunction into an incident and scattered wave of infinite extent. In other words the singularity is not in the scattering wave function but in the asymptotic form. Of course infinite waves of precisely defined momentum are unphysical, in strong field experiments the incoming electron wave packet has dimensions measured in tens of Bohr. This is precisely the point at which a connection to a strong field model can be made. One option is to regularize the cross section as discussed in [13], using the information about the incident wave packet from a strong field model. The R-matrix method gives us the full scattering wavefunction [7], this suggests another approach, that is to use the scattering wavefunctions to create an incident wave packet of size and shape informed by a strong field model, then calculate the flux at some appropriate distance from the molecule. This would have the benefit of allowing ideas of electron holography to be explored, which rely on the interference of incident and scattered parts of the wavepacket explicitly excluded in the standard definition of a cross section. $FNQRZOHGJHPHQWV This work was performed as part of an EPSRC funded project to study ultrafast molecular imaging. We acknowledge helpful discussions with all members of the consortium and in particular Jon Marangos, Misha Ivanov, Olga Smirnova and Jonathan Underwood.

5HIHUHQFHV 1. A. Harvey, J. Tennyson, J. Mod. Opt. , 1099 (2007) 2. P.G. Burke, K.A. Berrington (eds.), $WRPLF DQG 0ROHFXODU 3URFHVVHV DQ RPDWUL[ $SSURDFK (IOP Pub., Bristol, 1993) 3. J. Tennyson, C.J. Noble, P.G. Burke, Int. J. Quantum Chem. , 1033 (1986) 4. J. Colgan, D.H. Glass, P.B. K. Higgins, Comput. Phys. Commun. , 27 (1998) 5. P.G. Burke, J. Colgan, D.H. Glass, K. Higgins, J. Phys. B , 143 (2000) 6. A.G. Harvey, J. Tennyson, J. Phys. B , 095101 (2009) 7. J. Tennyson, Phys. Rep. , 29 (2010) 8. L.A. Morgan, J. Tennyson, C.J. Gillan, Comput. Phys. Commun. , 120 (1998) 9. L.A. Morgan, C.J. Gillan, J. Tennyson, X. Chen, J. Phys. B , 4087 (1997) 10. A.G. Harvey, Electron re-scattering from aligned molecules using the R-matrix method. Ph.D. thesis, University College London (2010) 11. B. Sheehy, R. Lafon, M. Widmer, B. Walker, L.F. DiMauro, P.A. Agostini, K.C. Kulander, Phys. Rev. A , 3942 (1998) 12. Z. Chen, A.T. Le, T. Morishita, C.D. Lin, Phys. Rev. A , 033409 (2009) 13. V.G. Baryshevskii, I.D. Feranchuk, P.B. Kats, Phys. Rev. A , 052701 (2004) 14. J. Tennyson, At. Data Nucl. Data Tables , 253 (1996) 15. Y. Mairesse, J. Levesque, N. Dudovich, P.B. Corkum, D.M. Villeneuve, J. Mod. Opt. , 2591 (2008) 16. J.D. Gorfinkiel, J. Tennyson, J. Phys. B , L343 (2004) 17. J.D. Gorfinkiel, J. Tennyson, J. Phys. B , 1607 (2005)

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18. S. Graefe, M.Y. Ivanov, J. Mod. Opt. , 2557 (2008) 19. P.G. Burke, V.M. Burke, J. Phys. B , L383 (1997) 20. H.W. van der Hart, M.A. Lysagt, P.G. Burke, Phys. Rev. A , 043405 (2007)

9LVXDO $QDO\VLV RI 4XDQWXP 3K\VLFV 'DWD Hans-Christian Hege, Michael Koppitz, Falko Marquardt, Chris McDonald, and Christopher Mielack

$EVWUDFW During the past two decades data visualization has matured as an own sub-discipline in computer science. Its methods are successfully applied in almost all areas of science, engineering, and medicine, in order to depict and visually analyze data—both from experiment and simulation. The goal of data visualization is to achieve a better understanding of data by intuitive, perceptually efficient and interactively steerable depictions of the data. For this specific data analysis methods are combined with visualization techniques that utilize modern computer graphics. Quantum physics, however, so far remained largely omitted as application area, in particular due to the high dimensionality of the phenomena. However, the situation is not hopeless; on the contrary, there are many ways to visualize quantum mechanical phenomena. In this paper, this will be demonstrated by means of visualizations of simulation data from quantum chemistry and high-harmonic generation.

 ,QWURGXFWLRQ In the past, research in quantum physics and quantum chemistry primarily aimed at improving our understanding of fundamental and basic quantum effects. SophisHans-Christian Hege Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, e-mail: IFHF![JCEF Michael Koppitz Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, e-mail: LPQQJU[![JCEF Falko Marquardt Department of Mathematics, FU Berlin, Arnimallee 6, 14195 Berlin, Germany, and Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, e-mail: NBSRVBSEU![JCEF Chris McDonald University of Ottawa, 150 Louis Pasteur, Ottawa, Canada, e-mail: DNDEP!VPUUBXBDB Christopher Mielack Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, e-mail: NJFMBDL![JCEF A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_6, © Springer Science+Business Media, LLC 2011

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ticated experiments and computations have been performed, dealing mainly with quantum systems with only few quantum degrees of freedom. Research on fundamental questions of quantum physics, particularly since its resurgence in the nineties, continuously reveals exciting results about fundamental issues, like nonlocality, quantum correlations, decoherence and entanglement. In parallel physicists, chemists and engineers, who are more practically oriented, are dealing with increasingly complex quantum objects, trying to handle and utilize quantum effects in applications, e.g., in quantum information processing, quantum communication and molecular sciences. The impressive development of experimental techniques and the substantial advancement of simulation techniques lead to increasingly complex data that need to be understood. This is, where computer-based visualization and visual data analysis comes in. Visualization of quantum mechanical phenomena, however, has mostly been used for didactical purposes in quantum physics courses. Here typically oneor two-particle systems are considered, primarily living in one space dimension. Thus simple plots and coloring schemes suffice. The visualization of more realistic quantum dynamics beyond the level of the pioneering textbooks [1–3], however, is an rarely tilled field—mainly due to the high number of degrees of freedom and the associated high dimensionality of the data that make visualization difficult. On the other side, in quantum physics H[SHULPHQWDO visualization techniques are being developed. It took more than eighty years from its discovery until the most fundamental object in quantum mechanics, the wave function, could be determined and visualized experimentally [4]. Now, new experimental techniques aiming at visualization of quantum phenomena are being developed—most prominently molecular orbital tomography based on high harmonic generation [5], which allows us to measure the amplitude of an electronic wave function. More sophisticated approaches are underway, see, e.g., the recent work in [6–8]. Although techniques for intramolecular imaging are still in their infancy, it is foreseeable that specific techniques of image reconstruction and image analysis need to be developed to extract as much information as possible from such experimental data. Thus almost all classical image sciences become involved in quantum physics. In this paper we will shortly present an overview on the image sciences and then particularly on interactive data visualization and its role in quantum physics.

 ,PDJH 6FLHQFHV DQG 9LVXDOL]DWLRQ  7KH 9DOXH RI ,PDJHV Information and knowledge can be represented in different ways; two major classes are verbal and nonverbal conceptualizations. Examples of the latter are visual representations, formal systems, and mathematical theories. With each knowledge representation specific mental operations are associated: verbal, visual and formal reason-

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ing. While philosophers, writers and historians typically are strong verbal thinkers, logicians and algebraists typically are strong formal thinkers. By contrast, geometricians, physicists and engineers tend to be strong visual thinkers. Famous examples of outstanding visually thinking physicists were Faraday, Maxwell, Boltzmann, Einstein and Bohr. From this attribution it can be concluded that images are an important if not indispensable means for science. This has at least three reasons: 1. Images serve as a major source of information due to the wide variety of imaging techniques that nowadays cover almost all length scales. Just think of the many types of microscopy1 and telescopy,2 as well as of techniques like tomography3 and holography.4 With all these imaging techniques classical fields of informatics, like image reconstruction, image processing and image analysis are connected, and contribute methodically. 2. Data visualizations and graphic depictions are essential in knowledge creation: they help to understand complex phenomena, to create hypotheses, to form and to harmonize mental images and thereby to achieve a consensual understanding. 3. Images support communication between researchers and provide indispensable help in education as well as in promotion and popularization of science.

 ,PDJH 6FLHQFHV The universe of modern sciences dealing with digital images and the relations between its various branches can be explained easily on the basis of the diagram depicted in Fig. 1. We distinguish between the following types of data: non-visual data and visual data (images). Methods that get images as input and produce images as output belong to the domain of LPDJH SURFHVVLQJ. Methods that take as input nonvisual data and produce as output images belong to LPDJH V\QWKHVLV, one branch if which is GDWD YLVXDOL]DWLRQ; another branch, dealing specifically with the techniques of image generation is FRPSXWHU JUDSKLFV. Methods that get images as input, analyze these and produce non-visual representations of the image content, belong to LPDJH DQDO\VLV, which belongs to the field of FRPSXWHU YLVLRQ. Image acquisition and image processing often are summarized as LPDJLQJ.

1 Types of microscopes used today: optical, electron and scanning probe with more then 20 subtypes (from atomic force microscopes to scanning SQUID microscopes). 2 More than ten types of telescopes are used today: telescopes that detect electromagnetic radiation (radio, sub-millimetre, far-infrared, infrared, visible spectrum, ultraviolet, X-ray and Gamma-ray telescopes) or high energy particles (cosmic-ray, neutrino and energetic neutral atom telescopes), and, maybe in the future also gravitational wave telescopes. 3 More than 25 types of tomography are used today, ranging from quantum tomography to acoustic tomography (on geophysical scale) and neutral hydrogen tomography (on cosmological scale). 4 Types of holography used today: IR, visible light and X-ray, acoustic, as well as electron, neutron, and atomic holography.

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)LJ  The universe of images sciences and the relations among them.

 'DWD 9LVXDOL]DWLRQ Since understanding needs mental images, it is not an accidental coincidence that the art of scientific illustration and the art of systematically describing natural science started more or less with one person—the Italian Leonardo da Vinci (1452–1519). The first and in many respects unequaled visualizations for the purpose of better scientific understanding were created by him. As an intellectual genius and outstanding visual thinker with exceptional crafting and graphical abilities he used depictions to aid understanding. For instance, he developed techniques for representing flow patterns with the aim to classify and better understand flow phenomena, see Fig. 2. In his nature studies he did not look just at the visible, directly perceptible; instead he

)LJ  Self-portrait and sketches depicting various flow phenomena by Leonardo da Vinci (1452 –1519). Source: IUUQXXXESBXJOHTPGMFPOBSEPPSH

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tried to grasp the inherent laws behind the phenomena and then to produce graphical illustrations that show also invisible things. Aiming to explain, e.g., a cannon shot, Leonardo sketched several drawings showing how shrapnels leave the gun barrel, fly in curved trajectory and explode near the target. Similarly, he constructed explanations of the birds’ flight. What he showed here according to the kinetic principle, on which also film is based, cannot be substantiated solely by plain observation. This step from the reproduction of the purely visible to that of the natural laws constitutes a turning point in the history of understanding of nature. With all these accomplishments, Leonardo da Vinci is considered as the father of scientific visualization. Today, in the era of digital computers, LQWHUDFWLYH YLVXDOL]DWLRQ has become a standard tool. With interactive visually-based analysis, cf. Fig. 3, in contrast to batch processing, goals can be achieved that cannot be automated. If we have ‘the-man-inthe-loop, ’ i.e., if human cognition, human expertise and unformalized know-how is needed for the analysis, interactivity is necessary. This is the case, if we are seeking overview, if no definite questions are (yet) available, if the features to be identified are qualitative and not (yet) quantifiable, if the data are (yet) too complex for automated analysis, if unknown facts, relations, or correlations shall be discovered, or if meaningful hypotheses shall be created. Today, interactive visualization is used to study scientific phenomena, analyze data, visualize information, and to explore large amounts of multivariate data. It enables the human mind to gain novel insights by empowering the human visual system, encompassing the brain and the eyes. While it is believed that the process of creating interactive visualizations is reasonably well understood, the process of stimulating and enabling human reasoning with the aid of interactive visualization tools is still a highly unexplored field [9].

)LJ  Interactive visual data analysis: typically this is supported by semi-automatic analysis tools that are in parts interactively controllable.

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 9LVXDOL]DWLRQ 7HFKQLTXHV IRU 4XDQWXP 'DWD  6WDQGDUG 9LVXDOL]DWLRQ 7HFKQLTXHV For an overview on visualization and feature extraction methods see, e.g., the handbook [10] and the text book [11]. Today techniques are available for depicting— static and time-dependent, and mainly real—scalar, vector and tensor fields in two, three and sometimes four-dimensional domains. Furthermore, a lot of techniques have been developed to extract geometric, topological or application-specific features from such fields and visualize these. Examples for JHRPHWULF features, are, e.g., ridges in scalar fields, vortex regions in vector fields, or shape and orientation of vectors in tensor fields. Examples for WRSRORJLFDO features are, e.g., critical points in all types of fields, the contour tree and the Morse–Smale complex in scalar fields, the limit sets and topological skeletons in vector fields, or topological lines in tensor fields. Examples for DSSOLFDWLRQVSHFILF features are, e.g., vessel trees in medical image data, or separation and attachment lines or vortex core lines in fluid flows. Regarding visualization of molecular systems much work has been done for depiction of molecular systems, see for instance [12–14] and for visualization of classical dynamics, e.g., displaying molecular trajectories from molecular dynamics simulations and molecular conformations from conformation analysis [15]. For higher dimensions visualization techniques have been devised mainly to depict point sets as well as techniques to extract and depict multidimensional clusters, trends and outliers [16–20]. Unfortunately almost nothing is available for depicting fields with high-dimensional domains. Visualization of quantum mechanical data is challenging, due to the high dimensionality, high frequency and sometimes complex-valued quantities. Nevertheless, existing visualization techniques can be applied and adapted to create informative images, as will be demonstrated in the following.

 /RZ'LPHQVLRQDO 'DWD 9ROXPH 5HQGHULQJ For three-dimensional systems, the square of the wave function j j2 can be directly visualized using volume rendering [21]. In this visualization technique the scalar field is modeled as a cloudy density that emits and absorbs light [22]. For every density value y in the scalar field the amount of absorption of a virtual ray (of wavelength ) that travels through the domain can be defined. Such a mapping is called the opacity transfer function  .y/. For each pixel of the rendered image, rays are then cast through the domain of the scalar field. At each point the ray’s intensity I at a wavelength is given by the volume rendering integral:

Visual Analysis of Quantum Physics Data

 Z I .D/ D I0; exp  Z C 0

D

77

   y.t/ dt



0

D

 Z   L .s/ y.s/ exp 

D

   y.t/ dt ds ;

(1)

s

where I0; is the initial intensity of the ray, the integration parameter s describes the position along the ray, y.s/ is the density value at a given ray position,  is the attenuation caused by absorption of light by the medium (the opacity transfer function) and L is the emission of light by the medium caused by irradiation from external light sources. The spectral domain (wavelength ) is usually discretely sampled by just three sample points (RGB). The choice of transfer function ultimately determines the appearance of the scalar data in the rendered image. In the simplest case it consists of a RGB color map that assigns different colors to different scalar values, and an opacity map that falls off from higher to lower values, so that regions of low density appear more transparent in the visualization. As quantum mechanical wave functions (and densities) are smooth in most cases, introducing discontinuities or rough changes in the transfer function’s color and opacity dependence can greatly enhance visual contrast and let the viewer grasp the systems spatial properties much better, see Fig. 4. In cases where not only the (real valued) probability density is of interest but instead the complex wave function ( W R3 ! C), the colormap has to deal with complex numbers (or equivalently two real scalars). As suggested in [2] one can, for example, map the complex phase to the hue and the amplitude to the opacity and the saturation of the color. Fig. 5 depicts a complex 3D wavefunction of an HC 2 ion, using such a color map (also cf. Sect. 4.1).

(a)

(b)

)LJ  Visualization of bonding orbitals of the Fe2 S2 (SH)2 4 molecule [23]. The historical dataset (often referred to as ‘KLSLS’) has been used in the visualization community for over 20 years. Shown in (a) is a volume rendering with stepped colormap and continuous opacity map and in (b) nested semi-transparent isosurfaces created by volume rendering with spiked opacity map and continuous color map.

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)LJ  Volume rendering of a complex 3D wavefunction of an HC 2 ion using the same dataset as for Fig. 7. The complex phase is mapped to the hue and the amplitude to opacity and saturation. The internuclear distance R goes from bottom to top, the electronic axial position z from left to right, and the radial distance r from front to back.

,VRVXUIDFLQJ Given a continuous scalar field f .p/W Rd ‘ Œa; b, up to degenerate cases, the level sets S D fp j f .p/ D g are .d  1/-dimensional manifolds, called isosurfaces. These surfaces provide insight into the topology of the scalar field and are especially useful when displayed interactively, since then one can scroll through all possible isovalues and thereby grasp the structure of the scalar field. In practice the scalar field is mostly given as a discrete grid function, i.e., it is sampled at discrete locations of a more or less regular grid. Assuming that the underlying, unknown function is continuous, various interpolation methods (or ‘reconstruction methods’) are used to extend this grid function to a continuous function in the whole continuous domain. Then a marching cube type algorithm is used to create a linear approximation to the level set of that interpolant [24, 25]. An effect similar to isosurfacing can be produced when volume rendering is used in conjunction with an opacity map that is constantly zero, except for a few narrow spikes around one or few scalar values i . This yields isosurfaces of finite thickness for every i . The advantage of this approach is that volumetric isosurfaces are more visible in the areas tangential to the viewer’s direction than those where the surface is perpendicular, since in tangential regions the ray travels a longer distance through the volume compared to passing it perpendicularly. This outlines the edges of the isosurfaces and gives a hint to its overall geometry while being mostly transparent and showing other surfaces inside or behind the isosurface, as can be seen in Fig. 4(b).

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These techniques, however, are limited to three dimensional scalar fields, and hence to very basic quantum mechanical systems, such as a single particle or very small molecules with some symmetries. When more dimensions come into play, more sophisticated methods have to be developed. In particular, one has to define the structural features that are of interest and develop extraction methods as well as visualization techniques for these.

 0XOWLGLPHQVLRQDO 'DWD Here we shortly sketch a few techniques for depiction of point clouds in highdimensional spaces. 6FDWWHU 3ORW 0DWULFHV A well-known technique to depict points in high-dimensional space is to pick two coordinate axes of that space and project the data into the plane spanned [26]. Naturally the result depends strongly on the choice of the two axes. To minimize the loss of information one builds an n  n matrix of such plots, the so-called ‘scatter plot matrix’. Since this matrix is symmetric it does suffice to draw only the upper (or lower) triangle and use the freed space for zoomed-in renderings or a display of statistical data. One can, of course, additionally color the points by some other data set or vary their size with the (n-dimensional) distance to the coordinate plane. Such a matrix is a quite effective means for specific tasks; for instance, clusterings or outliers can be immediately recognized. Also variants have been proposed that provide support for interactive navigation in the multidimensional space. A recently proposed variant visualizes transitions between scatterplots as animated rotations in 3D space [27].  3DUDOOHO &RRUGLQDWHV Another technique frequently used is called ‘parallel coordinates’; here all coordinate axes are drawn parallel [28]. The coordinates of a high dimensional point will be marked on the axes as usual. These marks are then connected by straight lines that combine to a polyline in 2D space. This means each point in high-dimensional space is represented by a polyline. Since this is an unusual way of displaying data, for the unexperienced user it takes some time to read off interesting information. It is, however, very easy to spot linear dependencies between neighboring coordinates. Since the order of the dimensions is arbitrary, the user usually can swap them freely making it an interactive tool. Many extensions have been proposed, see [29]. Recent work specifically targets the visualization of multivariate scientific data by density-based parallel coordinates [30].

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 *UDQG 7RXU This is the most intricate of the methods mentioned here. The idea is to create a VHTXHQFH of 2D subspaces, project the data into them and identify the screen with these subspaces [31]. This creates the illusion of a tour through the high dimensional space. Interpolating in the Grassmannian space of it, it is possible to arrange the tour such that each subspace will be covered with the same probability so that eventually the entire space is covered. The idea is to identify interesting subspaces during display of the sequence that can later be explored individually. Usually these methods are combined in an interdependent tool box that allows the user explore the data interactively. For instance, a selection made in one part (‘brushing’) will have an effect also in the other parts (‘linked views’) [32]. Or one can zoom in or highlight some regions and apply additional tools, as for example a principal component analysis. In our software A MIRA [33], we have implemented such a toolbox (see Fig. 6). For the alternative of choosing a subset or a derived lower dimensional quantity there are of course infinite ways. The most obvious idea is to pick a low dimensional subspace to project the data into. This can be a simple 2D subspace spanned by coordinate axes as done in the scatter plot or any other subspace of dimension n  3. The challenge here is to find one that is suitable for uncovering the interesting properties. In multiparticle quantum systems the choice often made is to integrate out the degrees of freedom of one or more particles. In Z  .r i / D dr 1 dr 2    dr i 1 dr i C1    dr N  .r 1 ; r 2 ; : : : ; r N / : (2) the integration is taken over all but one particles. This yields a three-dimensional density interpreted as the probability distribution of single particles and makes it

)LJ  Scatterplot and parallel coordinates for a 3-dimensional quantum mechanical system evolved in a Henon–Heiles potential. Displayed is the 6D Wigner function after some evolution sampled by a Metropolis Monte Carlo procedure and colored by the value of the initial Wigner function.

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possible to view their time evolution in three dimensions using standard techniques mentioned above. However, particle interactions remain obscure, because by integrating their spatial distribution the other particles only contribute to the density with their averaged density. For the analysis of particle interactions, the degrees of freedom which are not being integrated over can also be chosen to be from different particles, e.g., the x-coordinates of the first two particles, Z  .x1 ; x2 / D dy1 dz1 dy2 dz2 dr 3    dr N  .r 1 ; r 2 ; : : : ; r N / : (3) resulting in a two-dimensional, so-called FRUUHODWLRQ GLDJUDP. This type of diagram can be used to visualize the average particle density of one particle along a certain direction, depending on the distribution of the second particle along the same or some other direction.

 7ZR 9LVXDOL]DWLRQ ([DPSOHV IURP 4XDQWXP 5HVHDUFK  'HSLFWLRQ RI &RXSOHG (OHFWURQ DQG 1XFOHL '\QDPLFV LQ H2C The first data set under study is the three-dimensional complex valued wave function  .R; z; r/ of the H2C molecular ion system discussed in [34]. It depends on the internuclear distance R and the two electronic cylindrical coordinates z and r, where the azimuth is neglected for symmetry reasons. The three-dimensional system can be visualized directly with volume rendering using a two-dimensional colormap and mapping the amplitude to the opacity, as discussed in Sect. 3.2 and shown in Fig. 5. The internuclear distance R goes from bottom to top, the electronic axial position z from left to right, and the radial distance r from front to back. One clearly sees the high oscillations of the phase in R direction, while the phase in z and r directions stays almost constant. Such kind of visualization is not easy to interpret, particularly if it is statically depicted; with interactive slicing, however, one can analyze the structure of the wave function. Another way to visualize this data, is to look at observables such as the density j j2 as well as the flux density j D .„=m/ Im.N r / and to transform the system back from its mathematical space into real space. Integrating the expanded density and flux density over the nuclear degrees of freedom then leeds to the electronic density and flux density, while integration over the electronic components gives the nuclear density and flux density respectively. This gives four quantities which are visualized simultaneously in Fig. 7. To determine the structure of the electronic density, nested semi-transparent isosurfaces are used. As a vectorial quantity the electronic flux density can be visualized most intuitively with arrows, whose orientation is given by the direction of the flux density and whose arrow length is proportional to the magnitude of the flux den-

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)LJ  Depiction of the coupled dynamics of electrons and nuclei of the HC 2 ion. The nuclear and electronic densities as well as flux densities are simultaneously visualized, with color codes as indicated.

sity. As the system is modeled such that the nuclei can only move in one direction, also the nuclear flux density has one direction. Both quantities are displayed with a rotational symmetric shape aligned along the one degree of freedom. The thickness of the shape is proportional to the nuclear density and the nuclear flux intensity is mapped with colors on this shape. For details regarding theory, model and results, we refer the reader to reference [34].

 9LVXDOL]DWLRQ RI 'DWD IURP D ' 6LPXODWLRQ RI +LJK+DUPRQLF *HQHUDWLRQ Here we report on the visualization of data from a 3D model analysis of multielectron correlation in high-harmonic generation. 6LPXODWLRQ For simulation the Multi-Configuration Time-Dependent Hartree Fock (MCTDHF) method [35] has been used. The high-dimensional (C3N ) N -electron wave function has been decomposed into a linear combination of n low-dimensional (C3 ) discretized basis functions i .r/ and coefficients Ai;j;::::

Visual Analysis of Quantum Physics Data

 .r 1 ; r 2 ; : : :/ D

n X

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Ai;j;:::  i .r 1 /  j .r 2 /     :

(4)

i;j;:::D1

The simulated system is an extended H2 molecule under the influence of a strong electric field E.t/ perpendicular to the molecule’s symmetry axis; the molecule’s hydrogen core potentials were aligned along the y-axis, the electric field was applied along the x-axis. The Hamiltonian of the system (in atomic units) is b .t/ D H

 2  2 X X 1 1  rj2  p 2  E.t/x j 2 xj C .yj C ci /2 C zj2 C a2 j D1 i D1 Cp

1 .r 1  r 2 /2 C b 2

(5)

where i is an index for hydrogen cores and j is an index for the electrons. a D 0:14 and b D 0:1 are shielding parameters that soften the otherwise singular Coulomb potentials, c1 D 2 and c2 D 2 are the positions of the protons along the y-axis, and rj D .xj ; yj ; zj / are the position eigenvalues of electron j . The time dependence of the electric field of the laser is given by E.t/ D E0 cos.!t/ sin2 .t!=8/ with E0 D 0:0534 and ! D 0:057, which amounts to an electric field of intensity I D 1014 W=cm2 and wavelength D 800 nm. For the duration of 440 a:u: (exactly one laser pulse length), the wave functions were calculated in steps of 1 a:u: and saved in steps of 5 a:u:. The results have been visualized using the techniques explained in Sect. 3.1.  3RVW3URFHVVLQJ Ten three-dimensional single particle configuration space basis functions i .r/ with the orthonormality property h j j i i D ıij are given. Due to the spin-independent Hamiltonian, see Eq. (5), and the antisymmetry of the coefficient matrix Aij , the two single-particle-densities (calculated by integrating over the position coordinates of one of the particles) must be equal. Therefore only one of the electrons needs to be visualized. The reduced 3D single particle density can be obtained by calculating the 6D density from the wave function and integrating over the spatial degrees of freedom of the second particle, Z  .r 1 / D dr 2 .r 1 ; r 2 / Z X D dr 2 Aij i .r 1 / j .r 2 /  Akl k .r 1 / l .r 2 /hsi ; sj jsk ; sl i D

X ijk

ijkl

Aij Akj i .r 1 / k .r 1 /ısi sk

(6)

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)LJ  The probability density of an electron in the H2 molecule in its ground state, visualized with volume rendering with a spiked opacity map as described in Section 3.2.

where the indices i; j; k; l run from 1 to 10, and in the last step the orthonormality of the basis functions, h i j j i D ıij , has been used.  9LVXDOL]DWLRQ In the following depictions all quantities are given in atomic units. The 3D visualizations were created with the data analysis and visualization system A MIRA [33]. The probability density is depicted by multiple isosurfaces for isovalues n D 10n , n 2 f6; 5; 4; 3; 2g. In Fig. 8 the density for the first time step (the molecule in its ground state) is shown. Looking at the whole time series, the deformations and dynamics induced by the laser pulse can be nicely seen in Fig. 9. They are similar to the ones explained and schematically depicted in [5]. The first few amplitudes of the electric field only slightly shift the wave back and forth. At t D 190 a:u: for the first time part of the density starts to seclude (very well visible in the top right illustration in Fig. 9) from the rest of the density just when the electric field changes sign again. This wave packet diffuses, decelerates, returns and then goes into superposition with the bound state wave function.

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)LJ  A visualisation of the electron dynamics induced by the oscillating electric field during a High Harmonic Generation process. The diagram on top shows the time dependence of the xcomponent of the electric field. The densities at the four time steps 170 a:u:, 190 a:u:, 240 a:u: and 300 a:u: in the x-y- and x-z-plane are depicted below.

 &RQFOXVLRQ Though the visualization of quantum systems is a challenging task, systems at least with few degrees of freedom can be depicted. Up to 3D + time, standard visualization techniques can be easily adapted to the needs of quantum physics. Higher dimensional data can be visualized, if they can be mapped to point sets in multidimensional spaces. Better toolkits that combine the various techniques and offer interactive methods for fast projection, integration over degrees of freedom, marginaliza-

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tion of densities, brushing together with linked views, are conceivable. They would allow the user to analyze more complex quantum systems. A largely unexplored domain (which is successfully applied in other sciences) is the possibility to define physically meaningful features of quantum systems that characterize structural aspects of these, but are easier to visualize. $FNQRZOHGJHPHQWV We are very grateful to J¨orn Manz and Beate Paulus for the fruitful cooperation, the provision of extensive expertise in quantum physics and quantum chemistry, the many stimulating discussions and the strong conviction that visualization of quantum systems can be of real value. Without them this work would not have been possible. We also greatly thank Guennaddi K. Paramonov, Ingo Barth, Hiroshi Ikeda and Anatole Kenfack for cooperation and the provision of the HC 2 simulation data. Furthermore, we are very grateful to Thomas Brabec for inspiring discussions during the Workshop “Quantum Dynamic Imaging” at the Centre de recherches math´ematiques (CRM) in Montreal, Canada, in Oct. 2009, as well as for cooperation and provision of the simulated HHG data. HCH and MK thank Andr´e D. Bandrauk for the invitation to this workshop and the great patience regarding this manuscript. Furthermore, we thank Andr´e Montpetit for careful editing and correcting of the manuscript. Financial support from the Center of Scientific Simulations (DFG) at Freie Universit¨at and the research center M ATHEON is also gratefully acknowledged.

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7KHRU\ RI '\QDPLF ,PDJLQJ RI 0ROHFXOHV ZLWK ,QWHQVH ,QIUDUHG /DVHU 3XOVHV C.D. Lin, Anh-Thu Le, and Zhangjin Chen

$EVWUDFW When a molecule is exposed to an intense laser pulse, electrons that are removed earlier in the laser field may be driven back by the oscillating electric field of the laser to recollide with the target ion. The rescattering of the returning electrons, analogous to the scattering by free electrons, offers the opportunities to probe the structure of the molecule. Since lasers of pulse duration of a few femtoseconds are already available, thus it is possible to use few-femtosecond lasers for dynamic imaging of transient molecules. The basic scattering theories for dynamic quantum imaging of molecules are discussed.

 ,QWURGXFWLRQ Imaging, or the determination of the structure of an object, has always played an important role in physical sciences. For microscopic systems, X-ray and electron diffraction are the conventional methods for achieving spatial resolutions on the order of Angstrom or less [1]. These methods, however, are not suitable for following the time evolution of a dynamic system, in particular, in a chemical reaction, which requires temporal resolutions of a few femtoseconds [2]. Although ultrafast electron diffraction (UED) and X-ray free-electron lasers (XFEL) that have come online, as well as others that are being developed at big facilities, are aiming at C.D. Lin Department of Physics, Cardwell Hall, Kansas State University, Manhattan, KS 66506, USA, email: DEMJO!QIZTLTVFEV Anh-Thu Le Department of Physics, Cardwell Hall, Kansas State University, Manhattan, KS 66506, USA, email: BUMF!QIZTLTVFEV Zhangjin Chen Department of Physics, Cardwell Hall, Kansas State University, Manhattan, KS 66506, USA, email: [KDIFO!QIZTLTVFEV A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_7, © Springer Science+Business Media, LLC 2011

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achieving temporal as well as spatial resolutions, it is important to investigate the possibilities of accomplishing the same goal using small-scale methods. In this contribution, we propose such an approach based on infrared or mid-infrared (MIR) lasers for dynamic imaging of molecules. Today such lasers have pulse durations of a few femtoseconds already, thus if they can achieve good spatial resolution, then in principle these lasers can be used for dynamic imaging of molecules. To study the structure of a molecule, the first step is to determine the bonding lengths and bond angles of all the atoms in the molecule. This is what X-ray diffraction and electron diffraction are good for. For objects that can be crystallized, X-ray diffraction is preferred. For molecules in the gas phase, electron diffraction is the standard method. The knowledge of bond lengths and bond angles (the geometry) in a molecule alone does not offer the full knowledge of the “chemistry” of a molecule, in particular, the bonding of the outer-shell electrons, as well as how the molecule will respond to an external perturbation. To probe the molecules at the next higher level, electrons and photons are used. Thus electron-molecule scattering where electrons with energies from a few eVs to hundreds of eVs are used. Similarly, photons from ultraviolet to soft X-rays are used to probe the outershells of a molecule. To understand the results of these measurements, advanced quantum mechanical many-body theories are needed since effects of exchange and electron correlation are known to play important roles in such situations. A large fractions of modern atomic and molecular physics since the 1960s have been devoted to these latter topics. Many powerful experimental and theoretical tools have been developed and a great deal of data base for each species have been documented. They form an essential component of our present knowledge of the microscopic world. Infrared or MIR lasers have wavelengths from fractions to a few microns. They are much longer than the typical interatomic separations in a molecule. However, when a molecule is exposed to a short infrared laser pulse, electrons that are removed at an early time of the pulse may be driven back by the oscillating electric field of the laser to recollide with the molecular ion, to incur processes like highorder harmonic generation (HHG), high-energy above-threshold-ionization (HATI) electrons and nonsequential double ionization (NSDI). According to the rescattering model [3, 4], they are similar to conventional electron collision processes with the molecular ions. Thus HHG is due to the photo-recombination of the returning electrons with the molecular ion with the emission of high energy photons, i.e., a time-reversed process of photoionization of a neutral molecule. Similarly, HATI spectra are due to elastic electron-ion collisions where the electrons are scattered into the backward directions, and NSDI is due to impact ionization by the returning electrons with the emission of another electron, that is, similar to the field-free .e; 2e/ process. Unlike low-energy photoelectrons that are removed by the tunneling process which occurs far away from the core of the molecule, these rescattering processes are by fast electrons that have been accelerated by the laser field, and these collisions occur close to the core of molecules. The energy of the returning electron is proportional to the laser intensity, and increases quadratically with the wavelength of the laser. Thus for a linearly polarized Ti-Sapphire laser with wavelength of 800 nm and peak intensity of 2  1014 W=cm2 , the maximum returning

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electron energy is about 38 eV. Using a MIR laser at wavelength of 1200 nm or 1800 nm, at the same peak intensity, the returning energies will be about 86 eV and 192 eV, respectively. Using IR or MIR lasers for probing the structure of a molecule, we rely on electron scattering. With the returning electron energies of 86 eV and 192 eV, their de Broglie wavelengths are 1.42 and 0.64 Angstroms, respectively. These are closer to typical molecular size, making MIR lasers capable of probing the structure of a molecule. The rescattering model, or the so-called three-step model described above, has been around since the early 1990s. Similarly, many papers have been published on laser induced electron diffraction since the pioneer work of Zuo et al. [5]. However, these earlier works have been used to “interpret” qualitatively the observed rescattering phenomena only, or the calculations are based on solving the time-dependent Schr¨odinger equation which cannot be easily extended to many-electron molecular systems. Recently we have developed a quantitative rescattering (QRS) theory [6,7] which can be used to calculate HHG, HATI and NSDI yields quantitatively. The applications of the QRS to these processes have been documented in a recent review article [7]. Unlike earlier works, essential to the QRS is the usual field-free scattering cross sections that have been well-investigated in energy-domain scattering theories. For this book chapter, we will focus on basic theories used in QRS instead of presenting many actual results. Thus, this article is written for those in the strong field community who are not familiar with quantum scattering theories. Readers interested in the development of QRS and earlier works are advised to look at [7] for more details. Clearly only simple theories can be presented here. Advanced manybody scattering theories developed in the past half a century are beyond the reach of this contribution. Atomic units will be used in this article unless otherwise indicated.

 7KHRUHWLFDO 7RROV IRU 6WXG\LQJ $WRPV DQG 0ROHFXOHV LQ 6WURQJ )LHOGV In this section we present the elementary scattering theories for the solution of time-dependent and time-independent Schr¨odinger equations. These basic theoretical tools serve as the common language for both theorists and experimentalists in understanding the interaction of light or electrons with atoms and molecules.

 6ROXWLRQ RI WKH 7LPH'HSHQGHQW 6FKUŽRGLQJHU (TXDWLRQ Based on quantum mechanics, the interaction of an intense laser pulse with an atom is described by the time-dependent Schr¨odinger equation (TDSE) i

@  .U; t/ D H  .U; t/ @t

(1)

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where H D H0 C Hi .t/ D  12 r 2 C V .r/ C Hi .t/ :

(2)

The atom-field interaction, in length gauge, is given by Hi D U  (.t/ ;

(3)

where (.t/ is the laser field. For simplicity, the atom is approximated by a model potential V .r/ D 

1 C a1 exp.a2 r/ C a3 r exp.a4 r/ C a5 exp.a6 r/ ; r

(4)

where the parameters ai ’s are obtained by fitting to the known energies of the ground state and the first few excited states of the atom. The model potential for a neutral atom can also be expressed as V .r/ D Vs .r/  1=r ;

(5)

where the Coulomb potential part is separated out and Vs .r/ is the remaining shortrange potential. The TDSE can be solved by expanding  .U; t/ in terms of eigenfunctions of H0 within a box of r 2 Œ0; rmax  X  .U; t/ D cnl .t/Rnl .r/Ylm .OU/ (6) nl

where the radial functions Rnl .r/ are expanded using the DVR (discrete variable representation) basis functions associated with Legendre polynomials, while the cnl are calculated using the split-operator method. At the end of the laser pulse t D tend , the photoelectron yield is computed by projecting out the total wavefunction onto eigenstates of a continuum electron with momentum N, D.k; /

@3 P D jh˚N j .t D tend ij2 @3 N

(7)

where the continuum state ˚N satisfies the following equation 

 1 2 k2   r C V .r/ ˚N D ˚ : 2 2 N

(8)

Here ˚N satisfies the incoming wave boundary condition. It can be expanded in terms of partial waves as 1 X l  O ˚N .U/ D p i expŒi.l C ıl /RE l .r/Ylm .OU/Ylm .N/ : k lm

(9)

Here, ıl is the lth partial wave phase shift due to the short range potential Vs .r/ in (5), and l is the Coulomb phase shift. The Y’s are the usual spherical harmonics.

Theory of Dynamic Imaging of Molecules with Intense Infrared Laser Pulses

For the continuum radial function RE l , it is energy normalized such that Z 1 RE l .r/RE 0 l .r/r 2 dr D ı.E  E 0 / ;

93

(10)

0

and has the asymptotic form r 1 2 RE l .r/ ! sin.kr  l=2   log 2kr C l C ıl / : r k

(11)

The details of such calculations can be found in [8].

 6WURQJ )LHOG $SSUR[LPDWLRQ In the strong field approximation (SFA), one treats the atomic potential as perturbation. In the first-order SFA, called SFA1 here, the ionization of the atom is calculated from Z 1 f1 .N/ D i dt h N .t/jHi .t/j0 .t/i : (12) 1

Here 0 is the initial state, Hi is the laser-electron interaction and N is the Volkov state. To describe rescattering, a second-order SFA (SFA2) [9–11] is needed. For SFA2, the scattering amplitude can be written as Z 1 Z 1 Z f2 .N/ D  dt dt 0 dS h N .t 0 /jV j S .t 0 /ih S .t/jHi .t/j0 .t/i : (13) 1

t

Here the electron is first ionized at time t and rescattered at time t 0 . The potential V is the electron-ion interaction, i.e., the model potential chosen. Integration over the momentum is carried out using saddle-point approximation.

 /DVHU)UHH (ODVWLF 6FDWWHULQJ 7KHRU\ In quantum mechanics, the scattering of an electron by a spherically symmetric potential V .r/ is governed by the time-independent Schr¨odinger equation Œr 2 C k 2  U.r/ .U/ D 0

(14)

where U.r/ D 2V .r/ is the reduced potential and p k is the electron momentum, related to the incident electron energy by k D 2E. For a short-range potential which tends to zero faster then r 2 as r ! 1, the scattering wave function satisfies the asymptotic outgoing wave boundary condition

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C.D. Lin et al. C



.U/r!1 D

1 exp.ikr/ exp.ikz/ C f ./ 3=2 r .2/

 (15)

where  is the polar angle measured from the incident direction. We choose the z-axis along the direction of the incident wave vector N. Here C .U/ satisfies the outgoing wave boundary condition and can be expanded in terms of partial waves, r 2 1 X l C  O .U/ D i exp.iıl /ul .kr/Ylm .OU/Ylm .N/ (16)  kr lm

where Ylm is a spherical harmonic. The continuum waves are normalized to ı.N  N0 /. The radial function ul .kr/ satisfies  2  d l.l C 1/ 2 C k   U.r/ ul .kr/ D 0 : (17) dr 2 r2 The above expressions are valid for short-range potential only. For a Coulomb potential, Vc D Z=r, its full wavefunction can be expanded as r 2 1 X l C  O .U/ D i exp.il /ucl .kr/Ylm .OU/Ylm .N/ (18) c  kr lm

where l D argΠ.l C 1 C i/

(19)

is called the Coulomb phase shift with  D Z=k. The scattering amplitude for the Coulomb potential can be obtained analytically using parabolic coordinates, fc ./ D  exp.2i0 /

expfi lnŒsin2 .=2/g : 2k sin2 .=2/

(20)

The atomic potential in (5) is an example of modified Coulomb potential. In such a potential the scattering amplitudes is the sum of Coulomb scattering amplitude and the amplitude from the short-range potential alone f ./ D fc ./ C fO./ ;

(21)

where the first term is the Coulomb scattering amplitude [(20)], and the second term is given by fO./ D

1 X 2l C 1 exp.2il / exp.iıl / sin ıl Pl .cos / k

(22)

lD0

where Pl .cos / is the Legendre polynomial, and ıl is the phase shift from the short-range potential. Due to the short range nature, the summation in (22) can be truncated after some number of partial waves, depending on the electron energy.

