Light-Matter Interaction of Molecules with Intense
Ultrashort Laser PulsesbyAndres Mora
B.S., Engineering Physics, 2019
B.A., Chemistry, 2019

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofMaster of Science
Department of Physics
2019This thesis entitled:
Light-Matter Interaction of Molecules with Intense Ultrashort Laser Pulses
written by Andres Mora
has been approved for the Department of Physics

Associate Research Prof. Agnieszka Jaroń-Becker

Prof. Andreas Becker

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Mora, Andres (MS Physics)
Light-Matter Interaction of Molecules with Intense Ultrashort Laser Pulses
Thesis directed by Associate Research Prof. Agnieszka Jaroń-Becker

Over the last few decades, laser technology has greatly advanced resulting in high intensity
ultrashort laser pulses operating at many frequencies, which have made a huge impact in thefield of Atomic, Molecular and Optical Physics. As a result of these advances, new phenomena
have been discovered and studied in atoms such as high harmonic generation, multiphoton andabove-threshold ionization, among many more. While methods have been developed to study suchphenomena, such as the strong-field ionization, and the tunneling ionization models, these models
fail to properly describe the interaction between high intensity ultrashort laser pulses and molecules
due to their complex multielectron nature, and extra degrees of freedom.
In this thesis we apply time-dependent density functional theory, optical Bloch equations, andFloquet theory to study the interaction of high intensity ultrashort laser pulses with molecules in thecontext of high harmonic generation, strong field ionization and nonadiabatic electron localization.
Based on our numerical results we analyze new features in high harmonic spectra of molecules suchas the ellipticity of generated harmonics in CO2 , as previously measured in experiments, and theappearance of Mollow sidebands in the respective spectra of N2+ and C2 H4+ . We also consider
the modification of harmonic spectra by the interaction of two linearly polarized pulses and two
circularly polarized pulses interacting with molecules. We then look into the effects of a laser
induced coupling of orbitals in the context of ionization and show that as a result of the coupling,ionization contributions from inner shell orbitals are greatly enhanced. We also consider how theaddition of a second linearly polarized pulse affects the ionization of molecules. Lastly, we study theeffects of electron localization via the laser induced coupling of orbitals for the interactions between
molecules and lasers in the context of time-dependent density functional theory and Floquet theory.

Dedication

To my family and friends.

v

Acknowledgements

I would like to thank and express my gratitude to Agnieszka Jaron-Becker, my researchadvisor. Without her help and patience I would not have been able to write this thesis.
In addition I would like to thank her husband Andreas Becker for all the guidance over theyears.
I would like to also thank all of the graduate students within the group, namely Tennesse
Joyce, Brynn Reiff, Joel Venzke, Spencer Walker, Yonas Gebre, and Lauren Bauerle. Thank you
for the help and guidance as well as the good memories I have gained.
I will also like to thank the JILA computing group for all of their help whenever I needed
insight into software and computational needs.

viContents

Chapter
1 Introduction

1

2 Theoretical Methods in Ultrafast Atomic, Molecular, And Optical Physics

4

2.1

Multi-Electron Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1.1

Hohenberg-Kohn Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.2

Density-Functional Theory and the Kohn-Sham Equations . . . . . . . . . . .

6

2.2

Runge-Gross Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3

Time-Dependent Density-Functional Theory (TDDFT) . . . . . . . . . . . . . . . . . 10

2.4

Spin Polarized Time-Dependent Density Functional Theory . . . . . . . . . . . . . . 10

2.5

Implementation of Time-dependent Density functional Theory . . . . . . . . . . . . . 11

2.6

Absorbing-Boundary Conditions in Time-Dependent Density-Functional Theory . . . 12

2.7

Optical Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7.1

2.8

Derivation of Optical-Bloch Equations . . . . . . . . . . . . . . . . . . . . . . 13

Floquet Theory and The Dressed-State Picture . . . . . . . . . . . . . . . . . . . . . 15

3 Multi-electron Effects in High Harmonic Generation
3.1

19High Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1

Motivation for High Harmonic Generation . . . . . . . . . . . . . . . . . . . . 19

3.1.2

A Semi-Classical Description of High Harmonic Generation . . . . . . . . . . 21

3.1.3

Features in High Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . 22

vii
3.2

3.3

Molecular High Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1

Ellipticity of Generated High-Harmonics . . . . . . . . . . . . . . . . . . . . . 23

3.2.2

Ensemble Alignment Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.3

Ellipticity of Harmonics From CO2 . . . . . . . . . . . . . . . . . . . . . . . . 27

Mollow Sidebands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1

Adiabatic and Non-Adiabatic Behavior of Laser Induced Dipole Moments . . 33

3.3.2

Open Shell Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.3

Appearance of Mollow Sidebands in High-Harmonic Generation . . . . . . . . 36

3.3.4

High Harmonic Generation Dependence On Pulse Envelope . . . . . . . . . . 42

3.3.5

Mollow Sidebands in Optical Bloch Equations . . . . . . . . . . . . . . . . . . 46

3.3.6

Comparison Between Theoretical Methods in the Appearance of Mollow Sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.7
3.4

3.5

Time-Frequency Analysis of High-Harmonic Generation. . . . . . . . . . . . . 54

Two-Color High-Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1

Two-Color Linearly Polarized Laser Pulses . . . . . . . . . . . . . . . . . . . 57

3.4.2

Two-Color Circularly Polarized Laser Pulses . . . . . . . . . . . . . . . . . . 61

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 Multi-Electron Effects in Strong-Field Ionization
4.1

Strong-Field Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1

4.2

4.3

66

Multiphoton Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Charge-Resonance Enhanced Ionization . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1

Charge-Resonance Enhanced Ionization in Multi-Electron Molecules . . . . . 70

4.2.2

Single-Color Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.3

Two-Color Ionization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

viii
5 Electron Localization in Molecules
5.1Electron Localization in the Presence of One-Color Laser Pulses
5.1.1

5.2

Single-Color Orbital Contributions to Electron Localization in Molecules . . . 87

Individual Orbital Contributions to Two-Color Electron Localization . . . . . 96

Floquet Picture of Electron Localization . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3.1

5.4

. . . . . . . . . . . 85

Electron Localization in the Presence of Two-Color Laser Pulses . . . . . . . . . . . 94
5.2.1

5.3

84

Two-Color Charge-Resonance Localization . . . . . . . . . . . . . . . . . . . . 105

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Summary, Discussion, and Outlook

112

6.1

Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Bibliography

115

Appendix
Appendices

118

A

119

ix

TablesTable
5.1

Transition dipole moments for orbitals coupled by laser field of given wavelength for
N2+ and C2 H4+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

xFiguresFigure2.1

Schematic diagram of a two-level system with energies Eg and Ee , representing states|gi and |ei respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1

Eadweard Muybridge’s The Horse in Motion, 1878. [19] . . . . . . . . . . . . . . . . 20

3.2

Diagram of the three-step model for high harmonic generation. Adapted from [46]. . 21

3.3

A schematic of a spectrum obtained from HHG due to the interaction of an atom ormolecule with a linearly polarized laser. . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4

Diagram illustrating the method for angle averaging. . . . . . . . . . . . . . . . . . . 26

3.5

Ellipticity as a function of harmonic order for CO2 interacting with an 800 nm pulseW
14 W . Various orientation
at intensities (a) I = 1.5 × 1014 cm
2 and (b) I = 2 × 10
cm2

angles are shown: θ = 20o , θ = 40o , and θ = 80o . . . . . . . . . . . . . . . . . . . . . 28
3.6

Ellipticity of harmonics of CO2 as a function of harmonic order and alignment angle
for different orbital contributions added step by step. No ensemble average is made.
W
The applied laser field is an 800 nm linearly polarized field with intensity 1.5×1014 cm
2 . 30

3.7

Ellipticity of harmonics of CO2 as a function of harmonic order and alignment angle.
Here an ensemble average is made. The applied laser field is an 800 nm linearlyW
31
polarized field with intensity 1.5 × 1014 cm
2. . . . . . . . . . . . . . . . . . . . . . . .

3.8

Ellipticity of harmonics of CO2 as a function of harmonic order and alignment angle
for different orbital contributions added step by step. No ensemble average is made.
W
The applied laser field is an 800 nm linearly polarized field with intensity 2 × 1014 cm
2 . 32xi
3.9

Ellipticity of harmonics of CO2 as a function of harmonic order and alignment angle.
Here an ensemble average is made. The applied laser field is an 800 nm linearlyWpolarized field with intensity 2 × 1014 cm
33
2. . . . . . . . . . . . . . . . . . . . . . . . .

3.10 Time-dependent dipole moments of N2+ in the presence of a 400 nm laser field withW
34
intensity I = 2 × 1014 cm
2 . (a) Adiabatic behavior, and (b) Non-adiabatic behavior. .

3.11 Molecular orbital diagram of (a) N2+ and (b) C2 H4+ . The purple arrow indicates aresonant coupling between the orbitals (a) 3σg and 2σu and (b) 1b3g and 1b2u induced
by a 400 nm laser pulse with polarization parallel to the molecular axis, and the redarrow indicates a resonant coupling between the orbital 3σg and 1πu induced by an
800 nm laser pulse with polarization perpendicular to the molecular axis. . . . . . . 35
3.12 Potential energy surfaces of the ground state Σg and two excited states Πu and Σuof N2+ and their respective diagrams showing the electron configuration for each state. 36
3.13 HHG spectra for N2+ interacting with (a) a 400 nm laser pulse of intensity I =W
5 × 1014 cm
2 with polarization parallel to the molecular axis, and (b) an 800 nm pulseWof intensity I = 2 × 1014 cm
2 with polarization perpendicular to the molecular axis.

Note: figure (b) shows the spectrum before the frequency cutoff to make visualizationof Mollow-sidebands easier to see. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.14 Time-dependent projections between coupled orbitals of N2+ , for the cases of (a)W
coupling between 3σg and 2σu by a 400 nm pulse of intensity I = 5 × 1014 cm
2polarized parallel to the molecular axis, and (b) coupling between 3σg and 1πu by anW
800 nm pulse of intensity I = 2 × 1014 cm
2 polarized perpendicular to the molecularaxis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.15 HHG spectra for N2 interacting with (a) a 400 nm laser pulse of intensity I =W
2 × 1014 cm
2 with polarization parallel to the molecular axis, and (b) an 800 nm pulseWof intensity I = 2 × 1014 cm
41
2 with polarization perpendicular to the molecular axis. .

xii
3.16 Time-dependent projections between coupled orbitals of N2+ , for the cases of (a)W
coupling between 3σg and 2σu by a 400 nm pulse of intensity I = 2 × 1014 cm
2polarized parallel to the molecular axis, and (b) coupling between 3σg and 1πu by anW
800 nm pulse of intensity I = 2 × 1014 cm
2 polarized perpendicular to the molecularaxis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.17 Plots of the laser fields with different pulse envelopes of wavelength λ = 400 nm in
the time and frequency domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.18 HHG spectra for C2 H4+ interacting with a 400 nm pulse of peak intensity I =W45
1 × 1014 cm
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.19 HHG spectra for H2+ obtained via the optical Bloch equations. Three cases areWshown: (a) R = 2.12 a.u. with laser parameters λ = 1064 nm, I = 1014 cm
2 , (b)W
R = 5 a.u. with λ = 1064 nm at I = 4 × 1013 cm
2 and (c) R = 10 a.u. with λ = 1064
W
nm and I = 1013 cm
49
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.20 HHG spectra for N2+ obtained via the optical Bloch equations. The laser pulse hasa wavelength of 400 nm and the intensity of the pulse is increased for each panel. . . 51
3.21 HHG spectra for N2+ obtained via the optical Bloch equations. The laser pulse hasa wavelength of 800 nm and the intensity of the pulse is increased for each panel. . . 52
3.22 Predicted (blue dashed line) vs. Numerically obtained (red circles) frequencies of
Mollow sidebands appearing in HHG spectra obtained from optical Bloch equations(panels a and b) and TDDFT (panels c and d) for the cases were the laser wavelengthis 400 nm and 800 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.23 Wavelet spectrum for N2+ interacting with a 400 nm laser pulse of intensity I =W
2 × 1014 cm
2 with polarization direction oriented perpendicular to the molecular axis.

Panel (a) shows the spectrum over the total length of the laser propagation, and(b) shows a zoom in from 5 to 10 femtoseconds and has labels showing where thetrajectory of the electron is short or long during the propagation of the laser.

. . . 55

xiii
3.24 Wavelet spectrum for N2+ interacting with a 400 nm laser pulse of intensity I =W
2 × 1014 cm. . . 57
2 with polarization direction oriented parallel to the molecular axis.

3.25 Schematic diagram for the N2+ interacting with (a) a 400 nm pulse with fixed polarization parallel to the molecular axis and an 800 nm pulse varied in angle θ = 0o toθ = 90o with respect to the molecular axis, and (b) an 800 nm pulse with polarization fixed perpendicular to the molecular axis and a 400 nm pulse with polarization
varied from θ = 0o to θ = 90o with respect to the molecular axis. . . . . . . . . . . . 58
W
3.26 HHG spectra for a 400 nm pulse at 2 × 1014 cm
2 polarized along the molecular axisWowith an 800 nm pump pulse at 1×1013 cm
2 with polarization direction rotated from 0

to 90o with respect to the molecular axis. Note: Intensities are scaled for comparison. 59
W
3.27 HHG spectra for a 400 nm pulse at 2 × 1014 cm
2 polarized along the molecular axis.

60

W
3.28 HHG spectra for an 800 nm at 2 × 1014 cm
2 probe pulse polarized along the molecularWaxis with an 400 nm pump pulse at 1 × 1013 cm
2 with polarization direction rotatedfrom 0o to 90o with respect to the molecular axis. Note: Intensities are scaled forcomparison.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.29 Plots of bichromatic laser fields of wavelengths 400 nm and 800 nm with intensity
W
I = 1 × 1013 cm
2 for cases where the fields are (a) co-rotating and (b) counter-rotating. 62

3.30 HHG spectra of H2 interacting with bichromatic circularly polarized pulses of waveW
lengths 400 nm and 800 nm with intensity I = 1 × 1013 cm
2 for the cases where thefields are (a) co-rotating and (b) counter-rotating. . . . . . . . . . . . . . . . . . . . 63
4.1

Figures depicting multiphoton ionization where (a) an electron is ionized into thecontinuum, and (b) an electron is excited into a higher energy state via multiphoton
absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2(a) Ionization rate of H2+ as a function of internuclear distance, when interactingWwith a linearly polarized 1064 nm laser with intensity I = 1014 cm
2 and (b) the

energy levels of H2+ as a function of internuclear distance. Adapted from [43] and [28]. 69

xiv
4.3

Ionization probability for two cases: (a) N2 in the presence of a 400 nm laser pulseWwith intensity I = 2 × 1014 cm
2 and polarization along the molecular axis, and (b)W
N2+ in the presence of a 600 nm laser pulse with intensity I = 2 × 1014 cm
2 andpolarization along the molecular axis. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4

Ionization probability for N2 while under the presence of an 800 nm laser pulse withW
72
intensity I = 2 × 1014 cm
2 and polarization perpendicular to the molecular axis. . . .

4.5

Ionization probability for N2+ while under the presence of a 400 nm laser pulse withW
intensity I = 2 × 1014 cm
74
2 and polarization parallel to the molecular axis. . . . . . .

4.6

Ionization probability for N2+ while under the presence of an 800 nm laser pulse withW
intensity I = 2 × 1014 cm
74
2 and polarization perpendicular to the molecular axis. . . .

4.7

Ionization probability for C2 H4+ while in the presence of a 400 nm laser pulse withW
intensity I = 1 × 1014 cm
75
2 and polarization perpendicular to the molecular axis. . . .

4.8

Total ionization probability as a function of polarization alignment for cases (1) and(2) respectively. In (a) the polarization of the 800 nm pulse is varied from 0o to 90owith respect to the molecular axis, and in case (b) the polarization of the 400 nm
pulse is varied from 0o to 90o with respect to the molecular axis. . . . . . . . . . . . 76

4.9

Ground state orbital projections onto time-dependent orbitals for the cases of N2+
Winteracting with a 400 nm pulse of intensity I = 2 × 1014 cm
2 with fixed polarization
W
along the molecular axis and an 800 nm pulse of intensity I = 1 × 1014 cm
2 withpolarization (a) at angle θ = 0o and (b) polarization at an angle θ = 90o withrespect to the molecular axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.10 Total ionization for each orbital as a function of angle θ for the case where N2+ interacts with two pulses, a 400 nm one with polarization fixed parallel to the molecularW
14 W
axis and intensity I = 2 × 1014 cm
2 , and an 800 nm one of intensity I = 1 × 10
cm2with polarization angle varied from 0o to 90o with respect to the molecular axis. . . 79

xv
4.11 Ground state orbital projections onto time-dependent orbitals for the cases of N2+
Winteracting with an 800 nm pulse of intensity I = 2 × 1014 cm
2 with fixed polarizationW
perpendicular to the molecular axis and a 400 nm pulse of intensity I = 1 × 1014 cm
2with polarization (a) at angle θ = 0o and (b) polarization at an angle θ = 90o withrespect to the molecular axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.12 Ionization probability for each orbital as a function of angle θ for the case whereN2+ interacts with two pulses, a 800 nm one with polarization fixed perpendicularW
to the molecular axis and intensity I = 2 × 1014 cm
2 , and a 400 nm one of intensityWoo
I = 1 × 1014 cm
2 with polarization angle varied from 0 to 90 with respect to themolecular axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.13 Ground state orbital projections onto time-dependent orbitals for the cases of N2+
Winteracting with an 800 nm pulse of intensity I = 2 × 1014 cm
2 with fixed polarizationW
perpendicular to the molecular axis and a 400 nm pulse of intensity I = 1 × 1014 cm
2with polarization at angle θ = 60o with respect to the molecular axis for both thespin up and down orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1

Visualization of total electron localization of N2+ along the molecular axis at different
laser wavelengths as a function of time: (a) 400 nm laser field of intensity I =W
14 W . In all cases the
2 × 1014 cm
2 , and (b) 800 nm laser field of intensity I =2 × 10
cm2

orientation of the polarization is parallel to the molecular axis. . . . . . . . . . . . . 86
5.2

Visualization of total electron localization of C2 H4+ interacting with a 400 nm laserW
field of intensity I = 1 × 1014 cm
2 with polarization parallel to the molecular axis.

