Theoretical studies on imaging of electron motion andexcitation on ultrafast time scales
byJ. Venzke
B.S., Drake University, 2016
M.S., University of Colorado at Boulder, 2020

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2021
Committee Members:
Andreas Becker, ChairJohn Bohn
Jose D’Incao
Michael Litos
Bengt Fornberg

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Venzke, J. (Ph.D., Physics)
Theoretical studies on imaging of electron motion and excitation on ultrafast time scales
Thesis directed by Dr. Andreas Becker

The development of ultrashort laser pulses with durations of a few tens of attoseconds (1 as= 10−18 s) has opened the possibility of observing and controlling the motion of electrons in matterat their natural time scale. However, the broad spectral bandwidth of the pulses pose the challengethat several linear and nonlinear pathways compete and interfere during the interaction of thosepulses with the target. The ionized electrons contain information about the target, the electronic
wavepacket prior to ionization, the laser pulse, and the various pathways to ionization. In thisthesis, we will use numerical and analytic tools to study and interpret these ionization spectra.
Additionally, we will examine electron excitation to Ryberg states driven by intense IR pulses anddiscuss the tools we are developing to study electron correlation.
First, we will give a brief review of ultrafast and ultrastrong physics literature. This is followed by an overview of the numerical and analytic methods used in this thesis including our recent
insights in frequency corrections for short pulses and work on using radial basis functions to solve
the time-dependent Schrödinger equation. We then focus on few photon ionization including effects
induced by ultrashort pulses, the development of novel generalized asymmetry parameters, and thereconstruction of electronic wavepackets undergoing attosecond motion. Next, we present theoretical studies concerning the analysis of excitation and stabilization effects related to Rydberg statesin the case of both linear and bi-circular laser pulses. Finally, we will discuss the development andfirst tests of a fully correlated two-electron code that utilizes a basis of hyperspherical harmonics.

Dedication

To my parents Jim and Kelli, siblings Tom and Andrea, brother-in-law Bobby, niece Jaidyn,
and nephew Robby.

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Acknowledgements

It is amazing how important a great support system is to achieving a PhD. There are many
more friends, family members, acquaintances, and colleagues who have helped me along the way
than I have space to mention here, but I will do my best. To start off, I would like to thank my PhD
advisor, Andreas Becker, for his never ending support, both professionally and personally, during
my PhD. I would also like to thank my committee members John Bohn, Jose D’Incao, Michael
Litos, and Bengt Fornberg for the fruitful scientific discussions along the way.
I would be admiss if I didn’t acknowledge my high school physics teacher David Sklenicka
for sparking my interest in physics. Without his influence, it is very unlikely I would have pursued
a PhD in physics. I would also like to extend many thanks my undergraduate community at
Drake University, most notably Klaus Bartschat and Nicolas Douguet for developing my interestin ultrafast physics and Timothy Urness for inspiring my interest in computer science.
I extend great gratitude to my fellow lab mates and collaborators both past and present:
Andrew Spott, Cory Goldsmith, Erez Shani, Brynn Reiff, Tennesse Joyce, Yonas Gebre, Spencer
Walker, Andres Mora, Bejan Ghomashi, Zetong Xue, Lucas Kolanz, Edward MacDonald, andAgnieszka Jaron-Becker. The countless scientific discussions and support through out my PhD
have been instrumental in developing the contents of this thesis.
Finally, I would like to extend my greatest thanks to my parents Jim and Kelli and siblings
Andrea and Tom for the endless support when there was the enviable setback, multi-week longhunts for bugs in my code, and when life threw curve balls. Without their support, there is no way
I would have been able complete my PhD.

Contents

Chapter
1 Introduction

1

1.1

Attosecond time scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Light sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.1

High harmonic generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

Free electron lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

2 Theoretical and experimental background7

2.1

Attosecond dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

Pump-probe spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3

Few photon ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4

Rydberg state excitations induced by strong fields . . . . . . . . . . . . . . . . . . . 11

2.5

Electron correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Modeling laser-atom interactions

15

3.1

Time-dependent Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2

Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1

3.3

Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1

Coordinate systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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3.3.2

Finite difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.3

Radial basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.4

Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.5

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.6

Bound state calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.7

Continuum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.8

Time propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.9

Convergence studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.10 Parallel Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4

Single-active electron potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5

Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6

Frequency shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6.1

Estimation of frequency shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6.2

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Imaging attosecond wave packet dynamics with photoelectrons
4.1

4.2

4.3

49

Short pulse effect in ionization of ground-state helium atom . . . . . . . . . . . . . . 51
4.1.1

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.2

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Generalized asymmetry parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.1

Generalized asymmetry parameters . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.2

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Wavefunction reconstruction method . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.1

Concept and theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.2

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.3

Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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5 Rydberg state excitations induced by intense near-infrared radiation
5.1

5.2

Linear polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.1.1

Angular momentum distributions . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1.2

Radiation spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Bi-circular polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.1

Excitation with co-rotating pulses . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2.2

Excitation with counter-rotating pulses . . . . . . . . . . . . . . . . . . . . . 113

5.2.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6 Towards studies of correlation effects in attosecond electron dynamics
6.1

96

118

TDSE in hyperspherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1.1

Hyperspherical harmonic definition . . . . . . . . . . . . . . . . . . . . . . . . 121

6.1.2

Jacobi coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.1.3

Electronic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2

Implementation and first tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7 Summary

133

Bibliography

135

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TablesTable
3.1

The relative error of the ground state energy for a hydrogen-like atom with a nuclear
charge Z. A grid spacing around 0.1 produces well converged results for Z = 1.
Excited states have lower relative error. . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2

Coefficients of SAE potentials in Eq. 3.33 for hydrogen, helium, and neon from [204]. 33

3.3

Several common analytic pulse envelopes. The conversion factor between N , asdefined in Eq. 3.45 (number of cycles within one standard deviation to either side of
the maximum), and the more typical NFWHM (number of cycles in the full-width halfmaximum of the electric field) is given. The last column shows the excess kurtosis
γ2 , defined in Eq. 3.47, which is independent of the pulse duration T . (Table from[236]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Figures

Figure1.1

Relevant space and time scales present in ultrafast physics. (Figure from [135]) . . .

1.2

Schematics of the three step model in which an electron tunnel ionizes, is acceler-3

ated in the continuum by the driving laser and recombines with the remaining ion
producing a high energy photon. (Figure from [197]) . . . . . . . . . . . . . . . . . .
1.3

Schematics of an FEL laser showing the electron bunch production, undulator, andbeam dump that are essential parts to generate an X-ray pulse. (Figure from [202])

2.1

5

An isosurface of the wavefunction of a helium atom in a 1s−2p+ superposition (|rΨ|)
undergoing attosecond motion. (Figure from [233]) . . . . . . . . . . . . . . . . . . .

3.1

4

8

The dipole oasis shows the laser parameters that ensure that the dipole and nonrelativistic approximations hold. (Figure from [151]) . . . . . . . . . . . . . . . . . . 15

3.2

Panels (a) and (b) show the convergence of the lowest energy bound states in Hydrogen without and with an attached Slater type orbital respectively for various radialspacing (dr). The solid blue line show the error in the 1s state and dashed lines show
the error for n = 2 states (2s, 2p−1 , 2p0 , and 2p1 ) in random order. Panel (c) shows
the minimum and maximum imaginary part of the calculated bound state energies.

25

x
3.3

The finite difference stencil near the edge of the grid. Panel (a) shows how in
the fourth order finite difference the stencil extends beyond the grid leading to aboundary condition being imposed at two points. Panel (b) shows the non-centered
4th order stencil that enforces a single boundary condition. . . . . . . . . . . . . . . 27

3.4

Convergence with respect to grid spacing for various orders of FD for the 1s (a), 2s(b), and 3s (c) states in hydrogen atom.

3.5. . . . . . . . . . . . . . . . . . . . . . . . 32

Scaling results for present Crank-Nicolson method. Two runs each for calculations
with 1 million (blue with squares) and 15 million grid points (green with triangles)
have been performed. Note that the scaling improves with increasing number of grid
points.

3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Temporal (left) and spectral (right) distributions of vector potential (solid lines) andelectric field (dashed line) for pulses with FWHM of one cycle (top), and three cycles(bottom) at central frequency ωA = 1.0 a.u. of the vector potential. Also shown is
the Gaussian envelope of the vector potential. (Figure from [236]) . . . . . . . . . . 37

3.7

Ratio ωE /ωA as a function of the normalized number of cycles N defined in Eq. 3.45.
The numerical results were calculated by maximizing Ẽ(ω) for Gaussian (dottedline), sech (dashed-dotted line), and sin4 (dashed line) envelopes and are comparedwith the simple analytic estimate (solid line) given in Eq. 3.43. The inset reveals aslight dependence on envelope shape, which can be attributed to the correction termin Eq. 3.43. (Figure from [236])

3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Photoelectron spectra P (k) as function of photoelectron momentum obtained forinteraction of hydrogen atom with laser pulses at central frequencies ωA = 2 (dashedline) and ωE = 2 (solid line) and duration of (a) 10 cycles and (b) 2 cycles FWHM.
(Figure from [236]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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3.9

Population in 2p state following one-photon excitation of hydrogen atom with alaser pulse as a function of ωA (left) and ωE (right) for different pulse lengths at
peak intensity 1012 W/cm2 . Each line represents results obtained for a fixed pulse
duration in terms of τ1 = 405 as. The results from time-dependent Schrödinger
equation (TDSE) calculations (a,b) and predictions within first-order perturbation
theory (PT) (c,d) are in excellent agreement. The vertical line marks the energy
difference between 2p and initial 1s state. The green dots indicate the maximum
excited state population for each pulse duration. (Figure from [236]) . . . . . . . . . 44

3.10 Numerical results for population in 2s state following two-photon excitation with
τ2 = 811 as. The vertical line marks half of the energy gap between 1s and 2s
representing the resonance condition for the two photon process. Other parameters
are the same as in Fig. 3.9. (Figure from [236]). . . . . . . . . . . . . . . . . . . . 45

3.11 HHG spectrum at driver wavelength 730 nm (ωcentral = 0.0625 a.u.) vs. number ofcycles NFWHM . In the upper plot the central frequency ωA = ωcentral while in thelower panel ωE = ωcentral . The vertical white dashed lines mark field-free transitionenergies between excited states and the ground state, while the green solid lines markthe harmonic energies np ωE with respect to the central frequency of the electric field.
(Figure from [236])
4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Schematic representation of one- (a), two- (b), and three-photon (c) ionization in
an ultrashort pulse. The Gaussian distribution shows the spectral width of theultrashort pulse centered about the energy of the 2p-state. The red arrows represent
the photon absorption pathways, and the black lines depict some of the resonant
structure of the helium atom. (Figure from [235]). . . . . . . . . . . . . . . . . . . 55

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4.2

Photoelectron angular distributions for helium atom ionized by laser pulses at central
photon energy ω0 = 22.2 eV, peak intensity I0 = 1011 W/cm2 , carrier-to-envelope
phase φ = 0 and three different pulse durations: N = 0.6 FWFM cycles (0.112 fs,
left column), N = 1.2 FWHM cycles (0.224 fs, middle column) and N = 1.8 FWHM
cycles (0.335 fs, right column). Angular distributions are obtained for three valuesof photoelectron energy E = 1.45(2ω0 − |E1s |) = 1.0 a.u. (top row), E = 2ω0 − |E1s |= 0.687 a.u. (middle row), and E = 0.58(2ω0 − |E1s |) = 0.4 a.u. (bottom row),where E1s is the energy of the 1s state in the SAE potential. (Figure from [235]) . . 56

4.3

Cross section σ(E) as a function of peak intensity for photoelectron emission atenergy E = 2ω0 − Ip = 18.7 eV in 0.5-cycle (one-photon process dominates), 1.0cycle (transition regime), and 1.5-cycle (two-photon process dominates). Central
photon energy ω0 = |E2p − E1s | = 22.2 eV and carrier-to-envelope phase φ = 0.
(Figure from [235])

4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Parameters β1 (a) to β4 (d) as function of pulse duration (FWHM) and peak intensity.
Central photon energy ω0 = |E2p − E1s | = 22.2 eV, carrier-to-envelope phase φ = 0
and photoelectron energy E = 2ω0 − Ip = 18.7 eV. (Figure from [235]) . . . . . . . . 58

4.5

Anisotropy parameters β1 (left) and β2 (right) as function of pulse duration andphotoelectron energy E for central frequencies ω0 = 1.1|E1s − E2p | (top row), ω0 =|E1s − E2p | (middle row) and ω0 = 0.9|E1s − E2p | (bottom row) at peak intensity of1011 W/cm2 and φ = 0. In each panel the dashed line corresponds to E = 2ω0 − Ip .
(Figure from [235])

4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Anisoptropy parameter β5 (a) as function of pulse duration and peak intensity at
central frequency ω0 = |E1s − E2p | and photoelectron energy 2ω0 − |E1s | and (b)as function of pulse duration and photoelectron energy at central frequency ω0 =0.9|E1s − E2p | and peak intensity of 1013 W/cm2 . The dotted lines correspond to
E = 2ω0 − Ip and E = 3ω0 − Ip . The other parameters are as in Figures 4.4 and 4.5,
respectively. (Figure from [235]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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4.7

Anisotropy parameters β1 (solid line), β2 (dashed line), β3 (dotted line) and β4(dashed-dotted line) as function of carrier-envelope-phase. The results averaged
using a Gaussian distribution for the CEP (c.f., Eq. (4.2)) with widths of (b) α = 0.2(in units of 2π) and (c) α = 0.4 are compared with the unaveraged results (a). Peakintensity: 1011 W/cm2 , central frequency: ω0 = |E1s − E2p |, pulse duration: 1.2
FWHM cycles and photoelectron energy 2ω0 − |E1s |. (Figure from [235])

4.8. . . . . . 62

Anisotropy parameters β1 (left) and β2 (right) as function of pulse duration of theGaussian window used in modeling the FEL pulses and photoelectron energy averaged over 10 (middle row) and 200 (bottom row) partially incoherent free electron
pulses as compared to a single shot result (top row). Peak intensity of the Gaussian window: I0 = 1011 W/cm2 and central frequency of the spectral distribution
ω0 = |E1s − E2p |. (Figure from [235])

4.9. . . . . . . . . . . . . . . . . . . . . . . . . . 64(a-c): Ionization pathways effective in different intensity and pulse length regimes.
(d) Photoelectron angular distributions for ionization of neon atom, prepared in
2p−1 − 3d2 superposition, as function of intensity and pulse length. (e) The same as(d) including additionally ionization from the 2p0 and 2p1 states. (Figure from [232]) 67
even/odd

4.10 Conceptual illustration of GAPs (for ∆m = 3). Left: Integrals I±, Eqs. (3,4),

are defined over regions of constructive and destructive inferference, indicated bydifferent colors (dark blue, light blue, dark red, light red), in xy-plane. Middle:
GAPs are constructed based on the parity of the parameter γ = le +me +lg +mg +Np
from the integrals over the regions denoted by a certain color (light blue, dark red).
Right: Examplary PADs displaying the asymmetry captured by the parameters foreven and odd γ. (Figure from [232]) . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xiv
4.11 Generalized asymmetry parameters A10 (solid lines) and A11 (dashed lines) as functionof intensity for ionization of superpositions of 2p−1 − 3d0 (a), 2p0 − 3d1 (b), and
2p1 − 3d2 (c) in neon atom with initial populatations of 0.5 for each state (top row).
Results in bottom row for superpositions with populations of P2p−1 = 0.98, P3d0 =0.02, P4s0 = 0.00057 (d), P2p0 = 0.94, P3d1 = 0.06 (e), and P2p1 = 0.88, P3d2 = 0.12(f). Results have been obtained for one-cycle (blue lines) and four-cycle (orange
lines) pulses at photoelectron energy E = 2ω − Ip . (Figure from [232]) . . . . . . . . 72
4.12 Same as Fig. 4.11 but for superpositions generated by a right-handed circularly
polarized pulse via one-photon transition in helium atom (a), three-photon transitionin neon atom (b), and one-photon transition in neon atom (c) (for details see text).
(Figure from [232]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.13 Asymmetry parameter A11 for ionization of helium atom (1s-2p1 ) with one- and fourcycle pulses at central frequency ω = 1.2ω0 . (Figure from [232]). . . . . . . . . . . 75

4.14 Asymmetry parameter A10 for ionization of helium atom (1s-2p1 ) with one-cycle
pulses at central frequency ω0 . Results of averages over different Gaussian distributions of CEP with width α are compared with those at fixed CEP (α = 0, solid line).
(Figure from [232]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.15 (a) Isosurfaces of the 1s − 2p1 wave function (|rΨ|2 is shown) evolving in time. (b)Ionization scheme in the x − y plane. The laser polarization along the x-axis is
depicted in red, the direction of the ionized electron is shown in orange, and thecharge migration is shown in yellow. Selection rules for ground state signal (c) andexcited state signal (d). The solid lines illustrate the transitions of photons absorbed
at the central frequency and the dashed lines show the pathways for short pulse effectdue to the large bandwidth of an ultrashort pulse. (Figure from [233]) . . . . . . . . 80

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4.16 Results of TDSE calculations for photoelectron angular distributions over one cycleof the probe pulse field for ionization of 1s − 2p1 superposition with ultrashort twocycle (left column) and ten-cycle (right column) laser pulse with peak intensities of1014 W/cm2 (top row), 1012 W/cm2 (middle row), and 1010 W/cm2 (bottom row).
The time delays ∆τ are given with respect to a reference time τ0 after the end of
the first pulse. (Figure from [233]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.17 Errors in amplitude (a, d) and phase (b, e) for the two-state reconstruction using
PADs generated with laser pulses at two, five and ten cycle pulse duration. Also
shown is the residual R(a) (c, f). The reconstructions are based on PADs at 20
time samples (∆t = τ /20) using PADs in the full x − y plane (upper row) andphotoelectron signals in forward-backward direction (lower row). (Figure from [233])86

4.18 Same as Fig. 4.17 but based on PADs at just two time delays. Errors in amplitude(filled symbols with solid lines) and phase (open symbols with dashed lines) are
shown for using PADs in the full x − y plane (blue squares) and in forward-backward
direction (orange circles). (Figure from [233]) . . . . . . . . . . . . . . . . . . . . . . 87
4.19 Comparison of amplitude error for (a) two and (b) five cycle probe pulse from calculations using full second order PT (blue), neglecting only the two-photon transition
from the excited state (orange) and neglecting both the one-photon transition fromthe ground state and the two-photon amplitude from the excited state (green) in thereconstruction. (Figure from [233]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.20 Amplitude (a, c) and phase (b, d) errors for ionization at a detuned photon frequencyof ω = 0.8ω0 (upper row) and ω = 1.2ω0 (lower row). The reconstructions are basedon full PADs in x − y plane at 20 time delays. (Figure from [233]) . . . . . . . . . . 90
4.21 Amplitude and phase error for the reconstruction of an arbitrary unknown two-statesuperposition. Reconstruction is based on full PADs in x − y plane at 20 time delays
with a five-cycle probe pulse. (Figure from [233]) . . . . . . . . . . . . . . . . . . . . 90

xvi
4.22 Comparison of (a) amplitude and (b) phase error for reconstruction using PADs atmachine precision (blue) and with accuracy limited to 1% and 10%. (Figure from
[233]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.23 Amplitude (a) and phase (b) error for variation of CEP only (blue squares with solidlines), intensity only (orange circles with dashed lines) and both CEP and intensity(green stars with dotted lines) at peak intensity of 1011 W/cm2 and pulse durationof 2 cycles. Reconstruction is based on full PADs in x − y plane at 20 time delays.
(Figure from [233]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.24 Amplitude (a, c) and phase (b, d) errors for the reconstruction of the superposition√(|1si + |2p1 i + |3p1 i)/ 3 in the helium atom (upper row: errors for 2p-state, lower
row: errors for 3p-state). (Figure from [233]) . . . . . . . . . . . . . . . . . . . . . . 93
5.1

Excited state distribution as function of n (vertical axis) and l (horizontal axis)
at the end of 20 cycle pulses with sine-squared envelope and peak intensities: (a)I0 = 3.4 × 1013 W/cm2 , (b) I0 = 6.0 × 1013 W/cm2 , (c) I0 = 8.6 × 1013 W/cm2 , and(d) I0 = 1.12 × 1014 W/cm2 . Left (right) column corresponds to cases in which the
Rydberg states are resonant with an even (odd) number of photons. (Figure takenfrom [237]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2

Same as Fig. 5.1 but for pulses with Gaussian envelope and 14 cycles FWHM. The
pulse duration approximately matches that for the sine-squared pulses in Fig. 5.1 at
FWHM. (Figure taken from [237]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3

The absorption pathways in an even (odd) photon process in the left (right) panel.
Green squares represent virtual states with energy N ω above the ground state andangular momentum l. The yellow squares represent real states labeled by the quantum numbers. Solid lines refer to open pathways while dashed lines represent pathways which are suppressed due to population trapping in a lower excited state.
(Figure taken from [237]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xvii
5.4

Excited state distribution as function of l (horizontal axis) and n (vertical axis) at
the end of 2 cycle (panels on left), 10 cycle (panels in middle), and 20 cycle (panels
on right) pulses, at low peak intensity I0 = 3.4 × 1013 W/cm2 (top row), and highpeak intensity I0 = 1.64 × 1014 W/cm2 (bottow row). (Figure taken from [237])

5.5. . 103

Population of excited states from a 20 cycle pulse with peak intensity of I0 = 1.64 ×
1014 W/cm2 and CEP of φA = 0.204π (a) and φA = 0.882π (b). (Figure taken from[237])

5.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Radiation spectra generated during the pulse (red dashed lines) and those including
line emissions after the pulse (black solid lines) are shown for peak intensities: (a)I0 = 3.4 × 1013 W/cm2 , (b) I0 = 6.0 × 1013 W/cm2 , and (c) I0 = 1.64 × 1014 W/cm2 .
The vertical gray lines show the field free energy differences between the np energy
levels (up to 14p) and the 1s ground state. (Figure taken from [237])

5.7. . . . . . . . 106

Excited state distribution as function of n (vertical axis) and ` (horizontal axis) for(a) m = −1, (b) m = −2, (c) m = −3 and (d) m = −4 at the end of 20 (at 800 nm)
cycle pulses (40 cycle at 400 nm) with sin squared envelope and total peak intensityof 1 × 1014 W/cm2 for co-rotating laser pulses of equal intensity. (Figure taken from[234])

5.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Absorption pathways in co-rotating laser pulses at frequencies ω and 2ω startingfrom a m = 0-state. Without lack of generalization it is assumed that both pulses
have left-handed helicity. Absorption of a photon at frequency ω and at frequency
2ω is represented by a red and blue arrow, respectively. The numbers in the boxes
denote the minimum number of photons to reach a certain level. (Figure taken from[234]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

xviii
5.9

Excited state distribution as function of orbital angular quantum number ` summed
over n ≥ 4 and m at (a) I400 = 5 × 1013 W/cm2 , I800 = 5 × 1012 W/cm2 , (b)I400 = 5 × 1013 W/cm2 , I800 = 1 × 1013 W/cm2 , (c) I400 = 5 × 1013 W/cm2 ,I800 = 5 × 1013 W/cm2 , and and (d) I400 = 1 × 1013 W/cm2 , I800 = 5 × 1013 W/cm2 .
Pulse durations: 20 cycles at 400 nm, 10 cycles at 800 nm. (Figure taken from [234]) 111

5.10 Excited state distribution as function of n (vertical axis) and m (horizontal axis)
summed over `. Laser parameters: 20 (800 nm) cycle pulses with sin squared envelope
and total peak intensity of 1 × 1014 W/cm2 for co-rotating laser pulses of equal
intensity. (Figure taken from [234]). . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.11 Distribution in magnetic quantum states for (a) ` = 2 and (b) ` = 3 and different
peak intensities of the 800 nm pulse. I400 = 5 × 1013 W/cm2 and other parameters
are as in Fig. 5.10. (Figure taken from [234]) . . . . . . . . . . . . . . . . . . . . . . 113
5.12 Same as Fig. 5.7 but for counterrotating laser pulses. (Figure taken from [234]). . 114

5.13 Absorption pathways in counter-rotating laser pulses at frequencies ω and 2ω startingfrom a m = 0-state. Without lack of generalization it is assumed that the pulses
at frequency ω has left-handed helicity, while the second harmonic pulse has righthanded velocity. Other symbols as in Fig. 5.8. (Figure taken from [234]). . . . . . 114

5.14 Excited state distributions as a function of m, summed over n ≥ 4 and ` (top),
and as a function of n and m, summed over ` (bottom), for the interaction with aleft-handed circularly polarized laser pulse at 800 nm (20 cycles) and a right-handed
circularly polarized laser pulse at 400 nm (40 cycles). Both pulses have the same
peak intensity of 5 × 1013 W/cm2 . (Figure taken from [234]). . . . . . . . . . . . . 115

5.15 Orbital angular momentum distributions in excited states induced by counter-rotating
laser pulses at 400 nm (20 cycles) and 800 nm (10 cycles) at peak intensities of (a)I400 = 5 × 1013 W/cm2 , I800 = 5 × 1012 W/cm2 , (b) I400 = 5 × 1012 W/cm2 ,I800 = 5 × 1013 W/cm2 , and (c) I400 = 5 × 1013 W/cm2 , I800 = 5 × 1013 W/cm2 .
(Figure taken from [234]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xix
6.1

Relation of the hyperradius (R) to the two single-electron sub coordinate systems(r1 , r2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.2

The Jacobi coordinates for three body interactions. Each coordinate can be obtained
from the previous by utilizing a Raynal-Revai coefficient. . . . . . . . . . . . . . . . . 123

6.3

Convergence of He atom first ionization potential (blue solid line) and excited states(blue dashed line) with RRC (left) and non-RRC (right) methods. Reference lines
for He-SAE (black), NIST energies (red), and literature convergence (green, [126])
are shown.

6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Rabi flopping between the 1s and 2p states in helium atom. The populations in the1s (blue), 2p (orange), and all other states (green) for the SAE code are plotted.
The black dots show the two electrons simulations using the RRC method with
Kmax = 48 agree well with the SAE results.

. . . . . . . . . . . . . . . . . . . . . . 131

Chapter 1
Introduction

In 1905, Einstein published a paper on the photoelectric effect [56], for which he later received
the Noble Prize. The effect shows when light is shined on an object an electron can be ionized froman atom, molecule, or solid by absorbing a photon with a sufficiently high energy. With the adventof the laser in 1960, it became possible to focus coherent light tight enough to allow the ionization
process to occur not only by absorbing one but many photons in rapid succession leading to aprocess known as multiphoton ionization [85]. Since then, laser intensities have increased to thepoint where the superposition of the electric field of the laser with the attractive potential produced
by a nucleus can create a barrier and allow the electron to tunnel and ionize from the atom [123].
Furthermore, depending on various laser parameters, the electron that tunneled through the barrier
can interact with the laser field and recombine with the residual ion. During the recombination
process, photons are released at harmonics of the driving laser’s central frequency. This process
can convert a large numbers of infrared (IR) photons into a single X-ray photon in a process known
as high-order harmonic generation (HHG) [71, 140, 40]. HHG is one method that allows for the
production of bright X-ray laser pulses with pulse durations as short as tens of attoseconds [256, 37](10−18 sec). Recent developments have allowed for the generation high-order harmonics using
not only linearly polarized lasers but also lasers or combination of lasers with other polarizations[64, 196, 125, 58, 100]. Such pulses provide the tools necessary for the imaging of electron dynamics
on their fundamental time scale [135].
Einstein’s photoelectric effect mostly considered continuous wave (CW) light that produced a2near delta function like energy distribution. These CW lasers are good for precision energy measurements, however, their time resolution it greatly limited. The recent advances in attosecond pulse
technology have strongly decreased the interaction time allowing for the imaging of (attosecond)
electron dynamics at the cost of energy bandwidths of the pulses on the order of several eVs. The
largely increased pulse bandwidth, however, leads to many competing pathways of ionization that
can involve different numbers of photons complicating the ionization process. The photoelectron
angular distributions (PAD) produced by such pulses contain complex shapes that encode featuresof the target’s initial wavefunction, ionization cross sections for each process, the ionizing pulse,
and the various competing pathways to ionization. This naturally raises the question of how can
one backtrack all of this information from such a PAD signal?
In this thesis, we will explore methods to understand and image the motion of electrons on
ultrashort time scales from ionization signals and photoelectron angular distributions. Then, we
will also analyze how intense IR light can cause that electrons are being trapped in highly excitedstates although the absorption of a single additional photon would lead to ionization. Finally, thefocus will be turned to a new numerical method that we are developing to study two fully correlated
electrons interacting with each other and an ultrafast laser pulse.

1.1

Attosecond time scale
The idea of an attosecond (1 as = 10−18 sec) is difficult to wrap ones mind around. Some

perspective can be gained by noting that there are more attoseconds in a single second than seconds
in the lifetime of the universe. At this ultrashort time scale, nearly all motion in an atom, moleculeor solid is frozen. Only the motion of electrons, particularly that in atoms as visualized in Fig. 1.1,
is non-negligible. With laser technology recently reaching pulse durations with sub 100 attosecond
pulse widths [256, 37], it is becoming possible to “image” electrons as they move around inside ofatoms and molecules. One approach to imaging electron motion will be presented in this thesis(see Chap. 4).

3

Figure 1.1: Relevant space and time scales present in ultrafast physics. (Figure from [135])

1.2

Light sources
The fields of ultrastrong and ultrafast physics rely on bright laser light to study and under-

stand the behavior of matter on short time scales. The light used ranges from intense IR radiation,that can bend atomic and molecular potentials enough to allow electrons to tunnel into the continuum, to hard X-rays that can ionize multiple electrons with a single photon. The intense IR lightis often produced using Ti:sapphire or similar laser setups. In the following subsections, we will
briefly describe the two major sources of ultrafast extreme ultraviolet (XUV) to X-ray sources andthe applications they are designed for.

1.2.1

High harmonic generation

High harmonic generation (HHG) is a process that converts low-energy photons of intense
IR laser pulses into high-energy photons of an X-ray laser pulse through a highly non-linear andnon-perturbative process. The process is often understood through the so-called three step model
shown in Fig. 1.2 [140, 40, 197]. First, the strong electric field of the IR laser bends the atomic
Coulomb potential allowing for the electron to tunnel ionize. Next, the electron interacts with the

4field gaining energy and reversing direction when the oscillating field changes direction. Finally,
the electron recombines with the ion releasing its energy in the form of a high-energy photon. For
linearly polarized laser pulses, the emitted electrons have energies corresponding to (2n+1)ω where
n is an integer and ω in the central frequency of the driving laser pulse.

Figure 1.2: Schematics of the three step model in which an electron tunnel ionizes, is accelerated in
the continuum by the driving laser and recombines with the remaining ion producing a high energy
photon. (Figure from [197])

Nowadays, HHG sources can produce laser pulses with arbitrary polarization [64, 196, 125, 58,100]. The theoretical method for generating circularly polarized HHG sources was proposed morethan two decades ago [53, 150]. By using two counter rotating cicularly polarized IR lasers withfrequencies ω and 2ω, a so called counter-rotating bi-circular laser pulse is produced. The three foldsymmetry of the final laser pulse allows for the electron to recombine with the parent ion, somethingthat is not possible with a circularly polarized pulse alone. The resulting HHG spectrum contains
circularly polarized harmonics at energies of (3n + 1)ω and (3n + 2)ω with alternating helices andno harmonic at energies of 3nω. Since then, non-collinear methods of circularly polarized HHG[100] and methods for elliptically polarized harmonics [12] have been demonstrated.
HHG sources have the advantage that they are compact table top setups. Additionally,
the light produced is coherent and carrier-to-envelope phase locking can be achieved in certain
cases, which make it possible to control the light produced in these sources. As a result, trains of5attosecond long pulses can be generated [128], and isolated pulses with a duration of only a few
tens of attoseconds have been demonstrated [256, 37]. For more details on HHG light sources, werefer to Ref. [197, 38]

1.2.2

Free electron lasers

Free electron lasers (FEL) differ significantly from HHG sources. Often located at large
shared user facilities [218], they start with a high energy electron beam from an accelerator. The
electron beam is then passed through an undulator consisting of alternating magnetic fields. The
electrons oscillate back and forth leading to a phenomenon known as micro bunching and thegeneration of light. After leaving the undulator, the electron beam is dumped while an X-ray laser
pulse propagates towards the experimental stations, as shown in Fig. 1.3. By tuning the spacing
between the magnets, the frequency of the light can be changed.

