Analytical Modeling and Numerical Simulations of Time
Delays in Attosecond Streaking of One- and Two-Photon
Ionization
by
Cory Scott Goldsmith
B.S., University of Georgia, 2012

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 Chemistry and Biochemistry2018This thesis entitled:
Analytical Modeling and Numerical Simulations of Time Delays in Attosecond Streaking of Oneand Two-Photon Ionization
written by Cory Scott Goldsmith
has been approved for the Department of Chemistry and BiochemistryProf. Andreas Becker

Prof. Agnieszka Jaroń-Becker

Date

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

Goldsmith, Cory Scott (Ph.D., Chemical Physics)
Analytical Modeling and Numerical Simulations of Time Delays in Attosecond Streaking of Oneand Two-Photon Ionization
Thesis directed by Prof. Andreas Becker

The generation of attosecond-duration (1 as = 10−18 s) coherent light through the processof high-order harmonic generation has opened the perspective for probing fundamental processes,
such as photoionization, on the natural timescale of electron dynamics in matter. One probing
technique is the attosecond streaking method, in which the momentum of the photoelectron ismeasured as a function of the time delay between the ionizing, attosecond extreme ultraviolet(XUV) pulse, and a weak, femtosecond near-infrared (NIR) pulse which streaks the momentum of
the photoelectron, known as a streaking trace. The observed trace contains time information about
the photoionization process in the form of a time offset to the vector potential of the streaking field,
known as the streaking time delay. Theoretical simulations show that for one-photon ionization
this time delay is accumulated by the photoelectron in the continuum when propagating away fromthe parent ion, whereas for resonant two-photon ionization there exists an additional absorption
delay which depends on the properties of the XUV pulse. In this thesis, we use both analyticaltechniques and numerical simulations to study the contributions of the total time delay observedin streaking, and further explore applications of the streaking time delay to gain insights into theelectron dynamics.
We first derive an analytical formula for the streaking time delay in one-photon ionization.
The predictions based on the model formula, which can be performed within seconds of computation time, are in good agreement with those of computationally extensive numerical simulations.
We demonstrate that the analytical formula not only allows deeper insight into the nature of thetime delay, but also offers the opportunity to effectively analyze other theoretical interpretations
and potential effects, such as the effect of a chirp in the ionizing attosecond pulse on the time

iv
delay measurement. We then apply time-dependent perturbation theory to derive an analyticalformula for the absorption delay in resonant two-photon ionization. We use the analytical formula
to demonstrate how the absorption delay can be controlled further by the attosecond pulse duration and central frequency in case of an isolated resonance. Furthermore, we show how multiple
resonances within the bandwidth of the ionizing pulses as well as the streaking field influence theabsorption delay in model systems as well as simple atoms and molecules. We conclude by exploring the option to apply isolated elliptically polarized attosecond pulses to obtain sub-attosecond
temporal information via the observation of photoelectron angular distributions as a function of
the ellipticity of the pulse.

Dedication

To my sister Rachel. We miss you dearlyAcknowledgements

There are too many people to acknowledge, but I’ll do my best in making sure not to leaveout too many. First and foremost I would like to thank my advisors Andreas and Agnieszka for
helping me achieve my dream of becoming a PhD. I certainly could not have done this without their
help. When I strayed too far, they were always there to guide me back to the right path. I also
thank them for always supporting my efforts, even when I became distracted by other happenings
in my life. I would also like to thank my committee members John Bohn, Carl Lineberger, Margaret
Murnane, and David Jonas for intriguing conversations during my tenure at CU Boulder.
As was alluded to in my Dedication, my sister Rachel passed away in March 2015, during thework leading to this thesis. This was without a doubt, and probably will be forever, the hardest
thing I have ever endured in my life. I would be lying if I said that it didn’t cross my mind, several
times, to quit graduate school during this time. However, the support of Andreas and Agnieszka,
my fellow group members, and my parents allowed me to continue pursuing my doctorate. I pushed
on, knowing that Rachel would want me to finish, and here I am. On that note, I would like tothank my parents for being the strongest people I know, and for giving me all of the opportunities
in life that allowed me to make it to this point. They are the people who are most responsible forthis work. For without them, I would be lost. I would also like to thank my grandmothers Therese
Goldsmith and Rita Schoenberg, my Aunts Jean Goldsmith and Leah Stitzer, my Uncles Richard
Youtz and Frederic Goldsmith, and my cousins Yaakov and Chaim Stitzer, David, Deborah, and
Liz Ross, as well as Shifra Stitzer and Andy Berman.
I would also like to express my deepest gratitude towards my fellow group members, past andvii
present: Jing Su, Hongcheng Ni, Andrew Spott, Michelle Miller, Yuqing Xia, Carlos HernándezGarcı́a, Dan Weflen, Brynn Reiff, Joel Venzke, Tennesse Joyce, Andres Mora, and Zetong Xue. I
would not have been able to make the jump to physics from chemistry if it were not for the help of
these talented physicists. Many of them also helped me to become a programmer, as I had almost
no programming experience coming into the group. Most importantly, they were always willing toassist when I needed to discuss anything, whether academic or personal. I truly consider all of you
my friends, and appreciate having you in my life.
Working in JILA was an extraordinary experience, and I would like to thank the staff and myfellow JILAns for providing such a welcoming environment. I also would like to thank Professors
Michael Duncan and Gary Douberly at the University of Georgia for mentoring me during myundergraduate education, and leading me on a path that resulted in my being here. Furthermore,
I was fortunate enough to do an internship for Ableton AG in Berlin during the summer of myfourth year. This was one of the best summers of my life, and I would like to thank Neil Burdock
and Ableton for the life-changing experiences I encountered there.
The friends I have made in Boulder and Denver after moving from Atlanta were extremely
helpful during the last six years. There are countless times I was encouraged by close friends
to continue when I felt like giving up, or made me happy when I was extremely depressed and/or
exhausted. Being involved in the Denver music scene and Denver Trance Family also greatly helpedwith this. At that, I really should thank music itself. If I weren’t composing music and playing
shows during my PhD, I surely would’ve lost it.
I end by listing the names of friends that were most instrumental in motivating me andhelping me to keep my eyes on the prize whilst completing this thesis, and for just making mylife better in general (hopefully not forgetting any): Phoebe Tengdin, Alex Hopkins, Vince andKara Casini, Devon Whitt, Zak Bissani, Kelsey Higgins, Timothy Watts, Patrick Farace, Joey
Hreha, Evan Cerwonka, Andy Clark, Jordan Brown, Blas Lopez, Jay and Lyndzey Miller, Capers
Hammond, Magnus and Spenser Fain, Chris Davis, Seth Hensel, Daniel Loyd, Niels Wright, Hayes
Capers, Matthew McCloskey, John Cameron, Zaak Kerstetter, Brittany Nelson, Tom Osmond,viii
Maxi Bethge, Duncan Newell, Luigi Palagano, Evan Dackerman, Kimberly Hall, Stephen Kirkwood,
Claire Hart, Steven Galloni, Stuart Mclellan, Craig Hughes, Rick Welsch, CJ Robinson, Ryan
Reuting, Britton Lee, Micki Pi, Nick Bell, Greg and Cayla Kimble, Ashley Kisner, Mitch Zonneveld,
Erik Sowers, and Brandon Heffernan.

ixContents

Chapter
1 Introduction

1

1.1

The attosecond time scale and electron dynamics . . . . . . . . . . . . . . . . . . . .

1

1.2

Attosecond pulse generation via high harmonic generation . . . . . . . . . . . . . . .

3

1.3

Attosecond spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.4

1.3.1

Attosecond measurements via photoemission . . . . . . . . . . . . . . . . . . 11

1.3.2

Attosecond measurements via absorption . . . . . . . . . . . . . . . . . . . . 17

Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Delays in Photoionization and Previous Theoretical Analysis

22

2.1

Experimental observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2

Theoretical analysis and interpretation of photoemission time delays . . . . . . . . . 27

2.3

2.2.1

The sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2Classical streaking and the delay as a field-weighted sum

The absorption delay in two-photon ionization . . . . . . . . . . . . . . . . . . . . . 32

3 Methods for solving the time-dependent Schrödinger equation
3.1. . . . . . . . . . . 30

36

Semi-classical formulation of the interaction between field and matter . . . . . . . . 37
3.1.1

Classical electron dynamics in a laser field . . . . . . . . . . . . . . . . . . . . 39

3.1.2

Semi-classical dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.3

Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 41

x

3.2

3.3

3.1.4

Single Active Electron potentials . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.5

The scattering phase and the WS time delay . . . . . . . . . . . . . . . . . . 46

Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1

Discretization and propagation of the TDSE

. . . . . . . . . . . . . . . . . . 50

3.2.2

Calculation of bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.3

Exterior complex scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Extraction of time delays from simulations of attosecond streaking . . . . . . . . . . 58

4 Analytical estimates of time delays in streaked photoionization of atoms
4.1

4.2

Short- and long-range contributions to the streaking delay1 . . . . . . . . . . . . . . 63
4.1.1

Field-free time delays and the cutoff Coulomb potential . . . . . . . . . . . . 63

4.1.2

Analytical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.3

Relation to the Wigner-Smith Delay . . . . . . . . . . . . . . . . . . . . . . . 77

Application of analytical streaking formula . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.1

Application to and analysis of recent theoretical proposal . . . . . . . . . . . 79

4.2.2

Influence of the attochirp on attosecond streaking delays2 . . . . . . . . . . . 85

5 Absorption time delays in resonant and non-resonant two-photon ionization
5.1

5.2

5.1.1

Time-dependent perturbation theory analysis . . . . . . . . . . . . . . . . . . 99

5.1.2

Comparison with numerical simulations . . . . . . . . . . . . . . . . . . . . . 101

Absorption delay in the case of multiple resonances . . . . . . . . . . . . . . . . . . . 109
Analysis using model potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Absorption delay in H+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2
5.3.1

1

97

Two-photon ionization through an isolated resonance3 . . . . . . . . . . . . . . . . . 98

5.2.1
5.3

62

Simulations of the absorption delay . . . . . . . . . . . . . . . . . . . . . . . . 118

The results of this section are also presented in C. Goldsmith, J. Su, A. Becker, and A. Jaroń-Becker, Phys.
Rev. A 96, 053410 (2017).
2
The results of this section are also presented in C. Goldsmith, A. Jaroń-Becker, and A. Becker, J. Phys. B 51,
025601 (2018).
3
The results of this section are also presented in C. Goldsmith, J. Su, A. Jaroń-Becker, and A. Becker, J. Phys.
B: At. Mol. Opt. Phys. 51, 155602 (2018).

xi
5.3.2

Role of the excited states and interfering pathways . . . . . . . . . . . . . . . 120

6 Single photon ionization with isolated elliptically polarized attosecond pulses
6.1

123

Unassisted XUV ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.1.1

TDSE simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2

Dependence of angular distributions on ellipticity . . . . . . . . . . . . . . . . . . . . 126

6.3

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7 Summary and perspective

133

Bibliography

135

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TablesTable
5.1

Results of simulations for the absorption delay (in as) in the model system as afunction of the pulse length T (in as) of a two-photon resonantly ionizing pulse andstreaking pulses at different wavelengths λ (in nm), intensities I (in W/cm2 ). Also
√
(res)
(res)
(non−res)shown are the predictions ∆tabs = T / π (in as) and ∆tabs + ∆tabs(in as)from perturbation theory (Taken from Ref. [1]). . . . . . . . . . . . . . . . . . . . . . 107

xiii

FiguresFigure1.1

Development and decrease of laser pulse duration. The minimum duration continuously decreased until the mid 1980s. Following marginal progress for more thana decade, the femtosecond barrier (100 on the y-axis) was broken in 2001 (Taken
from [2]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

4(a) HHG can be conceptualized via a three-step model instigated by the tunnel
ionization of an electric wavepacket, which then propagates and accelerates in theincident laser field, and finally recombines with the parent core. Upon recombination, the energy absorbed from the electric field during propagation is released asa high-energy photon (from [3]). (b) A prototypical example of an HHG spectrum,featuring radiation with consistent efficiency over an extended interval of frequencies. (c) Via a classical model, one can predict the photon energies contributing
to the HHG spectrum, indicating a time-dependent relationship between ionization,
recombination, and the energy of the photon released. Each color corresponds tothe same classical trajectory which is indicated in (d); Classical trajectories leading
to HHG, grey lines indicate trajectories for ionization events without recombination,
thus non-contributing to HHG (Taken from [4]). . . . . . . . . . . . . . . . . . . . .