The direction is parallel to the C-C bond.
5.3. . . . . . . . . . . . . . . . . . . . . . . 87

Visualization of orbital contributions to electron localization of N2+ in the presenceWof a 400 nm laser pulse with intensity I = 2 × 1014 cm
2 and polarization along themolecular axis. The orbitals are ordered, (a) 2σg , (b) 2σu , (c) 1πu , and (d) 3σg . . . . 88

xvi
5.4

Visualization of electron localization for the independent orbitals of N2+ in the presW
ence of a 400 nm laser pulse with intensity I = 2 × 1014 cm
2 and polarization along

the molecular axis. The orbitals are ordered, (a) 2g , (b) 2σu , (c) 1πu and (d) 3σg forthe spin up electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5

Time-dependent energy expectation values of each molecular orbital of N2+ during
the interaction with a 400 nm laser pulse. . . . . . . . . . . . . . . . . . . . . . . . . 90

5.6

Time-dependent energy expectation values of each molecular orbital of C2 H4+ during
the interaction with a 400 nm laser pulse. . . . . . . . . . . . . . . . . . . . . . . . . 91

5.7

Visualization of electron localization for the individual orbitals of C2 H4+ while in theWpresence of a 400 nm laser pulse with intensity I = 1 × 1014 cm
2 and polarization

along the molecular axis spin down electrons. . . . . . . . . . . . . . . . . . . . . . . 92
5.8

Visualization of electron localization for the individual orbitals of C2 H4+ while in theWpresence of a 400 nm laser pulse with intensity I = 1 × 1014 cm
2 and polarization

along the molecular axis for spin up electrons. . . . . . . . . . . . . . . . . . . . . . . 93
5.9Visualization of electron localization of the total density of N2+ interacting with aW
400 nm pulse of intensity I = 2×1014 cm
2 with polarization fixed along the molecularW
axis, and an 800 nm pulse of intensity I = 1 × 1014 cm
2 with polarization varied from

θ = 0o to θ = 90o with respect to the molecular axis. . . . . . . . . . . . . . . . . . . 96
5.10 Visualization of orbital contributions to electron localization of N2+ in the presenceWof a 400 nm laser pulse with intensity I = 2 × 1014 cm
2 and an 800 nm pulse withW
intensity I = 1 × 1014 cm
2 both with polarization along the molecular axis. The

orbitals are ordered, (a) 2σg , (b) 2σu , (c) 1πu , and (d) 3σg . . . . . . . . . . . . . . . 98
5.11 Visualization of orbital contributions to electron localization of N2+ in the presenceWof a 400 nm laser pulse with intensity I = 2 × 1014 cm
2 polarized along the molecularWaxis and an 800 nm pulse with intensity I = 1 × 1014 cm
2 with polarization at anangle θ = 50o with respect to the molecular axis. The orbitals are (a) 2σg , (b) 2σu ,(c) 1πu , and (d) 3σg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

xvii
5.12 Visualization of orbital contributions to electron localization of N2+ while in theWpresence of a 400 nm laser pulse with intensity I = 2 × 1014 cm
2 polarized along theWmolecular axis and an 800 nm pulse with intensity I = 1 × 1014 cm
2 with polarization

at an angle θ = 90o with respect to the molecular axis. The orbitals are (a) 2σg , (b)2σu , (c) 1πu , and (d) 3σg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.13 Visualization of electron localization for the 1πu orbital projected along the directionof the orbitals rather than the molecular axis for the cases when the 800 nm pulseWooof intensity 1 × 1014 cm
2 is oriented at angles (a) θ = 0 and (b) θ = 90 with respect

to the molecular axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.14 Electron localization for H2+ with internuclear distance R = 7 a.u. for the cases of(c) full 3-D TDSE, and (d) Floquet theory. Adapted from [44]. . . . . . . . . . . . . 106
5.15 Quasienergies 1 and 2 as functions of laser intensity. . . . . . . . . . . . . . . . . . 108
5.16 Time-dependent left- and right- populations for N2+ using the analytical solution
obtained from Floquet theory for the cases where the laser is (a) at 400 nm withW
16 W . . . . 109
intensity I = 1.5 × 1015 cm
2 and (b) at 800 nm with intensity I = 1.5 × 10cm2

5.17 Left- and right- time-dependent populations for C2 H4+ using the analytical solution
obtained from Floquet Theory for the case where the laser is 400 nm with intensity
W
I = 1.5 × 1015 cm
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Chapter 1

Introduction

In the world of physics there exist many disciplines that aim to help us understand the
behavior of nature ranging from phenomena related to the cosmos, to the behavior of sub-atomic
particles. One of those disciplines is Atomic, Molecular and Optical Physics (AMO Physics) which
includes the study of the interaction between matter and light. Within itself, AMO is a broad areaof physics that spans a variety of topics such as laser cooling and atom trapping, quantum optics,
and the study of ultrafast processes in atoms and molecules by ultrashort laser pulses.
Ever since the laser was invented in 1960 it has become a fundamental tool to study the dynamics
and response of atomic and molecular systems which in turn has expanded our knowledge of how
Wmatter interacts with light. We know that at low laser intensities I < 1013 cm
2 the dominatingprocess is related to emission or absorption of few photons and can be properly analyzed with theuse of perturbation theory. However, with technological advances in laser science we were able toincrease the intensity threshold beyond what perturbation theory can explain. In turn, a new realmWof light-matter interaction was opened. At intensities of I ≥ 1013 cm
2 the interaction between theelectrons in a system with the electric field of the laser pulse and the Coulomb interaction becomecomparable, leading to interesting highly nonlinear high order processes due to the absorption andemission of many photons, such as high harmonic generation (HHG), and above threshold ionization(ATI).
So far, the interaction between atomic systems and intense laser pulses has been extensively studied
and considered to be well understood. Many theoretical methods exist to describe such interactions

2and the results have so far agreed with those obtained via experiments. High-order processes suchas ATI have been successfully modeled by methods such as the Keldysh, Faisal, Reiss theory (KFR)
and quasi-static tunneling models [1,2,3], where it is assumed that only the most loosely boundelectron responds to an external field in an adiabatic manner. This is known as the single-active
electron approximation (SAE) and so far it has been shown that it works well for atomic systems
due to the large energy spacing between atomic orbitals.
However, unlike for atoms, the interaction between molecular systems and strong laser fields is lesswell understood. This is largely due to the ever increasing complexity brought upon by multielectron
effects, rotational and vibrational degrees of freedom, and multi-center nuclear potential surfaces,
among many others [4,5,6,7,8,9,10]. As a result, many previous approximations that are usefulwhen studying intense laser light interaction with atoms no longer apply and the theories must
be modified, as approximations like SAE begin to break down due to the closer spacing in energy
levels, and due to the multielectron effects present.
In this thesis we consider ab-initio methods and certain approximation models that allow us tounderstand the more complex dynamics occurring in the interaction of intense lasers and molecules.
The thesis is organized as follows: In chapter 2 we introduce the theoretical methods we use to
study our systems. These include time-dependent density functional theory (TDDFT), the optical
Bloch equations, and lastly Floquet theory.
In chapter 3 we investigate multielectron effects in high-harmonic generation, in the cases where
either one or two lasers are interacting with molecular systems. We describe the differences thatarise in HHG from molecules compared to what has been observed in atoms. We also analyze
the appearance of Mollow sidebands present in HHG spectra as a result of laser induced coupling
between molecular orbitals in N2+ and C2 H4+ by utilizing TDDFT and optical Bloch equations, andcompare both results from theories. We then analyze the ellipticity of harmonics from CO2 which
agrees with previously conducted experiments. We end the chapter by considering the interactionof bichromatic circularly polarized pulses interacting with H2 and describe the differences that arise
in HHG spectra as a result of the fields used.

3
In chapter 4 we consider the effects of laser induced resonances in multiphoton ionization, and study
how such couplings of orbitals can enhance ionization from inner shell orbitals in molecules like N2+
and C2 H4+ . We study the case for both one color interactions as well as two color interactions andexplain the differences between them.
We then analyze nonadiabatic electron localization arising from the coupling of orbitals in chapter
5, and show how such dynamics can be visualized from the models we utilize such as TDDFT and
Floquet theory. We again consider the cases of one and two-color laser interactions and explain
how they both differ.
Lastly, we summarize our findings in chapter 6.

Chapter 2Theoretical Methods in Ultrafast Atomic, Molecular, And Optical Physics

2.1

Multi-Electron Systems
Atoms and molecules are multielectron systems. A system of N electrons interacting witha laser field can be described by its wavefunction Ψ(r, t) (r = [r1 ,...,rN ]) which obeys the timedependent Schrödinger equation (note: we will be working in atomic units, e = m = ~ = 1, in thewhole thesis):ĤΨ(r, t) = i∂Ψ(r, t)
,
∂t

(2.1)

where the Hamiltonian H is written in the form
H(r, t) = T (r) + Vee (r) + Vne (r) + Vlaser (r, t),(2.2)

with T (r) being the kinetic energy of the electronsN

1X 2
T (r) = −
∇i .
2

(2.3)

i=1

The second term denoted Vee (r) is the electron-electron interaction which has the form
Vee (r) =

N
X
i<j=1

1
,
|ri − rj |(2.4)

Vne (r) corresponds to the nuclear-electron interaction given by
Vne (r) = −

X
i,k

Zk
,
|ri − Rk |

(2.5)

5
where Rk is the position and Zk is the charge of the k th nucleus. The last term Vlaser (r, t) includes
the interaction between the electrons and the applied laser field
Vlaser (r, t) = Eo f (t)sin(ωt)

NX!
ri · n̂ ,

(2.6)

i=1

where Eo is the amplitude of the electric field, f (t) is the envelope of the pulse, ω is the frequencyof the laser, and n̂ is the polarization direction of the field.
While the multielectron Schrödinger equation describes the system fully, it is computationallyexpensive to solve for Ψ(r, t) in the case of larger systems. Due to this, it is indeed currently
impossible to exactly solve the multielectron Schrödinger equation for systems with more than
two electrons interacting with intense laser pulses in a reasonable amount of time. Fortunately,
there exists an alternative approach to approximately solve multielectron systems known as Density
Functional Theory (DFT).

2.1.1

Hohenberg-Kohn Theorem

The basis of DFT originates from the Hohenberg-Kohn (HK) theorem [11,12,13] which statesthat the external potential given by:V̂ext = V̂ + Ŵ =N
X

v(ri ) +

i=1

N
1 X
1
,
2
|ri − rk |(2.7)

i,k,i6=kof the system is a functional of the electron density ρ(r), and that the ground state of a system canbe determined by its electronic density given byρgs (r) = N

XZZ
dx2 ...

dxN |Ψgs (r, σ, x2 , ..., xN )|2 .

(2.8)

σ

We can show this by considering a ground state wavefunction ψ(r) with an electron density ρ(r)for an external potential V (r), with a Hamiltonian H and energy E, and another ground state
wavefunction ψ 0 (r) corresponding to a second external potential V 0 (r) with a Hamiltonian H 0 andenergy E 0 , which gives the same electron density ρ(r). Due to the variational principle we can then6
say that:hψ|H|ψi <hψ 0 |H|ψ 0 i = hψ 0 |H 0 |ψ 0 i + hψ 0 |H − H 0 |ψ 0 i
Z0= E + (V (r) − V 0 (r))ρ(r)dr,(2.9)

which gives the following:
Z

0

E<E +

(V (r) − V 0 (r))ρ(r)dr,(2.10)

and also for the case of the primed and unprimed indices being reversed:

0

Z

E <E+

(V 0 (r) − V (r))ρ(r)dr.

(2.11)

If we then add equations (2.10) and (2.11) we get the following contradiction:

E + E0 < E + E0,(2.12)

so we see that the external potential V (r) is indeed a unique functional of the ground state electron
density ρ(r). Or in other words, the ground state wavefunction Ψgs (r) is a functional of the groundstate density ρgs (r), so it can be written as
Ψgs (r) = Ψgs [ρgs (r)],(2.13)

where square brackets denote functional dependence. This then leads to the fact that any observablecorresponding to the ground state is also a functional of the density ρgs (r)O[ρgs (r)] = hΨgs [ρgs (r)]|O|Ψgs [ρgs (r)]i.

2.1.2(2.14)

Density-Functional Theory and the Kohn-Sham Equations

Based on the HK theorem it has been established that the observables of a system arefunctionals of the density ρgs (r), we therefore need to consider ways to solve for the density. Thiscan be done using the Kohn-Sham equations.

7
Consider a system of noninteracting particles with a Hamiltonian given by:Ĥs = T̂ + V̂s =

N
X

!
∇2i
−
+ vs (ri ) ,2

i=1(2.15)

where Vs is an arbitrary external potential. We can write the associated total energy of the system
as a functional given by:Z
Evs [ρ] = Ts [ρ] +d3 rρ(r)vs (r),(2.16)

where ρ(r) is the electron density of the system.
Since the particles are non-interacting, a Slater determinant can be used to approximate the groundstate wavefunction Ψgs (r) of the N-particle system:
φ1 (r1 )φ1 (r2 )φ1 (r3 )...

φ1 (rN )φ2 (r1 )1Ψgs (r) = √..
N!
.

φ2 (r2 )..
.

φ2 (r3 )..
.

...

φ2 (rN )..
.

φN (r1 ) φN (r2 ) φN (r3 ) . . .

(2.17)φN (rN )where the orbitals φi (r) are solutions to the single particle Schrödinger equation1− ∇2 φi (ri ) + Vs (ri )φi (ri ) = i φi (r)i ,2(2.18)and the ground state density of the N electron system is given byρgs (r) =NX|φi (r)|2 .

(2.19)

i=1

In this case the HK theorem applies and there exists a one to one mapping between the groundstate density ρ(r) and the potential Vs (r). The energy of the system is given byZ
Es,gs [ρ] = Ts [ρgs ] +drρ(r)vs (r) =

N Z
X
i=1

1
drφ∗i (r)(− ∇2i )φi (r) +2Z
drρ(r)vs (r).

(2.20)

Now consider a system of interacting particles with an external potential denoted by Vext (r). The
ground state density ρgs (r) for this system can be obtained from the ground state density of a system

8of noninteracting particles with a potential denoted by Vs (r). This turns the problem of solving
a multielectron Schrödinger equation into multiple noninteracting single particle Schrödinger-likeequations. In turn, this greatly reduces the computational costs needed to numerically solve such
systems.
For a system of interacting particles the energy functional is given byE[ρgs ] = Ts [ρgs ] + VH [ρgs ] + Exc [ρgs ] + V [ρgs ].

(2.21)The second term VH [ρgs ] is the Hartree energy (classical Coulomb energy) which is given by1
VH [ρgs ] =
2

Z

Z
dr

dr0

ρgs (r)ρgs (r0 ),|r − r0 |(2.22)

while Exc [ρgs ] is the exchange correlation energy:

Exc [ρgs ] = T [ρgs ] − Ts [ρgs ] + U [ρgs ] −

1
2

Z

Z
dr

dr0

ρgs (r)ρgs (r0 ).
|r − r0 |(2.23)Taking the derivative of the exchange correlation energy, gives the exchange correlation potential
Vxc [ρgs ] =

δExc
.
δρgs

(2.24)

This potential has an unknown form, but can be approximated for different systems.
For a system of interacting particles the variational principle with respect to the electron density
results in0=δTs [ρgs ] δV [ρgs ] δVH [ρgs ] δExc [ρgs ]
+
+
+
.
δρgs
δρgs
δρgs
δρgs

(2.25)

Similarly, for a system of noninteracting particles the variational principle with respect to density
yieldsδExc [ρgs ]δTs [ρgs ] δVs [ρgs ]
=0=
+
.
δρgs
δρgs
δρgs

(2.26)

Comparing the variational principle for both the interacting and noninteracting system, we getvs (r) = vext (r) + vH (r) + vxc (r),(2.27)

9to achieve both minimizations resulting in the same electron density ρs (r) = ρ(r). This implies
that the density of an interacting system within an external potential vext (r) can be determinedby solving the Schrödinger equation for many non-interacting particles in a potential vs (r), only
if the exchange-correlation potential is known. Based on this assumption, the orbitals of thenoninteracting system can be obtained by solving the Kohn-Sham (KS) equation, given by1− ∇2 φi (r) + vs (r)φi (r) = i φi (r),2(2.28)and the electron density of the interacting system can then be obtained via
ρ(r) =NX|φi (r)|2 .

(2.29)

i=1

The KS equations must be solved self-consistently because both the terms vH (r) and vxc (r) dependon the density ρ(r), which depends on the orbitals φi (r), which in turn depend on vs (r). Due tothis, the calculation starts with an initial guess for the electron density ρ(r), then the potential vs (r)is obtained, which leads to a solution of the KS equation φi (r), which then yields a new density
ρ(r). This process is repeated until the system converges to the correct ground state density. Once
convergence is obtained, then observables for the system can be calculated.

2.2

Runge-Gross Theorem
The Schrödinger equation maps an external potential v(r, t) to a time-dependent wavefunction

Ψ(r, t) with a given initial state Ψo . However, there exists a second mapping between the density
ρ(r, t) and Ψ(r, t), which leads to the Runge-Gross theorem [11,12,13] which states that given
two electron densities ρ(r, t) and ρ0 (r, t), both obtained from an initial wave function Ψo which is
influenced by two different potentials v(r, t) and v 0 (r, t), will become different after a given time to .
This implies that there is a one-to-one correspondence between densities and potentials obtained
from a given initial wavefunction.

ρ(r, t) → V (r, t) → Ψ(r, t).

(2.30)

10

2.3

Time-Dependent Density-Functional Theory (TDDFT)
An extension of DFT beyond the static potential is TDDFT, which considers the systembeing subjected to a time dependent external potential, typically for laser-molecule interactions, ofthe form of equation (2.6). There are several assumptions made in this description(1) The field is treated classically since at high intensities the field has a large photon density.
(2) The dipole approximation is used, which holds when the wavelength of the laser field islarger than the size of the system.
The electromagnetic field is treated as purely electric field, and its magnetic component can beneglected for the considered range of laser frequencies and intensities.
In the case of a time dependent external potential, the Runge-Gross theorem states that there
is a one-to-one mapping between the electron density ρ(r, t) and the external potential v(r, t).
Therefore, instead of solving a many body time dependent Schrödinger equation, the density canbe found by using the time dependent Kohn-Sham equation given by1∂− ∇2 φi (r, t) + vks (r, t)φi (r, t) = i φi (r, t),2∂t(2.31)

where for the time-dependent case, the external potential includes the potential from the laser field
and is of the form:vks (r, t) = vext (r) + vH (r) + vxc (r) + vlaser (r, t).

(2.32)

By solving the time-dependent Kohn-Sham equations the time-dependent density is then given byρ(r, t) =NX|φi (r, t)|2 ,(2.33)

i=1

where φi (r, t) is the ith time-dependent Kohn-Sham orbital.

2.4

Spin Polarized Time-Dependent Density Functional Theory
A useful feature of DFT and TDDFT is that spin-dependent cases can be considered [13].

This can be implemented in both the time-independent and time-dependent Kohn-Sham equations

11
by adding a spin index σ =↑, ↓, which gives"1− ∇2 + vσ (r, t) +2Z#

3 0 ρ(r’, t)d r|r − r’|+ vXC,σ [ρ↑ , ρ↓ ](r, t) φi,σ (r, t) = i∂
φi,σ (r, t),∂t(2.34)

where upon solving, the total density is given by:ρ(r, t) =Xρσ (r, t) =

σ=↑,↓

2.5

N
X X

|φi,σ (r, t)|2 .

(2.35)σ=↑,↓ i=1

Implementation of Time-dependent Density functional Theory
The solution of the KS equation can be found numerically by discretizing both the time andspatial components via some finite difference method. This converts the KS equation into a matrix
equation. To obtain the initial wavefunction, the eigenvalue problem is solved self-consistently
with an initial guess for the ground state needed. After the ground state of the system has beenobtained, the wavefunction can be propagated in time. There are a variety of methods available tonumerically solve the KS equations [14].
Given some initial wavefunction for the j th orbital ψj (t = 0), time evolution states that after sometime ∆t the wavefunction is given by:
ψj (t + ∆t) ≈∆t
1 − i ∆t
2 H(t + 2 )∆t
1 + i ∆t
2 H(t + 2 )ψj (t).

(2.36)

This time propagation scheme is known as Crank-Nicolson and it is what was used to generate theresults presented in this thesis. The Hamiltonian operator H(t) can be evaluated by using a finite
difference scheme [14] where the wavefunction of the system is defined on a discrete spatial andtemporal grid.
For the results presented in this thesis, all of the simulations were performed using the ab-initio
TDDFT software Octopus [15,16,17]. The simulations for the molecules were performed at theirequilibrium bond length with a trapezoidal or sin2 (.) pulse shape at wavelengths 400 nm, 600 nm,

12
and 800 nm. In addition, the simulation grid used is spherically shaped with a varying radiusdepending on intensity of the pulse. The exchange correlational functional used in the simulations
was the local-density approximation (LDA) [17] and an average-density self-interaction correction
was applied [18].

2.6Absorbing-Boundary Conditions in Time-Dependent Density-Functional
TheoryA property of TDDFT is that the calculated time-dependent wavefunctions conserve theirnorm for all times, which can be given by:Z

d3 rρ(r, t) = N,

(2.37)

where N is the norm, or number of particles. This then implies probability conservation, meaningthat probability density cannot be created or destroyed.
In many situations in which a system is driven by a strong electric field, as in this thesis, ionization occurs, and boundary conditions at the edge of the numerical grid have to be considered.
Numerically speaking, if one is calculating the dynamics of a system in the presence of a strong
electric field via TDDFT, the outgoing wavefunctions driven by the field, will indeed return, if theboundary conditions are reflective. Fortunately, absorbing boundary conditions can be applied to
the real-space grid in which the calculation is taking place. This is done by adding a negative shortrange complex valued potential to the total potential in the asymptotic region which is located far
away from the system that is being ionized. If the outgoing wavepacket hits the absorbing boundary and is absorbed, the norm of the wavefunction will no longer be conserved and the outgoing
portion of the wavefunction is assumed to describe ionization.

2.7

Optical Bloch Equations
While TDDFT captures the full 3-dimensional dynamics of a system in the presence of astrong electric field, it can be computationally expensive and time-consuming. Therefore, it is13
useful to consider simple models which can capture some of the dynamics that one can investigate
using TDDFT. We apply the optical-Bloch equations, which are useful when analyzing the resonant
dynamics in two-level systems. Unfortunately, when using such simple models many aspects of the
dynamics are not seen. In the case of high-harmonic generation, which we will study closely in thisthesis, the optical Bloch equations fail to capture the ionization process of HHG as there is no way
to correctly include the continuum into the picture in a straight forward manner. Yet, the simple
picture gives insight into some of the dynamics we will investigate.

2.7.1

Derivation of Optical-Bloch Equations

Consider a two level system with a ground state labeled |gi and an excited state labeled |ei
with energies Eg and Ee , respectively. Figure 2.1 shows a diagram of such a system.

Figure 2.1: Schematic diagram of a two-level system with energies Eg and Ee , representing states|gi and |ei respectively.