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The elastic scattering DCS is then given by .k; /

dP D jfc ./ C fO./j2 : d˝

(23)

For high-energy scattering, if the scattering is treated to first-order, then the scattering cross section is given by the so-called plane wave Born approximation (PWBA) or first Born (B1) approximation. In B1, the DCS is calculated from PWBA .k; / D

1 jV .T/j2 ; 4 2

(24)

i.e., proportional to the square of the potential in the momentum space. Here T is the momentum transfer and its magnitude is q D 2k sin.=2/. In B1 the continuum electron wavefunctions are represented by plane waves. For electron-target ion collisions, B1 is not valid even at large collision energies since it neglects the effect of long-range Coulomb interaction as well as the strong short-range potential due to the atomic ion or molecular ion core.

 4XDQWLWDWLYH 5HVFDWWHULQJ 7KHRU\ For a one-electron atom, we can solve TDSE to obtain the photoelectron momentum spectra D.k; /. For a given model potential V .r/, one can also calculate accurate DCS, .kr ; r /, for a given kr . Both can be obtained essentially “exactly.” Since the rescattering model does not specify how to calculate the returning electron wave packet, we define it as the ratio W .kr ; r / D D.k; /=.kr ; r / :

(25)

If the rescattering concept is correct, then the ratio should not depend on the angle r . Before the ratio can be calculated, we need to establish the relation between .k; / and .kr ; r /. The kr is the momentum value after electron is scattered by the ion. Since the collision occurs in the laser field, it is still under the influence of the laser after the collision. As it exits from the laser field, an additional momentum along the direction of the laser polarization will be added. Thus we can write kz D k cos  D ˙Ar kr cos r ;

(26)

ky D k sin  D kr sin r :

(27)

The upper signs in (26) refer to the right-side (kz > 0) while the lower ones to the left-side (kz < 0) electrons. Here Ar =A.tr / is the vector potential at the time tr of collision. Note that the electron can return from the left or the right along the polarization axis to revisit the target ion. Based on the classical rescattering theory, the returning momentum kr is related to the vector potential Ar by Ar D kr =1:26. Note that this sets the maximum returning electron energy at 3:2 Up . Here Up is the

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ponderomotive energy, Up D I =4! 2 , with ! the angular frequency of the laser and I the peak intensity. With these conditions, we then solve sin r ; ˙.1=1:26  cos r / k 2 D kr2 .1:63  1:59 cos r / :

tan  D

(28) (29)

These two expressions relate the momentum of the photoelectron and the momentum of the electron in the field after elastically rescattered by the ion.

 7KH 5HVFDWWHULQJ (OHFWURQ :DYH 3DFNHW In Fig. 1 the relation between the two momentum vectors are shown. Take the photoelectron yield along the semicircle of kr =constant and divide it by the DCS, .kr ; r /, we obtain W .kr ; r /. Examples of such W .kr ; r / are shown in Fig. 2. For the upper frame ones, they are obtained from TDSE solutions and the exact DCS. We note that W .kr ; r / does not depend on r , thus we can drop the dependence on r and write it as W .kr /. This condition is essential if we are to call W .kr / a wave packet as in typical scattering theory. Note how W .kr / was defined. It is not a measurable quantity since it is defined when the laser field is still on. Using SFA2, we can obtain a similar wave packet. Here the DCS is taken from PWBA .kr ; r /, (24). Since in SFA2 the electron-ion interaction is treated to first order only, thus the PWBA cross section should be used for .kr ; r /. We note that the resulting W .kr / from SFA2, shown in the bottom row of Fig. 2, are essentially identical to the ones on the upper row. We comment that only the kr -dependence of the wave packet is

ky / A0

2.5

10Up

2

6Up

1.5

4Up

1

0 -2 -4 -6

2Up

k

0.5 θ

0 -2.5 -2 -1.5 -1 -0.5 0 0.5 kz / A0

kr

θr

1

1.5

2

2.5

)LJ  Typical 2D electron momentum distributions (in logarithmic scale): The TDSE calculation is for single ionization of Ar in a 5 fs laser pulse at the intensity of 1:0  1014 W=cm2 with the wavelength of 800 nm. Photoelectrons of a given energy are represented on a concentric circle centered at the origin. The elastic scattering of a returning electron with momentum kr in the laser field is represented by a partial circle with its center shifted from the origin by Ar D kr =1:26.

Theory of Dynamic Imaging of Molecules with Intense Infrared Laser Pulses

6

12

4

8

2

4

0

0

-9

-9 W(kr,θr) (10 a.u.)

-7 W(kr,θr) (10 a.u.) (d)

(c)

12

3

8

2

4

1

0

0.9

1

0 1.1

kr (a.u.)

180 175 170 165 160 155

θr (deg)

-5 W(kr,θr) (10 a.u.) (b)

(a)

1.2

1.3

1.2 1.3 1.4 1.5 1.6 1.7 1.8

180 175 170 165 160 155

θr (deg)

-6

-6 W(kr,θr) (10 a.u.)

97

kr (a.u.)

)LJ  Right-side wave packets extracted from the electron momentum distributions calculated using TDSE and SFA2, for single ionization of Ar in a 5 fs laser pulse with the wavelength of 800 nm. Left column: from TDSE (top) and SFA2 (bottom), at intensity of 1:0  1014 W=cm2 ; Right column: same, but at 2:0  1014 W=cm2 .

considered here. Their absolute values are different, reflecting that the ionization rates from TDSE and from SFA are different. The W .kr / obtained from SFA2 and from TDSE are identical is the most profound result of the QRS model. In fact, in hindsight, it should be expected if the rescattering picture is valid. After tunneling ionization, the motion of the electron is mostly away from the atomic ion. Thus W .kr / is determined mostly by the interaction of a free electron with the laser field. This interaction is treated exactly in SFA2, thus the W .kr / from SFA2 is very close to the one from TDSE. In SFA2, the tunneling ionization rate is not calculated correctly, nor the electron-ion scattering cross sections. The electron wave packet, W .kr /, however, is accurate. Thus in the quantitative rescattering (QRS) theory, we can obtain HATI spectra from D.k; / D W .kr /.kr ; r / ;

(30)

where W .kr / is the WSFA .kr / calculated from SFA2. Since SFA2 calculations are much faster than solving TDSE, and the calculation of .kr ; r / is the same as in field-free scattering cross sections, the QRS provides a very simple and accurate method of obtaining HATI spectra on the one hand, and on the other hand, allows one to extract field-free DCS, .kr ; r /, from experimental HATI spectra. The latter establishes the theoretical foundation for using infrared lasers for imaging the structure of a target. Since infrared lasers with pulse durations of a few femtoseconds are already available, it has the potential for dynamic chemical imaging with temporal resolution of a few femtoseconds.

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The analysis so far concerns with the interaction of an atom with a laser of welldefined intensity. In actual experiments, an intense laser does not have fixed intensity within the focus volume. Thus to compare with experimental HATI spectra, theoretical calculations must include the volume effects [12]. For a peak intensity I0 at the focal point, the yield of the photoelectrons with momentum N should be Z



I0

S.N; I0 / D %

DI .k; / 0

@V @I

dI

(31)

where % is the density of atoms in the chamber, DI .k; / denotes the momentum distribution for a single intensity I and .@V =@I /dI represents the volume of an isointensity shell between I and I C dI . In the QRS calculations, we obtain the volume-integrated returning electron wave packet using (31) since the DCS does not depend on the laser intensity. Consequently, (31) becomes S.N; I0 / D W I0 .kr /.kr ; r /

(32)

where W I0 .kr / is the volume-integrated wave packet at the peak intensity I0 Z W I0 .kr / D %



I0

WI .kr / 0

@V @I

dI

(33)

with WI .kr / being the wave packet for the laser pulse at a single intensity I .

 +LJK2UGHU +DUPRQLF *HQHUDWLRQ For the model one-electron atomic system, once the time-dependent wavefunction  .U; t/ is obtained from solving (1), one can calculate the induced dipole of the atom by the laser field either in the length or acceleration forms DL .t/ D h .U; t/jzj .U; t/i ; @V .r/ j .U; t/i : DA .t/ D h .U; t/j @z

(34) (35)

The HHG power spectra are obtained from Fourier components of the induced dipole moment D.t/ as given by ˇZ 2 ˇ ˇ d D.t/ i!t ˇ2 ˇ P .!/ / ja.!/j D ˇ e dt ˇˇ  ! 4 jD.!/j2 : dt 2 2

(36)

Since TDSE cannot be solved accurately for molecular targets, the strong field approximation, or the SFA2, has been employed for obtaining HHG since the 1990s. Here we show SFA2 for molecular targets. Assuming that the molecules are aligned

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along the x-axis, in a laser field E.t/, linearly polarized on the x-y plane with an angle  with respect to the molecular axis. The parallel component of the induced dipole moment can be written in the form 3=2  Œcos dx .t/ C sin dy .t/ " C i=2 0  Œcos dx .t  / C sin dy .t  /E.t  /  expŒiSst .t; /a .t/a.t  / C c:c: Z

Dk .t/ D i



1

d

(37)

where G.t/ GŒSst .t; /C$.t/, G.t / GŒSst .t; /C$.t / are the transition dipole R t moments between the ground state and the continuum state, and Sst .t; / D  t  $.t 0 /dt 0 = is the canonical momentum at the stationary points, with $ the vector potential. The perpendicular component D? .t/ is given by a similar formula with Œcos dx .t/ C sin dy .t/ replaced by Œsin dx .t/  cos dy .t/ in (37). The action at the stationary points for the electron propagating in the laser field is Z t  ŒSst .t; / C $.t 0 /2 Sst .t; / D C Ip dt 0 ; (38) 2 t  where Ip is the ionization potential of the molecule. In (37), a.t/ is introduced to account for the ground state depletion. According to the rescattering model, HHG occurs when the returning electrons photo-recombine with the parent ions, with the emission of high-energy photons. Following the QRS, we can anticipate a relation similar to (30) for HHG [13]. Since the phase of the HHG is important, we write the FRPSOH[ laser-induced dipole as jD.!/jei.!/ D jW .E/jei .E / jd.!/jeiı.!/ ;

(39)

where and  are the phases of the harmonic and the returning electron wave packet W .E/, respectively; d and ı are the amplitude and phase of the photorecombination transition dipole. The electron energy E is related to the emitted photon energy ! by E D !  Ip , with Ip being the ionization potential of the target. Note that the W(E) defined here is the complex electron wave packet amplitude, whereas W .kr / defined for HATI electrons are the “intensity.” In the equation above the transition dipole is similar to the one used in photoionization theory. For the transition from an initial bound state i to the final continuum state N due to a linearly polarized light field (in the length form) it is given by dN;Q .!/ D hi jU  QjN i :

(40)

Here Q is the direction of the light polarization and N is the momentum of the ejected photoelectron. For atomic targets, using the scattering wavefunction given by (9), we can evaluate the dipole transition matrix element. For example, for photoionization from the 3p shell of Ar, the transition dipole can be written as hi jzjNC i

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D p

1 3k



  exp.i.0 C ı0 / hR31 jrjRE 0 i=2  ei.2 Cı2 / hR31 jrjRE 2 i : (41)

For molecular targets, the calculation of the transition dipole is more complicated. Both the ground state and the continuum state wavefunctions have to be calculated using more advanced quantum chemistry packages. The photoionization differential cross section (DCS) can be expressed in the general form as d2  I 4 2 !k D jhi jU  QjN ij2 ; d˝N d˝Q c

(42)

where k 2 =2 C Ip D ! with Ip being the ionization potential, ! the photon energy, and c the speed of light. A more extended discussion for the calculation of the transition dipole moments of molecules can be found in Sect. II.D of Le et al. [13] where references to other works can be found as well.

 1RQVHTXHQWLDO 'RXEOH ,RQL]DWLRQ 'XH WR WKH 'LUHFW .H; 2H/ &ROOLVLRQ 3URFHVVHV Nonsequential double ionization (NSDI) is another interesting rescattering phenomenon when atoms and molecules are placed in the laser field. NSDI involves at least two electrons. According to the rescattering picture, the returning electron can knock out another electron in a process similar to the electron impact ionization of an atom, or the so-called .e; 2e/ process. Clearly (30) can be used to obtain the momentum distributions of the ionizing and the ejected electrons. Since electron impact ionization occurs in the laser field, the photoelectron momentum of each electron has to be shifted in the same way as in (26) and (27). According to QRS, the main additional ingredient needed is the differential cross sections for the .e; 2e/ collisions. Below we discuss the elementary field-free electron impact ionization theory. Consider a two-electron atomic system. Let U1 and U2 be the position vectors of the projectile and the bound state electron, respectively, the exact Hamiltonian for the whole system is 1 ZN 1 ZN 1 H D  r12   r22  C : 2 r1 2 r2 r12

(43)

This Hamiltonian can be rewritten approximately as 1 1 ZN Hi D  r12 C Ui .r1 /  r22  2 2 r2

(44)

where we assume that the HeC ion initially is in the ground state, and the charge of the nucleus is ZN D 2. Here Ui .r1 / is the initial state distorting potential which

Theory of Dynamic Imaging of Molecules with Intense Infrared Laser Pulses

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is used to calculate the initial state wave function for the projectile. Using the prior form, the direct transition amplitude for the .e; 2e/ collision process is fe2e .N1 ; N2 / D hN1 ;N2 jVi jNi i

(45)

where Vi is the perturbation interaction, Vi D H  Hi D

1 ZN   Ui .r1 / : r12 r1

(46)

In (45), N1 ;N2 is an exact solution of the three-body problem satisfying the incoming-wave boundary condition for two electrons with momentum vectors N1 and N2 . There is no exact analytical solutions for N1 ;N2 , thus various approximations have to be used. First consider the so-called BBK model [14] which is expressed as N1 ;N2 .U1 ; U2 / D .2/3 exp.iN1  U1 / exp.iN2  U2 /  C.˛1 ; N1 ; U1 /C.˛1 ; N2 ; U2 /C.˛12 ; N12 ; U12 / ;

(47)

where the Coulomb part of the wave function is defined as C.˛; N; U/ D exp. ˛=2/ .1  i˛/1 F1 Œi˛I 1I i.kr C N  U/

(48)

and N12 D ˛1 D 

1 .N1  N2 / ; 2

ZN ; k1

˛2 D 

U12 D U1  U2 ; ZN ; k2

˛12 D

1 : 2k12

(49)

In (47), the Coulomb interaction between the two outgoing electrons has been taken into account. If we set ˛12 D 0, then the electron-electron interaction is turned off and the two continuum electrons are then given by the product of two Coulomb functions, each electron seeing the charge ZN from the nucleus. In the equation above,  is the gamma function and 1 F1 is the confluent hypergeometric function. From the approximate initial state Hamiltonian Hi , we can write down the initial state wavefunction in the product form Ni .U1 ; U2 / D 'Ni .U1 / HeC .U2 /

(50)

where 'Ni .U1 / describes the incident electron and satisfies Π12 r12 C Ui .r1 /  12 ki2 'Ni .U1 / D 0 ;

(51)

here Ni is the incident momentum and HeC .U2 / is the ground state wavefunction of HeC . Due to the energy conservation, the wave vectors k1 and k2 satisfy 1 2 k 2 i

D 12 k12 C 12 k22 C Ip :

(52)

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The discussion up to now assumes that the two electrons are distinguishable. Including the exchange, the triply differential cross section (TDCS) for the electron impact ionization process is given by d3 e2e k1 k2  3 D .2/4 jfe2e .N1 ; N2 /  ge2e .N1 ; N2 /j2 d˝1 d˝2 dE ki 4  C 14 jfe2e .N1 ; N2 / C ge2e .N1 ; N2 /j2

(53)

where ˝1 .1 ; 1 / and ˝2 .2 ; 2 / are the solid angles of detection of the two electrons leaving the collision with momenta N1 and N2 , and ge2e .N1 ; N2 / is the exchange amplitude with ge2e .N1 ; N2 / D fe2e .N2 ; N1 /. The distortion potential Ui .r1 / in (51) is not determined so far. If we choose Ui .r1 / D 0, then the incident electron is given by a plane wave. On the other hand, if the distortion potential is set as Ui .r1 / D .ZN  1/=r1 , then it is described by a Coulomb wave. By calculating the TDCS using different approximate initial and final state wavefunctions, we can specify P-CC, P-CCC and C-CCC models for the .e; 2e/ processes; here the first letter indicates a plane wave (P) or a Coulomb wave (C) for describing the incident electron, the second string of letters indicate that electron-electron repulsion is not included, in CC, or is included, in CCC. By comparing the results of such calculations with experimental measurement, the effect of Coulomb interaction between the two electrons can be assessed. Note that this effect depends on the kinetic energy, as well as the angle between the momentum vectors of the two electrons.

 1RQVHTXHQWLDO 'RXEOH ,RQL]DWLRQ 'XH WR WKH ,QGLUHFW ([FLWDWLRQ7XQQHOLQJ ,RQL]DWLRQ 3URFHVVHV In NSDI, electron collision occurs in the laser field. If the core electron is excited by the returning electron to an excited state, the excited electron may be tunnel ionized by the laser field, resulting in the emission of two electrons. To calculate such processes, one first needs to obtain electron impact excitation cross sections, then evaluate the removal of the excited electron, where one can use the simple tunneling model. For electron impact excitation, the final state wavefunction is given by ˚Nf .U1 ; U2 / D 'Nf .U1 / f .U2 /

(54)

where 'Nf is the wave function used to describe the outgoing electron, which satisfies the differential equation  1 2   2 r1 C Uf .r1 / 'Nf .U1 / D 12 kf2 'Nf .U1 / : (55)

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For the present purpose, we set the distorting potential Uf .r1 / D .ZN  1/=r1 such that the scattered outgoing electron is described by a Coulomb wave. The final excited state f .U2 / satisfies  1 2 ZN  r2   "n f .U2 / D 0 (56) 2 r2 2 where "n D 0:5ZN =n2 is the energy of the excited state. The T-matrix element for a transition from an initial state to a final state is then given by fexci D h'Nf .U1 / f .U2 /jVi j'Ni .U1 / i .U2 /i ; (57)

where Vi is given by (46). The differential cross section (DCS) for this transition is given by  kf  3 dexci D .2/4 jf  gexci j2 C 14 jfexci C gexci j2 4 exci d˝ ki

(58)

where the exchange amplitude gexci is calculated from gexci D h'Nf .U2 / f .U1 /jVi j'Ni .U1 / i .U2 /i :

(59)

The .e; 2e/ and the excitation cross sections presented here are for the collisions of free electrons with an atomic ion. In the collisions by the rescattering electrons with the parent ion, one should preserve the spin of the total system. Thus if the target is helium, for example, the total spin of the returning electron and the target HeC ion should be spin singlet. In this case, the triplet contributions in (53) and (58) should be dropped.

 /DVHU,QGXFHG 0HGLXP (QHUJ\ (OHFWURQ 'LIIUDFWLRQ RI 0ROHFXOHV The scattering theories presented here are mostly for atomic targets. Similar theories can be and have been generalized to molecular targets. In most cases, electron scattering or photoionization of molecules are rather complicated and will not be described here. With the additional degrees of freedom for molecules, such calculation would be very time consuming. However, there are situations where such calculations are quite simple. It is the so-called Independent Atom Model (IAM). In IAM, a molecule is modeled as consisting of a collection of individual atoms fixed at 5i . Let fi be the complex scattering amplitude of the ith atom alone, according to IAM, the total scattering amplitude for a molecule fixed-in-space can be expressed as X F .k; ; 'I ˝L / D fi exp.i T  5i / ; (60) i

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where ˝L is the angle between the molecular axis with respect to the direction of the incident electron, and T D N  N0 is the momentum transfer. The incident electron momentum N0 is taken to be along the z-axis. The scattering cross section is then given by X Itot .; 'I ˝L / D IA C fi fj exp.i T  5ij / ; (61) P

i ¤j

where 5ij D 5i  5j , and IA D i jfi j2 . Here IA is the incoherent sum of scattering cross sections from all the atoms in the molecule. The second term, IM , is the molecular interference term. For electron scattering from a sample of randomly distributed molecules, the above expression is averaged over ˝L , and hItot i./ D IA C

X i ¤j

fi fj

sin.qRij / qRij

(62)

in which q and Rij are the moduli of T and 5ij , respectively. It is interesting to note that the molecular interference term does not vanish after the average, as pointed out by Fano and Cohen in 1967 [15]. According to IAM, we can define the molecular contrast factor (MCF) as D

1 X sin.qRij / fi fj : IA qRij

(63)

i ¤j

In the traditional gas-phase electron diffraction (GED) where molecules are randomly distributed, an inverse sine transform was used to derive the interatomic separation distributions from (62). In GED, typical incident electrons have energies in the tens and hundreds of keV, and the scattered electrons are collected in the forward directions. From Sect. 2.4, for HATI spectra, clearly the DCS can be extracted from the photoelectron momentum spectra generated by infrared lasers. However, the returning electron for a molecule in a laser field with Ponderomotive energy Up can only reach a maximum energy of about 3:2 Up . For typical 800 nm Ti-Sa lasers, the returning electrons have energies near or below 50 eV. Thus the standard GED theory cannot be applied. However, as shown in Fig. 3, the DCS of CO2 at large angles are well-described by the IAM model for incident electron energies above about 50 eV, for molecules that are randomly distributed. The results in Fig. 3 indicates that DCS extracted from the HATI spectra for molecular targets may be used to obtain the target information if the returning electron energies are above 50 eV. Such returning energies can be easily achieved using mid-infrared (MIR) lasers. Since the IAM is the basis of the GED, this implies that molecular structure can also be retrieved from diffraction images generated by MIR lasers. This further implies that MIR lasers can be used to study the time-dependent structural change of a molecule. The temporal resolution of such measurement is determined by the pulse durations which are in the order of few femtoseconds. Further development of dynamic chemical imaging of molecules with infrared lasers is awaiting for HATI electron spectra from aligned molecules.

Theory of Dynamic Imaging of Molecules with Intense Infrared Laser Pulses

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105





(a) E=20 eV

 



(b) E=50 eV











 

(c) E=100 eV

  



(d) E=200 eV

  

                6FDWWHULQJ$QJOHθGHJ

)LJ  Elastic scattering cross sections at large angles between electrons and isotropically distributed CO2 molecules. The experimental data [16, 17] are compared to the theoretical prediction of the Independent Atom Model, showing poor agreement at 20 eV, but general good agreement at higher collision energies.

 6XPPDU\ In this chapter we summarize the elements of basic scattering theory for dynamic chemical imaging with infrared lasers. These scattering theories have been developed in the conventional energy domain physics for studying the structure of atoms and molecules. Based on the quantitative rescattering (QRS) theory we demonstrated that the same field-free collision theory can be used for describing timeresolved studies of transient atomic and molecular systems. Clearly, even for the time-dependent systems, experimentally it is the energy, momentum or charge of the atomic and molecular systems that are measured. By taking advantage of the basic scattering theories developed over the years directly for the study of timedependent systems, one does not need to develop all the new tools for studying quantum imaging problems. In the meanwhile, by utilizing the coherent nature of the returning electron wave packet or the coherent harmonic light sources, studies in ultrafast atomic and molecular processes would reveal new features that cannot be revealed in the conventional energy-domain studies.

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5HIHUHQFHV 1. I. Hargittai, M. Hargittai (eds.), 6WHUHRFKHPLFDO $SSOLFDWLRQV RI *DV3KDVH (OHFWURQ 'LIIUDF WLRQ (VCH, New York, 1988) 2. A.H. Zewail, J.M. Thomas, ' (OHFWURQ 0LFURVFRS\ ,PDJLQJ LQ 6SDFH DQG 7LPH (Imperial College Press, London, 2009) 3. P.B. Corkum, Phys. Rev. Lett. , 1994 (1993) 4. J.L. Krause, K.J. Schafer, K.C. Kulander, Phys. Rev. Lett. , 3535 (1993) 5. T. Zou, A.D. Bandrauk, P.B. Corkum, Chem. Phys. Lett. , 313 (1996) 6. T. Morishita, A.T. Le, Z. Chen, C.D. Lin, Phys. Rev. Lett. , 013903 (2008) 7. C.D. Lin, A.T. Le, Z. Chen, T. Morishita, R. Lucchese, J. Phys. B , 122001 (2010) 8. Z. Chen, T. Morishita, A.T. Le, M. Wickenhauser, X.M. Tong, C.D. Lin, Phys. Rev. A , 053405 (2006) 9. W. Becker, F. Grasbon, R. Kopold, D.B. Miloˇsevi´c, G. Paulus, H. Walther, in $GYDQFHV LQ $WRPLF 0ROHFXODU DQG 2SWLFDO 3K\VLFV, vol. 48, ed. by B. Bederson, H. Walther (Academic Press, 2002), p. 35 10. E. Hasovic, M. Busuladˇzi´c, A. Gazibegovi´c-Busuladˇzi´c, D.B. Miloˇsevi´c, W. Becker, Laser Phys. , 376 (2007) 11. Z. Chen, T. Morishita, A.T. Le, C.D. Lin, Phys. Rev. A , 043402 (2007) 12. T. Morishita, Z. Chen, S. Watanabe, C.D. Lin, Phys. Rev. A , 023407 (2007) 13. A.T. Le, R.R. Lucchese, S. Tonzani, T. Morishita, C.D. Lin, Phys. Rev. A , 013401 (2009) 14. M. Brauner, J.S. Briggs, H. Klar, J. Phys. B , 2265 (1989) 15. H.D. Cohen, U. Fano, Phys. Rev. , 30 (1966) 16. D.F. Register, H. Nishimura, S. Trajmar, J. Phys. B , 1651 (1980) 17. I. Iga, M.G. Homem, K.T. Mazon, M.T. Lee, J. Phys. B , 4373 (1999)

$E ,QLWLR 0HWKRGV IRU )HZ DQG 0DQ\(OHFWURQ $WRPLF 6\VWHPV LQ ,QWHQVH 6KRUW3XOVH /DVHU /LJKW M.A. Lysaght, L.R. Moore, L.A.A. Nikolopoulos, J.S. Parker, H.W. van der Hart, and K.T. Taylor

$EVWUDFW We describe how we have developed an DE LQLWLR method for solving the time-dependent Schr¨odinger equation for multielectron atomic systems exposed to intense short-pulse laser light. Our starting point for this development is to take over the algorithms and numerical methods employed in the HELIUM code we formerly developed and which has proved highly successful at describing few-electron atoms and atomic ions in strong laser fields. We describe how we have extended the underlying methods of HELIUM to describe multielectron systems exposed to intense short-pulse laser light. We achieve this extension through exploiting the powerful R-matrix division-of-space concept to bring together a numerical method (basis set) most appropriate to the multielectron finite inner region and a different numerical method (finite difference) most appropriate to the one-electron outer region. In order for the method to exploit massively parallel supercomputers efficiently, we M.A. Lysaght School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: NMZTBHIU!RVCBDVL L.R. Moore School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: MNPPSF!RVCBDVL L.A.A. Nikolopoulos School of Physical Sciences, Dublin City University, Dublin 9, Ireland, e-mail: MBNQSPT OJLPMPQPVMPT!EDVJF J.S. Parker School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: KQBSLFS!RVCBDVL H.W. van der Hart School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: IWBOEFSIBSU!RVCBDVL K.T. Taylor School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK, e-mail: LUBZMPS!RVCBDVL A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_8, © Springer Science+Business Media, LLC 2011

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time-propagate the wave function in both regions by employing schemes based on the Arnoldi method, long employed in HELIUM.

 ,QWURGXFWLRQ Over the past decade and more we have developed in Belfast theoretical and computational approaches designed for the accurate and reliable description of one and two-electron atoms and molecules exposed to intense short-duration laserlight [1–7]. Most recently we have extended our efforts to develop methods that are capable of accurately calculating the dynamical response of multielectron atoms (and also molecules) to such light. Our developments were initially prompted, and continue to be stimulated, by the experimental interest [8–14] in strongly time-dependent atomic and molecular systems. (They were also stimulated by the appearance in the mid-90s of the first worthwhile massively-parallel computers, absolutely essential to progress in this area of research.) Advances in laser technology and in detection techniques [15, 16] steadily increase the possibilities and refinement of experimental measurement. For instance, angular information regarding the ionization of two electrons can now be gained and the Free Electron Lasers (FELs) that have become operational are making available unprecedented high laser-light intensities in the ultraviolet to soft X-ray regimes [17, 18]. Alongside advances in the short wavelength regime, the recent generation of light pulses with durations in the attosecond range (1 as D 1018 s) [19] heralds an exciting new era in ultrafast science. These pulses have opened up the possibility for the measurement and possible control of inner-shell electron dynamics inside atoms and molecules. A wide variety of attosecond experimental techniques have been demonstrated so far [20] which have led to the first real-time observations of the decay of an inner-shell vacancy [21] and of interference effects in the double ionization of Ne due to shake-up processes [22]. If theory is to play a meaningful role, and especially a predictive one, in such circumstances, then sophisticated methods of calculation are required. We therefore take the opportunity in this article to bring together various methods (from the R-matrix method onwards) that have been developed in Belfast over the past 40 years and that have recently come together in an efficient and genuinely multielectron treatment of a complex atom exposed to short-pulse laser light. Such a treatment must go after the time-dependent Schr¨odinger equation (TDSE) directly and thus a procedure which efficiently propagates an accurate solution of this equation forward in time (especially on massively-parallel supercomputers) is a first essential. To this end we first developed the Arnoldi method in its application to the TDSE some years ago [1]. Secondly, we found finite-difference (FD) methods invaluable in representing one- or two-electron wavefunctions over large radial distances and, moreover, combined FD methods with the Arnoldi propagator [1]. Thirdly, it is clear that in the ionization of complex atoms almost all the electrons

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remain within 20 bohr or so of the nucleus. But within this 20 bohr a multielectron representation is essential in order to allow for the various photon absorption and electron excitation and exchange processes that can occur there. We can make this multielectron representation appropriately via a close-coupling wavefunction form [23] but, on top of this, the R-matrix method [24] allows an efficient way of limiting this multielectron representation to the appropriate limited region surrounding the nucleus via basis set techniques. Indeed the separation of space by the R-matrix method allows further flexibility. Specifically it makes possible the use of the FD method for the one (or at most two) of the electrons located in the largeradial-extent outer region whilst still invoking a basis set form over the multielectron inner region. This combination taken together with the Arnoldi propagator applied to both regions brings us finally to the desired efficient and genuinely multielectron treatment of a complex atom exposed to short-pulse laser light. The presentation below will reflect such development stages. First an account is given of the Arnoldi propagator (Sect. 2) followed by a description in Sect. 3 of how it has been combined with FD methods to handle ionization of few-electron atoms in intense laser fields. In Sect. 4 we set out some basics of the R-matrix method and show how it can be combined with the Arnoldi propagator in handling just the inner multielectron region. This stage of development has also been reached elsewhere [25]. Finally in Section 5 we bring everything together into an efficient method for complex atoms. Illustrative results for ionization rates and other quantities will be included where appropriate.

 7LPH 3URSDJDWLRQ 8VLQJ 7D\ORU 6HULHV DQG $UQROGL 3URSDJDWRU 0HWKRGV Krylov subspace techniques were originally introduced by Lanczos and Arnoldi to obtain eigenstates of matrices. The techniques are readily adapted to the integration of differential equations. We have used these methods, which we call Arnoldi propagators [26], for time-propagating wavefunction solutions to the one-electron and two-electron TDSEs for hydrogen and helium respectively in an intense laser field. We have also begun recently to deploy them in handling the many-electron TDSE describing the response of a multielectron atom to an intense pulse of laser light. Popular alternatives such as the second-order Crank–Nicholson propagator tend to be either too inaccurate or too inefficient for our applications. The Crank– Nicholson for example has local truncation error (LTE) per time step ıt of order ıt 3 , (and global integration error of order ıt 2 ). In a typical example it may exhibit a LTE of 104 (on wavefunctions that have been normalized to one). We find (perhaps surprisingly) that in typical applications LTE of greater than 107 can result in unacceptably large integration errors. In many applications LTE  1010 is a requirement. This is true for example in calculations of nonsequential double ionization of two-electron atoms at low laser field intensities. To reduce a Crank– Nicholson LTE from 104 to 1010 requires a reduction in ıt by a factor of 100. By

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6 contrast, to reduce a 12th order integration p scheme by a factor of 10 in LTE would require a reduction of ıt by a factor of 10. Our primary test of the Arnoldi propagator has been through comparison with the results of an independently written Taylor series propagator. The Taylor series propagator proves to be particularly reliable but is 2–30 times slower than the Arnoldi one. The Taylor series is perhaps conceptually the simplest of explicit single-step propagators. The better known Runge–Kutta integrators become identical to the Taylor series when applied to time-independent linear differential equations, P D M  , but require more evaluations of M  than the Taylor series if the order of the integrator is greater than four. The Taylor series is formed by repeatedly differentiating the differential equation. In the limit in which the time derivatives of H can be neglected the kth derivative of  is just .iH /k  , and the Taylor series for  .t C ıt/ in terms of  .t/ is:

 .t C ıt/ D c0  .t/ C c1 H  .t/ C c2 H 2  .t/ C    C cn H n  .t/ ;

(1)

where ck are the Taylor series coefficients .iıt/k =kŠ, and H the Hamiltonian. The higher order time-derivatives of H.t/ can easily be included at negligible cost but neglect of these terms is found to introduce no detectable error. The Krylov subspace KnC1 is that spanned by the vectors ; H ; : : : ; H n  . Gram–Schmidt orthogonalization, with iterative refinement, is used to obtain an orthonormal set of vectors that span KnC1 which we write Q0 ; Q1 ; : : : ; Qn , where Q0 D =j j. Qk is obtained by calculating HQk1 and then orthonormalizing this vector with respect to Q0 ; : : : ; Qk1 . We write this as HQk D hkC1;k QkC1 C hk;k Qk C    :

(2)

If we define Q to be formed from the n C 1 column vectors (Q0 ; Q1 ; : : : ; Qn ), then the above equation in matrix form reduces to h D Q HQ. We see then that h is the Krylov subspace Hamiltonian, (i.e., H in the space spanned by Q0 ; Q1 ; : : : ; Qn ). It is clear from the above outline that h is calculated simultaneously with Q, at no extra cost. As Lanczos showed, the eigendecomposition of h can be used as the first step in an iterative scheme to calculate eigenvalues of H . The approach outlined above was first given by Arnoldi [26]. More recently, it has been appreciated that e D QhQ ) can be used as a replacement for H in a wide h (or more accurately H variety of applications [27–31], including the integration of differential equations. e is beneficial because the In our applications, the TDSE, replacing H with H e can be summed to effectively arbitrary order in negligible time Taylor series in H compared to that required for the calculation of ; H ; : : : ; H n  . This follows eıt D Qeihıt Q . The matrix h is typically from .QhQ /m D Qhm Q , so that eiH a 1616 tridiagonal matrix, so that its exponentiation through direct diagonalization of h is inexpensive. The method proves attractive for four reasons. First, despite the work required to orthonormalize the vectors, the computational overhead rises linearly with n. The work is almost entirely in the calculation of ; H ; : : : ; H n  . Even on a paral-

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lel machine, we have found the calculation of h and Q to be less than 10% of the total overhead. Since the method is explicit it obviates the need for costly inversion operations. Second, the method may be viewed as a means of constructing a unitary propagator that is correct to order n in ıt. Third, the Arnoldi formulation provides a very efficient means of obtaining eigenstates of the Hamiltonian. This feature is used in the generation of the ground state of helium and in other initial states of time-dependent calculations. Finally and perhaps most importantly the Arnoldi propagator demonstrates at least twice the efficiency of the Taylor series and the performance ratio improves linearly as the order of the method increases. For example, suppose we set as a constraint the requirement that local truncation errors remain less than 1010 . If we increase the order of the Arnoldi propagator from 8 to 12, then computational cost of each timestep increases by 50% whereas ıt can be doubled. If we increase the order of the Arnoldi propagator from 8 to 16, then computational cost of each timestep increases by a factor of 2 whereas ıt can be quadrupled. By contrast, the performance of the Taylor series remains constant as n is increased. In other words, if n is doubled (doubling the computational cost of each timestep), then ıt can at most be doubled. This scaling law for the Arnoldi propagator favors the highest order possible, but storage limitations for the Q vectors limits n to between 12 and 18.

 )LQLWH'LIIHUHQFH *ULG 0HWKRGV The finite-difference method is a long-established reliable technique for solving PDEs. Each spatial coordinate is discretized on a uniformly spaced grid of points. The evaluation of spatial derivatives at a specific point couples values of the function from neighboring points. If the spatial coordinate is of infinite (or semi-infinite) range as in our work then, since computer memory is finite, some finite upper-limit must be taken. We thus consider representation over a box of finite size. If the box size is chosen too small, the wavefunction can be reflected back from the boundary towards the nucleus. This can alter population, hence affecting ionization rates. If the reflected wavefunction actually reaches the region near the nucleus or nuclei, it can further cause spurious peaks in the harmonic spectra, and spurious ionization rates. Reflections from the edge of the box are eliminated by using a Gaussian masking function to split the wavefunction into two parts. This splitting operation is performed each timestep and ensures that the wavefunction being propagated goes gradually to zero as r ! Rmax . The total population removed by the mask is accumulated in arrays that store ionization yields. A finite-difference grid can be set up over the whole range Œ0; Rmax , but this is not a specific requirement of finite-difference methods. It is possible, and in some situations even advantageous, to divide the interval Œ0; Rmax  into subintervals, within which different representations of r may be used. An example of this is described in Sect. 5, which considers laser-driven multielectron atoms: the radial coordinate of one of the electrons is handled by a basis set method on the interval

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Œ0; Rb  and by a finite-difference grid on the interval ŒRb ; Rmax . But first we focus discussion, in Sect. 3.1 just below, on our approach to solving the TDSE for laser-driven helium. In this case, a finite-difference grid running over Œ0; Rmax  is employed for each of the two radial coordinates in the system.