6

xiv
1.3

Methods developed to isolate attosecond pulses from HHG, including: (a) filtering
cutoff frequencies [5], (b) varying the polarization of the electric field [6], (c) rotating
the spatial wavefront of the driving field [7], and (d) manipulating the phase matchingconditions of the emitted radiation [8] (Taken from [4]). . . . . . . . . . . . . . . . .

1.4

8(a) Two elliptically polarized beams with opposite helicity are focused into a gas
jet in a non-collinear fashion to generate an isolated attosecond pulse of arbitrary
polarization. The inset displays the electric field distribution on the focal plane –rotating local electric field vectors crossing the transverse direction. The driving
pulses are as short as one-cycle, generating isolated pulses. (b) Continuous spectra
taken when an analyzer selects its polarization along the semi-major (black line) andsemi-minor (gray dashed line) axes of elliptically polarized attosecond pulses withan average ellipticity of 0.43. The ellipticity is analyzed in three different photon
energy ranges, and are nearly identical. (Taken from [9]). . . . . . . . . . . . . . . . 10

1.5

Conventional (left, taken from [10]) and attosecond (right, taken from [11]) streak
cameras.

1.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
(nl,n0 l0 )(n,l)

Conceptual visualization of the streaking time delays ∆ts,expt , ∆ts(n0 ,l0 ), and ∆ts

for two different streaking traces as compared to that of a free particle. Panel (b)is an enlargement of the traces shown in panel (a) to clearly show the different timedelays.
1.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Schematic of the principle of the RABBITT technique, where for simplicity only two
harmonics are indicated. Two electron wave packets originating from different shells
are simultaneously created using the same comb of high harmonics. The outgoing
wavepackets are further probed with a weak IR field. The experimental harmonic
spectrum used is also shown (Taken from [12]).

. . . . . . . . . . . . . . . . . . . . 16

xv
1.8

Schematic of the attoclock, where an electron wavepacket is liberated via tunnel
ionization. The rotation of the polarization vector of the field acts as a minute handfor the attoclock, and the time-dependent emission can be mapped via the angulardependence of the distribution (Taken from [13]).

1.9. . . . . . . . . . . . . . . . . . . 18

Single atom (a) and macroscopic (b) calculations of the transient absorption in helium, compared with experiment (c). It can be seen that macroscopic effects playa role in determining the absorption spectrum (Figures taken from [14], experimentby [15]).

2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Schematic for the WS time delay in scattering, taken from Ref. [16]. The time delay∆tWS is defined as ∆tWS = tp (R) − t0 (R)|R→∞ [16]. The time tp (R) is the durationa particle (or wavepacket) spends in a region of space R within the potential (top),
and t0 (R) is the duration that the same particle or wavepacket spends in the same
region R, but without the presence of the potential, i.e., free particle (bottom), as
the region R extends to infinity.

2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

The raw attosecond streaking spectrogram data for neon atom from the experiment of Schultze et al. shows that the photoelectrons emitted from the 2p state(higher energy/top signal) are delayed with respect to those of the 2s state (lower energy/bottom signal) as a function of delay between the attosecond XUV and streaking IR pulses. (Captured from [17]). . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3(a) Kinetic energy diagram as presented in the work of Isinger et al. for direct
2s ionization and 2p ionization accompanied by 2p → 3p excitation (shake-up).
(b) Photoelectron spectra obtained with XUV only (blue) and XUV + IR (red).
(c) Energy-resolved intensity and phase of the two photon oscillation, obtained by
Fourier transform of the signal. The symbol H, e.g. H61, corresponds to the direct
signal from a high harmonic, where as S, e.g. S56, refers to a sideband signal (Taken
from Ref. [18]).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

xvi
2.4

2p
Time delay differences ∆t2s
R − ∆tR in neon from the experiment in Ref. [18] as afunction of photon energy. The yellow and red dots were obtained via two different
measurements. Theoretical calculations within many-body perturbation theory [19]agree very well with the experimental data (solid line). The obtained time delaysudifferences ∆t3p
− ∆t2pR
R between shake-up and 2p ionization (diamonds), as well as

the time delay obtained in the experiment from Ref. [17] (square), are also shown(Taken from Ref. [18]).
2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28(a) Streaking time delay simulations from Su et al. [20] as a function of XUV pulse duration for 3D helium atom. Three ionization processes are considered: non-resonant
two-photon ionization (TPI) (blue lines with circles), resonant TPI (green lines withsquares), and single photon ionization (SPI) (red lines with asterisks). (Taken from
Ref. [20]).

2.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Angular-resolved time delays from Ref. [21]. The measured atomic time delay (redsymbols) is shown as a function of electron emission angle for SB16. The green lines
show the calculated values in resonant (solid) and non-resonant (dashed) conditions.
The error bars indicate the standard deviation of the measurements. (Taken from
Ref. [21]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1

Three streaking traces for a resonant two-photon ionization from numerical streaking
simulations (blue solid curve), classical electron trajectory calculations (green dashedline), and the reference free-particle streaking formula kff ree (τ ) = k0 − As (τ ) (reddashed-dotted curve). Panel (b) is an enlargement of the traces in panel (a) to show
the streaking (∆ts ) and continuum (∆tc ) time delay. Laser parameters used in thesimulations are: IXUV = 1.0 × 1013 W/cm2 , ωXUV = 81.81 eV, TXUV = 550 as, andφXUV = −π/2; Is = 1.0 × 1011 W/cm2 , λs = 2400 nm, Ns = 3, and φs = −π/2. . . . 61

xvii
4.1

Comparison of predictions of analytical formula for the field-free time delay ∆tf f ,Eq. (4.13), with results extrapolated from ab-initio calculations for hydrogen atom
(Z = 1) [22] as a function of propagation time (Taken from [23]).

4.2. . . . . . . . . . 67

Convergence of the time delay shown as percent error of ∆ts as a function of the logof the timestep δt used for the case of the ionization of the hydrogen ground state
where the final kinetic energy of the photoelectron is kf = 1.5 a.u. (solid blue line),
kf = 2.0 a.u. (dashed orange), and kf = 2.5 a.u. (dotted green). Streaking fieldparameters were N = 3 cycles, λ = 800 nm, Is = 1 × 1012 W/cm2 . . . . . . . . . . . 68

4.3

Comparison of analytical estimates (solid lines) with results of ab-initio calculations(circles, [22]) of the streaking delay in hydrogen atom and helium ion. Parametersof the streaking field: peak intensity Is = 1 × 1012 W/cm2 , wavelength λs = 800 nm
and pulse duration of three cycles. In the analytical calculations, ionization of thephotoelectron at the peak of the pulse and the most probable location of the electronin the initial state has been assumed (Taken from [23]). . . . . . . . . . . . . . . . . 68

4.4

Comparison of analytical predictions (solid lines) with results of ab-initio calculations(symbols) for the streaking time delay in the case of photoionization of (a) helium(ab-initio data from [22]) and (b) neon atom as a function of the XUV photon
energy. Ab-initio results for Ne are from calculations using the present SAE potential(circles, [24]) and multi-electron B-spline R-matrix calculations (diamonds, [25]) are
shown. Streaking laser parameters as in Fig. 4.3 (Taken from [23]).

4.5. . . . . . . . . 70

Absolute difference ∆short as a function of the XUV photon energy for the contribution due to the short-range potential for (a) neon atom and (b) helium atom.
Streaking laser parameters as in Fig. 4.3 (Taken from [23]). . . . . . . . . . . . . . . 76

4.6

Absolute difference ∆phase for photoionization of (a) neon, (b) helium and (c) hydrogen. Streaking laser parameters as in Fig. 4.3 (Taken from [23]).

4.7. . . . . . . . . 76

Absolute difference ∆log for photoionization of (a) neon, and (b) helium. Streaking
laser parameters as in Fig. 4.3 (Taken from [23]).

. . . . . . . . . . . . . . . . . . . 76

xviii
4.8

Comparison of present analytical estimates (solid lines) for ∆tCLC with those of analternative analytical approximation (dashed lines, [26]). Streaking laser parameters
as in Fig. 4.3 (Taken from [23]).

4.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Pseudopotential as suggested in the work by Kiesewetter et al. [27]. (a) An electronis ejected at time t = 0 in the presence of a weak NIR field F (t) with period T .
For 0 ≤ t ≤ T /4, the electron has an average momentum of P1 and experiences
a constant restoring force along the laser polarization direction. For t > T /4, its
momentum is modeled as constant at P2 until after the next field maximum at
T /2 + t0 . After the net energy transfer has been established, the photoelectron
assumes its asymptotic momentum p0 . (b) Illustration of the corresponding spatial
dependence of the pseudo-potential along the field polarization. (Taken from Ref.
[27]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.10 Eq. (4.5) shown as a function of time tj for streaking wavelengths λs = 400 nm (a),800 nm (b), 1600 nm (c), and 2400 nm (d). For each panel, the results are shown for
final electron kinetic energies corresponding to XUV central frequencies of ω = 44.22
eV (blue), 55.27 eV (orange), 68.03 eV (green), 82.48 eV (red), 98.64 (purple) eV,
116.49 eV (brown), and 136.05 eV (pink). The times T10% and T1% (as described in
the text) are shown with stars and circles respectively. For all simulations shown, a
3 cycle streaking pulse of intensity Is = 1 × 1012 W/cm2 is used. . . . . . . . . . . . 82
4.11 The times T10% (left) and T1% (right) shown as a function of XUV central frequency
for different pulse durations; Ns = 3 cycles (circles), Ns = 4 cycles (stars), Ns = 5
cycles (diamonds), and Ns = 6 cycles (squares). For the case of T10% , we further show
T /2 (dashed light blue) and T /4 (solid green). The same streaking pulse intensity
as for the results in Fig. 4.10 is used for all calculations.

. . . . . . . . . . . . . . . 83

xix
4.12 Estimates of the streaking delay for Neon ((a) 2p, (b) 2s) and Argon ((c) 3p, (d)3s) given by Eq. (4.52), as a function of the central XUV frequency for times of thetransition in the continuum, ti = −τF W HM /2 (dash-dotted line), ti = 0 (solid line),
and ti = τF W HM /2 (dashed line) are compared. Also shown are results from abinitio calculations (diamonds), [25]. In all calculations a streaking pulse of wavelength
λs = 800 nm, sin2 -envelope consisting of 3 cycles, and peak intensity Is = 1 × 1012
W/cm2 has been considered (Taken from [28]). . . . . . . . . . . . . . . . . . . . . . 88
4.13 (a) Neon 2p photoionization cross section [29] and time averaged probability for (b)ω0 = 60 eV. The chirp is taken to be b = 0 eV/fs (solid line), b = 15 eV/fs (dashedline), and b = 25 eV/fs (dash-dotted) and τ = 450 as (Taken from [28]). . . . . . . . 89
4.14 Same as Fig. 4.13 but for (a) Argon 3p photoionization cross-section [29]; time averaged probability for (b) ω0 = 45 eV, (c) ω0 = 60 eV, and (d) ω0 = 80 eV (Taken
from [28]).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.15 Change in streaking time delay for photoionization of Ne from 2p (a) and 2s (b)shells and of Ar from 3p (c) and 3s (d) in chirped laser pulses (with respect to an
unchirped pulse) for b = 5 eV/fs (line with plus signs), b = 10 eV/fs (dashed), b = 15
eV/fs (dash-dot), b = 20 eV/fs (dotted), and b = 25 eV/fs (solid) (Taken from [28]).