For such a system we can write the Hamiltonian as:
Ĥ = ĤA − d̂ · Ê(ro ),(2.38)

where ĤA is the unperturbed atomic Hamiltonian of the system, dˆ is the transition dipole momentof the atom and Ê(ro ) is the electric field operator evaluated at the position of the dipole ro .
We make the assumption that the electric field is monochromatic and has a frequency close to the14transition frequency between the states |gi and |ei. We then find the matrix elements of the various
atomic operators which are:he|ĤA |ei = Ee ,hg|ĤA |gi = Eg
(2.39)

he|ĤA |gi = 0,

hg|ĤA |ei = 0,he|d̂|ei = 0,hg|d̂|gi = 0

and(2.40)
hg|ĤA |ei = (deg )∗ ,he|d̂|gi = deg ,where deg and (deg )∗ are the transition dipole moments between states and can be written as:

deg = dr + idi ,dge = dr − idi ,(2.41)

where dr and di are the real and imaginary components of the dipole moment. From here we cannow express the dipole operator in terms of Pauli matrices:d̂ = dr σ̂x − di σ̂y .

(2.42)

Similarly we can write the atomic Hamiltonian in terms of Pauli operators given by:
1
1
ĤA = (Ee + Eg )Iˆ + (Ee − Eg )σˆz ,22(2.43)

where Iˆ is the identity matrix. The full form of the Hamiltonian given in equation (2.38) is then:11Ĥ = (Ee + Eg )Iˆ + (Ee − Eg )σˆz − (dr · Ê)σ̂x + (di · Ê)σ̂y .
22(2.44)

If we consider the Heisenberg equation for some operator Ô given by:˙
i~Ô = [Ô, Ĥ],(2.45)

we then obtain three equations for the Pauli matrices given by:2σ̂˙ x (t) = −ωo σ̂y (t) + [di · Ê(t)]σ̂z (t),~(2.46)2σ̂˙ y (t) = ωo σ̂x (t) + [dr · Ê(t)]σ̂z (t),~(2.47)

1522σ̂˙ z (t) = − [dr · Ê(t)]σ̂y (t) − [di · Ê(t)]σ̂x (t),~~(2.48)

where ωo is the transition frequency between states:ωo =Ee − Eg.
~(2.49)We then introduce the expectation values of the Pauli operators given by:si (t) =< σ̂i >, i = x, y, z,(2.50)

so then our coupled equations can be written as:
2ṡx (t) = −ωo sy (t) + [di · Ê(t)]sz (t),~(2.51)2ṡy (t) = ωo sx (t) + [dr · Ê(t)]sz (t),~(2.52)22ṡz (t) = − [dr · Ê(t)]sy (t) − [di · Ê(t)]sx (t),~~(2.53)

which are the optical Bloch equations that describe the interaction of a two-level atom with anelectric field.

2.8

Floquet Theory and The Dressed-State Picture
There is yet another theoretical method that gives insight into the interaction of atoms andmolecules with laser pulses, known as Floquet Theory. In this method we consider an intense fieldinteracting with a system, whose interaction energy is labeled Hint (t), so the full time-dependent
Hamiltonian is of the form:

H(t) = Ho + Hint (t),(2.54)

16
where Ho is some field free Hamiltonian. We consider that the laser pulse is a CW monochromaticfield of intensity I such that Hint (t) is periodic in time. This then implies that the time-dependent
Hamiltonian in equation (2.54) is also periodic in time:

H(t) = H(t + τ ),(2.55)

where τ is the optical period of the applied laser field, equal to 2π/ω, where ω is the optical frequencyof the pulse. With this, the time-dependent Schrödinger equation (TDSE) can be written as:i∂|Ψ(t)i = H(t)Ψ(t) = [Ho + Hint (t)]|Ψ(t)i,∂t(2.56)

where Ho is the field-free Hamiltonian. According to Floquet theory, the solution to the TDSE forthis system can be written as:|Ψ(t)i = e−iEF t |PE (t)i,(2.57)

where, EF is the Floquet quasienergy, and |PE (t)i is some state vector that is periodic in time:|PE (t + τ )i = |PE (t)i.

(2.58)

We can then find the state vector |PE (t)i and the quasienergy EF by considering the TDSE to bea time-independent problem and write |PE (t)i as:|PE (t)i =∞Xe−inωt |Fn (E)i.

(2.59)n=−∞

Then the solution to the TDSE given in equation (2.57) can be written as:|Ψ(t)i = e−iEF t∞Xe−inωt |Fn (E)i.

(2.60)n=−∞

We can also expand the interaction term Hint (t) as a Fourier series resulting in:
Hint (t) =

∞
X
n=−∞

(Hint )n e−inωt .

(2.61)

17
We substitute equations (2.60) and (2.61) into the TDSE given in equation (2.56) and obtain thefollowing form:(Ho − nω)|F (E)n i +∞X(Hint )n−k |F (E)k i = E|F (E)n i,(2.62)k=−∞where n = 1, 2, 3, ... denotes the number of photons absorbed or emitted by the system. We canthen write the above equation in a simpler form as a time-independent Schrödinger equation:

HF |F (E)i = E|F (E)i,(2.63)

where HF is the Floquet Hamiltonian given by a matrix whose elements are time-independent
operators, and |F (E)i are the Floquet vectors. Recall that we assumed that the laser field was
monochromatic with frequency ω. We can therefore write our interaction Hamiltonian Hint (t) in
the form:

Hint (t) = H+ e−iωt + H− eiωt ,(2.64)

where H+ and H− are time-independent Hermitian operators. With this we can rewrite equation(2.62) in the form:(Ho − nω)|F (E)n i + H+ |F (E)n−1 i + H− |F (E)n+1 i = E|F (E)n i,(2.65)

which tells us that the Floquet Hamiltonian will be of the form:




..

.



· · ·
HF = 

· · ·


H+ Ho − (n − 1)ω
0

H+

H−

0

Ho − (n − 1)ω H−· · ·
.

0


..
.

(2.66)

This form of the Floquet Hamiltonian has a simple interpretation, the harmonic components|F (E)n i describe the state that the system becomes upon exchanging n-photons with the applied18
laser field. We refer to these Floquet states as ”dressed” states which are obtained via diagonalization. We apply this method to charge resonance enhanced ionization related to localization inchapter 5 of this thesis.

Chapter 3

Multi-electron Effects in High Harmonic Generation

3.1

High Harmonic Generation
High harmonic generation (HHG) is a phenomenon that has been observed and studied forseveral decades now. While many studies of HHG have been focused on atoms, it has been shownthat HHG can in fact occur in molecules.
In this chapter, we will study the cases of laser induced resonant coupling in molecules
which results in non-adiabatic electron dynamics that leads to Mollow sidebands appearing in
HHG spectra. We consider the cases of a single linearly polarized laser pulse interacting with thesystem, as well as the case of multiple linearly and circularly polarized pulses interacting with amolecule. In addition, we study the cases where generated harmonics are elliptically polarized.

3.1.1

Motivation for High Harmonic Generation

The demand for efficient coherent light generation at high frequencies has been somethingscientists and engineers have pushed for since the maser was first invented. In particular, thegeneration of coherent light in the extreme-ultraviolet (EUV) to x-ray regime has been a problem
for many years. An attractive and practical solution to this problem has been the generation ofradiation by non-linear processes such as the interaction of ultra-short high-intensity pulses withatoms, which as a result generate high harmonics [19,20]. A particularly important applicationof HHG has been the generation of attosecond (1 attosecond = 10−18 seconds) pulses, which areimportant within the discipline of ultrafast AMO.

20
To motivate HHG further we consider a historical experiment performed by Leland Stanford andEadweard Muybridge in 1872 [21]. Stanford asked the question if, at any point during a horses
gallop, were all of its feet in the air. Unfortunately, the human eye was not enough to determine
this. With the help of Muybridge, they arranged many threads along a race track, each connected
to an individual camera which captured a photograph when the threads were pulled or broken. The
horse proceeded to run along the track triggering each camera resulting in a series of pictures (seefigure 3.1). The mystery was then solved and it was determined that by time resolving the horse’s
motion, that all its feet were indeed in the air at a given time within its gallop.

Figure 3.1: Eadweard Muybridge’s The Horse in Motion, 1878. [19]Similar to Stanford’s and Muybridge’s experiment, HHG can act as a camera which capture electron
dynamics in atoms and molecules at their fundamental time scale, the attosecond. Hence, it is a
powerful tool within AMO, and gives us further insight into the behavior of light matter interactions.

21
3.1.2

A Semi-Classical Description of High Harmonic Generation

The mechanism of HHG was first proposed by Corkum and others in 1993 [22, 23]. Thismodel is referred to as the three step model. It gives a simple description of the dynamics involvedin HHG. In step 1 of the model, the bound electron is ionized into the continuum by the appliedfield, which in this case is a laser pulse. In step 2, the electron is accelerated by the field, untilin step 3 the switching of direction of the polarization of the field causes the electron to returnand recombine with the parent ion, and a photon of energy equal to the energy of the recombined
electron is emitted. There is a caveat in step 3 however, if the recollision of the ionized electron
is inelastic then above-threshold ionization occurs, and HHG is suppressed. Figure 3.2 shows adiagram of the three-step model.

Figure 3.2: Diagram of the three-step model for high harmonic generation. Adapted from [46].

22
3.1.3

Features in High Harmonic Generation

High harmonic spectra are often calculated by taking the Fourier transform of the timedependent dipole moment d(t), and then using the formula below:

P (ω) =

α3 ω 2d(ω) · d∗ (ω).
3

(3.1)

Such spectra have various notable features as can be seen in figure 3.3. The first regime of thespectra is regarded as the perturbative regime, which typically shows up to the 3rd -5th generated
harmonics. Following is the plateau, where high-order harmonics are generated at similar intensities.
Occurrence of this regime is a sign that there is a non-perturbative interaction between the electrons
in an atom or a molecule and the applied laser field. There is then a cutoff following the plateau
beyond which the intensity of the generated harmonics is greatly reduced.

Figure 3.3: A schematic of a spectrum obtained from HHG due to the interaction of an atom ormolecule with a linearly polarized laser.

The frequency at which the cutoff occurs can be determined using the following formula obtained
from classical analysis:

ωcutof f = |Ip | + 3.17Up ,

(3.2)

23
where Ip is the ionization energy of the bound electron and Up is the ponderomotive energy of anelectron interacting with a laser, given by:

Up =

Eo2
,
4ω 2

(3.3)

where e is the electron charge, Eo is the amplitude of the laser field, m is the mass of the electron,and ω is the frequency of the laser field.
It is typical for systems with inversion symmetry, i.e. systems with invariance under a pointreflection, to exhibit only odd harmonics in their respective spectra, while under the influence of alinearly polarized pulse. It is worthwhile to mention that HHG spectra do not necessarily have thefeatures shown in figure 3.3, modified properties of HHG can be seen when the polarization of thelaser field or fields is not linear, but rather circular, as will be discussed later on in this chapter.

3.2

Molecular High Harmonic Generation
Harmonics generated from atoms with the interaction of linearly polarized pulses are linearlypolarized as well. However, since molecular systems tend to be more complicated due to multielectron effects, their interaction with linearly polarized pulses is also more complex. An example ofsuch complex interactions is that elliptically polarized harmonics can be generated. This is largely
due to the non-spherical symmetry of molecular orbitals.

3.2.1

Ellipticity of Generated High-Harmonics

As described by the 3-step model, HHG is a result of tunnel ionization followed by accelerationof the wavepacket into the continuum and then recombination with the parent ion accompanied by
the release of energy as a harmonic. It is expected that the polarization of these harmonics is thesame as the polarization of the laser field. This is indeed the case for atoms due to their inversion
symmetry. However, harmonic generation from molecules has two components, one parallel andanother perpendicular to the polarization of the field. One would expect another component to be24along the propagation direction of the field, however it cannot be phase-matched, thus it is neglected[24]. This further complicates the use of the 3-step model to describe HHG from molecules.
HHG generated from molecules can encode various properties of the molecule such as structural
information [24], in particular if the molecule is aligned. Until recently, only the intensity of generated harmonics could be obtained as well. However, due to technological advances, the phaseof generated harmonics can now be measured. As a result, it is now possible to experimentally
determine elliptically polarized HHG emission from molecules interacting with a linearly polarized
pulse. It is also worth mentioning that the widely used theory known as the strong-field approximation (SFA) fails to predict elliptical polarization of molecular harmonics correctly. However via
the use of ab-initio methods such as TDDFT, ellipticity of HHG can be predicted properly.
Measuring the polarization of emitted harmonics is of particular interest for various reasons. Boththe parallel and perpendicular components depend on the optical frequency ω and the angle θ ofthe molecule with respect to the field polarization. These components are directly related to thefield induced dipole moment and have a ratio given by:r=|d⊥ (ω)|,|dk (ω)|(3.4)

which is independent of the angle-dependent ionization (see [24]). From there it is possible todetermine the phase difference between the parallel and perpendicular components given by:δ = arg[d⊥ (ω)] − arg[dk (ω)],(3.5)

where this relative phase is not dependent on the intrinsic phase accumulated during the ionization
and acceleration steps of the 3-step model because they are common for both the parallel andperpendicular components. While both r and δ are assumed to be independent of angle dependent
ionization in SFA, we do not make this assumption within TDDFT and in our calculations. In
the end, measuring the polarization of HHG should in theory allow us to investigate the ionization
and recombination steps in the 3-step model in deeper detail and thus extend our knowledge of the25mechanism of HHG in molecules.
By obtaining the HHG spectrum, we can then determine the ellipticity of a given harmonic within
the spectrum with the following equation:s=p
1 + 2r2 cos(2δ) + r4p,
1 + r2 + 1 + 2r2 cos(2δ) + r4
1 + r2 −(3.6)

The ellipticity has a range of 0 ≤  ≤ 1, where 0 corresponds to linear polarization, and 1 to circularpolarization and anything in between is considered elliptical polarization. From equation (3.6) we
can see that the maximum ellipticity occurs when r = 1 and δ = 90o , which means that boththe parallel and perpendicular components of the dipole moment must be of similar magnitude.
However, as a result of the strong oscillation of the electron due to the laser, the perpendicularcomponent tends to be smaller in magnitude than the parallel component, which is due to theshaking of the electron wavepacket happening in the parallel direction, so a larger time-dependent
dipole moment component is expected.

3.2.2

Ensemble Alignment Averaging

In this thesis we investigated the appearance of elliptically polarized harmonics generated
from the interaction of a linearly polarized laser and CO2 . It is typical in experiments that molecules
are aligned using a laser pulse, which creates a rotational wave-packet that synchronizes the motionof molecules. Then at a certain time during the experiment the molecules are aligned, however not
perfectly. The distribution of alignment is measured by doing rotational averaging.
Within our theoretical studies, we begin by solving the time-dependent Kohn-Sham equation (2.31)
for our system to obtain the time-dependent dipole moments d(t). This is done for each orientation
angle ranging from 0o to 90o . Then, in order to account for the distribution of alignment angles,
as done in the experiment, the dipole moments are averaged using the same averaging technique,hcos2 θi [24,26].
If we consider the direction of the probe laser pulse to be along the z-direction and the pumplaser field to lie in the xy-plane with a polar angle α, while orienting the molecule around the26
direction given by the vector (sinθm cosφm , sinθm sinφm , cosθm ), as shown in figure 3.4, where θis the intersection angle between the molecular axis and the pump laser field. As a result, thedistribution of the alignment of the molecules with respect to the pump laser is a function of theintersection angle θ. We can then solve for this angle via:cosθ = sinαsinθm sinφm + cosαcosθm .

(3.7)

We can then determine an angular distribution ρ(α, θm , φm ) for the molecular axis at any given
orientation angle given by (θm , φm ) with respect to the pump-probe angle α. With that the averaged
time dependent dipole is given by:

Z π Z 2πhdk/⊥ (t, α)i =dθm dφm dk/⊥ (t, θm , φm )ρ(α, θm , φm )sinθm .
0

(3.8)

0

Considering to the symmetry of molecules, we only performed calculations in which the polar angle
θm was varied from 0o to 90o as mentioned prior, while setting φm = 0

Figure 3.4: Diagram illustrating the method for angle averaging.

27
3.2.3

Ellipticity of Harmonics From CO2

In experiments involving N2 and CO2 interacting with a linearly polarized laser field, ellipticity of harmonics was observed for N2 but not for CO2 [24,25]. However, other theories based on
SFA which include multiple electronic continuum dynamics predict ellipticity of harmonics fromCO2 [24,25]. In other words, multi-electron effects in molecules can indeed lead to the productionof elliptically polarized harmonics in CO2 .
W
We investigate the cases of CO2 interacting with an 800 nm field of intensities (i) 1.5 × 1014 cm
2 andW(ii) 2×1014 cm
2 . The orientation angle of the molecule with respect to the laser polarization is varied

from 0o to 90o with a step of 10o . With the use of TDDFT we are able to calculate the responses
of a set of occupied orbitals to the driving field. Figures 3.5a-b show ellipticity as a function ofharmonic order for various orientation angles θ = 20o , θ = 40o , and θ = 80o for both intensities
discussed above. We can see that there is in fact elliptically polarized harmonic generation in CO2 ,and for some harmonic orders the ellipticity is quite large ranging from values of 0.5 to 0.8, which
contradicts the results found in experiments conducted in [24,25]. We investigate this further inthe following sections by looking at the single orbital contributions to ellipticity.

28
0.9

0.5
= 200

0.45

= 200

0.8

= 400

= 800

0.7

0.35

0.6

Ellipticity

Ellipticity

= 400

= 800

0.4

0.3
0.25
0.2

0.5
0.4
0.3

0.15

0.2

0.1

0.1

0.05
0

0
0

5

10

15

20

25

30

0

5

10

15

20

Harmonic Order

Harmonic Order

(a)

(b)

25

30

Figure 3.5: Ellipticity as a function of harmonic order for CO2 interacting with an 800 nm pulseW
14 W . Various orientation angles are shown:at intensities (a) I = 1.5 × 1014 cm
2 and (b) I = 2 × 10cm2

θ = 20o , θ = 40o , and θ = 80o .

3.2.3.1

Orbital Contributions to Ellipticity of Harmonics

As mentioned, multi-electron effects in molecules play a role in the generation of ellipticallypolarized harmonics. To further analyze, we look at the individual orbital contributions to thegeneration of elliptical harmonics. We begin by looking at an intensity plot of the ellipticity foreach orbital added step by step as a function of harmonic order and molecular alignment angle
as shown in figure 3.6a-f. For the sake of understanding these contributions, we do not perform
an ensemble average of the molecular alignment. From the figures we observe that each orbital isindeed influencing the generation of elliptically polarized harmonics. We observe that the ellipticityof harmonics is quite strong when we consider just the 1πu orbital. However upon adding other
orbital contributions the ellipticity becomes weaker largely due to interference.
Recall that HHG depends on ionization which in turn is dependent on the ionization potential andthe alignment angle of the molecule. When the molecule is oriented at 0o and 90o the contribution
to HHG from the 1πg orbital is greatly suppressed due to quantum interference, thus ellipticity is35

29
nearly zero at these angles. However when the alignment angle is between 0o and 90o ellipticityof harmonics in non-zero. In contrast the σg and σu orbitals exhibit strong ellipticity when thealignment is near these angles.
Realistically and experimentally we do not have the means of observing ellipticity without ensemble
averaging, or the ability to observe individual orbital contributions. As a result the actual ellipticity
of generated harmonics will be quite weak as is observed in figure 3.7. Interestingly enough, when
the intensity of the field is increased, ellipticity becomes more apparent, as can be seen in figure
3.9. This makes sense as the strength of the induced dipole is related to the strength of the field,
thus the generation of harmonics will be increased30(a) (3σg )2 (2σu )2 (4σg )2 (1πu )4 (3σu )2 (1πg )4(b) (2σu )2 (4σg )2 (1πu )4 (3σu )2 (1πg )4(c) (4σg )2 (1πu )4 (3σu )2 (1πg )4(d) (1πu )4 (3σu )2 (1πg )4(e) (3σu )2 (1πg )4(f) (1πg )4Figure 3.6: Ellipticity of harmonics of CO2 as a function of harmonic order and alignment angle
for different orbital contributions added step by step. No ensemble average is made. The appliedW
laser field is an 800 nm linearly polarized field with intensity 1.5 × 1014 cm
2.

31(a) (3σg )2 (2σu )2 (4σg )2 (1πu )4 (3σu )2 (1πg )4Figure 3.7: Ellipticity of harmonics of CO2 as a function of harmonic order and alignment angle.
Here an ensemble average is made. The applied laser field is an 800 nm linearly polarized field withW
intensity 1.5 × 1014 cm
2.

32(a) (3σg )2 (2σu )2 (4σg )2 (1πu )4 (3σu )2 (1πg )4(b) (2σu )2 (4σg )2 (1πu )4 (3σu )2 (1πg )4(c) (4σg )2 (1πu )4 (3σu )2 (1πg )4(d) (1πu )4 (3σu )2 (1πg )4(e) (3σu )2 (1πg )4(f) (1πg )4Figure 3.8: Ellipticity of harmonics of CO2 as a function of harmonic order and alignment angle
for different orbital contributions added step by step. No ensemble average is made. The appliedW
laser field is an 800 nm linearly polarized field with intensity 2 × 1014 cm
2.