 0L[HG )LQLWH'LIIHUHQFH DQG %DVLV 6HW 7HFKQLTXHV IRU 6SDWLDO 9DULDEOHV LQ 6SKHULFDO *HRPHWU\ ZLWK $SSOLFDWLRQ WR /DVHU'ULYHQ +HOLXP Handling the electron-laser interaction in the electric dipole velocity gauge, the Hamiltonian H for laser-driven helium takes the following form X 1 Z A.t/ @ 1 2 H D  ri   C (3) 2 ri ic @zi r12 i D1;2

where atomic units (e D me D „ D 1) have been used and A.t/Oz is the vector potential for the linearly polarized electric field of the laser. The single nuclear centre present makes a choice of spherical geometry with all its associated algebraic advantages and selection rules very convenient for this problem. With the restriction that the laser light is linearly polarized with the polarization axis chosen as the zaxis then the overall two-electron wavefunction is rotationally symmetric about this axis and its spatial part can be conveniently written as:  .r1 ; r2 ; t/ D

X l1 ;l2 ;L

1 fl ;l ;L .r1 ; r2 ; t/jl1 ; l2 ; Li r1 r2 1 2

(4)

where the jl1 ; l2 ; Li consist of one-electron spherical harmonics (with orbital angular momentum quantum numbers l1 and l2 ) vectorially coupled to yield an overall orbital angular momentum L with conserved projection M D m1 C m2 D 0 along the z-axis. The three quantum numbers l1 , l2 and L together with the explicit radial coordinates r1 and r2 on the RHS of (4) reflect the 5 degrees of freedom remaining in the electronic motion. The jl1 ; l2 ; Li form a basis set representation of the angular coordinates with the 2 radial variables r1 and r2 handled by finite-difference (FD) techniques over a 2dimensional grid which can be of the order of 3000  3000 points or more. Each of the three operators appearing in the Hamiltonian, r 2 , pz and 1=r12 , is in some way r1 and r2 dependent and must be cast in FD form. Each of these operators presents us with special P difficulties. The 1=r12 operator, for example, which we represent as the series l rlC1 Pl .cos 12 /, has a singularity at r1 D r2 , which falls exactly on the diagonal elements of the lattice. It is far from clear a priori that the series representation of 1=r12 will converge to the correct value as ır ! 0 and as the number of terms in the series increases. Care must be taken to ensure that convergence is possible, to verify that the truncated series approximates 1=r12

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adequately, and to estimate the truncation error. First, we consider two of the most basic parameters in the code, the FD lattice-point spacing, ır, and the timestep used in the propagation, ıt. The box-size (the radius of the integration volume) together with the associated wavefunction splitting techniques have already been addressed in Sect. 3 above. In intense field experiments ionizing electrons with energies well over 100 eV have been detected. We believe that the lattice, at a minimum in such circumstances, should be able to model excitations as high as 500 eV. We have measured excitations (plane waves representing the ionized electron) at energies as high as 270 eV (10 a:u:), and consider it prudent in such cases to allow for energies as high as 500 eV. The question now arises whether or not the largest value of ır we are typically using, namely 0:333 a:u:, is sufficiently small to meet the above requirements. To first approximation, the highest energy plane wave supported by the lattice has a de Broglie wavelength roughly equal to twice ır, 0:666 a:u:, which corresponds to an energy of 44:4 a:u: (1200 eV). We judge this to be adequate although far from ideal. Values of ır D 0:25 a:u: are routinely used, especially for high photon frequencies which are less computationally demanding. As well as the constraints on ıt coming from the numerical methods used, there is an upper limit set by the need to sample physical quantities on a time scale that is short compared to the period Tmin of oscillation of the highest energy excitation, where Tmin D 2=!max . For Emax D „!max D 18 a:u: (  500 eV) we have Tmin D 0:35 a:u:. We use ıt D 0:036 a:u:, hence there are about 10 timesteps per period Tmin , which is adequate. The second derivative operator, d2 =dr 2 , appears as a contribution to the r 2 operator acting on each electron, but applies only to the functions fl1 ;l2 ;L .r1 ; r2 / as defined in (4). The simplest FD representation for d2 =dr 2 is a 3-point formula. However, this results in a 30% error in the helium ground state. As ır ! 0, these errors should disappear. However, we are constrained to a relatively large value of ır and so the 3-point method is unacceptable. A 5-point representation has a smaller error, and it also has the desirable characteristic that it can be tuned to give the correct derivative at the boundaries of the radial lattice, even on the states with the largest finite-differencing errors. To see how this very important property comes about consider, for simplicity, a 1-dimensional FD lattice for a radial coordinate r. The lattice is laid out so that lattice points lie at the two boundaries, r D 0 and r D Rmax . At the two boundaries, r D 0 and r D Rmax , f is 0, so f need not be explicitly calculated there. But this fact is taken into account in the calculation of derivatives of f at points near the boundary. The largest error occurs in the s states, i.e., those states in which l is zero, due to the fact that they have a nonzero derivative at the r D 0 boundary (called the inner boundary). If a 5-point FD rule is used to calculate d2 f =dr 2 near the inner boundary then f must be given a fictional value at r D ır. Freedom is available to us in the choice of f .ır/ and this enables us to improve substantially the degree to which the FD lattice can model the atom, especially the lowest energy states that play a vital role in the problems of interest. The policy that worked best in general was to set

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f .ır/ in such a way that minimized finite-differencing error in the ground state of HeC . This turns out to minimize the error in the ground state of He as well, and as a result the ground states of both singly ionized helium and neutral helium are within 1% of the true values, even with a coarse grid, ır D 0:333 a:u:. The helium ground state energy is calculated to be 2:93 a:u: while the true value is 2:9037 a:u: In actual runs of the code additional corrections are performed to ensure that all bound state energies are correct to 4 significant figures. The quality of the numerical results is sensitive to these states because the ground state of neutral helium is typically the initial state in time-dependent calculations and because the ground state of singly ionized helium is a major component of the screening of the ionizing electron. All of the above theory, algorithms and numerical methods have been brought together in a f90/MPI massively-parallel computer code, HELIUM [1]. A very important application of this code has been to model the full electronic response of the helium atom exposed to intense laser light of 390 nm wavelength [5] which corresponds to frequency-doubled Ti:Sapphire light. In our first calculations for this wavelength [32], we calculated and compared with experiment [33] the ratio of HeCC to HeC yields, taking fully into account the temporal and spatial character of the laboratory laser pulse. Very good agreement was obtained upon applying a unilateral shift upwards by 50% of each experimental measurement of intensity. Calibration of laser intensity at high operating intensities has long been recognized by experimentalists as something notoriously difficult to achieve reliably. We believe that our computational method, by yielding reliable and accurate ionization rates for an experimentally accessible atom (helium), gave at that time the first opportunity for high-intensity lasers to be calibrated reliably in intensity. Calculations using HELIUM have also uncovered many previously unknown features of the electronic response at laser wavelengths of 200 nm and longer. Especially notable amongst these findings has been the discovery [34], subsequently confirmed by experiment [9], that the two simultaneously ionizing electrons can emerge on the same side of the nucleus with equal momenta. This discovery was made by subjecting the time-evolving wavefunction to scientific visualization analysis after first constructing an appropriate lower dimensionality cut in the 5 spatial dimensions. This processing of the calculated wavefunction, especially when a transformation to a different geometry is required, is a nontrivial task requiring a computational effort often comparable to that involved in the initial generation of the time-evolving wavefunction. In our more recent calculations, the spatial wave function gained by the end of the laser pulse has been transformed to momentum space. Accurate transformations to momentum space require retention of the wavefunction’s highest momentum components. We therefore turn off the absorbing boundary potential and consequently, to avoid reflections, the calculations require very large integration volumes. Typically we use finite difference grids extending to 1500 Bohr in each of the two radial coordinates. The increase in computational demand required to handle calculations of this magnitude has been met by the highest performance computers available in the UK. Figure 1 displays a quantitative comparison with laboratory experiment for

)LJ  Electron momentum distributions at 390 nm for singly and doubly ionized helium at (a) 81014 W=cm2 and (b) 11  1014 W=cm2 [5]. Numerical integration and experiment are shown as solid and dashed lines respectively. In the case of double ionization, plotted is the momentum of just one of the ionized electrons, determined independently of the other. )LJ  Joint-probability distribution in momentum space of the doubly ionizing electrons at the end of a 7 field period pulse of intensity 10  1014 W=cm2 [5]. Along the vertical line electron 1 is constrained to kinetic energy 1:9 Up . Along the white circular arc total kinetic energy equals 5:3 Up .

Probability Density (arbitrary units)

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1

(a) Double (x 7273)

0.1 Single 0.01

14

2

I = 8 x 10 W/cm

1 Double (x 3333)

(b)

0.1 Single 0.01 0

14

2

I = 11 x 10 W/cm 1

0.5

2 1.5 Momentum (au)

2.5

3

3.0

1.00

p 1 (au) 0.0

0.5

1.0

1.5

2.0

2.5

0.75

p2

0.50

0.25

0.00

momentum-resolved single- and double-ionization spectra at 390 nm at the two laser intensities 8  1014 W=cm2 and 11  1014 W=cm2 [5]. Figure 2 shows the joint-probability distribution in momentum space of the doubly ionizing electrons at the end of a 7 field period pulse of intensity 10  1014 W=cm2 . It is characterized by a notch, one edge of which is marked by a vertical line at 1:9 Up where one of the electrons exhibits a cutoff in momentum that is independent of the momentum of the other. This notch in the joint-probability distribution has subsequently been observed in laboratory experiment using an 800 nm laser of intensity 4:5  1014 W=cm2 [11].

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 R0DWUL[ %DVLV 6HW 7HFKQLTXHV IRU 6SDWLDO 9DULDEOHV LQ 6SKHULFDO *HRPHWU\ ZLWK $SSOLFDWLRQ WR 0XOWL(OHFWURQ $WRPV Theoretically it is a huge task to treat the exact time-dependent response of a multielectron system exposed to a strong laser field by DE LQLWLR methods. Although the full-dimensional finite-difference approach described in Sect. 3.1 has enjoyed much success at describing single and double ionization of He in strong laser fields, its application to treating complex atoms with more than two electrons is for the moment impracticable due to the increased dimensionality of the multielectron problem. One of the most sophisticated time-dependent methods for describing complex multielectron atoms, such as Ne and Ar, irradiated by short light fields adopts the single-active-electron (SAE) approximation [35, 36], in which the electrons are assumed to be effectively independent and only the electron that is ionized is assumed to be ‘active.’ The model has proven to be very useful in cases where multiple electronic excitations are insignificant. This is typically the case for single electron ionization of atoms irradiated by long wavelength light sources such as the Ti:Sapphire laser where typically the light field interacts most strongly with the outermost electron and multielectron excitations should not be important. However, in the case of multielectron atoms irradiated by short wavelength light sources such as attosecond light pulses or the short-wavelength light generated at FEL facilities, it is more probable that the inner electrons residing closer to the system’s core will interact more strongly with the light field. For example, due to the high frequency components of an attosecond pulse, shake-up states can be populated resulting in Auger transitions that occur on a femtosecond time-scale [22]. Since the light pulse duration is on the order of attoseconds, approaches are required that can reliably describe such ultrafast multielectron rearrangement dynamics. It is clear that such considerations demand approaches which go beyond single-active electron dynamics to accurately describe both the multielectron atomic structure and the multielectron response to the light field. One DE LQLWLR approach capable of treating multielectron systems is R-matrix theory, with the basic formulation appearing first in the context of nuclear theory, and later on applied in the field of atomic physics [23, 37, 38]. In the R-matrix method the position space occupied by the electrons is divided into two regions: An inner region (region I) surrounding the nucleus where a multielectron wave function is constructed and multielectron atom-laser Hamiltonian matrix elements are calculated explicitly, and an outer region (region II), chosen such that only one electron (or at most two) is present and the electron there, besides experiencing the laser field directly, is aware of the remainder of the atomic system only via longrange multipole interactions. Traditionally, R-matrix theory is a theory where time is not explicitly involved in the study of the collision or photoionisation processes. An early application of R-matrix theory to the description of intense laser fields interacting with multielectron atoms was made by combining the traditional timeindependent R-matrix method [24] with a Floquet expansion of the driven timedependent wave function. Over the last two decades the R-matrix Floquet (RMF)

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approach has been successfully employed to describe a wide range of atomic multiphoton processes [39–41]. It is an DE LQLWLR theory, which is fully nonperturbative and is applicable to arbitrary multielectron atoms and atomic ions, allowing an accurate description of electron-electron correlation effects. However, because this theory is based on the Floquet–Fourier ansatz, its applications are confined to treating laser fields consisting of many cycles of the field, typically exceeding several tens of femtoseconds in length. For the description of complex atoms in ultrashort light fields RMF theory is thus inappropriate and a direct integration of the TDSE is required. In this section, we describe an DE LQLWLR time-dependent R-matrix basis (RMB) approach, based on R-matrix theory and associated codes [24, 42, 43], which is capable of describing the multielectron atomic response to intense short light pulses. The method is a spatially limited variant of R-matrix theory which solves the TDSE within a box using R-matrix inner-region basis sets and employs the Arnoldi propagator described in Sect. 2 to propagate the wave function forward in time. This use of R-matrix basis sets within a time-dependent treatment of laser field interactions with multielectron atoms has recently also been developed elsewhere and has been used to study multiphoton single ionization of Ne and Ar [25, 44–46]. Although the basis set method described in this section is in itself capable of describing the interaction of a laser field with a multielectron atom, it should be viewed here as merely forming an initial step in our development of a more powerful time-dependent method which exploits the natural division of configuration space occurring in R-matrix theory and which is described in detail in Sect. 5.

 7KH 50% $SSURDFK Neglecting relativistic effects, the behavior of the N C 1 electron atomic system in the presence of the laser field is governed by the TDSE i

@  .;N C1 ; t/ D ŒHN C1 C DN C1 .t/ .;N C1 ; t/ ; @t

(5)

where the field-free Hamiltonian is HN C1 D

N C1 X i D1

N C1 X 1 2 Z 1  ri  C ; 2 ri rij

(6)

i >j D1

and where the dipole operator DN C1 .t/ is defined in the length gauge as DN C1 .t/ D (.t/ 

N C1 X i D1

Ui :

(7)

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We have taken the origin of the coordinates to be in the nucleus, which we assume has infinite mass. We have rij D jUi  Uj j where Ui and Uj are the vector coordinates of the ith and j th electrons, and we have written ;N C1 [1 ; [2 ; : : : ; [N C1 where [i Ui i are the space and spin coordinates of the electron. The use of the length gauge is in contrast to strong field calculations for one- and two-electron systems in which the velocity form of the dipole operator has been used with advantage. However, for the interaction of the laser field with a multielectron atom near the nucleus, we have found, from previous investigations using basis set methods, that the laser field is in practice best described using the length form of the dipole operator [44, 47]. In order to solve (5) we expand the wave function  .;N C1 ; t/ in terms of eigenstates of the field-free Hamiltonian for each accessible total angular momentum, L  Lmax : X  .;N C1 ; t/ D Ck .t/ k .;N C1 / : (8) k

The basis functions k have the form of a close-coupling expansion with pseudostates [24]. By projecting the eigenstates k .;N C1 / onto (5), the TDSE is reduced to a set of coupled equations for the coefficients Ck , X d Ck .t/ D H.t/kk 0 Ck 0 .t/ ; dt 0

(9)

H.t/kk 0 D ŒHN C1 C DN C1 .t/kk 0 :

(10)

i

k

where The Hamiltonian in the presence of the field has a block-tridiagonal structure, with the diagonal blocks containing the field-free Hamiltonian for each angular momentum, and the off-diagonal blocks containing the dipole couplings between the different angular momenta. We can rewrite (9) in matrix notation as i

d &.t/ D +.t/&.t/ ; dt

(11)

where Œ&.t/k D Ck .t/ and Œ+.t/kk 0 D Hkk 0 .t/. In order to propagate the wave function forward in time we assume that the Hamiltonian changes little over the interval ıt so that we can determine the coefficients &.t/ at time t C ıt as &.t C ıt/ D eiıt +.t / &.t/ :

(12)

The computation of &.t Cıt/ in (12) can be accomplished using several different approaches. As has already been pointed out, the Arnoldi propagator possesses several advantages over other propagators and in consequence we proceed to calculate &.t C ıt/ using the method outlined in Sect. 2. In order to set up the Hamiltonian matrix, +.t/ in (12), we employ the R-matrix II suite of codes [42, 43]. This set of codes provides eigenenergies for an atomic system within a box for a range of angular momenta and reduced dipole matrix elements between all the eigenstates.

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The code has recently been adapted by replacing the standard R-matrix continuum functions by a set of continuum functions based on B-splines [44]. In this R-matrix ‘inner region only’ method the wave function dies off to zero (throughout the propagation) before a suitably large inner-region boundary is reached. This ensures the Hamiltonian +.t/ is effectively Hermitian and also prevents spurious reflections from the boundary. As a proof-of-principle calculation we study the decay of population in the 2p6 1 Se ground state of Ne during its interaction with a 6-cycle laser pulse with a frequency of ! D 0:8 a:u:. We compare results obtained using the RMB method to those obtained using a separate R-matrix inner-region basis set method which employs a Crank–Nicholson scheme to propagate the wave function forward in time [44]. For both sets of calculations we use the R-matrix basis developed for single-photon ionization of Ne [38]. The box radius is set to 100 bohr and the set of continuum orbitals contains 100 B-spline functions of order kspl D 5 for each available angular momentum, l, of the continuum electron. The results presented have been obtained by including only the 2s2 2p5 2 Po ground state of NeC as a residualion state. Details on all the orbital functions used in the calculation can be found in [38]. The description of Ne includes all 2s2 2p5 "l channels up to L D Lmax where Lmax D 5. A pulse shape similar to that used in [44] has been used: it has a three-cycle sin2 turn-on of the electric field followed by a three-cycle sin2 turn off with a peak laser pulse intensity I0 D 1  1013 W=cm2 . For the RMB calculations the eigenbasis is truncated by removing eigenstates with energies Ec > 150 a:u:. This has no adverse effect on the accuracy of the calculations. The removal of these very high lying eigenstates, and the respective transitions between them, allows us to use a time step ıt D 0:01 a:u: with a Krylov subspace dimension of 12. The same time-step was used for the Crank–Nicholson-based calculations. Figure 3 shows the decay in the population of the ground state of Ne during the 6-cycle pulse calculated using the two independent methods. The excellent agreement between the results helps establish the reliability of the current RMB method as a means of accurately propagating the multielectron wave function forward in time.

)LJ  Comparison of the ground state population of Ne as a function of time calculated with R-matrix inner-region methods using an Arnoldi propagator (red dash) and using a Crank– Nicholson propagator (blue solid).

Ground state population

1.002 1 0.998 0.996 0.994 0.992 0

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Time-dependent basis set methods, such as the one described above, offer an accurate means of describing short pulse interactions with multielectron atoms which takes full advantage of sophisticated and detailed computer codes previously developed over many years for time-independent problems. However, there are several constraining factors which limit the applicability of pure basis set approaches. For example, as seen in Sect. 3.1, obtaining accurate energy spectra in the case of a long wavelength pulse interacting with an atom, requires large spatial regions, which need an extremely large set of basis functions to fully span. This renders a pure basis set approach to such a problem computationally impracticable. For this reason, basis set approaches are better suited to describing atoms irradiated by short, high frequency pulses, as is evidenced by their widespread use in this wavelength regime [48, 49]. However, even for the case of a short wavelength pulse interacting with a multielectron atom, the basis set approach has drawbacks. For example, in the case of a sequence of ultrashort pulses interacting with an atom, it has been shown that both a large box size and a large basis set are required to obtain the high momentum resolution needed to observe interferences between ejected wavepackets [45]. It is with such considerations in mind that we have recently initiated the development of a method that meets the requirements of describing complex multielectron systems irradiated by both short DQG long wavelength pulses. To this end we have borrowed from the central concept of R-matrix theory, i.e., the reduction of the laser-atom problem into a ‘complex’ inner region close to the nucleus, in which electron-electron interactions are fully described and for which the method based on R-matrix eigenstates described above appears to be both accurate and tractable, and a ‘simple’ outer region, in which an effective one-electron (or possibly twoelectron) problem is solved using the state-of-the-art grid-based technology outlined in Sect. 3.1. The basics of the method have recently been set out for the hydrogen atom [50], where a high-order Taylor propagator was employed and the feasibility of combining basis set methods with finite-difference methods for describing the laser-atom interaction was firmly established. In what follows we will describe how we have adapted the method to employ the Arnoldi propagator throughout both spatial regions to provide an efficient method for multielectron systems.

 0L[HG )LQLWH'LIIHUHQFH DQG R0DWUL[ %DVLV 6HW 7HFKQLTXH IRU 6SDWLDO 9DULDEOHV LQ 6SKHULFDO *HRPHWU\ ZLWK $SSOLFDWLRQ WR 0XOWL(OHFWURQ $WRPV Figure 4 displays the division of multielectron position space which underlies Rmatrix theory. In region I the time-dependent wavefunction is expanded over the R-matrix eigenstates of the field-free Hamiltonian. Region I is defined by all N C 1 electrons of the system having a radial coordinate rq  b, q D 1; : : : ; N C 1. Region II is defined by N electrons having a radial coordinate rq  b, q D 1; : : : ; N and one electron having its radial coordinate rN C1 > b. In region II the time-

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Finite differences r R−matrix basis rN Region I OUTER

b

ψII

INNER

ψI

Region II

II

I b−h b+h 0 r(0)

R r(i R )

b−2h b r(ib)

)LJ  Partition of the configuration space for the electron coordinate. In the internal region I an eigenstate expansion representation of the wavefunction is chosen, while in the external region II a grid representation is considered. The boundary of region I is at r D b and the outer boundary of region II is at r D R. The radial variable of the .N C 1/th electron is denoted as r. The grid points in region II are denoted by i, where ib is the grid point on the boundary with region I and iR is the grid point on the outer region boundary. The spacing between grid points is denoted by h.

dependent wavefunction is represented over a finite-difference grid by its values at equidistant grid points rN C1 .i / D ih, i D ib ; ib C 1; : : : ; iR . This section is organized as follows. In Sects. 5.1 and 5.2 we describe our approach to solving the TDSE in the outer region (region II) and in the inner region (region I) respectively. In Sect. 5.3 we present preliminary results pertaining to the single-ionization of helium and neon.

 2XWHU 5HJLRQ Following R-matrix theory [24], the time-dependent .N C1/-electron wavefunction beyond a certain distance can be written as  .;N C1 ; t/ D

X p

1 ˚p .;N I UO N C1 ; N C1 / fp .r; t/; r

r b

(13)

where the radial variable of the .N C 1/th electron is denoted as rN C1 D r. The ˚p .;N I UO N C1 ; N C1 / are channel functions formed by coupling the target states of the residual ionic system ˚p .;N / with angular and spin parts of the ejected electron wave function. The time dependence of the wavefunction is contained within the radial channel functions fp .r; t/ describing the radial motion of the ejected elec-

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tron (in the pth channel). The antisymmetrization operator is absent from the wavefunction expansion because only the .N C 1/th electron is present at large radial distances, thereby making it distinguishable from the remaining N electrons. Because the number of electrons in the outer region (region II) is limited to just one, the TDSE in the outer region is, for each state of the residual-ion, reduced in dimensionality to at most three, thus simplifying the computational problem considerably. The TDSE in the outer region is given by (5). By projecting the known target functions ˚p .;N I UO N C1 ; N C1 / onto the TDSE and integrating over all spatial variables except the radial coordinate of the ejected electron, the following set of coupled partial differential equations for the radial channel functions fp .r; t/ is obtained: i

@ fp .r; t/ D hIIp .r/fp .r; t/ @t X  C WEpp0 .r/ C WDpp0 .t/ C WPpp0 .r; t/ fp0 .r; t/

(14)

p0

where 1 d2 lp .lp C 1/ Z  N C  C Ep ; 2 2 dr 2r 2 r ˇN ˇ

 ˇX 1 N ˇˇ 1 1 ˇ 0 WEpp0 .r/ D r ˚p ˇ  ˇr ˚p ; jr  r j r hIIp .r/ D 

j D1

j

ˇ ˇ  N X ˇ ˇ 1 ˇ ˇ WDpp0 .t/ D r ˚p ˇ(.t/  UL ˇr ˚p0 ; i D1 ˇ ˇ ˝ ˛ WPpp0 .r; t/ D r 1 ˚p ˇ(.t/  Uˇr 1 ˚p0 :

1

(15) (16)

(17) (18)

WE is referred to as the long range potential in the R-matrix literature [24] and arises from the electron-electron and electron-nuclear potential terms in the Hamiltonian. WD and WP arise respectively from the interaction of the light field with the residual N -electron ion and from the interaction of the light field with the ejected electron and are defined in the above equations in the length gauge. Ep is the energy of the residual N -electron ion and lp is the angular momentum of the outgoing electron. To solve the set of coupled partial differential equations given in (14) we employ finite difference methods. The radial channel functions fp .r; t/ are discretized on an equidistant grid as depicted in Fig. 4. The second derivative operator in (14) is recast in a 5-point finite difference representation, in the same way as was done for the helium application discussed in Sect. 3.1 above. In this case however, the inner boundary of the finite-difference region is not at the nucleus, where r D 0, but at some distance from the nucleus at r D b. The 5-point finite-difference rule when applied to the radial channel function at the first grid point, i D ib , thus requires information on the value of the radial channel function at points i D ib  1 and i D ib  2. (The function value at i D ib  1 is also needed when the rule is

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applied to the function at grid point i D ib C 1.) These two grid points lie just inside the outer boundary of the inner region. Since the wavefunction is assumed to be one-electron in nature moving outwards from the boundary at r D b, then, so long as b is chosen large enough, it can also be assumed that the wave function has still a one-electron character at the slightly smaller radial distance corresponding to the inner region grid point at i D ib  2 (this can also be enforced). In the inner region, the wavefunction is given by (8). By projecting onto the target orbitals ˚p .;N I UO N C1 ; N C1 / and integrating over all spatial variables except r (the radial coordinate of the ejected electron), the following radial wavefunction for the ejected electron in the inner region can be defined: X fNp .r; t/ D Ck .t/fpk .r/; 0r b: (19) k

This equation is one of the main relations within our development because it links the time-dependent solutions of the inner and outer regions at the inter-region boundary (at r D b). By requiring fNp .b; t/ D fp .b; t/ we obtain the following relation: X fp .b; t/ D Ck .t/!pk .b/ (20) k

where !pk .b/ are the known channel amplitudes evaluated on the boundary. They are defined explicitly by (29) below. In an analogous fashion, we can obtain expressions for the radial wavefunctions at distances corresponding to the inner region grid points at i D ib  1 and i D ib  2 just inside the boundary at r D b: X fp .b  h; t/ D Ck .t/!pk .b  h/ ; (21) k

fp .b  2h; t/ D

X

Ck .t/!pk .b  2h/ :

(22)

k

Thus, knowing the basis expansion coefficients, Ck .t/ at time t means that the radial wavefunction of the ejected electron at inner-region points i D ib  1 and i D ib  2 can be obtained at time t. The finite-difference Hamiltonian may then be applied to the wavefunction in the outer region. To propagate the outer region wavefunction forward in time, from time t to time t C ıt, we use the Arnoldi propagator discussed in Sect. 2 above. We assume that at time t the wavefunction is known throughout the inner and outer regions, a valid assumption since the ground state wavefunction will be known initially at t D 0. The first step is to form the radial wavefunctions of the ejected electron in the inner region at r D b  h and r D b  2h using (21) and (22). This enables the outer region H  operation to be calculated. However, propagation involves computing not only H  but also H 2 ; H 3 ; : : : ; H n  where n is the maximum propagation order. Therefore at the points i D ib  1 and i D ib  2 it is not only the values of fp .r; t/ which must be established, but also the values of Hfp .r; t/; H 2 fp .r; t/; : : : ; H n1 fp .r; t/. One way in which these additional quan-

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tities may be calculated is by repeatedly applying the outer region Hamiltonian to fp .r; t/ at the grid points i D ib  1 and i D ib  2. This ought to be permissible as the wavefunction at these two grid points should already consist only of the wavefunction of the one ejected electron. Once these quantities are calculated, the outer region wavefunction may be propagated forward one step in time using the methods outlined in Sect. 2. We turn now to a discussion on the propagation of the inner region wavefunction.

 ,QQHU 5HJLRQ The TDSE in the inner region is given by (5). However, in the inner region, the Hamiltonian HN C1 is not Hermitian owing to the presence of the kinetic energy  12 ri2 term in (6) and the finite value of the wave function on the inner-region boundary. Consequently we introduce the Bloch operator LN C1

 N C1 1 X g0  1 d D ı.ri  b/  ; 2 dri ri

(23)

i D1

which is such that HN C1 C LN C1 is Hermitian in the internal region for any value of the arbitrary constant g0 . The TDSE in the internal region may then be written as i

@ I .;N C1 ; t/ D HI .t/I .;N C1 ; t/  LN C1  .;N C1 ; t/ ; @t

(24)

HI .t/ D HN C1 C LN C1 C DN C1 .t/

(25)

where and where I .;N C1 ; t/ is the wave function defined over region I in Fig. 4. Equation (24) is a key one to the method. The second term on the right-hand side compensates for the Bloch term introduced to make HI Hermitian. Note that it makes a contribution only at r D b and brings into play there  .;N C1 ; t/, a wave function form which we have defined throughout both regions. This term is central to any time propagation scheme in region I because it connects the wave function form I .;N C1 ; t/, which is multielectron in nature, with a wave function form that, at r D b, represents a single electron and which, numerically, is obtained from region II. By projecting (24) onto the eigenstates k .;N C1 / we obtain the evolution equations for the time-dependent coefficients Ck .t/: i

X d Ck .t/ D HIkk0 .t/Ck0 .t/  h dt 0

k jLN C1 j i

:

(26)

k

After some algebra we can rewrite the evolution equations for the time-dependent coefficients as

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125

ˇ X d iX @fp .r; t/ ˇˇ Ck .t/ D i HIkk0 .t/Ck 0 .t/ C !pk dt 2 @r ˇ k0

where

p

1 0 fp .r; t/ D h˚p rN C1 j irN C1 Dr ;

;

(27)

rDb

(28)

with the prime on the matrix elements denoting that the integral is carried out over space and spin coordinates of all N C 1 electrons except the radial coordinate rN C1 of the ejected electron. These reduced radial functions, fp .b; t/ are analytical continuations of the functions defined in Sect. 5.1. The surface amplitudes, !pk have also been mentioned in Sect. 5.1 and are defined by 1 !pk D h˚p rN C1 j

0 k irN C1 Db

:

(29)

In the rest of this subsection we present a numerical method for solving (27)) which enables the inner region wave function to be propagated forward one timestep from time t to time t C ıt. Equation (27) differs fundamentally from (9) governing the time-dependent coefficients of the pure basis set approach described in Sect. 4.1. The latter equation represents a homogeneous differential equation whereas (27) represents an inhomogeneous differential equation due to the inclusion of the Bloch surface terms on the R-matrix boundary. We rewrite (27) in matrix notation as

where

d &.t/ D i+I .t/&.t/ C i 6.t/ ; dt

(30)

ˇ 1X @fp .r; t/ ˇˇ Sk .t/ D !pk : 2 p @r ˇrDb

(31)

The inhomogeneous TDSE of the form given in (30) arises in many areas of quantum dynamics. In particular, it has been used in a time-dependent treatment of reactive scattering [51] and in optimal control theory using state dependent constraints [52]. To date, the inhomogeneous TDSE has been solved using splitpropagator schemes [53] or via full diagonalization of the Hamiltonian [52]. It has also recently been solved by employing a semi-implicit Crank–Nicholson method within an earlier time-dependent R-matrix framework [54]. However, it is only recently that attempts have been made at using high-order explicit methods to solve equations of the type given by (30). Recently, a formal solution of the inhomogeneous TDSE has been derived which has been adapted to a Chebyshev propagation scheme [55]. An alternative simplified version of the high-order propagator for the inhomogeneous TDSE can also be devised by approximating the formal solution explicitly by a Taylor expansion [55]. This is the method used for the single-electron implementation of the current method [50]. While the Taylor expansion approach has been shown to give accurate results for the case of hydrogen [50], it suffers from the same drawbacks as those outlined in Sect. 2. For this reason we tackle the prob-

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lem of solving the inhomogeneous TDSE using a Krylov subspace based approach similar to that outlined in Sect. 2. As shown in [55], the Taylor series for &.t C ıt/ in terms of &.t/ is: &.t C ıt/ D

1 X

ıt j

j D0

1 X .iıt+I /k 8j .t/ ; .k C j /Š

(32)

kD0

where

dj 1 6.t/ : dt j 1 Equation (32) can be rewritten in terms of so-called functions [56] as 80 .t/ D &.t/ ;

8j .t/ D i

&.t C ıt/ D exp.iıt+I /&.t/ C

1 X

ıt j j .iıt+I /8j .t/ ;

(33)

(34)

j D1

The j .iıt+I / functions are related to the exponentiation of the Hamiltonian matrix and are due to the time-dependent inhomogeneity in (30)). For a scalar argument z the j .z/ functions are defined by the integral representation 1 j .z/ D .j  1/Š

Z

1

e.1 /z  j 1 d ;

j 1:

(35)

0

For small values of j , these j .z/ functions are given by: 1 .z/ D

ez  1 ; z

2 .z/ D

ez  1  z ; z2

3 .z/ D

ez  1  z  z 2 =2 : (36) z3

The j .z/ functions satisfy the recurrence relation j C1 .z/ D

j .z/  1=j Š ; z

0 .z/ D ez ;

j D 0; 1; 2; : : :

(37)

and have the Taylor expansion j .z/ D

1 X kD0

zk : .k C j /Š

(38)

By suitably truncating the series in (34), we obtain the approximation &.t C ıt/  b &.t C ıt/ D exp.iıt+I /&.t/ C

nj X

ıt j j .iıt+I /8j .t/ : (39)

j D1

The first term on the right-hand side of (39) is the familiar solution to the homogeneous TDSE encountered in Sect. 4.1 for the pure basis set method. In order to calculate this term we use the same procedure as that outlined in Sect. 2. For the

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calculation of the second term on the right-hand side of (39) we also use a Krylov subspace based method to calculate the action of the j .iıt+I / functions on the 8j .t/ vectors passed from region II (see Fig. 4). The method is similar to that used to calculate the action of exp.iıt+I / on &.t/. We form a Krylov subspace for each of the vectors j .iıt+I /8j entering the summation of (39) using the same approach outlined in Sect. 2. We begin by approximating the vector j .iıt+I /8j in a Krylov subspace KnC1;j spanned by the vectors 8j ; +I 8j ; : : : ; +In 8j . Following the standard Arnoldi method we apply the Gram–Schmidt procedure to obtain an orthonormal set of vectors that span KnC1;j which we write as 40;j ; 41;j ; : : : ; 4n;j where 40;j = 8j =j8j j. Let 4j denote the matrix formed by the nC1 column vectors

(40;j ; 41;j ; : : : ; 4n;j ). Then the n-by-n matrix Kj D 4j +I 4j is the projection of the action of +I on the Krylov subspace KnC1;j expressed in terms of the basis f40;j ; : : : ; 4n;j g.

We now approximate j .iıt+I /8j by j .iıt4j Kj 4j /8j . Since 4j 4j D , and 4j 4j 8j D 8j we have j .iıt4j Kj 4j /8j = 4j j .iıtKj /4j 8j . The advantage of this formulation is that since the matrix Kj has size order n the evaluation of j .iıtKj / is much cheaper than that of j .iıt+I /. In recent years, there has been considerable effort made at developing efficient and accurate methods for computing the j .z/ functions. The most popular method for computing the matrix exponential (the 0 .z/ function) is the scaling and squaring method combined with the Pad´e approximation [57]. This method has recently been adapted to compute the closely related j .z/ functions with j  1 [56]. The first method to take advantage of the Krylov subspace approach to calculating the action of j .z/ functions on a vector was developed several years ago [58]. In this method, the Arnoldi algorithm was extended in order to compute the 1 .z/ function that appears in the solution of linear ordinary differential equations with constant inhomogeneity. This work has very recently been generalized to compute the action of j .z/ functions on a vector with j  1 [59]. In our current implementation the calculation of the j .z/ functions is tackled by following an idea set out in [60] and generalized in [58] whereby the reduced matrix Kj is augmented to form a larger matrix given by 2 3 Kj H1 0 KO j D 4 0 0 , 5 ; (40) 0 0 0 where H1 is the first vector in the standard basis and the identity matrix, ,, has j  1 rows. Following the augmentation of Kj , the top n entries of the last column of exp.iıt KO j / yield the vector j .iıtKj /H1 . As an initial approach to computing the j .z/ functions we have followed this augmentation method and in the final step we compute exp.iKO j ıt/ using a degree-6 Pad´e approximant for general matrices, combined with scaling and squaring. The procedure outlined above is repeated for each of the j .iıt+I /8j terms in the summation on the right-hand side of (39). Summing these nj terms and adding to the first term on the right-hand side of (39) then provides us with a means of

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propagating the wave function in the R-matrix inner region forward one step in time. By this stage the wavefunction is known at time t Cıt throughout regions I and II and we can progress further in time by repeating, for successive time steps ıt, the procedures described both in Sect. 5.1 above and in this section.

 5HVXOWV As a first step to verifying the accuracy of the new mixed R-matrix with timedependence (RMT) method we investigate electron wavepackets ejected from Ne irradiated by a laser pulse with a photon energy ! D 21:8 eV and compare results with those obtained using the RMB method described in Sect. 4.1. For both the RMT and RMB calculations, we use in the inner-region the R-matrix basis developed for single-photon ionization of Ne [38] as done in Sect. 4.1. In the case of the RMT calculations, we use an inner region radius, b D 20 bohr. The set of continuum orbitals contains 60 continuum functions for each available angular momentum l of the continuum electron. The results presented here have once again been obtained including only the 1s2 2s2 2p5 2 Po ground state of NeC as a target state. The description of Ne includes all 1s2 2s2 2p5 "l channels up to L D Lmax where Lmax D 5. For both sets of calculations the eigenbasis is truncated as in Sect. 4.1, once again making sure this has no adverse effect on the accuracy of our calculations. This allows us to use a time step ıt D 0:01 a:u: with a Krylov subspace dimension of 12. In the RMT outer region we propagate the wave function outwards to a radial distance of 100 bohr in order to prevent any reflections of the wave function from the outer region boundary. In the outer region, the finite difference mesh spacing (h in Fig. 4) is ır D 0:2 bohr. For the RMB calculations, we use a box size of 100 bohr with 100 continuum functions for each available angular momentum l of the continuum electron. The pulse shape is the same as that used in Sect. 4.1. The laser pulse has a peak intensity I0 D 1  1013 W=cm2 . Figure 5 shows the real part of the 1 S (l D 1) partial continuum wave function of Ne calculated 0.2 fs after the end of the 6 cycle 21:8 eV laser field. Ionization to the 1 S (l D 1) continuum channel occurs during two-photon absorption from the laser pulse. The blue and red solid thin lines show the inner and outer region parts of the wave function respectively obtained using the present RMT method. The RMT wavefunction is compared to that obtained using the RMB method (black dash line) described in Sect. 4.1. The vertical black dash-dot line at r D 20 bohr indicates the position of the R-matrix boundary, b. The comparison of the time-dependent wave functions obtained using the RMT method with those obtained using the accurate and well established RMB method represents one of the most stringent tests of accuracy for the new RMT method. The excellent agreement between the results of the two independent methods, as is evident in Fig. 5, helps verify the accuracy and reliability of the RMT method. We have also compared the wavefunctions for the

Real part of continuum wave function

$E ,QLWLR Methods for Few- and Many-Electron Atomic Systems

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−5

6

x 10

4

RMT outer region RMT inner region RMB

2 0 −2 −4 −6 0

20 40 60 80 100 Radial distance of outgoing electron (Bohr)

)LJ  Real part of the 1 S (l D 1) partial continuum wave function of Ne calculated 0:2 fs after the end of the 6 cycle 21:8 eV laser field. The wave function obtained using the present RMT method has its outer region part shown by a red solid line and its inner part by a blue solid line. The RMT wavefunction is compared to that obtained using the RMB method (black dash line) described in Sect. 4.1. The values of the wave function close to the nucleus (r  9:75 bohr) are not shown due to these values being much larger there than beyond the R-matrix boundary, r D b.

other ionization channels included in the calculation and have found the same level of agreement between the two methods. As a second means of demonstrating the accuracy of the RMT method, we calculate single-electron ionization rates for He and Ne irradiated by a laser pulse with a central wavelength of 248 nm corresponding to the fundamental wavelength of the KrF laser. We compare these rates with ionization rates obtained using the timeindependent RMF method and, in the case of He, also with the HELIUM code (described in Sect. 3.1 above). For Ne we use the same atomic structure basis as described above. For Helium we use the same basis as described in [61] with the Rmatrix boundary, b D 20 bohr. In both sets of calculations we use only the ground state of the residual ion as a target state and typically use Lmax D 11–15. In order to obtain RMT ionization rates that are comparable to the RMF results we irradiate He and Ne with pulses that typically consist of 20–30 cycles of the electric field, and therefore extend the outer region boundary to a radius of typically 500–1000 bohr. The ionization rate is obtained using the RMT method by calculating the decrease in the norm of the inner region wave function beyond the R-matrix boundary distance, b D 20 bohr for both He and Ne. Table 1 shows a comparison of helium single-electron ionization rates for four intensities. Away from resonances, the ionization rates calculated by RMT agree well (to within 10%) with those calculated by HELIUM [62] and by the RMF method [61]. At an intensity of 1:78  1014 W=cm2 there is a resonance with the 1s4p 1 Po and 1s4f 1 Fo states. At this intensity the HELIUM ionization rate is higher than the RMF rate by 22%. However, there is an uncertainty of ˙15% in the rate calculated by HELIUM at this intensity. This is because the time-evolution of the

M.A. Lysaght et al.

130

7DEOH  Single-ionization rates in atomic units obtained by RMT, by HELIUM and by the threestate approximation in the RMF approach for given values of the peak laser intensity. Intensity (1014 W=cm2 )

RMT

HELIUM

RMF

0.5 1.78 3.2 4.0

1:18  108 2:77  107 1:27  105 3:23  105

1:25  108

1:13  108 2:76  107 1:23  105 3:02  105

1:21  105 3:15  105

decay of the inner region norm is complicated, exhibiting long-period oscillations and thus requiring integrations over greater than a hundred field periods. To avoid this complication and hence obtain a more precise ionization rate with the RMT method, we measured the increase in population in the outer region beyond an extended radial distance of 250 bohr, a distance from the origin which is more than sufficient to contain the resonant states. The resulting RMT ionization rate is in very good agreement with the RMF rate. Figure 6 shows the corresponding ionization rates for Ne irradiated by a laser pulse similar to that used in the He calculations described above. The ionization rates are compared to those obtained using the RMF method. Agreement between the two sets of results is vey good, typically within 10% of each other away from resonance. Throughout our calculations we have also compared our results with those calculated using a separate time-dependent R-matrix method [54,63] and have found excellent agreement throughout. This time-dependent R-matrix method also divides configuration space into an inner and outer region. However, instead of using a finite-difference method in the outer region it employs an R-matrix-propagator technique based on Green’s functions. The method employs a Crank–Nicholson propagator to time propagate the wave function and has been successful at describing ultrafast correlated electron dynamics in multielectron atoms [64].

10

14

12

-1

)LJ  Comparison of ionization rates for Ne irradiated by 248 nm laser light as a function of intensity. Rates from the current RMT approach (solid red circles) are compared to those obtained using the R-matrix Floquet approach (solid black line).