92

4.16 Comparison of the contributions from the (a) Wigner-Smith and (b) Coulomb-laser
coupling terms for the change in the streaking delay for photoionization of Ar from
3p shell. Lines and symbols as in Fig. 4.15(c) (Taken from [28]). . . . . . . . . . . . 94
4.17 Difference of streaking time delays for photoemission (a) from 2p and 2s in Neon at
XUV central frequencies of 80 eV (solid), 90 eV (dashed), and 100 eV (dash-dotted)
and (b) from 3p and 3s in Argon at XUV central frequencies of 55 eV (solid), 60 eV(dashed), 70 eV (dash-dot), and 80 eV (dotted) as a function of the chirp parameterb (Taken from [28]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

xx
5.1

(res)

Resonant contribution to the absorption delay ∆tabs (in attoseconds) as a functionof detuning δr and pulse duration T (Taken from Ref. [1]).

5.2. . . . . . . . . . . . . . 102

Three streaking traces for a resonant two-photon ionization from numerical streaking
simulations (blue solid curve), classical electron trajectory calculations (green dashed(free)line), and the reference free-particle streaking formula kf(ti ) = k0 − As (ti ) (reddashed-dotted curve). Panel (b) is an enlargement of the traces in panel (a) to show
the streaking (∆ts ) and continuum (∆tc ) time delay. Laser parameters used in thesimulations are: IXUV = 1.0 × 1013 W/cm2 , ωXUV = 81.81 eV, T = 550 as, andφXUV = −π/2; Is = 1.0 × 1011 W/cm2 , λs = 2400 nm, Ns = 3, and φs = −π/2
(Taken from Ref. [1]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3

Absorption time delay from numerical simulations (blue circles) and model predic(res)(non−res)tions for ∆tabs + ∆tabs(res)(red solid line) and ∆tabs(black dashed line) as afunction of pulse duration for a Gaussian XUV pulse tuned into resonance with thefirst excited state of (a) the model system and (b) helium atom (data for numericalsimulations taken from [20]). Laser parameters: IXUV = 1.0 × 1013 W/cm2 (resonant), φXUV = −π/2; Is = 1.0×1011 W/cm2 , λs = 2400 nm, Ns = 3, and φs = −π/2
(Taken from Ref. [1]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4

(res)

Absorption delay ∆tabs as a function of pulse duration for detunings δr = 0.68 eV(solid line), 0.82 eV (dashed line), 0.95 eV (dash-dot line), 1.09 eV (stars), 1.36 eV(diamonds), 2.04 eV (boxes), and 2.72 eV (circles) (Taken from Ref. [1]).

5.5(non−res)

Absorption delay ∆tabs. . . . . . 108

as a function of pulse duration for detunings δr = 0.68

eV (solid line), 0.82 eV (dashed line), 0.95 eV (dash-dot line), 1.09 eV (stars), 1.36
eV (diamonds), 2.04 eV (boxes), and 2.72 eV (circles).
5.6. . . . . . . . . . . . . . . . 108

Absorption delay as a function of XUV pulse duration, for a detuning chosen suchthat the resonance is at FWHM of the XUV pulse (Taken from Ref. [1]).

. . . . . . 110

xxi
5.7

(a) TDSE calculations of the absorption delay ∆tabs (diamonds), the two-photon
streaking delay ∆ts (stars), and the continuum delay ∆tc (circles) for the model
system as a function of XUV central frequency. The pulse parameters used are
FWHM =
λs = 2400 nm, Ns = 3 cycles, Is = 1 × 1011 W/cm2 , φs = φXU V = −π/2, τXUV

350 as. For the simulation of the resonant two-photon ionization, an XUV intensity of IXUV = 1 × 1013 W/cm2 is used, whereas for the continuum delay ∆tc , anintensity of IXUV = 1 × 1014 W/cm2 is used to initiate the one-photon ionization.
In (b), the absorption delay from TDSE calculations (diamonds) are compared with
perturbation theory predictions, when including the third and fifth excited states asintermediates (dashed line) as well as when the first excited state is also included(solid line). The dipole moment products µf j µji for intermediate state |ji have beenused as (constant) fitting parameters.
5.8. . . . . . . . . . . . . . . . . . . . . . . . . . 113

The “dark” 2s state in He may be populated via absorption or stimulated emissionof an IR photon in addition to an XUV photon for a transition from the initial 1s
state (Taken from Ref. [14]).

5.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115(a) Absorption delay as a function of streaking pulse intensity for three different
XUV central frequencies. All other parameters remain fixed to the same values as inFig. 5.7. (b) Absorption delay as a function of ionizing pulse intensity for the caseof the dipole-forbidden fourth excited state minus a streaking photon (blue line plus
diamonds), and the case where resonant with the dipole-allowed third excited state(red line plus asterisks). All other pulse parameters remain fixed. (c) The streaking(blue dashed with circles), continuum (red dashed with diamonds), and absorption(solid green line with asterisks) delay as a function of streaking pulse intensity when
the ionizing pulse is tuned to resonance with the dipole-allowed third excited state.

116

5.10 Potential energy surface for the 1D model of H2+ . The ground state, and the first
four excited states are shown as a function of internuclear distance. Also shown isthe equilibrium internuclear distance of Req = 2.64 a.u. (dashed line). . . . . . . . . 118

xxii
5.11 Absorption delay as a function of XUV central frequency for the resonant two-photon
ionization of H+
2 . Field parameters for the streaking pulse used are the same as thosein Fig. 5.7. For the XUV pulse, a sine squared envelope is used with a pulse durationof TXU V = 3 fs, and the same intensity and CEP is used as in Fig. 5.7.

. . . . . . . 119

5.12 Resonant two-photon ionization of the 1D model of H+
2 calculated via time-dependent
perturbation theory (blue solid line) where (a) m = 1, (b) m = 1, 3, and (c) m = 1,
3, and 5 are considered in the sum in Eq. (5.3) as compared to TDSE simulations(red stars). The vertical lines mark the third (black) and fifth (purple) excited state
resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1

PADs for ionization from the ground state of the 2D hydrogen model for ellipticitiesof ξ = (a) 0.0 (linearly polarized), (b) 0.3, (c) 0.6, (d) 0.8, (e) 0.9, and (f) 1.0 (rightcircularly polarized). Other laser parameters are given in the text. . . . . . . . . . . 127

6.2

PADs along kmax as a function of angle. Ellipticities are the same as in Fig. 6.1.
The peak of the distribution (red vertical line) and the location corresponding tozero degrees rotation from the major axis (black vertical line) are shown.

6.3. . . . . . 128

Angle of maximum emission in the PADs from ionization of the 1s state of the 2Dhydrogen model, plotted as (a) an angle and (b) a time (see text for relation), as afunction of ellipticity ξ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.4

The first three eigenstates of the 2D model hydrogen atom are shown, where |gi
denotes the ground state, and since we only consider three eigenstates in this section,
we label the first two excited states as |px i and |py i without a principal quantum
number.

6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Vector components of the time-dependent dipole moment, Eq. (6.6), of the ring
current state, when allowed to freely evolve in time. We see that both dx (blue) anddy (orange) oscillate with a period of about 20 a.u.

. . . . . . . . . . . . . . . . . . 132

Chapter 1

Introduction

1.1

The attosecond time scale and electron dynamics
The understanding of electron dynamics has played a fundamental role in technology andthe advancement of atomic and molecular physics in the past century. Chemical reactions dependon the behavior of electrons, as does the response of atomic and molecular systems to external
fields. In this sense, electrons participate in nearly every aspect of modern technology, and thus
their dynamics are essential to understand. Considering the energy-time form of the uncertainty
principle, ∆E∆t ∼ ~, a time scale of tens to hundreds of femtoseconds (1 fs = 1 × 10−15 s) is
expected for energies of millielectron volts (1 meV = 1 × 10−3 eV), corresponding to the orderof magnitude of the difference between vibrational energy levels in molecules. The separation ofelectronic states in an atom or molecule is on the order of eV, corresponding to an attosecond (1as = 1 × 10−18 s) time scale. An alternative, intuitive picture of the time scale is given by the Bohr
model for the hydrogen atom. Within this model, the classical kinetic energy of the electron in the
ground state is Ek = v 2 /2 = 1/2, and therefore the period of the classical orbit is T = 2π/v = 2π
in Hartree atomic units (a.u.). Therefore the atomic unit of time is 24.19 as = 24.19 ×10−18 s,
which corresponds to the period of the classical electron circling in this first Bohr orbit divided by
2π.
In order to observe the dynamics of attosecond processes, it is necessary to use measurement
tools with resolution of this magnitude (e.g., [2, 10, 13]). A video camera captures processes on
millisecond time scales by taking snapshots with a shutter time quick enough to resolve the motion

2of most macroscopic objects. To study the real-time evolution of electron dynamics by taking
snapshots of its probability distribution, a “camera” with a shutter time on the attosecond timescale is required. It is thus expected that probing, and furthermore controlling electron dynamics
is possible using pulses of light of durations on this time scale. The recent advent of high-orderharmonic generation (HHG) in gases [30–40] has enabled us to generate isolated pulses and pulse
trains of attosecond duration, which provides such a tool for observing electron dynamics.
Among the many interesting physical processes which occur in the attosecond time regime,
photoionization is one which has a long history dating back to the beginning of quantum mechanics.
In 1905, Albert Einstein demonstrated the photoelectric effect [41] (translation [42]), which led tothe proposal of the existence of photons. Einstein showed that in order for the electron to escape
the potential of its parent ion, it needed to absorb a light quanta (photon) with enough energy toovercome the binding energy of this potential, and that the final kinetic energy Ek,f of such an
electron is expressed in atomic units (~ = me = e = 1) as:
Ek,f =q

2 (ω − Ip ),(1.1)

where ω is the energy of the photon and Ip is the ionization potential, also known as the binding
energy or work function. At the time when Einstein received the Nobel Prize in Physics, i.e.,
1921, the physical interpretation of the ionization process concerning its temporal resolution remained elusive. With the advent of attosecond pulses, we now have the capabilities of probing thephotoionization process on the natural time scale of the electron.
In 1931, the process of two-photon absorption was predicted by Maria Göppert-Mayer in her
doctoral dissertation [43]. In such a process, the light interacting with the matter is sufficiently
intense such that two-photons are absorbed in a transition. In relation to the photoelectric effect,
this means that if Ip > ω, the system may still be ionized by two photons if Ip < 2ω, and in sucha case the kinetic energy of the emitted electron is
Ek,f =q
2 (2ω − Ip ).

(1.2)

As we will discuss in this thesis, there are differences in the dynamics of one- and two-photon

3ionization which have important implications in the quest to use attosecond pulses to observe andcontrol electrons. These differences as they relate to attosecond experiments, and specifically theattosecond streaking experiment (see section 1.3.1.1) were first pointed out by a former colleague,
Jing Su [16, 20] in the proposal of a so-called absorption delay for a resonant two-photon ionization
process. The present work builds upon the work of Su, as well as suggests how the idea of theabsorption delay can be extended to different attosecond experiments.

1.2

Attosecond pulse generation via high harmonic generation
The effort towards studying dynamics in matter on decreasingly shorter time scales has

driven technological advances over the past decades. After the invention of mode-locking in 1965,
the minimum duration of laser pulses decreased continuously until 1986 when 6 fs pulses weregenerated [44]. As seen in Fig. 1.1, there was only marginal progress from 1986 until 2001, as
a radically different technology was needed to break the 6 fs benchmark, which took 15 years
to develop [38, 45–53]. This effort subsequently led to the development of ultrashort laser pulses
from such prominent sources as ultrashort electron pulses [54], X-ray free electron laser pulses [55]
as well as femtosecond laser pulses via the process of high harmonic generation [47, 48]. Strong
laser pulses with peak intensities on the order of 1013 - 1015 W/cm2 have field strengths of similar
magnitude to the internal electric field caused by the Coulomb force in atoms and molecules. With
such a high intensity, many non-linear processes can occur. One of these processes is high harmonic
generation (HHG) in gases, which occurs when an intense laser beam (usually a near-infrared (NIR)
Ti:Sapphire pulse) is focused into an atomic or molecular gas medium.
The HHG process has been most successfully described using a semi-classical three-stepmodel [32–34]. According to this model, the incident laser electric field is sufficiently intense suchthat the Coulomb field of the atomic system is tilted significantly, and an electron now has sufficient
probability of tunneling through the Coulomb barrier in a process called tunneling ionization, as
shown in Fig. 1.2(a). After tunnel ionization has occured, the electron wavepacket is accelerated in
the laser electric field first away and then, upon reversal of the field direction, back to the parent ion.