33(a) (3σg )2 (2σu )2 (4σg )2 (1πu )4 (3σu )2 (1πg )4Figure 3.9: Ellipticity of harmonics of CO2 as a function of harmonic order and alignment angle.
Here an ensemble average is made. The applied laser field is an 800 nm linearly polarized field withW
intensity 2 × 1014 cm
2.

3.3

Mollow Sidebands

3.3.1

Adiabatic and Non-Adiabatic Behavior of Laser Induced Dipole Moments

There are several types of dynamics which can occur as a result of the interaction between
molecules and laser pulses. One such type is regarded as adiabatic, meaning that the oscillation of
the electron follows the oscillation of the electric field. Typically, this results in HHG spectra withthe usual features as mentioned earlier in this chapter, at least for the case of linearly polarized
pulses. The other type of dynamics which can occur is regarded as non-adiabatic, meaning thatthe electron does not follow the oscillation of the laser pulse. This can occur as a result of laser
induced resonant couplings of orbitals. Typically this is done by setting the optical frequency ofthe laser pulse equal to the frequency between the orbitals. This type of dynamics gives rise to34
interesting phenomena that appears in HHG and electron localization, as we will see further inthis thesis. Figures 3.10a-b respectively show adiabatic and non-adiabatic dipole behavior for N2+ ’stime-dependent dipole in the presence of an laser field for the cases of a 400 nm pulse with intensity
W
I = 2 × 1014 cm
2 and polarization (a) perpendicular to the molecular axis and (b) parallel to themolecular axis.

0.4⟨ipole
⟩aser

1.5
1.0

0.2

⟨X⟩ (a.u.)⟨X⟩ (a.u.)

2.0⟨ipole
⟩aser

0.0

0.5
0.0

−0.5

−0.2

−1.0
−0.4

−1.5
0

2

4

6

8

10

Time (fs)
(a)

12

14

16

−2.0

0

2

4

6

8

10

Time (fs)

12

14

(b)

Figure 3.10: Time-dependent dipole moments of N2+ in the presence of a 400 nm laser field withW
intensity I = 2 × 1014 cm
2 . (a) Adiabatic behavior, and (b) Non-adiabatic behavior.

3.3.2

Open Shell Molecules

Within this thesis, non-adiabatic effects were studied for N2+ and C2 H4+ , which are both open
shell molecules, i.e. they contain valence shells which are not completely filled. In N2+ such effects
happen in two cases studied. The first case is a linearly polarized laser of 400 nm with polarization
oriented along the molecular axis. This case results in a laser induced resonant coupling between
the 3σg and 2σu orbitals. The second case is an 800 nm linearly polarized pulse whose polarizationis oriented perpendicular to the molecular axis, and for this case there is a laser induced resonant
coupling between the 3σg and 1πu orbitals. Figure 3.11 shows the molecular orbital diagram of N2+

16

35
along with the possible resonant couplings. Note, that we are able to specify in the TDDFT code
OCTOPUS the orbital with the electron hole, thus making it the open-shell orbital.

(a)

(b)

Figure 3.11: Molecular orbital diagram of (a) N2+ and (b) C2 H4+ . The purple arrow indicates aresonant coupling between the orbitals (a) 3σg and 2σu and (b) 1b3g and 1b2u induced by a 400 nm
laser pulse with polarization parallel to the molecular axis, and the red arrow indicates a resonant
coupling between the orbital 3σg and 1πu induced by an 800 nm laser pulse with polarization
perpendicular to the molecular axis.

While C2 H4+ is also an open shell molecule, it is far more complex than N2+ . Still, it is worthwhile
to investigate the non-adiabatic effects occurring due to a laser induced coupling of orbitals. Unlike
N2+ , where the open shell orbital is the highest occupied molecular orbital (HOMO), the open shellorbital in C2 H4+ is 1b3g , which is the HOMO-1, and a 400 nm linearly polarized laser pulse withpolarization oriented parallel to the molecular axis induces a resonant coupling with the 1b2u orbital(HOMO-3).
It is worthwhile to mention that we make a distinction between molecular orbitals and states inthis thesis. While we talk about coupling orbitals, there is another picture that can give us insight
into the dynamics occurring. We consider the ground state of N2+ labeled Σg and two excited36
states of N2+ labeled Πu and Σu in figure 3.12, which have the respective electron configurations,(2σg )2 (2σu )2 (1πu )4 (3σg )1 , (2σg )2 (2σu )2 (1πu )3 (3σg )2 , and (2σg )2 (2σu )1 (1πu )4 (3σg )2 . We make thedistinction that the ground state is the lowest energy configuration, while the two latter are thehigher energy configurations, hence the excited state. We can think as the orbital coupling of asuperposition wavepacket of the ground and excited states during the interaction with the laser.
However for the remainder of this thesis we will talk about coupling of orbitals and not the states.

Figure 3.12: Potential energy surfaces of the ground state Σg and two excited states Πu and Σu of
N2+ and their respective diagrams showing the electron configuration for each state.

3.3.3

Appearance of Mollow Sidebands in High-Harmonic Generation

As a consequence of laser induced resonant coupling between orbitals, certain features appear
in the respective HHG spectra of molecules. One of such features is the appearance of Mollow
sidebands [43], which unlike regular harmonics they do not appear at integer multiples of the
optical frequency of the laser field, but rather at a linear combination of the multiples of the laser

37frequency and the Rabi frequency which is given by:ΩRabi = µEo ,(3.9)

where µ is the transition dipole moment between the orbitals, and Eo is the amplitude of theapplied laser field. Figure 3.13 shows the respective HHG spectra for N2+ interacting with (a) a 400W
nm laser with intensity I = 5 × 1014 cm
2 with polarization parallel to the molecular axis and (b) anW
800 nm laser with intensity I = 2 × 1014 cm
2 polarized perpendicular to the molecular axis. As isevident from the figures, a laser induced resonance between orbitals causes additional features toappear in the HHG spectra in the form of the so-called Mollow sidebands. An interesting featureof these sidebands is that they are relatively similar in intensity to the odd harmonics. This canbe attributed to the large transition dipole moment between orbitals as well as the strong coupling
between the orbitals. We can identify the coupled orbitals by projecting the ground state orbital
onto the excited state orbital during the simulation. The results show the population transfer
between orbitals, and can be seen in figure 3.14. For the case of N2+ interacting with the 400 nm
pulse polarized parallel to the molecular axis we can see that there is a strong coupling between the
3σg and 2σu orbitals, and that during the interaction with the laser there is a population transfer
of about 80%. In contrast, for (b) the population transfer between 3σg and 1πu is about 40%.
We can therefore conclude that the appearance of Mollow sidebands is indeed caused by the laser
induced coupling between the orbitals.

38
10−2

Intensity (arb. units)

10−4
10−6
10−8

10−10
10−12

0

5

10

Harmonic Order

15

20

25

(a)

10−4
10−5

Intensity (arb. units)

10−6
10−7
10−8
10−9

10−10
10−11
10−12
10−13

0

5

10

15

20

Harmonic Order

25

30

35

40

(b)

Figure 3.13: HHG spectra for N2+ interacting with (a) a 400 nm laser pulse of intensity I =W
5 × 1014 cm
2 with polarization parallel to the molecular axis, and (b) an 800 nm pulse of intensity
W
I = 2 × 1014 cm
2 with polarization perpendicular to the molecular axis. Note: figure (b) shows the

spectrum before the frequency cutoff to make visualization of Mollow-sidebands easier to see.

39

0.40
0.35

0.8

Projection

Projection

0.30
0.6

0.4

0.25
0.20
0.15
0.10

0.2

0.05
0.0

0.00
0

2

4

6

8

10

Time (fs)

12

14

16

18

0

2

(a)

4

6

8

10

Time (fs)

12

14

16

18

(b)

Figure 3.14: Time-dependent projections between coupled orbitals of N2+ , for the cases of (a)W
coupling between 3σg and 2σu by a 400 nm pulse of intensity I = 5 × 1014 cm
2 polarized parallelto the molecular axis, and (b) coupling between 3σg and 1πu by an 800 nm pulse of intensity
W
I = 2 × 1014 cm
2 polarized perpendicular to the molecular axis.

It is worthwhile to mention, that due to the electron hole in the 3σg orbital of N2+ , the appearanceof Mollow sidebands in the ion is more prevalent than in the neutral counterpart N2 . This islargely attributed to the fact that the electron located in the 3σg must first be ionized in orderfor the electrons in inner shell orbitals to resonate between them, thus limiting the contribution
to Mollow sidebands appearing in the spectra. However, if one begins with the ionic molecule,there is no need for the electron in the 3σg orbital to first be ionized which then allows for theelectron in the coupled orbital to transition at a faster rate to the orbital where the electron holeis, thus making Mollow sidebands appear more prevalent in the HHG spectra. Figures 3.15a-b
Wshow the HHG spectra for N2 interacting with (a) a 400 nm laser of intensity I = 2 × 1014 cm
2W
polarized along the molecular axis, and (b) an 800 nm laser of intensity I = 2 × 1014 cm
2 polarized

perpendicular to the molecular axis. We can see that for case (a) there appears to be some peaks
forming next to the odd harmonics, albeit not of similar intensity as in N2+ , while in case (b) there
appear only odd harmonics with no trace of sidebands. This is largely due to the transition dipole

40
moment being much larger in the case where one couples the 3σg and 2σu orbitals with a 400 nm
pulse than when one couples the 3σg and 1πu orbitals, and while the Mollow sidebands appear low
in intensity, they are still noticeable for certain harmonic orders. Interestingly enough, the timedependent projection between orbitals (see figures 3.16a-b), shows that there is indeed a strong
population transfer between orbitals, in fact comparable to that of N2+ for the case of a 400 nm
laser pulse polarized parallel to the molecular axis. However, for the case of an 800 nm pulse withperpendicular polarization, the population transfer is quite strong compared to the ionic counter
part.

41

Intensity (arb. units)

10−5
10−7
10−9

10−11
10−13
10−15 0

5

10Harmonic Order

15

20

25

15

20

25

(a)

Intensity (arb. units)

10−5
10−7
10−9

10−11
10−13
10−15 0

5

10Harmonic Order(b)

Figure 3.15: HHG spectra for N2 interacting with (a) a 400 nm laser pulse of intensity I =W
2 × 1014 cm
2 with polarization parallel to the molecular axis, and (b) an 800 nm pulse of intensity
W
I = 2 × 1014 cm
2 with polarization perpendicular to the molecular axis.

42
1.0

0.8

Projection

Projection

0.8

0.6

0.4

0.2

0.6

0.4

0.2

0.0

0.0
0

2

4

6

8

10

Time (fs)

12

14

16

18

0

2

(a)

4

6

8

10

Time (fs)

12

14

16

18

(b)

Figure 3.16: Time-dependent projections between coupled orbitals of N2+ , for the cases of (a)W
coupling between 3σg and 2σu by a 400 nm pulse of intensity I = 2 × 1014 cm
2 polarized parallelto the molecular axis, and (b) coupling between 3σg and 1πu by an 800 nm pulse of intensity
W
I = 2 × 1014 cm
2 polarized perpendicular to the molecular axis.

3.3.4

High Harmonic Generation Dependence On Pulse Envelope

Up until now, every system we have considered has had the laser envelope be trapezoidal,
figure 3.17a, i.e., a flat top pulse. Realistically speaking, such pulses cannot be realized at the
femtosecond level, and mathematically the ’kink’ that occurs after the ramp up brings aboutproblems when considering the Fourier transform of the pulse. Despite that, they are useful wheninvestigating HHG due to their continuous wave like properties, resulting in cleaner and smoother
HHG spectra. Simply put, the electrons feel the peak intensity of the pulse for a majority of theinteraction. Fortunately enough we are not limited to this pulse envelope when using TDDFT.
In fact, we have complete freedom to choose what envelope we use throughout a simulation. We
present the laser field obtained by using a sin2 (.) envelope in figure 3.17c and from a first glance it isevident that the electrons in the system experience different intensities throughout the interaction,rather than a peak intensity for a longer period of time as in the trapezoidal case. Looking at the43
Fourier transform of the sin2 (.) envelope, figure 3.17d, also shows that the spectrum about thepeak frequency, which in this case corresponds to λ = 400 nm, is broader when using the sin2 (.)
envelope rather than the trapezoidal one and as a result the HHG can vary greatly as shown infigure 3.18, where we present HHG for the case where C2 H4+ interacts with a 400 nm laser polarized
parallel to the molecular axis and couples the 1b3g and 1b2u orbitals. We observe that when using
the trapezoidal pulse the overlap between Mollow sidebands and the odd harmonics is such thatthe peaks are distinguishable, however when using the sin2 (.) pulse the overlap between Mollow
sidebands is increased so one cannot identify the structures caused by the laser coupling of orbitals.
In the context of this thesis however, we are interested in distinguishing such features that arise
from non-adiabatic behavior, so we will consider trapezoidal pulses for most of the results presented.

44

Intensity (arb. units)

100

Amplitude (a.u.)

0.04
0.02
0.00

−0.02
−0.04

0.0

2.5

5.0

7.5

10.0

Time (fs)

12.5

15.0

Intensity (arb. units)

Amplitude (a.u.)

0.02
0.00

−0.02
−0.04
5.0

7.5

10.0

Time (fs)

12.5

15.0

(c) sin2 Envelope Time-Domain

10 6
10 8
0

1
2
3
4
Frequency (Normalized)

5(b) Trapezoidal Envelope Frequency-Domain

0.04

2.5

10 4

10 10

17.5

(a) Trapezoidal Envelope Time-Domain

0.0

10 2

17.5

10 1
10 4
10 7
10 10
10 13
10 16

0

1
2
3
4
Frequency (Normalized)

5(d) sin2 Envelope Frequency-Domain

Figure 3.17: Plots of the laser fields with different pulse envelopes of wavelength λ = 400 nm in
the time and frequency domains.

45

10−3

Intensity (arb. units)

10−5
10−7
10−9

10−11
10−13
10−15 05

1015

Harmonic Order

20

25

30

20

25

30

(a) Trapezoidal envelope.

10−3Intensity (arb. units)

10−5
10−7
10−9

10−11
10−13
10−15 05

1015

Harmonic Order(b) sin2 envelope.

Figure 3.18: HHG spectra for C2 H4+ interacting with a 400 nm pulse of peak intensity I = 1 ×W1014 cm
2.

46
3.3.5

Mollow Sidebands in Optical Bloch Equations

While the use of TDDFT allows us to capture full 3-dimensional multielectron dynamics inmolecules interacting with a laser field and get a good quantitative and qualitative description of
the features of HHG, it can computationally be quite time consuming. As a result we turn toother methods, which give us insight into what is occurring during HHG. One such way is based on
optical Bloch equations, as described in chapter 2. While not as effective as TDDFT in capturing
the full effects occurring in HHG, it paints a simple picture of the effect related to the appearanceof Mollow sidebands in HHG.
We begin by considering the wavefunction of a two-level system given by a superposition state:|ψ(t)i = a1 (t)|ψ1 i + a2 (t)|ψ2 i,(3.10)

where a1 (t) and a2 (t) are determined by solving the system of equations obtained from the TDSE,
given by: (assuming that hψ1 |ψ2 i = 0)ωoa1 (t) + V (t)a2 (t),2

(3.11)ωoa2 (t) + V (t)a1 (t),2

(3.12)

iȧ1 (t) = −iȧ1 (t) =where ωo is the frequency corresponding to the energy difference between states, V (t) = µE(t), andµ is the transition dipole moment between states. The optical Bloch equations are for this casedefined as equations for x(t), y(t), and z(t):x(t) = a1 a∗2 + a2 a∗1 ,(3.13)

y(t) = i(a2 a∗1 − a1 a∗2 ),(3.14)

z(t) = |a1 |2 − |a2 |2 ,(3.15)

47
which upon combination with the equations for the coefficients a1 and a2 lead to the following
equations:ẋ(t) = −ωo y(t),(3.16)

ẏ(t) = ωo x(t) + 2V (t)z(t),(3.17)ż(t) = −2V (t)y(t).

(3.18)

Solving the set of equations gives us the induced dipole moment d(t) = µx(t) and the population
inversion z(t).
Previous studies conducted by Bandrauk have shown that symmetric molecular ions have charge
resonant states that are coupled by a laser field with optical frequency equal to the energy difference
between the orbitals. However depending on wavelength and potential energy surface, it is also
possible to couple orbitals when the internuclear distance R is enlarged. Such is the case for
H2+ , as shown by Bandrauk and Zuo [27,28], where they created charge resonant (CR) pairs fromthe superposition of the 1σu and 1σg orbitals. They considered three different excitation regions
where the CR pairs could be created: (1) small R corresponding to multiphoton excitation, (2)intermediate R which is near resonant excitation, and (3) large R corresponding to the strong
coupling regime.
We consider similar conditions to those considered by Bandrauk and show that such simple models
like the optical Bloch equations do in fact predict Mollow sidebands. We consider the three following
cases of H2+ interacting with a continuous wave laser of the form E(t) = Eo cos(ωt): (1) R = 2.12
W
a.u. with laser parameters λ = 1064 nm, I = 1014 cm
2 , (2) R = 5 a.u. with λ = 1064 nm at
W
13 W . The HHG spectra
I = 4 × 1013 cm
2 and (3) R = 10 a.u. with λ = 1064 nm and I = 10
cm2

corresponding to these cases are shown in figures 3.19a-c, respectively. In each of the HHG spectra
we can see that Mollow sidebands are indeed appearing, though their intensities vary depending on

48
the internuclear distance of H2+ and also each of the three coupling regimes we considered.

49
10

0

Intensity (arb. units)

10 -5

10 -10

10 -15

10

-20

1

3

5

7

9

11

13

15

11

13

15

Harmonic Order(a) R = 2.12 a.u.

10 0Intensity (arb. units)

10 -5

10 -10

10 -15

10 -20

1

3

5

7

9

Harmonic Order(b) R = 5 a.u.

10 5Intensity (arb. units)

10 0

10 -5

10 -10

10 -15

10 -20

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

Harmonic Order(c) R = 10 a.u.

Figure 3.19: HHG spectra for H2+ obtained via the optical Bloch equations. Three cases are shown:
W(a) R = 2.12 a.u. with laser parameters λ = 1064 nm, I = 1014 cm
2 , (b) R = 5 a.u. with λ = 1064W
13 W .
nm at I = 4 × 1013 cm
2 and (c) R = 10 a.u. with λ = 1064 nm and I = 10cm2

50
While we do observe Mollow sidebands in the HHG of H2+ for different internuclear distances wemust consider that this system is quite simple compared to the others we have been investigatingin this thesis so the question we now ask is if the optical Bloch equations can indeed predict Mollow
sidebands in multielectron molecules such as N2+ .
We begin by considering the case were we couple 3σg and 2σu with a 400 nm laser pulse. We utilize
the transition dipole moment µ = 1.558a.u obtained from TDDFT calculations, and obtain theW
15 W .
respective HHG from cases where we vary the intensity from I = 1 × 1013 cm
2 to I = 5 × 10cm2

The HHG spectra are shown in figure 3.20a-f. At the lowest intensity we observe odd harmonics
ranging from order 0 to 5 with small peaks forming on the left and right side of each odd harmonic.
These are indeed Mollow sidebands caused by the coupling of orbitals. As the intensity is increased
the overlap between odd harmonics and Mollow sidebands decreases which is expected due to the
Rabi frequency being directly proportional to the field strength. Another noticeable feature is thatthe harmonic order increases as the intensity is raised, however due to the perturbative nature of
the optical Bloch equations, the cutoff is not where one would expect it.
The second case considered is when an 800 nm pulse couples the 3σg and 1πu orbitals. FigureW
3.21a-f shows the HHG spectra with intensities once again varying from I = 1 × 1013 cm
2 to I =W
5 × 1015 cm
2 . We again observe odd harmonics as well as Mollow sidebands whose overlap increases

as the intensity is raised. However, unlike the full 3-dimensional calculations, the cutoff appears
to occur at low harmonic orders. This is due to the fact that the continuum is not consideredin optical Bloch equations, hence there is no acceleration acceleration and recombination of theionized wave-packet resulting in a cutoff at higher orders.

51
100

Intensity (arb. units)

Intensity (arb. units)

100

10-5

10-10

1

5

9

13

10-10

17

13

W
(b) I = 5 × 1013 cm
2

17

100

Intensity (arb. units)

Intensity (arb. units)

9W(a) I = 1 × 1013 cm
2

1

5

9

13

10-5

10-10

17

1

5

9

13

Harmonic Order

Harmonic OrderW(c) I = 1 × 1014 cm
2W(d) I = 5 × 1014 cm
2

17

100

Intensity (arb. units)

100

Intensity (arb. units)

5Harmonic Order

10-5

10-5

10-10

1

Harmonic Order

100

10-10

10-5

10-5

10-10
1

5

9

13

17

1

5

9

13

17

21

Harmonic Order

Harmonic OrderW(e) I = 1 × 1015 cm
2W(f) I = 5 × 1015 cm
2

25

29

Figure 3.20: HHG spectra for N2+ obtained via the optical Bloch equations. The laser pulse has awavelength of 400 nm and the intensity of the pulse is increased for each panel.