Rates (s )

10

10

10

10

8

0

1

2

3

Intensity (10

4 14

5 -2

W cm )

6

$E ,QLWLR Methods for Few- and Many-Electron Atomic Systems

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 &RQFOXVLRQ We have reviewed recent progress made in Belfast towards the goal of developing accurate theoretical methods for the description of atoms and molecules exposed to intense short laser pulses. Our recent work builds on top of well-established DE LQLWLR methods developed in Belfast over the last 40 years. One of the most successful of these methods has proven to be the full-dimensional finite-difference method and its associated computer code, HELIUM [1]. When combined with the Arnoldi propagator the finite-difference method allows for the highly accurate description of the response of few-electron atoms and molecules to short intense laser pulses. The capability of the HELIUM code relies on its efficient implementation on massively parallel supercomputers and its success is clearly evidenced by the results presented here for the double-electron ionization of He exposed to intense 390 nm laser light. We went on to describe our developments leading towards an DE LQLWLR timedependent method for the description of multielectron systems irradiated by intense short laser fields. The method borrows from the powerful concept of R-matrix theory which divides the position space of the electrons surrounding the nucleus of the atom into a ‘complex’ inner region and a ‘simple’ outer region. We described how we first developed a spatially limited variant of the R-matrix method, which we adapted to allow for an explicitly time-dependent treatment of the laser-atom interaction. The method shares similarities with the close-coupling description of the multielectron wave function and uses R-matrix basis sets to describe the wave function. We have adopted the Arnoldi propagator (already used for more than a decade in the HELIUM code) for the time propagation of the multielectron wave function and have demonstrated the reliability of the method by studying the interaction of Ne with a short wavelength laser pulse and comparing results with a separate time-dependent R-matrix basis method. Nevertheless, we only begin to truly exploit the R-matrix concept when we combine the inner region with an R-matrix outer region which has been the most recent stage of our progress. In the outer region we use a grid representation of the wave function where propagation in space is obtained through the finite-difference method. In this region we also employ the Arnoldi propagator to propagate the wave function in the time-domain. Combining the inner-region with the outer region requires the solution of an inhomogeneous TDSE in the inner region, which we have solved by extending significantly the Arnoldi propagator used there in the spatially limited variant of the code. We have demonstrated the accuracy of the new method by finding excellent agreement between Ne continuum wave functions obtained using the current method at the end of a short wavelength laser pulse with those obtained using a separate time-dependent RMB method. We have also compared ionization rates for He and Ne irradiated by a 248 nm laser pulse obtained using the current method with results obtained using the time-independent R-matrix Floquet method and have found very good agreement. While the current method has so far proved successful at describing the singleelectron ionization of multielectron atoms, further development work is required. For example, we currently employ the length gauge in both spatial regions for the

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description of the laser-atom interaction. However, a transformation to a velocity gauge description in the outer-region is considered preferable due to the lower number of angular momenta required to obtain convergence in this gauge in this region. In the near future we intend to adapt the method so as to accurately describe the single-electron ionization of a multielectron molecule irradiated by intense short laser pulses. We also see the current method as forming a natural prerequisite for the development of a method that is capable of accurately describing the doubleelectron ionization of atoms and molecules exposed to intense short laser pulses. Undoubtedly, such large-scale computations will involve design of computer codes that run efficiently on massively parallel supercomputers. Since the underlying numerical methods in the inner-region and outer-region are considerably different, significant emphasis will be placed on the efficient load-balancing of the code throughout this development phase. $FNQRZOHGJHPHQWV MAL and JSP acknowledge funding from The Numerical Algorithms Group (NAG) Ltd. LRM, HWvdH and KTT acknowledge funding from the UK Engineering and Physical Sciences Research Council. LAAN acknowledges funding from the Science Foundation Ireland Stokes Lectureship Programme.

5HIHUHQFHV 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

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6WURQJ)LHOG ,RQL]DWLRQ RI 0ROHFXOHV 6LPSOH $QDO\WLFDO ([SUHVVLRQV Ryan Murray, Serguei Patchkovskii, Olga Smirnova, and Misha Yu. Ivanov

$EVWUDFW We use the method developed in [1] to derive simple expressions for strong-field ionization rates of molecules, as a function of their alignment with respect to the electric field. Our analytical expressions offer transparent physical interpretation, and demonstrate the connection between the geometry of the ionizing molecular orbital and alignment-dependent ionization rates, including the sensitivity of the angular patterns to the field strength. We also show how static tunnel ionization rates can be generalized to describe the subcycle ionization dynamics in oscillating laser fields.

 ,QWURGXFWLRQ We present simple analytical formulas describing ionization of molecules in static and strong low-frequency laser fields. Our approach allows us to achieve the same Ryan Murray Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada & Department of Physics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK, e-mail: SZBONVSSBZ!JNQFSJBMBDVL Serguei Patchkovskii Max-Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Strasse 2A, D12489 Berlin, Germany, e-mail: 4FSHVFJ1BUDILPWTLJJ!OSDDB Olga Smirnova Max-Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Strasse 2A, D12489 Berlin, Germany, e-mail: TNJSOPWB!NCJCFSMJOEF Misha Yu. Ivanov Department of Physics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK, e-mail: NJWBOPW!JNQFSJBMBDVL  On leave from the National Research Council of Canada, 100 Sussex Dr., Ottawa, ON K2A 0R6, Canada

A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_9, © Springer Science+Business Media, LLC 2011

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level of analytical simplicity for molecular ionization as for atomic ionization, while maintaining complete physical transparency of all essential characteristic features associated with molecular alignment and orbital geometry. In particular, we explicitly show how, when, and to what extent tunnel ionization maps the geometry of the ionizing orbital, and how the strength of the ionizing field affects this mapping. Ionization of an atom or a molecule is at the core of nearly every process in strong laser fields, from high harmonic generation (HHG) to laser induced electron diffraction (LIED), generation of high-energy electrons, correlated multiple ionization, etc. (see, e.g., [2]). The ionization of atoms in strong low-frequency laser fields has been studied theoretically and experimentally over several decades, and now appears well understood. Among the most important theoretical advancements are semianalytical models providing adequate quantitative description of the spectra of the so-called direct electrons, including the effects of the ionic core during and after ionization [3, 4] and the subcycle dynamics of nonadiabatic tunneling [4–9] for moderate   1 Keldysh parameters [10]. General understanding of atomic ionization has been extrapolated to molecules [11, 12]. However, it has now become clear that straightforward extrapolation is insufficient, stimulating theoretical efforts [13–19]. There are many important reasons for the renewed interest, including the advent of high harmonic generation spectroscopy of multielectron dynamics in molecules [20–22]. This spectroscopy requires one to accurately calculate relative contributions of several molecular orbitals (electronic states of the molecular ion) participating in the high harmonic generation process [23, 24]. We propose simple and physically transparent upgrade of the existing theories. In atoms, our approach gives results identical to the classic results of Perelomov, Popov and co-workers [5–7] (see [1] for details), later popularized by Ammosov, Delone and Krainov [25] and now commonly referred to as the ADK theory. In molecules, the well-known molecular ADK theory (MO-ADK) [12] can be obtained as a limiting case of our approach, within clearly defined approximations [26]. However, our general approach does not require spherical symmetry of the binding potential. In contrast to the standard tunneling approach of Perelomov, Popov and coworkers [5–7] and later work (ADK, MO-ADK) [12, 25], we explicitly keep track of the angle between the direction of the electric field and the characteristic direction of electron tunneling. The degree to which the tunneling electron can deviate from the direction of the electric field depends on the field strength, and is chiefly responsible for intensity-dependent features in the angle-resolved ionization rates for aligned molecules. Simple analytical formulas can be derived using the additional assumption of a purely Coulombic asymptote of the core potential far from the nucleus. With this assumption, we are able to approximate the ionization rate as a product of two factors. The first is the tunneling factor, associated with the transmission amplitude during the electron motion in the classically forbidden region. The second is the geometrical factor, which reflects the geometry of the molecular orbital and the interference of tunneling currents originating from different lobes of the ionizing orbital. Within

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this formalism, the generalization to tunneling in oscillating laser fields becomes fairly straightforward, and is discussed in the conclusion of the paper.

 *HQHUDO $SSURDFK Let z be the direction of the electric field that induces tunnel ionization. The ionization rate is given by the total current through the plane orthogonal to z: Z 1 (1) dx dy   .x; y; z/pOz .z/ .x; y; z/ C c:c:  D 2 where atomic units e D me D „ D 1 are used, pOz is the electron momentum operator orthogonal to the x-y plane,  and   are the wave function of the tunneling electron and its complex conjugated. The continuity equation ensures that the total current is z-independent after the tunneling electron exits the potential barrier. To calculate the rate, we use our approach described in [1], where it has been verified for atomic systems. At some point z0 in the classically forbidden region, sufficiently far from both the entrance zin and the exit zex from the tunneling barrier (see Fig. 1), we rewrite the wave function of the tunneling electron  .x; y; z0 / in the mixed coordinate-momentum representation Z Z 1  .x; y; z0 / D (2) dpx dpy eixpx iypy ˚.px ; py ; z0 / 2 where, ˚.px ; py ; z0 / D

1 2

Z

Z dx

dy eixpx Ciypy  .x; y; z0 / :

(3)

Energy

in

ex

)LJ  Schematic of tunneling, with zin and zex the entrance and the exit point from the barrier; z0 is the matching point.

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The mixed representation wave function ˚.px ; py ; z0 / can be effectively propagated under the barrier using the semiclassical (WKB) method. A key approximation, previously used in the classic papers by Popov and co-workers [5–7], is that of small deviations between the tunneling trajectories and the z axis. Then, the wave function at a point z is approximated as [1]: ˚.px ; py ; z/ D ˚.px ; py ; z0 /aT .z0 ; z/ ; s  Z z  .px2 C py2 /T jpz .z0 /j 0 0 aT .z0 ; z/ D exp  pz .z / dz  : jpz .z/j 2 z0

(4)

The tunneling amplitude, which has been derived previously [1, 5–7] is the WKB solution describing the motion of a particle under a 3D barrier, with the core (ionic) potential treated perturbatively compared to the electric field. This approximation explains the 1D-like form of the expression for aT .z0 ; z/, with the 3D nature of the motion appearing only through the momenta px ; py . In the classically forbidden region the electron momentum is imaginary, which is why the absolute value is used for pz . The matching point z0 is supposed to be chosen pwell before the exit of the barrier, and hence we approximate jpz .z0 /j '  D 2Ip , where Ip is the ionization potential. Finally, T is the tunneling time between z0 and z, Z z dz 0 T D : (5) 0 z0 pz .z / Given that z0  zex , the tunneling time is approximately independent of z0 and for z D zex one finds T ' 2zex = ' =F . With the approximate expression for ˚.px ; py ; z/ now at hand, the wave function in the coordinate space becomes  Z z r   .x; y; z/ ' exp  pz .z 0 / dz 0 jpz .z/j z0 Z 1 2  dpx dpy eixpx iypy ˚.px ; py ; z0 /ep? T =2 (6) 2 2 with p? px2 C py2 . Thus, 3D tunnel ionization is the combination of what looks like 1D tunneling along the electric field with Gaussian filtering in the momentum space orthogonal to the direction of tunneling. This tunneling filter is G.p? / D 2 expŒp? T =2. Using the convolution theorem, we can write the wave function as

p  Z z     1 2  .x; y; z/ ' p exp  pz .z 0 / dz 0  .x; y; z0 / exp   2T jpz .z/j z0 (7) where 2 D x 2 C y 2 and we took into account that the inverse Fourier transform of the Gaussian filter G.p? / is g./ D .1=/ expŒ2 =2T . Equation (7) can now be substituted into the general expression for the tunneling rate (1). As always with asymptotic tunneling theories, one has to carefully deal with matching the bound wave function  .x; y; z0 / at the point z0 to the in-field semi-

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classical tunneling amplitude from z0 to z. The potential problems hidden in this approach have recently been highlighted in [18, 19], and ways to correct for the potential errors outlined in [18, 19] have been discussed in [1]. For atomic ionization, one has to match spherically symmetric bound state with the cylindrically symmetric behaviour at large distances from the origin, imposed by the electric field. The molecular case is less straightforward, as it lacks the spherical symmetry of the bound wave function. The procedure we use generalizes the approach developed by Popov and co-workers for the atomic case [5–7], and introduces additional corrections that turn out to be important in the molecular case. These corrections are absent in the direct generalization of the atomic theory for molecules, known as MO-ADK [12]. Their essence is to incorporate the angular width of the tunneling wavefunction in the direction perpendicular to the direction of tunneling z. If  is the angle with the electric field (i.e., with the laboratory zaxis), then standard approximation of the tunneling theories is to replace sin    , cos   1 and 1  cos   0, while we attempt to keep all terms proportional to  2 and hence do not neglect 1  cos  . More detailed discussion follows below. In contrast to the MO-ADK [12], the tunneling rate can be calculated without expanding the wave function into the single-center basis of spherical harmonics. On the other hand, such expansion can always be done in (7) if desired. Then, one can show [26] that the MO-ADK result emerges from our method if (i) single-center Coulomb potential is assumed at z > z0 and (ii) standard approximations sin    and cos   1 are used for the tunneling wavefunction.

 6LPSOH $QDO\WLFDO ([SUHVVLRQV We now derive simple, closed form analytical expressions for tunnel ionization of different orbitals in a linear molecule, with the electric field aligned at an angle L relative to the molecular axis. Let ionization create the molecular cation in some final electronic state. We shall denote  .x; y; z/ the corresponding Dyson orbital, i.e., the overlap between the initial N -electron wave function of the neutral and the final N  1 electron wave function of the electronic state of the ion. In the asymptotic region z0 zin we write  .x; y; z0 / as:  .x; y; z0 / ' C  3=2

er0 .r0 /Q= fM .M ; M / : r0

(8)

Here spherical angles M and M refer to the molecular frame, i.e., they are measured relative to the molecular axis ZM ; z0 r0 cos M . The function fM .M ; M /, also given in the molecular frame, incorporates the geometry of the orbital that, in turn, also reflects the shape of the binding potential. Deviations from the singlecenter Coulomb potential, which are obviously very significant near the core, are responsible for how fM .M ; M / looks like in the asymptotic region. Apart from

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Θ ΘM

ZL

ΘL ZM

)LJ  Schematic representation of a molecular orbital (shaded blue) and the two Z-axes, molecular ZM and laboratory ZL , which define the plane of tunneling (X-Z). The tunneling angle  is relative to the lab axis.

fM .M ; M /, the radial asymptotic behavior corresponds to a purely Coulombic tail of the potential Q=r, with Q the effective charge. This asymptotic form is quite adequate for a large range of molecules and orbitals, including H2 , N2 , O2 , CO2 , HCl, HF, etc. [27]. For molecules lacking the inversion centre, the cation’s dipole moment will no longer vanish, and the asymptotic form may emerge further away from the nuclei than for the symmetric molecules. The angle between the molecular axis and electric field is denoted as L . The plane M D 0 (i.e., the X -Z plane) is set to coincide with the plane defined by the molecular and laboratory axes ZM and ZL , see Fig. 2. In what follows, angles  and refer to the lab frame, while M ; M refer to the molecular frame. The steps of the calculation are (i) transformation of fM .M ; M / into the lab frame, which yields fL .; / after rotation by L in the X-Z plane, (ii) calculation of the convolution between the transformed function and the tunneling ‘filter’ g./, and (iii) calculation of the tunneling current. Substituting (8) into (7), (1), and calculating the tunneling integral between z0 and z for the potential Q=r following [5–7] (i.e., using the eikonal approximation to match the asymptotic form of the radial wave function), we obtain the following expression for the tunneling rate  D A;s R.L / :

(9)

Here the first term A;s D

   2 2 3 =.3F / 2 3 2Q= C e T F

(10)

is the standard tunneling rate for an atomic s-orbital with spherically symmetric angular structure. The second term incorporates all aspects of the orbital geometry, including the interference of the tunneling currents coming from the different lobes of the orbital. It is expressed as

Strong-Field Ionization of Molecules

R.L / D

1 R.0/

Z

1

141

 d e

2 = T

0

Z

2

d jfQL .; I f /j2

(11)

0

where  D r0 sin  and fQL is the result of tunnel-filtering the original fL .; / in the momentum space, which is expressed via the convolution integral: fQL D

Z

1

0

0 2 =2T 02 =2z0

 d e

e

0

1 2

Z

2

d 0 e

0 cos. 0 /= T

fL . 0 ; 0 / : (12)

0

Finally, the normalization factor R.0/ is obtained by performing the same calculation with fL .; / D 1, i.e., for an atomic s-orbital. In what follows, we focus on orbitals with ˙ and ˘ symmetry. For the latter, ionization is dominated by the ˘x -orbitals. Orthogonal to it ˘y orbital has a nodal plane in the x-z plane and hence its ionization in that plane is suppressed. For ˙ and ˘x orbitals, the angular dependence fM .M ; M / has the following form fM .M ; M / D F .cos M ; sin M cos M / (13) (while for the ˘y orbital sin M cos M would be replaced with sin M sin M ). For convenience, we will use the notation F .u; v/, where u D cos M and v D sin M cos M . The transformation between the molecular and the lab frame is cos M ! cos L cos   sin L sin  cos ; sin M cos M ! cos L sin  cos C sin L cos 

(14)

with  and the angles in the lab frame, and L the angle between the molecular axis and the electric field. Substituting these expressions into F .u; v/, we can calculate all the integrals using the fact that the angle  between the tunneling electron and the field is small, and expanding F .u; v/ in Taylor series up to the second order with respect to  . For the moment, in contrast to the standard approximation cos  D 1 [5–7], we shall keep the small parameter 1  cos    2 =2. Small tunneling angles also imply that 0 =T < 1, allowing one to use it as a small parameter while calculating the convolution integral over 0 in (12). The final result is  2 1 R.L ; z0 / ' F0  .1  cos T .z0 //F2 C sin2 T .z0 /F3 4 1 1 C F2 2T . C z0 =T /2 1

(15)

s

where T .z0 / D

2 . C z0 =T /z0

(16)

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is the characteristic angular width of the tunneling wave function at the matching point z0 . The coefficients Fk are related to the function F .u; v/ and its derivatives taken with respect to u; v and calculated at u D cos L and v D sin L : F0 D F .cos L ; sin L / ; F1 D Fv cos L  Fu sin L ; F2 D Fu cos L C Fv sin L ; F3 D Fvv cos2 L C Fuu sin2 L  Fuv sin 2L :

(17)

So far, our result for the geometrical factor maintains its dependence on the matching point z0 . However, we can simplify (15) further and remove this dependence using one of several methods. The traditional way is to use the assumption that z0  zex which is valid only for weak fields. Taking into account that T D 2zex , we see that, formally, z0 =T  . Then, the dependence of the coefficient in front of F1 (the last term in the expression (15) on the matching point z0 disappears. Now, let us look at the terms 1 cos T .z0 / and sin2 T .z0 /. Since we expect z0 =T  , we see that 1  cos T  sin2 T  T2  1=z0 . Given that the matching point is supposed to be sufficiently far, z0 1, the terms proportional to F2 and F3 have a small parameter 1=z0  1 compared to F0 . This leads to simplified expression R.L ; z0 / ! R.L / ' F02 C

1 F2 2T  2 1

(18)

where the dependence of R.L ; z0 / on the matching point is removed, as common in the asymptotic tunneling theories. Applying the result (18) to the atomic s- and p-orbitals leads to the results identical with standard expressions [5–7, 18, 25]. While (18) and the overall result for the molecular tunneling ionization rate use the same approximations as the standard tunneling theories, our expressions are simpler than the standard expressions of MO-ADK [12]. Instead of expanding the wavefunction into the spherical harmonics, as done in the MO-ADK, all one needs in (18) is the first derivative in the direction perpendicular to the electric field, which yields F1 . In relatively weak fields, where the requirement 1  z0  zex holds well, the term proportional to F1 becomes important only near the zeroes (nodal planes) of F0 . However, in most practical situations we are interested in tunnel ionization for relatively strong fields of a few Volts per Angstrom, with ionization lifetimes about 10 fsec, i.e., with substantial ionization probabilities during the few cycles of the laser pulse. Under such conditions the rather strong requirement 1  z0  zex can no longer be met. The F1 -term becomes important in a broader range of angles, not only where F0 D 0. The terms proportional to F2 and F3 are also no longer negligible. One can easily check by expanding 1  cos T .z0 /  T .z0 /2 =2 and sin2 T .z0 /  T .z0 /2 that the terms proportional to F2 and F3 are of the same order as that proportional to F1 . The associated corrections are also of the same order as long as F0 ¤ 0.

Strong-Field Ionization of Molecules

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To calculate ionization rates in this case, we can no longer use the traditional method of ignoring the terms proportional to T2 .z0 /, which means that the z0 dependence in (15) should be handled differently. To deal with this problem, we keep all the terms proportional to T2 .z0 / and calculate the characteristic tunneling angle T .z0 / at the exit point zex : s 2 T .zex / D (19) . C zex =T /zex with zex  Ip =F . The tunneling angle approaches 20–30ı as the field strength ˚ reaches Volts per A(intensity approaching 1014 W=cm2 ). Thus, setting z0 ! zex in (15), we obtain the final expression for the molecular orbital geometry-dependent term, better suited for strong electric fields and high ionization rates  2 T2 .zex / T2 .zex / R.L / D R.L ; zex / ' F0  F2 C F3 2 4 1 1 C F2 : (20) 2T . C zex =T /2 1 Our result shows to what extent the alignment-dependent tunneling rate  .L / directly maps the ionizing orbital. The leading term in R.L / (20) is F02 , and it is indeed given by the wave function in the direction of tunneling. Corrections to this simple picture are determined by F1 ; F2 ; F3 . These terms are particularly important in the vicinity of nodal planes, where F0  0.

 ([DPSOHV Let us now apply the above expressions. Analytical function describing the asymptotic behavior of the orbitals is suggested in Sect. 10.2 of [27]) for ˙ and ˘ orbitals. In case of linear symmetric molecules this function is (in the molecular frame) F .cos M ; sin M cos M / D cosh. cos M /.1 C c 2 cos2 /.cos M /n .sin M cos M /m

(21)

where for ˙ orbitals m D 0 and for ˘x orbitals m D 1. The parameters ; c and n; m, as well as the overall coefficient C in (8) are also tabulated in Sect. 10.2 of [27]), but they have to be taken with a grain of salt as they are based on rather old Harfree–Fock calculations. For example, for HOMO of N2 [27]) suggests D 2:66, while more accurate calculations of the Dyson orbitals for N2 yield D 1:8 in the relevant range of distances r D 5–8 a:u: from the core, with the Coulombasymptotic form well approximating the Dyson orbitals already at r D 4–5 a:u: from the origin in this case.

144 N2 at F=0.07 1E-3

Rate, a.u.

)LJ  Angle dependent ionization rate, in atomic units, for the three different Dyson orbitals of N2 . The field strength is F D 0:07 a:u:.

R. Murray et al.

HOMO

1E-4

HOMO-1

1E-5 1E-6

HOMO-2

0

30

60

90

Angle, deg

In particular, for the Dyson orbital corresponding to the ionization from HOMO the fitting parameters in (21) are D 1:8, n D 0, m D 0, c D 0, C D 0:5. For HOMO-1 of N2 the fitting parameters in (21) are D 1:8, n D 0, m D 1, c D 0, C D 0:5. For HOMO-2 of N2 the fitting parameters in (21) are D 1:7, n D 1, m D 0, c D 0, C D 0:72. The resulting angle-dependent rates for N2 , calculated using (20) for R.L /, are shown in Fig. 3, for the field strength F D 0:07 a:u:. They are consistent with recent numerical results of [15].

 &RQFOXVLRQ ([WHQVLRQ WR /RZ)UHTXHQF\ /DVHU )LHOGV We now discuss routes to generalizing these results for the case of ionization in lowfrequency laser fields. The key underlying modification is the change of the trajectory in the classically forbidden region, which now has to be found in the presence of oscillating rather than static electric field. Such trajectories were first introduced to strong-field ionization problems by Popov and co-workers [5–7]. Later, under the name ‘quantum trajectories’ or ‘quantum orbits’ (see, e.g., [28] and [29]), they were successfully used for the description of many strong-field phenomena such as harmonic generation, above-threshold ionization, correlated multiple ionization. Recent applications to ionization dynamics and spectra of the so-called ‘direct’ electrons can be found in [4]. Analysis of subcycle ionization dynamics in atoms [8, 9] relies on the same concept. Application of this concept suggests simple modifications of A;s and R.L / in the rate (9). We begin with A;s , which is determined by the imaginary component of the action accumulated in the classically forbidden region. The eikonal approximation with respect to the binding core potential U , used for z > z0 , separates the action into the sum of two terms. Consequently, the tunneling amplitude is a product of two terms, a.T/ D a.SFA/ a.CC/ , and A;s / jaT j2 D ja.SFA/ j2 ja.CC/ j2

(22)

Strong-Field Ionization of Molecules

145

The first term, a.SFA/ , is standard in the strong-field approximation (SFA). The second term introduces the so-called ‘Coulomb correction’: This is the term responsible for the factor Œ2 3 =F 2Q= in the ionization rate (10). Both amplitudes are modified in oscillating fields, where the problem has to be formulated directly in the time-domain [3, 4, 8]. The SFA contribution to the amplitude is simply replaced with the Yudin–Ivanov formula [8] for the electron ‘born’ at the phase b D !L tb of the oscillating field F cos !L t. In the static limit this expression yields the familiar static exponent, a.SFA/ D exp. 3 =3F /. We now turn to the Coulomb correction. In static fields, itRresults from the integral of the Coulomb potential along the under-barrier trajectory, U.z/ dz=jv.z/j, where v.z/ is the complex electron velocity. This integral can be rewritten in time domain using the usual relationship dt D dz=v.z/. It is, however, important to remember that in the classically forbidden region both velocity and time are purely imaginary, t D i, with the positive imaginary part  decreasing as the electron moves towards zex . The result is:  Z  ast.CC/ D exp i

tb

U.zst .t// dt

(23)

t0

Here t0 D t0 .z0 / D i.z0 / and tb D t.zex / D 0 are the moments of time when the trajectory zst .t/ crosses the matching point z0 and exits the barrier. The trajectory begins at the origin z D 0 at the moment tin D iT D i=F . A similar expression applies in the oscillating laser field [3,4]. The corresponding trajectory zdyn .t/ is parametrized by requiring zero velocity v D 0 at the ‘phase of birth’ b D !tb . The moment of ‘birth’ tb uniquely defines the complex time tin at which the electron enters the classically forbidden region [8]: Œsin !L tb  sin !L tin 2 C  2 D 0

(24)

In the static tunneling limit tin ! iT . Just as in static fields, at tin the trajectory starts at the origin, zdyn .tin / D 0, with imaginary velocity i. Thus, in the oscillating field we can write the Coulomb correction as follows R tb  3 Q= expŒi t .z U.zdyn .t// dt a.CC/ . / 2 .CC/ .CC/ dyn 0/ adyn . / D ast D : (25) R 0 F expŒi U.zst .t// dt ast.CC/ i.z0 /

Note that at the matching point both trajectories, in the static and in the oscillating electric field, must match the same initial wavefunction. Consequently, at the beginning of the under-barrier motion zdyn D zst and the ratio of the two exponents in (25) is independent of the small offset z0 . To estimate the ratio of the integrals, one can set the matching point z0 D 0, keeping in mind that both trajectories start at the same initial position with the same initial velocity. Then t.z0 / D tin and i.z0 / D iT . The singularity of the Coulomb potential is cancelled in the ratio of the two exponents. Thus, the atomic-like component of the molecular ionization rate becomes

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  3 2Q=  2 F2 2 A;s . b / D C exp 2 3 ˚.; b / T  F !L R tb expŒ2i tin U.zdyn .t// dt  R0 expŒ2i iT U.zst .t// dt

(26)

where ˚.; b / is defined in [8]. We now turn to the geometrical factor. Its generalization to oscillating fields requires two substitutions, T ! T . b / and zex D T =2 ! zex . b /, into the expression for R.L / (20). In the oscillating fields, the phase-dependent tunneling time T . b / is [3]: sin b coshŒ!L T . b / D p (27) P where p P D .1 C  2 C sin2 b  D/=2 ; D D .1 C  2 /2 C sin4 b C 2 sin2 b . 2  1/ : (28) The exit point zex . b / is [3] zex . b / D

F REŒcos b  cos in C sin b . b  in / !L2

where in D arcsin

p

P C i!L T . b / :

(29)

(30)

Our general results, applied here to tunnel ionization of N2 , are applicable to nearly any small molecule. Simple analytical expressions require fitting the Dyson orbital for the ionization channel to a Coulombic asymptote, with the complexity of the binding potential appearing through the angular shape of the ionizing orbital. Future work will include extension of simple expressions to polar molecules such as HCl. $FNQRZOHGJHPHQWV We acknowledge financial support of NSERC through the discovery grant to M.I.

5HIHUHQFHV 1. 2. 3. 4. 5. 6. 7.

R. Murray, W.K. Liu, M. Ivanov, Phys. Rev. A , 023413 (2010) F. Krausz, M. Ivanov, Rev. Mod. Phys. , 163 (2009) O. Smirnova, M. Spanner, M.Y. Ivanov, Phys. Rev. A , 33407 (2008) S.V. Popruzhenko, G.G. Paulus, D. Bauer, Phys. Rev. A , 53409 (2008) A.M. Perelomov, V.S. Popov, M.V. Terent0 ev, Sov. Phys. JETP , 924 (1966) A.M. Perelomov, V.S. Popov, M.V. Terent0 ev, Sov. Phys. JETP , 207 (1967) A.M. Perelomov, V.S. Popov, M.V. Terent0 ev, Sov. Phys. JETP , 336 (1967)

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G.L. Yudin, M.Y. Ivanov, Phys. Rev. A , 13409 (2001) M.Y. Ivanov, M. Spanner, O. Smirnova, J. Mod. Opt. , 165 (2005) L.V. Keldysh, Sov. Phys. JETP , 1307 (1965) J. Muth-B¨ohm, A. Becker, F.H.M. Faisal, Phys. Rev. Lett. , 2280 (2000) X.M. Tong, Z.X. Zhao, C.D. Lin, Phys. Rev. A , 33402 (2002) M. Awasthi, A. Saenz, J. Phys. B , S389 (2006) Y.V. Vanne, A. Saenz, J. Mod. Opt. , 2665 (2008) S. Petretti, Y.V. Vanne, A. Saenz, P. Decleva, Phys. Rev. Lett. , 223001 (2010) M. Abu-samha, L.B. Madsen, Phys. Rev. A , 23401 (2009) M. Spanner, S. Patchkovskii, Phys. Rev. A , 063411 (2009) I.I. Fabrikant, G.A. Gallup, Phys. Rev. A , 13406 (2009) G.A. Gallup, I.I. Fabrikant, Phys. Rev. A , 033417 (2010) O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, M.Y. Ivanov, Nature , 972 (2009) O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, M.Y. Ivanov, Proc. Natl. Acad. Sci. U.S.A. , 16556 (2009) Y. Mairesse, J. Higuet, N. Dudovich, D. Shafir, B. Fabre, E. M?evel, E. Constant, S. Patchkovskii, Z. Walters, M.Y. Ivanov, O. Smirnova, Phys. Rev. Lett. , 213601 (2010) A.T. Le, R.R. Lucchese, C.D. Lin, J. Phys. B , 211001 (2009) S. Haessler, J. Caillat, W. Boutu, C. Giovanetti-Teixeira, T. Ruchon, T. Auguste, Z. Diveki, P. Breger, A. Maquet, B. Carr´e, R. Ta¨ıeb, P. Sali`eres, Nature Phys. , 200 (2010) M.V. Ammosov, N.B. Delone, V.P. Krainov, Sov. Phys. JETP , 1191 (1986) R. Murray, M. Ivanov. In preparation A.A. Radzig, B.M. Smirnov, 5HIHUHQFH 'DWD RQ $WRPV 0ROHFXOHV DQG ,RQV, 6SULQJHU 6HULHV LQ &KHPLFDO 3K\VLFV, vol. 31 (Springer, Berlin–New York, 1985) P. Sali`eres, B. Carr´e, L. Le D´eroff, F. Grasbon, G.G. Paulus, H. Walther, R. Kopold, W. Becker, D.B. Miloˇsevi´c, A. Sanpera, M. Lewenstein, Science , 902 (2001) R. Kopold, W. Becker, D.B. Miloˇsevi´c, J. Mod. Opt. , 1362 (2002)

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$EVWUDFW In the past few years new and efficient algorithms have been developed to solve the time-dependent Schr¨odinger equation (TDSE) for few-electron systems. When coupled with the advances in and availability of high performance computing platforms, it is now possible to numerically calculate nearly exact solutions to the interactions of short, intense laser pulses with simple one and two-electron systems. In addition, somewhat less accurate treatments of the heavier rare gases and simple two-electron molecules are also becoming available. The proceedings from this workshop have provided a unique opportunity to describe the substantial numerical

Barry I. Schneider Office of Cyberinfrastructure and Physics Division, National Science Foundation, 4201 Wilson Blvd., Arlington, VA 22230, USA, e-mail: CTDIOFJE!OTGHPW Johannes Feist ITAMP, Harvard–Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA, e-mail: KGFJTU!DGBIBSWBSEFEV Stefan Nagele Institut f¨ur Theoretische Physik, Technische Universit¨at Wien, Wiedner Hauptstraße 8, 1040 Wien, Austria, e-mail: TUFGBOOBHFMF!UVXJFOBDBU Renate Pazourek Institut f¨ur Theoretische Physik, Technische Universit¨at Wien, Wiedner Hauptstraße 8, 1040 Wien, Austria, e-mail: SFOBUFQB[PVSFL!UVXJFOBDBU Suxing Hu Laboratory for Laser Energetics, University of Rochester, 250 E. River Road, PO Box 278871, Rochester, NY 14623-1212 USA, e-mail: TIV!MMFSPDIFTUFSFEV Lee A. Collins Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA, e-mail: MBD!MBOMHPW Joachim Burgd¨orfer Institut f¨ur Theoretische Physik, Technische Universit¨at Wien, Wiedner Hauptstraße 8, 1040 Wien, Austria, e-mail: CVSH!EPMMZXPPEJUQUVXJFOBDBU A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_10, © Springer Science+Business Media, LLC 2011

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and algorithmic progress that has been achieved over the past few years to solve the TDSE and to illustrate them on the He atom and HC 2 molecule.

1 Introduction In recent years, revolutionary new technologies have made coherent, ultrashort, and intense pulses in the vacuum and extreme ultraviolet (VUV-XUV) region available. These pulses are currently generated from two quite different types of sources. One are free electron lasers (FEL) [1–7]. Currently, there are two FELs in the VUV-XUV and X-ray regime (XFELs) in operation: FLASH at DESY in Hamburg, Germany [8, 9], and LCLS in Stanford, USA [10, 11]. FLASH has reached focused intensities of up to 1016 W=cm2 , and photon energies as high as 190 eV, while LCLS reaches even higher energies up to 8 keV and intensities of up to 1018 W=cm2 . The duration and temporal structure of the individual FEL pulses is not well known, but is of the order of 10–50 femtoseconds for FLASH. In addition, there have been a number of proposals aimed to decrease the duration of these pulses to a few hundred attoseconds [12–18]. The other approach to produce intense ultrashort pulses at XUV wavelengths is to use high harmonic generation (HHG) from a driving infrared (IR) laser [19–28]. This technique has been successfully used to create the shortest pulses available today, with durations down to 80 as [27]. With current technology, attosecond pulses are much less intense than FEL pulses. The focused intensities are not well known but typically do not exceed  1012 W=cm2 , although various ways to increase the maximally available intensity have been proposed [29–36]. The continuing development of these novel light sources has led to an increased interest in multiphoton processes at high photon energies. Simultaneously, the ultrashort duration of the pulses in the femtosecond (1 fs D 1015 s) or even attosecond (1 as D 1018 s) domain enables the study of time-resolved electron dynamics, starting the field of attosecond science [37–41]. In this contribution, we report on some of our recent theoretical and numerical investigations, motivated by the availability of these pulses. We study relatively simple systems (He and HC odinger equation including all rel2 ), for which the Schr¨ evant degrees of freedom can be fully solved. This necessitates the use of numerical approaches that take advantage of modern high-performance computing facilities. The work that is reviewed in this manuscript has been previously published in the diploma theses of Stefan Nagele [42] and Renate Pazourek [43] and the PhD thesis of Johannes Feist [44], as well as some journal publications [45–50]. We start by giving an overview of the numerical methods we are using to discretize the spatial (Sect. 2) and temporal (Sect. 3) degrees of freedom. We then comment on some numerical details of our implementation (Sect. 4). For the case of helium (Sect. 5), we focus on two-photon double ionization. Double ionization of helium has long been of great interest in atomic physics since it provides fundamental insights into the role of electronic correlation in the full three-

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body Coulomb break-up process. This simple, two-electron system gives crucial insight into the dynamics of more complex atoms and even simple molecules [51–62]. Until recently, the focus of these studies was on one-photon double ionization, where a single photon releases both electrons from the nucleus. In two-photon double ionization (TPDI) of atomic helium, two electrons are absorbed either simultaneously or sequentially, ejecting both electrons. This is one of the simplest multiphoton processes involving electron correlation, and has been the subject of intense studies in the past few years [42–49, 63–114]. We discuss both (i) cross sections in the nonsequential regime of TPDI, which require pulses of at least a few femtoseconds duration, and (ii) the possibility to probe and control correlation using ultrashort (attosecond) XUV pulses in the sequential regime of TPDI. In Sect. 6, we then discuss the hydrogen molecular ion HC 2 . We simulate its combined electronic and nuclear motion in attosecond pulses fully, i.e., without resorting to the Born–Oppenheimer approximation. We study energy and angular patterns in one-photon ionization. The conclusions we draw for this simple test case could be applied to e.g., attosecond photoelectron microscopy. The photoelectron distribution obtained through ionization by an attosecond pulse can provide information about the “frozen” position of the (slowly moving) nuclei after excitation through other means. We demonstrate a strong polarization dependence for the ionization probability, which disappears for high photon energies (& 170 eV). In addition, we find that the double slit interference pattern that is caused by the two distinct molecular centers in the half-scattering process of photoionization only follows the classical Young’s double slit angular distribution if the electron de Broglie wavelength is noticeably smaller than the internuclear distance.

2 An Introduction to the Finite Element Discrete Variable Representation (FEDVR) To solve the multidimensional, time-dependent Schr¨odinger equation, i„

@ b .t/i ; j .t/i D Hj @t

(1)

one requires an efficient spatial discretization approach, and a way to couple that to a time-propagation technique that can exploit the structure of that discretization. Finite difference methods offer great simplicity and have been used for decades to numerically solve a variety of problems involving partial differential equations in science and engineering. However, approximating derivatives by finite difference formulas is intrinsically inaccurate unless high order methods are employed. In recent years, an alternative approach, the FEDVR, has been developed [115–118], which offers the accuracy of a spectral method and much of the simplicity and sparsity of finite difference approaches. This approach, which we describe in some detail

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below, employs the standard finite element method, but replaces the usual polynomial basis in each element by a basis derived from a discrete variable representation.

2.1 Finite Elements The basic idea of any finite element (FE) method is to divide the underlying configuration space of a (partial) differential equation into small subdomains or finite elements. For complex geometries in more than one dimension, FE techniques are especially useful to handle the boundary conditions that need to be satisfied at the surface or internally for the equation under consideration. However, they are also useful for “simple” problems in one dimension. In that case, each variable is divided into segments or finite elements with the FE boundaries 0  x .1/ < x .2/ <    < x .N /  xmax :

(2)

The approach of the FEDVR is to expand the wave function in a basis of functions that are local to each finite element, fm.i / .x/ ; x 2 Œx .i / ; x .i C1/  :

(3)

The basis in a given element is defined to be zero outside of that element. The only remaining issue is how the basis functions at the edges of each element connect with adjacent elements. Since the equations under consideration contain at most second order spatial derivatives, it is sufficient to ensure function continuity at the edges of the boundary elements from a rigorous mathematical treatment [119]. This does not mean that adding additional constraints on the basis functions such as first or second derivative continuity would not produce a more accurate representation for a fixed number of basis functions, it is just not required and by only imposing basis function continuity, the problem is significantly easier to handle numerically. In each finite element, we choose basis functions obtained from a discrete variable representation (DVR) approach, which also provides a prescription for calculating matrix elements. Apart from the basic idea of splitting space into smaller elements, we do not use the further features of FE methods. A more detailed treatment of the subject can be found in, e.g., [120].