4

Figure 1.1: Development and decrease of laser pulse duration. The minimum duration continuously
decreased until the mid 1980s. Following marginal progress for more than a decade, the femtosecond
barrier (100 on the y-axis) was broken in 2001 (Taken from [2]).

5Upon recombining with the parent ion, the energy gained from the absorption of many photons
in the continuum from the driving field is emitted as a high energy photon. Using a quantum
mechanical picture, the three step model is reflected via a product of three probability amplitudes,
accounting for ionization, propagation, and recombination of the electron wavepacket. Due to thequasiperiodic repetition of this process every half cycle of the laser field, the resulting emission
spectrum is discrete, consisting of odd harmonics of the driving laser’s central frequency.
Some of the basic features of HHG spectra and the underlying mechanism can be understoodfrom a simplified classical analysis [56], i.e. by solving Newton’s equation in one dimension, forpropagation of a point-like electron in an oscillating linearly polarized electric field E (t):d2 x
= −E (t) x,
dt2

(1.3)

assuming that the electron is released with zero velocity at the location of the parent ion (i.e.,
x = 0). In Figure 1.2 (d), the corresponding classical trajectories are shown, emerging at a central
cycle of the driving field. The grey lines show classical trajectories which do not return to theparent ion, and therefore cannot be related to the generation of harmonics. The analysis of the
trajectories, that return to the parent ion, leads to the prediction of a cut-off energy in the HHG
spectrum of
Emax = Ip + 3.17Up ' Iλ20 ,(1.4)

where Ip is the ionization potential of the atom, Up = I/4ω02 is the ponderomotive potential, andI = E02 is the peak intensity of the laser pulse. This cutoff can clearly be seen in Fig 1.2(b).
The ionization potential contributes to the cutoff energy via the recombination step, wherein theelectron recombines to the ground state from the continuum, and the contribution related to theponderomotive energy can be well predicted through the classical analysis of the possible electron
trajectories contributing to HHG, neglecting the remaining Coulombic force from the parent ion. In
Fig. 1.2(c), the kinetic energy of an electron released into an electric field at different points in timeis plotted according to the simulated instant of ionization and predicted moment of recombination
[4].

6

Figure 1.2: (a) HHG can be conceptualized via a three-step model instigated by the tunnel
ionization of an electric wavepacket, which then propagates and accelerates in the incident laser
field, and finally recombines with the parent core. Upon recombination, the energy absorbed
from the electric field during propagation is released as a high-energy photon (from [3]). (b) A
prototypical example of an HHG spectrum, featuring radiation with consistent efficiency over an
extended interval of frequencies. (c) Via a classical model, one can predict the photon energies
contributing to the HHG spectrum, indicating a time-dependent relationship between ionization,
recombination, and the energy of the photon released. Each color corresponds to the same classicaltrajectory which is indicated in (d); Classical trajectories leading to HHG, grey lines indicate
trajectories for ionization events without recombination, thus non-contributing to HHG (Taken
from [4]).

7Furthermore, there are several conclusions which can be drawn from the analysis giving rise topanels (c) and (d) in Fig. 1.2: for every energy except the cut-off frequency, electron propagation
along two different trajectories contribute to each photon energy, which follow short and longexcursion distances, respectively. Only the electrons ionized in the second and fourth quarters ofan optical cycle of the laser field may be driven back to the parent ion for the recombination step.
This temporal confinement of the underlying mechanism of HHG results in the generation of a train
of attosecond pulses spaced by half of the optical period of the driving laser pulse.
For many applications, and for some techniques explored in this thesis, it is preferable to use
an isolated attosecond pulse (IAP) as opposed to an attosecond pulse train (APT). The generationof IAPs is currently possible in a few different ways, which are summarized in Fig. 1.3. Usinga time-frequency analysis [4], as well as according to the classical analysis within the three-step
picture, it can be shown that the highest energies in an HHG spectrum are selectively emitted near
the peak of the electric field. A method for generating IAPs that takes advantage of this fact isshown in Fig. 1.3 (a), in which one selects photon energies near the cutoff that are only emitted
through one recombination event, by using a filter to absorb lower energy photons [5]. A second
method for generating IAPs is called polarization gating. In this technique the polarization of thedriving laser pulse is changed as a function of time (Fig. 1.3 (b)) [6]. Through this technique, nearlinear polarization is achieved only near the center of the laser pulse. In this narrow window, theelectron can recombine efficiently with the parent ion, releasing an attosecond pulse. The remaining
electron wavepackets which are released at different ionization times, when the pulse is elliptically or
circularly polarized, are driven away from the position of the parent ion, preventing recombinationfrom taking place, and consequently an isolated pulse is emitted [6, 56, 57]. A third technique isshown in Fig. 1.3(c), where one introduces a time-dependent rotation of the wavefront direction of
an intense, few-cycle laser field, providing a method for spatially separating the attosecond pulses
in an APT [7, 58]. The emitted light acquires a time-dependent propagation direction based on the
moment of production from the rotated intense field. Since the spatial position of each pulse in the
train is well defined, the directional extraction of individual pulses may be achieved. Finally, we8Figure 1.3: Methods developed to isolate attosecond pulses from HHG, including: (a) filtering
cutoff frequencies [5], (b) varying the polarization of the electric field [6], (c) rotating the spatial
wavefront of the driving field [7], and (d) manipulating the phase matching conditions of the emitted
radiation [8] (Taken from [4]).

9mention the generation of IAPs in systems featuring high intensities and midinfrared light sources,
such as the case in Fig. 1.3(d) [8,40]. This technique relies upon the manipulation of phase-matchingconditions for the emitted radiation in order to create an IAP. It is found that increasing the pulse
duration and gas pressure allows the index of refraction of the target gas to change during thecourse of the light pulse, such that emission near the peak of the laser field can be favored andisolated from light emission occuring at different times.
Very recently, it has become possible to control the polarization state of the generated attosecond laser pulses [9]. This will come to our advantage towards the end of this thesis, when
proposing a technique to probe time delays using isolated attosecond pulses of elliptical polarization
to ionize atomic species. In Fig. 1.4, the technique for producing such pulses is shown. Huang etal. have shown that by using two ultrashort (down to a single cycle) counter-rotating ellipticallypolarized fundamental beams in a non-collinear geometry one can generate isolated elliptically polarized attosecond pulses [9]. By varying the ellipticity of the counter-rotating few-cycle incident
pulses, one gains full control of the polarization states of isolated attosecond pulses.

1.3

Attosecond spectroscopy
Attosecond laser pulses have an intrinsically large bandwidth in energy, and are not yet

intense enough to be used for attosecond pump, attosecond probe measurements. Therefore, anumber of experiments using a combination of attosecond pulses and more intense femtosecond
pulses have been used in the attempt of characterizing ultrafast phenomena. In this section, wesummarize some of the more common techniques, putting an emphasis on those studied in thisthesis. In general there are two different ways of extracting information about ultrafast dynamics
using attosecond pulses; namely by measuring electrons ionized by the attosecond pulse, or byobserving the absorption spectrum of the system. This means either detecting the photoelectrons,or the light before and after the propagation of the pulses through a target system.

10

Figure 1.4: (a) Two elliptically polarized beams with opposite helicity are focused into a gas jet
in a non-collinear fashion to generate an isolated attosecond pulse of arbitrary polarization. The
inset displays the electric field distribution on the focal plane – rotating local electric field vectors
crossing the transverse direction. The driving pulses are as short as one-cycle, generating isolated
pulses. (b) Continuous spectra taken when an analyzer selects its polarization along the semi-major(black line) and semi-minor (gray dashed line) axes of elliptically polarized attosecond pulses withan average ellipticity of 0.43. The ellipticity is analyzed in three different photon energy ranges,
and are nearly identical. (Taken from [9]).

11
1.3.1
1.3.1.1

Attosecond measurements via photoemission
Attosecond streaking

The concept of the streaking camera has been around for decades with the intention ofcharacterizing the temporal structure of a short light pulse [59,60]. The scheme for this conventional
streak camera is shown on the left side of Fig. 1.5. In such a measurement photoelectrons are emitted
from a photocathode after interaction with a short light pulse. The time-dependent probabilityof the emission is proportional to the temporal intensity of the pulse. During the propagation
towards the detector, the electron crosses a time-dependent voltage gradient V (t), and therefore
the electrons are deflected to a degree which is dependent on the moment of release of the electrons.
Thus, the spatial distribution of the photoelectrons on the screen may be directly converted to atemporal pattern corresponding to the temporal structure of the incident pulse. Such a conventional
streak camera can achieve this goal with subpicosecond (1 ps = 10−12 s) resolution for a voltage
ramp that is within a fraction of a nanosecond (1 ns = 10−9 s) [59].
The attosecond streak camera originally shared the same goal as the conventional streak
camera, but aimed to measure the temporal structure of an even shorter pulse, on the scale ofattoseconds [53, 61–63]. The technique involves using two laser pulses: an attosecond XUV pulse
which initiates an ultrafast process, typically ionization, and a few-cycle infrared streaking pulsethat is used to modulate the final momentum of the electron according to its release instant (rightside of Fig. 1.5). That is, photoelectrons ionized by the XUV pulse at different instants along
the streaking field have different final momenta, which can be estimated by considering a classicalelectron propagating in a field, whilst neglecting the Coulomb potential of the ion core. Assumingthat a free electron with momentum k0 is released into the continuum at an instant in time ti ,the particle interacts with the streaking field Es (t) as it propagates, and the final momentum is
modified by the vector potential of this field As (t) as
Z T(free)kf(ti ) ' k0 −Es (t) dt = k0 − As (ti ) ,

(1.5)

ti

where the streaking pulse has duration T such that As (T ) = 0 given that the pulse starts at

12

Figure 1.5:
cameras.

Conventional (left, taken from [10]) and attosecond (right, taken from [11]) streak

13
t = 0. The dynamics of the electron in the continuum however, depend on its interaction with the
Coulomb potential of the residual ion, as well as the initial bound state |n, li from which it was
ionized. As a result, numerical simulations reveal that the final electron momentum as a functionof release time shows a temporal offset with respect to the vector potential given by
(n,l)

kf
(n,l)

where ∆ts(ti ) = k0 − As (ti + ∆t(n,l)
),
s

(1.6)

denotes the offset obtained for an electron which initially populated the bound state|n, li of the atom. Throughout this thesis, we refer to this offset as the “streaking time delay”, or“streaking delay” for short. We note that because the XUV pulse initiates the transition of theelectron into the continuum in a streaking setup, the asymptotic momentum of the electron in Eqs.
p(1.5) and (1.6) is typically defined in terms of the XUV photon energy as k0 = k̂ 2 (ω − Ip ), where
ω the XUV photon energy, and Ip is the ionization potential corresponding to the initial state |n, li.
In an attosecond streaking experiment however, the exact form of the vector potential for thefield is not measured. Instead, temporal information may be gained only through the comparisonof traces resulting from photoelectrons exposed to the same streaking field, but which populateddifferent orbitals initially. That is, in experiment one measures the quantity(nl,n0 l0 )

0 0

,l )
∆ts,expt = ∆t(n,l)− ∆t(nss(1.7)

for photoelectrons emitted from two different inital states |n, li and |n0 , l0 i, as is demonstrated inFig. 1.6. The first measurement of such a quantity was in an experiment by Schultze et al. [17],(2p,2s)where they showed that ∆ts,expt for neon atom is non-zero, which started a long debate in thetheoretical literature as is discussed in the next Chapter. In this thesis, we will primarily focus(n,l)

on the theoretical determination of the streaking delay ∆ts

for various atomic species. In theremaining Chapters, the initial state is always implied in the context of the problem, and therefore
we simply refer to this quantity hereforth as ∆ts .

14(nl,n0 l0 )(n,l)(n0 ,l0 )Figure 1.6: Conceptual visualization of the streaking time delays ∆ts,expt , ∆ts , and ∆ts
for two different streaking traces as compared to that of a free particle. Panel (b) is an enlargementof the traces shown in panel (a) to clearly show the different time delays.