52
100

Intensity (arb. units)

Intensity (arb. units)

100

10-5

10-10

1

5

9

13

10-10

17

9

13

W
(a) I = 1 × 1013 cm
2W(b) I = 5 × 1013 cm
2

17

100

Intensity (arb. units)

Intensity (arb. units)

5Harmonic Order

10-5

1

5

9

13

10-5

10-10

17

1

5

9

13

Harmonic Order

Harmonic OrderW(c) I = 1 × 1014 cm
2W(d) I = 5 × 1014 cm
2

17

100

Intensity (arb. units)

100

Intensity (arb. units)

1

Harmonic Order

100

10-10

10-5

10-5

10-10

10-5

10-10
1

5

9

13

17

1

5

9

13

Harmonic Order

Harmonic OrderW(e) I = 1 × 1015 cm
2W(f) I = 5 × 1015 cm
2

17Figure 3.21: HHG spectra for N2+ obtained via the optical Bloch equations. The laser pulse has awavelength of 800 nm and the intensity of the pulse is increased for each panel.

53
3.3.6

Comparison Between Theoretical Methods in the Appearance of Mollow
Sidebands

The appearance of Mollow sidebands is clearly seen in HHG spectra obtained from both
TDDFT and optical Bloch equations, however many differences arise when comparing both. Whenperforming the full 3-dimensional calculation via TDDFT one is considering multi-electron effects
as well as the electrons being ionized into the continuum and then recombining with the parent
ion, which gives rise to an extension of harmonics to higher orders of frequency, whereas in optical
Bloch equations the continuum is not included, thus the cutoff for harmonic generation occurs at
lower frequencies than that obtained in TDDFT.
In addition, HHG spectra obtained from optical Bloch equations appear to be cleaner and so it iseasier to distinguish Mollow sidebands from the odd harmonics. In TDDFT multielectron effects
give rise to many interferences and overlaps between harmonics so obtaining a cleaner HHG spectra
is sometimes more difficult.
For example, problems can arise when using TDDFT to model Rabi oscillations, as is the case inlaser induced orbital coupling. When one utilizes adiabatic functionals the potential energy surfaces
will change as the electron density changes, and so the system will be driven out of resonance. In
other words, Rabi oscillations cannot be fully captured when using the adiabatic approximation as
the instantaneous dependence of the state brings about a time-dependent detuning.
We look at the predicted versus measured Rabi frequencies given by the following equation:ΩM ollow = nω ± Ωrabi .

(3.19)

where nω is the n − th odd harmonic in the corresponding HHG spectra, and the second term is the
Rabi frequency. Figures 3.22a-d show the comparisons for both the optical Bloch equations and
TDDFT. We see that for both theories the measured frequencies agree with what is predicted withthe given equation above. In conclusion we see that both theories do in fact capture the effects of
the Rabi oscillations occurring as a result of the laser induced coupling between orbitals.

54
8

8
Predicted
Measured

Predicted
Measured7

Rabi Frequency (Normalized)

Rabi Frequency (Normalized)

7

6

5

4

3

2

6

5

4

3

2

1

1
1

3

5

7

1

3

5

7

Harmonic Order

Harmonic Order(a) Optical Bloch 400 nm(b) Optical Bloch 800 nm

3550
Predicted
Measured

Predicted
Measured45

Rabi Frequency (Normalized)

Rabi Frequency (Normalized)

30

25

20

15

10

40
35
30
25
20
15
10

5
5
0

0
1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

1

5

9

13

17

21

25

29

33

Harmonic Order

Harmonic Order(c) TDDFT 400 nm(d) TDDFT 800 nm

37

41

45

49

Figure 3.22: Predicted (blue dashed line) vs. Numerically obtained (red circles) frequencies of
Mollow sidebands appearing in HHG spectra obtained from optical Bloch equations (panels a andb) and TDDFT (panels c and d) for the cases were the laser wavelength is 400 nm and 800 nm.

3.3.7

Time-Frequency Analysis of High-Harmonic Generation.

When we obtain HHG spectra by performing a Fourier transform of the time-dependent
dipole moment we lose all information in the time-domain, per the Heisenberg uncertainty principle.
Fortunately, there exist a tool by which we are able to examine how HHG changes with time known
as time-frequency analysis, or wavelet analysis. The time-frequency spectra is obtained using the55transform equation below:Zd(ω, t) =d(t)exp(−(t − τ )2 /2σ 2 )exp(−iωτ )dt,

(3.20)

where the Gaussian function is a time window which constraints the FFT of the dipole to be applied
only in the size defined by the function.
We begin by looking at a nonresonant case of N2+ interacting with a 400 nm laser pulse of intensity
W
I = 2 × 1014 cm
2 with polarization direction oriented perpendicular to the molecular axis, which

does not induce a resonance between orbitals. The corresponding wavelet spectrum is shown infigures 3.23a-b , where (a) shows the total spectrum and (b) shows the spectrum zoomed in from
5 to 10 femtoseconds.

0.00400.00350.00300.00250.00200.00150.00100.0005Harmonic Order

14
12
10
8
6
0

5

10
Time (fs)

15

(a)

(b)

Figure 3.23: Wavelet spectrum for N2+ interacting with a 400 nm laser pulse of intensity I =W
2 × 1014 cm
2 with polarization direction oriented perpendicular to the molecular axis. Panel (a)shows the spectrum over the total length of the laser propagation, and (b) shows a zoom in from
5 to 10 femtoseconds and has labels showing where the trajectory of the electron is short or long
during the propagation of the laser.

From the spectrum we can see where there is an HHG burst during the propagation of the laser

56
pulse. We can also tell for each harmonic order the recombination times which are related toclassical trajectories of electrons. We see in figure 3.23b that at about 6 and 8 femtoseconds theelectron recombines with the parent ion resulting in an HHG burst of the 9th harmonic. While, thisharmonic is emitted at multiple times, the smaller time corresponds to a shorter electron trajectory
as a result of the interaction with the laser pulse, and the later time to the long trajectory.
For the nonresonant case there are modifications to the semi-classical description of HHG which arealso present in the time-frequency analysis. We consider the case where N2+ interacts with a 400W
nm laser of intensity I = 2 × 1014 cm
2 with polarization direction oriented parallel to the molecularaxis, shown in figure 3.24. We know from previous discussion that this induces a coupling between
the 3σg and 2σu orbitals, and as a result we expect this nonadiabatic behavior to be present in
the wavelet spectrum. Unlike the nonresonant case presented above, the generation of harmonics
in the resonant case is far more complex, and the use and interpretation of classical trajectories
become difficult when describing what is occurring in the spectrum. Instead, we see modulation
due to the nonadiabatic behavior induced by the laser coupling and the spectrum does not exhibit
the same periodic nature of the nonresonant case.

57

0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005

Harmonic Order

14
12
10
8
6
0

5

10
Time (fs)15

Figure 3.24: Wavelet spectrum for N2+ interacting with a 400 nm laser pulse of intensity I =W
2 × 1014 cm
2 with polarization direction oriented parallel to the molecular axis.

3.4

Two-Color High-Harmonic Generation
The interaction of molecular ions with a single color linearly polarized pulse brings about

interesting features in HHG spectra in the cases of a laser induced resonant coupling between
orbitals. However what happens when one considers adding another laser to the system? Will theeffects we have encountered before in the single color linearly polarized case still be present, and if
so will they be enhanced or diminished? What about changing the polarization of the fields, will
there be noticeable changes to the HHG spectra? We consider each of these cases in this part of
the thesis.

3.4.1

Two-Color Linearly Polarized Laser Pulses

We begin by considering two cases, (a) N2+ interacting with a 400 nm pulse with polarization
fixed parallel to the molecular axis, and an 800 nm pulse with polarization varied from 0o to 90o58with respect to the molecular axis, and (b) N2+ interacting with an 800 nm pulse with polarization
fixed perpendicular to the molecular axis, and a 400 nm pulse with polarization varied from 0o to
90o with respect to the molecular axis. Figure 3.25 shows a schematic diagram of both cases (a)and (b). Recall that for case (a) the 400 nm pulse will induce a coupling between 3σg and 2σu , andfor (b) an 800 nm pulse will induce a coupling between 3σg and 1πu . Will the addition of the otherpulses to the system affect coupling between orbitals thus drastically changing the HHG spectra?

(a)

(b)

Figure 3.25: Schematic diagram for the N2+ interacting with (a) a 400 nm pulse with fixed polarization parallel to the molecular axis and an 800 nm pulse varied in angle θ = 0o to θ = 90o withrespect to the molecular axis, and (b) an 800 nm pulse with polarization fixed perpendicular to themolecular axis and a 400 nm pulse with polarization varied from θ = 0o to θ = 90o with respect tothe molecular axis.

The HHG spectra for case (a) and all angles θ are shown in figure 3.26. If we compare case (a)W
to a single color case where N2+ interacts with a 400 nm laser pulse of intensity I = 2 × 1014 cm
2,

as shown in figure 3.27 we can see that the HHG spectra show similar signatures and behavior,
however there are slight differences. For example, the 3rd harmonic in the single color case seems
to have a larger magnitude than harmonics of lower order, whereas in the two-color case the 3rd
harmonic is lower in magnitude than those before. In addition, the Mollow sidebands seem to59become more broad in the two-color case, which is due to the second 800 nm pulse interfering withthe resonant effects. A noticeable difference occurs in the plateau regime of HHG for the two-color
case. We see that unlike the single color case the harmonics of higher magnitude extend further
in harmonic order, which is a result of the 800 nm pulse, that leads to longer harmonic spectrum
than the single 400 nm pulse.

1013
Intensity (arb. units)

108

θ=0
θ = 10θ = 20θ = 30θ = 40θ = 50θ = 60θ = 70θ = 80θ = 90

103
10−2
10−7

10−12
10−17
1

3

5

7
9
11
Harmonic order

13

15

W
Figure 3.26: HHG spectra for a 400 nm pulse at 2 × 1014 cm
2 polarized along the molecular axisWoowith an 800 nm pump pulse at 1 × 1013 cm
2 with polarization direction rotated from 0 to 90 withrespect to the molecular axis. Note: Intensities are scaled for comparison.

60
10 2

Intensity (arb. units)

10 0

10 -2

10 -4

10 -6

10 -8

1

3

5

7

9

11

13

15

17

Harmonic OrderW
Figure 3.27: HHG spectra for a 400 nm pulse at 2 × 1014 cm
2 polarized along the molecular axis.

W
We then consider case (b) where the 800 nm pulse of intensity I = 2 × 1014 cm
2 is the one withpolarization direction fixed perpendicular to the molecular axis. When comparing the spectrum toW
the single color 800 nm case of intensity I = 2 × 1014 cm
2 and polarization direction perpendicular

to the molecular axis, as shown in figure 3.13b, we see many notable differences. The sidebands
due to the resonant coupling are more defined for the lower order harmonics, but become smaller inmagnitude as the angle of the polarization direction of the 400 nm is varied from θ = 0o to θ = 90o .
Also, as the harmonic order is increased, the resolution of the sidebands becomes worse and so it is
not clear if there are indeed Mollow sidebands occurring as a result of the resonant coupling by the
800 nm pulse, or if they are even harmonics resulting from a breaking of inversion symmetry due
to the second laser being present. Regardless of the details, it is safe to assume that the additionof a second pulse does not enhance the emission of Mollow sidebands, but rather suppresses them.
Also, while some features of HHG for a single color are present in case (a) we see that it is not so
in case (b) so we can say that features in HHG spectra are indeed modified when adding a second
pulse to interact with our system.

61

1014
1010
Intensity (arb. units)

106

θ=0
θ = 10θ = 20θ = 30θ = 40θ = 50θ = 60θ = 70θ = 80θ = 90

102
10−2
10−6

10−10
10−14
10−18
1

5

9

13 17 21 25 29 33 37 41 45 49
Harmonic orderW
Figure 3.28: HHG spectra for an 800 nm at 2 × 1014 cm
2 probe pulse polarized along the molecularWooaxis with an 400 nm pump pulse at 1 × 1013 cm
2 with polarization direction rotated from 0 to 90

with respect to the molecular axis. Note: Intensities are scaled for comparison.

3.4.2

Two-Color Circularly Polarized Laser Pulses

As we have seen molecules interacting with strong linearly polarized pulses exhibit interesting
phenomena such as the appearance of Mollow sidebands in HHG spectra as a result of resonant
coupling, and the generation of ellipticaly polarized harmonics when the symmetry of the system isbroken. However, we ask the question of what occurs when the laser field has a circular polarization?
Will the generated harmonics be circularly polarized as well? The answer to this question is actuallyquite straight forward, no harmonics will be generated. The reason being that a strong single
circularly polarized pulse will ionize the wavepackets and cause no recombination with the parent
ion, hence no generated harmonics. However, if a second circularly polarized pulse is added to thesystem, then there will in fact be harmonic generation as was shown by Eichmann et al. [29] inwhich two cases were considered, the first being two circularly polarized pulses of frequencies ω and2ω, respectively, with the electric field vectors co-rotating in the same plane. In this case it was

62seen that all harmonics of frequency nω (n = 0, 1, 2, ...) were emitted with elliptical polarization.
The second case considered the same fields but counter-rotating and it was seen that harmonics
of frequency (3n ± 1)ω were emitted with circular polarization with their helicity changing signs
from one harmonic to the next. This is in fact quite different to the case of atoms and moleculesinteracting with a linearly polarized pulse, where only odd harmonics appear in the respective HHG
spectrum.
In this thesis we consider the cases of neutral H2 and ionic H2+ interacting with bi-chromatic coW
and counter- rotating fields of wavelengths 400 nm and 800 nm at intensity I = 1 × 1013 cm
2.

The fields are shown in figure 3.29a-b where (a) corresponds to the co-rotating case, and (b) the
counter-rotating case.

(a)

(b)

Figure 3.29: Plots of bichromatic laser fields of wavelengths 400 nm and 800 nm with intensity
W
I = 1 × 1013 cm
2 for cases where the fields are (a) co-rotating and (b) counter-rotating.

We first consider H2 under the presence of bichromatic pulses for both the cases mentioned above.
We obtain HHG spectra by performing calculations using TDDFT. Figure 3.30a-b shows the HHG
spectra for the cases of (a) co-rotating and (b) counter-rotating bichromatic fields. By inspection of
the respective spectra, we can see that harmonic generation is greatly suppressed when the fields are

63
co-rotating and the intensity of harmonics diminishes greatly as harmonic order increases, however,
harmonics of order nω seem to be present as discussed in [30]. When the fields are counter-rotating
the intensity of generated harmonics falls off slower than in the co-rotating case and higher ordersof harmonics are present in the HHG spectra. In addition, we observe that harmonics generated
are of order (3n ± 1)ω as discussed in [30].

10 -2

10 -2
10 -4

10 -4

10 -6

Intensity (a.u)

Intensity (a.u)

10 -6

10 -8

10 -10

10 -10
10 -12

10 -12

10 -1410 -8

10 -14

1

3

5

7

9

11

13

15

17

19

10 -16

1

3

5

7

9

11

Harmonic order

Harmonic order

(a)

(b)

13

15

17

19

Figure 3.30: HHG spectra of H2 interacting with bichromatic circularly polarized pulses of waveW
lengths 400 nm and 800 nm with intensity I = 1 × 1013 cm
2 for the cases where the fields are (a)co-rotating and (b) counter-rotating.

In conclusion, HHG spectra obtained from molecules are highly dependent on the type of laser
used. As we saw the differences between linearly and circularly polarized pulses producing HHG
vary drastically. This tells us the three-step semi-classical model can be heavily modified dependingon the laser used, and that in molecules the prediction of HHG patterns is not as simple as in atomic
systems.

64

3.5

Summary
In this chapter we investigated the appearance of Mollow sidebands appearing in HHG spectraof open shell molecules such as N2+ and C2 H4+ . We found that the sidebands were caused by alaser induced coupling between orbitals. We illustrated the appearance of the phenomena for bothparallel (σ-σ) transitions and perpendicular (σ-π) transitions. We further analyzed the appearanceof Mollow triplets by considering a simple two level system given by the optical Bloch equations andshowed that Rabi oscillations could indeed be captured by both theories, though certain differences
arise in HHG due to the complexity and approximations of the theories. We also considered how
different pulse envelopes affect the appearance of Mollow sidebands and showed that the resolutionof the sidebands in HHG spectra are highly dependent on the envelope used. We then looked at
the time frequency analysis to give insight in the generation of harmonics during the interaction
with the pulse, which we connected to the timescale of the trajectories of the electron due to theinteraction with the pulse.
We also considered the effects of HHG when molecules interact with two linearly polarized pulses.
We considered the following two cases: (a) a 400 nm pulse with polarization fixed parallel to themolecular axis and (b) an 800 nm pulse with polarization varied at angles θ = 0o to θ = 90o withrespect to the molecular axis. We found that while some features of single color HHG remain incase (a), the features are greatly modified for case (b), and the addition of the second laser does
indeed interfere with the nonadiabatic effects which we observed for the single color case.
Furthermore, we investigated how ellipticaly polarized harmonics can be formed from the interaction
between CO2 and linearly polarized pulses with orientation varied from θ = 0o to θ = 90o withrespect to the laser polarization. We considered the single orbital contributions to ellipticity andcompared to experiments reported in [24,25]. We showed that results from experiments and theory
agree.
Lastly, we looked at the cases where H2 and H2+ interact with bichromatic circularly polarized
pulses at optical frequencies ω and 2ω respectively for the cases of co- and counter rotating pulses.

65
We compared to previous theoretical results performed in [30] and confirmed that the respective
HHG spectra are indeed different than those resulting from interactions with linearly polarized
pulses, which in turn leads to the question of whether the three-step semi-classical model of HHG
must be modified in the case of pulses with non-linear polarization, and that such pulses lead tointeresting features in HHG spectra.

Chapter 4

Multi-Electron Effects in Strong-Field Ionization

Ionization is a fundamental process that is often studied in AMO, and it can act as a way toobtain information about the dynamics of atomic and molecular systems. While methods such as attosecond streaking and reconstruction of attosecond harmonic beating by interference of two-photontransitions (RABBITT) [31,32,33,34,35,36] allow for the experimental measurement of ionizationof electrons at their fundamental timescale, and give insight into the dynamics occurring during
the interaction of ultrafast, high intensity laser pulses with atoms and molecules it is useful to
use theories such as TDDFT to see if we can extract the same information as in the experiments.
While the tunneling ionization model helps paint a picture of the dynamics involved in strong-field
ionization of atoms, it lacks the modifications necessary to describe the mechanism in molecules,
which implies that there are fundamental differences in the strong-field ionization mechanism of
atoms and molecules.
Earlier experimental studies showed that atoms and molecules with similar ionization potentials
exhibit similar behavior, e.g. their ionization yields were similar when hit with laser pulses of equalpeak intensities [37,38,39]. Later experiments discovered that it is not always the case. Ionization
experiments using O2 and Xe, which have similar ionization energies, differed greatly, and this was
greatly due to the orbital structure in O2 [40]. This in fact showed that the tunneling formula was
not enough to describe ionization in molecules due to the more complex properties present and orcomplex dynamics occurring.
In this chapter, we study the ionization effects of molecules in the cases where a laser induces

67
resonant couplings between orbitals, which as shown in previous chapters can lead to non-adiabatic
behavior of electron dynamics, also seen in HHG via the occurrence of Mollow sidebands. In addition
we study the effects of ionization when multiple lasers interact with a molecule, in particular, if
ionization can be enhanced or suppressed.

4.1

Strong-Field Ionization
Since the early days of quantum mechanics, people have studied ionization in such systems,

but it was not until 1979 that Agostini et al. [41] observed ionization in the non-perturbative lightmatter interaction regime. Before this, data obtained by ionization experiments was well describedby low-order perturbation theory. However, in the strong-field regime, i.e the non-perturbative
regime, it was observed that an atomic or molecular system could in fact absorb multiple photons
above the ionization threshold with comparable probabilities and thus the result is called AboveThreshold Ionization (ATI). Typically such a process as ATI occurs in the intensity regime aboveW
1013 cm
2 . In this intensity regime atoms and molecules are significantly or even completely ionized

before the pulse reaches its maximum. To avoid this, it is typical that systems with high ionization
potentials, or in the case of this thesis, ultrashort pulses, in the femtosecond regime are used.
Up to this point, strong-field ionization in atoms has been well understood, in particular, in thehigh-intensity and long-wavelength regimes as described by the so called Keldysh parameter γ < 1[42] given by:
γ=

ω
,
ωt

(4.1)

where ω is the laser frequency and ωt is given by the equation:Eoωt = p ,
2Ip

(4.2)

where Eo is the amplitude of the laser field, and Ip is the ionization potential. With this, strong-field
ionization of atoms can be described as the electron tunnel-ionizing out of the Coulomb potential
as a result of the interaction with the laser pulse. This process is the same as the first step inthe semi-classical three-step model mentioned earlier in this thesis in the chapter devoted to HHG.