Figure 1.3: Schematics of an FEL laser showing the electron bunch production, undulator, andbeam dump that are essential parts to generate an X-ray pulse. (Figure from [202])

Due to the large amount of energy in an electron bunch, FEL sources are much brighter than
HHG sources in the same frequency range. FEL can produce arbitrary polarized laser pulses [109]and pulses with a duration less than a femtosecond have been demonstrated [158]. The laser pulses,

6
however, are only partially coherent and their shape varies widely from shot to shot limiting theirapplication for effects that depend significantly on the control of the pulse parameters. For moredetails on FEL technology, see [218].

1.3

Organization of this thesis
The remainder of this thesis is organized as follows. In Chap. 2 we cover the relevant the-oretical and experimental background. Chap. 3 provides a detailed description of the analyticaland numerical methods used in this thesis to model ultrafast laser-atom interactions. In Chap. 4
we present results of computations showing the impact of ultrashort laser pulses on the ionization
process and how the ionized electrons can be used to analyze electron motion on the attosecond
time scale. Next, in Chap. 5 various mechanisms by which electrons are excited into Rydberg states
when interacting with intense IR laser pulses are discussed. Finally, in Chap. 6 we provide details
on the development of a two-active electron code using hyperspherical harmonics including its current status, first tests, and possible paths forward. In this thesis, atomic units (~ = e = m = 1)
are used throughout unless noted otherwise.

Chapter 2Theoretical and experimental background2.1

Attosecond dynamics
Understanding and imaging motion on fast time scales has been of interest for more thana century. In 1878, Eadweard Muybridge invented high speed photography to find out if all ofa horse’s feet leave the ground at a full gallop [174]. Since then, high speed photography hasreached time scales of ∼ 10−5 s, a long ways off of the attosecond (10−18 s) time scale of electron
motion. On attosecond time scales, only electrons in atoms and molecules display significant motion(see Fig. 1.1) rendering electronic or mechanical shutters, from traditional photography, useless.
Additionally, the electrons being imaged are quantum objects further complicating the imaging
process. Before we go into the methods used for imaging electron motion, lets dive into what wemean when we say a quantum object is moving.
For a time independent system, such as an atom or molecule in vacuum, there exists a set of
stationary states that are eigenfunctions of the system’s Hamiltonian (for details see Sec. 3.1). In
the case of an atom or molecule, these stationary states are the bound or continuum states which
we will call |ψj (r)i. These states depend on space, and evolve in time like eiEj t where Ej is the√energy of the jth stationary state, i = −1 is the imaginary number, and t is time. This showsthat only the phase of the state is changing in time. As the global phase of a wavefunction is notobservable, an electron occupying a single stationary state displays no motion.
For attosecond motion to occur, either the system’s Hamiltonian must become time dependent, e.g. by introducing a laser pulse, and/or multiple stationary states must be populated forming

8
a superposition. The former is often required to image the motion of electrons, but the latter issimpler to understand. We will, therefore, start with the motion produced by an electron in asuperposition of more than one stationary state. In principle, one or more of the states could becontinuum states, but for simplicity we restrict ourselves to bound states only in the following
discussion. The resulting electronic wavefuction can be written as|Ψ(r, t)i =

N
−1
X
j=0

aj ei(φj +Ej t) |ψj (r)i(2.1)

where aj is a non-negative real number, φj is a phase, Ej is the energy of the state, and thesuperposition contains N total states each labeled by the subscript j. The global phase can be
accounted for by removing the time dependent phase of the ground state by setting E0 = 0. The
relative phases of the excited states (j > 0) changes in time like (Ej − E0 )t resulting in dynamics if
Ej 6= E0 . For a two state system, the dynamics will be periodic with a period of T = 2π/|E1 − E0 |.
Fig. 2.1 shows the dynamics of the wavefunction for a helium atom in a 1s − 2p+ superposition as
it rotates with a period of approximately 200 attosecond.

0

65∆t (as)130

Figure 2.1: An isosurface of the wavefunction of a helium atom in a 1s − 2p+ superposition (|rΨ|)
undergoing attosecond motion. (Figure from [233])

Adding additional states to the superposition allows for arbitrarily complicated field freedynamics to occur. In order to “image” the wavepacket, all of the values for aj and φj must beq
P −1 2
obtained at some time t0 with the restrictions that a0 = 1 − N
j=1 aj and φ0 = 0 to enforce

9
normalization and removing the global phase, respectively. Then using Eq. 2.1, the time dependenceof the system is known. A method for imaging simple superpositions is presented in Sec. 4.3.
When the system’s Hamiltonian becomes time dependent, however, the above approach does
not capture the dynamics completely. If the time dependent operator is weak, relative to thetime independent operators, the dynamics maybe captured by allowing aj and φj to become timedependent quantities. When effect of the time dependent operator becomes comparable or larger
than the independent operators, the stationary states, |ψj (r)i, may no longer be a good basis todescribe the system. Imaging the dynamics of such a system requires techniques beyond the scopeof this thesis.

2.2

Pump-probe spectroscopy
Pump-probe spectroscopy is a powerful tool for imaging ultrafast processes. The general

concept is quite simple. Two isolated ultrashort pulses are produced. The first pulse, known as
the pump pulse, prepares the system by starting some dynamical process. Some time after theend of the first pulse, the second (probe) pulse images the target. By delaying the probe pulse
with respect to the pump pulse, the impact of the dynamics on a given observable can be observed
experimentally. This gives a glimpse at how the system evolves over time. Although conceptuallyeasy to understand, attosecond pump-probe spectroscopy is an extremely difficult experiment toperform for the following reasons.
In many cases, it is required to have both angle- and/or energy-resolved spectra of the ionized electrons. With the development of experimental techniques, such as velocity map imaging(VMI) [131, 210] or cold target recoil ion momentum spectroscopy (COLTRIMS) [229], the detection of angle-resolved emission of the photoelectron following few-photon ionization of atomic
and molecular targets has become possible [153]. Additionally, one requires the ability to produce
well controlled isolated ultrafast laser pulses. In the recent past, it has become possible to generate isolated attosecond pulses [70]. This has opened the door for the application of pump-probe
spectroscopy to attosecond physics.

10
In 2001, the first experimental observation of attosecond electron dynamics was made witha pump-probe setup including a few cycle visible light pulse and an attosecond soft-X-ray pulse[95]. The bandwidth of attosecond laser pulses allows for both bound and continuum states to bestudied [161] and, more recently, charge dynamics in Tryptophan on the sub 4fs time scale have been
observed showing that attosecond pump-probe spectroscopy may be applied to complex molecules[137]. As experimental techniques evolve, the application of attosecond pump-probe experiments
will continue to expand. In this thesis, we look at how pump-probe spectroscopy can be used tounderstand the ionization of electrons in superposition states in Sec. 4.2 and reconstruct attosecond
electron motion in Sec. 4.3.

2.3

Few photon ionization
Multiphoton ionization was proposed by Göppert-Mayer in 1931 [85]. Thirty years later, twophoton ionization was demonstrated experimentally, shortly after the advent of the laser [120]. As
we approach a century since the proposal of multiphoton ionization, research in this area is still
progressing. With recent advances in HHG and FEL laser pulse sources in the extreme-ultraviolet(EUV) wavelength regime, studies involving few photon ionization in the perturbative intensity
regime have been of both experimental and theoretical interest [177, 230, 220, 193, 65, 214, 86,
66, 112, 113, 153, 207, 82, 47, 103, 104, 25, 243]. Using tools like VIM and COLTRIMS allows
for both the direction and momentum of the ionized electrons to be detected. At a given electronenergy, the probability distribution of an electron being ionized in a specific direction is known asa photoelectron angular distribution (PAD).
PADs contain information containing about the phase and amplitude of the various partial
waves produced during the ionization process. These signals have been used to study the competition between resonant and non-resonant two-photon ionization pathways [112, 113, 153] andbetween one- and two-photon ionization channels [82, 47, 25]. These tools are also used to investigate the impacts of many other physical effects, such as a significant circular dichroism via theasymmetry in the forward-backward electron emission from bromocamphor molecules induced by11
circularly polarized light has been identified [26]. Observation of the breakdown of the symmetry
in the photoelectron emission of argon has been shown in the region of the Cooper minimum [110],molecular vibrations and chirality have been studied [75] and other applications range from studies
of coherent control [199] to the characterization of ultrashort laser pulses [35].
Through the contents of Chap. 4 we explore the impacts of few photon ionization using
ultrashort pulses (Sec. 4.1), proposing new asymmetry parameters for understanding few photon
ionization of electrons in superpositions (Sec. 4.2) and how few photon ionization can be used toreconstruct attosecond electron motion (Sec. 4.3).

2.4

Rydberg state excitations induced by strong fields
Excited states in atoms and molecules have played an important role in the understanding ofmany physical phenomena in strong field physics such as resonant enhancement in the populationof excited states [42, 119], structures in the energy spectrum [72, 190, 3], and in energy-resolved
angular distributions [209] of photoelectrons. When excited states become resonant with a one- or
multiphoton process from an initial state, the electron can populate the excited state. Dependingon the system and laser involved, the population in the excited state can be trapped or it canenhance the probability of another process occurring. The role of excitation was initially observed
via resonant enhancement in the population of excited states [42, 119] and structures in the energy
spectrum [72, 190, 3] and in energy-resolved angular distributions [209] of photoelectrons.
As the laser intensity increases, the ionization potential of an atom shifts into the continuum
due to the AC-Stark shift. Additionally, the resonances of highly excited states, known as Rydberg
states, follow the same trend. The AC-Stark shifted resonance (E 0 ) for highly excited states istherefore approximately
E 0 = E + Up ,(2.2)

where E is the field-free energy of the excited state, Up = I/4ω 2 is the ponderomotive energy, I is
the peak laser intensity, ω is the central frequency of the laser, and E 0 is the energy of the shifted12
resonance. The initial findings were explained examining the effects of multiphoton absorption
becoming resonant with the AC-Stark shifted states.
More recently, significant excitation of atoms has also been observed in the tunneling regime
and described by the frustrated tunneling ionization (FTI) model for linearly polarized lasers [180].
Similar to the three-step model of HHG, the FTI model starts with the tunnel ionization of anelectron. The electron then interacts with the trailing end of the laser pulse which is slowing
it down. Finally, the electron is recaptured by the ionic potential ending with a final energy
corresponding to that of a Rydberg state. This observation has renewed the general interest in themechanisms leading to the population of excited Rydberg states during the interaction of an atomwith an intense laser pulse (most recently, e.g., in Refs. [38, 145, 144, 260, 221, 32, 142, 59, 27, 73,
152, 219, 89, 176, 248, 19, 138, 261, 22, 74, 115, 109, 156, 247, 194]).
The FTI model is semi-classical and provides no predictive power for the angular momentum
distribution of the populated excited states. Therefore, recent theoretical studies of the excitation
mechanism in linearly polarized strong fields mainly consider the distribution of the population
as a function of the principal quantum number of the excited states [145, 144, 261, 247, 194]. It
was shown that the modulation of the excitation probability is related to the channel closing effect[134, 145, 144, 194]. The latter phenomenon occurs at threshold intensities at which the absorptionof one more photon is needed to ionize the atom due to the AC- Stark shift. The interpretation thatan increase in excitation can be understood as result of the shift of the first ATI peak below theionization threshold [145, 144] is in agreement with the explanation of earlier experimental results(e.g., [72, 119, 209]) via the resonance enhanced population of AC-Stark shifted excited states.
On the other hand, the literature provides very limited understanding of the angular momentum distribution in the populated Rydberg states by linearly polarized fields. Assuming the initialstate has even angular momentum, it has been shown that the populated Rydberg states have angular momentum with the same parity as Np − 1, with Np being the minimum number of photons
needed to ionize the AC-shifted ionization potential using Floquet theory for a monochromatic laser
field [134] and results of numerical calculations for laser pulses with a trapezoidal envelope [194].

13
Furthermore, the angular quantum number of the states with the largest population in numericalcalculations [145, 144, 194] agrees well with semiclassical estimations [6], initially performed for
low-energy angular resolved photoelectron distributions.
For more complex laser set-ups, such as bichromatic circularly polarized laser pulses, almost
no work on the population of Rydberg states has been done. Recently, in bicircular fields it has
been observed that the probability to ionize an atom is significantly enhanced if the two fields arecounter-rotating as compared to co-rotating fields [157]. In that work, the authors propose thatthe counter-rotating pulse’s ionization pathways allow for resonance enhanced ionization while the
co-rotating pulse’s do not due to selection rules. However, distributions over the quantum numbers(principal, angular momentum, magnetic) were not presented. Such analysis can shed further lighton the role of excited states in the pathways to ionization since excitation in a resonant multiphoton
process should rely on the spin-angular momentum selection rules for the absorption of circularly
polarized photons.
In this thesis, we examine the excitation of Rydberg states in Chap. 5. We study the angularmomentum distributions caused by linearly polarized lasers in Sec. 5.1 and the various mechanisms
present in the excitation of Rydberg states by bicircular laser pulses in Sec. 5.2.

2.5

Electron correlation
For many effects in ultrafast physics, the single-active electron approximation (SAE), where

one electron is allowed to move in an effective potential of all remaining electrons and nuclei, isvalid. However, for laser pulses with high energy photons, multiple electrons can interact leading
to effects caused by electron correlation. Boll et al. recently showed that the SAE approximation
breaks down in the interaction of helium with lasers at high central frequencies [25].
Electron correlation effects can manifest themselves in many ways. One such example canbe found in direct vs. autoionizing channels [39] where anisotropy and asymmetry parameters
have been used to study the impact of the 3s−1 np resonance in argon. When interacting with
an autoionizing resonance, a photon is absorbed by an electron with energy that would allow for14
ionization and the promotion of a second electron to an excited state. The first electron remains
in the resonant state long enough to share its energy via electron correlation with another electronthat is excited to a state of higher energy. The first electron is then ionized at a lower energy.
In addition to autoionizing resonances, two electrons can be ionized by a single photon ina process known as single photon double ionization [160]. The process proceeds via two main
mechanisms, non-sequential double photoionaization (NSDI) and shake-off. In NSDI, the firstelectron absorbs a high energy photon before it interacts with a second electron. Both electrons arethen ionized with similar final energies. The ionized electrons are correlated and have been shown to
produce a double slit pattern when double ionizing the H2 molecule [5]. When even higher photon
energies are used, the first electron can be ionized so quickly, that the ionic potential is rapidlychanged and a second electron is ionized with a vary small relative momentum in the shake-off
process [33]. At even higher photon energies, core electrons can be ionized leaving a hollow atom.
A valence electron can then relax into the hole left behind. To conserve energy, a second valence
electron is ionized in a process known as the Auger effect originally proposed in 1922 [166].
In order to study these problems in detail, fully correlated multi-electron calculations are
required. A solution to the time dependent Schrödinger equation (TDSE) in full dimensions requires
3Ne + 1 dimensions where Ne is the number of electrons and the last dimension is time. Fortime dependent problems, a full solution for two electrons is possible, but requires the use of
supercomputer resources [231]. The typical approach is to expand the wavefunction in bi-sphericalharmonics resulting in two radii that must be discretized to large distances increasing the size of
the wavefunction to be stored. In Chap. 6 we discuss our recent developments and first tests of a
two-electron code based on hyperspherical harmonics with the goal to potentially remove the needfor supercomputers since only a single radius needs to be discretized.

Chapter 3
Modeling laser-atom interactions

In order to model laser-atom interactions, the regime of laser parameters must be considered.
The studies presented in this thesis fall well within the so-called dipole oasis shown in Fig. 3.1 [151].
The dipole oasis shows the areas where the dipole and non-relativistic approximations are valid. The
right edge of the dipole oasis in Fig. 3.1 occurs when the wavelength of light becomes comparable
to the size of an atom, and the upper left limits correspond to the large amount of energy in a laser
pulse. In the latter case, the magnetic components of the field become non-negligible resulting in
the breakdown of dipole and non-relativistic approximations.

Figure 3.1: The dipole oasis shows the laser parameters that ensure that the dipole and nonrelativistic approximations hold. (Figure from [151])

To study the dynamics of an electron interacting with and intense laser pulse, we solve thetime-dependent Schrödinger equation (TDSE, Sec. 3.1). For the intense laser pulses used in this16
work, an analytic solution to the TDSE has not been found. Therefore, numerical solutions to the
TDSE, described in Sec. 3.3, are used for the majority of the studies in this thesis. Numericalsolutions to the TDSE are computationally intensive since they require tight sampling near theatomic core to ensure the ground state is properly represented, and large boxes to allow ionizedwavepackets to remain on the grid. As a result, it is sometimes useful to apply approximate
methods. At low intensities, typically below 1013 W/cm2 , the laser can be treated as a perturbation
to the atomic potential allowing for approximate solutions to be obtained via perturbation theory(Sec. 3.2). At the other extreme when the laser pulse is much stronger than the atomic potential,
the strong field approximation (SFA) can be used [133, 57, 205].
For the hydrogen atom, an analytic potential for the Coulomb interaction of the electronwith the nucleus is known. However, for multi-electron atoms, one must use a single-active electron(SAE) potential to solve the TDSE. The SAE potentials used in this work are described in Sec. 3.4.
Once a solution to the TDSE has been obtained, we use various observables, discussed in Sec. 3.5,
to analyze the final wavefunction. Finally, the laser pulse definitions and frequency correction for
ultrashort pulses is given in Sec. 3.5.

3.1

Time-dependent Schrödinger equation
The TDSE can be written asiwhere i =√∂ψ(r, t) = Ĥψ(r, t)
∂t

(3.1)

−1 is the imaginary number, Ĥ is the Hamiltonian which describes the system, andψ(r, t) is the time dependent wavefunction. The Hamiltonian for a system with Ne correlated
electrons interacting with a laser field and Nn fixed nuclei is given by
!
Ne
Nn
Ne
X
X
X
p̂2e
Zn
1
Ĥ =
+ Vlas (r, t) −+2re,nre1 ,e2
e=1

n=1

(3.2)

e1 <e2 =1

where pˆe = (−i∇e ) is the momentum operator of the eth electron, Vlas (r, t) is the laser operator,
Zn is the charge of the nth nucleus, and ri,j is the Euclidean distance between particles i and j. The

17
laser interaction is typically considered using both the dipole and non-relativistic approximation ineither the length gauge (Vlas = −E(t) · r) or velocity gauge (Vlas = −A(t) · p̂e /c). Since no analytic
solution for more than one electron in a central potential is known, as the problem is not separable,
numerical methods are required. Solving the TDSE accurately in full dimensions for more than
one electron is extremely difficult and next to impossible for three or more active electrons. For
most studies, we make the single-active electron approximation (SAE) where all except one ‘active’
electron and the nuclei are fixed in space. A SAE potential generated by the nuclei and remaining
electrons is then used to model the interaction of the active electron with the fixed electrons andnuclei. The SAE Hamiltonian of an atom interacting with a laser field therefore becomes
Ĥ =

p̂2
+ Vlas (r, t) + VSAE (r),
2

(3.3)

where VSAE (r) is the SAE potential. The exact form of the SAE potentials used in this work is given
in Sec. 3.4. Extensions towards two-active electron calculations beyond the SAE approximation are
discussed in Chap. 6. Finally, it is often convenient to split the problem into a time-independent
field-free Hamiltonian (Ĥ0 ) and the time dependent laser field such thatĤ = Ĥ0 + Vlas (r, t).

(3.4)

Eigenfunctions associated with H0 can be used to analyze the results of a simulation.

3.2

Perturbation theory
In the case of a weak laser field (typically for intensities below 1013 W/cm2 ) relative to theatomic potential the TDSE can be solved using perturbation theory (PT). In some cases, such as forthe hydrogen atom, the corresponding terms of the perturbation series can be solved analytically.
However, for many of the SAE potentials that we use there is no analytic solutions. In such a case,a numerical solution to the first-order terms of the series provides much of the same insight.
Therefore, it is advantageous to provide a general framework that can be used for bothanalytical and numerical solutions. We start by diagonalizing the field-free Hamiltonian H0 to18
discretize both the bound (Sec. 3.3.6) and continuum states (Sec. 3.3.7). The result can be written
as a Green’s function0G0 (t, t ) =

Z
X
g

0

e−iEg t |φg (t)i φg (t0 ) eiEg t ,(3.5)

where |φg (t)i are the field free states (both bound and continuum), Eg is the energy of the state,
and t denotes time. For some initial wavefunction |Ψ0 (r, t)i, the consecutive orders of the PT canbe obtained via|Ψi+1 (r, t)i =

Z t

−∞

dt1 G0 (t, t1 )Vlas (t1 ) |Ψi (r, t1 )i(3.6)

with i = 0, 1, 2, . . .. The total wavefunction in N th order can then be obtained via a sum such that(N )|Ψ(r, t)i

=

N
X
i=1

|Ψi (r, t)i .

(3.7)

By taking N → ∞, the resulting wavefunction is exact in the regime where the perturbation issmall as compared to other interaction terms. However, if the perturbation is of similar order ascompared to other terms in the Hamiltonian, the series will not converge to the correct answer
leading to the breakdown of PT. The wavefunction obtained using PT can be used to extractobservables (Sec. 3.5) in the same way as for a (full) numerical solution of the TDSE. Additionally,
for laser-matter interactions PT offers the option to analyze the wavefunction of a particular order
to extract further information about how many photons are involved in a process.

3.2.1

Selection rules

The wavefunction is often expanded in 3D spherical harmonics to take advantage of selection
rules for interactions with an electric field. We proceed by considering the selection rules for length
gauge (Vlas = −E · r). For simplicity, we consider a linearly polarized laser field along the z-axis,
i.e.
Vlas = −Ez z.

(3.8)

19
Noting that z = r cos(θ) we obtain
Vlas = −Ez r cos(θ)
r
π 0= −2Ez r
Y (θ, φ),
3 1

(3.9)
(3.10)

where Y`m (θ, φ) is a spherical harmonic. This expansion can then be used to calculate the angularportion of a matrix element between two wavefunctions with Vlas as a perturbation:
r D
D 0
π
0
m
m
Y`m
Y`0 Vlas |Y` i = −2Ez r
Y10 |Y`m i .
03
The integral over three spherical harmonics can be written assZ(2`1 + 1)(2`2 + 1)
m1 ∗ m2 ∗ M
h`1 , 0, `2 , 0|L, 0i h`1 , m1 , `2 , m2 |L, M i ,Y`1 Y`2 YL =4π(2L + 1)
4π
where h`1 , m1 , `2 , m2 |L, M i is a Clebsch–Gordan coefficient. Thus, we gets
D 0(2`1 + 1)(2`2 + 1) 0Y`m
Vlas |Y`m i = −Ez r` , 0, 1, 0 `, 0 `0 , m0 , 1, 0 `, m ,03(2L + 1)(3.11)(3.12)(3.13)

which is only non-zero for `0 = ` ± 1 and m0 = m. Therefore, the selection rules for a linearlypolarized laser along the z-axis are given by ∆` = ±1 and ∆m = 0. Similarly, for a linearlypolarized laser along either the x or y axis one gets ∆` = ±1 and ∆m = ±1. Finally, a circularly
polarized laser in the xy-plane leads to ∆` = ±1 and ∆m = 1 for absorbing a photon from aright-handed circularly polarized field and ∆` = ±1 and ∆m = −1 for absorption of a photon froma left-handed circularly polarized laser field. By taking advantage of these selection rules, one cangreatly reduce the size of a wavefunction used in a particular order of PT or in a numerical TDSE
solution.

3.3

Numerical methods
For much of the parameter regime we are interested in, perturbation theory fails as an approxi-

mation method. Therefore, we have developed and implemented numerical solvers of the TDSE.
As for most partial differential equations, numerical solutions to the TDSE have a relatively simple

20
general form with the complexity hidden in the details. First, one chooses a basis/discretization
to write down the solution, in our case the wavefunction. Next, the operators are expanded in
the basis and written as a matrix. Then, one obtains a set of initial conditions at the start of thesimulation, in our case typically a bound state of the atom or molecule. Finally, the initial condition
is propagated in time with updates to the operating matrix between time steps. In this section,
we will lay out the basis/discretization choices in Sec. 3.3.1, followed by a detailed discussion of
the implementation in the remaining subsections. For the sake of simplicity, we will focus on
single-active electron calculations in this section (for details on two-active electron calculation see
Chapter 6).

3.3.1

Coordinate systems

The solution to a single-active electron calculation is three dimensional in space. The simplest
and most general strategy would be to expand the wavefunction on a 3D grid in real space withthe atom or molecule placed in the center. Such a grid is generated by discretizing each dimension(1D problem) with N grid points and then taking a Kronecker product producing a 1D vector that
represents the full 3D Cartesian space. The derivative operators can then be discretized utilizing
finite differences (discussed in the next section) and stored in a matrix. The implementation isrelatively straight forward and there are no assumptions about the symmetry of the problem.
However, this method has its flaws. If a potential has a singularity, such as the Coulomb potential
at zero (V = −1/r), special care must be taken to avoid sampling that point. Additionally, the sizeof the calculation scales like N 3 when the number of grid points in each dimension (N ) increases.
Therefore, obtaining converged results for large atoms, which require smaller grid spacing to sample
the core properly, and high intensities, which require larger box sizes to keep the wavefunction on
the grid, becomes computationally impractical. It is, therefore, useful to consider the symmetriesof the problem in question to greatly speed up the calculations. Two standard approaches are toutilize cylindrical or spherical coordinates.
The problem of an atom or a linear molecule aligned along a linearly polarized laser is21cylindrically symmetric. Therefore, in this case using cylindrical coordinates is the natural choice.
By choosing the laser polarization (and internuclear axis) along the z-axis, the solution in the
angular coordinate φ is given by the initial condition and remains constant. Thus, one only needs
to find a solution in the z and ρ dimensions numerically, reducing the problem to two dimensions.
As a result, solutions scale like N 2 which significantly accelerates calculations as compared to a 3D
Cartesian code.
However, in our studies we have actually found that spherical codes are often even faster.
When written in spherical coordinates, the TDSE for a central potential (such as an atom) canbe separated into a radial equation that contains the potential, and an angular portion that is
independent of the potential. The angular solution can then be found analytically in the form ofspherical harmonics and the radial solution is all that remains. For the hydrogen atom an analytic
solution exists. However, in general for a radial potential it is not guaranteed that there is ananalytic solution. Therefore, we expand the wavefunction in spherical harmonics and use finite
difference to discretize the radial portion. Since the laser field only couples spherical harmonics
according to the selection rules ` → ` ± 1 and m → m, m ± 1 depending on the polarization, thenumber of spherical harmonics needed is rather small (typically `max < 50) with a weak dependence
on laser intensity and wavelength. The result is a code that scales like N `max for linear polarization
and N `2max for arbitrary polarization. Since `max is small, the code typically scales like N as thepulse intensity, wavelength, or target atom is changed.

3.3.2

Finite difference

To discretize the spatial derivatives, we use finite difference. When using finite differences(FD), derivatives are represented by a set of coefficients called a stencil. The stencil provides
the non-zero weights ci for various sample points of the function. We can then write the second
derivative of ψ known at various evenly spaced grid points labeled by n as
d2
1 X
ci ψi
ψn ≈
2
dx
∆x2
i

(3.14)

22
The non-zero ci coefficients are given in the table below for various orders of accuracy:
Order

cn−3

cn−2

cn−1

cn

cn+1

-1

2

-1

−1/12

4/3

−5/2

4/3

−1/12

−3/20

3/2

−49/18

3/2

−3/20

2nd
4th
6th

1/90

cn+2

cn+3

1/90

Although it is possible to hard code the stencil weights, it becomes quite tedious particularly
for high order stencils, non-uniform grids, and near boundaries where off-centered stencils areneeded. Therefore, in this work we utilize the Fornberg method [68] to obtain the necessary stencil
for a particular simulation. For the remainder of this discussion, we will assume that we are using
second order derivatives. However, it is straightforward to expand it to higher order derivatives byadding more off-diagonal elements in matrices discussed in this section.
We begin with a 1D wavefunction and later extend to N D wavefunctions. Let us assume
that we start with ψ(x) known at various points along the x-axis spaced by a grid step of dx. We
will label the points by n such that ψn = ψ(x0 + ndx) with x0 being the smallest x value. Inserting
this into Equation 3.14 we get
d2
1
ψn ≈
(−ψn−1 + 2ψn − ψn+1 ) .
dx2∆x2(3.15)For the point ψn+1 we can therefore write1d2
ψn+1 ≈(−ψn + 2ψn+1 − ψn+2 ) .
2
dx
∆x2

(3.16)

This holds until we reach the other end of the grid. If we consider as boundary condition thatψ(x0 − dx) = ψ(xN + dx) = 0, then the points at x0 − dx and xN + dx can be simply left outof the matrix, which automatically imposes this boundary condition. For higher order stencils, an
off centered difference formula is needed such that the boundary condition is only imposed at one

23
point. The operator becomes a system of linear equations in the form of a matrix:
 2 −1  ψ0 






−1 2 −1  ψ1  . 
d2 ψ
1 
.
.
.



..
..
..
. 
≈
 . 
dx2
∆x2 










−1
2
−1
ψ
N
−1






−1 2
ψN

(3.17)

One uses a tensor product to produce the two and three dimensional versions of the operator.

3.3.3

Radial basis functions

A disadvantage of grid based methods, such as 3D Cartesian codes, is the structure of thegrid. The required structured grids for FD do not give the freedom to refine the sampling inpoints of interest, such as near the nucleus, while sparsely sampling at large radii, due to the
Kronecker product. Radial basis functions (RBF), on the other hand, is a meshless method thatallows for sampling points, called nodes, to be placed anywhere [69]. The lack of required structure
significantly reduce the number of points required to approximate the wavefunction. In this work
the RBF method described below was applied to quickly obtain bound and continuum states.
However, more work will be needed to allow for time propagation of the wavefunction in a laser
field.
The RBF method starts with a set of N unique points in space called nodes that discretize
the problem space. A stencil of size n < N is then made for each node via its n−1 closest neighbors
by using a k-nearest neighbors algorithm. For a RBF wavefunction φ(r), one defines the matrix φ(|x1 − x1 |) φ(|x2 − x1 |) · · · φ(|x1 − x2 |) φ(|x2 − x2 |) · · ·
A=

..
..

.
.


φ(|x1 − xn |) φ(|x2 − xn |) · · ·

φ(|xn − x1 |) 
φ(|xn − x2 |) 
,

..

.


φ(|xn − xn |)(3.18)

where xj represents the coordinates of a node. It is then possible to calculate the weights ωj of an24
operator L evaluated at some point xc by solving












 −



 1


 x1



 y1



 z1

 .
 ..


e−αr1

|
|

A

|

x1
..
.

1

xn yn zn · · ·

−

−

+ −

···

1

|

···

xn

|

···

yn

|

···

zn
..
.

|

e−αrn

|

···

|

−

y1
..
.

−

0

z1
..
.

···

1
..
.

−

−

e−αr1









  ω0   Lφ(|x − x1 |)|x=xc 

 

.. 
..
 .  .   ..  
.


 


 


 

e−αrn   ωn  Lφ(|x − xn |)|x=xc 

 


 






−  −  
−


 


 

L1|x=xc  ωn+1  
= . (3.19)   ωn+2  Lx|x=xc

 


 


 

  ωn+3  
Ly|x=xc

 


 


 

Lz|x=xc  ωn+4  


 

 .  .
..
  ..  