2.2 Discrete Variable Representation In this section we provide a self-contained introduction (following [121–124]) to the polynomial discrete variable representation (DVR), which is closely related to the well-known concept of the spectral (finite-basis) representation of wave functions. In coordinate space a wave function in its spectral representation (SR)

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

hxj i D

N X

hxj˚m ih˚m j i D

mD1

N X

am hxj˚m i

153

(4)

mD1

is described by the expansion coefficients Z am D h˚m j i D h˚m jxihxj idx

(5)

in a given orthonormalized basis fj˚m ig of the Hilbert space, where usually the basis is complete for N ! 1. For N 6D 1, this representation is also called a finite basis representation (FBR). The basis functions are orthonormal and thus fulfill Z (6) h˚m j˚n i D h˚m jxihxj˚n idx D ımn : Inserting this ansatz for the wave function in the time-independent Schr¨odinger equation gives N X

b m iam D h˚n jHj˚

mD1

N X

b nm am D an E ; H

(7)

mD1

b mi bnm D h˚n jHj˚ which is a matrix eigenvalue problem once the matrix elements H have been calculated. Equation (7) may be derived from the Rayleigh-Ritz variational principle. The coefficients an can be understood as the variational parameters resulting from the application of that principle to the Schr¨odinger equation. As a consequence the eigenvalues (7) will always represent an upper bound to the true solution for N ! 1. Thus, the SR is sometimes also called variational basis representation (VBR). The idea behind the DVR is to start by choosing a basis fj˚m ig for which the overlap integrals in (6) can be evaluated exactly by numerical quadrature. The classical orthogonal polynomials up to order N  1 are such a set. For each of these basis sets, there is an associated Gaussian quadrature of order N in which the product of two of these functions can be integrated exactly. This arises simply because the product is itself a polynomial of maximum order 2N  2 which may be integrated exactly by an N th-order Gaussian quadrature at points xi with weights wi . Replacing the integral in (5) by its discrete approximation yields aQ m D

N X

wj h˚m jxj ihxj j i

(8)

j D1

with the FBR remaining orthonormal under the quadrature rule, h˚m j˚n i D

N X j D1

wj h˚m jxj ihxj j˚n i D ımn :

(9)

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At this point, the only approximation that has been made is the use of the FBR. What b m i will be calculated. we have not yet specified is how the matrix elements h˚n jHj˚ It is straightforward to show that it is possible to compute the matrix elements of the kinetic energy operators exactly for most, if not all, of the classical orthogonal functions. However, the matrix elements of the potential are another matter. These are often complex functions of x and all that can be said is that using the quadrature rule for the matrix elements, h˚m jb Vj˚n i D

Z

 ˚m .x/V .x/˚n .x/dx '

N X

 wj ˚m .xj /V .xj /˚n .xj / ;

(10)

j

needs to be examined for accuracy. There is another disadvantage of (10). It is not diagonal in the FBR basis. While this is not terribly serious for a one-dimensional problem, it still would be nice if one could find an approach which only required the value of the potential at the i t h quadrature point. Such a diagonal representation, suitably generalized to the multidimensional case, could have very large advantages in practical computations. Provided that the quadrature approximation is accurate, it is possible to make a basis set transformation which does render the potential diagonal in the transformed basis. This new basis, ffj .x/g, the DVR basis, has the property that the basis functions are zero at all but one quadrature point, i.e., ıij fj .xi / D p wi

(11)

and in analogy to (4) and (9) they define an orthonormal basis (a rigorous proof can be found in [121]), Z hfi jfj i D

N X

Š

fi .x/fj .x/dx D

wm fi .xm /fj .xm / D ıij :

(12)

wj hxj˚m ih˚m jxj ihxj j i :

(13)

mD1

The wave function (4) then reads hxj i D

N X

aQ m hxj˚m i '

N X N X mD1 j D1

mD1

The DVR basis functions can be obtained from, fj .x/ D

p

wj

N X

hxj˚m ih˚m jxj i :

(14)

mD1

The wave function is then expressed as, hxj i D

N X j D1

hxjfj ihfj j i D

N X j D1

Qj fj .x/ ;

(15)

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

155

where the coefficients Qj D hfj j i are also directly connected to the values of the p wave function at the grid points, Qj D wj  .xj /. This is the reason that the DVR can be seen as a bridge between spectral basis methods and grid-based approaches – the coefficients of the basis functions fj simultaneously give the values of the wave function at the grid points in coordinate space, which are chosen as the quadrature points of the underlying Gaussian quadrature. A further consequence of this is that the coordinate operator is strictly diagonal in the DVR basis, hfi jxjf O j i D ıij xi ; (16) as the product of two basis functions has maximum order 2N  2 and integrals up to order 2N  1 can be evaluated exactly by N -point Gaussian quadrature. Consequently, the functions fj .x/ can be referred to as coordinate eigenfunctions because they depend only on the chosen quadrature and the corresponding mesh (which are, in turn, related to the corresponding equivalent finite-basis representation). In a DVR, the wave functions are thus represented by a complete and orthonormal set of basis functions that are uniquely related to the chosen grid and quadrature. The FBR and DVR are strictly equivalent (isomorphic) if the FBR consists of orthogonal polynomials, as we have chosen here [125, 126]. A DVR basis function is effectively represented by an interpolating polynomial. For a given order this interpolating polynomial is unique [127] and we can express it without loss of generality by the Lagrange polynomials, Li .x/ D

Y x  xj xi  xj

(17)

j ¤i

which fulfill Li .xj / D ıij :

(18)

As the representation is unique, i.e., the Lagrange polynomials form the only polynomial basis that fulfills (18), they are identical to the basis functions fj .x/ (up to a p factor wj ). This is the reason why in many publications (following a series of papers by Baye et al. [128–130]) polynomial DVRs are referred to as Lagrange-mesh techniques. Last, but certainly not least, in the DVR, the potential matrix elements are diagonal ((11) and (16)) within the accuracy of the quadrature approximation, Š

hfi jb Vjfj i ' V .xi /ıij

(19)

and thus equal to the potential evaluated at the grid points. This does not imply that the potential matrix elements are calculated exactly by quadrature. The reason is that a N -point Gaussian quadrature just allows for an exact calculation of polynomial integrands of degree 2N  1 and the required matrix element will not, in general, posses that property. In practice, it has been found that for sufficiently large N , the use of the DVR rule (19) works well. Stated differently, the fundamental approximation of the DVR is that all matrix elements of coordinate operators are diagonal. In other words (referring to the isomorphic FBR) the quadrature is not exact

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for those components of b Vj˚i i (aliasing terms) that do not remain in the spectral basis [131]. In practice, those errors are removed implicitly by extending the basis size until numerical convergence is reached. However, the strict variational properties of (7) are lost in the DVR. A more detailed treatment of the problem is given in [125, 131].

2.2.1 Calculating Derivatives In the DVR basis the n-th derivative of a function  .x/ is given by X @n fj @n  .x/ D Qj .x/ ; @x n @x n N

(20)

j D1

@n f .x/

where the derivatives of the basis functions @xjn have to be computed just once initially. If the n-th derivative is to be evaluated at the DVR points one gets X @n fj p @n  .x / D .xi / !i Qj WD Dij Qj i @x n @x n N

(21)

j D1

which represents a matrix-vector multiplication. Thus, in a DVR basis derivatives are calculated by multiplying the vectors Qj by a differentiation matrix D. The matrix D is full and hence the matrix-vector product is computationally expensive when the basis size is large.

2.3 FEDVR In an FEDVR [115–118] the underlying configuration space of a problem is divided into elements in each of which the wave function is represented in a local DVR basis. Consequently, the main advantages of the two techniques are brought together:  Since the DVR basis functions are only defined on local grids, the kinetic energy matrix is not full as in the standard DVR approach (cf. (20)) but consists of several blocks (one for each FE) which overlap at only one point (see Fig. 1). Thus, the matrix becomes quite sparse in 1D. Matrix-vector products can be calculated efficiently when a direct product basis is used in N dimensions. This property also allows for a computationally efficient parallelization scheme in which the inherent latency due to communication between the matrix blocks is minimal. As long as the number of basis functions in a given element is not too small compared to the size of the element, this provides an accurate numerical expression for the derivatives [115].  For a local potential, the matrix is diagonal, thus avoiding the need to calculate complex matrix elements as in SR or FBR methods.

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

157

0

Fig. 1 Illustration of the block structure of the kinetic energy operator of the FEDVR in one dimension.

0 0.6

20 18

0.5

16 r(i) − r(i−1) [ a.u.]

r(i) [ a.u.]

14 12 10 8 6 4

0.4 0.3 0.2 0.1

2 0

0 0

10

20

30

grid point #i

(a) Grid point distribution

40

50

0

10

20

30

40

50

grid point #i

(b) Grid point spacing

Fig. 2 Mesh points of a typical FEDVR grid with eleven basis functions per finite element. The oscillatory structure in the grid spacing is a consequence of the Gauss–Lobatto quadrature.

Since the resulting grid is composed of many local sub-grids for each finite element we have to use a quadrature where the end point in each node coincides with the starting point for the neighboring finite element. This is necessary to impose continuity for the represented wave functions. For Gauss-Legendre quadratures the mesh points are given by the roots of the Legendre polynomials. As all these roots lie inside the sub-grids, we instead use a Gauss–Lobatto quadrature in each element. In a Gauss–Lobatto quadrature, the first and last points are explicitly chosen, which leads to a slightly reduced accuracy. By choosing the first and last points to lie exactly at the FE boundaries, we can connect the last basis function in each element with the first function in the following element, forming a “bridge” function. By employing a Gaussian quadrature we implicitly choose a polynomial basis in the equivalent SR. Figure 2 shows the first points of a typical FEDVR grid with eleven basis functions per finite element and the resulting grid spacing. The finite elements have a constant extension of 4 a:u:. The FEDVR parameters determine the maximum energy of electrons that can be well represented on the grid, which is around 6 a:u: for the parameters used here. Near the origin, the basis has to represent the Coulomb singularity of the nuclear potential, which is already done with good accuracy for the parameters given here (the ground-state energy of the one-particle Hamiltonian has a relative error of  108 ). In order to increase the accuracy, it would be possible to use smaller finite elements close to the origin.

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From a Lagrange basis for each element 8 .i / rrj
(22)

we thus obtain (in agreement with [115]) the normalized FEDVR basis functions 8 .i / .i 1/ L1 .r/CLM .r/ ˆ i ˆ r ˆ ˆ .i / .i 1/ ˆ w1 CwM ˆ i ˆ < .i / L .i / m .r/ p fm .r/ WD .i / ˆ w ˆ ˆ L.i / m.r/CL.i C1/ .r/ ˆ Mi 1 ˆ ˆ r ˆ : w .i / Cw .i C1/ Mi

mD1 m D 2; : : : ; Mi  1

(23)

m D Mi

1

which fulfill the orthonormality relation Z  hfm.i / jfn.j / i D fm.i / .r/fn.j / .r/ dr Š

'

Mi N X X



wk.l/ fm.i / .rk.l/ /fn.j / .rk.l/ / D ıij ımn :

(24)

lD1 kD1

In contrast to (12), the FEDVR basis functions are only approximately orthogonal because for each element the integrand in (24) is a polynomial of degree 2Mi  2 whereas the integration is only exact for polynomials up to degree 2Mi  3. This deficiency of FEDVR compared to the DVR has little effect in practice since it can be compensated by increasing the number of basis functions. The definitions for the bridge functions (at m D 1 and m D Mi ) in (23) ensure that fM.ii/ .r/ D f1.i C1/ .r/ and thus the continuity of the represented wave functions at the FE boundaries. In contrast, the first derivative is not continuous, since the finite elements have an overlap of only one grid point. As pointed out in [119] the matrix elements of the momentum operator (which is equivalent to the derivative operator) and the kinetic energy operator (which relies on the calculation of the second derivative) are nevertheless correctly defined. The integrals hfm.i / jd=drjfn.j / i for the first derivative operator can be directly calculated, while the second derivative integrals need to be performed by partial integration, such that only first derivatives remain to be evaluated. In analogy to (15) a wave function  .r/ can be written as  .r/ D

Mj N X X

.j / .j / Q k fk .r/

(25)

j D1 kD1

and the scalar product of two wave functions in FEDVR representation is given by

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

h'j i D

X

'Qk.i / fk.i / .r/

i;k

X

Q .j / f .j / .r/ dr ' l l

j;l

X

159

'Qk.i / Q k.i / :

(26)

i;k

Consequently also the potential matrix elements are approximately diagonal, cf. (19)  .i /  Š hfm.i / jb 9jfn.j / i ' V rm ıij ımn (27) and derivatives are calculated by deriving the basis functions, cf. (20) Mj

X X .j / @n f .r/ @n  .r/ k D Q k : @r n @r n N

.j /

(28)

j D1 kD1

Figure 3 shows the eigenvalues of the GLVFUHWH Laplace operator on a FEDVR grid as obtained from (28). The step-like behavior for higher eigenvalues is a consequence of the division into finite elements. For physical problems one has to ensure that the occurring energies are in the region where the eigenvalues grow quadratically (and mimic the dispersion relation of a free particle).  %RXQGDU\ &RQGLWLRQV The FEDVR method, as outlined above, provides a flexible and accurate way of discretizing differential equations. In addition it is also necessary to properly implement the boundary conditions. How to accomplish this is a function of the coordinate under consideration. For example, in spherical coordinates, the wave function must behave regularly at the left hand boundary or matrix elements of potentials

8

200

7

180 160

6 E(i) [ a.u.]

E(i) [ a.u.]

140 5 4 3

120 100 80 60

2

40 1

20

0

0 0

50

100 150 eigenvalue #i

200

250

(a) The first eigenenergies of a FEDVR grid grow quadratically.

0

100

200

300

400

500

eigenvalue #i

(b) For higher energies the spectrum contains “unphysical” steps due to the division into finite elements.

)LJ  Eigenvalues of the GLVFUHWH Laplace operator on a FEDVR grid with rmax D 200 a:u: and eleven basis functions per finite element. The finite elements have an extension of 4 a:u:. The resulting number of grid points is 501.

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such as the Coulomb potential will diverge. One way to ensure that this behavior is satisfied is to omit the first basis function from the DVR wave-function expansion. By keeping the last basis function at the right hand boundary, one may impose open boundary conditions on the wave function. If the last basis function is removed from the expansion, this is tantamount to requiring the wave function to vanish at the right most boundary. If a wave packet reaches rmax during propagation, it encounters “hard walls” in any case since  .r > rmax / D 0 is imposed implicitly due to the end of the grid. The resulting unphysical reflections have to be avoided by either extending the grid so that the boundary is never reached in computing any physically meaningful quantities or by implementing DEVRUELQJ ERXQGDULHV (see following section).

 7HPSRUDO 'LVFUHWL]DWLRQ With the help of the time evolution operator  Z b 8.t C t; t/ D T exp i

t Ct

b 0 / dt 0 +.t

 (29)

t

the solution of the TDSE for a given initial state j .t0 /i can formally be written as  Z b j .t0 C t/i D 8.t0 C t; t0 /j .t0 /i D T exp i

t0 Ct

 b 0 / dt 0 j .t0 /i +.t

t0

(30) where T denotes the time-ordering operator for the exponential with noncommuting b 1 /; +.t b 2 / ¤ 0). argument (Œ+.t Direct evaluation of (30) is cumbersome since the time evolution operator has to be expanded in a Dyson series to represent the time-ordering. However, for small time intervals t the Hamiltonian can be assumed to be constant, thus giving

and

b b 8.t C t; t/ ' exp.i+.t/t/

(31)

b j .t C t/i ' exp.i+.t/t/j .t/i :

(32)

The time evolution operator (29) is WUDQVLWLYH b 8.t C 2t; t/ D b 8.t C 2t; t C t/b 8.t C t; t/

(33)

and XQLWDU\ b 8 .t C t; t/ D b 81 .t C t; t/ ) h .t/j .t/i D h .t C t/j .t C t/i

8 t; t :

(34)

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

161

Arbitrary times can therefore be reached by successive application of (31). The norm of the wave function is conserved for all times. For many practical problems the exponential in (31) can not be evaluated exactly but there exist many propagation schemes that provide different approximations (see [132] for a comprehensive review of different time-stepping techniques). In our approach, we use the 6KRUW ,WHUDWLYH /DQF]RV (SIL) propagation scheme, which will be briefly outlined below.

 /DQF]RV 3URSDJDWLRQ The Lanczos algorithm relies on Krylov subspace techniques that were originally introduced to calculate eigenvalues and eigenvectors of (large) matrices [133]. In the form presented here, it is only applicable for Hermitian matrices. The procedure works as follows: b on A Krylov subspace of order N C 1 is generated by the repeated action of + an initial state j0 i (assumed to be normalized) KN C1 D fj0 i; j1 i; j2 i; : : : ; jN ig bk j0 i : jk i D +

(35) (36)

Orthonormalizing the basis vectors in the subspace by the Gram-Schmidt procedure produces a new basis, N C1 D fj0 i; j1 i; j2 i; : : : ; jN ig :

(37)

b to a previThe most time consuming step in the process is the application of + ously computed Lanczos vector. Given the sparse matrix representation inherent in the FEDVR, it is possible to reduce this to a set of small and structured matrixvector multiplies using the matrices of the one-dimensional FEDVR blocks. These are individually small, dense systems whose size depends on the number of FEDVR functions used in that block. Thus the scaling of the matrix-vector multiply is signifib is then approximated cantly reduced from the more general case. The Hamiltonian + b./ in the N C1 basis. N is chosen much smaller as a .N C 1/  .N C 1/ matrix + b (which can be up to 109 ), with than the dimension of the matrix representation of + typical values of 12  15 in our case. Direct diagonalization of this small matrix can be efficiently performed. In the limit N ! 1, the eigenvalues and eigenvectors of the transformed Hamiltonian converge to those of the full Hamiltonian, with the extreme (i.e., largest and smallest) eigenvalues converging first. The Lanczos algorithm is very effective because in practice it is not necessary to explicitly build up the Krylov space and perform the full orthonormalization to all b./ is tridiagonal and its elements can be obprevious vectors, since the matrix + tained from a three-term recursion relation. This construction proceeds analogously b replacing the coordinate opto the construction of orthogonal polynomials, with + erator.

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We thus get a WKUHHWHUP UHFXUUHQFH UHODWLRQ j0 i D j0 i b 0 i  ˛0 j0 i  jq0 i ˇ0 j1 i D +j b j i  ˛j jj i  ˇj 1 jj 1 i  jqj i ˇj jj C1 i D +j

(38) (39) (40)

where j0 i is assumed to be normalized and b ji ˛j D hj j+j q ˇj D hqj jqj i ;

(41) (42)

where both ˛j and ˇj are real. The Hamilton operator in the subspace N C1 is thus real and tridiagonal and is given by 0 1 ˛0 ˇ0 0    0 B :: :: :: :: C Bˇ0 : : : : C B C B C ./ : : : b b : : : : (43) + D h j +j i D O B i j : ij 0 C B0 : : C B : : : C @ :: : : : : : : : ˇN 1 A 0    0 ˇN 1 ˛N To perform time propagation, we replace the Hamiltonian in the time evolution operator by its approximation in the Krylov subspace of j .t/i ( j0 i), b b ./ t/ : 8./ D exp.i+

(44)

Consequently, b 8./ is restricted to the same Krylov subspace, where the exponential can be evaluated by direct diagonalization X   b 8./ D jZl i exp ih./ t hZl j : (45) l l

b./ with the eigenvalue h./ . Here, jZi i denotes the eigenvector of + j The approximation for the propagated wave function then reads j .t C t/i D b 8./ j .t/i D b 8./ j0 i D

N X

ak jk i

(46)

kD0

with

ak D hk jb 8./ j0 i D

X

  ./ hk jZl i exp ihl t hZl j0 i :

(47)

l

Since the jk i are linear combinations of the jk i which, in turn, are given by bk j0 i, (46) is effectively a N th-order polynomial expansion of the exjk i D + ponential in (31). Moreover, the Lanczos procedure generates a set of orthogonal

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

163

polynomials and for a finite -dimensional operator (such as a Hamiltonian in a DVR representation) the approximation gets H[DFW for N !   1 [132]. In contrast to a standard N th-order Taylor or Chebyshev expansion the coefficients ak are optimized to give the best approximation for a given j0 i and in addition unitarity is conserved. The Lanczos procedure can also be adapted for non-Hermitian operators using a biorthogonal basis. For complex symmetric operators, it can be more simply implemented, by performing the dot products without complex conjugation of the vectors. This also leads to a tridiagonal matrix. Another approach (known as the Arnoldi– Lanczos procedure [134]), transforms the original matrix to an upper Hessenberg matrix, which has zero entries below the first subdiagonal, in (43). This requires that the vectors have to be explicitly orthogonalized to all previous Krylov vectors, and entails considerably more numerical work. All of the above methods can be used to include complex absorbing potentials in the propagation scheme, as the Hamiltonian is then complex symmetric instead of Hermitian.  6SOLW 2SHUDWRU 3URSDJDWLRQ In some of the work described later in this article, we have employed split-operator techniques to propagate the time-dependent Schr¨odinger equation. These rely on approximating the exponential of the sum of two noncommuting operators b $ and b % by a product of exponentials of the operators, i.e., $Cb %/t $t =2 b $t =2 e.b D eb e%t eb C O.t 3 / ;

(48)

where the third-order error term contains commutators of b $ and b %. Doing this for parts of the Hamiltonian in the time propagation operator and neglecting the thirdorder part gives a second-order approximation to propagating the Schr¨odinger equation. The propagation operator is then reduced to individual exponentials of each of the operators in the Hamiltonian. A particular version of this is the Real-Space-Product (RSP) approach [135], which we use for propagating the Schr¨odinger equation for the HC 2 molecular ion (Sect. 6). While the FEDVR kinetic energy operators are block-diagonal, the blocks overlap. Diagonalizing the whole kinetic energy operator thus destroys the sparsity of the matrix. We instead split it into two new operators such that each of the two gets alternating, nonoverlapping blocks. These new operators can be diagonalized by diagonalizing each separate FEDVR block, retaining the sparsity of the kinetic energy operator. The split-operator method has also been used in the He code to prevent reflections at the grid boundary. By adding a FRPSOH[ absorbing potential of the form   r  rcut A.r/ D i˛.r  rcut / ln cos (49) rmax  rcut

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to the Hamiltonian, it is possible to dampen the wave function to zero before it reaches the boundary. The potential and its first derivative are continuous functions, while the second derivative is discontinuous at rcut . The form of this particular potential is similar to a cosine-shaped “absorbing edge” masking function (as also employed in, e.g., [90]) and proved to be suitable for our purposes. Since adding this potential to the full Hamiltonian would make it non-Hermitian, we use the splitoperator method, b b O 8.t; t C t/ D exp.i+.t/t  iAt/     t t b D exp iAO exp.i+.t/t/ exp i AO C O.t 3 / :(50) 2 2 This has the considerable advantage of not requiring a non-Hermitian SIL procedure and is sufficiently accurate for our purposes. The effectiveness of such potentials always depends on the energy and the form of the incoming wave packet. In general, it is hard to completely avoid reflections. However, it is possible to suppress them below a certain level.1  6WDELOLW\ DQG 8QLWDULW\ &RQVLGHUDWLRQV The Krylov subspace approximation (45) for the time evolution operator is explicitly unitary. Therefore, the Lanczos algorithm is XQFRQGLWLRQDOO\ VWDEOH and the propagation scheme is norm-conserving for Hermitian Hamiltonians. Consequently, the algorithm is also energy-conserving for time-independent operators. Even though the propagation is explicitly unitary regardless of the properties of the discrete Hamiltonian, its spectrum still affects the propagation because large spectral ranges require small time steps or high orders to get accurate results. Thus, smaller grid spacings and more FEDVR basis functions make the temporal propagation computationally more costly.

 5HPDUNV RQ WKH 1XPHULFV DQG &RPSXWHU ,PSOHPHQWDWLRQ The number of angular momenta, number of radial points, extent of the radial grids, and propagation time that are needed to extract converged transition probabilities and/or cross sections depends sensitively on many factors. Care has to be taken to ensure that the basis is chosen such that, e.g., the Coulomb potential singularities at small distances and the highest energy electrons are represented well. Additionally, the question of whether one is interested in total or differential quantities determines the necessary basis sizes. Suffice it to say that by suitably arranging the density of finite elements and the order of the basis in each element, one is able to adequately converge the systems described in this article. The details of how to accomplish this 1

See, e.g., [136, 137] for a detailed treatment of the problem.

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

165

are nontrivial, and will not be fully discussed in the current contribution. A further issue, which follows from the size of the basis set concerns the spectral range of the Hamiltonian. The highest eigenvalues of the Hamiltonian will fix the largest step size in time that can be used in the time propagation. While there are some ways that one can reduce the basis set by prediagonalization, the price that is paid is a loss of sparsity in the Hamiltonian matrix which is one of the most compelling reasons for using an FEDVR representation. Difficult choices have to be made but in general the computations can only be performed with substantial computational resources. The vector lengths may get as large as 109 for some cases and a parallel implementation using MPI on a large cluster with many cores is necessary. The calculations described here were performed using the US NSF TeraGrid systems at the National Institute for Computational Science at the University of Tennessee/Oak Ridge, the Texas Advanced Computer Center at the University of Texas/Austin, the Lobo and Coyote cluster at the Department of Energy Los Alamos National Laboratory, and the Vienna Scientific Cluster at the Vienna University of Technology. Without these computational instruments, much of the work described herein would have been impossible.

 +H The first application of the foregoing method we discuss is the He atom. In this case, the nonrelativistic Hamiltonian is given by S21 b S2 Z Z 1 b Db +.t/ C 2  C Cb 9I .t/ ; 2 2 r1 r2 r12

(51)

where the electron-field interaction operator in dipole approximation is b 9IL .t/ D E .t/  .b U1 Cb U2 /

or b 9IV .t/ D A.t/  .b S1 C b S2 / ;

(52)

with the superscript denoting the use of length (L) or velocity gauge (V ), and E .t/ and A.t/ denoting, respectively, the electric field and the vector potential of the electromagnetic pulse. For the velocity gauge expression, the term A 2 =2 has been removed by adding a global time-dependent phase to the wave function. We proceed by expanding the six-dimensional wave function  .U1 ; U2 / in coupled spherical harmonics, 1 X 1 X RlLM .r1 ; r2 ; t/ LM 1 ;l2  .U1 ; U2 ; t/ D Yl1 ;l2 .˝1 ; ˝2 / (53) r1 r2 L;M l1 ;l2

with YlLM .˝1 ; ˝2 / D 1 ;l2

X m1 ;m2

hl1 m1 l2 m2 jLM iYlm11 .˝1 /Ylm22 .˝2 / :

(54)

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B.I. Schneider et al.

Substituting (51) and (53) into (1), multiplying by YlLM , and integrating over all 1 ;l2 angles yields a system of coupled partial differential equations in .r1 ; r2 ; t/ , the time-dependent close coupling (TDCC) equations [138, 139] i

1 X 1 X @ LM b 1 l2 LM iRLM .r1 ; r2 ; t/ ; Rl 0 ;l 0 .r1 ; r2 ; t/ D hl10 l20 L0 M 0 j+jl l1 ;l2 @t 1 2

(55)

L;M l1 ;l2

where in practice the sums have to be truncated at certain maximum angular momenta .Lmax ; l1;max ; l2;max /. In what follows, we restrict ourselves to radiation polarized along the z-axis and initial states of the system with M D 0, where the b 1 l2 L0i projection of the angular momentum along the z-axis is zero. hl10 ; l20 L0 00 j+jl is given by b 1 l2 L0i hl10 l20 L0 0j+jl    1 @2 l1 .l1 C 1/ Z 1 @2 l2 .l2 C 1/ Z D ıLL0 ıl1 l10 ıl2 l20  C   C  2 @r12 r1 2 @r22 r2 2r12 2r22  LL0 C WlL .r ; r ; t/ ; (56) 0 0 .r1 ; r2 / C V l1 ;l2 ;l 0 ;l 0 1 2 1 ;l2 ;l ;l 1 2

1 2

with the electron-electron interaction given by q L 0 .1/ WlL .r ; r / D ı .2l1 C 1/.2l10 C 1/.2l2 C 1/.2l20 C 1/ 0 0 1 2 LL 1 ;l2 ;l1 ;l2   0 0 0 1   X l1  l10 l2  l20 L l2 l1  r<  .1/ C1 : (57) 0 0 0 0 0 0  l 1 l2 r> D0 In the length gauge, the electron-field interaction is 0

VlLL 0 0 .r1 ; r2 ; t/ 1 ;l2 ;l1 ;l2     L 1 L0 l 1 l0 l1 l 2 L D E.t/cLL0 r1 .1/l2 cl1 l10 1 1 ı0 0 0 0 0 0 0 L0 1 l10 l2 l2    l 1 l0 l2 l 1 L 0 C r2 .1/l1 cl2 l20 2 2 ı ; 0 0 0 L0 1 l20 l1 l1

(58)

while in the velocity gauge, it is 0

VlLL 0 0 .r1 ; r2 ; t/ 1 ;l2 ;l ;l

  p L 1 L0 D iA.t/ .2L C 1/.2L0 C 1/ 0 0 0      0 @ l10 .l10 C 1/ C l1 .l1 C 1/ l 1l l1 l2 L 0  .1/l2 cl1 l10 1 1 1 ı  0 0 0 L0 1 l10 l2 l2 @r1 2r1     0 0 0 l2 .l2 C1/Cl2 .l2 C1/ @ l 1l l 2 l1 L 0 C .1/l1 cl2 l20 2 2 ı  ; (59) 0 0 0 L0 1 l20 l1 l1 @r2 2r2 1 2

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

167

where cij D .2li C 1/.2lj C 1/. The angular momentum selection rules restrict the coupling to being tridiagonal in L. The problem then reduces to a set of coupled two-dimensional, radial partial differential equations.

 5HPDUNV RQ WKH &DOFXODWLRQ RI 7ZR (OHFWURQ ,QWHJUDOV b is represented in TDCC by The electron-electron interaction Hamiltonian : b hk 0 j:jki D

1 X r< b ang :k;k 0 ; ; C1 r > D0

(60)

b 0 is the angular part of the electron-electron interaction operator, k and where : k;k ; k 0 are the combined angular indices .L; l1 ; l2 / and .L0 ; l10 ; l20 /, r< is min.r1 ; r2 / and r> is max.r1 ; r2 /. The usual approximation for potentials in the FEDVR approach is to represent them as a diagonal matrix, with the entries just being the values of the potential at the grid points. However, this approach would entail a large error for the electron-electron interaction operator, as the radial part r< =r>C1 has a derivative discontinuity at r1 D r2 , which can not simply be represented in the FEDVR basis. This can be fixed following the recipe of McCurdy et al. [140], a short summary of which is given in the following. The general idea behind this is to evaluate one of the radial integrals over FEDVR basis functions analytically, instead of using the Gauss–Lobatto quadrature associated with the grid. The radial parts of the interaction operator are given by the integrals ˇ  ˇ

 ˇ r< ˇ ˇ ˇ 0 0 fj1 fj2 ˇ C1 ˇfj1 fj2 r> Z rmax Z rmax r< D dr1 dr2 fj1 .r1 /fj10 .r1 / C1 fj2 .r2 /fj20 .r2 / ; (61) r> 0 0 ang

where fj .r/ is the j th FEDVR basis function. Instead of directly employing Gauss– Lobatto quadrature, we define the function Z

rmax

y.r/ D r 0

Z

r

D 0

dr 0

dr 0

r< r>C1

fj2 .r 0 /fj20 .r 0 /

r 0 fj .r 0 /fj20 .r 0 / C r 2

Z

rmax r

dr 0

r C1 fj .r 0 /fj20 .r 0 / ; r 0C1 2

which satisfies the radial Poisson equation,  2  2 C 1 d . C 1/ fj2 .r/fj20 .r/  y.r/ D  2 2 dr r r

(62)

(63)

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B.I. Schneider et al.

 with the boundary conditions y.0/ D 0 and y.rmax / D rj2 =rmax ıj2 ;j20 . Expanding y.r/ in the basis of FEDVR functions, inserting into (63), solving the resulting matrix equation, and finally adding a solution of the homogeneous radial Poisson equation to satisfy the boundary conditions leads to

y.r/ D .2 C 1/ where

N X rj r C1 fi .r/ ŒT  1 ıj2 ;j20 C 22C1 ıj2 ;j20 ; p i;j2 rj2 wj2 rmax i D1

ˇ 

ˇ ˇ d . C 1/ ˇˇ  Ti;j D fi ˇˇ  2 C ˇfj dr r2

(64)

(65)

is twice the single-electron kinetic energy operator for angular momentum  in the FEDVR basis, ŒT  1 i;j is the element i; j of its inverse, and wj is the Gauss–Lobatto integration weight associated with grid point rj . Inserting this expression for y.r/ back into the original (61) and performing the integral using the Gauss–Lobatto quadrature gives the final result ˇ  ˇ

 ˇ r< ˇ ˇfj 0 fj 0 fj1 fj2 ˇˇ C1 ˇ 1 2 r> ! rj1 rj2 2 C 1  1 D ıj1 ;j10 ıj2 ;j20 ŒT j1 ;j2 C 2C1 ; (66) p rj1 rj2 wj1 wj2 rmax which remarkably is still diagonal in the FEDVR grid indices j1 ; j2 . The inverse matrices only have to be calculated once at the start of the program, which does not incur a large computational overhead. The improved precision of this expression for the electron-electron interaction is considerable. As an example, the error in the ground-state energy (i.e., the deviation from the “real” ground state of the nonrelativistic Hamiltonian) is only 5:3  105 a:u: using this improved expression, while the error is 1:2  102 a:u: when using the “naive” expression for the interaction operator. This example was calculated using typical parameters for our simulations (FEDVR elements of order 11 with 4 a:u: extension, l1;max D l2;max D 9).

 2EVHUYDEOHV We exploit the fact that our time-dependent approach allows propagation of the wave packet for long times after the conclusion of the pulse. Once the distance between the two electrons has reached a large enough value, we can neglect the ˇ ˇ1 b12 D ˇb electron-electron interaction term, + U1 b U2 ˇ , which becomes insignificant for asymptotic distances. Consequently, we approximate the continuum by the exact solution of the (separable) stationary Schr¨odinger equation with the Hamiltonian without electron-electron interaction,

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

S21 b S2 Z Z b0 D b + C 2  : 2 2 r1 r2

169

(67)

If both electrons are far away from the nucleus and/or have high energies, it would also be possible to neglect the electron-nucleus interaction when constructing final states for double ionization. However, as in almost all cases there is single as well as double ionization, we retain the electron-nucleus interaction term. This also ensures that the approximate single-continuum and double-continuum eigenstates are orthogonal to each other. The separable Hamiltonian (67) is just the sum of two independent one-particle Hamiltonians. Before constructing the two-electron product states for the single and double continuum, we summarize some of the properties of the (analytic) eigenfunctions of the single-particle Hamiltonian of a hydrogen-like atom, with S2 Zeff b1 D b +  : 2 r

(68)

As (68) is spherically symmetric, its eigenstates can be separated into a radial and an angular part, k;l .r/ l ˚k;l;m .U/ D Ym .˝/ ; (69) r where the angular part is described by the spherical harmonics Ylm .˝/. For the ERXQG VWDWHV, the radial part of the regular eigenfunction is given by [141] s p       Zeff .n  l  1/Š 2Zeff r lC1 2lC1 2Zeff r Zeff r n;l .r/ D Lnl1 exp  n .n C l/Š n n n (70) where L stands for the generalized Laguerre polynomial and n  1 is the main quantum number. The eigenenergies of the bound states are given by En D 

2 Zeff : 2n2

(71)

The regular solution for the XQERXQG VWDWHV is given by the regular radial &RXORPE IXQFWLRQ Fl . ; kr/ [141] r 2 k;l .r/ D Fl . ; kr/ ;

j .l C 1 C i /j ikr Fl . ; kr/ D 2l e=2 e .kr/lC1 F .l C 1  i ; 2l C 2I 2ikr/ (72) ; .2l C 1/Š with the confluent hypergeometric series F F .a; bI z/ D

1 X  .a C n/  .b/ z n ;  .a/  .b C n/ nŠ nD0

(73)

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which in the limit r ! 1 behave as   l

Fl . ; kr/ ! sin kr  ln 2kr  C l ; 2

(74)

with the &RXORPE SKDVH l D argΠ.l C1Ci /. In (72) we introduced the Coulomb parameter , which determines the strength of the Coulomb term in (68), D

Zeff : k

(75)

Inserting the radial part (72) back into (69) yields the VSKHULFDO &RXORPE ZDYHV ˚k;l;m .U/, which are eigenfunctions of (68), orthonormalized in momentum k, total angular momentum /2 , and the z-component Lz of angular momentum, h˚k;l;m j˚k 0 ;l 0 ;m0 i D ı.k  k 0 /ıl l 0 ımm0 :

(76)

Since k is a continuous variable, the wave functions are normalized to the Dirac delta function (or distribution), while the discrete quantum numbers l, m are orthonormalized to a Kronecker delta. Instead of normalizing in momentum space, we can also use HQHUJ\QRUPDOL]HG radial Coulomb functions. Due to ı.k  k 0 / D

dE ı.E  E 0 / ; dk

kD

p

2E ;

(77)

they are related to the momentum-normalized functions (72) according to E;l .r/ D

k;l .r/ p : k

(78)

If we are not interested in the angular momentum quantum numbers, but want to specify a 3-vector N  .k; ˝k / for the momentum, we can use the expansion N .U/ D

1 X l X

i l eil Y.l/ m .˝k /˚k;l .U/ :

(79)

lD0 mDl

These functions are the solutions of the Coulomb problem satisfying incoming scattering boundary conditions that converge asymptotically to eigenstates of linear momentum N. ,QFRPLQJ boundary conditions are the appropriate basis states for extracting ionization probabilities [142–144]. Using these RQHHOHFWURQ eigenfunctions, we construct the GRXEOH FRQWLQXXP wave functions as symmetrized product states of two unscreened Coulomb waves (79) with effective charge Zeff D 2, where the symmetrization is necessary to account for the indistinguishability of the two electrons. For N1 ¤ N2 , these states (in singlet spin symmetry) are given by 1 NDC .U1 ; U2 / D p Œ 1 ;N2 2

N1 .U1 /

N2 .U2 /

C

N1 .U2 / N2 .U1 /

:

(80)

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

171

The use of a product final state amounts to neglecting the effect of electron-electron interaction, which is a good approximation only in the asymptotic regime (r12 ! 1). It should be noted that in our approach of solving the TDSE, we only need the double-continuum wave functions in the asymptotic region for extraction of the final momentum distributions. The correlation is thus included in the calculation at each step, the only approximation is the identification of the momenta N1 , N2 in the product state with the asymptotic momenta of the two electrons. In other approaches, such as time-independent perturbation theory, it is much more crucial to use an accurate representation of the continuum. This can be achieved in a number of ways, e.g., by using so-called 3C wave functions [145, 146] which consist of a product of three two-body Coulomb functions. Another approach is to use the techniques of exterior complex scaling (ECS) in combination with formal scattering theory, which can be used to extract the double ionization amplitudes by a surface integral at the edge of the box, where the product of Coulomb waves is again a good approximation if the box is large enough [140, 147, 148]. A third approach to get the double ionization wave function is to use the J -matrix method to generate fully correlated multichannel scattering wave functions for the single continuum and then obtain the double ionization wave packet by subtracting the bound and singly ionized parts from the total wave function [80]. Inserting the partial-wave expansion (79) into (80) and switching to coupled spherical harmonics yields the double-continuum wave function in coordinate space, 1 X 1 X

NDC .U1 ; U2 / D 1 ;N2

L;M l1 ;l2

1 1 i l1 Cl2 ei.l1 Cl2 / ŒYlLM .˝k;1; ˝k;2 / p 1 ;l2 2 r1 r2

 Œk1 ;l1 .r1 /k2 ;l2 .r2 /YlLM .˝r;1 ; ˝r;2 / 1 ;l2 C k1 ;l1 .r2 /k2 ;l2 .r1 /YlLM .˝r;2 ; ˝r;1 / : 1 ;l2

(81)

In analogy, we construct the VLQJOH FRQWLQXXP as a symmetrized product state of a ERXQG VWDWH ˚n;l;m .U/ of the HeC ion and a Coulomb wave N .U/ with effective charge Zeff D 1, 1 SC n;l;m;N .U1 ; U2 / D p Œ˚n;l;m .U1 / 2

N .U2 /

C ˚n;l;m .U2 /

N .U1 /

:

(82)

Inserting the partial-wave expansion for the Coulomb wave and the coordinate representation of ˚n;l;m .U/ yields the single-continuum wave function in coordinate space. The bound state is not expanded into a function of a wave vector (as in (79) for the continuum wave) leaving a Clebsch–Gordan coefficient from switching to the coupled angular momentum representation, SC n;l;m;N .U1 ; U2 /

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B.I. Schneider et al.