68However, due to the multi-electron nature of molecules, the ionization mechanisms differ from thatin atoms.

4.1.1

Multiphoton Ionization

When the Keldysh parameter is small, γ < 1, we expect tunneling ionization to occur, however
when γ >> 1, meaning that the ionization potential is much greater than the ponderomotive energy
Up given in equation (3.3), which is common for low intensity fields, or high frequencies then thedominant form of ionization is multiphoton ionization (MPI). In such a process, n photons areabsorbed by the electron in the ground state interacting with the field, and the electron transitions
via some virtual state (see figure 4.1b) into an excited state or into the continuum (see figure 4.1a).
If we increase the strength of the field, many more photons are absorbed and we expect effects such
as HHG and ATI to occur.

(a)

(b)

Figure 4.1: Figures depicting multiphoton ionization where (a) an electron is ionized into thecontinuum, and (b) an electron is excited into a higher energy state via multiphoton absorption.

69

4.2

Charge-Resonance Enhanced Ionization
One fundamental difference between ionization of atoms and molecules is charge-resonanceenhanced ionization (CREI). First observed in 1995 for H2+ at an internuclear distance Rc , larger
than its equilibrium bond length [43]. At this critical internuclear distance Rc the 1σg and 1σu
orbitals in H2+ become close in energy, as shown in figure 4.2b, and a charge resonant pair ofstates is created. Figure 4.2a shows the ionization rate as a function of internuclear distance forthis study. We can see that at large internuclear distances, especially at Rc = 10 a.u. the rateof ionization reaches a maximum, and as a result of the study, it was determined that at thisinternuclear distance there is in fact a charge resonance occurring between the orbitals, which inturn leads to an enhancement in ionization.

(a)

(b)

Figure 4.2: (a) Ionization rate of H2+ as a function of internuclear distance, when interacting with+W
a linearly polarized 1064 nm laser with intensity I = 1014 cm
2 and (b) the energy levels of H2 as afunction of internuclear distance. Adapted from [43] and [28].

70
4.2.1

Charge-Resonance Enhanced Ionization in Multi-Electron Molecules

While the study conducted in [39] gave insight into differences in strong-field ionization of
atoms and molecules, it was still limited due to the one electron nature of H2+ . It is worthwhile tostudy strong-field ionization and CREI in multi-electron molecules.
Using TDDFT, we solved the time-dependent Kohn-Sham equations (2.31), and obtained the probability density in the ith orbital. From there we obtain the population of a given orbital using:nks
i (t) =

Z

d3 r|φi (r, t)|2 ,(4.3)

where φi (r, t) is the ith Kohn-Sham orbital, and nks
i (t) is the time dependent population of thegiven orbital. The resulting ionization probability of the ith orbital is given by:

Piks (t) = 1 − nksi (t).

(4.4)

Typically, one would expect that the highest occupied molecular orbital would in fact contributethe most towards ionization due to the smaller ionization potential. This is in fact the case forclosed shell molecules, as well as those in which there is no laser induced coupling between orbitals.
W
Such examples are shown below in figure 4.3, where a 400 nm pulse at I = 2 × 1014 cm
2 withpolarization along the molecular axis interacts with N2 (left), and a 600 nm pulse at I = 2 × 1014
Wwith polarization also along the molecular axis (right) interacts with N2+ . Due to N2 being a
cm2

closed shell molecule we expect the HOMO to contribute the most to ionization due to the smaller
ionization energy, and this is in fact the case, albeit the fact that the laser parameters cause aresonant coupling between HOMO and HOMO-2 in N2+ thus resulting in CREI. In the second case
we see that even though the molecule is open-shell we do not observe CREI since to the opticalfrequency of the laser is not equal to the frequency corresponding to the energy difference between
orbitals, therefore not inducing a resonant coupling of orbitals.

71

σg
2σu
1 πu x1 πu y3σg

Ionization Probability

0.10

2

0.08

,
,

0.06

0.04

0.02

0.00
0

2

4

6

8

10

12

14

16

18

Time (fs)(a)

0.0025σg ↓σu ↓1 πu x ↓1 πu y ↓
2σg ↑
2σu ↑
1 πu x ↑1 πu y ↑3σg ↑2
2

Ionization Probability

0.0020,,0.00150.0010,,0.00050.0000
0

2

4

6

8

10

12

14

16

18

20

Time (fs)(b)

Figure 4.3: Ionization probability for two cases: (a) N2 in the presence of a 400 nm laser pulse with+W
intensity I = 2 × 1014 cm
2 and polarization along the molecular axis, and (b) N2 in the presenceWof a 600 nm laser pulse with intensity I = 2 × 1014 cm
2 and polarization along the molecular axis.

In both cases we can see that the HOMO is the orbital that contributes the most to ionization.
Interestingly enough, we would expect the other orbitals ionization probability to be almost negligible, this is in fact not what we see. For both cases the HOMO-2 still has a significant contribution to

72
the ionization. This brings in the following question: is the ionization potential the only factor wemust consider? Well, the answer to this question is no, other factors such as orbital symmetry andmolecular orientation with respect to the laser pulse play a role in the ionization process. Consider
the HOMO-2 which is a 2σu orbital and the HOMO-1 which is a 1πu orbital in N2 and N2+ . The
HOMO-2 extends along the molecular axis, so when the polarization of the pulse is oriented along
the molecular axis, the ionized wavepacket interferes constructively along the polarization direction,
thus enhancing the ionization of the HOMO-2, while if the polarization of the pulse was aligned
perpendicular, the ionized wavepacket would destructively interfere and hence suppress ionization.
A similar phenomenon happens for the case of 800 nm in which a resonant coupling occurs between
HOMO and HOMO-1 when the polarization is perpendicular to the molecular axis. The HOMO-1
as a π orbital extends perpendicular to the molecular axis, hence it is more easily ionized when
the polarization is perpendicular to the molecular axis. In addition, in N2 it is possible that the
HOMO-1 contributes more to ionization than the HOMO as can be seen in figure 4.4.

σg
2σu
1 πu x1 πu y3σg
2Ionization Probability

0.04

,

0.03

,

0.02

0.01

0.00
0

2

4

6

8

10

12

14

16

18

Time (fs)Figure 4.4: Ionization probability for N2 while under the presence of an 800 nm laser pulse withW
intensity I = 2 × 1014 cm
2 and polarization perpendicular to the molecular axis.

73
4.2.2

Single-Color Ionization

We begin by first investigating ionization when a single laser field is applied to a system of
N2+ and C2 H4+ . In prior discussion we established that there is a laser induced resonance couplingin N2+ between the 3σg and 2σu orbitals when the laser pulse is at 400 nm and has a polarization
parallel to the molecular axis, and between the 3σg and 1πu orbitals when the laser pulse is at
800 nm and has a polarization perpendicular to the molecular axis. Similarly, in C2 H4+ a coupling
between the 1b3g and 1b2u orbitals occurs when the laser pulse is at 400 nm and polarized parallelto the molecular axis. As a result of such laser induced resonances we expect that there will be anenhancement of ionization.
The first case we look at is the application of a 400 nm laser pulse polarized in the direction of theWmolecular axis of N2+ at intensity I = 2×1014 cm
2 . The orbital contributions to ionization are shownin figure 4.5. We saw before that for the cases of neutral N2 and nonresonant laser frequencies
that the 3σg orbital would contribute the most to ionization, shown in figures 4.3a-b. However
we now see that as a result of the laser induced coupling of orbitals the 2σu orbital contributes
the most to ionization. This however brings about another question which is why does the spin
down electron in 2σu ionize more than the spin up electron in 2σu ? The answer to this questionis actually quite simple, the presence of a spin down hole makes the transition by the spin down
electron more probable, whereas for the spin up electron in the 2σu orbital to excite into the 3σgorbital, the spin up electron occupying it would have to be ionized, first due to the Pauli exclusion
principle. Hence we expect the ionization to be greater for the electron with the same spin as theelectron hole. The effect is the same for the case of N2+ interacting with an 800 nm pulse withpolarization perpendicular to the molecular axis. Indeed the 1πu orbital contributes the most to
ionization as a result of the laser induced resonance between 3σg and 1πu . Again we see one spinstate contributing more to ionization than the other.

74

2σg ↑
2σu ↑
1 πu ↑
1 πu ↑
3σg ↑
2σg ↓
2σu ↓
1 πu ↓
1 πu ↓

0.012

Probability

0.010
0.008
0.006
0.004
0.002
0.000

0

2

4

6

8

10 12 14

Time (fs)Figure 4.5: Ionization probability for N2+ while under the presence of a 400 nm laser pulse withW
intensity I = 2 × 1014 cm
2 and polarization parallel to the molecular axis.

2σg ↑
2σu ↑
1 πu ↑
1 πu ↑
3σg ↑
2σg ↓
2σu ↓
1 πu ↓
1 πu ↓

0.0006

Probability

0.00050.00040.00030.00020.00010.0000

0

4

8

12

Time (fs)16

20

Figure 4.6: Ionization probability for N2+ while under the presence of an 800 nm laser pulse withW
intensity I = 2 × 1014 cm
2 and polarization perpendicular to the molecular axis.

In C2 H4+ we saw that there is a laser induced resonance transition between the 1b3g and 1b2u

75
orbitals by a 400 nm laser pulse with polarization parallel to the molecular axis when there is anelectron-hole present in the 1b3g orbital. We then expect that as a result of this coupling that the
1b3g would contribute the most to ionization, and indeed this is what we observe as shown in figure
4.7, and the explanation is the same as for N2+ .

2ag ↑
2bu ↑
1b2u ↑
3ag ↑1b3g ↑
1b3u ↑2ag ↓2bu ↓1b2u ↓
3ag ↓
1b3u ↓

Probability

0.008

0.006

0.004

0.002

0.000
0

2

4

6

8

10

Time (fs)12

14

Figure 4.7: Ionization probability for C2 H4+ while in the presence of a 400 nm laser pulse withW
intensity I = 1 × 1014 cm
2 and polarization perpendicular to the molecular axis.

4.2.3

Two-Color Ionization

As we saw, CREI is indeed present when there is a laser induced resonance between orbitals.
However, it is worthwhile to investigate the effects of ionization when an additional laser is madeto interact with the system. We first consider two cases: (1) a 400 nm laser with fixed polarizationW
parallel to the molecular axis with intensity I1 = 2×1014 cm
2 , and an 800 nm laser with polarizationW
varying from 0o to 90o with respect to the molecular axis with intensity I2 = 1 × 1014 cm
2 , and(2) an 800 nm laser with fixed polarization perpendicular to the molecular axis with intensityWoo
I1 = 2 × 1014 cm
2 , and a 400 nm laser with polarization varying from 0 to 90 with respect to theWmolecular axis with intensity I2 = 1 × 1014 cm
2 . Figure 4.8 shows the total ionization for both cases

76

0.010

0.00225

0.009

0.00200

0.008

0.00175

0.007

Probability

Probability(1) and (2), respectively.

0.006

0.001500.001250.005

0.00100

0.004

0.000750.00050

0.003
0

20

40
60
Angle (Degrees)

80

(a)

0

20

40
60
Angle (Degrees)

80

(b)

Figure 4.8: Total ionization probability as a function of polarization alignment for cases (1) and(2) respectively. In (a) the polarization of the 800 nm pulse is varied from 0o to 90o with respect
to the molecular axis, and in case (b) the polarization of the 400 nm pulse is varied from 0o to 90owith respect to the molecular axis.

We can see that for case (1) the ionization probability is maximized when both the 400 nm
and 800 nm laser pulses have polarization parallel to the molecular axis, and minimized when thepolarization of the 800 nm pulse is oriented perpendicular to the molecular axis. For case (2) it is
not so straightforward, ionization is minimized when the 400 nm pulse is parallel to the molecularaxis, as in case (1), but we see that ionization is maximized when the 400 nm pulse is oriented at
60o with respect to the molecular axis, and not when it is 90o . In order to fully understand why
this is occurring we must look at the single orbital contributions to ionization.

4.2.3.1

Orbital Contributions to Ionization

As we saw, for case (1) ionization was enhanced when the polarization of the 800 nm pulse
was parallel to the molecular axis, and became further suppressed as we increased the polarization
angle to θ = 90o . Analyzing the single orbital contributions tells us which orbitals contribute most,77
but it still does not give us the insight as to why suppression of ionization occurs as we increase
the angle θ. We must then consider the time dependent projections between coupled orbitals,
which initially begin as orthogonal orbitals but become coupled during the interaction with thefield. Figures 4.9a-b show the projections between the orbitals 3σg and 2σu as well as 3σg and 1πu
for the cases where the polarization angles of the 800 nm pulse are (a) θ = 0o and (b) θ = 90o .
We see that for case (a) where the 800 nm pulse has polarization parallel to the molecular axis
the projection between 3σg and 2σu has more irregular oscillations than in case (b), which can beinterpreted that the projected orbitals, which are typically orthogonal, being more in resonanceduring the interaction with the laser field and so ionization is enhanced, whereas in case (b) we
see that the oscillations occurring in the time dependent projections of orbitals behave in a more
periodic manner with values ranging from 0 to 0.9, meaning that there are more times in which theorbitals become orthogonal, i.e. nonresonant, resulting in a suppression of ionization. From here we
can interpret that as the polarization angle of the 800 nm pulse is changed from θ = 0o to θ = 90othere are more instances of the coupled orbitals going out of resonance during the pulse, hence
reducing the total ionization. We also see, based on the projections, that the major contribution
to ionization for case (1) should be coming from the 2σu even as the angle θ is increased. We canconfirm this by looking at the peak ionization for each orbital as a function of angle θ, as shown infigure 4.10. We see that for all angles the 2σu is ionized the most just like in the single color case.
However the probability of ionization of each orbital is less than that in the single color case, which
can mean that the addition of the second pulse is indeed suppressing total and partial ionizationof orbitals.

78
1.0

1.0

|⟨1πu|⟨σg⟩|2|⟨1πu|⟨σg⟩|2|⟨2σu|⟨σg⟩|2|⟨2σu|⟨σg⟩|2
0.8

Projection

Projection

0.8

0.6

0.4

0.6

0.4

0.2

0.2

0.0

0.0
0

2

4

6

8

10

Time (fs)(a) θ = 0o

12

14

16

18

0

2

4

6

8

10

Time (fs)

12

14

16

18

(b) θ = 90oFigure 4.9: Ground state orbital projections onto time-dependent orbitals for the cases of N2+
Winteracting with a 400 nm pulse of intensity I = 2 × 1014 cm
2 with fixed polarization along theWmolecular axis and an 800 nm pulse of intensity I = 1 × 1014 cm
2 with polarization (a) at angle

θ = 0o and (b) polarization at an angle θ = 90o with respect to the molecular axis.

79

0.008

Probability

3σg ↑1 πu ↑
1 πu ↑
2σu ↑2σg ↑1 πu ↓
1 πu ↓
2σu ↓
2σg ↓

0.006

0.004

0.002

0.000
0

20

40

60

Angle (degree)

80

Figure 4.10: Total ionization for each orbital as a function of angle θ for the case where N2+ interacts
with two pulses, a 400 nm one with polarization fixed parallel to the molecular axis and intensity
W
14 W with polarization angle varied
I = 2 × 1014 cm
2 , and an 800 nm one of intensity I = 1 × 10
cm2

from 0o to 90o with respect to the molecular axis.

In case (2) we fixed the polarization of the 800 nm pulse and varied the polarization of the
400 nm pulse, and as we saw ionization becomes more complicated than in case (1). Figure 4.8b
tells us that there is minimum total ionization when the polarization angle of the 400 nm pulse isθ = 0o . We would expect that the explanation to this is analogous to case (1) where the angle of
the 800 nm pulse is at θ = 90o where we see that the projection between coupled orbitals resonates
between 0 and 1 meaning that at points during the pulse there is no resonance between orbitals
so CREI is not occurring at times, however that is not what we are seeing when looking at thetime dependent projections, shown in figure 4.11a-b . In fact we see that there is strong coupling
between 3σg and 2σu as well as 3σg and 1πu . The electrons in both the 2σu and 1πu orbitals want
to excite into the 3σg however due to Pauli blocking, only one is able to be ionized at a given time.
Therefore ionization coming from both orbitals is suppressed as CREI only occurs at shorter time

80
periods during the laser pulse. We also see from figure 4.12 that there is a small difference between
the contributions of ionization from the 1πu and 2σu orbitals unlike in the single color case where
almost no ionization came from 2σu . Again, this is explained by looking at the time dependent
projections between orbitals which tells us that there is a strong coupling between 3σg and 2σumeaning that ionization from that orbital will be enhanced. When the polarization angle of the 400
nm pulse is θ = 90o we see that there is only a resonance between 3σg and 1πu which is expected
because the 400 nm pulse does not couple any orbitals when the polarization is perpendicular to
the molecular axis. We also see that when the angle is θ = 90o this becomes more like the single
color case of an 800 nm pulse polarized perpendicular to the molecular axis, and the ionization iscoming mostly from the 1πu orbital and the ionization from the 2σu goes almost to zero.

|⟨1πu|⟨σg⟩|2|⟨2σu|⟨σg⟩|2

0.25

Population

0.8

Population|⟨1πu|⟨σg⟩|2|⟨2σu|⟨σg⟩|2

0.6

0.4

0.2

0.20
0.15
0.10
0.05

0.0

0.00
0

2

4

6

8

10

Time (fs)(a) θ = 0o

12

14

16

18

0

2

4

6

8

10

Time (fs)

12

14

16

18

(b) θ = 90oFigure 4.11: Ground state orbital projections onto time-dependent orbitals for the cases of N2+
Winteracting with an 800 nm pulse of intensity I = 2 × 1014 cm
2 with fixed polarization perpendicularW
to the molecular axis and a 400 nm pulse of intensity I = 1 × 1014 cm
2 with polarization (a) at angle

θ = 0o and (b) polarization at an angle θ = 90o with respect to the molecular axis.

810.00160.00143σg ↑1 πu ↑
1 πu ↑
2σu ↑2σg ↑1 πu ↓
1 πu ↓2σu ↓2σg ↓Probability

0.00120.00100.00080.00060.00040.00020.0000

0

20

40

60

Angle (degree)

80

(a)

Figure 4.12: Ionization probability for each orbital as a function of angle θ for the case where N2+interacts with two pulses, a 800 nm one with polarization fixed perpendicular to the molecular axis
W
14 W with polarizationand intensity I = 2 × 1014 cm
2 , and a 400 nm one of intensity I = 1 × 10
cm2

angle varied from 0o to 90o with respect to the molecular axis.

As we saw case (2) had a maximum ionization when the polarization angle of the 400 nm pulse
was θ = 60o with respect to the molecular axis. This is unlike case (1) where the ionization has
a maximum and minimum at θ = 0o and θ = 900 respectively. In figure 4.12 we see that at thisangle there is also a large contribution to ionization from the spin up electron in the 2σu orbital.
This indicates that there is strong coupling between the spin up states of 3σg and 2σu . In fact
for all angles up to θ = 60o there seems to be a large contribution from the 2σu orbital, which iscontrary to what we have seen because a laser induced coupling between 3σg and 2σu should only
occur when the 400 nm laser is polarized parallel to the molecular axis. This is an indicator thatthe addition of a second pulse is breaking the symmetry of the system such that we are able to still
couple 3σg and 2σu even at polarization not parallel to the molecular axis, in turn the ionizationis enhanced due to charge resonance. Indeed we can further confirm this by looking at the timedependent projections of the coupled orbitals shown in figure 4.13, and we see that the 3σg and 2σufor both spin up and down remain coupled at this angle which in turn should enhance ionization

82
from those orbitals and contribute to the total ionization.

|⟨1πu ↓ |⟨σg ↓ ⟩|2

0.8|⟨2σu ↓ |⟨σg ↓ ⟩|2

Population|⟨2σu ⟩ |⟨σg ⟩ ⟩|2
0.6

0.4

0.2

0.0
0

2

4

6

8

10

Time (fs)

12

14

16

18

(a)

Figure 4.13: Ground state orbital projections onto time-dependent orbitals for the cases of N2+
Winteracting with an 800 nm pulse of intensity I = 2 × 1014 cm
2 with fixed polarization perpendicularW
to the molecular axis and a 400 nm pulse of intensity I = 1 × 1014 cm
2 with polarization at angle

θ = 60o with respect to the molecular axis for both the spin up and down orbitals.