15
1.3.1.2

Reconstruction of attosecond beating by interfering two-photon transitions(RABBITT)

A similar experiment to streaking invokes an attosecond pulse train, as opposed to an isolated attosecond pulse, in conjunction with a long weak near-infrared pulse, and is known as thereconstruction of attosecond beating by interference of two-photon transitions, or RABBITT. In
this method, an atom is ionized using a comb of high harmonics. The additional presence of thefundamental laser with frequency ω induces the formation of sideband peaks in between adjacentharmonics due to two-photon transitions including the absorption or emission of an infrared photon [12]. There are two distinct interfering quantum paths contributing to the same sideband (c.f.,Fig. 1.7). Similar to attosecond streaking, the delay between the attosecond pulse train and thedriving laser field, which we call ti for consistency, is varied. It can be shown that the amplitudeof the sideband signal from a given shell |n, li is modulated as [51]
h 
i
(n,l)
S (ti ) = α + βcos 2ω ti − ∆tA − ∆tR
,

(1.8)

where α and β are two ti -independent constants, ∆tA is a term proportional to the difference inphase between consecutive harmonics which is related to the group delay of the attosecond pulses,(n,l)

and ∆tR

represents the atomic delay due to the two-photon ionization process. We call this lastterm the RABBITT delay to make the analogy to the streaking delay. In combination with themeasurement of the amplitudes of the harmonics, the phase information can be used to reconstruct
the temporal profile of the attosecond pulses and the electron wavepackets initiated by the pulse
train. In experiment, similar to streaking experiments, experimentalists typically observe a relative
RABBITT time delay for the emission of electrons from two different orbitals, which is given by(nl,n0 l0 )(n,l)∆tR,expt = ∆tR(n0 ,l0 )

− ∆tR

.

(1.9)

(3p,3s)As an example, in the case of Ref. [12] ∆tR,expt was measured for argon atom.
It has been shown [64] that in the limiting case of a continuous-wave, monochromatic field
and a periodic repetition of identical attosecond pulses in the train, the RABBITT time delay

16

Figure 1.7: Schematic of the principle of the RABBITT technique, where for simplicity only two
harmonics are indicated. Two electron wave packets originating from different shells are simultaneously created using the same comb of high harmonics. The outgoing wavepackets are further
probed with a weak IR field. The experimental harmonic spectrum used is also shown (Taken
from [12]).

17agrees with the streaking time delay. RABBITT is in general less demanding experimentally thanstreaking, as it is easier to produce attosecond pulse trains than isolated pulses, and the probing
infrared field can be weaker in RABBITT than in streaking [65]. Furthermore, since multiple
sidebands are present in a typical RABBITT signal, time delays can be measured for multiple
photoelectron kinetic energies. In contrast, a streaking experiment allows for the determination ofa single time delay only.

1.3.1.3

Attoclock: Angular streaking

The attoclock is another attosecond measurement technique, different from streaking and
RABBITT in the sense that no attosecond pulse is necessary to achieve attosecond time resolution.
The experiment involves the application of a femtosecond (two-cycle or shorter) nearly-circularpolarized laser pulse with matter. The rotation of the field acts as a “clock”, where the electricfield vector in the polarization plane serves as the clock-hand (see Fig. 1.8). The target is ionized bythis pulse, and the final emission angle of the photoelectron is related to the instant of ionization.
Typically, it is assumed that the ionization is initiated by a tunnel ionization event, and the
propagation of the electron wavepacket in the combined fields of the circularly polarized field andCoulomb field of the remaining ion must be taken into account.
This technique has been used to resolve the so-called tunneling time-delay [13, 66]. Thisquantity relates to the question, if an electron wavepacket takes a finite time to tunnel through abarrier, and is that time measurable on an attosecond time scale? In this method, peak positionsin the final angular distribution can be measured with an angular resolution of about 1◦ . For an
800 nm pulse typical in such an experiment, this corresponds to a temporal resolution of roughly
7.4 as [66].

1.3.2

Attosecond measurements via absorption

Another class of time-resolved attosecond measurements depends on measuring an absorption
spectrum, as opposed to a photoelectron spectrum, as a function of delay between a femtosecond

18

Figure 1.8: Schematic of the attoclock, where an electron wavepacket is liberated via tunnel
ionization. The rotation of the polarization vector of the field acts as a minute hand for theattoclock, and the time-dependent emission can be mapped via the angular dependence of the
distribution (Taken from [13]).

near-infrared pulse and an attosecond XUV pulse. In this thesis, we will show that such pump-probe
techniques can be used to compliment photoemission experiments.

1.3.2.1

Attosecond transient absorption spectroscopy

Attosecond transient absorption spectroscopy (ATAS) is characterized by the absorption of
the pump and probe light by a medium. The technique has some similarities to a classical Beer’s Law
absorption experiment in chemistry [14]. The amount of light which leaves the medium is different
from that which was incident upon the medium. The amount of light absorbed depends upon the
wavelength of the light and the system being probed. ATAS is different from classical absorption
spectroscopy in that temporal resolution is achieved by delaying the near-infrared femtosecond
pulse and the attosecond XUV pulse. The experiment can be performed where the infrared pulse
precedes (conventional pulse sequence) or follows (unconventional pulse sequence) the XUV pulse.
The response of the medium to the light at the single-atom/molecule level can be calculated as [14]hi∗S (ω, τ ) = 2Im Ẽin(ω, τ ) d˜(ω, τ ) ,(1.10)

19
where Ẽin is the Fourier transform of the combined NIR-XUV electric fields, and d˜ is the Fouriertransform of the time-dependent dipole d (t) = − hΨ (r; t) |r| Ψ (r; t)i. Including the effects of macroscopic propagation via Maxwell’s equations can improve agreement with experiment, as is seenin Fig. 1.9 where a comparison is shown between theoretical simulations with (b) and without(a) macroscopic propagation and experiment (c) for the transient absorption spectrum of helium
atom [14]. In experiment, the spectrum is obtained by subtracting the spectrum of the combined
fields leaving the sample from the original spectrum of the fields prior to traversing through thesample.

Figure 1.9: Single atom (a) and macroscopic (b) calculations of the transient absorption in helium,compared with experiment (c). It can be seen that macroscopic effects play a role in determining
the absorption spectrum (Figures taken from [14], experiment by [15]).

1.4

Organization of this thesis
In the last section, we discussed how time delays in photoemission may be observed using thestreaking or RABBITT techniques. In this context, we wish to contribute to the understanding how
to time resolve simple quantum mechanical processes, such one- and two-photon ionization, using
attosecond pulses. In our quest to understand such dynamics, we have used analytical models
to explore the time delays measured in attosecond streaking experiments like that reported inRef. [17]. We have considered the original application of streaking measurements related to onephoton ionization as well as their extensions to resonant two-photon ionization, involving an isolated20
resonance as well as multiple resonances. Additionally we have explored the perspective to use therecently developed technique of generation of isolated elliptically polarized attosecond pulses toobtain information on a sub-attosecond time scale in an extension of the attoclock concept. The
structure and organization of this thesis is given as follows:
Recently, it has been predicted [20, 67] that a resonant multiphoton process leads to an additional phase shift and therefore time delay, called the absorption time delay. We contribute tothis discussion by analytically predicting such a delay in the case of an isolated resonance, andshowing that it has a dependence upon the pulse duration and central frequency of the attosecond
pulse. This additional delay is sensitive to the structure of the target, particularly in the case wheremultiple resonances are within the bandwidth of the ionizing attosecond pulse. In the overlapping
resonance case, we show that transient absorption spectroscopy provides complementary information which sheds light on the mechanism. Additionally, we have begun the effort of extending theconcept of the attoclock to isolated elliptically polarized attosecond pulses. To this end, we propose
a new method where a single attosecond pulse is used to gain temporal information.
In Chapter 2, we review previous work regarding photoemission time delays in recent years,
particularly as it relates to attosecond experiments such as streaking. To this end, we will briefly
mention some of the key experimental observations of photoemission delays using attosecond techniques, and then continue with a discussion of past theoretical interpretations of the streaking
delay for one-photon ionization. We further discuss recent work related to the absorption delay inresonant two-photon ionization in the context of both streaking and RABBITT methods.
In Chapter 3, we review the analytical and numerical models used in this thesis to solve
the relevant time dependent Schrödinger equation (TDSE). We begin by formulating the approximations made in the semi-classical description of the interaction of attosecond laser pulses withmatter. We then review time-dependent perturbation theory and scattering phases which provides
the basis for the analytical models developed in the thesis. We conclude Chapter 3 by presenting
our numerical method for solving the TDSE, including techniques for obtaining initial field-freestates, propagation of the wavefunction in the time domain, absorbing boundary conditions, using

21
single active electron (SAE) potentials, and finally the extraction of the time delay from simulationsof attosecond streaking.
In Chapter 4, we develop an analytical formula to further analyze and predict time delays in
one-photon ionization. Our formula accurately reproduces, in a few seconds of computation time,
the results of high-level TDSE simulations. Furthermore, we show that our analysis provides new
insight into the separation of the streaking delay in the so-called sum rule. We then apply ourformula to study recently proposed theoretical interpretations and potential other effects on theattosecond streaking measurements. For example, we show that the effect of the attochirp can besignificant in a frequency regime close to the Cooper minimum in the photoionization cross sectionof argon atom.
Chapter 5 includes our studies regarding the absorption delay in resonant two-photon ionization, and in particular the difference between the case of an isolated resonance and overlapping
resonances. We first develop an analytical formulation of the absorption delay in the case of an
isolated resonance using time-dependent perturbation theory. The results obtained with this ansatz
demonstrate the dependence of the absorption delay upon both the duration and the central frequency of the ionizing attosecond pulse. We then extend our studies to the case when multiple
resonances are excited within the bandwidth of the attosecond XUV pulse, and discuss the potential influence of the streaking field on the absorption time delay in model systems as well as simple
atoms and molecules.
In Chapter 6, we propose the use of isolated elliptically polarized XUV attosecond pulses tofurther study dynamics in photoemission. As a first step, we show that the angular distribution of
the emitted photoelectron can be controlled with the ellipticity of the pulse. We then explore thepotential application of isolated elliptically polarized attosecond pulses to so-called “ring-current”
states. We then end with a summary of the work over these last six years.

Chapter 2

Delays in Photoionization and Previous Theoretical Analysis

As mentioned in the previous Chapter, the measurement of photoemission time delays canprovide temporal information on the attosecond time scale. The concept of a time delay in quantum
mechanics was first introduced in scattering theory, and dates back to the PhD work of Eisenbudin the late 1940s [68]. It was further developed by Wigner and Smith in the next decade [69, 70].
Given the assumption of a central potential, Eisenbud, Wigner, and Smith derived a group delay
for the partial wave with angular momentum l of a scattered particle, evaluated asymptoticallygiven that the potential decays faster than 1/r as:∆tscatWS (E, l) = tp (R) − t0 (R)|R→∞ =

∂φl (E),∂E R→∞(2.1)

where φl (E) is the scattering phase for the lth partial wave at electron kinetic energy E, and WS
refers to “Wigner-Smith”. The time delay is most simply defined as the difference of the time tp ,that the particle spends crossing the region R with the potential, and the time t0 that a free particle
would spend crossing the same region (see Fig. 2.1).
For the scattering off an ionic potential containing a short range part and a Coulomb term
−Z/r for nuclear charge Z, defining the time delay in this manner becomes problematic. It is wellknown [71] that the delay in this case may be decomposed as∆tscatWS (E, l) = ∆tshort (E, l) + ∆tphase (E, l) + ∆tlog (E)(2.2)

where the first term ∆tshort is the contribution from the short range part of the atomic potential,
while ∆tphase and ∆tlog are the well known “Coulomb phase” and “logarithmic“ terms, respectively.