 


 

−αr
ωn+m
Le
|x=xc

The matrix A imposes the RBF restrictions while the additional rows and columns allow forpolynomials up to some degree (1, x, y, z, x2 , xy, · · · ) to be attached, improving the orderof accuracy for the stencil. We can also attach other functions such as a Slater type orbital1(e−αr ) to treat the discontinuity in the hydrogen atom wavefuction at r = 0. The weights ωj for
j = 1, . . . , n are used as the matrix elements in the operator matrix and weights with j > n are
discarded. When solving a PDE, it is often useful to take xc = x1 for each stencil such that theweights can be reused during time propagation. However, one can freely choose xc allowing for thenode placements to adapt to the PDE solution over time.
For the solution to the hydrogen atom we use 7th order polyharmonic spline functions (φ(r) =r7 ). The node set is then chosen to emulate the standard spherical codes that are used. To do this,
we discretize the radius at values rj = (j + dr)/2 for j = 0, . . . , jmax . A maximal determinate (MD)node set [211] for each radius is added to the total RBF node set. The MD node sets simulate theuse of spherical harmonics, which drastically reduces the required number of nodes for the problem
at hand.
1
The method was developed in collaboration with Dylan Abrahamsen and Bengt Fornberg (Department of Applied
Mathematics, University of Colorado Boulder).

10−3

10−3

10−6
10−9
10−12
10−15

(a)
10−1

dr (a.u.)

100

0.002

10−6
10−9
10−12
10−15

(Δ)
10−1

dr (a.u.)100

m[ΔE]|minΔmax (a.u.)100|ΔE| (a.u.)|ΔE| (a.u.)

25
100

0.001

no Slater type orbital
Slater type orbital

0.000−0.001−0.002(c)
10−1

dr (a.u.)

100

Figure 3.2: Panels (a) and (b) show the convergence of the lowest energy bound states in Hydrogen
without and with an attached Slater type orbital respectively for various radial spacing (dr). The
solid blue line show the error in the 1s state and dashed lines show the error for n = 2 states (2s,2p−1 , 2p0 , and 2p1 ) in random order. Panel (c) shows the minimum and maximum imaginary partof the calculated bound state energies.

Choosing a fixed order of MD nodes for all values of r leads to two major issues. At small r,
the nodes on each shell are much closer than the distance to the neighboring shells leading to the
obstacle that stencils do not couple in the radial direction. This can be avoided by reducing theorder of the MD node sets at small r so that the radial grid spacing (dr) is less than or equal tothe average distance between neighboring MD nodes within a shell. The order of the MD node set
then increases as r increases to the set maximum order. At large r, a similar issue occurs, where
only nodes at fixed angles (θ and φ) with varying radii are chosen for a given stencil. To avoid this,each MD node set is quasi randomly rotated before being added to the total node set.
A final implementation issue is caused when attaching Slater type orbitals (e−αr ) to thestencil. Since e−αr becomes small quickly for increasing r, it can have variations below machine
precision in a given stencil. The result is a linear dependence with the constant term producing poor
stencils. To remedy this issue, e−αr is only attached for stencils centered at radii with e−αr1 > 10−10 .
Convergence of the lowest five bound states using the RBF code are shown in Fig. 3.2. Results
were obtained on a mesh with an extension of 30 a.u. in radial direction, 8th order MD node set,
7th order PHS RBF, attached polynomials up to 3rd order, and a stencil size of 54 nodes. Panels(a) and (b) shows the error in the states energy without and with a Slater type orbital e−r being
attached. The addition for the Slater type orbital give machine precision for the ground state (blue26line) and improves the 2s (orange line) energy until nonphysical states appear with dr < 0.1. The
2pm states see little benefit from inclusion of the Slater type orbital.
Fig. 3.2(c) shows the maximum and minimum imaginary part of the energy of the five calculated states. The splitting occurs for the states with m 6= 0 when meshes other than a Cartesianlattice are used. The states with Im[E] < 0 lead to unstable time propagation for the CrankNicolson method used in this work. Further work on mesh generation, hyperviscoisty, and/or time
propagation methods will be needed to study time dependent problems with the RBF method.

3.3.4

Numerical integration

For equally spaced grids used in our finite difference methods, either the midpoint or trapezoid
rule are used for numerical integration. For both methods the integration is performed byZx

f (x0 )dx0 ≈nX

ci f (xi )(3.20)

i=1

where f (xi ) is the value of the function at xi and x is the integration interval. The midpoint rule isused if the grid is placed at half grid points on the interval (i.e. xi = (2i+1)dx/2 for i = 0, 1, 2, ..., n).
In this case ci = 1/dx for all grid points. For the trapezoid rule the grid points sample the ends ofthe integration range resulting in c0 = cn = 1/2dx and ci = 1/dx for i = 1, . . . , n − 1.
In the case of the RBF code, the MD node set comes with a provided integration weight for
the angular integration and the radial integration is done via the midpoint rule. For the sphericalcode, the angular integration is done analytically since spherical harmonics are orthonormal.

3.3.5

Boundary conditions

The wavefunction at large r must decay to zero for it to be normalizable. Therefore, we set
the wavefunction to zero at rmax and increase the value of rmax until a converged result is obtained(Sec. 3.3.9). Exterior complex scaling (ECS), which is discussed in the next paragraph, insures thatthe wavefunction decays prior to hitting the boundary. For the Cartesian code, this is the onlyboundary condition needed. For the cylindrical and spherical codes, another boundary condition

27
cn−2

cn−1

cn

cn+1

cn+2

cn−2

cn−1

cn

cn+1

cn+2

cn−2

cn−1

cn

cn+1

cn+2

..

..

..

..

.

.

(a)

.

.

..

cn−1

cn

cn+1

cn+2

cn+3

cn−2

cn−1

cn

cn+1

cn+2

cn−2

cn−1

cn

cn+1

cn+2

..

..

..

..

.

.

.

.

(b)

.

..

.

Figure 3.3: The finite difference stencil near the edge of the grid. Panel (a) shows how in the fourthorder finite difference the stencil extends beyond the grid leading to a boundary condition being
imposed at two points. Panel (b) shows the non-centered 4th order stencil that enforces a single
boundary condition.

at r = 0 is required. The cylindrical code the grid is chosen such that the grid points are given by
xj = (2j + 1)dx/2 with j = 0, 1, 2, . . . , n and the boundary condition of ψ(−x) = ψ(x) is used. In
the spherical code, however, xj = jdx with j = 1, 2, 3, . . . , n and the boundary condition ψ(0) = 0
is used. To enforce a boundary condition a “ghost node” is implemented. The ghost node is a
grid point that falls outside of the grid. If the ghost node is set to zero, the coefficient is left off
the matrix, and the boundary condition is met. If the stencil extends beyond the grid by morethan one point, such as in 4th order finite difference, the stencil is shifted so that only one point
is beyond the grid and it retains the same width (see Fig. 3.3). For the cylindrical case, the ghost
node is taken at ψ(−dx/2). Since ψ(−x) = ψ(x), the stencil coefficient at ψ(−dx/2) is added tothe coefficient for ψ(dx/2) which imposes the boundary condition.
To absorb any outgoing wave packets, we utilize an exterior complex scaling method (ECS,[92]) on the outer portion of the grid. This is equivalent to changing the spatial step used in finite
difference from xn+1 = xn + dx to xn+1 = xn + eiη dx. In test calculations we have found that η =π/4.0 works well. The result is an exponential decay of the wavefunction in the absorbing region.
Obtaining the finite difference weights in the transition and ECS region is done by implementing
the Fornberg method [68] using complex numbers.

28
3.3.6

Bound state calculations

The bound states of the system are simply the eigenstates of the Hamiltonian with energies
less than zero. Therefore they satisfy the equationĤφi = Ei φi(3.21)

where Ĥ is the Hamiltonian, Ei is the energy, and φi is the bound state assuming Ei < 0. To solve
for the bound states, we use the Krylov-Schur method that is implemented in the SLEPc library[97, 208]. When obtained using SLEPc, the states have a random complex phase associated withthem. The phase is physically irrelevant unless one is creating a superposition or comparing resultsof calculations that use different bound state sets such as a TDSE and PT result. To avoid issues,
it is best to use a single bound state calculation for each project. The populations obtained via
projections onto excited bound states at the end of a simulation are, for example, useful to analyze
and verify selection rules for a given laser polarization.
The coordinate system used for bound state calculations will control which states are generated. For spherical calculations, each ` block can be calculated independently and the results are
independent of m. Likewise, results of cylindrical 2D calculations are obtained for fixed m quantum
number. Finally, Cartesian 3D calculations will provide all n, `, and m states in one calculation
with the states being oriented at a random angle.

3.3.7

Continuum states

The continuum has states at all levels with energy larger than zero in the physical problem.
However, when we solve the TDSE numerically, due to the finite extension of the grid the continuum
is represented via a discrete set of states. For example, in spherical coordinates the TDSE solution
on the grid requires ψ(0) = ψ(rmax ) = 0 (for the radial coordinate). The boundary condition is thesame as for an infinite square well with a bottom in the shape of V (r). The continuum states are,therefore, a distorted solution to the infinite square well problem. We can obtain these states by
the same method as used in the bound state calculations providing a discrete set of energies that29
represent the continuum. This method is used for the numerical PT solution discussed in Sec. 3.2.
When calculating a photoelectron (energy or angular) spectrum, however, it is advantageous
to be able to select the energy of the continuum wave. For this case, we utilize the shooting method[215] to obtain a continuum wave at a given energy. The method is given via an ODE which one
solves as follows. The wavefuction at r = 0 is set to zero by the boundary condition. The firstpoint at r0 = dr is set to an arbitrary value (in this case 1) giving
φkl (r0 ) = 1.
with k =√(3.22)

2E. The second point is then obtained by
`(` + 1)
+ 2V (r0 ) − 2E + 2 .
r2(3.23)

`(` + 1)
+ 2V (ri ) − 2E + 2 − φkl (ri−1 ).
r2(3.24)φkl (r1 ) = φkl (r0 ) dr2Each additional point can be found by

φkl (ri+1 ) = φkl (ri ) dr2

Once the wavefunction has been obtained, we match the asymptotic solution to that for a hydrogenlike atom



lπ Z
φkl (r  1) → sin kr −+ ln(2kr) + δkl .
2
k(3.25)

The continuum state oscillates between -1 and 1 at r  1 leading to a normalization of






1


A = s2
φ0kl (r)2|φkl (r)| + (k+ Z )kr.

(3.26)

r=rmaxFinally, we extract the phase shift δkl by
φ0kl (r)iφkl (r) + (k+
Z
lπ 
)


kr 
δkl = arg  − kr + 2(2kr)iZ/k.

(3.27)

r=rmaxThis continuum state can then be used to obtain the photoelectron spectrum by
P (k, θ, φ) ∝

X
l,m

∗
Yl,m
(θ, φ)Z2e−iδkl (i)l φkl (r)∗ ψ(r, t)dr .

(3.28)

30
3.3.8

Time propagation

Time propagation is often performed using the split-operator method where the Hamiltonian(Ĥ) is split into its spatial dimensions, e.g. along (z) and perpendicular (ρ) to the polarization
direction of a linearly polarized laser. Similarly, the problem can be split along x, y, and z for an
arbitrary laser polarization. The resulting propagation scheme is given by∆t∆tψ(r, t + ∆t) ≈ e−iĤρ 2 e−iĤz (t)∆t e−iĤρ 2 ψ(r, t),(3.29)

wheree−iĤ∆t ≈1 − i ∆t2 Ĥ1 + i ∆t
2 Ĥ.

(3.30)

This produces a set of tridiagonal matrices which can be solved with O(N ) operations and O(N )memory. However, parallelization of such a method on a modern supercomputer with distributed
memory can be cumbersome, requiring multiple all-to-all Message Passing Interface (MPI) messages
during each time step which strongly reduces its scalability.
Instead, we avoid splitting the Hamiltonian and propagate the total Hamiltonian in timeusing a second order Crank-Nicolson scheme whereψ(r, t + ∆t) ≈ e−iĤ∆t ψ(r, t).

(3.31)

We note that the Crank-Nicolson method is not tridiagonal (due to a tensor product) and a direct
solution would require O(N 3 ) operations and O(N 2 ) memory which is significantly more than in
the split operator method. However, the system of equations in the full Crank-Nicolson method
is sparse and iterative methods can be used to vastly accelerate the time propagation. We utilize
the Generalized Minimal Residual Method (GMRES), implemented in PETSc [8, 9], which solves
the sparse system of linear equations in O(N log(N )) operations and O(N ) memory. The PETSc
library makes it straightforward to parallelize the Crank-Nicolson method on modern supercomputers with distributed memory. On a local supercomputer (Summit, CU Boulder), we achieved
super-linear scaling up to 3,000+ cores (see Fig. 3.5) allowing us to complete simulations in a matterof hours that would take weeks running on a high-end workstation.

dx
4.000e-01
2.000e-01
1.000e-01
5.000e-02
2.500e-02
1.250e-02

Z=15.995e-03
3.035e-04
1.041e-05
2.225e-07-2.663e-09-7.101e-10Z=26.217e-02
5.995e-03
3.035e-04
1.041e-05
2.225e-07-2.672e-09

Z=31.590e-01
2.612e-02
1.841e-03
7.810e-05
2.290e-06
2.992e-08

Z=42.568e-01
6.217e-02
5.995e-03
3.035e-04
1.041e-05
2.225e-07

Z=53.416e-01
1.086e-01
1.392e-02
8.331e-04
3.204e-05
8.341e-07

31

Table 3.1: The relative error of the ground state energy for a hydrogen-like atom with a nuclear
charge Z. A grid spacing around 0.1 produces well converged results for Z = 1. Excited states
have lower relative error.

3.3.9

Convergence studies

Numerical solutions to the TDSE require a convergence study to ensure that the final results
are reliable. One does this by varying a free parameter, such as grid spacing, spherical harmonic
expansions, time step, etc. until the observable in question remains within some tolerance. It is
possible to use an error calculation and reduce it until a numerical tolerance is met. However,graphical convergence, when two lines plotted on top of each other are indistinguishable, is often
a more practical convergence criteria. In Table 3.1 we show how the error of the ground stateenergies of a hydrogen-like atom with difference charge (Z) converges with decrease of the grid
spacing when using 4th order finite difference.
For a fixed target, it is advantageous to use the highest order finite difference that is stable.
Fig. 3.4 shows the convergence of the 1s, 2s, and 3s energy levels of the hydrogen atom using
2nd through 20th order FD. The slope of the convergence for each curve matches the order of themethod. 20th order FD achieves a converged result with the largest grid spacing leading to a shorter
calculation. It is therefore advantageous for bound state calculations. However, the Hamiltonian forhigher order methods become increasingly non-Hermitian near the boundaries leading to unstable
time propagation for 6th or higher order FD. Therefore, we have used 4th order for the vast majorityof calculations that we have performed in this work.

32

time per time-step (s)Figure 3.4: Convergence with respect to grid spacing for various orders of FD for the 1s (a), 2s(b), and 3s (c) states in hydrogen atom.

101
100

10−1
102
103
processors

Figure 3.5: Scaling results for present Crank-Nicolson method. Two runs each for calculations with
1 million (blue with squares) and 15 million grid points (green with triangles) have been performed.
Note that the scaling improves with increasing number of grid points.

3.3.10Parallel Computing

Many simulations can be performed on a laptop or small desktop, however, the largest calculations, such as those done in Cartesian coordinates or utilizing bicircular laser polarizations,benefit from supercomputing resources. Since the code is implemented utilizing the PETSc libraryand through the use of MPI, we are able to take advantage of the distributed memory of a modern supercomputer. Linear scaling, i.e. the computational time is cut in half when twice as many
processors are used, is often the benchmark for code scaling. Our code obtains super-linear (better
than linear) scaling on the Summit supercomputer up to 3,000+ processors (see Fig. 3.5). This is

Atom
H
He
Ne

z
1
1
1

Zc
1
9

c
2.03290.8870a1
0.3953-9.9286b1
6.18051.3746a2-5.9950b2
3.796333

Table 3.2: Coefficients of SAE potentials in Eq. 3.33 for hydrogen, helium, and neon from [204].

possible due to the increased size of cache memory as the number of CPUs increases. This allows
for calculations that would take a month on a desktop to be completed in a few hours on a supercomputer. Additionally, the supercomputing systems come with a large increase in the available
RAM (often on the order of terabytes) significantly reducing the memory limitations on calculation
size.

3.4

Single-active electron potentials
For many atoms, full treatment of Eq. 3.2 is impractical, if not impossible. Since in many

ultrafast laser-atom interactions, primarily a single valence electron is involved, in this case it
becomes advantageous to treat only the single-active electron in the potential produced by freezing
the remaining N −1 electrons in their ground state reducing the problem from 3N -dimensions down
to three dimensions. The resulting Hamiltonian becomesĤ = −

p2+ VSAE (r) + Vlas (t) ,2(3.32)

where Vlas (t) is the laser operator and VSAE (r) is the SAE potential. The SAE potentials utilized
in this work take the form [204]z Zc e−cr X
VSAE (r) = − −−an e−bn r .
rr
n(3.33)

The coefficients are obtained by fitting to a potential obtained through density functional theory(DFT). In Table 3.2 we show the coefficients for SAE potentials used in this thesis.

34

3.5

Observables
The wavefunction, given as numerical solution of our calculations, is not experimentallyobservable. It is therefore necessary to extract experimentally relevant information via observables.
For a given observable Ô, the expectation value is obtained by calculating hΨ| Ô |Ψi. In this work wehave studied high order harmonic generation (HHG) spectra, ionization, bound state populations,and photoionization (energy and angular) distributions.
HHG spectra are obtained by first calculating the dipole acceleration using the Ehrenfest
theoremâj =∂V (r),∂j(3.34)

where j ∈ x, y, z are the Cartesian coordinates. For the hydrogen atom, âj = j/r3 where r is the
distance from the nucleus. Then the HHG spectrum is obtained by taking a Fourier transform suchthat
ÔHHG (ω)j =

Z ∞

−∞

2

e−i2πωt hΨ(t)| âj |Ψ(t)i dt.

(3.35)

Bound state populations are obtained using
Pφi = |hφi |Ψi|2 ,(3.36)

where φi is the field-free bound state and Ψ is the wavefunction.
At the beginning of each simulation, the wavefunction is normalized to 1 such thathΨ(t0 )|Ψ(t0 )i = 1 .

(3.37)

Ionization probabilities can then be calculated in two manners. In the first method one propagates
the wave function after the pulse allowing the ionized part to be absorbed at the ECS boundary.
The ionization probability is then obtained by taking
Pion = 1 − | hΨ|Ψi |2(3.38)

since hΨ(t0 )|Ψ(t0 )i = 1. This method is slow to converge and can lead to numerical round off errors
due to extended time propagation. The preferred method for calculating the ionization probability

35
isPion = 1 −Xi|hφi |Ψi|2(3.39)

where φi is a bound state and the sum over i includes all states the grid supports with a real partof the energy that is less than zero. Therefore, the ionization probability can be obtained at theend of the pulse reducing the computational expense and numerical error.
The photoelectron spectrum (energy and angular) can are obtained at the end of the pulse
by calculating
P (k, θ, φ) ∝

X
l,m

∗
Yl,m
(θ, φ)

Z

2

e

−iδkl

l

∗

(i) φkl (r) ψ(r, t)dr,

(3.40)

where φkl (r) is a continuum wave, δkl is a phase shift, and Yl,m (θ, φ) is a spherical harmonic. The
angular distribution is obtained by plotting the angular dependence (θ and φ) for a given energy.
The energy distribution is obtained by integrating over θ and φ (see Sec 3.3.7 for details on obtaining
φkl (r) and δkl ).

3.6

Frequency shift2
Few-cycle laser pulses are used in many interesting strong-field applications (for reviews, seee.g., [197, 225, 238, 187, 242, 249]). For example, high-order harmonics and (isolated) attosecond
pulses are generated, ultrafast atomic and molecular dynamics as well as charge transfer and exciton
dynamics can be induced and time resolved, molecular structure can be imaged on ultrashort timescales, or chemical reactions may be controlled. Therefore, light sources generating ultrashortintense laser pulses in different regions of the spectrum, at extreme ultraviolet [79], ultraviolet[48], optical [67], near-infrared [259, 178, 179] and infrared wavelengths [111, 200, 176], have beendeveloped over the past decades. The simulation of the time-dependent response of matter to a
few-cycle pulse, e.g. via the numerical solution of the corresponding time-dependent Schrödinger
equation (TDSE), can however crucially depend on the definition of the electric field E(t) used.
To achieve quantitative agreement between theory and experiment, the potential issues present in
2

The content of this subsection has been also published in J. Venzke, T. Joyce, et al. PRA 98, 063409 (2018).

36
both numerical calculations and experiment must be well understood and minimized.
As pointed out by Chelkowski and Bandrauk [35], the representation of E(t) via an envelope
function times a trigonometric function may lead to a non-vanishing potential A(t) at the end of
the pulse. This inconsistency can be resolved by first defining the magnitude of the vector potential
A(t) as (we use Hartree atomic units: e = me = ~ = 1) [35]:A(t) =

cE0
f (t) cos(ωA t + φA ),ωA(3.41)

where f , ωA , c, E0 and φA are the envelope function, central frequency, speed of light, peak electricfield amplitude, and carrier envelope phase of the vector potential, respectively. The prefactors
censure that the peak intensity of the laser field is I = 8π
E02 . The magnitude of the electric field isthen obtained via the derivative:
E(t) = −

1∂
A(t)
c ∂t

E0 ∂f (t)=E0 f (t) sin(ωA t + φA ) −cos(ωA t + φA ).
ωA ∂t(3.42)The expression for the electric field, Eq. 3.42, includes a term that depends on the time
derivative of f and, hence, may have significant effects in the case of few-cycle pulses. As we willshow below, a direct implication is that the central frequency of the electric field spectrum, ωE , isnot equal to ωA . The frequency shift |ωE − ωA | is a purely electromagnetic effect that is directly
related to the definition of the electric field via the vector potential given in Eq. 3.42. For longpulses, the frequency shift is small, but it increases as pulse duration decreases. Note, that inexperiment typically the central laser frequency of the electric field ωE is measured and reported,
while in numerical calculations it is useful to define the laser pulse via the vector potential and therelated central frequency ωA . It is therefore important to study what impact, if any, the difference
between the central frequencies has on numerical calculations, or on the interpretation of theoreticalresults.
By solving the time-dependent Schrödinger equation both numerically, as well as within
perturbation theory, we show below that theoretical results obtained for linear processes, such as
excitation and ionization, involving few-cycle pulses with the same value for the central frequencies

37
ωA and ωE do not coincide. Furthermore, we find that the frequency shift is also noticeable fornonlinear processes, such as two-photon excitation and high-order harmonic generation, and it
scales with the number of photons involved. Our results also demonstrate that it is the central
frequency of the electric field ωE that is the physical relevant quantity for the interpretation of laser
driven quantum mechanical processes. Therefore, the frequency shift should be taken into accountwhen setting the central frequency ωA of the vector potential in a numerical calculation or, moregenerally, a theoretical analysis.

field (arb. units)

(a)

0.5

0.5

1.00 (b)

0.01

0.75

0.00

0.50
0.25

−0.01
0

0.00
0.0

10
20
time (a.u.)(c)field (arb. units)

field spectrum (arb. units)

Estimation of frequency shift

field spectrum (arb. units)

3.6.1

1.0

1.5

2.0

1.0

1.5

2.0

ω (a.u.)1.00 (d)

0.01

0.75

0.00

0.50
0.25

−0.01
0

25
50
time (a.u.)

75

0.00
0.0

ω (a.u.)Figure 3.6: Temporal (left) and spectral (right) distributions of vector potential (solid lines) andelectric field (dashed line) for pulses with FWHM of one cycle (top), and three cycles (bottom) at
central frequency ωA = 1.0 a.u. of the vector potential. Also shown is the Gaussian envelope of thevector potential. (Figure from [236])

The frequency shift due to the difference in central frequencies is illustrated in Fig. 3.6, wherethe vector potential A and the electric field E, obtained from Eqs. (3.41) and (3.42) for a GaussianNameGaussian
Cos2
Cos4
Sech

Envelope Function f (t)
2

(
cos2
0
(
cos4
0

e−(t/T )t− π2 ≤ Tt ≤ π2Totherwiset− π2 ≤ Tt ≤ π2Totherwise
sech( Tt )N/NFWHM

γ2

0.849

0

0.723

-0.5940.777-0.381

1.19

2.00

38

Table 3.3: Several common analytic pulse envelopes. The conversion factor between N , as definedin Eq. 3.45 (number of cycles within one standard deviation to either side of the maximum), andthe more typical NFWHM (number of cycles in the full-width half-maximum of the electric field) is
given. The last column shows the excess kurtosis γ2 , defined in Eq. 3.47, which is independent of
the pulse duration T . (Table from [236])

envelope f (t), are compared in both the time and frequency domain. While the temporal behavior
is satisfactory, the spectral distributions reveal different central frequencies. We define the central
frequency ωA (ωE ) as the location of the maximum in the spectral density |Ã(ω)| (|Ẽ(ω)|). The
discrepancy is much greater for the one-cycle full-width at half-max (FWHM) pulse (top), than for
the three-cycle FWHM pulse (bottom).
For the further analysis, we note that in order for the central frequencies to be well-definedand consistent with the definition of ωA in Eq. 3.41, we make several assumptions about theenvelope f (t):• f (t) is nonnegative and continuously differentiable,• f (t) falls off at least exponentially for large |t|,
• f (t) contains no appreciable Fourier components larger than ωA .
These assumptions could be relaxed significantly, but they are sufficient for the present discussion
and all practical purposes. The ratio of the central frequencies is given by the leading terms of an39
expansion in 1/N as3 :
ωE
ωA

=
≈

p

1 + 4(πN )−2γ2+ 4 N −4 + O(N −6 )2
6πp
1 + 1 + 4(πN )−2.
21+(3.43)(3.44)

Here
ωA
N≡
π

sR ∞

2
−∞ (t − t0 ) f (t)dt
R∞.
−∞ f (t)dt(3.45)

is the number of cycles within one standard deviation to either side of the pulse center, withR∞

t0 ≡ R−∞∞tf (t)dt−∞ f (t)dt,

and lastly γ2 is the excess kurtosis of the envelope
hR
i hR
i
∞
∞
4 f (t)dt
(t
−
t
)
f
(t)dt
0
−∞
−∞
γ2 ≡
− 3.
i2
hR
∞
2 f (t)dt
(t
−
t
)
0
−∞

(3.46)(3.47)

Note that N is proportional to the more typical NFWHM (number of cycles in the FWHM of f (t)),
but the ratio N/NF W HM depends on the shape of the envelope (c.f., Table 3.3).
Eq. 3.43 indicates that the ratio ωE /ωA depends on the number of cycles and the pulse shape,
but it is independent of peak intensity, carrier envelope phase, ellipticity, and ωA itself. In fact, thefirst term of the expansion, Eq. 3.43, is a very accurate estimate even for single cycle pulses, showingthat the shift is nearly independent of pulse shape. This can be seen from the comparison between
the predictions based on Eq. 3.43 with the exact frequency shift for a variety of pulse shapes inFig. 3.7. The exact shift was calculated by numerically maximizing the analytic expressions for|Ẽ(ω)|. The slight dependence on pulse shape is visible in the inset; the differences are described
well by the correction term in Eq. 3.43, which involves the excess kurtosis γ2 .
In the next section, we demonstrate that ωE is the physically observable and relevant central
frequency. Therefore, when modeling the interaction with a pulse using Eqs. (3.41) and (3.42), one
3

The analytic work for obtaining the frequency correction and perturbation theory calculations was preformed
by T. Joyce.

40

Figure 3.7: Ratio ωE /ωA as a function of the normalized number of cycles N defined in Eq. 3.45.
The numerical results were calculated by maximizing Ẽ(ω) for Gaussian (dotted line), sech (dasheddotted line), and sin4 (dashed line) envelopes and are compared with the simple analytic estimate(solid line) given in Eq. 3.43. The inset reveals a slight dependence on envelope shape, which canbe attributed to the correction term in Eq. 3.43. (Figure from [236])

should determine ωA such that it corresponds to the correct ωE . There are two different methods
to do this. The first one is to specify N , and use Eq. 3.43 to obtain ωA . However since N depends
implicitly on ωA through Eq. 3.45, the envelope f (t) must be stretched in time by the same factor
such that N remains unchanged. The second method is to specify f (t) instead of N . In that case,substituting Eq. 3.45 into Eq. 3.43 and solving for ωA yields
ωA ≈ ωE −R∞

f (t)dt
R ∞−∞.
ωE −∞ (t − t0 )2 f (t)dt(3.48)

Greater accuracy could be obtained in either case by including the correction term in Eq. 3.43;
however, the results in Fig. 3.7 show that this is in general not necessary.

3.6.2

Applications

In this section we present results of numerical calculations and first-order perturbation theory
which exemplify effects of the frequency shift on observables related to excitation, ionization and41high harmonic generation induced by short laser pulses. The frequency shifts obtain using firstorder perturbation theory and numerical calculations agree well with our analytical predictions
shown in Section 3.6.1. To this end, we solved the 3D one-electron time-dependent Schrödinger
equation (TDSE) in velocity gauge:
 2

∂
p
A(t) · pi ψ(r, t) =−+ V (r) ψ(r, t)∂t2
c(3.49) 2∂pψ(r, t) =+ E(t) · r + V (r) ψ(r, t)∂t
2(3.50)and length gaugeifor atomic hydrogen with a soft-core Coulomb potential
V (r) = − √1
r2 + α 2.

(3.51)

We consider a linearly polarized laser pulse within the dipole approximation, so A(t) = A(t)ẑ andE(t) = E(t)ẑ. Taking advantage of azimuthal symmetry, the wavefunction can be represented in
2D cylindrical coordinates ρ and z. We used the second order finite difference method for spatial
derivatives and the fully implicit second order Crank-Nicholson method for time propagation. The
laser field magnitudes A(t) and E(t) were defined as in Eq. 3.41 and Eq. 3.42, with a Gaussianenvelope function f (t). Specifically,
A(t) =E0 c
2sin(ωA t)e−(2t/τ ) ln 2 ,ωA(3.52)

where τ is the FWHM pulse duration. The results presented below were obtained in velocity gauge,
but additional test calculations in length gauge have confirmed that the results are gauge invariant.
In all calculations for single photon ionization, excitation and high harmonic generation weused α = 0.029 a.u., a grid spacing of 0.1 a.u., and a time step of 0.1 a.u. The soft core parameter isused to match the ground state energy to that of atomic hydrogen, given by E1s = −0.5001 a.u. on
the grid, and an excited state energy of E2p = −0.12504 a.u. To ensure the wavefunction remains
on the grid for our calculation of the photoelectron spectrum, the grid extended 500 a.u. in theρ-direction and 1000 a.u in the z-direction, with an exterior complex scaling absorbing boundary in42
the outer 50 a.u for single photon ionization (Sec. 3.6.2.1). The excitation (Sec. 3.6.2.2) and highharmonic generation (Sec. 3.6.2.3) calculations were preformed on a grid that extended over 100
a.u. in the ρ-direction and 200 a.u. in the z-direction, with an absorbing boundary over the outer
5 a.u.

3.6.2.1

Single photon ionization

First, we consider single photon ionization of the hydrogen atom by a few-cycle laser pulse
with peak intensity 1013 W/cm2 and central frequency ωcentral = 2.0 a.u. The central frequency isimplemented either by setting ωA = ωcentral or setting ωE = ωcentral , using the method describedin the previous section. Photoelectron momentum spectra P (k) were obtained by the following
procedure: the TDSE was propagated for five times the FWHM pulse duration plus an additional
100 a.u. in time, then all bound states with principle quantum number n ≤ 8 were projected out,
and lastly the remaining unbound wavepacket ψionized was projected onto Coulomb wave functions
up to lmax = 5. That is,
P (k) =

lX
max
l=0

Z

2

Fl (− k1 , kr)Yl0∗ (r̂)ψionized (~r)d3~r.