4.3

Summary
In this chapter we investigated the effects of strong field ionization in multielectron systems

such as N2+ and C2 H4+ . We considered the scenarios where a linearly polarized pulse couples twoorbitals, and observed that the coupling leads to a charge resonance enhanced ionization. We firststudied the case where our system interacts with a single linearly polarized laser pulse.
Later on we looked at how the addition of a second linearly polarized case affects the ionization of
our systems. We again considered the two following cases: (1) a 400 nm pulse with polarization
fixed parallel to the molecular axis and an 800 nm pulse with polarization varied at angles θ = 0oto θ = 90o with respect to the molecular axis. We saw that for case (1) the behavior of ionization
was almost analogous to that of the single color 400 nm case observed earlier. We then showedthat for case (2) the addition of a second laser pulse further complicates the dynamics, and that
resonant induced coupling between orbitals remains present even as the polarization angle of one of

83
the pulses was changed, which is unlike the single color case and case (1), leading to enhancement
and suppression of ionization from certain orbitals.

Chapter 5

Electron Localization in Molecules

An indication of multielectron effects occurring in molecules can be seen by studying the
behavior of electrons within them while under the influence of a laser pulse. A useful way toinvestigate such effects is to look at the electron localization, which provides a way to study how
said electrons respond to the laser field.
With TDDFT it is possible to visualize electron localization. We do this by solving the timedependent Kohn-Sham equation (2.31), and by obtaining either the total time-dependent density
given by equations (2.33) and (2.35) for the cases in which no spin polarization or spin polarization
are implemented in TDDFT, respectively. It is also possible to look at the individual orbital
contributions to the electron localization, which gives further insight into how each given electron
in a molecule responds to the applied laser field.
This can be done using one of the following equations for spin unpolarized and spin polarized cases,
respectively:ρ(r, t) =NX|φi (r, t)|2 ,

(5.1)

i=1

ρ(r, t) =Xσ=↑,↓ρσ (r, t) =N
X X

|φi,σ (r, t)|2 ,(5.2)σ=↑,↓ i=1

where the index i corresponds to the ith molecular orbital.
Upon solving for the total and/or individual orbital density, the three-dimensional densities are85then integrated over the spatial dimensions except that along the molecular axis.

Zρ(z, t) =dxdyρ(r, t).

(5.3)

One thing to note is that this specific dimension is chosen because it allows us to see the dynamics
clearly, in any other dimension chosen it might be difficult to visualize.
After integrating onto the desired spatial dimension and obtaining ρ(z, t), the ground state density
ρ(z, t = 0) is subtracted:∆ρ(z, t) = ρ(z, t) − ρ(z, t = 0).

(5.4)

The resulting density difference is then plotted and the electron localization visualized using surface
plots. In certain cases the density difference is small so the time propagated density is shown.
In this chapter we will first consider electron localization in the presence of a one-color laser pulse
for both N2+ and C2 H4+ . We will then consider the case were a second laser is added to our system
and investigate the differences observed in comparison to the one-color case. We then finish bylooking at a different description of electron localization by Floquet Theory and connect to thatobtained by TDDFT.

5.1

Electron Localization in the Presence of One-Color Laser Pulses
We begin by looking at the case in which a molecule interacts with a single linearly polarizedlaser pulse. As before the molecules considered in our study are N2+ and C2 H4+ . Recall that for N2+
there is a resonant coupling of the 3σg and 2σu orbitals by a 400 nm pulse, and a resonant couplingof the 3σg and 1πu orbitals by an 800 nm pulse. Figures 5.1a-b show the electron localization
obtained from the total density for two separate cases: (a) a 400 nm laser pulse, and (b) an 800
nm laser pulse, all with polarization oriented along the molecular axis and with peak intensity
W
I = 2 × 1014 cm
2 . We begin by considering the total electron density, and later look at the orbital

contributions to the nonadiabatic effects arising in electron localization. Recall that within our field,

86
nonadiabatic behavior corresponds to the electron not following the oscillation of the electric field
during the time propagation and adiabatic behavior means that the electron follows the oscillationof the electric field.

4

0.6

0

0.4

−2

0.2

0.5

0.3

0.6
0.5

2
Z (a.u.)2
Z (a.u.)

4

0.7

0.4

0

0.3
0.2

−2

0.1

0.1

−4

0

5

10
Time (fs)

15

0.0

−4

0

5

10
15
Time (fs)

(a)

0.0

(b)

Figure 5.1: Visualization of total electron localization of N2+ along the molecular axis at differentWlaser wavelengths as a function of time: (a) 400 nm laser field of intensity I = 2 × 1014 cm
2 , andW(b) 800 nm laser field of intensity I =2 × 1014 cm
2 . In all cases the orientation of the polarization isparallel to the molecular axis.

It is evident that as the optical frequency gets closer to the resonant frequency between orbitals,
then the appearance of non-adiabatic dynamics becomes more visible. In the cases shown in figure5.1, the interaction with the 400 nm pulse makes the localization of the electron behave more
non-adiabatically, hence not following the laser field. As we increase the wavelength to 800 nm,
the electron dynamics behaves adiabatically. Visualizing electron localization gives insight into the
non-adiabatic behavior brought upon when coupling the 3σg and 2σu orbitals of N2+ .
Such behavior is also evident in C2 H4+ . Recall that the laser coupled orbitals are 1b3g and 1b2u
by a 400 nm pulse with polarization parallel to the molecular axis. As a result we expect electron
localization to be nonadiabatic, and indeed it is as seen in figure 5.2.

87

4

0.6
0.5

Z (a.u.)

2

0.4
0.3

0

0.2
0.1

−2
−4

0.0

0

5

10
Time (fs)

15

−0.1

Figure 5.2: Visualization of total electron localization of C2 H4+ interacting with a 400 nm laserW
field of intensity I = 1 × 1014 cm
2 with polarization parallel to the molecular axis. The direction isparallel to the C-C bond.

This nonadiabatic process occurring as a result of the laser induced coupling can be broken down
into two competing processes: first, there is an adiabatic response by the electrons to the laser field,
second the 400 nm laser couples the two orbitals which leads to an oscillation of the electrons at
the Rabi frequency Ωr resulting in nonadiabatic electron localization at certain times during thelaser pulse.

5.1.1

Single-Color Orbital Contributions to Electron Localization in Molecules

While the electron localization for the total density shows that non-adiabatic effects areindeed occurring when there is a laser induced resonant coupling of orbitals it is necessary to lookat the individual orbital contributions to see if the coupled orbitals are indeed the ones that arecontributing the most to this effect, or if other orbitals are also involved.
W
We first consider the case of N2+ interacting with a 400 nm laser pulse at intensity I = 2 × 1014 cm
2.

Figures 5.3a-d show the independent orbital contribution to electron localization for spin down
electrons and figure 5.4a-d for the spin up electrons.

88

0.5

4

2

0.4

2

0.3

0

z (a.u)z (a.u)

0.7
4

0.2

−2

0.6
0.5
0.4

0
0.3
−2

0.2

0.1
−4

0.1

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.0

0

2

4

6

(a) 2σg ↓810Time (fs)

12

14

16

18

0.0

(b) 2σu ↓

0.7
4

4

0.35

0.6

0.30

0.20

0

0.5

2

0.25
z (a.u)z (a.u)

2

0.4
0
0.3

0.15
−2

0.10
0.05

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.00

−2

0.2
0.1

−4
0

2

(c) 1πu ↓

4

6

8
10
Time (fs)

12

14

16

18

0.0

(d) 3σg ↓Figure 5.3: Visualization of orbital contributions to electron localization of N2+ in the presence ofW
a 400 nm laser pulse with intensity I = 2 × 1014 cm
2 and polarization along the molecular axis. The

orbitals are ordered, (a) 2σg , (b) 2σu , (c) 1πu , and (d) 3σg .

4

0.5

4

2

0.4

2

0.3

0

0.5
0.4

0

0.3

0.2

−2

0.7
0.6

z (a.u)z (a.u)

89

−2

0.2

0.1
−4

0.1

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.0

0

2

4

6

(a) 2σg ↑810Time (fs)

12

14

16

18

0.0

(b) 2σu ↑

0.7
4

0.4

2

0.6
2

0.3

0
0.2
−2

z (a.u)z (a.u)

4

0.5
0.4

0

0.3
−2

0.2

0.1
−4

0.1

−4
0

2

4

6

8
10
Time (fs)12(c) 1πu ↑

14

16

18

0.0

0

2

4

6

8
10
Time (fs)

12

14

16

18

0.0

(d) 3σg ↑Figure 5.4: Visualization of electron localization for the independent orbitals of N2+ in the presenceWof a 400 nm laser pulse with intensity I = 2 × 1014 cm
2 and polarization along the molecular axis.

The orbitals are ordered, (a) 2g , (b) 2σu , (c) 1πu and (d) 3σg for the spin up electrons.

Figures 5.3 and 5.4 (b) and (d) correspond to the 2σu and 3σg orbitals for both the spin down andspin up electrons respectively which are coupled by the 400 nm laser pulse. We can see that the
nonadiabatic effects are present in those two orbitals while all the others exhibit more adiabatic
dynamics. This confirms the idea that there are two competing processes occurring in nonadiabatic
electron localization as mentioned before. Interestingly enough we see in figure 5.4c that the 1πu90orbital exhibits some nonadiabatic signature during certain instances of time. We can explain thisby looking at the time-dependent energy expectation values hφi (0)|H(t)|φi (0)i of each molecularorbital φi during the propagation with the pulse shown in figure 5.5. We see that there are instancesin which there is an energy crossing by the 1πu and 3σg orbital energies, which indicate that at
those times there is a short lived orbital coupling, giving rise to nonadiabatic dynamics in theelectron localization of the 1πu orbital.

−0.4
−0.5
Energy Eigenvalue

−0.6

3σg
1πu, y1πu, x
2σu
2σg

−0.7
−0.8
−0.9
−1.0
−1.1

0

2

4

6

8
Time (fs)

10

12

14

16

(a)

Figure 5.5: Time-dependent energy expectation values of each molecular orbital of N2+ during theinteraction with a 400 nm laser pulse.

By the same argument as for N2+ , we then expect similar dynamics to occur in C2 H4+ , however
when looking at the independent orbital contributions for both spin down and spin up, figures
5.7 and 5.8a-f respectively, we see that nonadiabatic effects are present not only in the coupledorbitals 1b3g and 1b2u but also in other orbitals. This brings about the question of whether or not
nonadiabatic electron localization is indeed only caused by laser induced coupling of orbitals. By
similar analysis to that in N2+ we consider the time-dependent energy expectation values of C2 H4+

91
during the interaction of the laser pulse to provide us with an explanation of this phenomena as
shown in figure 5.6. We see that during the pulse there are many crossings of the energy values formany orbitals meaning that at certain times during the pulse there exist couplings between other
orbitals which will in turn result in the appearance of nonadiabatic behavior for different orbitals.

−0.6
−0.7
Energy Eigenvalue

1b3u
1b3g
3ag
1b2u
2bu
2ag

−0.8
−0.9
−1.0
−1.1
0

5

10

Time (fs)

15

20

(a)

Figure 5.6: Time-dependent energy expectation values of each molecular orbital of C2 H4+ during
the interaction with a 400 nm laser pulse.

The conclusion is then that other features such as orbital geometry and energy spacing between
orbitals must be considered when investigating such effects so it is expected that nonadiabatic
response in electron localization for C2 H4+ can indeed appear from orbitals that one would not
expect to be coupled by the laser pulse.

92
6

0.35

0.5

4

0.30

4

2

0.25

2

0.4

0

0.3

−2

0.2

−4

0.1

0.20

0

z (a.u)z (a.u)

6

0.15
−2
0.10
−4
−6

0.05

0

2

4

6

8
10
Time (fs)

12

14

16

18

0.00

−6

0

2

4

6

(a) 2ag ↓810Time (fs)

12

14

16

18

(b) 2bu ↓

6

6
0.40

4

0.30

0.20
0.15

−2

0.10

−4

0.05
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.4

2
z (a.u)z (a.u)

0.25

0

0.5

4

0.35

2

−6

0.0

0.00

0

0.3

−2

0.2

−4

0.1

−6

0

2

4

6

(c) 1b2g ↓810Time (fs)

12

14

16

18

0.0

(d) 3ag ↓

6

6
0.4

4

0.30

4

0.25

0
0.2
−2

0.20
0
0.15
−2

0.1

−4
−6

2

0.3
z (a.u)z (a.u)

2

0

2

4

6

8
10
Time (fs)12(e) 1b3g ↓

14

16

18

0.0

0.10

−4
−6

0.05

0

2

4

6

8
10
Time (fs)

12

14

16

18

0.00

(f) 1b3u ↓Figure 5.7: Visualization of electron localization for the individual orbitals of C2 H4+ while in theWpresence of a 400 nm laser pulse with intensity I = 1×1014 cm
2 and polarization along the molecularaxis spin down electrons.

93
6

6
0.5

4

2

0.4

2

0

0.3

−2

0.2

−2

−4

0.1

−4

0.0

−6

−6

0

2

4

6

8
10
Time (fs)

12

14

16

18

z (a.u)z (a.u)

0.5
4

0.4

0.3

0

0.2

0.1

0

2

4

6

(a) 2ag ↑

0.4

16

18

0.0

0.2
−2
0.1

−4

0

2

4

6

8
10
Time (fs)

12

14

16

18

0.4

2

0.3

0

0.5

4

z (a.u)z (a.u)

2

0.0

0.3

0
−2

0.2

−4

0.1

−6

0

2

4

6

(c) 1b2u ↑8
10Time (fs)

12

14

16

18

6

4

0.4

2
0

0.2
−2

z (a.u)

0.3

0.35

4

0.30

2

0.25
0.20

0

0.15
−2

0.1

−4

0

2

4

6

8
10
Time (fs)12(e) 1b3g ↑0.0(d) 3ag ↑6z (a.u)

14

6

4

−6

12

(b) 2bu ↑

6

−6

8
10
Time (fs)

14

16

18

0.0

0.10

−4
−6

0.05

0

2

4

6

8
10
Time (fs)

12

14

16

18

0.00

(f) 1b3u ↑Figure 5.8: Visualization of electron localization for the individual orbitals of C2 H4+ while in theWpresence of a 400 nm laser pulse with intensity I = 1×1014 cm
2 and polarization along the molecularaxis for spin up electrons.

94
While the interaction of molecules with ultrashort laser pulses showed interesting behavior occurringin electron localization we now ask the question of what will happen to electron localization if
another laser pulse is added to our system.

5.2

Electron Localization in the Presence of Two-Color Laser Pulses
While one color interactions bring about interesting nonadiabatic features in electron local-

ization as a result of two competing processes, it is worthwhile to ask if one can control theseeffects by the addition of a second laser pulse. We consider the case shown in figure 3.25a where
we have N2+ interacting with a 400 nm pulse with polarization fixed along the molecular axis, andan 800 nm pulse with polarization varied along the molecular axis with angle θ = 0o to θ = 90o inincrements of θ = 10o . For this case the pulse with the fixed polarization is designed to control the
coupling effects while the one whose polarization varies is designed to either enhance or suppress
such effects. We only consider the case with the 400 nm pulse with fixed polarization because as
mentioned, electron localization is difficult to see if we consider projecting along dimensions thatare not the same as the molecular axis. We present the total electron localization for each angle
θ in figures 5.9a-j. It is evident that nonadiabatic electron localization is neither suppressed norenhanced, in fact it seems to remain the same for all angles presented. This could be due to theWintensities we have chosen for the pulses. For the 400 nm pulse the intensity was I = 2 × 1014 cm
2,W
while for the 800 nm pulse the intensity was I = 1 × 1014 cm
2 , which are not too different, however

because the intensity of the 400 nm pulse and the transition dipole moment are stronger we expect
the orbital coupling between the 3σg and 2σu to be stronger than anything coupled by the 800 nm
pulse. Indeed this is the case as seen in figures 4.9a-b, where for θ = 0o we expect that only the 3σg
and 2σu are coupled and for the case of θ = 90o where we also expect a coupling between the 3σg
and 1πu orbitals in addition to the other one. Thus we can confirm that the greatest contribution
to the nonadiabatic effects is a result of the interaction with the 400 nm laser pulse.

95

4

0.8

2

0.6

0

0.4

z (a.u)z (a.u)

4

0.2

−2

0.8

2

0.6

0

0.4
0.2

−2

0.0

0.0

−4

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0

2

4

6

(a) θ = 0o

12

14

16

18

(b) θ = 10o

4

4

0.8

2

0.6

0

0.4

z (a.u)z (a.u)8
10Time (fs)

0.2

−2

0.8

2

0.6

0

0.4
0.2

−2

0.0

0.0
−4

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0

2

4

6

(c) θ = 20o

12

14

16

18

(d) θ = 30o

4

0.8

0.4

0

0.2

−2

z (a.u)0.6

2
z (a.u)8
10Time (fs)

4

0.8

2

0.6

0.4

0

0.2
−2
0.0

0.0
−4

−4
0

2

4

6

8
10
Time (fs)12(e) θ = 40o

14

16

18

−0.2

0

2

4

6

8
10
Time (fs)12(f) θ = 50o

14

16

18

−0.2

4

0.8

4

2

0.6

2

0.4

0

z (a.u)z (a.u)

96

0.8
0.6
0.4

0

0.2

0.2
−2

−2
0.0

0.0
−4

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

−0.2

−0.2

0

2

4

6

12

14

16

18

(h) θ = 70o

4

0.8

4

2

0.6

2

0.4

0

0.2

−2

z (a.u)z (a.u)(g) θ = 60o

810Time (fs)

0.8

0.6

0.4

0

0.2
−2
0.0

0.0
−4

−4
0

2

4

6

8
10
Time (fs)12(i) θ = 80o

14

16

18

−0.2

0

2

4

6

8
10
Time (fs)

12

14

16

18

−0.2

(j) θ = 90oFigure 5.9: Visualization of electron localization of the total density of N2+ interacting with a 400W
nm pulse of intensity I = 2 × 1014 cm
2 with polarization fixed along the molecular axis, and an 800
Woo
nm pulse of intensity I = 1 × 1014 cm
2 with polarization varied from θ = 0 to θ = 90 with respect

to the molecular axis.

5.2.1

Individual Orbital Contributions to Two-Color Electron Localization

As we saw the presence of a second laser pulse does not enhance or suppress nonadiabatic
dynamics in electron localization for the total electron density. However it is still worthwhile toinvestigate the individual orbitals and see if it is only the coupled ones that are exhibiting such

97
effects or if others are indeed playing a role. We begin by looking at the case where the 800 nm
Wopulse of intensity I = 1 × 1014 cm
2 is oriented at an angle θ = 0 , figure 5.10. This case is identicalto the single color case when only the 2σu and 3σg orbitals bring about the nonadiabatic dynamics.
This is expected because the 400 nm pulse is the only one inducing a resonant coupling between
orbitals while the 800 nm pulse does not induce a coupling when the polarization is oriented along
the molecular axis.

98

4

0.5

2

0.4

0.7

4

0.3

0

2
z (a.u)z (a.u)

0.6

0.4

0

0.3

0.2

−2

0.5

−2

0.2

0.1
−4

0.1

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.0

0

2

4

6

(a) 2σg θ = 0o810Time (fs)

12

14

16

18

0.0

(b) 2σu θ = 0o

0.40

4

0.7

4

0.35

0.6

0.30

2

2

0.5

0

0.20
0.15

−2

z (a.u)z (a.u)

0.25
0.4

0

0.3
−2

0.2

0.10
0.05

−4
0

2

4

6

8
10
Time (fs)1214(c) 1πu θ = 0o

16

18

0.00

0.1

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.0

(d) 3σg

Figure 5.10: Visualization of orbital contributions to electron localization of N2+ in the presence of aW
14 W
400 nm laser pulse with intensity I = 2×1014 cm
2 and an 800 nm pulse with intensity I = 1×10cm2both with polarization along the molecular axis. The orbitals are ordered, (a) 2σg , (b) 2σu , (c)1πu , and (d) 3σg .

We must turn back to the question of whether or not other orbitals exhibit such an effect when theangle θ of polarization is nonzero. We present results for the following angles: θ = 50o and θ = 90o .
We only consider these angles because presenting all of the figures for each angle would be in fact
quite cumbersome and will not bring much into our discussion of what is occurring.

99
We consider the case where the angle is θ = 50o shown in figure 5.11. We know that at thisangle there is no induced coupling from the 800 nm laser however, we indeed see that the electron
localization from the 1πu orbital behaves adiabatically except near the end of the pulse where we
begin to observe electron densities overlapping, and again this might not be a result of an induced
coupling but rather perturbations from the ramping down of the laser pulse. The 3σg and 2σu
orbitals still exhibit the most nonadiabatic behavior as is expected due to the resonant couplingby the 400 nm pulse. We can say that we do not expect the behavior from those two orbitals tochange as we vary the angle of polarization of the 800 nm pulse.