D

1 X 1 X

lk X

L;M lk D0 mk Dlk

1 1 i lk eilk ŒYlmkk .˝k / hlmlk mk jl lk LM i p 2 r1 r2

 Œn;l; .r1 /k;lk .r2 /YlLM .˝r;1 ; ˝r;2 / 1 ;l2 C n;l .r2 /k;lk .r1 /YlLM .˝r;2 ; ˝r;1 / : 1 ;l2

(83)

 ,RQL]DWLRQ 3UREDELOLW\ 'LVWULEXWLRQV  )XOO\ 'LIIHUHQWLDO 3UREDELOLW\ 'LVWULEXWLRQV By projecting the single- and double-continuum functions constructed in the previous section onto the fully correlated final state wave function, we obtain momentum probability distributions. The electron momentum distribution for GRXEOH LRQL]DWLRQ is given by ˇ ˇ2 P DI .N1 ; N2 / D P DI .k1 ; k2 ; ˝k;1 ; ˝k;2 / D ˇhNDC j iˇ : 1 ;N2

(84)

Using the expression (81) for the double continuum and (53) for the calculated wave function (for which M D 0 because of cylindrical symmetry) yields P DI .N1 ; N2 / ˇ1 1 1 ˇX X l1 l2 i.l Cl / L0 D ˇˇ i e 1 2 Yl1 ;l2 .˝k;1 ; ˝k;2 / 2 L l1 ;l2 1Z 1

Z

dr1 dr2 RlL1 ;l2 .r1 ; r2 /Œk1 ;l1 .r1 /k2 ;l2 .r2 / 0 0 ˇ2 ˇ C k1 ;l1 .r2 /k2 ;l2 .r1 /ˇˇ ; 

(85)

where we used the orthonormality relation 0

0

hYlLM jYlL0 ;lM0 i D ıLL0 ıl1 l10 ıl2 l20 ıMM 0 1 ;l2 1 2

(86)

for the evaluation of the angular part of the integral in position space, and the fact that the Coulomb wave functions can be chosen real. Furthermore, we can use the exchange symmetry for the wave function j i RlL2 ;l1 .r2 ; r1 / D .1/.l1 Cl2 L/ RlL1 ;l2 .r1 ; r2 / and obtain for the double ionization probability distribution

(87)

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

173

ˇ2 1 X 1 ˇX ˇ l1 l2 i.l1 Cl2 / L0 L ˇ P .N1 ; N2 / D ˇ i e Yl1 ;l2 .˝k;1 ; ˝k;2 /Pl1 ;l2 .k1 ; k2 /ˇˇ (88) DI

L l1 ;l2

with PlL1 ;l2 .k1 ; k2 / D

p Z 2

1

0

Z

1

dr1 dr2 RlL1 ;l2 .r1 ; r2 /k1 ;l1 .r1 /k2 ;l2 .r2 / :

0

(89)

For the six-dimensional VLQJOH LRQL]DWLRQ probability distribution we find a similar expression, ˇ1 1 ˇX X P .n; l; m; N/ D ˇˇ SI

lk X

.i/lk eilk Ylmkk .˝k /

L lk D0 mk Dlk



ˇ2 ˇ

hlmlk mk jl lk L0iPlL1 ;l2 .n; k/ˇˇ

(90)

dr1 dr2 RlL1 ;l2 .r1 ; r2 /n;l .r1 /k;lk .r2 / :

(91)

with PlL1 ;l2 .n; k/

p Z D 2 0

1

Z 0

1

Instead of using distributions differential in PRPHQWXP, it is equally possible to use HQHUJ\ differential distributions, which can be described by the the same expressions, except for the use of energy-normalized Coulomb functions E;l (78) instead of k;l in (89) and (91). The fully differential probability distributions (88) and (90) contain all the information about the final state momenta of the electrons. Often, it is more interesting to look at lower-dimensional distributions, obtained either by integrating out some variables of the full distribution or by choosing specific FXWV in the six-dimensional space.  (QHUJ\ 3UREDELOLW\ 'LVWULEXWLRQV Integrating out the angles ˝1 ; ˝2 in (88) gives the joint energy probability distribution for the two (ejected) electrons in a double ionization process P DI .E1 ; E2 / D

1 X 1 ˇ ˇ2 X ˇ L ˇ ˇPl1 ;l2 .E1 ; E2 /ˇ :

(92)

L l1 ;l2

Further integrating over E1 or E2 gives the single-electron energy probability distribution for double ionization, Z 1 Z 1 DI DI P .E/ D P .E1 ; E2 /dE1 D P DI .E1 ; E2 / dE2 ; (93) 0

0

174

B.I. Schneider et al.

i.e., the probability for one detected electron to have the energy E.  $QJXODU 3UREDELOLW\ 'LVWULEXWLRQV More detailed information about a double ionization event is provided by angular differential distributions. The joint angular distribution is obtained by integrating over the energies of both electrons, Z 1Z 1 DI P .˝k;1; ˝k;2 / D P DI .E1 ; E2 ; ˝k;1; ˝k;2 / dE1 dE2 : (94) 0

0

This gives the distribution in angles, regardless of the energies of the electrons. By dropping the integration over one of the energies, one obtains the angle-energy probability distribution, which can reveal correlations between the angular and energy degrees of freedom. An additional observable of interest is the one-electron angular distribution, which can be characterized by the anisotropy parameters ˇj , as shown in the following. The one-electron probability distribution P DI .E1 ; ˝k;1 / for one electron with respect to the laser polarization axis is given by integrating (88) over E2 and ˝k;2 . Because of the total cylindrical symmetry in the system, the resulting one-electron probability distribution is independent of the azimuthal angle '1 . In the following, we therefore integrate over '1 , and consequently, P DI .E1 ; 1 / D 2 P DI .E1 ; ˝k;1 /, and ZZZ DI P .E1 ; 1 / D P DI .E1 ; E2 ; ˝k;1 ; ˝k;2 / dE2 d˝k;2 d'1 : (95) Due to the indistinguishability of the two electrons it follows that P DI .E1 ; 1 / D P DI .E2 ; 2 / D P DI .E; / :

(96)

This expression can be characterized by the angular anisotropy parameters2 ˇj (cf.,e.g., [92]) that are obtained by projecting P DI .E; / on Legendre polynomials Pl .cos /, 1 X P DI .E; / D P DI .E/ ˇj .E/Pj .cos / ; (97) j D0

where the energy differential ionization probability P DI .E/ has been factored out of the sum so that ˇ0 .E/ D 1, as the integral over the Legendre polynomials is zero for j ¤ 0. For ionization by a specified number of photons (i.e., if there is no interference between processes with different photon numbers), the parity of the wave function is well-defined, and the coefficients of Legendre polynomials with odd j vanish. This can be seen in (101) from the 3j-symbol containing j , L and L0 , with all magnetic quantum numbers equal to zero (the SDULW\ MV\PERO), which is zero for odd j C L C L0 . As L and L0 are either both odd or both even (L D 1 for 2

Often the labeling ˇ D ˇ2 and D ˇ4 is used instead [79].

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

175

one-photon transitions, L D 0; 2 for two-photon transitions, L D 1; 3; 5 for threephoton transitions, . . . ), L C L0 is always even. Therefore, j also has to be even. In addition, because of the triangle inequality in the parity 3j-symbol, the highest j occurring in an n-photon transition from a state with L D 0 is j D 2n. The anisotropy parameters can be expressed by inserting the electron momentum distribution for double ionization (88) into (95) and analytically performing the integration Z P .E1 ; 1 / D DI

1

0

Z D

1

0



Z

Z

2

P DI .E1 ; E2 ; ˝1 ; ˝2 / dE2 d˝2 d'1 Z

˝2

˝2

0

Z

0

ˇ1 1 2 ˇX X ˇ ˇ

il1 l2 ei.l1 Cl2 / YlL0 .˝1 ; ˝2 / 1 ;l2

L l1 ;l2

ˇ2 ˇ

PlL1 ;l2 .E1 ; E2 /ˇˇ

dE2 d˝2 d'1 :

(98)

For brevity we will use DlL1 ;l2 .E1 ; E2 / D il1 l2 ei.l1 Cl2 / PlL1 ;l2 .E1 ; E2 /

(99)

in the following formulas. This corresponds to an expansion in our coupled basis representation in energy space, thus expressing the doubly ionized wave function  DI .E1 ; E2 ; ˝k;1 ; ˝k;2 / in coupled spherical harmonics,  DI .E1 ; E2 ; ˝k;1 ; ˝k;2 / D

1 X 1 X

DlL1 ;l2 .E1 ; E2 /YlL0 .˝k;1 ; ˝k;2 / : 1 ;l2

(100)

L l1 ;l2

The integrals over the angles can be performed analytically. The final result for the angular probability distribution P DI .E1 ; 1 / is then q X X 0 P DI .E1 ; 1 / D .1/j LL l2 .2l1 C 1/.2l10 C 1/.2L C 1/.2L0 C 1/ j

L0 ;l10 L;l1 ;l2

 0   0 l1 l1 j j L L0 j l1 l10 .DlL0 ;l2 / DlL1 ;l2 0 0 0 0 0 0 l2 L0 L 1  Pj .cos 1 / : (101)  .2j C 1/

For j D 2 this formula coincides with the expression for ˇ parameters presented by Kheifets et al. [92] and an analogous expression for ˇ parameters of two-photon single ionization by Gribakin et al. [149].

176

B.I. Schneider et al.

 $QJXODU 'LVWULEXWLRQ IRU 6LQJOH ,RQL]DWLRQ Similar to the one-electron angular distribution for double ionization expressed in terms of the anisotropy parameters, we define the analogous probability distribution for single ionization by summing over the states of the bound electron XZ P SI .E; / D P SI .n; l; m; E; ˝k / d' ; (102) n;l;m

with P SI .n; l; m; E; ˝k / defined in (90). The derivation is similar to the one for the double ionization angular distribution, SI

D

.n; l; m; E; ˝k / 1 X 1 X

lk X

 lk ilk

.i/ e

Ylmkk .˝k /

L lk D0 mk Dlk

 l lk L P L .n; E/ ; m mk 0 l1 ;l2

(103)

with PlL1 ;l2 .n; E/ given in (91) and the square brackets denoting a Clebsch–Gordan coefficient. Instead of performing the integral over the continuum states of the second electron, we take the sum over the bound states Z dE2 d˝k;2 

l X X

:

(104)

n;l mDl

This yields P SI .E; k / D

X

ˇj .E/Pj .cos k / ;

(105)

j

with ˇj .k/ D

XX

0

.1/j LL l

q .2lk C 1/.2lk0 C 1/.2L C 1/.2L0 C 1/.2j C 1/

n;l L0 ;l 0 k



L;lk

   0 i 0 0 lk lk0 j j L L0 j lk lk0  .i/lk e lk PlL0 ;l .n; k/ 0 0 0 0 0 0 l L0 L k    .i/lk eilk PlLk ;l .n; k/ :

(106)

The different bound states .n; l; m/ are summed up incoherently in (106). For a fixed n; l; m, we get the probability distribution for the free electron associated with production of the HeC ion in different excited (shake-up) states (e.g., 1s; 2s; 2p; : : :).

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

177

 7RWDO &URVV 6HFWLRQV Integrating (85) over all variables including E1 and E2 gives the total double ionization yield. Up to prefactors, it also gives an approximation for the total double ionization cross section for a suitably chosen pulse. The dependence on the primary photon energy is only implicit through the electromagnetic pulse entering the propagation. Within a time-dependent calculation, the resulting double ionization (DI) probability depends on the spectral distribution, i.e., the shape and duration of the laser pulse, while the fundamental quantity of interest, the DI cross section (DICS) at fixed frequency of the ionizing radiation does not. Extraction of the DICS therefore requires special care. For the case of one-photon ionization, it is straightforward to relate the energy-dependent yield to the cross section. From a single pulse, one can thus calculate the cross section for all energies contained within the pulse [80, 102]. This is not possible (without additional approximations, such as used in [106]) for two- or multiphoton ionization, since the relation between cross section and yield contains an integral over intermediate energies. For the evaluation of this integral, the intermediate states and energies would have to be explicitly available. This is not easily possible in the current approach without losing the key advantage of the time-dependent method of not having to construct intermediate or final states explicitly. The simplest alternative is to use a sufficiently long pulse with narrow spectral width and calculate the cross section from the total yield with the approximation that it is constant over the width of the pulse. For this approximation to be valid, the spectral width of the pulse must be smaller than the energy width over which the cross section significantly changes. We can check the convergence by varying the pulse length. Figures 4 and 5 illustrate this for both the joint two-electron energy distribution P DI .E1 ; E2 / (Fig. 4) and the integral (Fig. 5) along lines of constant total energy E1 C E2 in Fig. 4 for three different pulses, with durations of T D 1 fs ( 10 cycles), T D 4 fs ( 40 cycles), and T D 9 fs ( 90 cycles). While the 4 fs pulse is sufficient to resolve the cross section a few eV above the threshold, the shorter pulse (frequently employed, see [72, 76, 80, 86]) results in averaging over the threshold region. Close to the sequential threshold, the 9 fs pulse, or even longer ones, are necessary to resolve the detailed structure of the cross section. Another requirement is that the pulse has to be weak enough such that lowest order perturbation theory is applicable and that ground-state depletion can be neglected. We therefore choose a peak intensity of I0 D 1012 W=cm2 . Variation between 1011 W=cm2 and 1013 W=cm2 results in deviations for the total cross section at 42 eV of less than 0:3%. For an intensity of 1013 W=cm2 , the two-photon yield is a factor of 104 higher than with 1011 W=cm2 . Another test for applicability of perturbation theory is the linear scaling of the yield with the total duration T of the pulse, i.e., the transition rate must be proportional to ˚.t/N , where ˚.t/ D I.t/=! is the photon flux and N is the minimum number of photons required for the process to take place. The double ionization yield is then given by

178

B.I. Schneider et al.

Z DI Pnonseq D

1 1

dt N ˚.t/N ;

(107)

where N is the total generalized N -photon cross section for double ionization of He. Accordingly, the cross section is given by  N

! I0

N

ZZZZ

1

dE1 dE2 d˝1 d˝2 P DI .E1 ; E2 ; ˝1 ; ˝2 / ;

Teff;N

(108)

where the effective time Teff;N for an N -photon process is defined as Z Teff;N D



1

dt 1

I.t/ I0

N :

(109)

For a sin2 pulse envelope and a two-photon process, Teff;2 is found to be 35T =128 [72, 80, 86]. (108) is valid for direct, i.e., nonsequential double ionization when no on-shell intermediate state is involved.

 'LIIHUHQWLDO &URVV 6HFWLRQV The triply differential cross section (TDCS) for emitting one electron with energy E1 into the solid angle ˝1 , while the second one is emitted into ˝2 follows from (108) as

80

N

Z

1

dE2 P DI .E1 ; E2 ; ˝1 ; ˝2 / :

Teff;N

10

80

50

6

40 4

30 20

2

8

E2 [eV]

E2 [eV]

60

60 50

6

40 4

30 20

2

10 0

0 0

2

4

6

8

10

E1 [eV]

(a) 1 fs sin2 pulse

10

80

70 prob. dens. [arb.u.]

70 8

(110)

70 8

60 50

6

40 4

30 20

2

10 0

0 0

2

4

6

8

E1 [eV]

(b) 4 fs sin2 pulse

10

prob. dens. [arb.u.]

! I0

E2 [eV]

10



prob. dens. [arb.u.]

d N D dE1 d˝1 d˝2

10 0

0 0

2

4

6

8

10

E1 [eV]

(c) 9 fs sin2 pulse

)LJ  Energy distribution after two-photon double ionization from three different laser pulses with a mean energy of h!i D 42 eV. All three pulses have a sin2 envelope for the vector potential, with total durations (a) 1 fs ( 10 cycles), (b) 4 fs ( 40 cycles), (c) 9 fs ( 90 cycles). The distributions are centered around the line E1 C E2 D 2 h!i  I1  I2  5 eV. The width of the distribution directly shows the energy uncertainty due to Fourier broadening.

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

41

43

45

47

P DI (Etot ) 1 |F(ω)|2

10 8

0.8

6

16

37

39

41

43

45

47

P DI (Etot ) |F(ω)|2

14 12

10 8

10

6

4

0.4 0.2

2

0

0

0

2

4

6

8

10

4

6 4

2

2 0

0

2

4

6

8

Etot [eV]

Etot [eV]

(a) 1 fs sin2 pulse

80

37

39

41

43

45

47

P DI (Etot ) |F(ω)|2

70 60

10 8

50

6

40

8

0.6

(×10−9 )

39

(×10−9 )

(×10−9 )

37

hω [eV] ¯

¯hω [eV]

hω [eV] ¯ 1.2

179

(b) 4 fs sin2 pulse

10

0

4

30 20

2

10 0

0

2

4

6

8

10

0

Etot [eV]

(c) 9 fs sin2 pulse

)LJ  Total energy distribution P DI .Etot / and Fourier spectra of 1 fs, 4 fs and 9 fs sin2 laser pulses. P DI .Etot / is the integral over lines with Etot D E1 C E2 from Fig. 4. The left and lower axes describe P DI .Etot /, the right and upper axes describe jF .!/j2 , the Fourier transform of A.t/. For the 4 fs and 9 fs pulses, the double ionization probability directly reflects the Fourier spectrum. For the shorter pulse the electron energy distribution is strongly influenced by the energy dependence of the cross section (cf. Fig. 6).

In the limit of an infinitely long laser pulse with well-defined energy (i.e., a deltalike spectrum), (110) becomes equivalent to 

! I0

N

1 Teff;N

Z dE2 P DI .E1 ; ˝1 ; ˝2 /ı.E0 C N „!  E1  E2 / ;

(111)

where E0 is the ground-state energy. It is worthwhile to mention that unlike the joint two-electron energy distribution, the TDCS as calculated by (110) is, within reasonable limits, insensitive to the pulse shape used in the time-dependent approach since the Fourier width of the pulse is accounted for by the integration over the energy of the second electron. Instead of specifying one of the energies and integrating over the other, it is also possible to specify energy (or momentum) sharing. For that purpose, we transform from the usual coordinates .E1 ; E2 / to .Etot ; ˛/, with Etot D E1 CE2 and tan.˛/ D E1 =E2 . For a fixed value of ˛, the integration is performed over the total energy Etot , i.e., along straight lines through the origin in Fig. 4. This results in the TDCS at fixed energy sharing (the frequently investigated case of equal energy sharing corresponds to ˛ D =2).

 5HPDUNV RQ 7ZR 3KRWRQ 'LUHFW DQG 6HTXHQWLDO ,RQL]DWLRQ RI +H Two-photon double ionization (TPDI) of helium is one of the simplest multiphoton processes involving electron correlation. Consequently, TPDI of atomic helium has been the subject of intense theoretical studies in the past few years [45–47, 63– 108, 114, 150]. Most of the existing literature deals with either (i) cross sections

180

B.I. Schneider et al.

in the QRQVHTXHQWLDO regime of TPDI, or with (ii) the effects of ultrashort (attosecond) XUV pulses in the VHTXHQWLDO regime of TPDI. Although the cross sections for nonsequential TPDI have attracted a significant amount of interest by theoreticians, the published results show large discrepancies. However, in the last few years, agreement has been observed between some quite different approaches for which the convergence has been extensively tested [45, 98, 106]. There are much fewer experimental studies as of yet, all of which are concerned with the nonsequential regime [99, 109–112]. For these, the experimental uncertainties are still too large to help in resolving the discrepancies in the theoretical results. Calculations for two-photon ionization employ either a time-independent (TI) or a time-dependent (TD) approach. TI methods involve either lowest-order perturbation theory (LOPT) or R-matrix Floquet theory, and are only applicable in the limit of (infinitely) long pulses. TD methods are based on a direct solution of the time-dependent Schr¨odinger equation and are therefore not restricted to any given order of the perturbation or pulse duration. The results we present in the following are mostly calculated at moderate intensities of the XUV field ( 1012 W=cm2 ). At this intensity, corrections to LOPT are expected to be small. The decisive advantage of TD methods here stems from a different aspect. Namely, TI calculations of processes involving correlated two-electron final states in the continuum, N1 ;N2 .U1 ; U2 /, require the knowledge of the final state in the entire configuration space in order to calculate the two-photon transition amplitude ti.2/ !N1 ;N2 . As the numerical or analytical determination of accurate correlated continuum final states remains a challenge, evaluation of ti.2/ !N1 ;N2 involves, inevitably, additional approximations that are difficult to control. Adding the time as an additional degree of freedom to the six spatial dimensions of the two-electron problem allows one to bypass the determination of N1 ;N2 . Instead, we propagate the wave packet for sufficiently long times such that we can extract the relevant dynamical information entirely from the asymptotic region where electron correlations become negligible. Moreover, residual errors can be controlled by systematically varying the propagation time. This advantage comes along with a distinct disadvantage: Results will, in general, depend on the time-structure imposed on the external perturbation, specifically on the duration and temporal shape of the XUV pulse. A comparison with TI calculations on the level of (generalized) cross sections therefore requires a careful extraction of information and checks of the independence from pulse parameters.

 6HTXHQWLDO DQG 1RQVHTXHQWLDO 5HJLPHV RI 73', The nature of the two-photon double ionization (TPDI) process depends strongly on the photon energy „!. In order to doubly ionize the helium atom, „! has to be large enough so that two photons can fully ionize the atom, i.e., 2„! > I1 C I2 D E0 , where I1 24:6 eV and I2 54:4 eV are the first and second ionization potential of helium, while E0 79 eV is the total ground-state energy. In a “long” pulse with an approximately delta-like energy spectrum, there are two distinct regimes of

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

181

TPDI, depending on the photon energy. In “real” pulses with finite spectral width, the photon energy is described by a distribution F.!/. The following arguments thus depend on the assumption that the width of that distribution is sufficiently small for the regime to be identified unambiguously. For „! > I2 , one photon has enough energy to ionize the HeC ion in its ground state. In this regime, an independent-particle picture is applicable: each electron absorbs one photon and electron-electron interaction is a priori not required for double ionization to occur. Therefore, the double ionization can proceed in two wellseparated steps, and this energy regime is called the VHTXHQWLDO regime. The first electron is ejected with energy E1 D „!  I1 , carrying with it the energy contained in the electron-electron interaction in the ground state. At a later time, when the first electron is well separated from the remaining ion, the second electron is ejected with the energy E2 D „!  I2 . In long pulses, this is the dominant process, leading to an electron energy spectrum with two sharp peaks at E1 and E2 . In the limit of low pulse intensities, where depletion can be neglected, the total yield is proportional DI to the square of the pulse duration (Pseq / T 2 ), taken to be the signature of the sequential (two-step) nature of the process. For high photon energies, different sequential pathways become accessible. The first photon absorption can produce VKDNHXS in the remaining HeC ion, leaving it in an excited state, with the second absorption proceeding from this excited state. These pathways are accessible for the sequential process when one photon provides enough energy to strip one electron from the atom and simultaneously excite the ion to a higher state, i.e., if „! > I1 C En , where En D .2  2=n2 / a:u: is the excitation energy to the nth shell of the HeC ion. In long pulses and for high photon energies, this leads to shake-up satellite lines in the electron energy spectrum [151]. While the first ionization potential is increased for shake-up ionization, the second ionization potential is decreased (I20 D I2 =n2 ). Consequently, the peak positions E10 D „!  I1 En , E20 D „!I2 CEn are different from those without shake-up, but the overall picture of sequential and independent photoionization events remains unchanged. There are, however, two reasons why some correlation between the electrons can be expected even for long pulses: for one, the electron that is emitted later is IDVWHU than the first electron in the shake-up pathway. If the electrons are emitted in the same direction, the second electron can thus collide with the first one, modifying the independent-particle behavior. In addition, the excited states of the HeC ion are (almost) degenerate in angular momentum, such that the HeC ion can remain in a superposition of excited states, with the coefficients depending on the emission angle of the first electron. This can also cause nonvanishing angular correlation between the electrons even in very long pulses. As the photon energy approaches the threshold for one-photon double ionization at „! D E0 , successively higher shake-up states become accessible. However, the probability for shake-up quickly decreases with the quantum number n of the intermediate excited state, such that typically, only the first few excited states play a role even if more are energetically accessible. If the photon energy „! is smaller than the second ionization potential I2 , the sequential process can not occur. The two photons still provide enough energy to

182

B.I. Schneider et al.

doubly ionize the helium atom, but only if the two electrons share the available energy. This regime is called the QRQVHTXHQWLDO regime. This implies that the process can only happen if both photons are absorbed almost simultaneously. The two consequences of this are that the energy of the intermediate state, which is only populated transiently, does not need to be „! C E0 (because of the quantum mechanical time-energy uncertainty), and that the electrons can interact and exchange energy. Consequently, the asymptotic energies of the electrons in the final state do not have to be E1 D „!  I1 and E2 D „!  I2 (where for „! < I2 , E2 would be negative and therefore not correspond to a free electron). Because the photons have to be absorbed quasi-simultaneously, the total double ionization yield in the DI nonsequential regime is linearly proportional to the pulse duration, Pnonseq / T as long as depletion can be neglected. It should be stressed that even in the sequential spectral regime, there are nonsequential contributions to the total double ionization which can be identified by their linear scaling with T . In particular, final states where the electron energies are not at the sequential peaks are only reached by nonsequential processes.

 5HVXOWV RQ 7ZR 3KRWRQ 'LUHFW ,RQL]DWLRQ RI +H In Fig. 6, we compare the present results for the total cross section with various published data. In order to achieve converged values, the spectral bandwidth of the laser pulse used to calculate the cross section (according to (108)) has to be sufficiently small. The spectral width of the pulse depends both on the pulse shape as well as on the total duration of the pulse. The laser pulses had a sin2 envelope, defined by   2 0 < t < T; sin . =T /t (112) f .t/ D 0 otherwise. As the threshold for sequential TPDI is approached, successively longer pulses are necessary to resolve the rapidly growing cross section. The present results in Fig. 6(a) were obtained with 4 fs pulses for „! 51 eV, 11 fs pulses for „! 53 eV and 20 fs pulses for „! D 53:5 eV and „! D 54 eV. The calculations were performed with different box sizes depending on the pulse duration, with rmax D 240 a:u: for the smallest boxes and rmax D 1400 a:u: for the largest boxes. The FEDVR elements contained 11 basis functions each and spanned 4–4:4 a:u:. The maximum angular momentum values used were Lmax D 3 for the total angular momentum and l1;max D l2;max D 7 for the individual angular momenta. The peak intensity was chosen as I0 D 1012 W=cm2 . The ionization yields were extracted at least 1 fs after the pulse. The projection error should be less than 2% [45]. For photon energies below around 50 eV, the total cross section for TPDI is a relatively smooth function of photon energy, showing an approximately linear increase. Above the threshold for sequential TPDI (54:4 eV), the cross section is not defined, as the yield then scales with the square of the pulse duration, whereas a cross sec-

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems 9 8

present results Palacios et al. [106] Horner et al. [98] Feng & van der Hart [73]

[99]

σ [10−52 cm4 s]

7

Ivanov & Kheifets Foumouo et al. (NC) Foumouo et al. (FC) Nikolopoulos et al.

183 [88] [80] [80] [86]

6 5 4 3 2 1

[111] [110]

0 40

42

44

46 48 Photon energy [eV]

50

52

54

50

52

54

(a) Longest available pulses 4

ten-cycle sin2 pulse Foumouo et al. (NC) [80] Hu et al. [76] (×128/70) Laulan & Bachau [72] Guan et al. [100]

3.5

σ [10−52 cm4 s]

3 2.5 2 1.5 1 0.5 0 40

42

44

46 48 Photon energy [eV]

(b) Ten-cycle pulses )LJ  Comparison of the total two-photon double ionization (TPDI) cross sections, obtained from (108), with Teff D 35T =128. In (a), “present results” labels the data obtained with the longest sin2 pulse that we calculated at each energy (see text), all with a peak intensity of 1012 W=cm2 . For the results of Foumouo et al. [80], (NC) labels the results obtained by projecting onto uncorrelated Coulomb waves, while (FC) labels the results obtained using the J -matrix method. (b) shows the results obtained with ten-cycle pulses compared to other approaches using the same pulses. The results of Hu et al. [76] were rescaled by a factor of 128=70 in order to include the correct Teff .

tion requires linear scaling with pulse duration. In order to extract the cross section close to the threshold, it is therefore necessary to ensure that the spectral width of the pulses is small enough such that the total yield only contains negligible contributions from the sequential process. By using successively longer pulses, we were able to resolve the threshold behavior up to less than one eV below the threshold, with the result for 53:5 eV being converged for T D 20 fs. In order to resolve the behavior for energies even closer to the threshold, still longer pulses would have to

184

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be used, which becomes prohibitively expensive. Close to the sequential threshold, the sequential process is DOPRVW possible and the electrons only have to exchange very little energy, leading to the observed rise in the cross section. This has been called the signature of the “virtual” sequential process [93]. The present results show a more pronounced variation with photon energy than other results obtained by direct integration of the time-dependent Schr¨odinger equation. This can be easily explained by the fact that most previous work employed tencycle pulses. At photon energies of 42–54 eV, this corresponds to about 1 fs total duration, and consequently, a spectral width (FWHM) of about 6 eV (for sin2 pulses). The results are therefore an average over a rather large energy window. In contrast, we use pulses of up to 20 fs duration with a narrower spectrum (FWHM  0:3 eV). To facilitate comparison with previous calculations, we have also performed a calculation using ten-cycle pulses (Fig. 6(b)) for which we indeed find better agreement. The pulse duration dependence becomes, in particular, critical near the threshold for sequential ionization at 54:4 eV. We compare our results with data from both time-dependent and time-independent approaches. Laulan and Bachau [72] solved the TDSE by means of a B-spline method and an explicit Runge–Kutta propagation scheme. The double ionization probability was obtained by projecting onto uncorrelated Coulomb functions. They also included first-order correction terms in the representation of the double continuum (thus partly taking into account radial correlations). However, they found little difference with respect to the uncorrelated functions, as expected from our investigations. Hu, Colgan, and Collins [76] solved the time-dependent close-coupling equations using finite-difference techniques for the spatial discretization and the real-space product formula as well as a leapfrog algorithm for temporal propagation. The double ionization probability was also extracted by projection onto uncorrelated Coulomb waves. Guan, Bartschat and Schneider [100] used an approach very similar to ours, employing the FEDVR and using the Lanczos method for time propagation. Palacios et al. [106] also used an FEDVR basis, combined with a CrankNicholson time propagator. They extracted the double ionization yields by application of exterior complex scaling (ECS) and a formal propagation to t ! 1. The volume integral for extraction of the momentum distribution can then be rewritten as a surface integral, performed at asymptotic distances to the core. The resulting amplitudes thus also include correlation. Foumouo et al. [80] employed a spectral method of configuration interaction type (involving Coulomb–Sturmian functions) and an explicit Runge–Kutta time propagation to solve the TDSE. The double ionization probability is calculated by closure, i.e., by subtracting the singly ionized states from the total wave function and taking the remaining probability as the double ionization probability. The singly ionized states were constructed by using the J -matrix method, which should contain angular and radial correlations to the full extent (labeled FC in Fig. 6). In addition, they also performed projection on the uncorrelated product of Coulomb waves (labeled NC in Fig. 6). The results of Ivanov and Kheifets [88] are based on the time-dependent convergent close-coupling (CCC) method, taking into account correlations in the

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final state to some degree. Nikolopoulos and Lambropoulos [86] solved the TDSE using an expansion in correlated multichannel wave functions. Within the time-independent methods, Nikolopoulos and Lambropoulos [68] applied lowest-order non-vanishing perturbation theory (LOPT) to determine the generalized cross sections. Feng and van der Hart [73] employed R-matrix Floquet theory in combination with B-splines basis sets. The data from Horner et al. [93, 98] also result from LOPT calculations. They solved the Dalgarno–Lewis equations for two-photon absorption in LOPT employing exterior complex scaling (ECS) and also account for correlation in initial, intermediate, and final states. Overall, our results are in reasonable to good agreement with those of [72, 73, 76, 80, 98, 100, 106] while sizable discrepancies exist in comparison with those of [68, 86] as well as those of [80] in which corrections due to final-state correlations are included. Clearly, the degree of convergence of the present results on the few percent level preclude any change of cross section by a factor of 5  10, which would be necessary to obtain values of the same magnitude as [68,80,86]. This conclusion can be supported by analyzing the radial wave packet without converting to momentum space (not shown, cf. [44, 45]). The only other calculations approaching the threshold for sequential TPDI are those by the Berkeley group [93, 98, 106], which also show the strong increase of the total cross section as the threshold is approached. In particular, the data of Palacios et al. [106] agree with ours almost perfectly up to 52 eV. The deviation at 53 eV can be explained by the fact that they used pulses of total duration T D 3 fs, while we used longer pulses up to T D 20 fs close to threshold. The experimental values of Hasegawa, Nabekawa et al. [109, 110] at 41:8 eV and of Sorokin et al. [111] at 42:8 eV (cf. Fig. 6) are compatible with most of the theoretical data. Antoine et al. [99] provide an experimental lower bound for the cross section at 41:8 eV, which is right at the value of the cross section obtained by most time-dependent approaches. Due to the experimental uncertainties (e.g., the harmonic intensity in [109,110] or the assumptions on the pulse shape and focusing conditions in [111]), the currently available data are not sufficient to strongly support or rule out any of the theoretical results. We turn now to the triply differential cross section (TDCS), the quantity most sensitive to the level of the underlying approximations. The present results show qualitative agreement with the published data [76, 88, 98, 106], but there are some pronounced quantitative differences. While the prominent back-to-back emission lobe (anti-)parallel to the laser polarization direction is well reproduced in most calculations (Fig. 7), the angular distribution for less favored emission directions (e.g., 1 D 90ı ) differs significantly from other calculations. One reason is the sensitivity to the partial wave expansion. In contrast to the WRWDO cross section, the TDCS needs a larger number of angular momentum combinations .L; l1 ; l2 / in the expansion of the wave function to converge. In order to resolve angular correlations in the TDCS, it is necessary to use large expansions in single electron angular momenta. More specifically, good convergence of the TDCS is only reached for values as high as l1;max D l2;max D 7 (cf. [45]), which exceeds the angular momentum content of most other calculations [76, 88].

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present results 128/70×ref. [76] ref. [88] ref. [98] 0.8×ref. [106]

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)LJ  Comparison of triply differential cross sections (TDCS) at 42 eV photon energy. Our data are obtained from (110), at E1 D 2:5 eV, i.e., equal energy sharing, using a 4 fs sin2 laser pulse. In comparison, the results of Hu et al. [76], Ivanov and Kheifets [88], Horner et al. [98], and Palacios et al. [106] are shown. The vertical gray line shows the ejection angle 1 of the first electron. The angular momentum expansion used values of Lmax D 4 and l1;max D l2;max D 9. The radial box had an extension of 400 a:u:, with FEDVR elements of 4 a:u: and order 11.

The data by Palacios et al. [106] were obtained from a 550 as pulse, for which the extracted total cross section is somewhat larger than in the converged case. After rescaling by a factor of 0:8 to account for this difference, the data agree almost perfectly with ours, owing to the fact that they used a similarly large angular momentum expansion. Horner et al. [98] solve the Dalgarno–Lewis equations for LOPT using an exterior complex scaling technique. In order to produce converged results, a small imaginary part has to be added to the photon energy in the first step of the calculation. The obtained results then have to be extrapolated to zero imaginary part, leading to some uncertainty in the relative phases of different contributions, which strongly influence the TDCS. Ivanov and Kheifets [88] take correlation in the final states into account using a convergent close-coupling (CCC) method. While the magnitude of their results is similar to those presented here, the shape differs considerably. In particular, they find significant probability for emission of both electrons in the same direction (1 D 2 ), where the mutual repulsion of the electrons should be strongest. Foumouo et al. [105] calculated the TDCS for equal energy sharing at 45 eV photon energy using two different methods. The results obtained by projecting the final wave function on products of Coulomb waves resemble ours (not shown here for 45 eV, but the behavior is similar as for 42 eV). However, when correlation in the final state is taken into account using their J -matrix method, the results are

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much larger in magnitude (as for the total cross section, cf. Fig. 6) and display a shape reminiscent of the one obtained by Ivanov and Kheifets [88], with the same surprising feature of emission in the same direction at equal energy sharing.

 5HVXOWV RQ 7ZR 3KRWRQ 6HTXHQWLDO ,RQL]DWLRQ RI +H The (two-photon) sequential ionization yield can be written as Z 1 Z 1 DI Pseq D dt 1 ˚.t/ dt 0 2 ˚.t 0 / ; 1

(113)

t

where 1 is the one-photon cross section for single ionization of He, and 2 is the one-photon cross section for ionization of the HeC ion. Using the symmetry of the integrand yields DI Pseq

1 D 1 2 2

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2 1 2 I02 dt˚.t/ D .Teff;1 /2 ; 2 2! 1 1

(114)

which is proportional to the square of the total pulse duration. It is therefore impossible to define a cross section in the usual sense and one needs to examine angular and energy distributions to understand the physical process at play. The effect of the duration of the laser pulse on the double ionization is shown in Fig. 8. In this part, we label the pulses by the FWHM of the electric field envelope, which is the more common value for ultrashort pulses.

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)LJ  (a) Double ionization (DI) rate P DI .E /=T (i.e., DI probability divided by the pulse duration) for TPDI by an XUV pulse at „! D 70 eV with different FWHM pulse durations T . For sufficient pulse duration, the DI rate converges to a stable value except near the peaks of the sequential process, indicating a nonsequential process. Close to the energies of the shake-up peaks, the Fano-like structures indicating interference between the sequential and nonsequential contributions can be observed. (b) and (c) show the two-electron energy spectrum P DI .E1 ; E2 / for (b) T D 300 as and (c) T D 750 as.

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)LJ  Two-photon double ionization electron spectra P DI .E1 ; E2 / at 70 eV photon energy for different pulse durations: (a) 150 as, (b) 450 as, (c) 3000 as FWHM. The sides show the spectrum integrated over either energy, i.e., the one-electron energy spectra P DI .E1 / D P DI .E2 /.

The effect of the pulse duration can be seen even more clearly in the joint probability P DI .E1 ; E2 / distribution in the plane of electron energies, This clearly reveals the breakdown of the sequential ionization picture with decreasing pulse duration (Fig. 9). For long pulses (T D 3 fs), two distinct peaks signifying the emission of the “first” electron with energy E1 D „!  I1 (with I1 the first ionization potential) and the “second” electron with E2 D „!  I2 are clearly visible. Contributions from shake-up satellites with energies E10 D „!  I1  En and 0 E2 D „!I2 CEn are below the one-percent level and barely discernible. For pulses of the order of one hundred attoseconds a dramatically different picture emerges: The two peaks merge into a single one located near the point of symmetric energy sharing Es D „!  .I1 C I2 /=2. It should be noted that this is not simply due to the Fourier broadening of the pulse. Instead, the close proximity in time of the two emission events allows for energy exchange between the two outgoing electrons representing a clear departure from the independent-particle behavior. Differently stated, the time interval between the two ionization events is too short for the “remaining” electron to relax to a stationary ionic ground (or excited) state. In the limit of ultrashort pulses the notion of a definite time ordering of emission processes loses its significance, as does the distinction between “sequential” and “nonsequential” ionization. The attosecond-pulse-induced dynamical electron correlation becomes more clearly visible in the joint angular distribution P DI .12 ; 1 / (Fig. 10), where 1 is the polar emission angle of one electron, chosen in the following to coincide with the polarization axis of the XUV pulse (1 D 0ı ), and 12 is the angle between the two electrons (here and in the following we choose coplanar geometry with 1 D 2 D 0). In the limit of “long” pulses .T & 3 fs), the joint angular distribution is the product of two independent Hertz dipoles, each of which signifies the independent interaction of one electron with one photon. Consequently, also the conditional angular distribution P DI .12 ; 0/ corresponds to a Hertz dipole. With decreasing pulse duration, P DI .12 ; 0/ displays strong deviations and develops a pro-

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)LJ  Conditional angular distributions P . 12 ; 1 / at 1 D 1 D 2 D 0ı of ejected electrons for different pulse lengths at 70 eV photon energy. The innermost (solid blue) line is for 75 as FWHM pulse duration, with successive lines for 150 as, 300 as, 450 as, 750 as, 1500 as, and 3000 as FWHM pulse duration. The outermost line is the dipolar (cos2 . 1 /) distribution expected in the independent-particle limit of very long pulses. The distributions have been normalized to a maximum value of one for better comparison. DI

nounced forward-backward asymmetry. The conditional probability for the second electron to be emitted in the same direction as the first is strongly suppressed. For T of the order of a hundred attoseconds, the two electrons are emitted back to back. It is worth noting that the strong preference for emission in opposite directions persists after integration over the emission energies. Nevertheless, approximately equal energy sharing dominates (see Fig. 9). Thus, the dominant break-up mode induced by an attosecond pulse corresponds to ejection of the two electrons at 180 degrees, the so-called “Wannier ridge” configuration [152]. It is now instructive to inquire into the strong electron correlation observed for short pulses. Three different sources can be distinguished: (i) Initial-state correlations in the helium ground state. Due to Coulomb repulsion, the electrons in the ground state are not independent of each other. For extremely short pulses, two-photon double ionization can thus be interpreted as a pump-probe setup that maps out the position (and momentum) of the second electron before it has time to relax to a spherically symmetric one-electron state. (ii) Induced dipole polarization in the intermediate bound-free complex. When the first electron leaves the core, its electric field induces polarization of the remaining ion, leading to an asymmetric probability distribution of the second electron. (iii) Final-state electron-electron interaction in the continuum. After the second electron has been released within the short time interval T as well, the mutual repulsion may redirect the electrons towards a back-to-back configuration. While the dividing line between those mechanisms is far from being sharp, the present time-dependent wave packet propagation allows to shed light on their relative importance since they occur on different time scales. Relaxation of the ground-state correlations (i), i.e., of the deviation of the joint angular distribution P DI .˝1 ; ˝2 / from a product of two spherically symmetric distributions is expected to occur on the time scale of the orbital period of the residual electron. As the re-

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maining one-electron wave function will be mostly in the n D 1 and n D 2 shells, the uncertainty principle shows that the relevant timescale is approximately 16 as. Therefore, ground-state correlations will become clearly visible only for pulses with durations much shorter than one hundred attoseconds, shorter than those investigated here. The time scale for induced dipole polarization (ii) can be estimated by the time the first electron takes to escape to a distance larger than the radial extent of the residual wave function forming a dipole (hri 3 a:u: in n D 2). Choosing a somewhat arbitrary distance of 10 a:u:, the time necessary for the first electron to reach this distance after absorbing a 70 eV photon is about 120 as and thus of the order of the pulse lengths T considered. For higher photon energies, the first electron escapes to the same distance in a shorter time, decreasing the importance of this effect. In order to verify this, we have performed calculations at photon energies of 70; 91; 140, and 200 eV for a pulse duration of 75 as FWHM. Fig. 11(a) demonstrates that for higher energies, the asymmetry of the joint angular distribution is indeed strongly reduced. Long-range Coulomb interactions in the continuum (iii) extend over much longer timescales which also strongly depend on the relative emission angles and energies of the electrons, i.e., jN1  N2 j. For example, for two electrons ejected in the same direction and with similar energies, the interaction will last much longer than for ejection in opposite directions. This can be verified by using an ultrashort pulse to start a two-electron wave packet in the continuum and observing the evolution of the joint angular distribution after the laser pulse is switched off (Fig. 11(b)). Directly after the pulse, the distribution of the electrons shows a decreased probability for ejection on the same side of the nucleus (primarily because of (ii)), but the lobes in forward and backward direction still mostly retain the shape expected from a dipole transition. As continuum final-state interactions persist, the joint angular distribution develops a pronounced dip at equal ejection angle as time passes. The change at larger relative angles is almost negligible. One remarkable feature of the conditional angular distribution is the persistence of the nodal plane at  D 90ı . While correlation effects strongly perturb the shape

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)LJ  Conditional angular distributions of ejected electrons after an XUV pulse. (a) For a duration (FWHM) of 75 as for different photon energies. From inside to outside: 70 eV, 91 eV, 140 eV, and 200 eV. The amount of asymmetry decreases with increasing pulse energy. (b) For different times after a 150 as (FHWM, total duration 300 as) pulse at 70 eV photon energy. The times (from the outermost to the innermost line) are 25 as, 350 as, 1000 as, and 1800 as after the end of the pulse.