(3.53)The results in Fig. 3.8 show that in fact the photoelectron spectra for central frequency
ωA = 2 a.u. (solid lines) and ωE = 2 a.u. (dashed lines) do not agree due to the frequency shift.
As expected, the discrepancy is larger for 2 cycle FWHM (panel (b)) than for 10 cycle FWHM
(panel (a)) pulses. These results however raise the question whether the central frequency of thevector potential ωA or the central frequency of the electric field ωE is the relevant quantity forfurther physical interpretation or a comparison with experimental data. To address this question,
we consider the resonant population transfer between bound states by analyzing results of numericalcalculation as well as those obtained using first-order perturbation theory in the next subsection.

yield (arb. units)

43
102
101
100

(a)

65
60
55
1.72 1.73 1.74 1.75

yield (arb. units)

10−1
(b)

100

10−1
10−2
10−3
10−4

ωA
ωE
1.0
1.5
2.0
photoelectron momentum (a.u.)

Figure 3.8: Photoelectron spectra P (k) as function of photoelectron momentum obtained for interaction of hydrogen atom with laser pulses at central frequencies ωA = 2 (dashed line) and ωE = 2(solid line) and duration of (a) 10 cycles and (b) 2 cycles FWHM. (Figure from [236])

3.6.2.2

Excitation

Next, we examine transitions to the n = 2 orbitals in the hydrogen atom as a function ofboth pulse length and central frequency of the vector potential (ωA ) and the electric field (ωE ).
Typically, the excitation probability is largest when the central frequency of the laser matches theresonant frequency for np photon absorption, given byωres = |Efinal − Einitial |/np ,(3.54)

where Efinal and Einitial are the energy of the final and initial state, respectively. In view of thepredicted frequency shift between the central frequencies ωA and ωE , we therefore expect that theresults for resonant excitation will provide insights into the physical relevance of ωA vs. ωE .
Within first order perturbation theory, the excitation probability is given by: µω 2res
Pi→f = |Ã(ωres )|2
c


i2
π
µE0 ωres τ 2 h −(ωA −ωres )2 τ 2 /(16 ln 2)2 2=e− e−(ωA +ωres ) τ /(16 ln 2) ,
ln 2
4ωA

(3.55)

Here we have used the pulse defined in Eq. 3.52 and µ = 27.5 3−5 is the transition dipole for the44hydrogen 1s → 2p transition. Eq. 3.55 shows that the excitation probability Pi→f is not maximized−2
at ωA = ωres , due to the prefactor of ωA. This prefactor originates from Eq. 3.41, where it ensuresthat the peak intensity is held fixed as ωA changes.

pulse duration (in units of τ1)

03

04

05

07

10

2p population 2p population

10−1 TDSE
10−2 (a)(b)

10−1 PT
10−2 (c)

(d)
0.35
0.40
ωA (a.u.)

0.35
0.40
ωE (a.u.)

Figure 3.9: Population in 2p state following one-photon excitation of hydrogen atom with a laser
pulse as a function of ωA (left) and ωE (right) for different pulse lengths at peak intensity 1012
W/cm2 . Each line represents results obtained for a fixed pulse duration in terms of τ1 = 405 as.
The results from time-dependent Schrödinger equation (TDSE) calculations (a,b) and predictions
within first-order perturbation theory (PT) (c,d) are in excellent agreement. The vertical line
marks the energy difference between 2p and initial 1s state. The green dots indicate the maximum
excited state population for each pulse duration. (Figure from [236])

We have also obtained the population in the 2p state due to single photon excitation bydirect numerical solution of the TDSE. These numerical results are presented together with thoseobtained within first order perturbation theory, Eq. 3.55, in Fig. 3.9 as function of ωA (left column)and ωE (right column) for various pulse lengths. The pulse length τ is measured in multiples of

45
the period associated with the resonant frequency
τnp =

2πnp
2π
=
,
ωres
|Efinal − Einitial |(3.56)

such that τ1 ≈ 405 attoseconds, τ2 ≈ 811 attoseconds and so on for the 1s → 2p transition. For
long pulses the peak in the population (marked by a green dot) occurs at the expected frequency
ωres (marked by vertical line) for a resonant transition in all distributions. When the pulse lengthis decreased, the peak in the distributions as a function of ωE (right column) remains at ωres . In
contrast, the peak shifts significantly towards lower frequencies in the distributions as a function of
ωA (left column) due to the frequency shift. We note that the results of our numerical calculations(upper row) are in excellent agreement with those obtained using first-order perturbation theory(lower row) and indeed both reveal the same dependence on the central frequencies ωA and ωE .
Thus, the differences in the distributions as functions of ωA and ωE are not a numerical artifact but
are due to the frequency shift. Furthermore, the observed shift agrees with our estimate obtainedin the previous section. Thus, we can conclude that the central frequency of the electric field is thephysically relevant quantity for interpreting laser induced excitation processes.

pulse duration (in units of τ2)

2s population

03

04

05

07

10

10−4
10−5 (a)

(b)
0.175
0.200
ωA (a.u.)

0.175
0.200
ωE (a.u.)

Figure 3.10: Numerical results for population in 2s state following two-photon excitation with
τ2 = 811 as. The vertical line marks half of the energy gap between 1s and 2s representing theresonance condition for the two photon process. Other parameters are the same as in Fig. 3.9.
(Figure from [236])

These conclusions are further supported by our numerical results for two-photon excitation

46
from the 1s to the 2s state in Fig. 3.10. Whereas the peak of the population as a function ofωE (panel b) occurs at ωres , independent of the pulse duration, the peak of the population as afunction of ωA once again shifts to lower frequencies in Fig. 3.10(a). We note that, if the population
as function of ωA in Figs. 3.9 and 3.10 were used to determine the energy difference |Efinal − Einitial |,the error caused by the frequency shift would be twice as much in the two-photon case as in the
one-photon case, accounting for the difference in τnp . This indicates that multiphoton processes
may be affected by the frequency shift even more than few-photon processes. To further illustrate
this point, we examine high harmonic generation.

3.6.2.3

High harmonic generation

Finally, we consider a highly nonlinear laser induced process. High harmonic generation(HHG) in atoms can be described as absorption of an odd number of photons leading to theexcitation of a electron, followed by the emission of a single photon as the electron recombines
into the ground state. Based on the results above, we expect that in this nonlinear process thefrequency shift ∆ω between ωA and ωE will lead to a shift of the energy of the np th harmonic bynp ∆ω. In our calculations the HHG spectrum has been obtained by a Fourier transformation of
the time dependent dipole acceleration along the laser polarization direction. A Hanning filter wasused to return the dipole acceleration to zero at the beginning and end of the simulation.
Fig. 3.11 shows the various harmonics in a HHG spectrum as a function of the number of cycles
in the driving laser pulse at a peak intensity of 1×1014 W/cm2 and central frequencies ωA = 0.0625
a.u. (upper panel) and ωE = 0.0625 a.u. (lower panel), corresponding to a wavelength of 730 nm.
The spectrum consists of odd harmonics and additional emission lines due to the population of
excited states during the interaction with the laser pulse. While we will focus on the generationof harmonics, we note that the emission lines occur at photon energies between the 5th and 9th
harmonics. The corresponding field-free energy differences between the excited states and theground state in our numerical model of the hydrogen atom are marked, as reference, by whitevertical dashed lines.

47

NFWHM

10.0

(a)

7.5
5.0
2.5
3

5
7
9
11
energy (in units of ωcentral)

NFWHM

10.0

13

(b)

7.5
5.0
2.5
3

5
7
9
11
energy (in units of ωcentral)

13

0.040
0.035
0.030
0.025
0.020
0.015
0.010

0.040
0.035
0.030
0.025
0.020
0.015
0.010

Figure 3.11: HHG spectrum at driver wavelength 730 nm (ωcentral = 0.0625 a.u.) vs. number ofcycles NFWHM . In the upper plot the central frequency ωA = ωcentral while in the lower panel
ωE = ωcentral . The vertical white dashed lines mark field-free transition energies between excitedstates and the ground state, while the green solid lines mark the harmonic energies np ωE withrespect to the central frequency of the electric field. (Figure from [236])

In the spectrum as function of multiples of ωA (panel a) one can see that the centers of
the harmonics do shift to energies larger than np ωA , as the pulse duration decreases. In fact,
the energies of the harmonics follow the analytical predictions for np ωE (green solid lines). As
expected, the shift increases with larger the harmonic number. In contrast, in the HHG spectrum
obtained as multiples of ωE (panel b) the centers of the harmonics remain at the same energy,
i.e. np ωE (green solid lines), as the pulse duration decreases. This confirms the importance ofthe shift between the central frequencies of the vector potential and the electric field in nonlinear
processes driven by ultrashort pulses. Furthermore, the HHG results confirm that the central
frequency of the electric field ωE is the physical relevant quantity for the interpretation of light
induced processes. Consequently, if in a numerical simulation or theoretical analysis the vector

48
potential is set via Eq. 3.41, it is necessary to consider the frequency shift between ωA and ωE to
avoid a misinterpretation of the results. Our analytical estimates of the frequency shift in Eq. 3.43
and Eq. 3.48 provide formulas to obtain ωA from the physically relevant ωE .
To summarize, we have shown that the definition of the electric field of a laser pulse via thederivative of the vector potential, which guarantees that both quantities vanish at the beginningand end of the pulse, implies that the central frequencies of the spectral distributions of the vector
potential and electric field do not coincide. In our analysis we have derived an analytical estimateof the frequency shift, which shows that the shift mainly depends on the number of cycles in thepulse and becomes most relevant for few-cycle pulses. Utilizing results of numerical simulations
and first-order perturbation theory we have analyzed how the frequency shift affects excitation,
ionization and high harmonic generation induced by short laser pulses. The effect is found to be
most noticeable in nonlinear strong-field processes since the frequency shift scales with the numberof photons involved. Both, numerical results and predictions based on first-order perturbation
theory agree well with each other and confirm that the central frequency of the electric field isthe physically relevant quantity for the interpretation of the light induced processes. Thus, theshift should be taken into account when setting the central frequency of the vector potential innumerical simulations to avoid potential misinterpretation of the theoretical results, specificallywhen compared to experimental data. Eq. 3.43 and Eq. 3.48 provide formulas to obtain ωA fromthe physically relevant ωE .

Chapter 4
Imaging attosecond wave packet dynamics with photoelectrons

In the previous chapter, we provided an overview of the theoretical methods used to study
ultrafast laser-matter interactions. In this chapter, we will apply these tools to the process of atomic
ionization. We will focus on the impacts of ultrashort pulses on few photon ionization, asymmetries
that appear in the angular dependence of the emission of photoelectrons due to ionization of atoms
in single states and superposition states, and a method to obtain amplitude and phase informationof a wavefuction representing attosecond motion using photoelectron angular distributions (PAD).
In these studies we will consider interaction with lasers similar to experimentally available
ultrabright light sources such as free-electron lasers [218] and table-top laser systems based on highorder harmonic generation [197, 38] that deliver high-intensity pulses of few- or sub-femtosecond
duration. Nowadays, laser pulses with a duration as short as a few tens of attoseconds havebeen achieved experimentally [256, 37]. Isolated attosecond pulses or trains of attosecond pulses
have been generated from the vacuum ultraviolet to the soft X-ray wavelength regime and thepolarization of such pulses can nowadays be controlled. This recent progress in ultrafast laser
pulse technology makes it possible to probe, steer and control the dynamics of electrons in atoms,
molecules and solids [238, 187, 31, 249, 201, 188]. To name a few examples, the time-resolved
measurement of the electron emission in the photoelectric effect has been realized [34, 216, 129,
226, 114]. Using an isolated attosecond pulse or a train of extreme-ultraviolet attosecond pulses
to ionize an atom along with an infrared laser field interferograms have been measured to obtain
information about phases of electron wavepackets, an important milestone towards reconstructing

50
the wavefunctions of atoms [206, 161]. By extracting the phase and amplitude via application ofphotoelectron spectroscopy recently the birth of a photoelectron through a Fano resonance has
been observed on the attosecond time scale [83].
The production of extreme-ultraviolet (EUV) pulses using high harmonic generation (HHG)
and free-electron lasers (FEL) has led to a resurgence of multiphoton ionization studies in the perturbative intensity regime recently [177, 230, 220, 193, 65, 214, 86, 66, 112, 113, 153, 207, 82, 47,
103, 104, 25, 243]. Along with experimental techniques, such as velocity map imaging [131, 210]
or cold target recoil ion momentum spectroscopy [229], the detection of angle-resolved emissionof the photoelectron following few-photon ionization of atomic and molecular targets has become
possible [153]. Probing atoms and molecules in their ground or excited states with ultrashort laser
pulses opens a new regime where several linear and nonlinear ionization pathways compete andinterfere [112, 153, 82, 47, 104, 25, 243, 235]. PADs are determined by the amplitudes and phasesof the partial waves of all pathways contributing to the emission at a given energy into a given solid
angle. PADs and related anisotropy parameters are therefore useful observables to obtain quantitative insights in the relative strength of the different pathways and their competition. This has
been demonstrated in the past for the competition between resonant and non-resonant two-photon
ionization pathways [112, 113, 153] and for one- and two-photon ionization channels [82, 47, 25].
In the simple case of photoionization from a single state, anisotropy and asymmetry parameters
have been used in the past to identify and analyze interesting physical effects. For example, asignificant circular dichroism via the asymmetry in the forward-backward electron emission frombromocamphor molecules induced by circularly polarized light has been identified [26]. Observationof the breakdown of the symmetry in the photoelectron emission of argon has been shown in theregion of the Cooper minimum [110]. Interference between resonant and non-resonant pathways[112] or direct and autoionizing channels [39] can be identified via anisotropy and asymmetry parameters. Other examples can be found in double photoionization [160] or molecular vibrations andchirality [75] and applications range from studies of coherent control [199] to the characterizationof ultrashort laser pulses [35].

51
In this chapter, we examine the PADs of electrons which are emitted when atoms are ionized
by few photon processes during the interaction with ultrashort pulses with the goal to image
attosecond dynamics. In Sec. 4.1 we study the impact of the increased bandwidth of these ultrashortpulses on the PAD for a two-photon process including intensity and pulse jitter studies. In thisinitial ionization study from a single state (i.e., the ground state of the atom), we will perform theanalysis with conventional asymmetry and anisotropy parameters. Next, in Sec. 4.2 we extend thesestudies to the more general case of ionization from a superposition state. Since in these situations
application of the conventional parameters fail, we will present a new set of generalized asymmetry
parameters (GAPs) that is an extension of the standard front-back asymmetry parameter. It allows
to study interference in the emission from states with differing magnetic quantum numbers thatcan be prepared via excitation with circularly and elliptically polarized laser pulses. Finally, we
will consider the time dependent ionization of helium atom, prepared in the 1s − 2p1 superposition,and present tools capable of reconstructing dynamics that repeat over a 200 attosecond period fromthe PADs in Sec. 4.3.

4.1

Short pulse effect in ionization of ground-state helium atom1
Two-photon ionization of atoms, specifically the helium atom, has been studied in theory andexperiment in the past, first concerning total ionization yields [177, 230, 220, 193, 65, 214, 66] and,more recently, with a focus on photoelectron angular distributions [112, 113, 153, 25]. It has beenshown, that the PADs strongly depend on the pulse duration or, equivalently, the spectral width of
the EUV pulses. Initially, theoretical studies by Ishikawa and Ueda revealed that the competition
between resonant and nonresonant pathways depends on the pulse width [112, 113]. Changes in the
PADs for 1 - 21 fs pulses at photon energies below the first ionization energy were characterized by
the even anisotropy parameters β2 and β4 . Via these parameters the relative phase between the sand d-wave packets for the photoelectron emission can be determined. Related experimental workat the SPring-8 Compact SASE Source test accelerator verified the co-presence of the two pathways
1

The content of this section has been also published in J. Venzke et al., J. Phys. B. 53, 085602 (2020).

52[153]. Very recently, Boll et al. extended these studies, covered a much wider photon energy range
and explored the impact of radial and angular electron correlation [25]. They showed that in the
low-energy regime a single-active electron picture is valid while at higher energies in the regime of
autoionizing states angular correlations play a major role for the understanding of the PADs.
The ongoing quest in shortening pulses at extreme- and deep-ultraviolet wavelengths towards
the single-cycle regime in duration are achieved via large spectral widths. Such broadband energy
pulses give rise to competition between one- and two-photon (as well as three-photon) ionization
processes for emission at a given energy. As mentioned in Ref. [25], in helium atom this leads to theadditional interference of the s- and d-wave packets due to two-photon absorption with a p-wave
packet from single photon absorption. In this section, we study the transition regime between
single photon ionization and two-photon absorption. We first show how in this regime the PADs
quickly change over a small window of pulse duration, about 1 cycle in full width at half maximum(FWHM). The variation is reflected in the anisotropy parameters βi (i = 1, 2, 3 and 4), whichshow the transition as well as the interference between the ionization pathways. The dependence
on pulse duration, photon energy as well as photoelectron energy is studied. We further consider
how variation of pulse parameters, such as the carrier-envelope-phase (CEP), the peak intensity,and the partial coherence of FEL pulses, do impact the anisotropy parameters. Finally, we briefly
study at which intensities three-photon ionization, which occurs for these large bandwidth pulses
as well, may become observable. In order to accurately account for the different pulse parameters,
we have performed simulations based on the time-dependent Schrödinger equation. Since a recent
study has shown that for pulses with central photon frequency below the first ionization potentialof the helium atom the single-active electron (SAE) picture remains valid [25], we have used a SAE
potential for our numerical simulations.
We solve the time dependent Schrödinger equation for the interaction of a electron in a
spherically symmetric SAE potential with the electric field E of a linearly polarized laser pulse(along the ẑ-direction) in dipole approximation and length gauge as described in Chapter 3. We
have expanded the wavefunction in spherical harmonics up to lmax = 20 with m = 0 due to the53symmetry of the problem. The radius is discretized using fourth order finite difference method witha grid spacing of 0.05 a.u. and a maximum radius of 300 a.u. We utilized the exterior complex
scaling method on the outer 30 a.u. of the grid with a time step of 0.01 a.u. and apply a SAE
potential given in Eq. 3.33. Our numerical simulations have been performed for laser pulses withcentral frequencies at or near to the energy difference between the field-free 1s and 2p state of thehelium atom. In our SAE model of helium the ground state energy is -0.944 a.u., while the 2p
energy is -0.128 a.u. To ensure that the electric field integrates to zero [35], we define the vector
potential following the procedure described in Sec 3.6.
As result from our ab-initio numerical calculations we obtain the wavefunction at the end of
the interaction with the laser pulse. Via projection on continuum states we can then determine theionization probability as well as photoelectron energy and angular distributions. If more than one
pathway, such as one- and two-photon ionization, contribute to the emission of the photoelectron at
a given energy in a given direction, our numerical simulations do not allow to separate the respective
probabilities for each channel easily. This is in contrast to calculations based on perturbation theory,in which the amplitude for each pathway is evaluated and can be analyzed separately. In order toanalyze the dominant pathways for photoemission as functions of intensity, photoelectron energy
and pulse duration, we therefore determine anisotropy parameters (β-parameters), which provide
quantitative insights in the different pathways involved as well as the interference between them[82, 47, 25].
It has been shown [136] that the photoelectron angular distribution for an isotropic target isgiven by:nX
σ(E) 
P (E, θ) =1+
βj (E)Pj (cos(θ))
4π

(4.1)

j=1

where θ is the angle of electron emission with respect to the polarization direction of the laser
field, while σ(E) is the ionization probability at E and Pj are the Legendre polynomials. The
anisotropy parameters βj depend on the amplitudes of the different pathways leading to emissionof the photoelectron (for explicit expressions, see [82]). For combinations of one- and two-photon

54ionization processes, that will be the focus of our study, contributions from the first four βj parameters are expected. The contributions related to the odd polynomials indicate interference
between partial waves resulting from one- and two-photon absorption. We also note that the
impact of three-photon ionization, that we will briefly consider below, is expected to show up in
the anisotropy parameter β5 and higher order parameters.
We have determined the anisotropy parameters in our calculations using Equation (4.1) as
follows. At the end of the simulations of the time-dependent Schrödinger equation we obtained thephotoelectron angular distribution at a given momentum k by projecting the wavefunction onto thefield-free continuum states on the numerical grid. The numerically obtained PAD has been thenprojected onto the j-th Legendre polynomial Pj (j = 1, . . . , 5) and the results have been normalizedusing the total ionization probability σ.

4.1.1

Results and Discussion

Following a general discussion of the processes involved in Sec. 4.1.1.1, we present PADs andrelated anisotropy parameters for pulses having a Gaussian envelope. The results are analyzed,
in view of a competition between one- and two-photon ionization, as a function of pulse duration,
central frequency and peak intensity in Sec. 4.1.1.2. Impact of shot-to-shot fluctuations of thecarrier-envelope-phase, the peak intensity and the partial coherence of FEL pulses are studied inSec. 4.1.1.3.

4.1.1.1

Impact of one-, two-, and three-photon processes in ultrashort pulses

To first illustrate the impact of the one-, two- and three-photon ionization pathways on theemission of the photoelectron in ultrashort laser pulses, let us consider an emission induced by apulse with central frequency ω0 tuned to the 1s − 2p energy gap. The broad energy spectrum of
an ultrashort Gaussian pulse is schematically depicted in the panels of Figure 4.1 centered about
the energy of the 2p-state. Absorption of two photons by an electron in the ground state at thecentral frequency ω0 (c.f., Figure 4.1(b)) will lead to its emission at energy E = 2ω0 − |E1s |. If

Ip

Ip

(a)
E0

EnergyEnergyEnergy

55

(b)
E0

0
1
2
` quantum number

Ip

(c)
E0

0
1
2
` quantum number

0
1
2
3` quantum number

Figure 4.1: Schematic representation of one- (a), two- (b), and three-photon (c) ionization in an
ultrashort pulse. The Gaussian distribution shows the spectral width of the ultrashort pulse centered about the energy of the 2p-state. The red arrows represent the photon absorption pathways,
and the black lines depict some of the resonant structure of the helium atom. (Figure from [235])

the bandwidth of the pulse is broad enough (i.e., the pulse duration is sufficiently short) ionizationof a photoelectron with the same photoelectron energy can occur via absorption of one photonwith energy 2ω0 (c.f., Figure 4.1(a)) or via absorption of three photons with energy 32 ω0 (c.f.,Figure 4.1(c)). We note that besides the two-photon and three-photon pathways with absorptionof photons of equal energies, there exist more such pathways to a final photoelectron energy E forabsorption of photons with unequal energy as long as the sum of the photon energies is equal to2ωE .
The occurrence of the one- and three-photon processes requires a broad spectral bandwidthof the pulse. That means, for long pulses we expect that only the two-photon ionization process ispresent. If the pulse is shortened to a few optical cycles, both the one- and three-photon channels
become effective, since the probability of photon absorption at energies away from the central
frequency increases. In the perturbative intensity regime the probability for a n-photon processscales as I n , where I is the intensity of the pulse. We therefore expect that at a given ultrashortpulse duration at low intensities the one-photon process dominates over the two-photon process
in an ultrashort laser pulse. As the intensity increases, first the two-photon process should thenbecome dominant, before at significantly higher intensities contributions from the three-photon

56
process will become detectable. In this section, we mainly study the transition from one- to twophoton processes, while the influence of the three-photon process will be given minor attention.

Figure 4.2: Photoelectron angular distributions for helium atom ionized by laser pulses at central
photon energy ω0 = 22.2 eV, peak intensity I0 = 1011 W/cm2 , carrier-to-envelope phase φ = 0 andthree different pulse durations: N = 0.6 FWFM cycles (0.112 fs, left column), N = 1.2 FWHM
cycles (0.224 fs, middle column) and N = 1.8 FWHM cycles (0.335 fs, right column). Angular
distributions are obtained for three values of photoelectron energy E = 1.45(2ω0 − |E1s |) = 1.0
a.u. (top row), E = 2ω0 − |E1s | = 0.687 a.u. (middle row), and E = 0.58(2ω0 − |E1s |) = 0.4 a.u.
(bottom row), where E1s is the energy of the 1s state in the SAE potential. (Figure from [235])

4.1.1.2

One- vs. two-photon ionization

In Figure 4.2 we present examples of PADs that show the transition from single photon
ionization, illustrated by the p-wave character of the distribution (panels (d, g, h)), to an emission following two-photon absorption, highlighted by the dominating d-wave character in the PAD(panels (b, c, f)). The results are obtained for ionization by a laser pulse at a central photon
energy ω0 = 22 eV, which corresponds to the energy difference between the 1s- and 2p-states in the
single-active electron potential for helium used in the calculations, peak intensity I0 = 1011 W/cm2

57
and carrier-to-envelope phase φ = 0. At a given photoelectron energy, the transition occurs for avariation of the pulse duration, e.g. at E = 2ω0 −|E1s | = 0.687 a.u. from 0.112 fs to 0.335 fs (middlerow). Conversely, for a fixed pulse duration, the transition is seen for increase of the photoelectronenergy, e.g. at τ = 0.224 fs from 0.4 a.u. to 1.0 a.u. (middle column). In the transition regime the
PADs exhibit the interference between the one- and two-photon ionization processes.

Figure 4.3: Cross section σ(E) as a function of peak intensity for photoelectron emission at energy
E = 2ω0 − Ip = 18.7 eV in 0.5-cycle (one-photon process dominates), 1.0-cycle (transition regime),and 1.5-cycle (two-photon process dominates). Central photon energy ω0 = |E2p − E1s | = 22.2 eV
and carrier-to-envelope phase φ = 0. (Figure from [235])

Further insights in the transition between the two processes can be found via the cross section
σ(E) and the first four anisotropy parameters βi (i = 1, . . . , 4) . In Figure 4.3 the results for the
cross section are shown as function of peak intensity at fixed photon and photoelectron energies.
The cross section for a 0.5 cycle pulse has a slope of one corresponding to the dominance of one
photon ionization over the whole intensity regime. In contrast, the cross section obtained for the
1.5 cycle pulse is dominated by the two photon process due to the reduced bandwidth of the pulse.
In the data for the 1-cycle pulse the transition from the one- to the two-photon process is seen to
occur near 1012 W/cm2 .
In Figure 4.4 the results for the β-parameters are shown as function of pulse duration and peakintensity at fixed photon and photoelectron energies. The even β-parameters show the transition
from dominant one-photon ionization (β2 > 0, β4 ≈ 0) to dominant two-photon ionization (β2 ≈ 0,

58

Figure 4.4: Parameters β1 (a) to β4 (d) as function of pulse duration (FWHM) and peak intensity. Central photon energy ω0 = |E2p − E1s | = 22.2 eV, carrier-to-envelope phase φ = 0 andphotoelectron energy E = 2ω0 − Ip = 18.7 eV. (Figure from [235])

β4 > 0), while the odd β-parameters exhibit the regime of interference between the two processes(β1 < 0 and β3 < 0 for φ = 0). The latter peak where the magnitudes of the two transition
amplitudes coincide. At a given peak intensity the interference regime extends over a change ofpulse duration of about 0.5 cycles at FWHM. For a fixed pulse duration, the interference regime
extends over a variation of peak intensity by a factor of 2-5. We can therefore expect that theresults are rather stable with respect to intensity fluctuations of the pulse. The impact of otherpotential pulse fluctuations in the experiment on the β-parameters will be discussed in subsection
4.1.1.3.
The transition probabilities of the one- and two-photon processes, σ1 and σ2 , scale linearlyand quadratically with intensity. Assuming that the two-photon ionization process involves pre-59
dominantly the absorption of photons at equal energies, the peak intensity at maximum interference
can be estimated as Iinf er ∝ D(1) (E)f (Ω)/(D(2) (E)f 2 (ω)), where D(i) (E) is the square of the transition dipole moments for the one- and two-photon process, f is the spectral distribution of thepulse and Ω = 2ω = E + |E1s |. Since ω = ω0 for the results in Figure 4.4, f (ω) = f (ω0 ) = 1 andIinf er ∝ f (Ω). Thus, we expect that the interference between the two processes requires a larger
spectral bandwidth, i.e. shorter pulse duration, the larger the peak intensity of the pulse. The
results in Figure 4.4 confirm this expectation.

Figure 4.5: Anisotropy parameters β1 (left) and β2 (right) as function of pulse duration and photoelectron energy E for central frequencies ω0 = 1.1|E1s − E2p | (top row), ω0 = |E1s − E2p | (middlerow) and ω0 = 0.9|E1s − E2p | (bottom row) at peak intensity of 1011 W/cm2 and φ = 0. In eachpanel the dashed line corresponds to E = 2ω0 − Ip . (Figure from [235])

60
Next, we consider how the transition between the two pathways depends on the photoelectronenergy and the central photon frequency at a given peak intensity I0 = 1011 W/cm2 and carrierto-envelope phase φ = 0. To this end, we show in the Figure 4.5 the anisotropy parameters β1(left column) and β2 (right column) as a function of photoelectron energy and pulse duration for
central frequencies below (top row), on (middle row) and above (bottom row) resonance with the
1s − 2p transition in the helium single-active electron potential. We restrict ourselves to β1 and β2
since the higher order β-parameters contain equivalent information (see Figure 4.4). Also shown in
Figure 4.5 is the photoelectron energy E = 2ω0 − Ip , corresponding to a two-photon transition at
the central frequency ω0 .
The comparison shows that the transition regime depends on the central frequency, since
the regime shifts for a given pulse duration to larger photoelectron energies as the central frequencies increases (from bottom to top). At a given central frequency, the pulse duration, at which
the transition between the one- and two-photon ionization pathways occurs, increases as the photoelectron energy decreases. Qualitatively, this dependence can be understood using the relation
Iinf er ∝ D(1) (E)f (Ω)/(D(2) (E)f 2 (ω)). For photoelectron energies larger than E = 2ω0 −Ip (dashedline), both ω and Ω are larger than the central frequency. Consequently, at large photoelectronenergies a broader spectral distribution (shorter pulse duration) is required for the same value off (Ω)/f 2 (ω) than at photoelectron energies below E = 2ω0 − Ip (dashed line), where ω < ω0 whileΩ > ω0 .
Finally, we briefly mention that in the limit of short pulses and high intensities, both the oneand three-photon processes can interfere since the three-photon process scales with I 3 . The results
for the anisotropy parameter β5 in Figure 4.6 confirm this expectation. At a photoelectron energy of2ω0 − |E1s | (panel (a)) a peak intensity of about 1014 W/cm2 and a rather large spectral bandwidthis required to facilitate the three-photon process with significant probability since the corresponding
photon energies are smaller than the central frequency. The distribution over photoelectron energy(panel (b)) shows that the impact of the three-photon ionization process is indeed more visible atenergies near E = 3ω0 − |E1s |.