23
Note that the logarithmic term is independent of l, as the effect of the long-range part of the
Coulomb potential on scattering is independent of angular momentum. These last two terms result
from scattering off the −Z/r part of the ionic potential for nuclear charge Z. However, whereas∆tphase converges as R → ∞, the term ∆tlog diverges. Indeed, this term is responsible for thewell known “logarithmic divergence” of the time delay in the case of a Coulomb potential [71, 72].
Therefore, one typically refers to the first two terms alone when discussing the “Wigner Smith”(WS) delay in the context of scattering off an ionic core with a long range Coulomb potential [22,73]:∆tWS (E, l) = ∆tshort (E, l) + ∆tphase (E, l) .

(2.3)

Since photoionization can be modeled as a “half-scattering” process, the discussion of aphotoemission delay in Ref. [17] reignited theoretical interest in the relevance of the WS delayin this context (e.g. [29, 74–79]). Specifically, the time delays measured in both streaking and
RABBITT experiments (c.f., section 1.3.1) have been attributed to the sum of the WS delay andan additional term. Yet, the interpretation of the measured time delay and the extraction of the
WS delay in attosecond experiments have been a source of much controversy over the past sevenyears, particularly due to the disagreement of results of numerical simulations with data of thestreaking measurement of Ref. [17]. Furthermore, as will be discussed in section 2.3, extension of
the original attosecond time delay concept, as for example in Refs. [20] and [80], showed that thetime delay measured in attosecond streaking and RABBITT respectively, includes an additional
contribution, called the absorption delay, in the case of resonant two-photon ionization processes(as compared to non-resonant ionization). In this Chapter, we will review some of the experimental
results important to photoemission delays in section 2.1. Then, we will review past theoretical workwith respect to the different contributions to the streaking delay in sections 2.2 and 2.3.

2.1

Experimental observations
The attosecond streak camera, an established experiment for pulse characterization in at-

tosecond science [53, 61–63], was used by Schultze et al. [17] to demonstrate the possible use of

24

Figure 2.1: Schematic for the WS time delay in scattering, taken from Ref. [16]. The time delay∆tWS is defined as ∆tWS = tp (R) − t0 (R)|R→∞ [16]. The time tp (R) is the duration a particle (orwavepacket) spends in a region of space R within the potential (top), and t0 (R) is the durationthat the same particle or wavepacket spends in the same region R, but without the presence of thepotential, i.e., free particle (bottom), as the region R extends to infinity.

25
this technique to study temporal dynamics of electrons during XUV photoionization of atoms. The
fundamental question posed was whether or not there was a time delay between the photoemissionof electrons from different orbitals, specifically in this case of the 2s and 2p orbitals in a neon atom.
In the experiment, two streaking traces (c.f., Fig. 2.2) are recorded simultaneously for emission
from the 2s and 2p orbitals in neon atom. The 200 attosecond pulse used had a central frequencyof 100 eV, which is sufficient to ionize an electron from either one of the orbitals. A 4 fs IR pulse
is used to “streak” the momentum as described in the previous Chapter.
In comparing the two streaking traces for the ionization of 2s and 2p electrons, a time delay
between the two events can be extracted using the frequency resolved optical gating for complete
reconstruction of attosecond bursts (FROG-CRAB) algorithm [17]. In the study of Schultze etal., a 21 ± 5 as offset between the two streaking traces was observed (c.f., Fig. 2.2), which has
been interpreted as a time delay in the photoemission of the 2p electron with respect to thatof the 2s electron. While the work of Schultze et al. has not been repeated directly by anotherattosecond streaking experiment, the 2p−2s photoemission time delay in Neon was recently revisited
experimentally by Isinger et al., but using the RABBITT technique with an attosecond pulse
train [18].
The experiment of Isinger et al., and specifically the interpretation of their results, depends
upon being able to achieve sufficient spectral resolution to distinguish between the signals of two
processes separated by 7.2 eV, which is on the order of the bandwidth of a typical attosecond
pulse (the bandwidth of the attosecond pulse used in Ref. [17] was ∼ 14 eV). In order to achieve
such standards in resolution, they invoke a method used by Ye and co-workers in Ref. [81]. Thismethod has been demonstrated to circumvent the natural trade-off between temporal and spectral
resolution in the visible spectrum by using high-resolution frequency combs based on phase-stablefemtosecond pulse trains [81]. In Ref. [18], Isinger et al. extended this method to an XUV attosecond
pulse train, and they achieved a train of attosecond pulses with harmonic comb teeth of only 1
eV in bandwidth. They used this pulse train in a RABBITT setup (c.f., section 1.3.1.2), and indoing so suggest that the experiment of Schultze et al. did not have sufficient spectral resolution

26

Figure 2.2: The raw attosecond streaking spectrogram data for neon atom from the experiment of
Schultze et al. shows that the photoelectrons emitted from the 2p state (higher energy/top signal)
are delayed with respect to those of the 2s state (lower energy/bottom signal) as a function of delay
between the attosecond XUV and streaking IR pulses. (Captured from [17]).

27
to disentangle the signal from direct 2s ionization and that of the so-called “shake-up” process inwhich a 2p electron is ionized while a second electron is excited to the 3p state (c.f., Fig. 2.3 (a)),(2p,2s)and that therefore the time delay ∆ts,expt obtained in Ref. [17] was distorted by the overlap ofthese two signals. Isinger et al. show that their resolution is sufficient that they can distinguishbetween the two sideband oscillation amplitudes, as shown in Fig. 2.3 (b), such that they were alsoable to obtain the phase (c.f., Fig. 2.3 (c)), and therefore the RABBITT time delays corresponding
to these processes, as shown in Fig. 2.4. The authors of Ref. [18] state that the shake-up harmonic
oscillates out of phase with the sideband, which causes a sudden drop in the energy-resolved phase.
Since the phases due to direct and shake-up contribution can be distinguished in their RABBITT(2p,2s)(2p,3p )signal, the time delays ∆tR,expt and ∆tR,exptsu , where 3psu refers to the signal resulting from the
shake up process, may be obtained. Once being able to eliminate the possibility of contributions(2p,2s)from the shake-up pathway, the obtained ∆tR,expt achieved agreement with theoretical calculations
implementing many-body perturbation theory [18, 19, 64], as shown in Fig. 2.4.

2.2

Theoretical analysis and interpretation of photoemission time delays
As mentioned in the introduction, in theoretical analysis typically the time delay for emission

from one particular orbital is considered, which in the streaking technique is defined as the offset ofthe streaking curve for emission from one orbital as compared to the streaking of a free electron (or,equivalently, the vector potential, see section 1.3.1.1). This time delay has been widely attributed
to the sum of two terms [22, 24, 25, 73], which we refer to as the “sum rule”. The first term is the
Wigner-Smith delay (2.3), and the second term is due to the coupling of the long-range tail of the
Coulomb potential of the residual ion and the streaking laser field upon the propagation of theelectron. In this section we review the past work done in justifying this rule.

2.2.1

The sum rule

As a first step, Schultze et al. [17] proposed to apply Eq. (2.1) in order to interpret themeasured time delay, despite its divergence for Coulomb potentials, which raised concerns from

28

Delay (as)Figure 2.3: (a) Kinetic energy diagram as presented in the work of Isinger et al. for direct 2s
ionization and 2p ionization accompanied by 2p → 3p excitation (shake-up). (b) Photoelectron
spectra obtained with XUV only (blue) and XUV + IR (red). (c) Energy-resolved intensity andphase of the two photon oscillation, obtained by Fourier transform of the signal. The symbol H,
e.g. H61, corresponds to the direct signal from a high harmonic, where as S, e.g. S56, refers to a
sideband signal (Taken from Ref. [18]).

0
-10
-20
-30
-40
-50
-60
-70
60

70

80

90

100

110

Photon Energy (eV)2p
Figure 2.4: Time delay differences ∆t2s
R − ∆tR in neon from the experiment in Ref. [18] as afunction of photon energy. The yellow and red dots were obtained via two different measurements.
Theoretical calculations within many-body perturbation theory [19] agree very well with the ex3psu
perimental data (solid line). The obtained time delay differences ∆tR− ∆t2pR between shake-up
and 2p ionization (diamonds), as well as the time delay obtained in the experiment from Ref. [17](square), are also shown (Taken from Ref. [18]).

29
some theorists [73, 82–84]. Nagele et al. [73] and Zhang et al. [84] discovered independently thatthe non-divergent WS delay defined by Eq. (2.3) does not account for the streaking delay obtainedin calculations alone, unless the potential is very short ranged. In the case of Ref. [73], Nagele etal. calculated the streaking trace classically, and they confirmed that indeed the calculated traceshows an offset from the original streaking formula for a free particle, i.e. kf (ti ) ' k0 − As (ti ).
Nonetheless, this time offset, or time delay, in general did not agree with the WS time delay. It wasthen concluded that there is an additional contribution to the streaking delay due to the interplayof the long-range tail of the Coulomb potential and the streaking laser, and we refer to this term
throughout the thesis as the “Coulomb-Laser Coupling” ∆tCLC .
Inspired by this result, Ivanov et al. [85] derived an analytical formula for ∆tCLC by using an
eikonal approximation-based quantum approach. Within the eikonal approximation, they predictedthat the short-range part of the potential also coupled to the streaking field. Therefore, it was
proposed that ∆tCLC did not only arise from the long-range Coulomb potential, but the short-range
part as well, and this short-range contribution was on the order of ∆tWS . They then proceeded touse their formula for ∆tCLC to calculate ∆ts = ∆tWS +∆tCLC for both the 2p and 2s photoemissionin neon atom, using calculations of ∆tWS from Ref. [86], and obtained a time delay of 18 as, which iswithin the uncertainty of the measured streaking delay of Schultze et al. 21±5 as. However, Nagele
et al. pointed out that the prediction of a short-range contribution to ∆tCLC must be incorrect,
because for short-range potentials the streaking delay is found to be equal to the WS delay [24].
Therefore, it was concluded that the coupling delay ∆tCLC has no relation to the short-range partof the atomic potential, and arises solely from the coupling effect of the streaking field with thelong-range tail of the Coulomb potential. In Ref. [24], it was further concluded that because ∆tCLC
only depends on the long-range Coulomb tail of the potential, it must be the same for all singleelectron systems with the same parent ion charge. The streaking delay written as the well known
sum rule:∆ts = ∆tWS + ∆tCLC

(2.4)

30
can therefore be obtained for all systems with the same asymptotic behavior as the hydrogen atom,
by calculating ∆tWS via Eq.(2.3) and calculating ∆tCLC = ∆ts − ∆tWS for the hydrogen atom. It is
important to note that the sum rule is supported by numerical streaking simulations and analysis oflaser assisted photoionization using perturbation theory (see review [64]). To this end, in Chapter
4, we present an alternative way to justify the sum rule.
In the time since, many publications in the literature have made use of Eq. (2.4) (e.g.,[17, 18, 22, 24–26, 29, 64, 73, 85–91]), using the ∆tCLC calculations obtained by Nagele et al. forhydrogen atom [24] in the manner described above and using Eq. (2.3) for ∆tWS . It has beenused to theoretically analyze additional effects such as multielectron effects [25, 29], the effect of an
attochirp [91] and the possibility of a multielectron “shake-up” process [25], however no agreementwith the original experiment of [17] was obtained. As we mentioned in section 2.1 though, using
the RABBITT technique and the contribution of a shake-up process may explain the disagreementwith the original measurement [18].