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems (b)

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)LJ  Angle-energy distribution P DI .E1 ; 12 ; 1 D 0ı / in coplanar geometry at 70 eV photon energy for different pulse durations: (a) 150 as, (b) 450 as, (c) 3000 as FWHM. The side plots show the distribution integrated over, respectively, energy and angle.

of the independent-particle dipolar shape, the nodal plane expected for the angular distribution of two electrons absorbing one photon each is approximately preserved. This is in contrast to one-photon double ionization, where necessarily only one electron absorbs energy from the photon and electron ejection at normal angle to the polarization axis is indeed observed [67, 153, 154]. Additional insights can be gained from a different projection of the two-electron momentum space distribution onto the energy-angle plane, Z DI ı P .E1 ; 12 ; 1 D 0 / D P DI .E1 ; E2 ; ˝1 ; ˝2 / dE2 ; (115) in coplanar geometry .1 D 2 D 0ı ) and for 1 D 0ı . While for long pulses the energy of the emitted electrons (E1 ) is independent of the relative emission angle (Fig. 12(c), strong energy-angle correlations develop for short (T 500 as) pulses. The dominant emission channel is the back-to-back emission at equal energy sharing (E1 30 eV). This corresponds precisely to the well-known Wannier ridge riding mode [152], previously observed in e-2e ionization processes [155] and also invoked in the classification of doubly-excited resonances [156]. Because of the large instability of the Wannier orbit its presence is more prevalent in break-up processes than in quasi-bound resonances. A second subdominant but equally interesting channel opens for short pulses at 12 D 0ı , i.e., emission in the same direction. One of the electrons is slowed down while the other one is accelerated, i.e., the fast electron is “pushed” from behind by the slower electron. The slower electron thus transfers part of the energy absorbed from the photon field to the faster electron. This is the well-known SRVWFROOLVLRQ LQWHUDFWLRQ [157–159] first observed by Barker and Berry in the decay of autoionizing states excited through ion impact [160].

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 +C 2  ,QWURGXFWLRQ The question of the validity of the Born-Oppenheimer approximation when molecules are exposed to ultrashort laser pulses is of considerable interest. For molecules containing more than one electron, the situation can become exceedingly complex as the absorption of photons, the relative energies of the electrons, the electronic interaction and the nuclear motion play against one another in a subtle manner. For a one electron molecule, HC 2 , there are many simplifications. Since only one electron is ejected, the inter-electronic aspects of the ionization are not present and the question reduces to whether the electron can escape from the molecular core sufficiently quickly that the absorption of the photons and the dynamics of the nuclei are basically decoupled. Clearly, if the electron is ionized with very small kinetic energy, there is the possibility of significant nonadiabatic effects. Conversely, if the electron moves out of the interaction region rapidly, the nuclei are likely to impulsively fly apart and be governed primarily by simple classical dynamics. To examine this question, the four dimensional .x; y; z; R/, time-dependent Schr¨odinger equation (TDSE) was solved using the FEDVR/Real-Space-Product (RSP) technique [135], including the dynamical motion of the nuclei. Specifically, we exposed the target to linearly polarized light at arbitrary angles with respect to the molecular axis. Calculations were performed at different angles and photon energies („! D 50 eV–630 eV) to investigate the energy and orientation dependence of the photoionization probability [50, 161]. While we have not yet seen any major evidence of non-Born–Oppenheimer behavior, we have uncovered some quite interesting physics. At a photon energy of „! D 50 eV, there is a strong orientation dependence of the photoionization probability of HC 2 . At this energy, the ejected photoelectron is emitted “tilted” with respect to the molecular axis. This ionization anisotropy then appears to vanish at higher photon energies („!  170 eV). When these higher-energy XUV pulses are polarized perpendicular to the internuclear axis, a “double-slit-like” interference pattern is observed. However, we find that the diffraction angle only approaches the classical formula n D sin1 .ne =R0 / when ne becomes less than 65% of the internuclear distance R0 , where n is the diffraction order and e is the wavelength of the released electron. The observations are explained by the different geometric “cross-sections” seen by the photo-ejected electron in the half-scattering process of leaving the molecule. The results illustrate the possibility of employing attosecond pulses to perform photoelectron microscopy of molecules.

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems

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 5HVXOWV DQG 'LVFXVVLRQV  7KH *URXQG 6WDWH As an initial condition, we choose the ground state of 4D HC 2 , determined using imaginary time (ITW t ! i ) propagation of (1). Under the transformation to imaginary time, an initial arbitrary wave packet will evolve into the ground-state wave function for sufficiently long propagation times. As shown in Fig. 13(a) and (b), the total energy and the expectation value of the internuclear distance hRi approach E0 ' 0:59 a:u: and hRi ' 2:02 a:u: respectively. The errors are within 2% of the analytical values (E0 ' 0:603 a:u: and hRi ' 2:00 a:u:). The ground-state probability density is plotted in the xy-plane and the xR-plane in Fig. 13(c) and (d), with the internuclear distance along the x-axis.  3KRWRLRQL]DWLRQ $QLVRWURS\

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We first consider linearly-polarized, attosecond XUV pulses in the xy-plane at a photon energy of „! D 50 eV at any angle  with respect to the molecular axis R. The driving radiation is a sin2 pulse with a total duration of 500 as and a field strength of about 0:53 a:u:. The case of  D 0ı ( D 90ı ) corresponds to the laser polarization parallel (perpendicular) to the molecular axis. Figure 14 traces the dy-

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)LJ  The ground state of 4D HC 2 obtained from imaginary-time propagation: (a) The system energy as a function of the propagation time, (b) the expectation value of the internuclear distance hRi versus propagation time, (c) the probability density profile in the xy-plane, and (d) the probability density profile in the xR-plane.

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Population on the ground state

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)LJ  Time-dependent ground-state population of HC 2 driven by attosecond XUV pulses („! D 50 eV, T D 500 as) polarized parallel to the molecular axis [x-axis] (dash-dotted/red line), perpendicular to the molecular axis (dashed/blue line), and at D 45ı relative to the molecular axis in the xy-plane (solid/green line), respectively. A strong orientation dependence of the groundstate depletion is observed for such conditions.

namics of the ground-state population for three cases: parallel ( D 0ı ) [red/dashdotted line], perpendicular ( D 90ı ) [blue/dashed line], and “tilted” ( D 45ı ) [green/solid line]. As the ground-state population is gauge-dependent, one has to be careful concerning the physical interpretation of features seen in the figures. We have chosen to use the length gauge, which typically shows much larger oscillations than the velocity gauge. The ground-state populations adiabatically follow the field variation due to polarization effects, but with a doubled frequency because both the positive and negative oscillations of the field polarize the ground state in the same way. At the end of the XUV pulse, we observe three times more ground-state depletion from the perpendicular as compared to the parallel orientation, with the “tilted” ( D 45ı ) case lying in between. While the intermediate polarization (the “dips”) behaves similarly in all three cases, the ionization probabilities differ considerably. In the parallel case, the field-driven wave function returns almost entirely to the ground state. In the perpendicular configuration, the ground state is more strongly depleted, as shown by the continuously decreasing population peaks [blue/dashed line]. To better understand the dependency on orientation, we calculated the total energy change .hH0 i  E0 / while the pulse was interacting with the molecule. Figure 15 shows the results for the three cases. The high-frequency oscillations are a numerical artifact and should be ignored. The perpendicular orientation substantially enhances the energy exchange between the field and the molecule, yielding more than three times the energy absorption of the parallel ( D 0ı ) case. Greater energy absorption at  D 90ı implies a larger excitation/ionization probability, which is consistent with the larger ground-state depletion seen in Fig. 14.

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)LJ  The molecular system energy change [hH0 i  E0 ] during the attosecond pulse interaction („! D 50 eV, T D 500 as), for polarization parallel (dash-dotted/red line), perpendicular (dashed/blue line), and at 45ı relative to the molecular axis in the xy-plane (solid/green line), respectively. Note, the very high frequency oscillations are a numerical artifact.

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)LJ  Electron probability densities of HC 2 at the end of the attosecond XUV pulse („! D 50 eV) for (a) polarization parallel to the molecular axis and (c) perpendicular to the molecular axis. The corresponding momentum distributions of the ejected photoelectrons are shown in (b) and (d), respectively. The ionization probability is three times higher for perpendicular polarization than for parallel polarization.

Snapshots of the probability density in the xy-plane of configuration space are plotted in Fig. 16(a) and (c) at the end of the XUV pulse for  D 0ı and 90ı . The corresponding momentum distributions are shown in Fig. 16(b) and (d), respectively. By integrating the electron probability in regions greater than  5 bohr from the molecular center, one finds, not unexpectedly, that the ionization probability

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)LJ  (a) Pulse-ending snapshot of the electron probability density of HC 2 for the case of “tilted” polarization ( D 45ı ) with respect to the molecular axis (x-axis); (b) corresponding photoelectron momentum distribution in the xy-plane. It is noted that the ejected photoelectron peaks off the polarization axis, which is a result of the orientation dependence of attosecond photoionization of HC 2 .

becomes more than three times higher in the perpendicular than the parallel case. Overall, the ejected electron momentum spectrum is dominated by single-photon absorption, i.e., a peak at „!  IP D EK ' 20 eV. Therefore, the observed ionization difference between orientations is independent of the XUV pulse intensity. Although electrons are ejected in line with the XUV pulse polarization for these two cases, slightly different features arise. For example, small-amplitude features appear at large angles in the parallel case [Fig. 16(a) and (b)], associated with the (half-)scattering of the electron between the two scattering centers. This feature is absent in the perpendicular case although the more extended distribution in the momentum spectrum may mask weak scattering signatures. The orientation dependence of HC 2 photoionization observed above has a consequence for a generally “tilted” pulse polarization in that the direction of electron ejection and the XUV polarization can differ. To illustrate this, we have plotted the probability density snapshots in Fig. 17(a) for the case of  D 45ı as well as the corresponding ionization momentum spectrum [Fig. 17(b)]. Instead of aligning with the XUV polarization direction, the probability density for the ejected electron now peaks at  ' 82ı . The linearly-polarized field ( D 45ı ) can be equally decomposed into x- and y-components, for which the y-component of the field induces more ionization than the x-component. Thus, the overall ionization of the wave packet tends to “bend” toward the y-axis. Our calculations appear contrary to a recent XUV experiment of molecular tunneling ionization in H2 in intense optical fields [162, 163] which found slightly more ( 30%) ionization in the parallel orientation. We have also performed a few 4D calculations for HD C in the optical regime (not shown), which find behavior similar to that in the cited experiments. In the optical regime, the ejected electron has a wide range of energy from a few eV to a few hundred eV due to multiphoton ionization by the intense field. Very lowenergy electrons have de Broglie-wavelengths much larger than the molecular size

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and show no directional preference. On the other hand, when electron are ejected with high energy, ionization along the parallel polarization direction is slightly preferred. Thus, the overall tunneling ionization, given by the sum over all electron spectra, shows a less dramatic anisotropy than for single-photon ionization in the XUV regime. To gain further insights into the orientation dependence of the ionization, we note that for a photon energy of „! D 50 eV, the ejected electron (EK ' 20 eV) has a de Broglie wavelength of e ' 5:2 bohr, more than twice the internuclear distance. Consequently, for the perpendicular direction (y-axis), the ejected electron cannot distinguish the two molecular centers and experiences a much larger geometrical cross-section for half-scattering than the case of parallel polarization. For the latter situation, the electrons always encounter aligned single-center half-scattering. We should therefore intuitively expect a more extended momentum distribution for perpendicular polarization. To characterize the extent of the momentum distribution, we take as a measure the angular spread ˚, estimated by the angle at which the probability drops to 1=e of its peak value. We find ˚ ' 35ı for perpendicular polarization, clearly exceeding ˚ ' 24ı for parallel polarization [cf. Fig. 16(b) and (d)]. A large cross section for scattering facilitates efficient photon absorption, thereby resulting in a higher ionization probability. If the above explanation is correct, such an orientation dependence of molecular photoionization should disappear once the de Broglie wavelength decreases (increasing „!) sufficiently that the photoelectron can distinguish both scattering centers. At that point the half-scattering should take on a more uniform distribution. To test this hypothesis, we have performed calculations at „! D 170 eV for both perpendicular and parallel polarizations with a 250 as XUV pulse but with a peak field strength of  3:76 a:u: in order to get a noticeable ionization probability. The results are shown in Figs. 18 and 19, respectively, for the ground-state depletion history and the total system energy change. From Fig. 18, we see that the final population left in the ground state is roughly similar for both situations although the parallel polarization results in a slightly higher ionization probability. A similar conclusion emerges from the field-molecule energy exchange as shown by Fig. 19. Finally, we plot in Fig. 20(a) and (c) the probability density snapshots for the parallel and the perpendicular polarization cases, as well as the corresponding momentum distributions of the ionized electron wave packet in Fig. 20(b) and (d). We find that the ionization probability is roughly the same for these two cases and that the dramatic orientation dependence of HC 2 photoionization, observed at the low photon energy of „! D 50 eV, disappears at „! D 170 eV. Once again, this is because the released “fast” electron (EK ' 140 eV) now has a de Broglie wavelength of e ' 1:96 bohr, which is slightly smaller than the internuclear distance. The ejected electron can now distinguish the two molecular “scattering” centers, with the consequence that the ionization no longer depends drastically on the direction of the ejected direction.

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Population on the ground state

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)LJ  Time-dependent ground-state population of HC 2 driven by high photon-energy attosecond XUV pulses („! D 170 eV), whose polarizations are either parallel (dash-dotted/red line) or perpendicular (dashed/blue line) to the molecular axis. 7 Parallel Perpendicular

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)LJ  Molecular system energy change [hH0 i  E0 ] versus the interaction time („! D 170 eV), for parallel polarization (dash-dotted/red line) and perpendicular polarization to the molecular axis (dashed/blue line).

 ,QWHUIHUHQFH (IIHFWV We further examine the double-slit-like interference patterns that appear in perpendicular polarization [Fig. 20(b) and (d)]. All of the interference peaks reside within the momentum circle dictated by energy conservation (EK D „!  IP ). Basically, the photoelectron wave packet released from each center interfere, producing electrons ejected in specific directions. This behavior was discussed by Cohen and Fano almost a half a century ago [164]. More recently, Walter and Briggs [165], modeled

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems 10–2

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)LJ  Electron probability densities of HC 2 at the end of the attosecond XUV pulses („! D 170 eV) for (a) polarization parallel to the molecular axis and (c) polarization perpendicular to the molecular axis. The corresponding momentum distributions of the ejected photoelectrons are shown in (b) and (d), respectively. The “double-slit-like” interference patterns are observed for the perpendicular polarization case. The orientation dependence of the yield in attosecond photoionization, observed at „! D 50 eV, disappears in this case.

this phenomenon using time-independent methods within the Born-Oppenheimer approximation. Their calculations were performed at slightly larger photon energy („! D 250 eV) [166, 167] than the current calculations and were compared with measurements using circularly polarized light [168]. Our time-dependent results show that the first interference peak appears at an angle of  ' 52ı relative to the polarization y-axis. Young’s double-slit formula, R0  sin.n / D ne used in time-independent studies [166], predicts the first peak at 1 ' 75ı for the associated photon energy. Here, R0 is the internuclear distance, e is the de Broglie wavelength of the ejected electron, and n is the order of the interference peak. To examine the long-range Coulomb effects, we have freely propagated the wave packets to large distances (> 60 bohr) but still find the first interference angle little changed. To resolve the discrepancy between our calculated diffraction angles and the predictions of the classical formula, we increased the photon energy of the XUV pulses from „! ' 210 eV to 630 eV and performed a series of calculations, the results of which are shown in Table 1. We observe that the differences between the calculations and the classical formula decreases as the XUV photon energy increases. Above „! ' 350 eV, the classical first-order diffraction angle (1 ) is exactly recovered from our TDSE calculations. Two such examples are shown in Fig. 21(a) and (b), in which the electron probability densities are plotted for „! ' 350 eV and

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7DEOH  The “double-slit” interference angles comparison between the “classical” double-slit (DS) formula and our TDSE calculations, as the photon energy („!) varies. „! (eV)

e (bohr)

170 210 250 300 350 430 630

1.9587 1.7274 1.5258 1.4104 1.2955 1.1588 0.9462

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' 52:0ı ' 49:5ı ' 46:5ı ' 42:3ı ' 39:9ı ' 34:8ı ' 28:4ı ' 60:7ı

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)LJ  The electron probability densities of HC 2 at the end of the attosecond XUV pulses for polarization perpendicular to the molecular axis, for (a) „! D 350 eV and (b) „! D 630 eV. Different field strengths are applied in order to obtain noticeable ionization. Classical “double-slit” interference angles are recovered whenever ne  65%R0 is satisfied (see, more discussions in text).

„! ' 630 eV, respectively. Interestingly, at the highest photon energy (630 eV), the second-order diffraction patterns appear. However, the resulting angle (2 ) again disagrees with the classical double-slit prediction (see Table 1) although 1 remains in good agreement with the classical formula. These results indicate that the validity of the classical double-slit prediction requires that ne 0:65R0 . If the interference path difference (ne ) becomes comparable to the internuclear distance R0 , the paths are no longer independent, and the classical double-slit condition (  d , where d is the separation of two slits) is not satisfied. e We conclude that the results of our calculations could guide the proper choice of photon energy for attosecond photoelectron imaging. Finally, to have a sense of the energy sharing between the electron and the nuclei during the attosecond ionization, we computed the nuclear momentum spectrum for the wave packets associated with the ionized electron. The result, presented in Fig. 22 for the parallel polarization cases with two different field strengths E D 0:53 a:u: and E D0:053 a:u: at „! D 50 eV, show a transfer of less than  0:1 eV of

Simulation of the TDSE for Short Pulses Interacting with Three-Body Systems 3

E ~ 0.053 atomic units E ~ 0.53 atomic units Spectrum (arbitrary units)

)LJ  The nuclear momentum spectrum for polarization parallel to the internuclear axis at „! D 50 eV.

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photon energy to the nuclear motion during the attosecond photoionization process. The higher the field, the more “vertical” is the photoionization. The same behavior is also seen in the case of perpendicular polarization.

 6XPPDU\ In conclusion, we have presented an overview of our numerical approaches to solve the time-dependent Schr¨odinger equation for two different three-body systems. We have demonstrated that these approaches provide detailed and reliable information about three-body breakup processes, both for the two-photon double ionization of He and the one-photon ionization (and break-up) of HC 2.

 +HOLXP In the case of helium, we have presented a detailed study of the dynamics of the two-photon double ionization process, for photon energies both in the nonsequential and sequential regime, and for a wide range of pulse durations (150 as to 9 fs). We have shown how electron-electron interaction, and thereby correlation, influences the observed energy spectra and angular distributions. In particular, we have determined well-converged results for the total and triply differential (generalized) cross sections for nonsequential TPDI. Additionally, we have investigated the pulse duration dependence of the extracted cross sections. As the sequential threshold is approached, ever longer pulses are necessary to obtain converged values for the cross section.

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For photon energies above the sequential threshold, the one-electron ionization rate P DI .E/=T converges to a stable value with increasing pulse duration for electron energies away from the peaks associated with sequential ionization, giving rise to a well-defined (direct) differential double ionization cross section. Near the peaks, P DI .E/=T grows with T . We have thus observed a nonuniform scaling of the double ionization probability with pulse duration. If the photon energy is large enough to allow for shake-up ionization, i.e., ionization of He and simultaneous excitation of the remaining HeC ion, a new kind of interference effect can be observed. In this spectral range, both the direct and sequential processes co-exist, giving rise to interferences which are induced by the short time correlation between the two emission events. This interference occurs between the nonsequential contributions of the channel without shake-up and the sequential shake-up channel, where the intermediate state after one-photon absorption is an excited state of the HeC ion. In attosecond pulses, the channels can not be distinguished, while in long pulses (longer than the 9 fs used here), the sequential shake-up channel will dominate. For pulse durations of a few femtoseconds, as obtained in X-ray free-electron lasers, the two channels are similarly important, and the observed interference may allow one to measure the duration of ultrashort XUV pulses. More information about the dynamics of the system is encoded in the angular distributions of the electrons. The electrons at the primary sequential peaks are essentially uncorrelated, while strong correlation is present for all other electron energies. Between the sequential peaks, i.e., close to equal energy sharing, the electrons are almost exclusively emitted in a back-to-back configuration. Outside the main peaks, the situation is reversed and the electrons are emitted preferentially in the same direction. For both of these cases, the two-lobed structure of a dipole transition from an s state is still visible, most clearly in the strong suppression of emission at an ejection angle of 90ı to the laser polarization axis. In addition, we have shown that attosecond XUV pulses can be used to probe, induce, and control electron correlation in two-photon double ionization. In such pulses, the scenario for “sequential” two-photon double ionization breaks down. Due to the small time interval between the two photoabsorption processes dynamical electron-electron correlations can be tuned by the pulse duration T . The angular and angle-energy distributions reveal the signatures of electronic correlation induced by the Coulomb interaction in the intermediate bound-free complex and in the final state with both electrons in the continuum. In ultrashort pulses, where the distinction between sequential and nonsequential processes breaks down, two wellknown scenarios, the Wannier ridge riding mode and the post-collision interaction process, are simultaneously present in the two-electron emission spectrum. The favored emission channel is the Wannier ridge riding mode of back-to-back emission at equal energies.

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 +C 2 We have investigated the attosecond XUV pulse ionization of HC 2 by numerically solving the 4D time-dependent Schr¨odinger equation. Our results indicate that the single-photon ionization of HC 2 has a strong orientation dependence at low photon energies, for which the released electron has a de Broglie wavelength much longer than the internuclear distance. The “half-scattered” electron sees a larger geometrical cross-section in the perpendicular direction, which therefore facilitates more ionization. With increasing photon energy, this photoionization anisotropy disappears due to the fact that the “fast” outgoing electron can distinguish the nuclear scattering centers. As a consequence, single-electron “double-slit-like” interference patterns emerge in the perpendicular polarization configuration. A series of calculations with high photon energies have been performed to explore the validity of the classical Young’s double-slit condition. We found that to recover the diffraction angle (n ) predicted by the classical Young’s double-slit formula, the identified condition of ne 0:65R0 needs to be satisfied. Namely, the electron wavelength is required to be less than  65% of the internuclear distance for the first diffraction angle to be same as the classical double-slit prediction, which guarantees each scattering pathway to be independent. These results provide a useful guide to attosecond photoelectron imaging of molecules. Finally, it is noted that the results from fixednuclei calculations (performed in 3D) are very similar to those from the current 4D calculations. This suggests that non-Born–Oppenheimer effects are negligible under the conditions used in this study. The capacity of our 4D code will be explored in the future by examining non-Born–Oppenheimer effects in intense field-molecule interactions. $FNQRZOHGJHPHQWV JF acknowledges support by the National Science Foundation through a grant to ITAMP. JF, SN, RP, and JB acknowledge support by the FWF-Austria, grants No. SFB016 and P21141-N16. The authors acknowledge the support of the NSF TeraGrid computational facilities at the Texas Advanced Computing Center (TACC) and at the National Institute for Computational Science (NICS), as well as computing time at the Vienna Scientific Cluster at the Vienna University of Technology and Institutional Computing resources at Los Alamos National Laboratory. The Los Alamos National Laboratory is operated by Los Alamos National Security, LLC for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396.

5HIHUHQFHV 1. 2. 3. 4. 5.

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,QIRUPDWLRQ RI (OHFWURQ '\QDPLFV (PEHGGHG LQ &RXSOHG (TXDWLRQV IRU )HPWRVHFRQG 1XFOHDU :DYHSDFNHWV Kazuo Takatsuka

$EVWUDFW We explore an interrelationship between nuclear wavepacket dynamics arising from the close-coupling equations based on the Born–Huang expansion, in which the electronic-state basis functions is free of time variable, and the nonadiabatic electron wavepacket dynamics that is propagated within on-the-fly scheme. The former usually describes femtosecond nuclear dynamics, while the latter is mainly used to represent attosecond electron dynamics. A question is then how they can be compatible with each other. In this note I particularly place a focus on how one can identify the information of electron wavepacket from the close-coupling equations for the femtosecond nuclear wavepackets such as those in our studies of time-resolved photoelectron spectroscopy and velocity map imaging.

 ,QWURGXFWLRQ It has been shown theoretically that time-resolved photoelectron spectroscopy [1–5] can give quite clear signals detecting the passage of nuclear (vibrational) wavepackets across the so-called avoided crossing (as in NaI [6, 7]) or conical intersection (as in NO2 [8, 9]), since the bifurcating wavepackets quite often yield individually different photoelectron kinetic energy signals. Further, not only the energy distribution but the angular distribution of photoelectrons gives much information about the relevant nuclear dynamics. Theoretically, the information of nuclear wavepackets reflected in photoelectrons through pump-probe photoelectron spectroscopy is extracted by faithfully solving the total Schr¨odinger equation for a total wavefunction (1), which is reduced to the coupled nuclear wavepacket dynamics as in (2) in the spirit of Born–Oppenheimer view of molecule. Interestingly, in this study, the nuclear dynamics of one femtosecond order is well represented through the phoKazuo Takatsuka Department of Basic Sciences, Graduate School of Arts and Sciences, The University of Tokyo, Komaba, 153-8902, Tokyo, Japan, e-mail: LB[UBL!NOTDVUPLZPBDKQ A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2_11, © Springer Science+Business Media, LLC 2011

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toelectron dynamics, whose time-scale must be usually in the order of 10–100 attoseconds. On the other hand, the current progress of “attosecond” laser drives us to develop nonadiabatic electron wavepacket dynamics, which is described later in this note [10–16]. In this theory, the time variable is explicitly included in the electronic wavefunctions, which is in marked contrast to the time-independent representation of electronic states in the Born–Huang expansion [17]. (Recall that in the time-dependent approach of electron dynamics, time step to integrate the electronic equations of motion has to be much shorter than 1 attosecond, irrespective of the presence of an external laser field.) So, we face a puzzle about the above two time scales: femtosecond electronic information as given by pump-probe photoelectron spectroscopy and attosecond direct dynamics of electronic wavepackets. Or, one may ask what is their relationship? Seeking for an answer to the question, we below explore whether one can retrieve information about the time evolution of electronicstate mixing from the available experimental data, particularly the photoelectron velocity map images. In so doing, we briefly review in Sect. 2 the general theory of femtosecond pumpprobe photoelectron spectroscopy, in which nuclear wavepacket dynamics arising from chemical reaction is monitored in terms of energy and angle distributions of photoelectrons. We show an example of the so-called photoelectron velocity map image for NO2 molecule, which is utilized to track the evolution of the wavepackets passing through a conical intersection. Then, we consider an attosecond electron dynamics in molecules in Sect. 3, outlining our general theory of nonadiabatic electronic wavepackets to be propagated in time along smoothly branching nuclear paths, the paths which are also quantized into the form of nuclear wavepackets in the end. The electronic wavepacket includes time variable explicitly, and therefore, we compare to make a correspondence between the space-time representation of thus constructed electronic-nuclear wavepackets and that of the Born-Huang representation, which is shown in Sect. 4. Then we finally show in Sect. 5 how these theoretical and experimental quantities are mutually correlated to give a unified view about femtosecond nuclear dynamics and attosecond electronic dynamics in the mixing process of the electronic states to an ion.

 3KRWRHOHFWURQ 9HORFLW\ 0DS ,PDJLQJ 5HI OHFWLQJ )HPWRVHFRQG 1XFOHDU :DYHSDFNHW '\QDPLFV Femtosecond time-resolved photoelectron spectroscopy is a quite powerful and versatile probe of ultrafast dynamics in molecules and has been applied in recent years to a variety of systems and processes [18–22]. It is particularly well suited to the study of wavepacket dynamics in nonadiabatic systems where the nuclear and electronic modes are coupled. Time-resolved photoelectron spectra probe the entire configuration space spanned by the evolving wavepacket and hence, in principle, can provide information along all energetically accessible internuclear geometries. In-

Electron Dynamics Embedded in Nuclear Wavepackets

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deed, we have previously demonstrated the time-resolved photoelectron spectra in a series of papers tracking fundamental wavepacket dynamics in different scenarios: vibrational motion across a one-dimensional double-well potential in an excited state of Na2 [1–5], wavepacket bifurcation at an avoided crossing in NaI [6, 7]. For more details, see [8, 9].

 7KHRU\ RI )HPWRVHFRQG 3XPS±3UREH 3KRWRHOHFWURQ 6SHFWURVFRS\ Our formulation of time resolved pump–probe photoelectron spectroscopy is fully discussed in earlier papers [2, 7] and here we present just a brief outline. The wavefunction of the total system,  .r; R; t/, is expanded in the electronic wavefunctions relevant to the pump–probe arrangement, which is referred to as the Born–Huang expansion [17], as Z X ./ .r; R; t/ D i .R; t/˚i .rI R/ C dk k .R; t/˚k .rI R/ ; (1) i

where i labels the electronic adiabatic or diabatic wavefunctions, ˚i .rI R/, of the neutral molecule and ˚k./ .rI R/ is the wavefunction of the final state (ion plus photoelectron). The latter is labeled with the continuous photoelectron wave vector k and R is the set of internal nuclear coordinates, r electronic coordinates, and t time. Molecular rotation is neglected in this study, but it is an easy task to take the rotational average if necessary. Thus, i .R; t/ and k .R; t/ are identified with vibrational wavefunctions in the neutral and ionized systems, respectively. We here consider only two low lying electronic bound states, which is actually considered in our pump-probe photoelectron study of the wavepacket passage of a conical intersection between the adiabatic ground state (X) and the first excited state (A) of NO2 molecule (i D X; A in (1)). Coupled equations for the vibrational wavefunctions on these two potential surfaces (actually we chose a diabatic representation) can then be written as      @ X VX .R/ Vpu .R; t/ X b i„ D 7C Vpu .R; t/ VA .R/ A @t A ! Z Vpr.X/ .R; t/ C dk k ; (2) .A/ Vpr .R; t/ and i„

X @ bR C Vion .R/ C "k  k C k D ŒT Vpr.i / i ; @t

(3)

i DX;A

bR is the diagonal kinetic energy operawhere atomic units are used throughout. T tor for the nuclear coordinates, Vi .R/ the potential energy surfaces of the neutral

212

K. Takatsuka

molecule, Vion .R/ a potential energy surface of the molecular ion, and "k the photoelectron energy (labeled by the photoelectron wave number k). b 7 is the 2  2 kinetic energy matrix operator in the adiabatic electronic basis, where off-diagonal terms appear because of the nonadiabatic coupling (CI) between the two electronic states of this system. We resort to quasi-diabatization to handle this term in the numerical computation, as explained in the next section. Thus the coupling of the surfaces enters through the potential energy term and the kinetic energy operator is made diagonal. Vpu .R; t/ is the pump–pulse interaction coupling the ground and excited neutral states in the dipole approximation as Vpu .R; t/ D Epu fpu .t/ cos.!pu t/12 .R/ ;

(4)

where Epu is the strength of the pump field, fpu the pump pulse envelope, !pu the pump frequency, and 12 .R/ the transition dipole amplitude along the polarization b is the probe pulse of the pump pulse. The complex function Vpr.i / .k; R; t; T; ˝/ interaction, to be discussed further shortly. This interaction is by the probed neutral electronic surface, i , the delay time from the center of pump pulse, T , and the b angles between the probe pulse polarization and the molecular axis, ˝. The electronic wavefunction of the ion state is written as an antisymmetrized product of an ion wavefunction, ˚C .rI R/, and a photoelectron orbital, k./ .rI R/, ./

./

˚k .rI R/ D A.˚C .rI R/  k .rI R// ;

(5)

./ O with kO being the angular and k .rI R/ is expanded in spherical harmonics, Yl .k/, part of k, X ./ ./  O k .rI R/ D i l ei l Yl .k/ kl .rI R/ : (6) l ./

In (6) r indicates the electronic coordinates in the molecular frame, kl .rI R/ is a partial wave component of the photoelectron orbital in the molecular frame with momentum k, the projection of l in the molecular frame, and l the Coulomb phase shift [23–28]. In the dipole approximation, the probe interaction, Vpr , becomes Vpr D Epr fpr .t  T / cos.!pr t/D ; where the dipole operator, D, is r DD

4 X 1 b r D0 .˝/Y1 .Or/ ; 3 

(7)

(8)

for the linearly polarized case. Here Epr is the probe field strength, fpr .t  T / the probe pulse envelope, !pr the probe frequency, T the delay time from the center of the pump pulse, r and rO the magnitude and angular part of r, respectively, and

Electron Dynamics Embedded in Nuclear Wavepackets

213

b orient the probe polarization with respect to the molecule through the the angles ˝ 1 rotation matrix (D0 ). (See [23–28] for more details.) The ion vibrational wavefunction, k .R; t/, is also expanded in spherical harmonics as X O ; k .R; t/ D kl .R; t/Yl .k/ (9) l

and integration over k in (2) becomes an integration over k and summation over l and . Integration over k is handled by a quadrature (with weights wj ) over discrete points kj (j D 1; 2; : : : ; Nk ), where the integration is terminated at a maximum wave number kNk . With discretization of both the wave number and angle, the ion vibrational wavefunction is represented by a set of wavefunctions f kj l .R; t/g, each associated with different photoelectron energies and angles. For Nl sets of .l; / included in the calculation, the number of coupled equations of motion is thus .2 C Nk Nl / for the two neutral states and the discretized final state. Equations (2) and (3) are then discretized and solved numerically. After propagation of the vibrational wavepackets for a delay time T , the final ion population, Pion .T /, can be obtained by integrating over k, Z Z Pion D dk dR j k .R; tf /j2 

Nk X

Z wj kj2

j D1

X

Z

Nk



j D1

wj

Z

2

sin k dk

0

0

Z

2

ˇ ˇ2 ˇX ˇ dR ˇˇ kj l .R; tf /Yl .k ; k /ˇˇ l



sin k Dk Pkj .k ; k / ;

d k 0

Z



d k

0

(10)

where tf is the time after the probe pulse interaction is over and kO D .k ; k /. The photoelectron kinetic energy distribution, P ."k /, and energy-resolved molecular frame photoelectron angular distribution, Pkj .k ; k / are given by the integrands of (10). The angular coordinates .k ; k / and the ion partial waves can be transformed to a frame aligned with the probe polarization, X l b l ; e lm D Dm .˝/ (11) 

where the tilde over variables denote the transformed frame. Integration over the Q angle around the polarization axis of the probe results in averaged angular distributions of the form: ˇ ˇ2 Z 2 Z ˇX ˇ 2 Q ˇ Q Q Q Pkj .k / D kj d k dR ˇ e kj lm Ylm .k ; k /ˇˇ 0

lm

214

K. Takatsuka

Z D kj2

ˇ ˇ2 ˇX ˇ dR ˇˇ e kj lm Ylm .Qk ; 0/ˇˇ :

(12)

lm

This averaging results in angular distributions from a set of molecules with their molecular axes aligned along one direction.

 9HORFLW\ 0DS ,PDJLQJ We now present the photoelectron velocity map image to directly monitor the quantum wavepackets passing through the conical intersection between the first two 2 A0 states of NO2 (Fig. 1 [9]). As stated above, the Hamiltonian explicitly includes the pump-pulse interaction, the nonadiabatic coupling due to the conical intersection between the neutral states, and the probe interaction between the neutral states and discretized photoelectron continua. Conical intersections (CIs) are ubiquitous in polyatomic molecules and are among the most important of nonadiabatic processes in these systems. They play a fundamental role in the excited-state dynamics of simple polyatomic molecules and are also believed to be responsible for the underlying photostability of DNA under ultraviolet radiation [29–31]. Femtosecond time-resolved photoelectron spectroscopy can be expected to be a versatile probe of wavepacket dynamics in and around CIs. In fact, internal conversion in polyatomic molecules was among the earliest suggested application of this technique [32], which was subsequently realized experimentally [33]. Very recently, Bisgaard et al. [34] and Horio et al. [35] could track the change in electronic character with vibrational motion in an excited state from measured photoelectron angular distributions. The scheme for our studies of the pump–probe photoelectron spectra in NO2 is outlined as follows: The molecule is placed in the xy-plane, and for C2v geometry, the two O atoms lie parallel to the x-axis, and the N atom lies on the negative y axis. The pump pulse is polarized along the x-axis and the polarization of the probe is taken parallel to that of the pump. NO2 is first transiently aligned using short laser pulses [34, 36], and it is then pumped to an excited electronic state by a femtosecond pulse. Because of the ultrafast time-scale of the associated dynamics, we employ pulses of a full-width at half-maximum (FWHM) of 8 fs in these studies. Wavepacket motion on the excited state, as well as on the ground electronic state that is coupled to the excited state by the CI, is probed with a time-delayed femtosecond pulse that directly ionizes the molecule. The photoelectrons are then energyand angle-resolved for signatures of the wavepacket motion. The CI between the first two 2 A0 states of the NO2 molecule is known to lead to an extremely complex absorption spectra [37–39]. For C2v geometries the two surfaces (2 A1 and 2 B2 ) intersect at a bond angle that depends on the bond length and form a one-dimensional CI seam [40]. The seam is located close to the bottom of the excited state and is readily accessible by a vibrational wavepacket launched onto the excited electronic surface from the Franck–Condon region of the ground state. In the present computa-

Electron Dynamics Embedded in Nuclear Wavepackets

215

Fig. 1 Femtosecond time resolved photoelectron images: The vertical axis is the photoelectron momentum along the polarization axis of pump and probe in atomic units. (Courtesy of Dr. Yasuki Arasaki.)

tional setup, the wavepacket pumped to the first excited state passes through the CI between 4 and 12fs, and the second passage is observed between 20 and 28fs. The

216

K. Takatsuka

probe laser shined with a delay time ionizes these bifurcating wavepackets to the two low lying cation states, namely, 1 A1 and 3 B2 (in C2v notation), thus producing the velocity map images. We draw Pk .Qk / of (12) in Fig. 1 in a color code, which is the so-called velocity map image, representing the above NO2 dynamics during the (delay) time 4 fs to 27 fs. In these maps, the vertical axis indicates the photoelectron momentum along the direction of the pump and probe polarizations in atomic units, and the horizontal axis the momentum perpendicular to it. The intensities (Pk .Qk /  104 of (12)) are indicated by color and brightness. These images offer a compact representation of the angular distribution of photoelectrons. The bright innermost ring at k D 0:2 at early times, mostly in direction parallel to the probe pulse (photoionization from the unexcited initial state). The k D 0:3 and k D 0:4 rings correspond to the 1:4 eV and 2:1 eV, and we see the build-up in intensity in the direction parallel to the probe as the wavepacket passes through the CI between 4 and 12 fs. At later times (20 to 28 fs), the peak intensity lies away from the probe polarization. In this way, the photoelectron velocity map imaging makes it possible to track the evolution of the wavepackets as population is transferred in nonadiabatic regions.

 *HQHUDO )UDPHZRUN RI 1RQDGLDEDWLF (OHFWURQLF :DYHSDFNHW '\QDPLFV 'ULYHQ E\ 1XFOHDU 0RWLRQ We next shift our view point from nuclear dynamics to nonadiabatic electron wavepacket dynamics. Our approach to construct coupled electronic and nuclear quantum wavepackets takes the following steps; (i) first establishing the exact mixed quantum (electronic) and classical (nuclear) dynamics, which naturally leads to the notion of nuclear-path branching, (ii) finding appropriate and reasonable approximations to this dynamics and performing electronic-state mixing to propagate electron wavepacket dynamics, and (iii) finally quantizing these non-Born–Oppenheimer nuclear paths. Incidentally, the first step mentioned above is equivalent to formulating the general dynamical theory for mixed quantum-classical system. The general solutions emanating from these mixed quantum and classical subsystems indicate, therefore, how quantum effects affect the dynamics in such classical subsystems. For more complete presentation of the theory along with numerical illustrations, we refer to [13–16].