61

Figure 4.6: Anisoptropy parameter β5 (a) as function of pulse duration and peak intensity at
central frequency ω0 = |E1s − E2p | and photoelectron energy 2ω0 − |E1s | and (b) as functionof pulse duration and photoelectron energy at central frequency ω0 = 0.9|E1s − E2p | and peakintensity of 1013 W/cm2 . The dotted lines correspond to E = 2ω0 − Ip and E = 3ω0 − Ip . The
other parameters are as in Figures 4.4 and 4.5, respectively. (Figure from [235])

4.1.1.3

Impact of pulse fluctuations

For the discussion in the previous subsection we have considered results obtained at fixed
pulse parameters. The results in Figures 4.4 and 4.5 show that the transition regime occurs overa rather small window of pulse duration while it appears to be rather stable for variations of thepeak intensity up to half an order of magnitude. The photon energies in the extreme ultraviolet canbe nowadays generated using high-order harmonics or free electron lasers. In view of the technicaldifficulties to control the carrier-to-envelope (CEP) of ultrashort pulses as well as the fluctuating
pulse shapes in the self-amplified spontaneous emission (SASE) mode of free electron lasers, wehave studied the impact of these variations on the anisotropy parameters.
To study the dependence on the CEP we have averaged the results about a given value φ0
using a Gaussian distribution of width α as:(α)βj (φ0 ) =hi
1 |φi −φ0 |exp−
σ(φi )βj (φi )
i
2
2πα
h

i
,
P
1 |φi −φ0 |
σ(φ
)
exp
−
i
i
2
2πα

P

(4.2)

where σ(φi ) is the total ionization probability at CEP φi . The results for averages with (b) α = 0.2
and (c) α = 0.4 are compared in Figure 4.7 with the unaveraged results (panel (a)). As expected,
the odd β-parameters, which reflect the interference between the one- and two-photon processes,
strongly depend on CEP, while the even parameters are independent of it. Thus, the transition

62

Figure 4.7: Anisotropy parameters β1 (solid line), β2 (dashed line), β3 (dotted line) and β4 (dasheddotted line) as function of carrier-envelope-phase. The results averaged using a Gaussian distribution for the CEP (c.f., Eq. (4.2)) with widths of (b) α = 0.2 (in units of 2π) and (c) α = 0.4
are compared with the unaveraged results (a). Peak intensity: 1011 W/cm2 , central frequency:
ω0 = |E1s − E2p |, pulse duration: 1.2 FWHM cycles and photoelectron energy 2ω0 − |E1s |. (Figure
from [235])

from one-photon to two-photon ionization can be observed via β2 and β4 even in pulses without
CEP stabilization. Although the odd parameters depend on CEP, the corresponding results appear
to be indicative for the transition up to fluctuations of about π/2.
We have further studied the impact of fluctuations of temporal FEL laser pulse shapes [171].
We note that current FEL technology does not provide the bandwidth required to generate pulses
down to the one- or two-cycle limit at the photon energies considered here. We may still attempt togive some theoretical insights concerning the robustness of the signal against the major fluctuations
present in an FEL pulse. To this end, we have applied a partial coherence method, which has beenused before to model longer FEL pulses [191]. We note that future technological progress towards
generation of ultrashort FEL pulses in the EUV regime may necessitate to extent the current
approach.
To generate the FEL pulses used in the numerical simulations, the spectrum of a vector
potential with Gaussian envelope corresponding to peak intensity I0 and FWHM pulse duration τ0
is used as an input. Each spectral component is then multiplied with a random phase factor andan inverse Fourier transform is taken producing A0 (t), which is then normalized and windowed intime to give:
AF EL (t) = A0 f (t)Re [A0 (t)]max |Re [A0 (t)] |

(4.3)

63
where A0 = cE0 /ωA in Equation (3.41) and f (t) represents the envelope, here a Gaussian envelope.
The electric field of the FEL pulse is then obtained using Equation (3.42). The resulting pulse
simulates the partially coherent nature of the SASE pulses produced by an FEL. The averageof results from numerical simulations over an increasing number of shots is expected to resemble
results similar to those produced during an FEL beam-time. Here, the β-parameters are calculatedusing a weighted average
βjF EL (E) =PNshotsi

σi (E)βj,i (E),PNshotsσi (E)
i(4.4)

to account for the photoelectron yield shot to shot.

In Figure 4.8 we compare the results for β1F EL (left) and β2F EL (right) as function of photoelectron energy and pulse duration of the Gaussian window used in modeling the FEL pulses,
averaged over 10 (panels (c, d)) and 200 (panels (e, f)) FEL shots for each pulse length (i.e., 6,200
calculations in total) with the exemplary results from a single shot (panels (a,b)). A robust distribution for β2F EL emerges as the number of shots increases, clearly showing the transition froma one- to a two-photon process (the same conclusion holds for β4F EL , not shown). However, theinterference in the photoelectron angular distributions cannot be determined via FEL laser pulses,
since the results for β1F EL and β3F EL (not shown) average to zero in the transition regime. We notethat the generated FEL pulses are not transform-limited and the pulse duration of the Gaussian
window used in modeling the pulses does not correspond to that of Gaussian pulses used in theprevious subsection 4.1.1.2. Comparing the data presented in Figure 4.8(f) with those in Figure
4.5(d) we estimate that the effective pulse duration of a bandwidth-limited Gaussian pulse with aspectrum corresponding to the average spectrum of the generated FEL pulses is about 0.6 times
shorter than that of the Gaussian window.

4.1.2

Summary

In this Section we have provided theoretical results, obtained from numerical solutions of
the time-dependent Schrödinger equation, for the competition between one-photon and two-photon
ionization in an ultrashort extreme-ultraviolet laser pulse. We have shown that the transition

64

Figure 4.8: Anisotropy parameters β1 (left) and β2 (right) as function of pulse duration of theGaussian window used in modeling the FEL pulses and photoelectron energy averaged over 10(middle row) and 200 (bottom row) partially incoherent free electron pulses as compared to asingle shot result (top row). Peak intensity of the Gaussian window: I0 = 1011 W/cm2 and central
frequency of the spectral distribution ω0 = |E1s − E2p |. (Figure from [235])

between the two processes can be observed in the photoelectron angular distributions and therelated anisotropy parameters βi (i = 1, . . . 4) as function of pulse duration, peak intensity, central
photon frequency and photoelectron energy distribution. While the even β-parameters exhibit thetransition via a change from zero to a finite value, the odd parameters indicate the interference
regime. At given photon and photoelectron energies this regime extends over about 0.5 FWHM
cycles in duration and a variation by a factor of 2-5 in peak intensity. The impact of three-photon

65
ionization, which becomes available at these broadband pulses as well, is seen at high intensities
and large photoelectron energies. Finally, we have considered typical variations in the CEP ofultrashort pulses, e.g. as produced in high harmonic generation, and fluctuating pulse shapes infree-electron laser pulses. It is found that the transition between one- and two-photon ionization
can be observed via the even β-parameters in FEL pulses and pulses without CEP stabilization.

4.2

Generalized asymmetry parameters2
While studies of quantum systems in a single state are important, very interesting physics

arises from the systems in superposition states. Nowadays the most prominent example of a twolevel quantum mechanical system is a qubit with its important applications in quantum computation and quantum simulations [213]. Yet, also the internal motion of quantum mechanical systems,
whether it is rotational, vibrational or electronic, is determined via superposition states. In ultrafast science the observation and resolution of such dynamics and, hence, the observation of atoms or
molecules in superposition states has always played a central role. Currently, it is the superpositionof atomic or molecular electronic states and the related attosecond electron dynamics that is the focus of studies in the field [80, 161, 105, 246]. Perhaps the simplest case of such dynamics is a helium
atom in a superposition of 1s and 2p1 state which results in a wavepacket rotating in a plane aroundthe nucleus with a period of ∼ 200 attoseconds. The dynamics in such quantum systems in superposition states can be probed via ionization with an ultrashort laser pulse. Unlike for the ionizationof a quantum system prepared in a single state, e.g. the ground state, conventional anisotropy andasymmetry parameters fail to provide comprehensive tools for the analysis of photoionization fromatomic superposition states. As shown in the previous section, the simplest case of a competition
between one- and two-photon ionization processes can be analyzed using asymmetry parameters
if the atom is prepared in the ground state (see also, [112, 153, 82, 47, 104, 25, 243, 235]). In
contrast, these analysis tools are either not applicable or do not provide a straightforward interpretation for the same processes if the atom is in the superposition of two states. Thus, an extension
2

The content of this section has been also published in J. Venzke et al., Sci. Rep. 10, 16164 (2020).

66
of the toolbox for the characterization of the states and the identification of competing pathways
is desirable. In this section, we propose a new set of generalized asymmetry parameters which aresensitive to interference effects in the photoionization of atomic systems in superposition states.
As we will show these new parameters can be used to identify the interplay of competing linear
and nonlinear pathways at low and high intensities, as well as at ultrashort pulse durations. The
application and relevance of the parameters is tested using state-of-the-art numerical solutions of
the time-dependent Schrödinger equation. Our method provides a new approach to the analysis ofexperiments dedicated to resolving attosecond electron dynamics.

4.2.1

Generalized asymmetry parameters

We consider a prototypical example in ultrafast science and, more general, in atomic physics,
which is schematically shown in Fig. 4.9. An atom in a superposition of two quantum states withdifferent magnetic quantum numbers m, say the ground (g) and an excited (e) state is probed via
ionization by an ultrashort linearly polarized laser pulse. The photoelectron emission is induced
by a pulse with central frequency ω tuned to the energy gap of the two states. The broad energy
spectrum of an ultrashort Gaussian pulse is schematically depicted in the panels of Fig. 4.9 centeredabout the energy of the excited state. There are three competing pathways leading to photoelectron
emission with energy 2ω − |Eg |, where Eg is the ground state energy of the atom: (a) absorptionof one photon at ω from the excited state, (b) absorption of two photons with sum frequency 2ω
from the ground state and (c) absorption of one photon at 2ω from the ground state. While theionization from the excited state (a) is the dominant pathway at low peak intensities and long pulse
duration, the transitions from the ground state will interfere at higher peak intensities (two-photon
process, (b)) and if the bandwidth of the pulse is broad enough (i.e., the pulse duration is sufficiently
short, (c)). The exemplary results obtained from the solutions of the time-dependent Schrödinger
equation in Fig. 4.9 for the interaction of neon atom, prepared in the superposition of 2p−1 - and3d2 -states (d) or prepared in the same superposition but including additionally ionization from the
2p0 and 2p1 ground states (e), with a linearly polarized laser pulse show that the photoelectron

Ip
Ee
Eg

(a)

Ip
Ee
Eg

−2 −1

0

1

EnergyEnergyEnergy

67

(b)
−2 −1

2

∆`

Ip
Ee
(c)

Eg
0

1

−2 −1

2

∆`

0

1

2

∆`

Intensity W/cm2



1014

1012

1010
(d)

(e)
1

4

Pulse length (cycles)

1

4

Pulse length (cycles)

Figure 4.9: (a-c): Ionization pathways effective in different intensity and pulse length regimes. (d)Photoelectron angular distributions for ionization of neon atom, prepared in 2p−1 − 3d2 superposition, as function of intensity and pulse length. (e) The same as (d) including additionally ionizationfrom the 2p0 and 2p1 states. (Figure from [232])

angular distributions (PADs) vary significantly as function of pulse duration and peak intensity.
The forward-backward asymmetry, often used in the past, cannot be applied to identify the impactof the different pathways in these PADs. Therefore, a new set of asymmetry parameters is needed
for photoionization from superposition states with ultrashort laser pulses.

68
4.2.1.1

Definition

We start by expanding the PAD as a coherent sum of spherical harmonics, assuming theatom is ionized by a linearly polarized pulse aligned along the z-axis as
P (θ, φ) =

X

2
m
m
C` g Y` g (θ, φ) + C`me Y`me (θ, φ)

(4.5)

`

where mg (me ) are the magnetic quantum numbers of the ground (excited) state in the superposition, C`m is the complex amplitude and Y`m (θ, φ) is the spherical harmonic. The asymmetry in
the PADs due to the interference between different channels is related to the relative phase andamplitude of the spherical harmonics. For each spherical harmonic, the sign of the phase is symmetric (asymmetric) across the xy-plane when ` + m is even (odd), while in the xy-plane the phaseis proportional to eimφ .
In the xy-plane there are regions of destructive and constructive interference between thetransition amplitudes from the ground and excited state, as illustrated on the left of Fig. 4.10. The
regions can be labeled by(φ − φ0 )∆m
ci =
π



(4.6)

where bc is the floor function, φ0 is a reference angle, and ∆m = |mg − me | (∆m = 3 in Fig. 4.10).
Setting φ0 at an angle where the interference switches from constructive to destructive, regions ofdestructive interference signal are labeled by even ci while odd ci denote regions of constructive
interference. Next we define the following integrals:
even
I±
=

X Z

P (θ, φ) dΩ±
ci

(4.7)

P (θ, φ) dΩ±
ci

(4.8)

even ci
odd
I±
=

X Z

odd ciwhere dΩci is the solid angle for the region ci in the positive (z > 0) or negative (z < 0) hemispherewith respect to the xy-plane. In the example in Fig. 4.10 (left) each integral represents the total
photoelectron signal in the regions of a specific color (dark blue, light blue, dark red, light red).
We now define general asymmetry parameters (GAPs) that account for the relative difference in

69
the regions of constructive and destructive interference:even +I even )−(I odd +I odd )

(I+

−
+
−


I
 +even +I−even +I+odd +I−odd max(φ0 )



∆m
Ap ≡





even
odd )−(I odd +I even )+− (I+even+I−even

odd
odd
I+

+I−

+I+ +I−max(φ0 )even γ(4.9)
odd γ

where γ = `e + me + `g + mg + Np with `g (`e ) are the quantum numbers of the ground (excited)state in the superposition, and p = (γ mod 2) is the parity of γ. Each parameter A∆m
is related topa certain total number of absorbed photons, Np = Ng +Ne , where Ng (Ne ) is the number of photons
absorbed in the transition from the ground (excited) state. We note that GAPs cannot only be
defined for ionization of superposition states with a linearly polarized laser pulse along the z-axis,
but the definition can be extended, for example, to ionization with a circularly polarized ionizing
laser pulse in the xy-plane. In that case γ = `e +me +`g +mg , the number of photons is not included
as both ` and m change by 1 for each photon absorbed. Using ∆m = |(mg ± Npg ) − (me ± Npe )|for right (+) and left (-) handed circular polarized probe pulses, processes with different number ofphotons involved can be studied by analyzing the signals with different ∆m. Here, in the further
discussion and applications we, however, focus on the case of a linearly polarized probe pulse.
To exemplify the significance of the asymmetry parameters we consider the ionization of Ne
atom, initially prepared in the 2p−1 − 3d2 superposition (i.e., `g = 1, mg = −1, `e = 2, me = 2;
and, hence, lg + mg + le + me = 4), by an ultrashort intense laser pulse. As discussed at theoutset, we expect interferences between two kind of pathways depending on the peak intensity
and the pulse duration. For an ultrashort pulse at low intensities one-photon transitions fromthe ground (Fig. 4.9(c), Ng = 1) and the excited states (Fig. 4.9(a), Ne = 1) will interfere, with
Np = Ng + Ne = 2, γ = 6 and p = 0. An examplary PAD, obtained via numerical TDSE solution
in the relevant intensity and pulse duration regimes, is shown at the top of the right column in
Fig. 4.10. For even γ the corresponding parameter A30 relates to the difference between the total
photoelectron signals in the dark red and light blue shaded regions, as depicted at the top of the
middle column of Fig. 4.10. Comparison with the PAD shows that this difference indeed accounts

Asymmetry

Integrals

PAD

70

even γodd γeven/odd

Figure 4.10: Conceptual illustration of GAPs (for ∆m = 3). Left: Integrals I±, Eqs. (3,4),
are defined over regions of constructive and destructive inferference, indicated by different colors(dark blue, light blue, dark red, light red), in xy-plane. Middle: GAPs are constructed based on theparity of the parameter γ = le + me + lg + mg + Np from the integrals over the regions denoted bya certain color (light blue, dark red). Right: Examplary PADs displaying the asymmetry capturedby the parameters for even and odd γ. (Figure from [232])

for the asymmetry in the PAD induced by the interference of the transition amplitudes for the
one-photon processes from the ground and the excited states. In contrast, due to the dependenceof multiphoton transition amplitudes on the intensity of the pulse at longer pulse duration andhigher intensities it is expected that the one-photon transition from the excited state (Fig. 4.9(a),Ne = 1) interferes with the two-photon absorption from the ground state (Fig. 4.9(b), Ng = 2),
giving rise to Np = 3, γ = 7 and p = 1 for ionization of Ne(2p−1 − 3d2 ). The regions relevant in
the calculation of the asymmetry parameter A31 and the corresponding exemplary PAD for the Ne
atom are presented in the bottom row of Fig. 4.10. The comparison indicates the significance of
the asymmetry parameter A31 for the detection of the interference at high intensities.

71
4.2.1.2

Application via numerical simulations

For the application of the GAPs we have considered certain superpositions of two atomic
states which are first prepared by a pump pulse and then probed by a linearly polarized pulse at a
set time delay such that the relative phase between the two states is determined. In the calculations
we have therefore simulated the interaction with the probe pulse only by using numerical solutions of
the time-dependent Schrödinger equation (TDSE) for the interaction of a electron in a single-active
electron (SAE) potential with the electric field E of a linearly polarized laser pulse (aligned along
the quantization ẑ-axis) in dipole approximation and length gauge with a frequency corrected laser
as described in Chapter 3. The present calculations are performed utilizing single active electron
potentials for He atom and Ne atom [204] with the electron initially prepared in a superpositionof the ground and an excited state. The TDSE has been solved by expanding the wavefunctionin spherical harmonics (up to `max = 50 for all relevant m values), as described in Sec. 4.1. The
computations have been performed on a radial grid of 300 a.u. with a grid spacing of 0.05 a.u.
using exterior complex scaling on the outer 15 a.u. of the grid. A time step of 0.05 a.u. has been
used.

4.2.2

Results and discussion

For our applications we have considered individual superposition states in neon (Fig. 4.11)
and realistic superposition states considering all possible initial mg states of neon and helium atoms(Fig. 4.12). The central frequency of the applied electric field was set to the energy difference of theinitially populated field-free states (ω0 = |Eg − Ee |). To analyze the relevance of both the shortpulse parameter A∆m
and the high-intensity parameter A∆mcalculations have been performed
01
for one- and four-cycle probe pulses (FWHM pulse durations) as function of the peak intensity ofthe pulse. At the end of each simulation of the time-dependent Schrödinger equation we obtained
the photoelectron angular distribution at a given momentum k (corresponding to a photoelectronenergy of E = 2ω − |Eg |) by projecting the wavefunction onto the field-free continuum states on the

72
numerical grid. The asymmetry parameters A∆m
and A∆m
are then determined from the numerical0
1
PADs using Eq. (4.9).
0.4

0.4

A N=1
A N=1
A N=4
A N=4
1

0.3

0.4

(a)

0

(b)

1

(c)

0.3

1

0.3

1

1

0.1

0.0
10
10

1

Ap

1

1

Ap

Ap

1

0

0.2

0.2

0.1

10

11

10

12

10

13

10

14

10

0.0
10
10

15

0.1

10

11

2

12

10

13

10

14

10

0.0
10
10

15

1

Ap

1

Ap

0.3

0.2

0.1

0.1

12

10

13

10
2

14

10

15

0.0
10
10

10

14

10

15

(f)

0.3

0.1

10

13

0.4

0.2

Intensity (W/cm )

10

0.5

0.2

11

12

0.6
(e)

0.4

0.3

10

2

0.5

0.4

10

11

Intensity (W/cm )

0.6
(d)

0.5

0.0
10
10

10

Intensity (W/cm )

0.6

1

10

2

Intensity (W/cm )

Ap

0.2

10

11

10

12

10

13

10
2

Intensity (W/cm )

14

10

15

0.0
10
10

10

11

10

12

10

13

10

14

10

15

2

Intensity (W/cm )Figure 4.11: Generalized asymmetry parameters A10 (solid lines) and A11 (dashed lines) as functionof intensity for ionization of superpositions of 2p−1 − 3d0 (a), 2p0 − 3d1 (b), and 2p1 − 3d2 (c)in neon atom with initial populatations of 0.5 for each state (top row). Results in bottom row
for superpositions with populations of P2p−1 = 0.98, P3d0 = 0.02, P4s0 = 0.00057 (d), P2p0 =0.94, P3d1 = 0.06 (e), and P2p1 = 0.88, P3d2 = 0.12 (f). Results have been obtained for one-cycle(blue lines) and four-cycle (orange lines) pulses at photoelectron energy E = 2ω − Ip . (Figure from
[232])To exemplify the application and provide basic insights in the relevance of the GAPs, we havefirst considered superpositions of individual sublevels in different shells of the neon atom as initialstate. Fig. 4.11 shows results for the GAPs as a function of peak intensity for interaction of a neon
atom, prepared in (2p−1 − 3d0 ) (a,d), (2p0 − 3d1 ) (b,e), and (2p1 − 3d2 ) (c,f) superpositions. For the
results in the top row equal population in the two initial states has been considered, while for thosein the bottom row populations, as produced via one-photon excitation by a 50 cycle, 1013 W/cm2
right handed circulary polarized pump pulse, are used for the initial state (states with populations
larger than 10−5 are considered, see figure caption for values). The central frequency of the ionizing
probe laser pulse is tuned to the energy gap between the field-free ground and excited states.

73
Overall, except for the results in panel (a) which we will discuss below separately, the shortpulse parameter A10 (solid lines) is large at low intensities and for the shorter of the two pulse
durations. This is expected from our discussion above and confirms that the parameter is anindicator for the interference of the one-photon signal from the ground state and the one-photon
signal from the excited state. Since both of these signals are first order in intensity, their relative
strengths and, hence, the A10 parameter are independent of intensity at low intensities for a given
initial superposition state. As the intensity increases, the relative amplitude of the two-photon
ionization channel increases and finally dominates over the one-photon channel from the groundstate. This trend is reflected in both the short-pulse and the high-intensity parameters. The
increase in A11 , resulting from the interference between the one-photon excited state signal and thetwo-photon ground state signal, shows in which intensity regime the two-photon signal starts toovertake the short pulse signal. Simultaneously with the increase of A11 we observe a decrease of theA10 signal, in agreement with our physical interpretation of the relevance of the different pathways.
Parenthetically, we note the onset of an increase of the A10 parameter at high peak intensities islikely due to pathways involving higher order processes with an even number of total photons.
While the general trends of the two parameters are similar in most of the cases, there is one√
exception where the details differ significantly. In the case of the (2p−1 − 3d0 )/ 2 superposition(panel a), A10 shows a different trend. It remains constant at low intensities before increasing
at higher intensities. As the laser pulse and relative populations are fixed in the top row (ac) of Fig. 4.11, the change in amplitude and shape of A10 and A11 are caused by changes in the
cross-sections of the various states. We, therefore, attribute the different trend in the A10 signal
in panel (a) to be due to the dominance of the one-photon transition from the 3d0 state at all
intensities, since m = 0 states are, in general, easier to ionize with a linearly polarized pulse
than those with |m| > 0. This interpretation is further supported by the results in Fig. 4.11(d)for a superposition with much lower initial population in the 3d0 -state. Despite the differencein magnitude of transition amplitudes, now the interference between the one-photon pathways iseffective and the general expected trend for the short-pulse parameter A10 is present.

74
Comparison of the results in the two rows of Fig. 4.11 shows that the intensity at which
the transitions in the parameters occur depends on the population in the two states in the initialsuperposition. Since in the bottom row the populations in the excited states are low, the relative
magnitudes of the corresponding one-photon transition amplitudes from the excited states as compared to those from the ground state levels are weaker than for the equally populated superpositions(top row). Therefore, we observe the impact of the two-photon pathway from the ground state atlower intensities in the results in the bottom row. Additionally, the A11 signal becomes sensitive tochanges in pulse length due to the weak signal from the excited state. Thus, the GAPs may also be
useful to detect the population ratio in the superposition states. Here, we do not further analyze
this feature, but focus on the more general application of the parameters to identify the presenceof different ionization pathways.
0.4

0.4

A m NΔ1A m NΔ1A m NΔ4
A m NΔ4

0.4

Δ

0

0.3

Δ

(a)

0.3

1

(b)

0.3

(c)

Δ

1

0.1

0.0
10
10

1

Ap

3

Δ

0.2

Ap

Ap

1

0

0.2

0.1

10

11

10

12

10

13

10
2

Intensity (W/cm )

14

10

15

0.0
10
10

0.2

0.1

10

11

10

12

10

13

10
2

Intensity (W/cm )

14

10

15

0.0
10
10

10

11

10

12

10

13

10

14

10

15

2

Intensity (W/cm )Figure 4.12: Same as Fig. 4.11 but for superpositions generated by a right-handed circularly polarized pulse via one-photon transition in helium atom (a), three-photon transition in neon atom(b), and one-photon transition in neon atom (c) (for details see text). (Figure from [232])

So far, in Fig. 4.11 we considered simple albeit somehow artificial superposition states restricted to certain sublevels of two shells in the neon atom. Now, we extend our analysis to threedifferent initial states as they can be generated with right-handed circularly polarized pump laser√
pulses: (a) one-photon excitation of helium atom to the 2p1 state leading to (1s−2p1 )/ 2 superposition, (b) three-photon excitation of neon atom to the 3d shell, leading to an initial state consistingof a superposition with equal population in 2p−1 and 3d2 along with populated 2p0 , and 2p1 states(note, that the latter two 2p states cannot be excited by the absorption of three right-handed cir-

75
cularly polarized photons to the 3d shell), and (c) one-photon excitation of neon atom to the 3d
shell leading to the combination of the three superposition states analyzed separately in the bottom
row of Fig. 4.11. In Fig. 4.12, we present results for the GAPs of the corresponding calculations.
The initial superpositions in case of the helium data (a) and the neon data in panel (b) consist ofjust one state in the excited shell and show the same general trends as those in the majority ofthe panels of Fig. 4.11 with the decrease of A∆malong with the increase of A∆moccurring in a01certain intensity regime. The transitions in helium occur at a higher peak intensity due to the more
tightly bound helium ground state, which is harder to ionize. The overall trends of the results in
Fig. 4.12(c) for the more complex initial superposition, consisting of six states in the 2p- and the
3d-shell of neon. The one-cycle pulse data are similar to those discussed before, providing the same
information about the impact of the two-photon transition from the ground state. However, the A11parameter, particularly at the longer pulse duration, shows an additional interference structure in
the transition regime. It is likely that this is due to an interference between transitions originatingfrom two excited states differing by ∆m = 1, here between one-photon ionization signals from 3d0
and 3d1 as well as from 3d1 and 3d2 . Although the number of absorbed photons in these transition
is not 3, the resulting γ has the same parity as the signal from a 2pm − 3dm+1 state.

N=1
N=4

0.10

1

0.06

A

1

0.08

0.04

0.02

0.00
10

10

10

11

10

12

10

13

10

14

10

15

2

Intensity (W/cm )Figure 4.13: Asymmetry parameter A11 for ionization of helium atom (1s-2p1 ) with one- and fourcycle pulses at central frequency ω = 1.2ω0 . (Figure from [232])

The high-intensity asymmetry parameter A∆m
is not only indicative of the general features1

76
concerning the interferences between the pathways discussed above, but also provides insights inmore subtle aspects. In the top row of Fig. 4.11 the maximum of the asymmetry parameter A∆m1
occurs at the same intensity independent of the pulse duration. This is due to the fact that in thecalculations the central frequency was set to the energy difference of the initially populated states.
In this case the center of the photoelectron energy distributions due to the one-photon ionizationfrom the excited state and the two-photon transition from the ground state nearly coincide. If thefrequency is detuned from the resonance, the position of maximum interference strongly depends
on the pulse duration. Due to the larger spectral width of the pulse at shorter pulse duration, theinterference between the channels becomes most effective at higher intensities than the longer pulse
duration results, at which the photoelectron distributions from the two channels have less overlap.
This effect is seen in the results in Fig. 4.13 for photoionization at a central frequency of 1.2 times
the energy difference between the 1s- and 2p1 -states in helium atom.

α=0
α = 0.11

α = 0.2
α = 0.5

0.04

A

0

0.06

0.02

0.00
10
10

10

11

10

12

10

13

10

14

10

15

2

Intensity (W/cm )Figure 4.14: Asymmetry parameter A10 for ionization of helium atom (1s-2p1 ) with one-cycle pulses
at central frequency ω0 . Results of averages over different Gaussian distributions of CEP withwidth α are compared with those at fixed CEP (α = 0, solid line). (Figure from [232])

Finally, it is important to consider variations in the laser parameters relevant for an application of the generalized asymmetry parameters in an experiment. Typically, the peak intensityof the applied laser pulse may vary from shot to shot as well as over the interaction volume. The
results in Fig. 4.11 and Fig. 4.12 show that the two GAPs vary rather slowly as a function ofpeak intensity. Thus, we can expect that variations in intensity in an experiment will not impact

77
the results significantly. Another parameter that is usually difficult to control is the CEP of alaser pulse, which typically becomes most important for ultrashort pulses. In the present study
the high-intensity A11 parameter is independent of variations in CEP. To study the dependence on
the CEP for the short-pulse low-intensity asymmetry parameter A10 we have obtained results as afunction of the CEP and averaged the results about a given value ϕ0 using a Gaussian distributionof width α (in units of 2π) as:A∆mp (I, T ; α) =hi
1 |ϕi −ϕ0 |exp−
σ(ϕi )A∆mp (ϕi , I, T )
i
2
2πα
h

i
,
P
1 |ϕi −ϕ0 |σ(ϕi )
i exp − 22παP

(4.10)

where σ(ϕi ) is the total ionization probability at CEP ϕi . In the calculations we have assumed
that the pump pulse (for the preparation of the initial state) has the same CEP as the probe pulse.
The comparison of the results for different averages in Fig. 4.14 with the results for fixed CEP(solid line) shows that for He atom the short-pulse asymmetry parameter A10 is indicative for theinterference up to fluctuations of about π/2 in the CEP.

4.2.3

Summary

In summary, in this section we have introduced a set of generalized asymmetry parameters (GAPs) which characterize the interference of linear and nonlinear pathways to ionization of
atoms, prepared in superposition states, due to the interaction with a linearly polarized intense
ultrashort laser pulse. These parameters may provide a new tool to analyze data in attosecond
experiments. The relevance of the parameters is demonstrated via the results of numerical simulations of ionization of helium and neon atom. The impact of short pulse and nonlinear effects, as
they arise in experiments with free-electron lasers and high-order harmonic generation, is shown.
The dependence on the central frequency of the applied laser pulse and the impact of variationsof laser parameters, such as the peak intensity and the carrier-to-envelope phase, are analyzed anddiscussed.

78

4.3

Wavefunction reconstruction method3
As mentioned above, one of the simplest cases of a superposition of atomic states withdifferent orbital angular momentum and magnetic quantum numbers is that of a helium atomin the 1s and 2p1 states. It results in an ultrafast electron dynamics given by a wave packet
rotating in a plane around the nucleus with a period of ∼ 200 attoseconds. We now use thisdynamics as a prototype example to study the reconstruction of the corresponding wave packet
via ionization with an ultrashort linearly polarized probe laser pulse. To this end, we analyze thephotoelectron angular distributions as a function of time delay from the instant of preparationof the atomic superposition states. The concept is based on the idea to utilize quantum beating
signals, where the imaged wave packet is interfered with a reference wave, to reconstruct a wave
function [185, 173, 161, 130, 198, 116]. Accounting for the phase accumulation, ionization crosssections, and characterization of the reference wave packet that is used in the measurement is a
non-trivial task. In this section we present a method for wave packet reconstruction based on
quantum beating signals that utilizes perturbation theory. Since the method requires knowledgeof the average values for intensity, carrier-to-envelope phase and pulse duration of the probe pulse
we perform a sensitivity study to show the impacts of noise produced in an experiment and thelimits in peak laser intensity on the reconstruction method. Finally, we consider the extension of
the method to imaging of wave packets consisting of superpositions of more than two states.
The rest of the section is organized as follows: In section 4.3.1 we first present and discuss
the ionization scheme on which the reconstruction method is based. Then we briefly review themethods for the numerical methods used for the reconstruction algorithm. In section 4.3.2 wepresent the application of the method to the reconstruction of a circular wave packet in helium
atom. Aspects of the error analysis with respect to the accuracy of the observation of the PADs
and the variation of the laser parameters will be presented. Furthermore, the impact of different
pathways on the reconstruction process will be studied. Finally, we briefly discuss the extension of3

The content of this section has been also published in J. Venzke et al., PRA. 103, 042808 (2021).

79
the method to superpositions of more than two states along with results for a specific three-statesuperposition. The section ends with a summary.

4.3.1
4.3.1.1

Concept and theoretical methods
Pump-probe scheme

In this study, we focus on the reconstruction of a wave function and the related imaging of
the dynamics via photoelectron angular distributions (PAD). To this end, we start our simulations
in a superposition state of the atom. An experiment will require a pump pulse to generate thesuperposition. There are a few requirements for the pump pulse itself to apply the reconstruction
scheme. Since the scheme relies on the measurement of the PADs at various time delays between
the pump and the probe pulse, the pump must be a reproducible pulse with a fixed carrier envelope
phase (CEP) such that the superposition generated for each measurement is very similar from shot
to shot if not the same. However, the form of the electric field of the pump pulse does not need to be
known. Next, the delay between the pump and probe pulses must be controlled on the attosecond
time scale to allow for measurements as a function of the delay. The shot-to-shot measurements
should be taken for time delays within one cycle of the probe pulse giving rise to the requirement of
an attosecond control over the delay. The total delay is irrelevant as long as the two pulses do not
overlap in time and the state to be imaged does not decay. Furthermore, photoelectrons generated
by the pump pulse have to be separated from those produced by the probe pulse. It might betherefore useful to limit ionization by the pump pulse.
The method studied here can be applied to simple superpositions, consisting of populationin two or three atomic states. As prototypical example we consider a helium atom in a 1s −2p1 superposition, which represents a circular wave packet rotating around the nucleus. This
superposition with equal population in the two levels can be created with a ∼ 23 cycle laser pulse
at 1014 W/cm2 tuned to the resonance frequency. Fig. 4.15(a) shows the dynamics via isosurfacesof the wave function (i.e., |rΨ|2 ) rotating in the x − y plane on an attosecond time scale.