2.2.2Classical streaking and the delay as a field-weighted sum

Motivated by studies showing that the streaking delay calculated by TDSE simulations wasin good agreement with classical simulations [73], Su and coworkers further analyzed the solutionof the Newton equation for the propagation of the photoelectron in the streaking field [92, 93]:dk (t)= −Es (t) − ∇V (r) ,
dt

(2.5)

where k (t) is the momentum of the photoelectron, Es (t) is the streaking field, and V (r) is thepotential of the residual ion. Considering a linearly polarized field, the simultaneous interactionof the electron with the laser and Coulomb fields takes effect along the direction of the laser
polarization, typically chosen as the z−axis:dkz (t)dV= −Es (t) −
.
dt
dz

(2.6)

31
Su et al. then integrated this equation directly to solve for the momentum kz as z → ∞ (denotedkf,z ), and set it equal to the z−component of the streaking equation Eq. (1.6):sZTskf,z (ti ) =

2 −2
k0,z

Es (t)kz (t)dt = k0,z − As (ti + ∆ts )

(2.7)

ti

where kz (t) in the integrand is the z−component of the momentum of the electron in the combined
streaking field and Coulomb potential, k0,z is the streaking-field-free asymptotic momentum, andti is the time of transition into the continuum, which often is assumed to coincide with the peak of
the NIR pulse. Note that the top bound of the integral on the left hand side of the equation is set
to the duration of the streaking pulse Ts as opposed to ∞, because Es (t) = 0 for t > Ts , given thatthe pulse starts at t = 0. Su et al. proceeded by making the approximation that one can expand
As (ti + ∆ts ) ' As (ti ) + ∆ts Es (ti ) for small ∆ts on the right hand side of Eq. (2.7). In addition,
they reasoned that the second term under the square root on the left hand side is typically small,and therefore expanded the square root to first order. After these two approximations, they solve
for the streaking delay as
1∆ts 'Es (ti )

Z Ts
ti



kz (t)
Es (t) 1 −
dt.
k0,z

(2.8)

Su et al. then made this equation more instructive by further rewriting this equation as a discretesum via assuming that the streaking field and electron momentum are approximately constant in
the time interval [tj , tj + δt], i.e. Es (t) ' Es (tj ) and kz (t) ' kz (tj ), to get:
∆ts '



N
kz (tj )1 X
Es (tj ) 1 −δt
Es (ti )
k0

(2.9)

j=1

=

N
X
Es (tj )j=1

Es (ti )(j)∆tfield-free .

(2.10)(j)In this equation, ∆tfield-free is the piecewise field-free time delay that the electron accumulates
during its propagation in the time interval [tj , tj + δt] and over the related finite region [zj , zj + δz]of the potential V (r) without the laser field as compared to the propagation of a free particle overthe same distance in space. This formula provided an interesting interpretation of the streaking
delay which we will refer to throughout this thesis. That is, the streaking delay can be viewed as32a sum of piecewise field-free time delays weighted by the streaking field strength present as theelectron wavepacket propagates over the corresponding part of the potential. In Chapter 4, we willshow, by extending this interpretation to a quantum mechanical picture, under which conditions
this formula can be further approximated by the sum rule.
A critical note is that per our discussions in sections 2.2.1 and 2.2.2, we see that it has been
determined in two different formalisms that, for one photon ionization, the streaking delay is solely
accumulated during propagation of the photoelectron in the continuum [20]. For this reason werefer to the streaking delay ∆ts in the context of one-photon ionization as the “continuum delay”
∆tc .

2.3

The absorption delay in two-photon ionization
In the previous section, we discussed the status of the theoretical analysis for the attosecond

streaking of one-photon ionization. In this section we discuss past work going beyond the onephoton absorption, where in some of these scenarios it has been shown that neither the sum rule
nor Eq. (2.8) is sufficient to account for the streaking time delay in theoretical calculations. For
example in resonant two-photon ionization, once accounting for both the WS and CLC time delays,an additional delay, which is dependent upon the properties of the ionizing pulse, can be observedin the attosecond streaking of resonant two-photon processes. This was discovered by Su andcoworkers in their comparison of streaking delays in one- and two-photon non-resonant ionization
vs. resonant two-photon ionization [20], and independently by Martı́n and coworkers [80] whenstudying RABBITT (c.f., 1.3.1.2) through a Fano resonance embedded in the continuum.
In the work of Ref. [20] it was shown that the streaking delay for resonant two-photon
ionization could be expressed as∆ts = ∆tabs + ∆tc ,(2.11)

where ∆tc is the “continuum delay” discussed at the conclusion of the last section. The additional
absorption time delay ∆tabs was found to be zero for one-photon as well as non-resonant two-33
photon transitions. Additionally as shown in Fig. 2.5, the streaking delay resulting from resonant
two-photon ionization shows a strong, linear dependence upon the pulse duration of the ionizing
pulse, whereas the same dependence is not seen for the streaking delays corresponding to onephoton and non-resonant two-photon ionization. In order to obtain the absorption time delay, Su
et al. show that it is possible to obtain the contribution using the continuum delay ∆tc in two
ways. One can either calculate ∆tc via a classical analysis, or using the streaking delay for the
one-photon process of the same final kinetic energy as the two-photon process. The latter can beused because by definition, the streaking delay and continuum delay are equivalent (∆ts = ∆tc )for one-photon ionization. The absorption delay ∆tabs is then calculated via an equivalent form ofEq. (2.11):∆tabs = ∆t(2ω)− ∆tc ,s(2ω)

where we use ∆ts(2.12)

to denote the streaking delay for the two-photon process.

Figure 2.5: (a) Streaking time delay simulations from Su et al. [20] as a function of XUV pulse
duration for 3D helium atom. Three ionization processes are considered: non-resonant two-photon
ionization (TPI) (blue lines with circles), resonant TPI (green lines with squares), and single photon
ionization (SPI) (red lines with asterisks). (Taken from Ref. [20]).

Around the same time, Martı́n and coworkers [80] found via TDSE simulations that in a
RABBITT setup as a harmonic traverses a resonance, both the phase shift and frequency of the

34

Figure 2.6: Angular-resolved time delays from Ref. [21]. The measured atomic time delay (redsymbols) is shown as a function of electron emission angle for SB16. The green lines show thecalculated values in resonant (solid) and non-resonant (dashed) conditions. The error bars indicate
the standard deviation of the measurements. (Taken from Ref. [21])35
sideband beating significantly varies with photon energy. They additionally showed that the sideband signals persist even when the pump and probe pulses do not overlap, which is contrary to
the findings of non-resonant RABBITT. Further studies were done, resulting in the conclusion that
there must be an additional delay when traversing through a resonance. Argenti along with Martı́n
and others [67] parameterized the RABBITT delay for resonant two-photon transitions as the sumof the time delay from the non-resonant process, plus an additional delay resulting from the resonance. Thus, they independently deduced from their results a similar conclusion as Su et al. forthe resonant two-photon ionization process. A year later, Cirelli et al. were able to measure theangle-resolved RABBITT time delay of argon experimentally in the vicinity of the 3s−1 np autoionizing resonances, and found that the presence of the resonance affected the angular dependence of
the RABBITT delay [21]. Using a theoretical model based on the work of [67], the authors wereable to confirm this interpretation, and therefore the existence of the additional absorption delay
predicted by Argenti et al. This is demonstrated in Fig. 2.6, where the time delays predicted by
the model both with and without inclusion of the resonance are compared as a function of angle.
It is clear in the figure that the experimental data are in better agreement with the theoreticalprediction when including the resonant term. This additional delay in RABBITT measurements is
not the focus of this thesis, however it is analogous to the absorption delay in attosecond streaking
discussed above.

Chapter 3

Methods for solving the time-dependent Schrödinger equation

Within this thesis, we study the interaction of ultrashort laser pulses with either one-electron
systems like hydrogen atom and H2+ , or multielectron systems approximated by single active
electron (SAE) potentials (c.f., section 3.1.4). Therefore, we need to solve the time-dependent
Schrödinger equation (TDSE) for a single-electron wavefunction Ψ (r, t). The TDSE is expressedasi∂Ψ (r, t) 
= Ĥ0 (r) + V̂I (r, t) Ψ (r, t) ,
∂t

(3.1)

where r is the position of the electron, Ĥ0 is the field-free (time-independent) Hamiltonian of thesystem, V̂I describes the interaction of the system with the laser field, and the total Hamiltonianis therefore given by Ĥ (r, t) = Ĥ0 (r) + V̂I (r, t). The goal is to determine Ψ (r, t) throughout thetime of the simulation, given an initial condition. There exist few examples where Eq. (3.1) may besolved accurately via analytical methods. It is common to, when appropriate, use approximations
like perturbation theory. These approximations can simplify the problem sufficiently in order togain insight via analytical expressions. For the interaction with an intense laser pulse, when theexact solution is needed, we must turn to numerical solvers.
In this Chapter, we review the analytical and numerical techniques relevant to calculating
observables in the interaction of a system with an attosecond pulse and/or a femtosecond (midinfrared) laser pulse. In the first section 3.1, we discuss the analytical formulation of both field
and matter, as well as their interaction within both a classical and semi-classical framework. We
also show specific methods for analytically modeling the dynamics of the interaction, including

37
time-dependent perturbation theory, single active electron potentials, the partial-wave expansionof scattering theory, and the classical Newton equation describing the motion of the electron in anelectromagnetic field. In section 3.2, we describe our methods for solving Eq. (3.1) numerically on a
finite grid. Lastly, we conclude with a discussion regarding how to obtain time delays in numericalsimulations of attosecond streaking in section 3.3.

3.1

Semi-classical formulation of the interaction between field and matterIn order to write down V̂I in Eq. (3.1), we need to describe the electromagnetic field interactingwith the atom or molecule of interest. In general throughout this thesis, we will assume that thefield can be described classically by an electric field E (r, t) and a magnetic field B (r, t), which
satisfy Maxwell’s equations (without sources):∇ · E = 0,(3.2)

∇ · B = 0,
∂B
,
∂t
1 ∂E
,
c2 ∂t

∇×E = −
∇×B =where c is the speed of light in vacuum. We can generate the fields E and B via the use of scalarand vector potentials φ (r, t) and A (r, t) respectively by
E = ∇φ −

1 ∂A
c ∂t

B = ∇ × A.

(3.3)
(3.4)

Since only the fields E and B have a unique physical significance as opposed to the potentials,
Maxwell’s equations (3.2) must be gauge invariant with respect to the scalar and vector potentials(for arbitrary real, differentiable function f (r, t)):A → A0 = A + ∇f,

(3.5)

∂f
.
∂t

(3.6)

φ → φ0 = φ −38
As a general statement, in this thesis we use the radiation, or Coulomb gauge, which is defned by
the condition∇ · A = 0.

(3.7)

This gauge is typically chosen when no sources are present, implying that φ = 0 and
1 ∂A
c ∂t

(3.8)

B = ∇ × A.

(3.9)

E = −

For a finite laser pulse, the solution of Eq. (3.2) is given by a superposition of monochromatic planewaves with the appropriate amplitudes, frequencies and phases. In the case where each plane wave
component propagates in the same direction k̂L in a field with linear polarization direction ˆ, thevector potential in the Coulomb gauge can be written as [94]Z ∞A0 (ω) sin (kL · r − ωt − φCEP (ω)) dω,A (r, t) = ˆ

(3.10)

0where φCEP (ω) is the carrier envelope phase associated with the angular frequency ω and propagation vector kL = (ω/c) k̂L . The corresponding electric field is thenZ ∞E0 (ω) cos (kL · r − ωt − φCEP (ω)) dω,E (r, t) = ˆ

(3.11)

0with E0 (ω) = ωc A0 (ω). In general, these fields may be written as the product of a carrier wave
and an envelope function E0 (t, kL , r, T ) with pulse duration T = 2πN
ω for an N -cycle pulse along apolarization direction ˆ, such that
E (r, t) = ˆE0 E0 (t, kL , r, T ) cos (ωt − kL · r + φCEP ) .

(3.12)

It follows that a laser pulse of arbitrary polarization with ellipticity parameter ξ can be readily
described in a similar manner. For two orthogonal unit basis vectors ˆ1 and ˆ2 (each mutuallyorthogonal to k̂L ), the electric field is now [94]E (r, t) =E0 E0 (t, kL , r, T )p[ˆ1 cos (ωt − kL · r + φCEP (ω))
1 + ξ2−ξˆ2 sin (ωt − kL · r + φCEP (ω))] .

(3.13)

39
Note that linearly polarized fields can be recovered from Eq. (3.13) via setting ellipticity to zero.
A further simplification can be made, as the dependence of the field upon the spatial coordinate r may be neglected by invoking the so-called dipole approximation, which states that because
the size of the atom (on the order of 1Å = 10−10 m) is much smaller than the wavelength of theradiation, we may assume that the electron experiences approximately the same field in all space
relevant for the interaction. This approximation consists of neglecting the “retardation effect”
due to the wave vector k [95]. Thus, we take only the first term in the expansion of the spatial
modulation factor
eik·r ' 1 + k · r + ...,

(3.14)

namely unity. Therefore the field in Eq. (3.13) becomes
E0 E0 (t, T )E (t) = p[ˆ
1 cos (ωt + φCEP ) − ξˆ
2 sin (ωt + φCEP )] , .
1 + ξ2(3.15)All of the electric fields used in this thesis are of this form.