Electron Dynamics Embedded in Nuclear Wavepackets

217

 %UDQFKLQJ 1XFOHDU 3DWKV 'XH WR 1RQDGLDEDWLF ,QWHUDFWLRQ  (OHFWURQLF 6WDWH 0L[LQJ DV D :DYHSDFNHW '\QDPLFV DORQJ 1XFOHDU 3DWKV To formulate the electron wavepacket dynamics on a clear basis, we begin by representing the total Hamiltonian operator in electronic Hilbert and nuclear configuration spaces using a basis set fj˚I .5/ij5ig such that [10–12] H.5; elec/

 2 X X 1X b .el/ k P k  i„ j˚I iXIJ h˚J j C j˚I iHIJ h˚J j : (13) 2 IJ

k

IJ

An extension of this representation to more general form so as to include the vector potential of classical electromagnetic field is easy [16]. One can readily reduce H.5; elec/ to a mixed quantum-classical representation by replacing the nuclear b k with its classical counterparts Pk such that momentum operator P X X 1X .el/ k HQ .5; 3; elec/ .Pk  i„ j˚I iXIJ h˚J j/2 C j˚I iHIJ h˚J j : (14) 2 IJ

k

IJ

This is the stating point of the present work. The dynamics of electron wavepackets is determined by the time dependent variational principle  Z @ e .5; 3; elec/ j˚.5; t/i D 0 : ı dth˚.5; t/j i„  H (15) @t P For an electron wavepacket expanded in a basis set as j˚i D I CI .t/j˚I i, this variational principle (with averaging the left and right variations) gives  X .el/ X @ „2 X k k k i„ CI D HIJ  i„ RP k XIJ  .YIJ C YJI / CJ : (16) @t 4 J

k

k

This expression seems to be similar to the standard representation of the semiclassical Ehrenfest theory, except that (16) includes (nontrivial) corrections as the third term in its right-hand side.  *HQHUDOL]HG )RUFH :RUNLQJ RQ 1XFOHL LQ 1RQDGLDEDWLF 5HJLRQ With the above Hamiltonian, (14), an analogy to purely classical mechanics brings about the “canonical equations of motion” for nuclear classical variables .5; 3/ as e d @H Rk D dt @Pk

and

e d @H Pk D  : dt @Rk

(17)

218

K. Takatsuka

As noticed immediately, the dynamics of .5; 3/ in these equations are essentially different from those of the purely classical counterpart in that the actions include the electronic ket-bra vectors. Therefore the quantities .d=dt/Rk and .d=dt/Pk should work as transition operators for the electronic states. To emphasize this difference, P k .d=dt/Rk and R Rk we express our quantities with the calligraphic fonts as R 2 2 .d =dt /Rk . After some manipulation, we find 

X @ d2 k .el/ R D .j˚I iHIJ h˚J j/ dt 2 @Rk IJ  XX  @ @ l k P  i„ Rl .j˚I iXIJ h˚J j/  .j˚I iXIJ h˚J j/ : (18) @Rk @Rl IJ

l

These “classica” operators can appear as physically useful quantities only after the electronic states are specified to be operated on. This implies that the force operators should be transformed to a matrix form by sandwiching two electronic states, say, h˚I j and j˚J i as k R k j˚J i FIJ h˚I jR  X X  @X l @H .el/ @X k .el/ .el/ k k IJ D ŒXIK HKJ HIK XKJ  IJ Ci„ RP l  IJ ; (19) @Rk @Rk @Rl K

l

which is equivalent to   X  @X l @X k @H .el/ k IJ FIJ D C i„ RP l  IJ ; @Rk IJ @Rk @Rl

(20)

l

k if the basis set used is complete. We refer FIJ to the force matrix. Obviously, the first term in the right-hand side of (20) comes from the electronic energy, while the second one represents the recoil from the delayed (kinematic) response of electrons to nuclear motions. This last term represents a force acting in the directions perpendicular to the velocity vector fRP k g as the magnetic Lorentz force does [11, 41]. Therefore it appears only in a multidimensional nonadiabatic system. The mathematical origin of the Lorentz-like force is already apparent in the form of (13). In case where no nonadiabatic interaction is present due to k XIJ D0;

(21)

it is most convenient to adopt the adiabatic wavefunctions f H .el/

˛

D E˛ad

Then we have k F˛ˇ D ı˛ˇ

˛

@Eˇad @Rk

:

˛ g,

satisfying (22)

(23)

Electron Dynamics Embedded in Nuclear Wavepackets

219

subject to the condition (21). Therefore the force matrix in the adiabatic limit reproduces the Born-Oppenheimer forces in its diagonal elements. In the above derivation we have simply classicalized the nuclear momentum operators from (13) to (14), which might be somewhat questionable. In this conjunction, Hanasaki and Takatsuka have developed a path integral formalism for nonadiabatic dynamics, which also sets a theoretical foundation for mixed quantum and classical dynamics, since the nuclear path integration is to be carried out within cnumber domain from the outset in the path integration [42]. After all, it turns out that the force matrix and the equations for electronic state mixing have been all rederived as above, and yet it has revealed new aspects of the theoretical structure of nonadiabatic dynamics [42].

 6PRRWKO\ %UDQFKLQJ 3DWKV WR 5HSUHVHQW WKH :DYHSDFNHW %LIXUFDWLRQ We next survey the characteristic features of the rigorous solutions of the coupled equations, (16) and (19) and how they can be approximated.  'LDEDWL]DWLRQ LQ 7HUPV RI WKH (LJHQVWDWHV RI WKH )RUFH 0DWUL[ The electronic wavepacket are to be carried along nuclear paths that are driven by the force matrix. Suppose we have an electronic wavepacket ˚.UI 5.t// at a phase space point .5; 3/, where 5 D 5.t/ and 3 is its conjugate momentum. To materialize an electronic-state mixing among given basis functions f˚I .UI 5/g (either adiabatic or any diabatic basis), we first integrate (16) for a short time, say, t to give a new set of fCI .t/g. Next we want to run a path using the force matrix F .5/ again for a short time t. However, the presence of the off-diagonal elements in the force matrix can induce additional (unnecessary) electronic-state mixing. To avoid this additional mixing, we diabatize the electronic basis for this time interval by diagonalizing the force matrix at 5 such that 1 0 f1 .5/ 0    B C 8.5/F.5/8.5/1 D @ 0 f2 .5/ A (24) :: :: : : with the associated electronic basis-set transformation 0 1 0 1 ˚1 .UI 5/

1 .UI 5/ B C B C 8.5/ @˚2 .UI 5/A D @ 2 .UI 5/A : :: :: : :

(25)

220

K. Takatsuka

The electronic wavepacket obtained as above at 5 may be reexpanded in the eigenfunctions f K .UI 5/g such that X ˚.UI 5.t// D DK .t/ K .UI 5/ : (26) K

Then, each electronic component DK .t/ K .UI 5/ is carried by its own path that is to be driven by the eigenforce fK , without the electronic mixing among f L .UI 5/g, to reach a new point after t. (Note however that mixing among f L .UI 5/g can occur in the electron dynamics even in L .UI 5/-representation of (16).) Different eigenforces make different paths even if they start from a single phase-space point .5; P/ in such a way that .5; P/ ! .5K ; 3K / : (27) Therefore a path at 5 is branched to as many pieces as the number of electronic states involved in the nonadiabatic coupling. The electronic-state mixing should be considered again at the individual points .5K ; 3K /, regarding the corresponding component DK .t/ K .UI 5/ in (26) as the renewed condition to integrate (16). Again, the path branching is followed at each .5K ; 3K /. Hence, the cascade of path-branching should continue as long as the nonadiabatic coupling cannot be effectively ignored. The full details of the entire procedure have been summarized elsewhere [11].  3KDVH6SDFH $YHUDJLQJ RI 7R%H%UDQFKLQJ 3DWKV WR ([WUDFW D )HZ 5HSUHVHQWDWLYH 3DWKV It is obvious that tracking every series of branching paths is extremely cumbersome and technically impossible, even though the branching feature reflects the essential physics behind nonadiabatic interaction. However, it is anticipated that those branching paths should not geometrically deviate much from each other in phase space in somewhat a short time. In other words, they should localize along a representative path forming a tube-like structure. Therefore, we extract such a representative path by taking an average of phase-space points in the following manner: (i) Suppose we have a path ending at .h5.t/i; h3.t/i/ in phase space. At this point, diagonalize the force matrix, F .h5i/j K .h5i/i D j K .h5i/ifK .h5i/

(28)

to obtain the eigenforces ffK g and its eigenstates fj K ig. The wavepacket ˚.UI h5.t/i/ is expanded in terms of these eigenstates as in (26). (ii) The Kth eigenforce drives a path starting from .h5.t/i; h3.t/i/ for a short time t in terms of the Hamilton canonical equations of motion as 5K .t C t/ D h5.t/i C 5K ; 3K .t C t/ D h3.t/i C 3K :

(29) (30)

Electron Dynamics Embedded in Nuclear Wavepackets

221

(iii) Now we average them into the form h5.t C t/i D h5.t/i C

X jDK .t/j2 5K

(31)

K

h3.t C t/i D h3.t/i C

X

jDK .t/j2 3K ;

(32)

K

which makes the next point .h5.t C t/i; h3.t C t/i/ of the representative path. (iv) With this averaged point, we calculate F .h5.t C t/i/j K .h5.t C t/i/i D j K .h5.t C t/i/ifK .h5.t C t/i/

(33)

and turn anew to the step (ii). The successive applications of the procedure (i)–(iii) gives a single finite path.  %UDQFKLQJ $IWHU $YHUDJLQJ The above approximation should be valid only when the averaging time is short enough for the branching paths not to separate from one another widely, which should depend on the averaging time and/or intensity of the nonadiabatic coupling elements. In case where a simple averaging approximation fails due to such hard branching, one may allow paths to further branch and make another averaging. This additional “branching and averaging” may be performed at several places, compromising with additional computational tasks. Suppose we are tracking one of the   average paths, say, the Kth path, 5K .t/; 3K .t/ . To emphasize that every force is generated along this path, we rewrite (24) explicitly as 0 1 f1 .5K / 0  B f2 .5K / C 8.5K /F .5K /8.5K /1 D @ 0 (34) A ; :: :: : : where the dependence of the force matrix on 5K has been stressed. The right-hand side of this representation reminds that other eigenforces, say, fLare also calculated  along 5K .t/. Therefore, at a point on the path 5K .t/; 3K .t/ , one may switch  the  force from fK to fL to  emanate DQRWKHU path, such that 5K .t/; 3K .t/ ! 5L .t C t/; 3L .t C t/ in terms of fL . If one continues using fK at the same     point, it follows that 5K .t/; 3K .t/ ! 5K .t C t/; 3K .t C t/ . Thus this procedure allows branching. At such a branching point with different forces, one can resume the averaging procedure using the procedure described as above, if necessary. In this way, we can make a hierarchy (cascade) of paths of branching and averaging. Illustrative

222

K. Takatsuka

examples, including the practice and timing, and other technical matters, are shown in [12, 15, 16]. We refer to this mixed quantum-classical approach as the PSANB (phase-space averaging over natural branching paths).  %UDQFKLQJ RI WKH $YHUDJHG 3DWKV $V\PSWRWLFDOO\ 7RZDUGV WKH $GLDEDWLF 3RWHQWLDO (QHUJ\ 6XUIDFHV When a molecule enters a coupling-free region after passing through a nonadiabatic region, the force matrix smoothly becomes a diagonal matrix, whose diagonal elements represent the forces arising from the individual adiabatic potential energy surfaces. Also, the adiabatic wavefunctions become the eigenfunctions of the force matrix. At the same time, the mixing of electronic states is turned off gradually. Therefore we stop taking the average as in (31) and (32), and simply let the individual components   DK .t/ K UI 5K .t/ (35) run being driven by their own forces. Then the coefficients DK .t/ coherently carry the information of transition amplitudes including the electronic phases. In this way, a path can naturally branch towards the asymptotic region.

 4XDQWL]DWLRQ RI WKH 1RQ%RUQ±2SSHQKHLPHU %UDQFKLQJ 3DWKV So far, we have established a method to propagate electronic wavepackets along branching non-Born–Oppenheimer paths. Our next target is to quantize thus generated nuclear paths. The obvious difficulty here is that WKRVH QRQ%RUQ±2SSHQKHLPHU SDWKV DUH QHYHU FODVVLFDO 1HZWRQLDQ WUDMHFWRULHV, and therefore most of semiclassical theories resting on classic trajectories cannot be applied. We therefore our developed semiclassical wavepacket method, called the action decomposed function   (ADF) [43, 44]: A wavefunction 5; tI 5.t/ localized at a center 5.t/ is represented as      i  5; tI 5.t/ D F .5  5.t/; t/ exp S 5  5.t/ ; (36) „   where 5.t/ is a point running on a path and S 5  5.t/ is an action function around 5.t/, at which the action integral is to be evaluated along the relevant path. Among many possibilities to apply this theory, we choose the normalized variable Gaussian (NVG) as F .55.t/; t/, which is a semiclassical Gaussian approximation to ADF [45].

Electron Dynamics Embedded in Nuclear Wavepackets

223

 1RUPDOL]HG 9DULDEOH *DXVVLDQ $')19* The function F .5  5.t/; t/ in (36) evolves in time in such a way that  X  1 i„ 2 F .55.t Ct/; t Ct/ D exp t  r vC r F .55.t/; t/ ; (37) 2 2 k

where v is the velocity of the path at 5.t/. We next approximate F .5  5.t/; t/ in terms of a normalized Gaussian function as  N=4   1=2  F .5  5.t/; t/ D Œdet &1=4 det &.t/ C i'.t/ 2   T  1    exp  5  5.t/ &.t/ C i'.t/ 5  5.t/ : (38) The inverse Gaussian exponent adopted in this function is particularly useful, since the time evolutions of &.t/ and '.t/ are given explicitly as Z &.t C t/ D exp

t Ct

Z b : dt &.t/ exp

t

t Ct

b dt :

T ;

(39)

t

where T indicates the transposition of a matrix, and Z '.t C t/ D exp

t Ct

Z b : dt '.t/ exp

t

t Ct

b dt :

T C 2„t, ;

(40)

t

b is a matrix defined as respectively, and : 2 b kl .t/ D @ W .5/ D @Pl : : @Rk @Rl @Rk

(41)

at a trajectory point 5.t/, where W is the Hamilton’s characteristic function. Also, the differential equations for them are given as

and

P b b &.t/ D :.t/&.t/ C &.t/:.t/

(42)

P b b C 2„, : '.t/ D :.t/'.t/ C '.t/:.t/

(43)

The exponent &.t/ is responsible for describing only the velocity gradient,  12 r  v in (37), while '.t/ reflects the dynamics not only for the velocity gradient but arising from the quantum diffusion term, .i„=2/r 2 of (37). The Planck constant appears only in the quantum diffusion term, and therefore only '.t/ is to be scaled to the magnitude of „. The details of this theory is reported elsewhere [45]. It should be noted that no stability matrix is needed in this theory, the stability matrix which is the principal origin of most of the difficulties in semiclassical mechanics. Moreover, although assuming the presence of the Hamilton-Jacobi equation

224

K. Takatsuka

implicitly in the above derivation, the final driving equations of the Gaussian do not need it explicitly: Only the presence of a flow in .5; P/ space suffices to propagate the NVG without respect to how the phase-space flow lines are constructed. With these wavepackets generated for nuclear paths along with the electronic wavefunctions obtained as above, we are now ready to build a total electronic and nuclear within an on-the-fly scheme. Note that in this representation it is possible to utilize as many NVGs as one can to represent a global nuclear wavefunction on each nuclear path for the higher approximation. Nevertheless, we have shown numerically that even single Gaussian approximation applied to individual branching path is already remarkably good [15,16]. We here refer to the present electronic and nuclear simultaneous quantum wavepacket method as PSANB-ADF-NVG.

 %2 5HSUHVHQWDWLRQ DQG (OHFWURQLF1XFOHDU ZDYHSDFNHW 5HSUHVHQWDWLRQ We then try to tie up the total wavefunction in the Born-Huang expansion as in (1) and our on-the-fly total wavefunction, which will be defined in (44).

 7KH 7RWDO (OHFWURQLF DQG 1XFOHDU ZDYHIXQFWLRQ Suppose we have a set of those branching paths, branching due to nonadiabatic coupling and/or optical interactions, that is, f5Jj .t/g indicating a j th path running on a J th electronic state. Accumulating all the dynamical components along the individual paths as studied above, we can now build a total electronic and nuclear wavepacket along the non-Born–Oppenheimer paths as tot OTF .U; 5; t/ D

path state X X    CJj .t/˚J UI 5Jj .t/ J

j



 FNVG .5  5Jj .t/; t/ exp

  i  S 5  5Jj .t/ ; „

(44)

where the suffix OTF stands for on-the-fly, and ˚J may be adiabatic or diabatic. This representation of the total wavefunction should be compared with the Born– Huang expansion, which is tot QM .U; 5; t/ D

state X

˚J .UI 5/ QM J .5; t/ :

(45)

J

To do so, we rewrite our represented wavefunction of (44) by assuming that FNVG .5  5Jj .t/; t/ is spatially well localized, which allows us to approximate

Electron Dynamics Embedded in Nuclear Wavepackets

225

tot OTF .U; 5; t/ as

tot OTF .U; 5; t/ Š

state X

˚J .UI 5/

path X

J

CJj .t/FNVG .5  5Jj .t/; t/

k

   i   exp S 5  5Jj .t/ „ state X ˚J .UI 5/ OTF J .5; t/;

(46)

J

with an obvious definition OTF J .5; t/ D

path X j

   i  CJj .t/FNVG .5  5Jj .t/; t/ exp S 5  5Jj .t/ : „

(47)

Now it is possible to compare with the quantum mechanical counterpart, QM J .5; t/, and indeed this is a way of our comparison between PSANB-ADF-NVG with the full quantum theory in numerical presentation. Recall that CJj .t/ included in the nuclear wavepacket OTF J .5; t/ of (47) have originally emerged from the electronic wavepacket bifurcation along the branching paths. Also, for numerical comparison QM between OTF J .5; t/ and J .5; t/, we refer to [15, 16], which have shown a very good agreement.

 )RU &RQWLQXXP 6WDWHV In case where an electronic continuum is involved in the total wavefunction such as that for photoionization, which has the following form tot QM .U; 5; t/ D

state X

Z ˚J .UI 5/ QM J .5; t/ C

dN ˚N./ .UI 5/ QM N .5; t/ ;

(48)

J

our on-the-fly electronic-nuclear representation of the total wavefunction should look like tot OTF .U; 5; t/

Š

state X J

˚J .UI 5/

path X j

CJj .t/FNVG .5  5Jj .t/; t/

   i   exp S 5  5Jj .t/ „

226

K. Takatsuka

Z C

./

dN ˚N .UI 5/

path X

CjN .t/FNVG .5  5jN .t/; t/

j

   i   exp S 5  5jN .t/ ; „

(49)

and therefore we formally have OTF N .5; t/ D

path X

 CjN .t/FNVG .5  5jN .t/; t/ exp

j

  i  S 5  5jN .t/ „

(50)

for the continuum state, where 5jN .t/ denotes a path running on the ion potential energy surface with a photoelectron of wave-number vector N. We further need an additional suffix when the ion manifold is composed of plural electronic states as in our treatment of NO2 , but in what follows we simply assume that the ion state consists of a single state as in (48). Equation (49) is not as straightforward as it tot formally looks, since the wavepacket wavefunction OTF .U; 5; t/ is expanded in the ./ stationary-state scattering wavefunctions ˚N .UI 5/, which spatially extend in the asymptotic (infinite) regions.

 6HSDUDWLRQ RI WKH $WWRVHFRQG 2VFLOODWRU\ )DFWRUV It would be convenient to use the adiabatic representation for the electronic bases, which leads to the following separation (the so-called interaction picture)   i 0 CJj .t/ D cJj .t/ exp  EJ t ; (51) „ and OTF J .5; t/ D

  i 0 cJj .t/ exp  EJ t FNVG .5  5Jj .t/; t/ „ j    i   exp S 5  5j .t/ : „

path X

(52)

Obviously the factor expŒ.i=„/EJ t is responsible for the fast oscillation arising from the electronic motion of the order of 10 to 100 attoseconds. Recall that the simple relation 1 „ t  (53) 2 E suggests that an energy difference as small as E D 1:0 eV already gives t D 330 0 as. Therefore even if cJj .t/ is not quickly varied, CJj .t/ certainly is, provided that

Electron Dynamics Embedded in Nuclear Wavepackets

227

E is large. Since the nuclear wavepacket dynamics is numerically integrated in the time scale of femto second, it seems difficult to retrieve attosecond information from 0 CJj .t/. However, the change in cJj .t/ may be much slower and well reproduced in the nuclear wavepacket calculations. This is indeed the case, as we observed above in the velocity map image of NO2 and other femtosecond time-resolved photoelectron spectroscopy.

 ,QIRUPDWLRQ RI (OHFWURQ '\QDPLFV LQ WKH 1XFOHDU :DYHSDFNHWV Here in this section we try to resolve the physical factors involved in the timeresolved photoelectron spectra and velocity map images. We start with an assumption that the photoionization process could be solved in terms of the PSANB-ADFNVG. (We are not claiming that this method is already applicable to ionization processes from a many-electron polyatomic molecule. Rather, it is still one of the most difficult problems.) Assuming the presence of such a PSANB-ADF-NVG wavefunction, we below represent the quantities relevant to photoelectron velocity map imaging.

 ,PDJLQJ DQG 3KDVH For the sake of simpler discussion, we proceed further on a qualitative basis adopting a single NVG approximation, which uses only one NVG for OTF N .5; t/, as OTF N .5; t/

   N  .„k/2  i D c .t/ exp  Eion 5 .t/ C t FNVG .5  5N .t/; t/ „ 2m    i  N  exp S 5  5 .t/ (54) „ N0

It is expected that 5N .t/ should not depend much on N. Therefore we may factorize OTF N .5; t/ into two parts (strongly N dependent)(almost N independent)       N  i .„k/2 i N0 OTF .5; t/ D c .t/ exp  t exp  E 5 .t/ t i on N „ 2m „    i  FNVG .5  5N .t/; t/ exp S 5  5N .t/ : „ Further we note that



  i .„k/2 c .t/ exp  t „ 2m N0

is a term that directly contains the information about N, while

(55)

(56)

228

K. Takatsuka

   i  N N FNVG .5  5 .t/; t/ exp S 5  5 .t/ „

(57)

can contribute indirectly to Pk .Qk /. Then the photoionization spectrum (the velocity map) is approximated by the expression (recall (10) and (12)) Pk .Qk / ' k 2

Z

ˇX ˇ2 ˇ ˇ k0 N Q ˇ e d5 ˇ cQlm .tf /F NVG;lm .5  5 .tf /; tf /Ylm .k ; 0/ˇˇ

(58)

lm

where the coincidence between the electronic state mixing and nuclear wavepacket motion is clearly identified. This suggests that a useful piece of information will be taken out by coincident experiments of photoelectrons and the product ions. Note that no phase information is retained in the above expression. However, the even trivial looking phase in (56), expŒ.i=„/.„k/2 =.2m/t, is important in that this is vital to identify the position of photoelectrons, as we observe 

i .„k/2 i „k i i tD k tD k.vt/ D  kr.t/ ; „ 2m 2 m 2 2

(59)

where r.t/ is the position of a photoelectron with a constant classical velocity. Obviously, no information about the three-dimensional spatial distribution of electrons is provided by the velocity map imaging technique. Therefore, it would be important to attempt to recover the phase information H[SHULPHQWDOO\ as much as possible.

 5HFRQVWUXFWLRQ RI WKH (OHFWURQ:DYHSDFNHW 9LHZ IURP WKH &ORVH&RXSOLQJ (TXDWLRQV Let us repeat to emphasize that nuclear wavepackets in the Born–Huang expansion contain the information simultaneously both of the nuclear dynamics and the QM electronic-state mixing in a vague fashion. Suppose that QM J .5; t/ and N .5; t/ are numerically available in the femtosecond study of nuclear wavepackets as we did in NaI and NO2 cases. Besides, assume that we can introduce an approximation to determine the nuclear paths 5jN .t/ and their associated FNVG .5  5jN .t/; t/. Then we bring them together as QM J .5; t/ '

   i  FNVG .5  5Jj .t/; t/ exp S 5  5Jj .t/ „ j   i 0  cJj .t/ exp  EJ t ; „

path X

(60)

0 where only the electronic mixing coefficients cJj .t/ are unknown. However we can 0 invert, theoretically at least, to obtain cJj .t/ in such a way that

Electron Dynamics Embedded in Nuclear Wavepackets

11 :: : : B    C B 0 C B C Bc .t/C D B FNVG .5i  5Jj .t/; t/ exp .i=„/S 5i  5Jj .t/ C C @ Jj A B  expŒ.i=„/E t J @ A :: : : : :: :: 0 1 :: B QM : C C B (61) @ J .5i ; t/A : :: : 0

:: :

1

0

229

::

Likewise, the electronic mixing coefficients for the continuum states may be taken as 0 :: 1 :: :    B B N0: C B FNVG .5i  5jN .t/; t/ exp .i=„/S 5i  5jN .t/ Bc .t/C D B       @j A B  exp .i=„/ Eion 5jN .t/ C .„k/2 =.2m/ t @ :: :: : : 0 1 :: B QM : C C B @ N .5i ; t/A : :: : 0

11

::

:

::

:

C C C C A

(62)

0 Once we can approximate cJj .t/ and cjN0 .t/, we should be able to reproduce the electron wavepacket dynamics at an instantaneous nuclear position 5.t/ (in the semiclassical sense) by bringing those quantities back to (1). This is meaningful in that we can reproduce the electron wavepacket dynamics from the stationary-state electronic scattering wavefunctions of photoionization. And eventually, one can visualize how the electronic state is deformed and how one of the electrons is ejected out of the molecule.

 &RQFOXVLRQ We have studied the interrelationship between the femtosecond wavepacket dynamics and attosecond electron wavepacket dynamics behind that. As far as the femtosecond molecular events are concerned, the nuclear wavepacket dynamics based on the Born–Huang expansion gives a nice view quantitatively and is suitable for reproducing and predicting the quantities to be observed experimentally. On the other hand, electron wavepacket dynamics may not be as straightforward, but it can more directly reveal the dynamical electron picture of chemical reactions such as the electron flux induced by reaction dynamics [46–48]. It is of no need to emphasize though that the essentially attosecond dynamics induced by a ultrashort pulse laser

230

K. Takatsuka

of, say 10 attoseconds, can be tracked practically only with the electron wavepacket dynamics. Even in such an attosecond regime, both native and laser induced nonadiabatic coupling should be properly taken into account. $FNQRZOHGJHPHQWV The author deeply thanks Dr. Y. Arasaki, Professor V. McKoy, and Dr. K. Wang for long-standing joint studies on time-resolved photoelectron spectroscopy. In fact, the content of Sect. 2 is mostly an excerpt from of our joint work documented in [9]. He is also grateful to invaluable discussions with Dr. T. Yonehara and Dr. S. Takahashi about nonadiabatic electron wavepacket dynamics and semiclassical theory, respectively. This work was supported in part by a Grant-in-Aid for Basic Science from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

5HIHUHQFHV 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

Y. Arasaki, K. Takatsuka, K. Wang, V. McKoy, Chem. Phys. Lett. , 363 (1999) Y. Arasaki, K. Takatsuka, K. Wang, V. McKoy, J. Chem. Phys. , 8871 (2000) K. Takatsuka, Y. Arasaki, K. Wang, V. McKoy, Faraday Discuss. , 1 (2000) Y. Arasaki, K. Takatsuka, K. Wang, V. McKoy, J. Electron. Spectrosc. Relat. Phenom. , 89 (2000) Y. Arasaki, K. Takatsuka, K. Wang, V. McKoy, J. Chem. Phys. , 7941 (2001) Y. Arasaki, K. Takatsuka, K. Wang, V. McKoy, Phys. Rev. Lett. , 248303 (2003) Y. Arasaki, K. Takatsuka, K. Wang, V. McKoy, J. Chem. Phys. , 7913 (2003) Y. Arasaki, K. Takatsuka, Chem. Phys. , 175 (2007) Y. Arasaki, K. Takatsuka, K. Wang, V. McKoy, J. Chem. Phys. , 124307 (2010) K. Takatsuka, J. Chem. Phys. , 064111 (2006) K. Takatsuka, J. Phys. Chem. A , 10196 (2007) T. Yonehara, K. Takatsuka, J. Chem. Phys. , 134109 (2008) K. Takatsuka, Int. J. Quant. Chem. , 2131 (2009) K. Takatsuka, T. Yonehara, Adv. Chem. Phys. , 93 (2009) T. Yonehara, S. Takahashi, K. Takatsuka, J. Chem. Phys. , 214113 (2009) T. Yonehara, K. Takatsuka, J. Chem. Phys. , 244102 (2010) M. Born, K. Huang, '\QDPLFDO 7KHRU\ RI &U\VWDO /DWWLFHV (Oxford Univ. Press, Oxford, 1954) I.V. Hertel, W. Radloff, Rep. Prog. Phys. , 1897 (2006) T. Suzuki, Annu. Rev. Phys. Chem. , 555 (2006) M. Wollenhaupt, V. Engel, T. Baumert, Annu. Rev. Phys. Chem. , 25 (2005) A. Stolow, A.E. Bragg, D.M. Neumark, Chem. Rev. , 1719 (2004) H. Satzger, D. Townsend, M.Z. Zgierski, S. Patchkovskii, S. Ullrich, A. Stolow, Proc. Natl. Acad. Sci. U.S.A. , 10196 (2006) R.R. Lucchese, D.K. Watson, V. McKoy, Phys. Rev. A , 421 (1980) R.R. Lucchese, G. Raseev, V. McKoy, Phys. Rev. A , 2572 (1982) S.N. Dixit, V. McKoy, J. Chem. Phys. , 3546 (1985) R.R. Lucchese, K. Takatsuka, V. McKoy, Phys. Rep. , 147 (1986) K. Wang, V. McKoy, J. Chem. Phys. , 4977 (1991) K. Wang, V. McKoy, Annu. Rev. Phys. Chem. , 275 (1995) R.V. Bensasson, E.J. Land, T.G. Truscott, in ([FLWHG 6WDWHV DQG )UHH 5DGLFDOV LQ %LRORJ\ DQG 0HGLFLQH (Oxford Univ. Press, Oxford, 1993), chap. 5 C.E. Crespo-Hern´andez, B. Cohen, P.M. Hare, B. Kohler, Chem. Rev. , 1977 (2004) S. Ullrich, T. Schultz, M. Zgierski, A. Stolow, Phys. Chem. Chem. Phys. , 2796 (2004) M. Seel, W. Domcke, J. Chem. Phys. , 7806 (1991) V. Stert, P. Farmanara, W. Radloff, J. Chem. Phys. , 4460 (2000)

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34. C.Z. Bisgaard, O.J. Clarkin, G. Wu, A.M.D. Lee, O. Geßner, C.C. Hayden, A. Stolow, Science , 1464 (2009) 35. T. Horio, T. Fujii, Y. Suzuki, T. Suzuki, J. Am. Chem. Soc. , 10392 (2009) 36. F. Rosca-Pruna, M.J.J. Vrakking, Phys. Rev. Lett. , 153902 (2001) 37. A. Delon, R. Jost, M. Lombardi, J. Chem. Phys. , 5701 (1991) 38. A. Delon, R. Jost, J. Chem. Phys. , 4300 (1991) 39. A. Delon, R. Jost, M. Jacon, J. Chem. Phys. , 331 (2001) 40. G. Hirsch, R.J. Buenker, C. Petrongolo, Mol. Phys. , 835 (1990) 41. T. Yonehara, K. Takatsuka, J. Chem. Phys. , 154104 (2008) 42. K. Hanasaki, K. Takatsuka, Phys. Rev. A, , 052514 (2010) 43. A. Inoue-Ushiyama, K. Takatsuka, Phys. Rev. Lett. , 1404 (1997) 44. A. Inoue-Ushiyama, K. Takatsuka, Phys. Rev. A , 3256 (1999) 45. S. Takahashi, K. Takatsuka. to be published 46. M. Okuyama, K. Takatsuka, Chem. Phys. Lett. , 109 (2009) 47. K. Nagashima, K. Takatsuka, J. Phys. Chem A, , 15240 (2009) 48. T. Yonehara, K. Takatsuka, Chem. Phys. , 115 (2009)

Index

A Ab-initio method, 107–132 Acceleration, 14, 24–29, 31–33, 57, 98 Adiabatic, 51, 52, 211, 212, 218, 219, 222, 224, 226 Ammosov–Delone–Krainov (ADK) theory, 136 Angular distribution, 56, 151, 174, 176, 185, 188–191, 201, 202, 209, 213, 214, 216 Argon (Ar), 14, 16–18, 21, 96, 97, 99, 116, 117 Arnoldi–Lanczos scheme, 14, 16, 20, 163 Arnoldi propagator, 108–111, 117–120, 123, 131 Attosecond pulse, 116, 150, 151, 188, 189, 192, 195, 202 Attosecond-time scales, 14 B Bifurcation, 211, 219–222, 225 Born-Oppenheimer, 192, 199, 219 Boundary condition, 92–94, 101, 152, 159–160, 168, 170 Branching, 210, 216–225 C CH3–CN, 9–10 Chemical reaction, 89, 210, 229 Close-coupling (CCC), 15, 102, 109, 118, 131, 166, 184, 186, 228–229 Collision, 16, 55–69, 95, 100–102, 105, 116, 191, 202 Complex absorption, 214

Control, 21, 38, 108, 125, 151, 180, 202 Crank-Nicholson, 184 Current, 137 Current tunnelling, 140 D Differential cross section, 58–63, 65–68, 100, 102, 103, 178–179, 185, 186 Dimensionality, 72, 76, 114, 116, 122 Dipole, 14, 15, 25–27, 29, 34, 41–43, 57, 98–100, 112, 117, 118, 165, 188–190, 202, 212 Discrete variable representation (DVR), 92, 152–156, 158, 160, 163 Discretization, 151, 160–164, 184, 213 Double ionization, 13–21, 90, 100–103, 108, 109, 115, 116, 150, 151, 169, 171–182, 184, 187–189, 191, 201, 202 Double-slit, 192, 198–200, 203 DVR. See Discrete variable representation (DVR) Dyson orbital, 41, 139, 141, 143, 144, 146 E Electron-dynamics, 85, 108, 116, 130, 136, 150, 209–230 Electron-harmonics, 29–31, 112 Electron-motion, 24, 29, 31, 136 Electron recollision, 24, 30, 55 F FEDVR. See Finite element discrete variable representation (FEDVR)

A.D. Bandrauk and M. Ivanov (eds.), Quantum Dynamic Imaging: Theoretical and Numerical Methods, CRM Series in Mathematical Physics, DOI 10.1007/978-1-4419-9491-2, © Springer Science+Business Media, LLC 2011

233

234

Femtosecond, 19, 24, 37, 39, 46, 89, 90, 97, 104, 116, 117, 150, 151, 202, 209–230 Finite-difference, 108, 111–116, 120–131, 184 Finite element, 151–160, 164 Finite element discrete variable representation (FEDVR), 151–161, 163–165, 167, 168, 182, 184, 186, 192 Flux, 69, 81, 82, 177, 229 Frequency, 23–34, 38, 51–53, 76, 96, 114, 116, 119, 120, 135, 136, 144–146, 177, 194, 195, 212 G Gabor wavelet, 26, 27 Grid, 1–11, 25, 66, 78, 111–115, 155 Ground state, 16, 17, 29, 56, 57, 63, 65, 84, 92, 99–101, 111, 113, 114, 119, 123, 128, 129, 157, 168, 177, 179–181, 189, 190, 193, 194, 197, 198, 211, 215 H H2+, 56, 59–64, 77, 78, 81–82, 150, 151, 163, 192–201, 203 Hamiltonian, 1, 14, 15, 39, 83, 100, 101, 110–112, 116–120, 122–126, 157, 160–165, 167–169, 214, 217 Hanning filter, 28 Harmonic, 14, 23, 32, 37–53, 59, 72, 90, 111, 136, 150 Helium (He), 14, 18–21, 103, 108, 109, 111–115, 121, 122, 129–131, 150, 151, 179, 180, 182, 189, 201–202 Hermitian, 119, 124, 161, 163, 164 Highest occupied molecular orbital (HOMO), 41, 49, 143, 144 High order harmonic generation (HOHG), 23, 24, 90, 98–100 HOHG. See High order harmonic generation (HOHG)

Index

I Imaginary-time, 193 Infrared imaging, 89–105 Intense-laser field, 18, 109, 116 Intense-laser pulses, 13–21, 23, 37, 89–105, 131 Interference, 56, 58, 62, 67–69, 104, 108, 120, 136, 140, 151, 174, 187, 192, 198–203 Interpolation, 78 Ionic, 17, 121, 136, 138, 188 Isosurfaces, 77, 78, 81, 84 Iterative, 2, 10, 110, 161 K Keldysh-parameter, 136 Krylov-subspace, 109, 110, 119, 126–128, 161, 162, 164 L Lanczos, 7, 10, 109, 110, 161–164, 184 Lanczos propagation, 161–164 Laser, 13–21, 23, 37, 55, 83, 89–105, 107–132, 135, 150 Laser induced electron diffraction (LIED), 91, 136 Lifetime, 142 Lowest order perturbation theory (LOPT), 177, 180, 185, 186 M Manybody, 90, 91 Matrix, 1, 15, 29, 43, 59, 79, 99, 110, 152, 212 Matrix-vector, 1, 2, 7–10, 156, 161 MO-ADK, 136, 139, 142 Molecular high order harmonic generation (MHOHG), 24, 25 Molecular orbital (MO), 38, 41–43, 53, 72, 136, 140, 143 Morlet wavelet, 26 Multidimensional, 3, 4, 6, 76, 79–81, 85, 151, 154, 218 Multiphoton, 13–21, 117, 150, 151, 177, 179, 196

Index

N N2, 38, 39, 43–47, 49, 51–53, 140, 143, 144, 146 Ne. See Neon Neon (Ne), 14, 16, 17, 21, 108, 116, 117, 119, 128–131 Nonadiabatic, 216–224 Non-Born-Oppenheimer, 192, 203, 216, 222–224 Nonlinear, 23, 50 Nonperturbative, 1, 23, 117 Nonproduct, 1–11 Nonsequential ionization, 188 Normal coordinate, 5–7 Nuclear, 23–24, 38, 81 Nuclear dynamics, 24, 209, 210, 216, 228 O Orbital, 15, 16, 24, 38, 41–44, 46, 49–51, 53, 72, 77, 112, 119, 123, 128, 136, 139–144, 146, 189, 212 Orbital tomography, 24, 72 Orientation, 44, 55–69, 76, 81, 192, 194, 196, 197, 199, 203 Orthogonal, 3, 15, 110, 137, 138, 141, 153–155, 158, 161–163 P Parallel-computing, 108, 114 Perturbation, 90, 93, 101, 171, 177, 180 Phase, 25, 45, 46, 49, 50, 55–69, 77, 78, 81, 90, 92, 94, 99, 104, 132, 145, 146, 165, 170, 186, 212, 219–222, 224, 227–228 Photoelectron, 90, 92, 95, 96, 98–100, 104, 151, 192, 195–200, 203, 209–216, 226–228 Photon, 13, 24, 41, 99, 109, 150 Polarization, 25, 38, 41–50, 52, 53, 57, 95, 99, 112, 151, 174, 185, 188–191, 193–203, 212–216 Polyatomic, 57, 214, 227 Power spectra, 25–29, 31, 34, 98

235

Probability, 16, 17, 19, 42, 77, 80, 84, 115, 151, 172–177, 179, 181, 184–190, 192–197, 199, 200, 202 Profile-time, 24, 27, 29–34, 40 Propagation-time, 14, 16, 20, 109–111, 124, 131, 151, 162–165, 180, 184, 193 Proton harmonics, 29–31 Proton motion, 28, 34 Pump-probe, 14, 37–39, 189, 209–214 Q Quadrature, 1–11, 153–155, 157, 167, 168, 213 Quantum chemistry, 71, 100 Quantum dynamics, 34, 72, 125 Quantum imaging, 105 Quantum scattering, 91 Quantum trajectory, 144 Quantum transition amplitude, 41 R Reaction-chemical, 89, 210 Recollision, 24, 27, 30, 31, 55, 56, 68 Rendered-imaging, 76, 77 Rendering-volume, 76–78, 81, 84 Representation, 2, 15, 44, 46, 47, 72, 73, 92, 109, 111–113, 121, 122, 126, 131, 137, 138, 140, 151–161, 163, 165, 171, 175, 184, 210, 211, 216, 217, 220, 221, 224–227 Return-electron, 27, 29–31, 34 Revival, 45, 46, 49, 52 R-matrix, 14, 18, 20, 55–69, 108, 109, 180, 185 R-matrix basis (RMB), 116–131 Rotational-wavepacket, 40, 42, 45 RSP, 163, 184, 192 S Schröedinger-equation, 1–11, 14, 15, 31, 91–93, 108, 149–203, 209 Sequential ionization, 179–180, 184, 187–191, 202

236

Index

SFA. See Strong field approximation (SFA) Short-iterative Lanczos (SIL), 161, 164 Single ionization, 14, 16–18, 96, 97, 117, 121, 130, 173, 175, 176, 187 Smolyak, 4–10 Spatial resolution, 89, 90 Spectra, 5, 10, 29, 31, 51, 90, 95, 97, 98, 104, 111, 115, 120, 136, 144, 179, 188, 197, 201, 210, 211, 214, 227 Split-operator, 25, 92, 163, 164 Split-operator propagator, 163–164 Stability, 16, 164, 223 State, electronic, 40, 57, 136, 139, 210, 212, 214, 216–220, 222, 224, 226, 229 Strong field, 40, 56, 68, 69, 91–105, 135–146 Strong field approximation (SFA), 93, 97, 98, 145 Sub-cycle, 136, 144 Symmetry, 21, 25, 34, 43, 44, 46, 49, 50, 53, 56, 57, 59–68, 81, 83, 136, 139, 141, 170, 172, 174, 187

TPDI. See Two-photon direct ionization (TPDI) Trajectory, classical, 144, 222 Trajectory, quantum, 144 Trajectory, tunneling, 138, 145 Tunnel ionization, 136–139, 142, 146 Two-photon direct ionization (TPDI), 151, 179–187, 201 U Ultrafast-laser, 89 Ultrashort-pulse, 52, 120, 150, 187, 188, 190, 202, 229 Unitary, 160, 164 Unitary propagator, 111

T TDSE. See Time dependent Schröedinger equation (TDSE) Temporal resolution, 89, 90, 97, 104 Time dependent Schröedinger equation (TDSE), 14, 31, 91–93, 108, 149–203 Time-frequency analysis, 23–34 Time-profile, 24, 27, 29–34, 39 Time propagator, 184 Total cross section, 59, 62, 64, 65, 81, 177–178, 182, 185–187

X X-ray, 73, 89, 90, 108, 150, 202 XUV, 14, 18, 20, 21, 150, 151, 180, 187, 188, 190, 192–200, 202, 203

V Velocity-map, 210–216, 227, 228 Vibrational-spectra, 5, 10 Virtual, 45, 76, 184 Visual analysis, 71–86 Visualization, 29–31, 72–85, 114 Volume-rendering, 76–78, 81, 84 W Wave packet, 96 WKB, 138

Y Yudin–Ivanov formula, 145