80

Figure 4.15: (a) Isosurfaces of the 1s − 2p1 wave function (|rΨ|2 is shown) evolving in time. (b)Ionization scheme in the x − y plane. The laser polarization along the x-axis is depicted in red, thedirection of the ionized electron is shown in orange, and the charge migration is shown in yellow.
Selection rules for ground state signal (c) and excited state signal (d). The solid lines illustrate thetransitions of photons absorbed at the central frequency and the dashed lines show the pathways
for short pulse effect due to the large bandwidth of an ultrashort pulse. (Figure from [233])

To characterize the motion, and in turn reconstruct the wave function, we expand the initialwave function in the eigenbasis of stationary states and write the wave function at some time t as:Ψ(r, t) = a0 ψ0 (r, t) +

N
−1
X
j=1

aj eiθj ψj (r, t)(4.11)

81
where aj is a positive real number, θj is a relative phase, and {ψj = Ψnlm ; j = 0, . . . , N − 1} is the
eigenstate basis. While the basis may contain an infinite number of eigenstates, for practical meansin the numerical calculations and any application it has to be truncated at some finite number
N . Thus, the wave function depends on N − 1 phases and N amplitudes since the global phase
is not a physically relevant quantity. In this framework, the time dependence is contained in thestates |ψj (t)i and the initial wave function is completely reconstructed if all aj and θj for j ≥ 1 are
obtained since a0 is fixed by the normalization and we set the phase of ground state |ψ0 i to zero.
If the wave function generated by the pump pulse can be reconstructed at a given time, the fulltime dependent motion of the wave function has also been obtained.
In the reconstruction scheme that we consider the electron dynamics is probed via ionization
by an intense ultrashort laser pulse as a function of time delay after the end of the pump pulse. In
the scheme predictions of first- and second-order perturbation theory (PT) are used to determine theunknown phases and amplitudes from the photoelectron angular distributions (PADs). The latter
may be determined in an experiment but in this theoretical study we utilize ab-initio numericalcalculations. For the case of a helium atom in a 1s−2p1 superposition, we apply a probe laser pulse
linearly polarized along the x-axis and detect the photoelectron angular distribution in the x − yplane, i.e. for θ = π/2, as a function of the angle ϕ from the x-axis as shown in Fig. 4.15(b). The
absorption of a photon will induce transitions with ` → ` ± 1 and m → m ± 1 due to the selection
rules. The pathways from the ground (c) and the excited state (d) are shown in Fig. 4.15. The solid
arrows represent the pathways for the transitions at the central frequency of the probe pulse. Ip isthe ionization potential and the ovals indicate possible resonances. Since we analyze photolelectron
emission induced by ultrashort pulses, we need to consider additional pathways due to the broad
bandwidth of the pulse (Sec. 4.1). These pathways are the ionization from the ground state witha single photon and the two-photon transition from the excited state, which are represented by
the dotted arrows in Fig. 4.15. Since states in the continuum with the same energy and quantum
numbers can be reached via different pathways, the photoelectron angular distribution will change
in both shape and yield as the relative amplitudes and phases of each signal varies.

82
4.3.1.2

Numerical solution of time-dependent Schrödinger equation

In order to test the applicability and limits of the reconstruction method we use numericalsolutions of the time-dependent Schrödinger equation (TDSE) as substitute for actual measurements of the photoelectron angular distributions. We consider the TDSE in length gauge andsingle-active electron approximation as described in Chapter 3. We use a Helium SAE potential
giving a ground state energy of -0.944409 a.u. and a 2p energy of -0.12847 a.u. The potential has
been constructed for benchmark tests between TDSE and time-dependent density functional theory(TDDFT) calculations [204]. The energies match those of the corresponding DFT potential while
the experimental values for the ground and the 2p state in helium are -0.9037 a.u. and -0.12382
a.u.
For the solution we have expanded Ψ(r, t) in spherical harmonics up to lmax = |mmax | =20 and discretized the radius using fourth order finite difference. The wave function has been
propagated in time with a time step of 0.01 a.u., on a grid with spacing of 0.05 a.u., maximum
radius of 300 a.u., and exterior complex scaling on the outer 30 a.u. of the grid utilizing the CrankNicolson method for time propagation. The numerical code has been tested against previously used
codes as well as results from numerical calculations published in the literature [217]. The electricfield is a frequency corrected sine squared pulse. Once the wave function has been propagated tothe end of the pulse we have obtained the photoelectron angular distribution by projecting onto
numerical continuum states.

4.3.1.3

Perturbation theory

To reconstruct the initial wave function, i.e. determine amplitudes and phases, we invert thephotoelectron angular distributions (here, TDSE results) as a function of time delay by utilizing
perturbation theory (PT) described in Sec. 3.2. To evaluate the PT wavefunction, we use the boundstates of the TDSE field free Hamiltonian for the initial states, and expand the Green’s function for
200 dipole allowed states per ` and m block. The states represent the bound part of the spectrum

83
and a discretized continuum up to an energy of ∼ 59 eV. The matrix elements and time integrals
are performed numerically.
The PT results are then used to obtain a residual
2

R(a) =

Xk,φ,τ

P
PPT (a)P TDSE− P|PTDSE ||PPT (a)|k,φ,τ

(4.12)k,φ,τ

where PTDSE and PPT are the TDSE and PT photoelectron distributions for all utilized final
momenta, time delays, and detection angles. The state vector a that minimizes R(a) gives thereconstructed wave function. The normalization maintains the relative yields at each final momentum. To minimize R(a) for the two (three) state system considered below, 1000 (250) evenly
spaced samples were used. For the current applications, this minimization technique is sufficient.
For superpositions involving more states the dimensionality of the minimization space increases.
It is likely that for such studies more advanced minimization methods such as stochastic gradientdescent will be needed. We also note that the method is applied to image field-free wave packets,
i.e. a is time independent. For cases, in which a changes with time, an extension of the present
method is needed.

4.3.2
4.3.2.1

Results and discussion
Reconstruction of two-state superposition: Intensity limits and number ofsamples

To apply the minimization and therefore reconstruct the wave packet, the photoelectron
angular distributions (PADs) have to vary as a function of time delay. It is therefore first interesting
to see at which laser parameters a significant variation in the PADs can be observed. To this end,in Fig. 4.16 we show how the PAD varies within one optical cycle of the pump pulse. The delay ∆τin the Figure refers to some fixed reference time τ0 after the end of the first pulse. Comparison of
the TDSE results for a two-cycle (∼ 190 as FWHM, left column) and a ten-cycle (∼ 950 as FWHM,
right column) laser pulse at three peak intensities ranging from 1010 W/cm2 (bottom row) over 1012
W/cm2 (middle row) to 1014 (top row) W/cm2 is shown. In these calculations the photon energy

84

0.6

0
(a)
0.2 0.4 0.6 0.8
ΔτΔ(opticalΔcycles)φΔ(rad)

π

−2π
−π

(c)
0.2 0.4 0.6 0.8
ΔτΔ(opticalΔcycles)φΔ(rad)

1.0

π

0.0

−π

1.0

π

0.4
(e)
0.2 0.4 0.6 0.8
ΔτΔ(opticalΔcycles)

0.4(b)0.2 0.4 0.6 0.8
ΔτΔ(opticalΔcycles)

0.6
0.4
(d)
0.2 0.4 0.6 0.8
ΔτΔ(opticalΔcycles)

0.0

−π

0.2
0.0

1.0
0.8
0.6

0

0.2

0.0

0.8

π
2

−2π

0.2

1.0

0

0.2

0.6

0

0.6

π
2

−2π

0.8

π
2

−π

−π

0.4

π

−2π

0.0

0.6

0

0.8

0

0.2

0.8

π
2

1.0

π
2

−2π

φΔ(rad)−π

0.4φΔ(rad)

0.8

π
2

−2π

π

1.0

φΔ(rad)φΔ(rad)π0.4(f)0.2 0.4 0.6 0.8
ΔτΔ(opticalΔcycles)

0.2
0.0

Figure 4.16: Results of TDSE calculations for photoelectron angular distributions over one cycleof the probe pulse field for ionization of 1s − 2p1 superposition with ultrashort two-cycle (leftcolumn) and ten-cycle (right column) laser pulse with peak intensities of 1014 W/cm2 (top row),
1012 W/cm2 (middle row), and 1010 W/cm2 (bottom row). The time delays ∆τ are given withrespect to a reference time τ0 after the end of the first pulse. (Figure from [233])

ωE has been set to the energy difference between the initially equally populated field-free 1s and√2p1 states in helium atom ((|1si + |2p1 i)/ 2). The angular distribution is taken at photoelectronenergy E = 2ωE − Ip , where ωE is the central photon energy and Ip is the ionization energy.
Although the laser pulse has a FWHM of similar or even longer duration than the charge

85
migration in the present application, the time delayed PAD shown in Fig. 4.16 has significant
enough variation for the minimization technique to converge. This is due to the coherent natureof the applied laser light. Since the laser is tuned near the resonance, the oscillation of the electricfield approximately agrees with that of the superposition to be imaged. This allows for subcycle
time dependence to be encoded in the PAD signal.
At the lowest intensity (Fig. 4.16, bottom row) we observe that the two-cycle signal varies
significantly. This is due to the interference between the one-photon ionization channels from theexcited state and the ground state. The pathway from the ground state provides a significant
contribution to the electron emission due to the broadband spectrum of the ultrashort two-cycle
pulse (Sec. 4.1 and Sec. 4.2). Thus, as the time delay varies, the relative phase of the groundand excited states changes producing a quantum beating. In contrast, for the longer ten-cycle
pulse the spectrum of the pulse is more narrow. Consequently, the contribution from the onephoton pathway from the ground state is much smaller. Hence, the PAD for the ten-cycle pulse at
the lowest intensity shows minimal variation as a function of time delay since it is dominated byionization of the excited state.
As the intensity increases, the relative contribution of the two-photon ionization pathway fromthe ground state increases because the power dependence of the signals scales with the number ofphotons absorbed. Since the two-photon channel depends on the absorption of photons at thecentral energy the significance of the contribution does not depend on the pulse duration. As aresult, we observe that at the largest intensity considered in Fig. 4.16 (top row) both the two- andthe ten-cycle PADs vary strongly as function of the time delay. This is due to the interference
between the one-photon ionization from the excited state and the two-photon ionization from theground state.
To demonstrate the reconstruction method the errors in reconstructing the amplitude (a,√d) and phase (b, e) of the superposition state (|1si + |2p+ i)/ 2 in helium atom as a functionof peak intensity are shown in Fig. 4.17. The error is obtained via ∆a = a0 − a where a is theamplitude or phase of the original wave function and a0 is the corresponding amplitude or phase

0.10

4
8

N Δ2
N Δ5
N Δ10

1011

0

−π8
1012 1013 1014
intensity (W/cm2)

0.10

1015

−π4 1010

1012 1013 1014
intensity (W/cm2)

1015
(e)

π

Δϕ1 (rΔd)

8

0.00

−0.05

−π8
1011

1012 1013 1014
intensity (W/cm2)

1015

0

−π4 1010

(c)

N =2
N =5
N =10

10−1
10−3
10−5
10−7
10−9
10−11 1010

1011

1012 1013 1014
intensity (W/cm2)

1015
(f)

R(a)

Δa1

1011

4

0.05

−0.10 1010

N Δ2
N Δ5
N Δ10

π

(Δ)

10−1
10−3
10−5
10−7
10−9
10−11 1010

R(a)

0.00

−0.10 1010(b)πΔϕ1 (rad)

Δa1

0.05

−0.05

86

π

(a)

1011

1012 1013 1014
intensity (W/cm2)

1015

1011

1012 1013 1014
intensity (W/cm2)

1015

Figure 4.17: Errors in amplitude (a, d) and phase (b, e) for the two-state reconstruction using
PADs generated with laser pulses at two, five and ten cycle pulse duration. Also shown is theresidual R(a) (c, f). The reconstructions are based on PADs at 20 time samples (∆t = τ /20) using
PADs in the full x − y plane (upper row) and photoelectron signals in forward-backward direction(lower row). (Figure from [233])of the reconstructed wave function. The reconstruction is based on PADs taken at 20 time delays
over one period of quantum beating τ = 2π/|E2p1 − E1s |. Also shown are the final results of theminimization for R(a) (panels (c) and (f)).
Reconstructions of two cases are compared in which either the full PAD in the x − y plane
via signals at 6,285 equally spaced angles (upper row) or only the signals in the forward-backward
direction along the polarization axis of the probe pulse (lower row) are used. The results in in
the upper row show that the reconstruction method reproduces the initial state up to peak laser
intensities of about 1013 W/cm2 if the full x − y plane and 20 time samples are used. The increaseof the errors at intensities larger than 1013 W/cm2 indicates that higher-order processes begin tocontribute leading to the breakdown of the present second-order reconstruction method. For a
two-state superposition just two amplitudes and the relative phase have to be determined. Using
the full PAD with several thousand signals is oversampling for the reconstruction method. This isconfirmed by the results for the reconstruction (d-f) using the PAD signals in forward-backward

87
direction only.
π

0.05

π

0.00

0

−0.05

−Δπ

−0.10

2
Δ

Δϕ1 (rad)

Δa1

0.10

1011

1013
intensity (W/cm2)

1015 −2πFigure 4.18: Same as Fig. 4.17 but based on PADs at just two time delays. Errors in amplitude(filled symbols with solid lines) and phase (open symbols with dashed lines) are shown for using
PADs in the full x − y plane (blue squares) and in forward-backward direction (orange circles).
(Figure from [233])

Next, we have studied if one can limit the number of PAD samples over time. To this end,
we present in Fig. 4.18 the errors in amplitude and phase based on a reconstruction using justtwo PADs over one period of quantum beating τ = 2π/|E2p1 − E1s |. If the full PAD in the x − yplane is used (blue squares) the reconstruction method remains successful over the same intensity
regime as before. However, when the available signals are limited to the forward-backward direction(orange circles), the data are insufficient for a one-to-one mapping between the initial state andthe obtained photoelectron signal. Consequently, the reconstruction fails at nearly all peak laser
intensities, since for just two time samples the normalization leads to a greater chance that multiple
values for amplitudes and the relative phase produce a given signal.
Thus, for the basic case of equal population in a two-state superposition and probing witha pulse having a central frequency equal to the field-free energy difference of the state, our results
show that the reconstruction method is successful for intensities in the perturbative regime up to1013 W/cm2 . If the PADs can be measured over a broad range of angles probing at two times over
one cycle of the probe pulse field is sufficient. In contrast, if only the forward-backward asymmetry
signals are measured more time samples are needed.

88
4.3.2.2

0.10

(Δ)

0.05

0.05

0.00

0.00

Δa1

Δa1

0.10

Impact of different pathways

(Δ)

−0.05

−0.05

−0.10 1010 1011 1012 1013 1014 1015
intensity (W/cm2)−0.10 1010 1011 1012 1013 1014 1015
intensity (W/cm2)

Figure 4.19: Comparison of amplitude error for (a) two and (b) five cycle probe pulse from calculations using full second order PT (blue), neglecting only the two-photon transition from theexcited state (orange) and neglecting both the one-photon transition from the ground state andthe two-photon amplitude from the excited state (green) in the reconstruction. (Figure from [233])

The reconstruction depends on the effective interference of at least two amplitudes to havea quantum beating. In the full reconstruction we consider the contributions from four pathways(one- and two-photon transitions from ground and excited states). It is interesting to ask which
pathways contribute effectively besides the one-photon transition from the excited state. To study
this question, we have deliberately neglected individual amplitudes in the reconstruction. In Fig.
4.19 we compare the results of these calculations with those of the full reconstruction (blue bars)
based on the full x − y plane and 20 time samples. We show the results for the amplitude error
only and note that those for the phase error lead to the same conclusions.
We performed two set of test calculations: First, we utilized all amplitudes except the twophoton signal from the excited state (orange bars). Next, we have removed the one-photon signal
from the ground state and the two-photon signal from the excited state from the reconstruction(green bars). Both transition amplitudes are expected to have an impact at ultrashort durations
due to the broad bandwidth of the pulses. Indeed, we do see noticeable changes for the two-cycle
data (a) while there appears to be insignificant impact for the five-cycle pulse (b) at all intensities.
For a two-cycle probe pulse, at intensities less than 1013 W/cm2 , the reconstruction without

89
the two-photon amplitude from the excited state matches the full data (Fig. 4.17(a)). At higher
intensities, the results deviate which agrees with the expectation. Removing also the one-photontransition from the ground state from the reconstruction causes the results to deviate from the full
results at 1010 W/cm2 and above 1013 W/cm2 . At the lowest intensity it is the one-photon signal
from the ground state that provides a dominant contribution to the PAD, leading to the deviation
at 1010 W/cm2 . At intensities above 1013 W/cm2 , the additional exclusion of the one-photon signal
leads to a better reconstruction than for the full data and when the two-photon pathway from theexcited state is neglected. This can indicate that inadvertently the two ultrashort amplitudes havean opposite effect on the reconstruction or that an even higher-order amplitude is more dominant
than the lower-order processes.

4.3.2.3

Variation of central frequency and population

So far, we have considered a special case with equal population in the two states and thecentral frequency tuned to the energy difference between the two field-free states. We will now study
deviations from this special case. First, we consider a laser pulse detuned from the energy difference
between the two equally populated states. We have calculated the photoelectron distribution
at E = ωE − |Ee |, where Ee is the energy of the excited 2p1 -state. Given a sufficiently broad
spectral range, i.e. sufficiently short pulse, the one-photon ground state signal (peaking at aboutE = ωE − |Eg |) at low intensities will interfere with the excited state signal creating the same
quantum beating as before allowing for the reconstruction process to work. In Fig. 4.20 we show thereconstruction results for a laser with central frequency of ωE = 0.8ω0 (upper row) and ωE = 1.2ω0(lower row). The results lead to the same conclusion as for the resonant case, namely that thereconstruction method can be applied in the perturbative intensity regime up to 1013 W/cm2 .
To show that the reconstruction works for a two-state superposition with arbitrary unknown
amplitudes and phases, we have performed a ‘blindfold experiment’ where the population of thestates were chosen randomly. The random values were held unknown and were only accessed toobtain the final error at the end of the reconstruction procedure. We have used this approachΔϕ1 (rΔd)

π

1013
1011
intensity (W/cm2)

(a)

4

0
−π4
−π2 1010

1011 1012 102 13
intensity (W/cm )

(b)
1014

π

2
π

Δa1

0.10
0.05
0.00
−0.05
−0.10

2

N Δ2
N Δ5
N Δ10

Δϕ1 (rΔd)

Δa1

0.10
0.05
0.00
−0.05
−0.10

90
π

102 131011
intensity (W/cm )

(c)

4

0
−π4
−π2 1010

1011 1012 102 13
intensity (W/cm )

(d)
1014

Figure 4.20: Amplitude (a, c) and phase (b, d) errors for ionization at a detuned photon frequencyof ω = 0.8ω0 (upper row) and ω = 1.2ω0 (lower row). The reconstructions are based on full PADs
in x − y plane at 20 time delays. (Figure from [233])
π

0.05

π

0.00

0

−0.05

−4π

−0.10 1010

2
4Δϕ1 (rΔd)

Δa1

0.10

Δϕ1
Δa1
π
11
14 −2
12
13
10
10
10
10
intensity (W/cm2)

Figure 4.21: Amplitude and phase error for the reconstruction of an arbitrary unknown two-statesuperposition. Reconstruction is based on full PADs in x−y plane at 20 time delays with a five-cycle
probe pulse. (Figure from [233])

91
in several calculations and for probe pulses of different duration, with results leading to the sameconclusion. As an example, one set of results for this ‘blindfold’ experiment is presented in Fig. 4.21,
showing that the reconstruction is successful with the same low errors as in the other cases studied.

4.3.2.4

π

8

(a)

0.05

16

0.00

0

0Δ
1Δ
10Δ
−0.10 1010 1011 1012 1013 1014 1015
intensity (W/cm2)
−0.05

(b)

π

Δϕ1

Δa1

0.10

Detector accuracy and variation of laser parameters

−16π
−8π

0Δ
1Δ
10Δ
10
10 1011 1012 1013 2 1014 1015
intensity (W/cm )Figure 4.22: Comparison of (a) amplitude and (b) phase error for reconstruction using PADs atmachine precision (blue) and with accuracy limited to 1% and 10%. (Figure from [233])

For the reconstructions so far we have used results from TDSE calculations up to machine
precision. Since detectors in experiment do not operate with the same precision we have studied how
less accurate data may impact the success of the reconstruction. To this end, we have deliberately
added random noise at a certain percentage level of the maximum signal in the PADs to the TDSE
data. In Fig. 4.22 we compare the results for the reconstruction with added noise at the 1% and
10% level with those at machine precision using the signals from the full x − y plane and 20 timesamples. It is seen that an accuracy of detection at the 1% level is sufficient to reconstruct the
wavefunction with similar error as in the full calculation with TDSE data at machine precision.
Similar results and conclusions have been obtained for the other cases presented in Figs. 4.17 and4.18. We expect that this limit can be achieved in a measurement.
We have further considered variations in the laser parameters relevant for an application of
the reconstruction method in an experiment. Typically, the peak intensity of the applied laser

92

Δa1

0.05

π
8

(a)

ΔEP
intensity
ΔEP and intensity

0.00

−0.05
−0.100(Δ)π
16Δϕ1 (rad)

0.10

0

π
−16

5

10
15
laser noise (%)

20

25

−π8

0

5

10
15
laser noise (%)

20

25

Figure 4.23: Amplitude (a) and phase (b) error for variation of CEP only (blue squares with solidlines), intensity only (orange circles with dashed lines) and both CEP and intensity (green stars
with dotted lines) at peak intensity of 1011 W/cm2 and pulse duration of 2 cycles. Reconstruction
is based on full PADs in x − y plane at 20 time delays. (Figure from [233])

pulse may vary from shot to shot as well as over the interaction volume. Another parameter thatis usually difficult to control is the carrier-to-envelope phase (CEP) of a laser pulse, specifically for
ultrashort pulses. To study the impact on the reconstruction, CEP, peak intensity, and both CEPand peak intensity have been varied randomly within a certain percentage in the TDSE results.
The results in Fig. 4.23 show that the reconstruction method still works well up to 10% variationin both parameters before significant errors occur. Perhaps most surprising is the rather large
acceptable tolerance in the CEP even for the shortest pulses, since CEP variations often lead tolarge changes in experimental observables. We assume that the shot to shot variation is ‘averaging
out’ over the various time delays leading to a quality in the reconstruction similar to fitting a linear
regression to noisy data.

4.3.2.5

Extension to superpositions of more than two states

Finally, we consider the extension of the reconstruction method to a superposition of morethan two states. The ionization scheme shown in Fig. 4.15(c) and (d) can be applied to each
excited state in the superposition separately, taking the waves generated via one- and two-photontransitions from the ground state as reference. For the reconstruction measurement of PADs at
separate energies for each excited state in the superposition is required. Based on PT results,

93
residuals R(a) (Eq. 4.12) have been obtained, taking into account both final momenta to reconstruct
the wave function.
π

(b)

2

N Δ2
N Δ5

N Δ10

1011
10213
intensity (W/cm )Δϕ1 (rΔd)

π

4

0
−π4
−π2

(c)

1011
102 13intensity (W/cm )

π

(d)

2
π

Δa2

0.10
0.05
0.00
−0.05
−0.10

(a)

Δϕ2 (rΔd)

Δa1

0.10
0.05
0.00
−0.05
−0.10

1011
10213
intensity (W/cm )

4

0
−π4
−π2

1011
102 13intensity (W/cm )Figure 4.24: Amplitude√ (a, c) and phase (b, d) errors for the reconstruction of the superposition(|1si + |2p1 i + |3p1 i)/ 3 in the helium atom (upper row: errors for 2p-state, lower row: errors for
3p-state). (Figure from [233])

To test the extension of the method we have considered the helium atom in the (|1si + |2p1 i +√|3p1 i)/ 3 superposition. A probe laser pulse at central photon energy tuned halfway between the|2p1 i- and |3p1 i-state and measurement of PADs at both E = ω −|E2p | and E = ω −|E3p | have been
applied. The results of the reconstruction, based on PADs in the full x − y-plane taken at 20 timedelays, are shown as function of peak intensity for different pulse durations in Fig. 4.24. The results
show that a similar degree of accuracy in the reconstruction as for the two-state superposition isachieved within the same intensity limits. The example indicates that the present method may be
extended to even more complex wave packets and electron dynamics.

94
Extension of the reconstruction method to more complex superpositions will however require
attention to the interference with the ground state signal and the increased dimensionality of theminimization space. Since the minimization process appears to be convex, a standard minimization
algorithm, like stochastic gradient descent, may be sufficient for an efficient reconstruction. For
each additional state in the interference scheme, the signal at the photoelectron energy chosenmust include interference with at least one other state. This can be achieved if all transitions
from the excited states interfere with that from the ground state by using a short pulse. If only
longer pulses are available, coupling over many pairs of states reduces the required bandwidth fora reconstruction. Finally, due to the selection rules some initial states do not generate a signal
in the xy-plane. In this case it may be required to utilize a full 4π-PAD for reconstruction of thecorresponding superposition.

4.3.3

Summary and outlook

In this section we have studied the use of perturbation theory to reconstruct an atomic
wave function representing ultrafast field-free charge migration. In the reconstruction scheme thephotoelectron angular distributions obtained by ionizing an atom, prepared in a superposition of twoor three states, are used. The electron dynamics are encoded in the PAD signal via interference ofthe ground and excited state ionization signals. Time delayed measurements allow for the extractionof all amplitudes and phases of the initial wavefuction that fully characterizes the charge migration.
Results of applications based on TDSE calculations show that the reconstruction is highly accurate
for probe pulse intensities below 1013 W/cm2 while allowing for CEP (intensity) variations of 10%(20%) from shot to shot. Additionally, a detection accuracy for the PADs of about 1% for areconstruction with small error is required.
These results may provide some basic guidelines for future experimental progress in imaging
electron motion on its fundamental time scale via the detection of photoelectron angular distributions. However, the results also show that experimental challenges remain. Controlling the creationof ultrashort laser pulses, especially those with attosecond pulse duration, is a non-trivial task. For

95the current method to be applied, average values for intensity, CEP and pulse duration of theelectric field of the probe pulse must be obtained and shot-to-shot variations must be limited up
to a certain degree. Given the rate of advancement in attosecond laser pulse technology over thelast decade, this level of pulse control may be, however, achieved in the coming years.
Additionally, precise knowledge of one- and two-photon transition dipoles is required to perform perturbation theory calculations for the target atom with high accuracy. Over the last decades
several standard theoretical methods, such as multielectron theories, density functional theory or
calculations using accurate single active electron potentials, have been developed to perform suchcalculations. As theoretical methods for bound and continuum wave functions further advance, thecurrent and other reconstruction methods will become increasingly accurate.

Chapter 5
Rydberg state excitations induced by intense near-infrared radiation

In the previous chapter, we have studied ionization of atoms involving few photons. We will
now turn our attention to processes in strong-field physics where many-photon processes driven byintense IR lasers are studied. When large numbers of photons are absorbed, the discrete natureof the multi-photon picture begins to blurs, allowing for processes to be understood in a complementary way through classical trajectory analysis and tunneling processes. Fundamental processes
in strong-field atomic physics are above-threshold ionization (ATI) [4], high harmonic generation(HHG) [71, 164, 62], and non-sequential double ionization [239]. These highly nonlinear processes
are induced by the absorption of multiple photons from the laser field, which in the limit of a large
photon number can also be described as a tunneling process. Excitation of the atom is known toplay an important role in each of the processes. It has been initially observed via resonant enhancement in the population of excited states [42, 119] and structures in the energy spectrum [72, 190, 3]
and in energy-resolved angular distributions [209] of photoelectrons. These resonance effects havebeen explained by multiphoton absorption through Rydberg states, which are AC-Stark shifted in
the presence of a laser field. More recently, significant excitation of atoms has also been observedin the tunneling regime and described by the frustrated tunneling ionization model [180].
For linearly polarized lasers, the latter observation has renewed the general interest in themechanisms leading to the population of excited (Rydberg) states during the interaction of an atomwith an intense laser pulse (most recently, e.g., in Refs. [38, 145, 144, 260, 221, 32, 142, 59, 27, 73,
152, 219, 89, 176, 248, 19, 138, 261, 22, 74, 115, 109, 156, 247, 194]). The important role of Rydberg

97states in various strong-field ionization processes and harmonic generation has been discussed. For
example, resonant enhancement of below-threshold harmonics [227, 38, 32], emission from excitedstates via free induction decay [32, 19], high harmonic emission through ionization from excitedstates and recombination to the ground state [23, 19] have been predicted and observed.
Recent theoretical studies of the excitation mechanism in linearly polarized strong fields
mainly consider the distribution of the population as a function of the principal quantum numberof the excited states [145, 144, 261, 247, 194]. It was shown that the modulation of the excitation
probability is related to the channel closing effect [134, 145, 144, 194]. Theoretical analysis of theangular momentum distribution in the populated Rydberg states by linearly polarized fields is lessadvanced. Predictions of Floquet theory for a monochromatic laser field [134] and results of numerical calculations for laser pulses with a trapezoidal envelope [194] yield that the angular momentumof the excited Rydberg states has the same parity as Np − 1, where Np is the minimum numberof photons needed to ionize the atom. Furthermore, the angular quantum number of the stateswith the largest population in numerical calculations [145, 144, 194] agrees well with semiclassicalestimations [6], initially performed for low-energy angular resolved photoelectron distributions.
Besides interactions with linear polarized laser pulses, recently studies on the interaction of
atoms and molecules with intense fields generated by the superposition of two circularly polarizedlaser pulses have seen an upsurge in activity in strong-field experiment and theory. For the mostpart, the renewed interest results from the capability to control the polarization of emitted lightin high-order harmonic generation with such pulses. The physical principle has been proposed
and applied first two decades ago [53, 150]. More recently, efficient phase matching of circularly
polarized high-order harmonic beams in the EUV and soft X-ray regime using bichromatic beams
with counter-rotating circular polarization has been demonstrated [64, 196, 125, 58, 100]. Since
then, much experimental and theoretical work on high-harmonic generation [169, 165, 170, 36, 17,98, 149, 159, 10, 203, 181, 46, 63, 258, 195, 18, 139, 7, 43, 12, 108, 117, 99, 184, 143], ionization andphotoelectron momentum distributions [175, 254, 167, 157, 168, 156, 189, 29, 101, 146, 1, 30, 50, 141,
87, 49, 148, 55, 76, 124, 2], double ionization [155, 51, 21, 44, 252, 107, 154] and other strong-field

98
processes [253, 28, 84] driven by bichromatic circularly polarized laser pulses has been performed.
One interesting aspect in these kind of strong-field interactions is the control of ionization via thehelicity of the applied bichromatic pulses [167, 157, 148]. Such studies complement related work on
the dependence of the ionization rate by one-color circularly pulse on the relative helicity between
the pulse and the electron in the atomic orbital [14, 122, 96, 15, 16, 13, 182, 90, 47, 244, 241, 255,
52, 147, 228].
For the interaction with bichromatic circularly polarized laser pulses resonant excitation
also plays an important role. It has been observed that the probability to ionize an atom issignificantly enhanced if the two fields are counter-rotating as compared to co-rotating fields [157].
The experimental observations were interpreted as due to the increased density of excited statesaccessible for resonant enhanced multiphoton ionization in the case counter-rotating fields. Resultsof numerical solutions of the time-dependent Schrödinger equation in Ref. [157] did confirm a close
relation between the ratios of total excitation and ionization probabilities for counter-rotating andco-rotating circularly polarized laser pulses. However, the results for excitation of the atom were
not further resolved by distributions over the quantum numbers (principal, angular momentum,
magnetic). Such analysis potentially can shed further light on the role of excited states in thepathways to ionization since excitation in a resonant multiphoton process should rely on the spinangular momentum selection rules for the absorption of circularly polarized photons (∆l = ±1 and∆m = ±1).
In this chapter, we present results for excitation of atoms with linearly polarized (Sec. 5.1)
and bi-circular (Sec. 5.2) laser pulses. Interpretation of the distributions of excitation over thestates with different quantum numbers will be given using multiphoton selection rules.