4

0.5

4

2

0.4

2

0.3

0

z (a.u)z (a.u)

100

0.2

−2

0.6
0.5
0.4

0
0.3
−2

0.2

0.1
−4

0.1

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.0

0

2

4

6

(a) 2σg θ = 50o

810Time (fs)

12

14

16

18

(b) 2σu θ = 50o

4

0.7

4

0.35

0.6

0.30
2

0.25
0.20

0

z (a.u)z (a.u)

2

0.5
0.4

0

0.3

0.15
−2

0.10
0.05

−4
0

2

4

6

8
10
Time (fs)1214(c) 1πu θ = 50o

0.0

16

18

0.00

−2

0.2
0.1

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.0

(d) 3σg θ = 50o

Figure 5.11: Visualization of orbital contributions to electron localization of N2+ in the presence ofW
a 400 nm laser pulse with intensity I = 2 × 1014 cm
2 polarized along the molecular axis and an 800
Wonm pulse with intensity I = 1 × 1014 cm
2 with polarization at an angle θ = 50 with respect to themolecular axis. The orbitals are (a) 2σg , (b) 2σu , (c) 1πu , and (d) 3σg .

On the other hand we know that there is an induced coupling between 3σg and 1πu by an 800
nm pulse with polarization oriented perpendicular to the molecular axis. Oddly enough, we do
not observe strong nonadiabatic effects arising in the electron localization of the 1πu orbital when
the angle is θ = 90o with respect to the molecular axis. However this could be due to projecting

101
the three-dimensional time-dependent electron density along the axis of the molecular axis. We
stated before that this was the ideal axis onto which to project as it would allow us to observe
the full features of electron localization, however this is not entirely true since the 1πu orbital
is not oriented along the molecular axis like the σ orbitals but rather perpendicular. Therefore,
we consider the projection onto an axis perpendicular to the molecular axis and look at just theelectron localization of the 1πu orbital. Figures 5.13a-b presents such cases and indeed we cansee that there is non-adiabatic electron localization present from the 1πu orbital as a result of thesecond 800 nm laser pulse oriented (b) perpendicular to the molecular axis, compared to (a) when
the 800 nm laser is oriented parallel to the molecular axis. Interestingly enough, while the casein which the 800 nm pulse has parallel polarization (a) exhibits adiabatic effects, the amount ofelectron density does not stay uniform. This could be a result of interferences occurring due to theother laser being present.
The results we presented confirm that both pulses are inducing separate resonant couplings, albeitof different strengths when the 400 nm pulse is parallel to the molecular axis and the 800 nm pulse
is perpendicular. We then conclude that two-color interactions do indeed enhance nonadiabatic
effects in electron localization, however with the caveat that certain effects can only be observed
when projecting along specific dimensions as we have shown.

102

0.5

4

2

0.4

2

0.3

0

z (a.u)z (a.u)

0.7
4

0.2

−2

0.6
0.5
0.4

0
0.3
−2

0.2

0.1
−4

0.1

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.0

0

2

4

6

(a) 2σg θ = 90o

810Time (fs)

12

14

16

18

(b) 2σu θ = 90o

4

0.7

4

0.35

0.6

0.30
2

0.25
0.20

0

z (a.u)z (a.u)

2

0.5
0.4

0

0.3

0.15
−2

0.10
0.05

−4
0

2

4

6

8
10
Time (fs)1214(c) 1πu θ = 90o

0.0

16

18

0.00

−2

0.2
0.1

−4
0

2

4

6

8
10
Time (fs)

12

14

16

18

0.0

(d) 3σg θ = 90oFigure 5.12: Visualization of orbital contributions to electron localization of N2+ while in theWpresence of a 400 nm laser pulse with intensity I = 2 × 1014 cm
2 polarized along the molecular axisWoand an 800 nm pulse with intensity I = 1 × 1014 cm
2 with polarization at an angle θ = 90 withrespect to the molecular axis. The orbitals are (a) 2σg , (b) 2σu , (c) 1πu , and (d) 3σg .

103
0.030

0.05

4

4
0.025

0.04
2

0.020

0

0.015
0.010

−2

y (a.u)y (a.u)

2

0.03
0
0.02

−2
0.01

0.005
−4

−4
0

2

4

6

8

10

12

Time (fs)

14

16

18

0.000

0

2

4

6

(a) 1πu , θ = 0o

8

10

12

Time (fs)

14

16

18

0.00

(b) 1πu , θ = 90oFigure 5.13: Visualization of electron localization for the 1πu orbital projected along the directionof the orbitals rather than the molecular axis for the cases when the 800 nm pulse of intensityWoo
1 × 1014 cm
2 is oriented at angles (a) θ = 0 and (b) θ = 90 with respect to the molecular axis.

5.3

Floquet Picture of Electron Localization
Aside from full 3-d calculations performed with Octopus, Floquet theory can provide a simple

picture of localization. We do so by approximating the laser field as a CW field, and by consideringa Hamiltonian for a two-level system that has the form:

H(t) =∆0[σuu − σgg ] − µgu E(t)[σgu + σug ],
2

(5.5)

where ∆0 is the energy difference between orbitals |gi and |ui, σij = |iihj| (i,j ∈ u,g), respectively,µgu is the transition dipole moment between orbitals, and E(t) is the applied laser field.
We consider an electric field that has a vector potential given by:
A(t) = Ao sin(ωt + φ) + Ã,(5.6)

where Ao is the amplitude of the vector potential, ω is the optical frequency of the field, φ is somephase shift, and Ã is an arbitrary shift to the vector potential that has no effect on the physical104
observables. This vector potential has a corresponding electric field given by:

E(t) = −dA(t)= −Eo cos(ωt + φ),
dt

(5.7)

where the field strength is given by Eo = ωAo . We then rewrite the above equations with adimensionless coordinate τ = ωt + φ, and obtain a TDSE given by:i∂|Ψ(τ )i = H̃(τ )|Ψ(τ )i,
∂τ

(5.8)

where H̃(τ ) = H(t(τ ))/ω. As is required by Floquet theory, we impose that the Hamiltonian be
2π periodic, so we can then say that this TDSE has a solution of the form:|Ψ(τ )i = e−ij τ |φFj (τ )i, j = 1, 2.

(5.9)

where j are the quasienergies of the Floquet states |φFj i which are eigenstates of the Floquet
Hamiltonian described in equation (2.63). We can expand the quasienergies by making the approximation that the optical frequency of the field ω is much larger than the energy difference ∆0
between states and/or the coupling between states is strong. The expansion is as follows:∆0J0 (ζ),
2ω

(5.10)∆0J0 (ζ),
2ω

(5.11)

1 = −

2 =where J0 (ζ) is the zeroth-order Bessel function of the first kind, and ζ =2µgu Eo, which is a ratioωof the Rabi frequency and the optical frequency of the field. With this we can then approximate
the corresponding Floquet states given by (see [45] for a complete and detailed derivation):∆0∆0cos(φ(τ ))ξs (τ ) + i sin(φ(τ ))ξa (τ )]|gi+
ω
ω

∆0
∆0
i sin(φ(τ )) +cos(φ(τ ))ξa (τ ) −sin(φ(τ ))ξs (τ )]|ui,ωω|φF1 (τ )i =[cos(φ(τ )) + i(5.12)105∆0∆0cos(φ(τ ))ξa (τ ) +sin(φ(τ ))ξs (τ )]|gi+
ω
ω

∆0
∆0
cos(φ(τ )) − i cos(φ(τ ))ξs (τ ) − i sin(φ(τ ))ξa (τ )]|ui,ωω|φF2 (τ )i =[i sin(φ(τ )) + i(5.13)

whereφ(τ ) = −µgu [A(τ ) − Ã],

ξa =

∞
X

J2k+1 (ζ)

k=0

ξs =

∞
X
k=1

(5.14)

cos[(2k + 1)τ ],2k + 1(5.15)

sin[2kτ ].
2k(5.16)

J2k (ζ)With all of the above the solution to equation (5.8) can be be expressed as a superposition of the
Floquet states (5.12) and (5.13) given by:|Ψ(τ )i = c1 e−i1 τ |φF1 (τ )i + c2 e−i2 τ |φF2 (τ )i,(5.17)

where c1 and c2 are coefficients that are constant in time.

5.3.1

Two-Color Charge-Resonance Localization

As we have discussed, ionic diatomic molecules in the presence of a laser field can undergo alaser induced coupling when the optical frequency of the pulse is close to the resonant frequency
between orbitals and the resulting electron density can oscillate between the two protons. We canextract the populations of the charge-resonant states given by the solution |Ψi either from full 3-D
calculations performed in Octopus or by using the two-state model described above in equation 5.5.
We consider the superposition which describes the electron localization at one of the protons as:
1|Li = √ (|gi + |ui),2(5.18)

1|Ri = √ (|gi − |ui),2(5.19)

106
we can then obtain the time-dependent local populations using:
PL =|hL|Ψi|2,|hL|Ψi|2 + |hR|Ψi|2(5.20)

PR =|hR|Ψi|2.
|hL|Ψi|2 + |hR|Ψi|2(5.21)

We know that in H2+ we observe charge resonance pairs between 1σu and 2σg orbitals at large
internuclear distances R. We then expect to see nonadiabatic effects in electron localization, andindeed this was shown by Takemoto and Becker [44] where they considered the internuclear distance of H2+ to be R = 7 a.u.. Figure 5.14 shows their results for electron localization for boththe (c) full 3-D TDSE and (d) Floquet theory. We see that Floquet theory indeed captures theessence of nonadiabatic electron localization. However many of the features included in the full 3-D
calculations are lost such as ionization of the electron into the continuum, similar to what occurs
when using the optical Bloch equations to study HHG. Nevertheless, Floquet theory gives us someinsight as to how the populations of electrons move around a molecule when there is a laser induced
coupling between orbitals.

Figure 5.14: Electron localization for H2+ with internuclear distance R = 7 a.u. for the cases of (c)full 3-D TDSE, and (d) Floquet theory. Adapted from [44].

While the full 3-D TDSE and Floquet picture of electron localization seem to agree for the case of
H2+ we must remind ourselves that this is a one electron system and that we are really interested
in looking at the multielectron effects in nonadiabatic dynamics in molecules. Unfortunately such

107
simple models as the one considered by the Hamiltonian in equation (5.5) fail to capture effects
from electrons not in the orbitals we consider. However, even a simple model can provide some
information about what is occurring and provide a simple picture that is otherwise too complicatedto explain when looking at full 3-D TDDFT calculations.
We compare the dynamics of the two-level system to those obtained by performing full 3-D calculations on Octopus for N2 + and C2 H4+ . We begin by first extracting the transition dipole moments
shown in table 5.1 by performing a DFT calculation in Octopus. This is done by taking the groundstate wavefunctions and using the integral below to calculate the transition dipole moment.

Zµ=Laser Wavelength (nm)
400
800
400

Ψ∗2 (~r)~rΨ1 (~r)d3~r.

Transition dipole momenth3σg |x|2σu ih3σg |y|1πu i
h1b2u |x|1b3g ispin down (a.u)
1.558
0.28682-1.7231(5.22)

spin up (a.u)
1.5691
0.236591.7431

Table 5.1: Transition dipole moments for orbitals coupled by laser field of given wavelength for N2+
and C2 H4+ .

Before going further we must consider the correct parameters, such that the quasienergies undergoa crossing, i.e., are degenerate. This is done by selecting an appropriate intensity of the pulse. In
the figures 5.15a-c respectively, we show the quasienergies as a function of intensity for the casesof N2+ interacting with a 400 nm and 800 nm laser field, and C2 H4+ with a 400 nm field. From
there we can choose an appropriate intensity in which there is a crossing, and set our laser fields
W
accordingly. We can see that for N2+ there is a crossing at about I = 1.5 × 1015 cm
2 for 400 nm,+
W
16 W for 400 nm.
I = 1.5 × 1016 cm
2 for 800 nm, and for C2 H4 at I = 4 × 10
cm2

108
0.6
0.4
0.4

ϵ1

0.0

ϵ2

Quasienergy

Quasienergy

0.2

-0.2

0.2

ϵ1

0.0

ϵ2

-0.2
-0.4

-0.4
-0.6
0

2 × 1015

4 × 1015

6 × 1015

8 × 1015

0

1 × 1016

5.0 × 1015

Intensity

1.0 × 1016

1.5 × 1016

2.0 × 1016

Intensity(a) N2+ λ = 400 nm(b) N2+ λ = 800 nm

0.4

Quasienergy

0.2

ϵ1

0.0

ϵ2

-0.2

-0.4

0

5.0 × 1016

1.0 × 1017

1.5 × 1017

2.0 × 1017

Intensity(c) C2 H4+ λ = 400 nm

Figure 5.15: Quasienergies 1 and 2 as functions of laser intensity.

We then use the chosen intensities and look at the left and right populations for each of theWmolecules. We first consider N2+ interacting with (a) a 400 nm pulse of intensity I = 1.5 × 1015 cm
2W
and (b) an 800 nm pulse of intensity I = 1.5 × 1016 cm
2 . Figures 5.16a-b show the respective left

and right populations for cases (a) and (b). We see that during the propagation of the pulse theelectron is localized at either the right or left side of the molecule. We note that the behavior of theoscillations of PL (t) and PR (t) is clearly nonadiabatic as there are instances during the propagation
where the populations stay more localized on one of the sides of the molecule.

109
0.8

Population

0.7
0.6
0.5

PL (t)

0.4

PR (t)

0.3
0.2
0

20

40

60

80

Time (a.u.)(a) N2+ λ = 400 nm

Population

0.7

0.6

0.5

PL (t)PR (t)

0.4

0.3
0

20

40

60

80

Time (a.u.)(b) N2+ λ = 800 nm

Figure 5.16: Time-dependent left- and right- populations for N2+ using the analytical solution
obtained from Floquet theory for the cases where the laser is (a) at 400 nm with intensity I =W
16 W .
1.5 × 1015 cm
2 and (b) at 800 nm with intensity I = 1.5 × 10
cm2

We now look at the time-dependent left and right populations of C2 H4+ interacting with a 400 nm
Wlaser at the chosen intensity I = 4 × 1016 cm
2 . Figure 5.17 shows both PL (t) and PR (t) for said case.

It is evident that the behavior is nonadiabatic however we see more localization along the middleof the molecule at certain times. This is similar to the effects seen in the full 3-D TDDFT orbital
contributions to localization for C2 H4+ (figure 5.7), where at certain times during the propagationof the pulse the electron is localized between the carbon nuclei.

110
0.8

Population

0.7
0.6
0.5

PL (t)

0.4

PR (t)

0.3
0.2
0

20

40

60

80

Time (a.u.)(a) C2 H4+ λ = 400 nm

Figure 5.17: Left- and right- time-dependent populations for C2 H4+ using the analytical solutionWobtained from Floquet Theory for the case where the laser is 400 nm with intensity I = 1.5×1015 cm
2.

In conclusion, we see that a simple two-level system can in fact predict electron localization inmolecules when there is a laser induced resonance between orbitals. However, we note that theperspective we gain by using this model is different than when using TDDFT, and we must analyze
electron localization using of left- and right- localized orbitals/states (i.e., charge resonance states)and populations which tell us that the population of laser induced charge resonance pairs is localizedon either side of the molecule and that each population behaves nonadiabatically when there is aresonance between orbitals. While the model helps predict some of the mentioned effects we musttake into consideration that we are neglecting full multielectron effects and ionization into thecontinuum that we can otherwise observe with the use of TDDFT. Nonetheless, the model helps
to simplify the effects occurring in nonadiabatic electron localization.

5.4

Summary
In this chapter we investigated the appearance of nonadiabatic effects in electron localization

for N2+ and C2 H4+ . We first considered the case where one laser pulse induces a coupling between
orbitals, and studied at the total electron localization and the orbital contributions of localization.
We determined that there are several competing processes that contribute to nonadiabatic electron
localization: (1) the oscillation of the electrons with the electric field and (2) the oscillation at

111
the Rabi frequency of the electrons coupled by the field. We then investigated the case where
our system interacts with two linearly polarized pulses, and exhibited the differences that arise
in nonadiabatic behavior of electron localization as a consequence of the addition of the second
pulse. Finally we investigated nonadiabatic electron localization with the use of Floquet theory
for a two-level system and analyzed how the population of the coupled orbitals varied between
the left and right sides of the molecule as a function of time and confirmed that in the case of aresonant coupling nonadiabatic behavior was evident. We also saw that while the way we looked
at localization using Floquet theory was different as compared to full TDDFT calculations, it gave
a simple description of the more complex dynamics occurring in the full 3-D model.

Chapter 6

Summary, Discussion, and Outlook

6.1

Summary and Discussion
In this thesis we investigated how multielectron effects play a role in the interactions between

molecules and intense ultrashort pulse lasers in the context of high harmonic generation, strongfield ionization, and electron localization. We considered various theories such as time-dependent
density-functional theory, optical Bloch equations, and Floquet theory to analyze our results andcompared each theory with each other.
In chapter 3 we studied the multielectron effects in molecules for HHG. We saw that a laser induced
coupling leads to the appearance of Mollow sidebands in HHG spectra for the cases of single-color
and two-color linearly polarized laser interactions for N2+ . We compared both TDDFT and optical
Bloch equations and found that while both theories vary greatly in complexity, they both capture
the essence of Rabi oscillations and hence the appearance of Mollow sidebands. We also analyzed
how the pulse envelope affects the resolution of the Mollow sidebands for C2 H4+ and found that
their appearance is greatly dependent on the laser pulse envelope used. We then investigated thegeneration of elliptically polarized harmonics from CO2 for various intensities and investigatedthe orbital contributions to ellipticity. Furthermore we compared the results to those obtained inexperiment and found that the theory and experiment agree. Finally we considered the interactionof H2 and H2+ with circularly polarized bichromatic pulses of commensurate frequencies for thecases where they are co- and counter-rotating and found that the HHG spectra show features of
the three-step model breakdown meaning that it must be modified for situations where the pulse

113
is not linearly polarized.
In Chapter 4 we considered the effects of strong-field ionization for both N2+ and C2 H4+ in the cases
where there was a laser induced coupling between orbitals and observed that the coupling enhances
ionization from inner shell orbitals that are coupled by the field in the form of charge-resonanceenhanced ionization. We then considered the case of interactions of N2+ with two-color linearlypolarized pulses where one pulse had fixed polarization and the other had polarization varied from
θ = 0o to θ = 90o . We found that the case where a 400 nm pulse had fixed polarization parallel tothe molecular axis and an 800 nm pulse had polarization rotated behaved almost analogous to thesingle-color 400 nm pulse case. In contrast, in the case where the 800 nm pulse had polarization
fixed perpendicular to the molecular axis and a 400 nm pulse had polarization varied behaved
different than what we have predicted based on previous results and that coupling between orbitals
remained even at polarization orientations where we would expect there not to be any coupling,
thus enhancing ionization.
Finally in chapter 5 we investigated the appearance of nonadiabatic behavior in electron localization
for both N2+ and C2 H4+ in the cases were a laser pulse induced a coupling between orbitals. We
observed such behavior in the cases of single color interactions and saw that it is apparent in the
total electron density and the single orbital density, which led us to the conclusion that there are twounderlying competing processes: (1) the electrons oscillate with the field, and (2) the electrons in
the coupled orbitals oscillate at the Rabi frequency which leads to nonadiabatic behavior in electron
localization. We then considered the case of two-color interactions similar to the other chaptersand observed that the addition of a second laser pulse does not enhance or suppress nonadiabatic
behavior in electron localization. Finally we looked at the Floquet picture of electron localizationin the context of how the population varied from side to side of the molecule and found that while
it gives a different picture than that obtained with TDDFT, the idea that laser induced coupling
between orbitals leads to nonadiabatic behavior is the same as that observed in TDDFT.

114

6.2

Outlook
While the analysis for many of our results were conclusive, there are certain open ques-

tions that remain when looking at the cases were there is a two-color laser pulse interaction withmolecules. In particular a question remains in the case where the 400 nm laser pulse had its polarization direction rotated from θ = 0o to θ = 90o with respect to the molecular axis and the 800 nm
laser pulse had polarization direction fixed perpendicular to the molecular axis. There appears to
be a coupling of the 3σg and 2σu orbitals even when the polarization direction of the 400 nm laser
pulse was not parallel to the molecular axis. Perhaps with more allotted time, we could have looked
further in to this to get a more detailed answer than what was provided in previous discussion inthis thesis. Other interesting cases could have been considered such as studying molecules different
than CO2 , N2+ , and C2 H4+ and applying similar scenarios as in this thesis and see if any of theeffects discussed change or if they remain the same or similar. Lastly, we could have further changed
the way we performed our calculations and investigated how different, if at all, our results would
have been. Such examples include trying different time-propagators aside from Crank-Nicolson,
trying different functionals, and even different space-time grids.

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Appendix A

This work utilized the RMACC Summit supercomputer, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder,and Colorado State University. The Summit supercomputer is a joint effort of the Universityof Colorado Boulder and Colorado State University [47]. In addition to Summit, the work presented in this thesis utilized the JILA computing clusters for calculations.