5.1

Linear polarization1
As mentioned before, for the interaction with linear polarization earlier work for monochro-matic laser fields [134] and laser pulses with a trapezoidal envelope [194] have been performed.
1

The content of this section has been also published in J. Venzke et al., Physical Review A 98, 043434 (2018).

99
Here, we extend these previous analysis by considering excitation in laser pulses with more realistic sine-squared and Gaussian envelopes. This gives us the opportunity to study if the parity ofthe populated angular momentum states in such pulses agrees with the selection rules obtained formonochromatic fields and how the results depend on the pulse length at low and high intensities. In
Sec. 5.1.2 we then investigate how the population of angular momentum states leaves its footprints
in the radiation generated via transitions from the excited states to the ground state.
For our studies we have made use of results of numerical solutions of the time-dependent
Schrödinger equation (TDSE) for the interaction of the hydrogen atom with an intense linearlypolarized laser pulse. Specifically, we are solving the time-dependent Schrödinger equation (TDSE)
for the interaction of an atom with a linearly polarized intense laser pulse in the velocity gauge(Eq. 3.3) for the Hydrogen atom (VSAE (r) = −1/r). The code utilizes cylindrical coordinates andsecond order finite difference. In all calculations presented below we used a grid ranging up to ρ =750 a.u. and z = ±750 a.u. with a grid spacing of ∆ = 0.1 a.u. in both directions and a time stepof ∆t = 0.15 a.u. We use the outer 37.5 a.u. of the grid in both directions for the ECS potential.
This grid can easily fit bound states up to n = 14 within the real portion of the grid.

5.1.1
5.1.1.1

Angular momentum distributions
Parity effect in pulses

We study the angular momentum distribution in Rydberg states of the hydrogen atom. To
this end, we consider processes during which these states are shifted into resonance with multiphoton
absorption from the initial ground state of the atom (with energy Ei ). Assuming that the energy
shift of high lying excited states under the influence of an intense laser field is approximately
equal to the ponderomotive shift [3] and the shift occurs instantaneously during the pulse [93], theintensity for a Np photon process to resonantly excite approximately with an excited state withenergy En is then given by:2
I = 4ωE(Np ωE + Ei − En ).

(5.1)

100
In our calculations we have considered peak intensities such that the n = 8 states are in resonanceat central frequency ωE , corresponding to a wavelength of 800 nm, with Np = 10, 11, . . . , 15 photon
processes. Since we study the interaction with laser pulses, instead of monochromatic fields, morethan one manifold of states can be resonantly excited. For example, the bandwidth of a 20 cycle
sine squared pulse at 800 nm covers all excited states n ' 6 within the same Np photon process.
We further note that in the present cases, the n = 3 and n = 2 states are approximately resonant
via Np − 1 and Np − 2 photon processes, respectively, assuming that the energy shift of these statesequals the ponderomotive energy as well.
14
13
12
11
10
9
8
7
6
5
4
3
2
1

10−5

n

n

10−6

14
13
12
11
10
9
8
7
6
5
4
3
2
1

10−7

0
1
2
3
4
5
6
7
8
9
10
11
12
13

l

10−6

10−3

10−4

10−5

(d)
0
1
2
3
4
5
6
7
8
9
10
11
12
13

10−5

14
13
12
11
10
9
8
7
6
5
4
3
2
1

n

n

10−4

(c)

10−7

l

l

14
13
12
11
10
9
8
7
6
5
4
3
2
1

(b)
0
1
2
3
4
5
6
7
8
9
10
11
12
13

10−8

0
1
2
3
4
5
6
7
8
9
10
11
12
13

(a)

10−6

l

Figure 5.1: Excited state distribution as function of n (vertical axis) and l (horizontal axis) at theend of 20 cycle pulses with sine-squared envelope and peak intensities: (a) I0 = 3.4 × 1013 W/cm2 ,(b) I0 = 6.0 × 1013 W/cm2 , (c) I0 = 8.6 × 1013 W/cm2 , and (d) I0 = 1.12 × 1014 W/cm2 . Left(right) column corresponds to cases in which the Rydberg states are resonant with an even (odd)number of photons. (Figure taken from [237])

The population in each quantum state (n, l) at the end of the pulse is shown in Fig. 5.1(20 cycle pulses with sine-squared envelope) and Fig. 5.2 (14 FWHM cycle pulses with Gaussianenvelope) for intensities between 1013 and 1014 W/cm2 . The results shown in the left (right)
column correspond to cases in which the Rydberg states are resonant with an even (odd) numberof photons. In all cases we see that the highest angular momentum states (l > 7) are not vary

14
13
12
11
10
9
8
7
6
5
4
3
2
1

10−4

101

10−5

n
10−7

0
1
2
3
4
5
6
7
8
9
10
11
12
13

(a)

10−8

ℓ

(b)
ℓ

n

10−4

10−5

10−3

10−4

10−5
0
1
2
3
4
5
6
7
8
9
10
11
12
13

(d)

0
1
2
3
4
5
6
7
8
9
10
11
12
13

(c)

10−6

14
13
12
11
10
9
8
7
6
5
4
3
2
1

n

14
13
12
11
10
9
8
7
6
5
4
3
2
1

10−6

0
1
2
3
4
5
6
7
8
9
10
11
12
13

n

10−6

14
13
12
11
10
9
8
7
6
5
4
3
2
1

ℓ

ℓ

Figure 5.2: Same as Fig. 5.1 but for pulses with Gaussian envelope and 14 cycles FWHM. The
pulse duration approximately matches that for the sine-squared pulses in Fig. 5.1 at FWHM.
(Figure taken from [237])

populated, in agreement with previous studies [194] and semiclassical estimates [6]. In the results
we indeed observe signatures of the selection rules resulting in dominant population of states withan even (odd) angular momentum quantum number for the absorption of even (odd) photons in
the plots on the left (right). This shows that in the intensity regime of 1013 to 1014 W/cm2 theparity selection rules, previously studied for monochromatic fields and flat-top pulses, are effective
for long pulses of about 20 cycles. Since the parity effect is found to occur independent of theform of the envelope we restrict ourselves below to pulses with sine-squared envelope. However,
we observe that the contrast between the population in even and odd states is stronger at lower
intensities. We will further discuss this aspect in subsection 5.1.1.2.
The results also show that for an odd parity process (right column) predominantly one angularmomentum state (l = 5) is populated. This is in agreement with the results presented by Li etal. [144], who conjectured that electrons in the low angular momentum states more easily absorb
additional photons resulting in suppression of population in these states due to ionization. However,
our results for even parity processes exhibit a pattern, alternating in l, showing that both low and102high angular momentum states, except for the s-states, remain populated at the end of the pulse.
Np

8s

8p

8d

Np − 1 3s3p3dNp − 2 2s

2p

8f

8g

8h

...

...
...
...

Np − 3

l=0

1

2

8s

8p

8d

Np − 1 3s3p3dNp − 2 2s

2p

4

5

...

8g

8h

...
...

...
...
...

Np − 438f

Np − 3...

Np − 4

Np

l=0

1

2

3

4

5

...

Figure 5.3: The absorption pathways in an even (odd) photon process in the left (right) panel. Greensquares represent virtual states with energy N ω above the ground state and angular momentum l.
The yellow squares represent real states labeled by the quantum numbers. Solid lines refer to open
pathways while dashed lines represent pathways which are suppressed due to population trappingin a lower excited state. (Figure taken from [237])

We therefore put forward an alternative explanation. To this end, in Fig. 5.3 we show thevarious pathways leading to a resonant population in the n = 8 (and the other Rydberg) states
for absorption of an even (top panel) or an odd (bottom panel) number of photons, termed by
Np . Starting from the 1s ground state, the absorption of successive photons proceeds throughvirtual states (depicted via empty green squares in Fig. 5.3), following the selection rule ∆l = ±1.
As mentioned above, for the hydrogen atom and a pulse at a central wavelength of 800 nm the(shifted) n = 3 and n = 2 states are approximately resonant via Np −1 and Np −2 photon processes,
as shown in Fig. 5.3 (real states are depicted by yellow squares labeled with quantum number nl).
Assuming that the resonant transitions to lower lying energy levels cause a trapping of populationin these states, further absorption from these states will be suppressed. Such suppressed pathways
are denoted by dashed lines in Fig. 5.3, while other (open) pathways are represented by solid lines.
Following the pathways, we see that in the manifold of the Rydberg levels for an even number
photon process (left panel) the l = 2 state is partially and the l = 4, 6, . . . states are fully accessible,
while the l = 0 (s-state) should be suppressed, which is in agreement with the pattern, alternating
in l, that we observe in Figs. 5.1 and 5.2 (left columns). In contrast, for an odd number photon

103
process (right panel) the pathways to the l = 1 states are strongly and those to the l = 3 states are
partially suppressed, making the states with l = 5 state the first fully accessible states among the
Rydberg levels, in agreement with the results presented in Figs. 5.1 and 5.2 (right columns).

Short vs. long pulses and CEP effects

10−8

10−7

(c)

ℓ
13

10−3

12

10

−3

10

−4

10

−5

11
10
9

10−4

8
7
6
5
4
2

13

12

9

8

10

7

5

1

11

(f)
4

ℓ

3

3

ℓ

10−5

2

0
1
2
3
4
5
6
7
8
9
10
11
12
13

(e)

0
1
2
3
4
5
6
7
8
9
10
11
12
13

(d)

10−6

ℓ

1

n

10−5

10−8

14

n

10−4

14
13
12
11
10
9
8
7
6
5
4
3
2
1

0

ℓ
14
13
12
11
10
9
8
7
6
5
4
3
2
1

(b)
0
1
2
3
4
5
6
7
8
9
10
11
12
13

0
1
2
3
4
5
6
7
8
9
10
11
12
13

(a)

10−7

10−6

6

10−8

14
13
12
11
10
9
8
7
6
5
4
3
2
1

10−6

n

n

10−7

14
13
12
11
10
9
8
7
6
5
4
3
2
1

0
1
2
3
4
5
6
7
8
9
10
11
12
13

10−6

n

14
13
12
11
10
9
8
7
6
5
4
3
2
1

n

5.1.1.2ℓFigure 5.4: Excited state distribution as function of l (horizontal axis) and n (vertical axis) at theend of 2 cycle (panels on left), 10 cycle (panels in middle), and 20 cycle (panels on right) pulses,
at low peak intensity I0 = 3.4 × 1013 W/cm2 (top row), and high peak intensity I0 = 1.64 × 1014
W/cm2 (bottow row). (Figure taken from [237])

The time-frequency uncertainty relation yields that a pulse spectrum broadens as the numberof optical cycles decreases. Consequently, in an ultrashort pulse an excited state can be reached
via absorption of different number of photons. Due to this mix of even and odd number of photon
processes one may expect that in such pulses a separation in population of odd vs. even angularmomentum quantum states cannot be achieved. This is confirmed by the results of our numericalcalculations at low and high intensities, shown in Fig. 5.4(a) and (d). For a 2 cycle pulse there is asmooth distribution over the lower angular momentum states for each principal quantum number
at low (3.4 × 1013 W/cm2 , panel (a)) and high (1.64 × 1014 W/cm2 , panel (d)) peak intensity.

104
In contrast, the narrowing of the energy spectrum as pulse duration increases does not necessarily lead to an increase of the population in the angular momentum states of one parity over theother. For the low intensity 10-photon resonant process (3.4 × 1013 W/cm2 , Fig. 5.4, upper row)
we observe in the numerical results that Rydberg states with a well defined angular momentum
parity are predominantly populated when the pulse duration is increased to 10 and 20 cycles. On
the other hand, at the higher intensity (1.64 × 1014 W/cm2 , Fig. 5.4, lower row) we observe some
contrast between the population in the angular momentum states of different parity for the 10 cycle
pulse (panel (e)), while the differences in the population of the different channels further blurs when
the duration is increased to 20 cycles (panel (f)).
Within the interpretation of resonant excitation the loss of the parity effect for long pulses
with high peak intensities can be understood as follows. At the higher peak intensity, the highly
excited states are shifted into resonance for 10-14 photon processes over the rising and trailing partsof the pulse, before they are resonantly excited with a 15 photon process at the peak intensity.
Although the excitation probability raises with an increase of the intensity during the pulse, thetime intervals over which the states are in resonance with a certain photon-order process increase
with the pulse length. Thus, there is a significant excitation of the Rydberg states due to theabsorption of odd as well as even numbers of photons in a pulse at high peak intensity and long
duration.

n

10−3

10−4

10−4

(b)
0
1
2
3
4
5
6
7
8
9
10
11
12
13

10−5

10−3

0
1
2
3
4
5
6
7
8
9
10
11
12
13

(a)

14
13
12
11
10
9
8
7
6
5
4
3
2
1

n

14
13
12
11
10
9
8
7
6
5
4
3
2
1

ℓ

ℓ

10−5

Figure 5.5: Population of excited states from a 20 cycle pulse with peak intensity of I0 = 1.64×1014
W/cm2 and CEP of φA = 0.204π (a) and φA = 0.882π (b). (Figure taken from [237])

Furthermore, excitation channels driven by absorption of different numbers of photons with

105
the same parity may interfere, leading to the enhancement or suppression of population in certain
angular momentum states and the loss of the parity effect. Interference effects in excitation havebeen studied extensively in the few-photon regime. Among others, it has been shown that theexcitation probabilities can be controlled via the carrier envelope phase (CEP) of the pulse, evenin the long pulse limit (e.g., [257, 245]). We observe in the results presented in Fig. 5.5 thatthe population distribution in the angular momentum states driven by the present more complex
multiphoton processes indeed changes significantly with variation of the CEP at high intensities,
indicative of potential interference effects.
We note that there may exist alternative explanations for the loss of the parity effect in the
distribution at long pulses with high peak intensity. For example, in the tunneling regime themechanism of frustrated ionization has been explored [180]. According to that mechanism highly
excited states are populated during the trailing edge of the pulse by deceleration of the electron
over many laser cycles and electron recapturing once the laser pulse ceases. Since excited statesof any angular momentum can be populated during the recapturing, the loss of the parity effectfor long pulses at intensities in the tunneling regime is consistent with the frustrated ionization
mechanism as well.

5.1.2

Radiation spectra

It has recently been discussed [38, 32, 19] that the population of excited Rydberg states leads
to the emission of radiation due to the transition from the excited states into the ground state,
which can be observed in the below-threshold part of high harmonic spectra. In the case of atomic
hydrogen such emission is associated with an excitation of the np states. In view of the parity effectin the excitation of the Rydberg states according to the absorption of an even or odd number ofphotons, we expect that the emission may be enhanced or suppressed in certain intensity regimes.
To allow for the separation between the field induced and field free radiation we have calculated

106
the high harmonic spectrum from the dipole acceleration a(t) such thatfHHG (ω; tf ) ∝

0.8
0.6
0.4
0.2
0.0

7

8
9
harmonic order (in 0.057 a.u.)

4 5 ...

0.6
0.4
0.2
7

8
9
harmonic order (in 0.057 a.u.)

3

1.0

(b)

0.8

0.0

(5.2)

−∞
3

1.0

(a)
intensity (arb. units)intensity (arb. units)

4 5 ...

0a(t0 )e−iωt dt0 .

intensity (arb. units)

3

1.0

Ztf

4 5 ...
(c)

0.8
0.6
0.4
0.2
0.0

7

8
9
harmonic order (in 0.057 a.u.)

Figure 5.6: Radiation spectra generated during the pulse (red dashed lines) and those including
line emissions after the pulse (black solid lines) are shown for peak intensities: (a) I0 = 3.4 × 1013
W/cm2 , (b) I0 = 6.0 × 1013 W/cm2 , and (c) I0 = 1.64 × 1014 W/cm2 . The vertical gray lines show
the field free energy differences between the np energy levels (up to 14p) and the 1s ground state.
(Figure taken from [237])
In Fig. 5.6 we present the radiation spectrum in the region of the 7th and 9th harmonics
produced by a 20 cycle, sin squared, 800 nm pulse at intensities (a) I0 = 3.4 × 1013 W/cm2 , (b)I0 = 6.0 × 1013 W/cm2 , and (c) I0 = 1.64 × 1014 W/cm2 . In each panel the red dashed line
shows the spectrum generated during the pulse (tf = τ ) while the black solid line includes freepropagation after the pulse has ended (tf = 2τ ). The difference between the two spectra exhibits
the radiation related to the transitions from the populated excited np states at the end of the pulse
to the ground state, which has previously been identified in both experiment [38] and macroscopic
simulations [32, 19].
Comparison of the spectra obtained at the two intensities clearly exhibits the expected difference in the features related to the parity effect in the population of the Rydberg states. At
an intensity of 3.4 × 1013 W/cm2 (Fig. 5.6a), the (shifted) Rydberg energy levels are resonantly
excited with a 10 photon process leading to population of the even angular momentum states.
Consequently, the spectrum is free of line emission from the almost unpopulated Rydberg p-states
while we find a strong line corresponding to the strongly populated 3p state, which is near resonantwith a 9 photon process. In contrast, by shifting the states by the energy of an additional photon,

107
at an intensity of 6.0 × 1013 W/cm2 (Fig. 5.6(b)) we observe a rich line emission spectrum justbelow the 9th harmonic line emissions from the np-states, that are resonantly populated by a 11
photon process. We note that the 6p and 7p states are the most populated np states and as a result
have the strongest emission lines. On the other hand the line corresponding to the 3p state, whichcannot be populated by absorption of 10 photons, is strongly suppressed.
As discussed in section 5.1.1.2, at the highest intensity I0 = 1.64×1014 W/cm2 , the population
distribution in the angular momentum well defined parity begins to blur. This leads to a relatively
even population in states with principle quantum number 4 to 14 seen in (Fig. 5.4(f)). As a result,
we see in Fig. 5.6(c) line emission from all populated np states with similar strengths for the states
resolved in our spectrum.

5.1.3

Summary

We have analyzed in this section the angular momentum distribution in the Rydberg statesof a hydrogen atom excited by a short strong laser pulse, based on numerical results of the timedependent Schrödinger equation, which we solved using a second order Crank-Nicolson scheme. At
low intensities the pattern in the population across the angular momentum states can be explained
via a parity effect due to the selection rules in multiphoton absorption along with the suppression ofpathways due to population trapping in low excited states, if the applied laser pulse is not too short.
At high intensities the parity effect gets lost even for long pulses, potentially since the Rydberg
states are in resonance with photon processes of different orders for a significant time during therising and trailing parts of the pulse. We predict that the parity effect can be observed via thestrengths of the line emissions from the np states emitted after the end of the pulse.

108

5.2

Bi-circular polarization2
The time-dependent Schrödinger equation for the interaction of a electron in the potential of

the hydrogen atom with a superposition of two circularly polarized intense laser pulses in dipole
approximation and velocity gauge is given in Eq. 3.33 . The wavefunction Ψ is expanded in sphericalharmonics up to lmax = 45 and mmax = 45. The radius is discretized using fourth order finite
difference method with a grid spacing of 0.1 a.u. and a maximum radius of 750 a.u. with exterior
complex scaling on the outer 38 a.u. of the grid. The wavefunction is propagated in time using the
Crank-Nicolson method with a time step of 0.05 a.u. The choice of gauge was based on the fasterconvergence of calculations in the velocity gauge for expansions of the wavefunction in sphericalharmonics [41, 88].
The interaction with the bicircular laser pulse is implemented via the total vector potentialas:A(t) = Aω (t) + A2ω (t)

(5.3)

where
2

AΩ (t) = A0,Ω sin



πt
τΩ



× [sin (Ωt) x̂ + Ω cos (Ωt)ŷ]

(5.4)

√

Ω
for Ω = ω and 2ω, respectively. A0,Ω = c ΩIΩ , τΩ = 2πNΩ , and c is the speed of light, where IΩ

is the peak intensity and NΩ denotes the number of cycles. Ω = ±1 denotes the helicity of thefundamental and 2nd harmonic pulse, respectively.
We have performed numerical calculations for the interaction of hydrogen atom with co- andcounter-rotating bicircular pulses operating at the central wavelengths of 800 nm and 400 nm.
Intensities of the two pulses were varied to study the distribution in the excited states as function
2

The content of this section has been also published in J. Venzke, Y. Gebre, et al., Phys. Rev. A 101, 053425(2020).
3
The corresponding numerical code has been developed by another graduate student (Y. Gebre). Results of thatcode have been tested well against results obtained using length gauge with the numerical program described in thisthesis.

109of all quantum numbers, principal, angular momentum and magnetic. For the results presented
below we have used pulses with the same pulse duration in time.

5.2.1

Excitation with co-rotating pulses

Figure 5.7: Excited state distribution as function of n (vertical axis) and ` (horizontal axis) for (a)m = −1, (b) m = −2, (c) m = −3 and (d) m = −4 at the end of 20 (at 800 nm) cycle pulses(40 cycle at 400 nm) with sin squared envelope and total peak intensity of 1 × 1014 W/cm2 forco-rotating laser pulses of equal intensity. (Figure taken from [234])

Selection rules for (single-)photon absorption from circularly polarized light are given by∆l = ±1 and ∆m = ±1, where the change in the magnetic quantum number is positive (negative) if
the helicity of the light is right- (left-)handed. Extending the concept to multiphoton absorption, thesimultaneous change in both quantum numbers puts distinct constraints on the parity and helicityof the accessible excited states in the atoms upon absorption of multiple photons. Specifically, it
is expected that states in which ` and m are either both even or both odd are being populated
during the interaction with the field. This selection holds for the interaction with a single circularly
polarized pulse as well as for the case of a superposition of two (or more) of such fields, independentof the relative helicity of the two pulses.
In Fig. 5.7 we show examples of the population in the excited states of hydrogen atom
as a function of n and ` for various m values at the end of the interaction with bichromatic corotating left-handed circularly polarized pulses. The results clearly confirm the expected population
distribution in states with either odd or even parity for a given value of m according to the selection
rules upon multiphoton absorption. The present results have been obtained for interaction withequal peak intensities I400 = I800 = 5 × 1013 W/cm2 .

110
Due to the correlation in changes of m and ` the observed pattern is independent of total
peak intensity, ratio of peak intensities and pulse duration, as long as the dipole approximation
holds. In our studies we have verified this up to intensities of 1 × 1014 W/cm2 . This is different
from the case of linear polarization [237], where selective population concerning the parity of the
populated excited states is observed for long pulses and low peak intensities only. In that casethe restriction to a given m-channel and a broad energy spectrum (for short pulse durations) or asignificant Stark shift of the excited states (at high peak intensities) leads to a mixing of population
over the states with odd and even orbital angular momentum quantum numbers.

Np (ω) : 4

4

Np (ω) : 3

3

2

3

2

Np (ω) : 2

2

Np (ω) : 1

1
1

Np (ω) : 0
m : −4

0
−3

−2

−1

0

1

2

3

4

Figure 5.8: Absorption pathways in co-rotating laser pulses at frequencies ω and 2ω starting froma m = 0-state. Without lack of generalization it is assumed that both pulses have left-handed
helicity. Absorption of a photon at frequency ω and at frequency 2ω is represented by a red andblue arrow, respectively. The numbers in the boxes denote the minimum number of photons toreach a certain level. (Figure taken from [234])

In co-rotating bicircular laser pulses all the photons have the same spin (either +1 or −1),
consequently the magnetic quantum number always changes either by ∆m = +1 or by ∆m = −1
upon absorption of each photon. For our studies we have chosen left-handed helicity for both pulses
and, hence, only excited states with negative m can be populated upon absorption of photons fromthe ground state with m = 0 (c.f., Fig. 5.8). Therefore, as already mentioned in Ref. [156], only
Rydberg states with high orbital angular quantum number ` are accessible. For example, for
excitation of Rydberg states (with n ≥ 4) in the hydrogen atom, the absorption of at least 4

111
photons in laser field at 400 nm or at least 8 photons at 800 nm is required. Thus, Rydberg stateswith ` < 4 (and m > −4) cannot be populated just via photon absorption alone.

Figure 5.9: Excited state distribution as function of orbital angular quantum number ` summed
over n ≥ 4 and m at (a) I400 = 5 × 1013 W/cm2 , I800 = 5 × 1012 W/cm2 , (b) I400 = 5 × 1013
W/cm2 , I800 = 1 × 1013 W/cm2 , (c) I400 = 5 × 1013 W/cm2 , I800 = 5 × 1013 W/cm2 , and and (d)I400 = 1 × 1013 W/cm2 , I800 = 5 × 1013 W/cm2 . Pulse durations: 20 cycles at 400 nm, 10 cycles at
800 nm. (Figure taken from [234])

Accordingly, the angular momentum distribution in the Rydberg states is controlled via therelative intensity of the two fields at the fundamental and second harmonic frequency. This isdemonstrated in Fig. 5.9, where the excited state distribution as a function of `, summed for n ≥ 4
and all m, is shown. For large ratio of I400 /I800 = 10 (panel (a)) the distribution is centered, as
expected, about ` = 4. As the intensity ratio decreases, high orbital angular momentum states get
increasingly populated due to the impact of the laser pulse at 800 nm.
Another interesting feature in Fig. 5.9 is that the population in angular momentum stateswith ` < 4 increases significantly when the intensities of the two pulses are similar. Further
insight can be gained by the distribution over the magnetic quantum number, which is displayedin Fig. 5.10(c) for the case of equal intensities. It is clearly seen that Rydberg states with magnetic
quantum numbers between m = 0 and m = −3 are populated. In view of the number of photons
needed to reach the Rydberg levels, the population in these states cannot be explained by absorptionof photons only.
Instead, we propose the following mechanism: Initially, Rydberg states with ` ≥ 4 are populated via the absorption pathways shown in Fig. 5.8. Then a redistribution of population occurs
via Raman-type Λ-transitions (c.f., [81, 60]). In the present bichromatic laser field the Λ-process

112

Figure 5.10: Excited state distribution as function of n (vertical axis) and m (horizontal axis)
summed over `. Laser parameters: 20 (800 nm) cycle pulses with sin squared envelope and total
peak intensity of 1 × 1014 W/cm2 for co-rotating laser pulses of equal intensity. (Figure taken from[234])

leads to a change in the magnetic quantum number, if photons from both fields are involved. For
the absorption of one 400 nm photon and emission of two 800 nm photons the magnetic quantum
number between initial and final state changes by ∆m = +1.
The order of absorption and emission may vary, i.e., the redistribution process can either
proceed via the continuum (absorption first, Fig. 5.10(a)) or via a lower excited state (emission
first, Fig. 5.10(b)). A larger change in m is achieved either via a sequence of these Λ-processes orby higher order processes (e.g., absorption of two 400 nm photons followed by emission of four 800
nm photons leading to ∆m = +2). We note that similarly the absorption of two photons at 800
nm and the emission of a 400 nm photon will lead to a change of ∆m = −1 in the present set-up
and, hence, contribute to population of states with higher ` and m.
Our interpretation is further supported by the results in Fig. 5.11, which shows how the
population in states of certain quantum numbers for (a) ` = 2 and (b) ` = 3 changes as function of
the relative intensity of the two pulses. It is clearly seen that the population in these quantum states,
which are not accessible via direct absorption of photons from the ground state, increases as theintensity of the pulse at 800 nm increases. Thus, these results provide further indications that the113

Figure 5.11: Distribution in magnetic quantum states for (a) ` = 2 and (b) ` = 3 and different
peak intensities of the 800 nm pulse. I400 = 5 × 1013 W/cm2 and other parameters are as inFig. 5.10. (Figure taken from [234])

presence of the redistribution process depends on the impact of both pulses and its effectivenessincreases with increase of the total intensity, in agreement with our interpretation of a Λ-type
process. The redistribution of population in Rydberg states via higher-order Raman-like process
induced by co-rotating bi-circular laser fields has been further studied in a project led by anothergraduate student [77].

5.2.2

Excitation with counter-rotating pulses

As discussed in the previous subsection, the selection rules by which only states with ` andm either both even or both odd hold independent of the relative helicity of the two pulses. This

114

Figure 5.12: Same as Fig. 5.7 but for counterrotating laser pulses. (Figure taken from [234])

is confirmed by the results that we obtained for the interaction with two counterrotating pulses at
equal intensities and equal pulse duration presented in Fig. 5.12. Depending on whether m is evenor odd, the distribution over the orbital angular momentum shows population in states with evenor odd parity. As in the case of corotating pulses, the observed pattern is found independent of
total peak intensity, ratio of peak intensities and pulse duration.

Np (ω) : 4

4

Np (ω) : 3

3
3

Np (ω) : 2

2
2

Np (ω) : 1

3
1

1

Np (ω) : 0
m : −4

2

0
−3

−2

−1

0

1

2

3

4

Figure 5.13: Absorption pathways in counter-rotating laser pulses at frequencies ω and 2ω startingfrom a m = 0-state. Without lack of generalization it is assumed that the pulses at frequency ω
has left-handed helicity, while the second harmonic pulse has right-handed velocity. Other symbols
as in Fig. 5.8. (Figure taken from [234])

Since in counter-rotating bicircular laser pulses photons of the two fields have opposite spin,
starting from the ground state with m = 0, excited states with both positive and negative magnetic
quantum numbers can be populated. The absorption pathways for the set-up chosen in the present
studies, namely right-handed helicity for the 800 nm pulse and left-handed helicity for the second
harmonic, are shown in Fig. 5.13. As can be seen from the Figure, the magnetic quantum num-115
ber reflects the difference between the number of 400 nm photons and that at 800 nm absorbed.
Furthermore, it can be seen that for a given total photon energy absorbed states with magnetic
quantum numbers separated by ∆m = ±3 are populated.

Figure 5.14: Excited state distributions as a function of m, summed over n ≥ 4 and ` (top), and as
a function of n and m, summed over ` (bottom), for the interaction with a left-handed circularly
polarized laser pulse at 800 nm (20 cycles) and a right-handed circularly polarized laser pulse at
400 nm (40 cycles). Both pulses have the same peak intensity of 5 × 1013 W/cm2 . (Figure takenfrom [234])

These features are clearly present in the population distributions as function of m, summed
over n and ` (top), and of n and m, summed over ` (bottom), in Fig. 5.14, which show the results
for counter-rotating pulses of equal peak intensity. In the Rydberg manifold (n ≥ 4) the highest
populated m-states differ by ∆m = ±3, other states show some but lower population as the manifold
is AC-stark shifted during the interaction with the pulses. In view of the nonlinearity of multiphoton
processes, it is likely that the states showing the largest probability are being populated near the
peak of the pulses at which the highest total intensity is present. Overall, the strongest populationis seen for states with negative magnetic quantum numbers, leading to the conclusion that it most
likely that either five (for excited states with m = −5) or two (for excited states with m = −2)
more 800 nm photons with left-handed helicity than 400 nm photons with right-handed helicity arebeing absorbed.
The distributions in Fig. 5.14 do not extend much beyond |m| = 5, which is consistent with

116

Figure 5.15: Orbital angular momentum distributions in excited states induced by counter-rotating
laser pulses at 400 nm (20 cycles) and 800 nm (10 cycles) at peak intensities of (a) I400 = 5 × 1013
W/cm2 , I800 = 5 × 1012 W/cm2 , (b) I400 = 5 × 1012 W/cm2 , I800 = 5 × 1013 W/cm2 , and (c)I400 = 5 × 1013 W/cm2 , I800 = 5 × 1013 W/cm2 . (Figure taken from [234])

the results shown in Fig. 5.15 showing that there appears to be a highest orbital angular momentum
number `max beyond which the population in the states drops off quickly. This is in agreementwith previous studies for Rydberg state excitation [194, 237] and low energy angular momentum
distributions [6]. In Ref. [6] a random walk analysis of the absorption pathways between theaccessible quantum states is used to obtain the classical orbital angular momentum for a electron√with zero energy in a laser field has been derived as L = 2Zα0 where Z is the charge of theresidual ion and α0 is the quiver radius. Relating classical orbital angular momentum and theorbital angular momentum quantum number by ` ≈ L − 1/2 we have estimated the maximum` gained in the bicircular counter-rotating pulse. The estimates, shown by the solid red lines inFig. 5.15, are in good agreement with the cut-offs seen in the numerical results. We note that the
random walk analysis of Ref. [6] can be applied in the case of counter-rotating pulses, since in eachabsorption step ∆m = ±1 and hence, in general, ∆` = ±1 is possible. In contrast, for co-rotating
pulses the changes in magnetic and angular quantum number are determined in each absorption
step (∆m = −1, ∆` = +1) and hence a random walk analysis is not applicable and a cut-off cannot
be derived.

117
5.2.3

Summary

In this section we have studied the distributions over the orbital angular momentum (`)
and magnetic (m) quantum numbers in Rydberg states due to the interaction with bichromatic
circularly polarized laser pulses. Multiphoton selection rules lead to population of states in which` and m are either both even or both odd, independent of relative helicity, peak intensity and pulse
duration. In the case of co-rotating pulses the results show that the distribution over the magnetic
quantum number can be controlled via the intensities of the two pulses. Furthermore, we proposethat the states are populated via direct absorption from the ground state and via Λ-type transitions
between Rydberg states of different ` and m, involving two photons at the fundamental wavelengthand one photon at the second harmonic. For bicircular laser pulses with opposite helicities Rydberg
states with magnetic quantum numbers that differ by ∆m = ±3 are predominantly populated. The
pattern allows to gain insights into the relative number of photons absorbed from the two fields. The
distribution is however restricted by the maximum orbital angular momentum quantum numberthat can be estimated by classical considerations.

Chapter 6
Towards studies of correlation effects in attosecond electron dynamics

Solving the TDSE for two active electrons is a computationally difficult task. A naive approach would be to discretize the six spatial dimensions using finite difference. This approach
would provide a straightforward solution and runs well on supercomputing systems, however, thesize of the wavefunction scales like N 6 where N is the number of grid points in each dimension.
Therefore, if one were to use a typical grid of 1000 points in each dimension, the wavefunction
would be 1018 points in size. Merely writing down the wavefunction in double precision floating
point numbers would require 16 Exabytes of RAM, approximately 100 times more RAM than exists
on the largest supercomputer in the world (Summit, Oak Ridge Leadership Computing Facility).
Instead, it is more convenient to choose a coordinate system and basis that greatly reduces thesize of the problem. One common approach is to use bi-spherical harmonics (e.g. [231]). In thiscase, the position of each electron is expanded in 3D spherical coordinates. The radius is discretizedwith finite difference, B-splines, finite-element discrete variable representation (FEDVR), or similar
method, and the two angular coordinates are expanded in the standard 3D spherical harmonics inthe same way a single-active electron code would do. When taking the tensor product of the twoelectrons, the electron-electron repulsion term is then expanded in spherical harmonics such that
∞

m=`

`
X X
r<
1
1
Y ∗ (θ , φ )Y (θ , φ ).
= 4π`+1 `m 1 1 `m 2 2|r1 − r2 |2` + 1 r>(6.1)`=0 m=−`

This results in a representation of the 6D space that can be used to solve the problem of twoelectrons in a molecular or atomic potential interacting with a laser.

119
In the bi-spherical codes, one must discretize two radii. As the grid increases in size, the code
still scales like N 2 . To circumvent this problem, we have implemented a code that contains a single
hyper-radius (R) and five angles that are expanded in 6D hyperspherical harmonics. Therefore, as
the grid size is increased, the code scales like N and the number of spherical harmonics is controlled
by the laser parameters. In the following section, we will describe the details to utilize two different“coordinate systems” that solve the two active electron problem. The first is based on the useof Jacobi coordinates, which allows for the finite mass of the nucleus to be accounted for. The
second is applied when using the infinite mass approximation for the nucleus, which allows that the
distance between the two electrons to be accessed more directly. The two problems can be solved
with similar sized wavefunctions, however, the time to calculate matrix elements is greatly reducedin the second method.
This chapter covers the background for the TDSE program developed in hyperspherical coordinates in Sec. 6.1 including the definition of hyperspherical harmonics in Sec. 6.1.1, the applicationof Jacobi coordinates in Sec. 6.1.2, and the application of electron coordinates in Sec. 6.1.2. A discussion of the implementation of the code and first tests will be presented in Sec. 6.2 while Sec. 6.3
provides a brief overview of the challenges that remain and useful next steps which could make itpossible to run fully correlated two-electron calculations on desktop machines.

6.1

TDSE in hyperspherical coordinates
The TDSEi∂Ψ = ĤΨ∂twith Hamiltonian written in hyperspherical coordinates is given by
"
#


1 1 ∂
K̂ 2 (Ω5 )
W (Ω5 )
5 ∂Ĥ = −
R−+,
5
2
2 R ∂R∂R
R
R
where R =

(6.2)

(6.3)

qP

2
i xi is the hyperradius with xi being a Cartesian coordinate, K̂ is the angularmomentum operator, Ω5 is the solid angle in 6D (5 angles), and V (R) = W (Ω5 )/R being thepotential. We will use the hyperspherical harmonics which contain two-3D sub spaces as described120
in Sec. 6.1.1. The resulting volume elements aredτ = R5 drdΩ5(6.4)

dΩ5 = cos2 α sin2 αdαdω(6.5)

dω = sin θr1 dθr1 dφr1 sin θr2 dθr2 dφr2

(6.6)

where θi and φi denote the angles in the i = r1 and i = r2 sub coordinate systems and α is the angle
produced by the right triangle with sides R, r1 , and r2 as shown in Fig. 6.1. r1 and r2 will denote
the Jacobi Coordinates, if a finite mass nucleus is considered, or the single electron coordinates
when the infinite mass approximation is made.

R
r2
α
r1
Figure 6.1: Relation of the hyperradius (R) to the two single-electron sub coordinate systems (r1 ,r2 ).

As for the 3D TDSE, it is beneficial to remove the first derivative in the radial equation.
This is accomplished by substituting Ψ = ψ/R5/2 to get as Hamiltonian"#
1 ∂2
K̂ 2 (Ω5 ) + 15/4
W (Ω5 )
Ĥ = −
−
+
.
2
2
2 ∂R
R
R

(6.7)

The wavefunction can now be written as
ψ=X

UK (R)YK (Ω5 )

(6.8)

K

with UK (R) being the hyperradial part of the wavefunction and YK (Ω5 ) being the 6D hypersphericalharmonics described in Sec. 6.1.1. Since hyperspherical harmonics are eigenfunctions of K̂ 2 witheigenvalue K(K + 4), the Hamiltonian becomes
1 ∂2
K(K + 4) + 15/4
W (Ω5 )
ĤK = −
−
+
.
2
2
2 ∂R
R
R

(6.9)

121
We solve this equation using finite difference to discretize the hyperradial function UK (R). For
W (Ω5 ) = 1 the equation is the 6D hydrogen atom (i.e. an electron interacting with a Coulomb
potential in 6 spacial dimensions, rather than the typical 3 spacial dimension) with an analytic
solution that can be used to test the hyperadial portion of the code (see [222] for energy levels).
The angular portion of the equations is then expanded in hyperspherical harmonics. The details of
the expansion are the focus of the remainder of this section.

6.1.1

Hyperspherical harmonic definition

There are multiple ways to define hyperspherical harmonics in 6D. Since they are equivalent
up to a unitary rotation, it is best to choose those that match the symmetry of the problem. In ourcase, we choose spherical harmonics that contain a hyperangle α that connects two-3D spaces asshown in Fig. 6.1. The 3D spaces belong to coordinates r1 and r2 with radii r1 and r2 respectively.
These spaces will either represent a Jacobi coordinate (Sec. 6.1.2) or the space of each electron(Sec. 6.1.3). The resulting 6D spherical harmonics are given by`r ,`r2`r ,`r12(Ω5 ) =NK 1
YK,L,M×(cos(2α)) cos`r1 (α) sin`r2 (α)
hiXhL, M |`r1 , mr1 , `r2 , mr2 i Y`r1 ,mr1 (r̂1 )Y`r2 ,mr2 (r̂2 )`r ,`rNK 1 2 =(α,β)(6.10)(6.11)

mr1 ,mr2

withwhere Pn(`r +1/2,`r2 +1/2)

Pn 1

s

2(K + 2)(n!)Γ(n + `r1 + `r2 + 2).
Γ(n + `r1 + 3/2)Γ(n + `r2 + +3/2)(6.12)

is the Jacobi polynomial, hL, M |`r1 , mx , `y1 , my i is a Clebsch–Gordan coefficient(CGC), Γ is the gamma function, Y`r ,mr (r̂) is the standard 3D spherical harmonics, Ω5 is the
6D solid angle containing five angular dimensions, and n = (K − `x1 − `y1 )/2. K, L, M , `r1
and `r2 are the quantum numbers with K = 2n + `x1 + `y1 being the grand angular momentum,`r1 ≥ 0 and `r2 ≥ 0 being the angular momentum for the r1 and r2 coordinates, respectively,|`r1 − `r2 | ≤ L ≤ `r1 + `r2 and −L ≤ M ≤ L. We define K = {K, L, M, `r1 , `r2 } allowing for thesimplified notation YK (Ω5 ).

122
The hyperspherical harmonics contain three parts, each depending on one free parameter α,
r1 , and r2 . By defining`r ,`r2P̃n 1`r ,`r2(α) = NK 1(`r +1/2,`r2 +1/2)

Pn 1(cos(2α)) cos`r1 (α) sin`r2 (α)(6.13)

hyperspherical harmonics become`r ,`r`r ,`r2

1
2
YK,L,M(Ω5 ) = P̃n 1(α)

X

mr1 ,mr2hihL, M |`r1 , mr1 , `r2 , mr2 i Y`r1 ,mr1 (r̂1 )Y`r2 ,mr2 (r̂2 ) ,(6.14)

which highlights the three main components including the dependence of the angles in the spaces
r1 and r2 as well as the dependence on the hyperangle α that couples the two spaces.

6.1.2

Jacobi coordinates

Jacobi coordinates are often used for studying many body interactions. They are definedby choosing a particle and defining a relative coordinate between the chosen particle and a second
particle. The next particle is defined by a coordinate connecting the center of mass of all theprevious particles to the new particle. The pattern continues until all particles have been included.
The result is a set of coordinates that has dimension 3(N − 1) assuming the exact location of thecenter of mass Rcm in space is irrelevant. For a three particle system, the center of mass is given
by
Rcm =

m1 r1 + m2 r2 + m3 r3Mwith mi being the mass of the ith particle, ri being its spatial coordinate, and M =three convenient sets of Jacobi coordinates are1/4
m2 m3 Mx1 =(r3 − r2 );m1 (m2 + m3 )2

1/4
m3 m1 Mx2 =(r1 − r3 );m2 (m3 + m1 )2

1/4
m1 m2 Mx3 =(r2 − r1 );m3 (m1 + m2 )2

(6.15)
P3

i=1 mi . The

1/4 

m1 (m2 + m3 )2
m2 r2 + m3 r3
y1 =
r1 −
m2 m3 M
m2 + m3
1/4
m2 (m3 + m1 )2
m3 r3 + m1 r1
y2 =
r2 −
m3 m1 M
m3 + m1
1/4
m3 (m1 + m2 )2
m1 r1 + m2 r2
y3 =
r3 −
.
m1 m2 M
m1 + m2(6.16)(6.17)(6.18)The three coordinate systems are depicted in Fig. 6.2. For the Helium atom we choose the nucleus
as m1 and the two electrons as m2 and m3 . For the rest of the section, we set m1 → ∞ and123
m2 = m3 = m such that1r2 + r3
x1 = β1 (r3 − r2 ); y1 =
r1 −
β1
2


1
r3 + r1x2 = β2 (r1 − r3 ); y2 =
r2 −
β2
2


r1 + r21
r3 −,x3 = β3 (r2 − r1 ); y3 =β32

(6.19)(6.20)(6.21)√
where β1 = 1/ 2 and β2 = β3 = 1 within the infinite mass approximation. By applying a
Raynal-Revai coefficient (RRC) it is possible to transition between the three different coordinate
systems. The transformation conserves the angular momentum quantum numbers K, L, and M of
the hyperspherical harmonics.
Coordinate: 1
m1

Coordinate: 2m2

Coordinate: 3
m3

RRC

RRC

y1

y2

y3

PRRC = 1
m3

x1

m2

PRRC = 2
m1

x2

m3

m2

m1

x3

Figure 6.2: The Jacobi coordinates for three body interactions. Each coordinate can be obtained
from the previous by utilizing a Raynal-Revai coefficient.

The RRCs for rotating between coordinate system i and j can be written explicitly as
" 4
#
X
Y
(−1)ni +nj< `xj , `yj |`xi , `yi >K,L = q niλ2 −λ1 +`yi −`yj
(2λk + 1)
nijC`x `y C`x `y λ1 ,λ2 ,λ3 ,λ4
k=1
j

j

i

i

× λ1 0λ3 0 `xj 0 hλ2 0λ3 0|`xi 0i λ2 0λ4 0 `yj 0 hλ1 0λ4 0|`yi 0iλ3 λ1 `xj × sgn(a12 )λ1 sgn(a21 )λ2 sgn(a11 )λ3 sgn(a22 )λ4 λ2 λ4 `yj lxi lyi L

×

X
ν,µ

(−1)µ |a12 |2µ+λ1 +λ2 |a12 |2ν+λ3 +λ4 Cλµ1 λ2 Cλν3 λ4(6.22)

where sgn(aij ) is the sign of aij , between spherical harmonics with quantum numbers K = 2ni +`xi +`yi = 2nj +`xj +`yj , the sums go over all possible combinations of K = 2(µ+ν)+λ1 +λ2 +λ3 +λ4 with λ3 λ1(λi , µ, ν = 0, 1, 2, · · · ), λ1 0λ3 0 `xj 0 is a Clebsch Gordan coefficient, λ2 λ4lxi lxy

124` xj 

is a Wigner
`yj 


L

(2µ+α+β+1)!
µ9j symbol, Cαβ= (µ)!(µ+α+β+1)!(2(µ+α)+1)!!(2(µ+β)+1)!!
, a11 = cos(ξki ), a22 = cos(ξki ), a12 = sin(ξki ),pand a21 = − sin(ξki ) with ξki = arctan (−1)PRRC M mj /(mi mk ) where mi is the mass of the ith

particle, M is the total mass and P is the number of permutations of m1 → m2 → m3 → m1 asshown in Fig. 6.2. As the value of K increases, RRCs become extremely computationally expensive
limiting the usefulness of this approach.
Utilizing the Jacobi coordinates, denoted by ‘1’ in Fig. 6.2, the Helium potential becomes
V (R) =Z
Z
1−−
r23 r12 r13(6.23)

which can be rewritten as
√
√
1
Z 2
Z 2
W (Ω5 ) = RV (R) = √ −−.
x1 2 |ŷ1 + x̂1 | |ŷ1 − x̂1 |(6.24)

By using the hyperspherical harmonics defined in Eq. 6.14, with r1 = x1 and r2 = y1 , the first term
is easy to calculate through a 1D integral over α written explicitly below. However, the next two
terms would require a 5D integral which is expensive and numerically unstable if computed directly.
Instead, we will exchange particles in the coordinate system to allow the Jacobi coordinate xi tocorrespond to the distance between the two interacting particles. The result is a 1D integral over αthat is similar to the electron-electron repulsion term. To achieve this, we utilize RRC (Eq. 6.22)
which allow us to write`x ,`yii
YK,L,M=X`xj `yj`x ,`yjj< `xj , `yj |`xi , `yi >K,L YK,L,M(Ω5j ) ,(6.25)

where Ω5i and Ω5j are the solid angles in the Jacobi coordinates labeled by i, j. We note that K,
L and M are conserved in this rotation allowing us to remove any subscripts.
For practical reasons, we need to calculate the matrix elements hK0 | V (R = 1, Ω5i ) |Ki =hK0 | W (Ω5i ) |Ki as the term 1/R can be factored out of V (R, Ω5i ) removing its dependence on R.
We can then store the results of hK0 | W (Ω5i ) |Ki in a hash table to be looked up when building thematrix [126].

125
The electron-electron repulsion term of W (Ω5 ) becomesK 0 `0x1 `0y1 L0 M 0 √1
1|K`x1 `y1 LM i = √ δ`0x ,`x1 δ`0y ,`y1 δL0 ,L δM 0 ,M
(6.26)
1
2 cos α12 1Zπ/2 0 0`x ,`y`x ,`y×P̃n0 1 1 (α1 )P̃n 1 1 (α1 ) sin2 (α1 ) cos(α1 )dα1 ,0(6.27)
where the delta functions arise from the integral over dω.
The Coulomb potential for the electron with mass m3 in W (Ω5 ) becomesK 0 `0x1 `0y1 L0 M 0 − Z

XR|K`x1 `y1 LM i =< `x2 , `y2 |`0x1 , `0x1 >K 0 ,L0 < `x2 , `y2 |`x1 , `x1 >K,L
r13
`x2 ,`y2(6.28)
× K 0 `0x2 `0y2 L0 M 0−Z|K`x2 `y2 LM icos α2(6.29)

with PRRC = 1 andK 0 `0x2 `0y2 L0 M 0 √1
1(6.30)|K`x2 `y2 LM i = √ δ`0x ,`x2 δ`0y ,`y2 δL0 ,L δM 0 ,M
2
2 cos α22 2Zπ/2 0 0`x ,`y`x ,`y×P̃n0 2 2 (α2 )P̃n 2 2 (α2 ) sin2 (α2 ) cos(α2 )dα20(6.31)and the Coulomb potential for the other electron (with m2 ) of W (Ω5 ) becomesK 0 `0x1 `0y1 L0 M 0 − Z

XR|K`x1 `y1 LM i =< `x3 , `y3 |`0x1 , `0x1 >K 0 ,L0 < `x3 , `y3 |`x1 , `x1 >K,L
r12
`x3 ,`y3(6.32)
× K 0 `0x3 `0y3 L0 M 0−Z|K`x3 `y3 LM icos α3(6.33)

with PRRC = 2 andK 0 `0x3 `0y3 L0 M 0 √1
1|K`x3 `y3 LM i = √ δ`0x ,`x3 δ`0y ,`y3 δL0 ,L δM 0 ,M
(6.34)
3
2 cos α32 3Zπ/2 0 0`x ,`y`x ,`y×P̃n0 3 3 (α3 )P̃n 3 3 (α3 ) sin2 (α3 ) cos(α3 )dα3 .
0(6.35)

126
Next, the laser potential for a linearly polarized laser aligned along the z-axis needs to be
calculated. The potential term is given by
Vlas (R) = −Ez z2 − Ez z3 = −2Ez

z2 + z3,2(6.36)

where z2 and z3 are the z-components of the two electrons and Ez is the magnitude of the zcomponent of the electric field. Therefore, the laser couples to the z-component of the center ofmass of the two electrons namely
r2ecm =

r2 + r3.
2(6.37)

Noticing that y1 from Eq. 6.16 contains r2ecm , the laser term only appears in the ŷ1 sphericalharmonics. Next, we need to calculate the Vlas (R) matrix element as
K 0 `0x1 `0y1 L0 M 0 − 2Ez z2ecm |K`x1 `y1 LM i .

(6.38)In the infinite mass approximation, Rcm = y1 and therefore
z2ecm = β1 y1 · ẑcm

(6.39)

y1 cos(θy1 )√2rπ
y1 Y1,0 (ŷ1 )
=2
6
r
π
=2
Rsin(α1 )Y1,0 (ŷ1 )6=(6.40)(6.41)(6.42)

Using this in Eq. 6.38 we obtainK 0 `0x1 `0y1 L0 M 0 − 2Ez z2ecm |K`x1 `y1 LM i
= −2Ez K 0 `0x1 `0y1 L0 M 0 2(6.43)rπ
Rsin(α1 )Y1,0 (ŷ1 ) |K`x1 `y1 LM i6

127
which becomesK 0 `0x1 `0y1 L0 M 0 − 2Ez z2ecm |K`x1 `y1 LM is(2`0y1 + 1)
= −2Ez δ`0x `x1 δM 0 M R
12(2`y1 + 1)
Zπ/2 0 0`x ,`y`x ,`y×P̃n0 1 1 (α1 )P̃n 1 1 (α1 ) sin3 (α1 ) cos2 (α1 )dα1 
0

×

×

X

m0x ,m0y ,mx ,my`0y1 , 0, 1, 0 `y1 , 0`0x1 , m0x , `0y1 , m0y L0 , M 0`0y1 , m0y , 1, 0 `y1 , myh`x1 , mx , `y1 , my |L, M i(6.44)

Since the M quantum number does not change and M = 0 for the ground state of helium atom,
the sum over m0x , m0y , mx , my simplifies to a single sum over − min(L0 , L) ≤ m ≤ min(L0 , L) with
m0x = mx = −m and m0y = my = m as we get δm0y ,my from the second CGC, δm0x ,−m0y (whenM 0 = 0) from the third CGC, and δmx ,−my (when M = 0) from the forth CGC. If M 6= 0 is needed,a different simplification is required.

6.1.3

Electronic coordinates

To remove the need for RRCs due to their numerical complexity for high K values, we will
use the nucleus as the origin and label the coordinates from the proton to the i-th electron as ri
where i = 1, 2. Therefore, we have
r1 =R cos(α)(6.45)

r2 =R sin(α)(6.46)dΩ5 = cos2 (α) sin2 (α)dαdωr1 dωr2 .

(6.47)

and a solid angle`r ,`r1
2
We will use spherical harmonics with labels YK,L,M(Ω5 ) where `ri is the spherical harmonic for the

3D space coordinate ri . The electron-nucleus terms take the form −Z/ri and the matrix elements

128
becomeZZ− Ψ|Ψi = − δL0 ,L δM 0 ,M δ`0r ,`r1 δ`0r ,`r2
1
2
r1
R
0

Zπ/2

P̃n0 1Zπ/2

P̃n0 1`0r ,`0r2`r ,`r2(α)P̃n 1(α) sin2 (α) cos(α)dα(6.48)(α) sin(α) cos2 (α)dα.

(6.49)

0

and
Z
Z
− Ψ
|Ψi = − δL0 ,L δM 0 ,M δ`0r ,`r1 δ`0r ,`r2
1
2
r2
R
0

`0r ,`0r2`r ,`r2(α)P̃n 10Note that the term 1/R can be factored out allowing the integral over α to be performed once andstored in a lookup table to improve computational performance.
The electron-electron term is less straight forward. We start by using∞

X X̀ 4π r`
1<=Y ∗ (r̂ )Y (r̂ )`+1 `,m 2 `,m 1|r1 − r2 |2` + 1 r>(6.50)`=0 m=−`and then after some extensive algebraic work we obtains00(2`0r1 + 1)(2`0r2 + 1)
δδ1
L ,L M ,MΨ0|Ψi =|r1 − r2 |R
(2`r1 + 1)(2`r2 + 1)

(6.51)

min(|`r1 +`r2 |,|`0r1 +`0r2 |)×X`0r1 , 0, `, 0 `r1 , 0`0r2 , 0, `, 0 `r2 , 0(6.52)`=max(|`r1 −`r2 |,|`0r1 −`0r2 |)




×

Zπ/4

P̃

`0r1 ,`0r2
n0`r ,`r2(α)P̃n 1`+2(α)

sin (α)dα +cos`−1 (α)

0

Zπ/2

P̃

`0r1 ,`0r2
n0`r ,`r2(α)P̃n 1π/4(α)

cos`+2 (α)
sin`−1 (α)




dα

(6.53)
`0r1

×
×

X

L0 , M 0 `0r1 , m0r1 , `0r2 , m0r2

(6.54)

m0r1 =−`0r1 ,m0r2 =M 0 −m0r1`r1X

mr1 =−`r1 ,mr2 =M −mr1 ,m=mr1 −m0r1

× `0r1 , m0r1 , `, m `r1 , mr1(−1)m hL, M |`r1 , mr1 , `r2 , mr2 i`0r2 , m0r2 , `, m `r2 , mr2 .

(6.55)(6.56)

Note the requirement for the last row gives m = mr1 − m0r1 and m = m0r2 − mr2 which gives thedelta function δM 0 ,M as well.
Finally, we need the laser operator
Vlas (R) = −Ez zr1 − Ez zr2 .

(6.57)

129
Since each electron has its own 3D space, it is easier to calculate the laser operator for each electron
independently. It is then convenient to definerπ
R sin(α1 )Y1,0 (r̂1 ),3rπVlas (r2 ) = −Ez zr2 = −2Ez zr2
R sin(α1 )Y1,0 (r̂2 ),3

Vlas (r1 ) = −Ez zr1 = −2Ez zr2

Vlas (R) = Vlas (r1 ) + Vlas (r2 ).
The resulting matrix elements ares(2`0r1 + 1)
δL0 ,L±1 δM 0 ,M δ`0r ,`r1 ±1 δ`0r ,`r2 `0r1 , 0, 1, 0 `r1 , 0
Ψ0 Vlas (r1 ) |Ψi = −Ez R
1
2(2`r1 + 1)×Zπ/2

`0r ,`0r2

P̃n0 1`r ,`r2(α)P̃n 1(6.58)(6.59)(6.60)(6.61)(α) sin3 (α) cos2 (α)dα(6.62)`0r1 , mr , 1, 0 `r1 , mr

(6.63)

0min[lr0 1 ,lr1 ,lr0 2 ,lr2 ]×X

mr =− min[lr0 1 ,lr1 ,lr0 2 ,lr2 ]× L0 , M `0r1 , mr , `r2 , −mr hL, M |`r1 , mr , `r2 , −mr i(6.64)

andΨ0 Vlas (r2 ) |Ψi = −Ez R
×

Zπ/2

s

(2`0r2 + 1)
δL0 ,L±1 δM 0 ,M δ`0r ,`r1 δ`0r ,`r2 ±1 `0r2 , 0, 1, 0 `r2 , 0
1
2(2`r2 + 1)

`0r ,`0r2

P̃n0 1`r ,`r2(α)P̃n 1(6.65)(α) sin3 (α) cos2 (α)dα(6.66)`0r2 , mr , 1, 0 `r2 , mr

(6.67)

0min[lr0 1 ,lr1 ,lr0 2 ,lr2 ]×X[

mr =− min lr0 1 ,lr1 ,lr0 2 ,lr2]

× L0 , M `r1 , mr , `0r2 , −mr hL, M |`r1 , mr , `r2 , −mr i .

(6.68)In this derivation, we used M = 0 therefore mr1 = −mr2 . For M 6= 0 adjustments to the sum areneeded.

130

6.2

Implementation and first tests
The RRC and non-RRC methods were both implemented in the same code, based on the

SAE-TDSE method. This allows for two electron calculations to be performed by simply changing
a few parameters in an input file.

Figure 6.3: Convergence of He atom first ionization potential (blue solid line) and excited states(blue dashed line) with RRC (left) and non-RRC (right) methods. Reference lines for He-SAE(black), NIST energies (red), and literature convergence (green, [126]) are shown.

To ensure the code is working properly, we have ran test on the convergences of the ionization
potential and low lying bound states shown in Fig. 6.3. Panel (a) shows the convergence of the RRC
method in blue as a function of the grand angular momentum using second order finite differencewith a grid spacing of 0.05 au. The convergence is in good agreement with the green lines from asimilar code [126]. Additionally, the NIST (red) and SAE potential (black) energy levels are shown.
Panel (b) shows the same data for the non-RRC method shown in blue. The non-RRC method has
more low lying bound states that are due to not separating out the eigenstates with two spin up
electrons. Taking account of it will reduce the runtime of the non-RRC method. As one can see,both methods produce highly accurate bound states quite quickly. The excited states, however, areslow to converge only outperforming the DFT based SAE potential energy levels with Kmax > 48.
As Kmax increases the matrix elements become increasingly difficult to calculate and the size of
the wavefunction grows rapidly. This greatly limits the usefulness of this method for two electron

131
calculations without the use of supercomputing resources.

Figure 6.4: Rabi flopping between the 1s and 2p states in helium atom. The populations in the1s (blue), 2p (orange), and all other states (green) for the SAE code are plotted. The black dotsshow the two electrons simulations using the RRC method with Kmax = 48 agree well with the
SAE results.

To test the stability and accuracy of the time propagation linearly polarized laser field, theRRC code was used to preform a 1s − 2p Rabi flopping with Kmax = 48 to closely match the 1s − 2p
energy gap to the SAE potential. The laser pulse had a sine squared envelope, peak intensity of 1014
W/cm2 , resonant central frequency of 0.812 au (0.822 au) for the SAE (RRC) code with varying
pulse lengths ranging from 1 to 34 cycles. Fig. 6.4 shows the SAE populations in the 1s (blue), 2p(orange), and all other bound/continuum states (green) for reference. The two electron populations
are shown in black. The agreement is quite good, and the small discrepancy are due to slightly
different energy gaps.

6.3

Outlook
The present methods still require improvement in order to perform converged two active

electron simulations on desktop systems. Switching from the RRC to the non-RRC method lead132
to orders of magnitude reduction in runtime for matrix generation. However, the need to perform
numerical integrals of Jacobi polynomials largely slows down the computation. Additionally, theslow convergence of excited states with respect to the grand angular momentum leads to large
wavefuctions. If the hyperangle α is discretized using finite difference, the integrals become integrals
over delta functions and the slow convergence with respect to the grand angular momentum (K)may be avoided.

Chapter 7
SummaryThis thesis covers theoretical approaches to understand ultrafast processes in photoionization, Rydberg state excitation, and electron correlation. In the first two chapters, an introduction
to ultrafast and strong-field physics was given with a review of relevant experimental and theoretical studies. The numerical and analytical methods that are used in this thesis were covered in
Chap. 3, including original studies using RBFs to solve the time independent Schrödinger equation(Sec. 3.3.3) and a frequency correction to ultrashort laser pulses (Sec. 3.6).
The focus was then turned to developing a method of imaging ultrafast electron wavepacket
dynamics using few photon ionization in Chap. 4. The extreme bandwidth of ultrafast pulses allows
for processes involving different numbers of photons from the same pulse to interfere. The impacts
on ionization of a helium atom were studied in Sec. 4.1 including the effects of pulse variations
present in experiment. Sec. 4.2 extends the typical forward backward ionization asymmetry fromground states to the ionization of wavepackets in superposition with different m quantum numbers.
To this end, we presented a novel set of generalized asymmetry parameters (GAP) to study theinterference of the ground and excited state ionization signals. Finally, a method for reconstructing
wavepackets undergoing attosecond electron motion using perturbation theory was presented inSec. 4.3. The impact of shot to shot variations and the laser parameter regime required for aquality reconstruction have been analyzed.
Next, Chap. 5 covers the excitation of Rydberg states by intense IR laser pulses. Linear
polarization results, in Sec. 5.1, show that the ` quantum number of the populated states has

134
the same parity as the sum of the number of photons involved in the process and the groundstates ` quantum number. The FID signals present in the HHG spectrum allows for experimental
verification of this effect. Sec. 5.2 extended this study to bi-circular driving laser pulses, including ananalysis of the pathways to available resonances and the occurence of high order Raman transitions
which have been further studied in a project led by another graduate student [77].
Finally, we turned to the development and first tests of a fully correlated two electron code in
Chap. 6. We used hyperspherical harmonics to reduce the number of radii that must be discretizedfrom two to one. The code produces ground and excited state energies consistent with experimental
results and has produced a Rabi flop between the 1s and the 2p state in He that matches our SAE
codes. Future development may allow for fully correlated two electron calculations to be performed
efficiently on desktop computers opening the door for more in depth studies of time dependent
multi-electron effects in ultrafast physics.

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