3.1.1

Classical electron dynamics in a laser fieldIn various scenarios, especially those where an electron is in the continuum, it has been shownthat certain aspects of its dynamics can be described classically. We therefore begin by writing
down the classical Hamiltonian for an electron in a radiation field without the influence of any
other potentials:
1
H (t) =22
1
p − A (t)c(3.16)

If we compare this Hamiltonian to that of the motion of a free electron, we see that the momentum
p has been replaced with p − A/c. This can be justified through the use of the Hamilton equations,which leads to the Lorentz equation governing the classical, non-relativistic dynamics of a freeelectron (of charge q = −e = −1) in a laser field as
dp= − [E (r, t) + p × B (r, t)] .
dt(3.17)In photoemission however, the electron is leaving an atomic potential V (r), and we must include
this potential in the Newton equation to fully describe the classical dynamics of a photoelectron

40
propagating away from a Coulomb potential. Furthermore, the magnitudes of the magnetic andelectric fields are related by the relation |B| = |E| /c, and for the intensities of the electric fields
used in this thesis, this relation implies that |B| << |E|. Therefore, we can neglect the contributionof the magnetic field to the force, and one has
dp= −E (r, t) − ∇V (r) .
dt(3.18)

It is common to gain insights in the interpretation of streaking measurements, and to complement
quantum simulations, by using classical analysis of the electron dynamics in the continuum basedon Eq. (3.18). For streaking simulations in particular, we may sample initial conditions for thesolution of Eq. (3.18) in Monte Carlo calculations, which will be described further in Chapter 5.

3.1.2

Semi-classical dynamics

In order to describe the dynamics quantum mechanically, we perform the usual prescription
for converting classical quantities to quantum operators, and obtain the semiclassical Schrödinger
equation:∂Ψ (r, t)
i
∂t

#
2
−i∇ − 1c A=
+ V (r, t) Ψ (r, t)2ih≡ Ĥ0 + V̂I Ψ (r, t) ,"(3.19)

which identifies the semiclassical electron-field interaction HamiltonianV̂I (r, t) =1
1
[2iA (r, t) · ∇ + i∇ · A (r, t)] + 2 A2 (r, t) ,c
2c(3.20)and the field-free part of the Hamiltonian isĤ0 = −

∇2
+ V (r) .
2(3.21)In Eq. (3.20), the i∇ · A term vanishes by definition of the Coulomb gauge, and therefore Eq.
(3.19) becomes∂Ψ (r, t)
∇2
i
1 2
i
= −
+ A (t) · ∇ + 2 A (t) + V (r) Ψ (r, t)
∂t
2
c
2c

(3.22)

41
which is known as the “velocity gauge” form of the TDSE. Under dipole approximation, we may
make the gauge transformation
Ψ (r, t) = e(i/c)r·A(t) ψ (r, t) ,(3.23)

and upon substitution into the TDSE (3.22) and using the definition of the electric field (E (t) =− 1c ∂A(t)∂t ), we get the so-called “length gauge” form of the TDSE:∂ψ (r, t)∇2i= −+ V (r) − r · E (t) ψ (r, t) .
∂t
2

(3.24)

Gauge invariance, by definition, implies that using either gauge in our calculations should givethe same results. However, in numerical simulations, the use of either gauge comes with a set ofadvantages and disadvantages, and so the choice of gauge can become important, depending on the
problem being solved numerically. This will be further discussed in the following section, however
for the moment, we choose the length gauge without loss of generality.
The solutions to Eq. (3.1) are, in general, not guaranteed to have analytical forms, and must
be determined numerically. However, under certain approximations and in some specific cases, theequation can be resolved analytically. One such approximation method is the so-called perturbation
theory, which becomes applicable when the field is sufficiently weak as compared to the binding
Coulomb potential.

3.1.3
3.1.3.1

Time-dependent perturbation theory
Time-evolution operator

In general, it is assumed that we know the state of our system at a given time t0 , and wish
to determine its state at a later time t. Therefore, the problem is determining the unitary timeevolution operator U (t, t0 ). As its name implies, this operator describes the evolution of the stateof the system in time in the Schrödinger representation. The evolution operator is completely
determined by the Hamiltonian Ĥ (t), and is defined by the integral equation
Z t
Û (t, t0 ) = 1̂ − i

Ĥ (t1 ) Û (t1 , t0 ) dt1
t0(3.25)

42
and the initial condition Û (t0 , t0 ) = 1. Due to the fact that Ĥ (t) is Hermitian, Û (t, t0 ) is unitary,and thereforeÛ † (t, t1 ) = Û (t1 , t) .

(3.26)

From this we get the equivalent definition of
Z t
Û (t, t0 ) = 1̂ − iÛ (t, t1 ) Ĥ (t1 ) dt1 .

(3.27)

t0

having replaced the evolution operator in Eq. (3.25) by its Hermitian conjugate and used Eq. (3.26).
Let us define the time-evolution operator for the field-free problem as Û0 (t, t0 ), which corresponds to Ĥ0 via a Schrödinger-like equation:i∂ Û0 (t, t0 )= Ĥ0 Û0 (t, t0 ) ,∂tÛ0 (t0 , t0 ) = 1.

(3.28)

Assuming that Û0 (t, t0 ) is known, we may determine Û if we can form the unitary operatorÛI (t, t0 ) = Û0† (t, t0 ) Û (t, t0 ) ,(3.29)

which is the evolution operator in the interaction representation. Furthermore, it can be shown [72]that the time dependence of ÛI is determined by the Hamiltonian
V̂I0 (t) ≡ Û0† (t, t0 ) V̂I (t) Û0 (t, t0 ) .

(3.30)

Equivalently, ÛI (t, t0 ) is defined by
Z t
ÛI (t, t0 ) = 1̂ − i

V̂I0 (t1 ) ÛI (t1 , t0 ) dt1 .

(3.31)

t0

The solutions to the integral equations (3.31) and (3.25) may be obtained formally via iteration,
Rti.e. substituting 1 − i t02 V̂I0 (t2 ) ÛI (t2 , t0 ) dt2 for ÛI (t1 , t0 ) in the right hand side of Eq. (3.31), weget
Z tÛI (t, t0 ) = 1̂ − i

V̂I0 (t1 ) dt1 + (−i)2t0

Z t Z t1
t0

V̂I0 (t1 ) V̂I0 (t2 ) Û (t2 , t0 ) dt2 dt1 .

(3.32)

t0

Continuing in this manner, defining(n)ÛI ≡ (−i)nZt>tn >tn−1 >...>t1 >t0

V̂I0 (tn ) V̂I0 (tn−1 ) ...V̂I0 (t1 ) dtn dtn−1 ... dt1 ,(3.33)

43
and using Eqs. (3.29) and (3.30), we get the following expansion for Û :Û (t, t0 ) = Û0 (t, t0 ) +∞XÛn (t, t0 ) ,(3.34)

n=1

where
Ûn (t, t0 ) ≡ (−i)nZ
Û0 (t, tn ) V̂I (tn ) Û0 (tn , tn−1 ) V̂I (tn−1 ) ...

(3.35)

t>tn >tn−1 >...>t1 >t0...Û0 (t2 , t1 ) V̂I (t1 ) Û0 (t1 , t0 ) dtn dtn−1 ... dt1 .
It is clear, since Eq. (3.34) is an infinte sum, that approximations are needed if we wish toproceed. In the case that the interaction term V̂I (t) (in our case, the interaction of the laser withthe system) is weak compared to the field-free Hamiltonian Ĥ0 (the atomic or molecular potential),
we may truncate the power series (3.34) to a given order.

3.1.3.2

Transition amplitudes and lowest-order perturbation theory

The method of the previous section is particularly applicable in cases where Ĥ0 is timeindependent, such as for a bound state of an atom or molecule. In addition, we assume that thesolutions for this Hamiltonian are known. The evolution operator Û0 is thenÛ0 (t, t1 ) = e−iĤ0 (t−t1 ) .

(3.36)At this point, we also introduce the notationωmn = Em − En

(3.37)

Vmn = hm |VI (t)| ni ,(3.38)

where states |mi and |ni are eigenstates of Ĥ0 with energies Em and En .
We assume that the system is in an eigenstate |ai of Ĥ0 at some initial time t0 . The goal is
to calculate the amplitudes whose square moduli are the time-dependent probabilities of measuring
the system in any bound state of Ĥ0 . Such an amplitude can be expressed asDE
cba = b Û (t, t0 ) a .

(3.39)

44
If we substitute the expansion Eq. (3.34) for the time evolution operator, we get
∞
∞ D
E
X
X
(n)
cba =
cba =
b Ûn a .
n=1

(3.40)

n=1

We can then write the successive contributions to the amplitude, e.g. the first order contribution
is
Z t

(1)
cba (t)

= −i
t0
Z t

= −i

e−iEb (t−t1 ) Vba (t1 ) e−iEa (t1 −t0 ) dt1(3.41)

eiωba t1 Vba (t1 ) dt1 ,t0

and the second-order contribution isZ
Z t Z t1
X
(2)
cba (t) = − m
Vbm (t2 ) eiωbm t2 Vma (t1 ) eiωma t1 dt1 dt2 ,
t0(3.42)

t0

where the sum with integral represents a sum over both discrete and continuous eigenstates of Ĥ0 .
Now we are prepared to consider the dynamics induced by a laser field, which is the caserelevant for this thesis. Let us consider a linearly polarized pulse with field strength E0 and arbitraryenvelope E0 (t, T ) of pulse durartion T , given by [94]E (t) = ˆE0 E0 (t, T ) cos (ωt + φCEP ) ,(3.43)and the corresponding interaction Hamiltonian is given in the length gauge byV̂I (t) = −iE0 E0 (t, T ) h i(ωt+φ)e+ e−i(ωt+φ) ˆ · d̂ ,2

(3.44)

where d = −r is the electric dipole moment operator for the system. At this point it is also useful
to introduce the transition dipole matrix element asµji = hj |ˆ · d| ii .

(3.45)We then get, to first order in the perturbation(1)

cf i (t) =iµf i E0
2

Z t
−∞



E0 (t1 , T ) ei(ωf i +ω)t1 eiφCEP + ei(ωf i −ω)t1 e−iφCEP dt1 ,(3.46)

45
and for nth order(n)cf i (t)


=

iE0
2

Z t

n X X
m1 m2

...

X

µf m1 µm1 m2 ...µmn−1 i(3.47)mn−1ondt1 E0 (t1 , T ) ei[(ω+ωf m1 )t1 +φCEP ] + e−i[(ω−ωf m1 )t1 +φCEP ]

×
−∞

Z t1

n
o
i((ω+ωm1 m2 )t2 +φCEP )−i[(ω−ωm1 m2 )t2 +φCEP ]dt2 E0 (t2 , T ) e
+e

×
−∞

Z tn−1
× ...

×

n
o
dtn E0 (tn , T ) ei[(ω+ωmn−1 i )tn +φCEP ] + e−i[(ω−ωmn−1 i )tn +φCEP ] .

−∞Note that the integral in Eq. (3.46) has two dominant contributions at ωf i + ω = 0 and ωf i − ω =0 (most easily seen in the case of a continuous wave field, i.e., E0 (t, T ) = 1 [94]), and these
correspond to stimulated emission and absorption of a photon, respectfully. Thus, for example,
one photon ionization may be modeled with first order time-dependent perturbation theory byneglecting the stimulated emission term and considering the first order transition of a bound state
to a continuum state. This process can be extended out to nth order to describe n-photon processes
in the perturbative regime (see Eq. (3.47)), and we will use it to model two-photon ionization in
Chapter 5.

3.1.4

Single Active Electron potentials

When studying a system with two or more electrons, the full numerical solution of the TDSE
becomes impractical. For processes in which the intense laser field predominantly interacts with
one electron only, it is useful to invoke single active electron (SAE) potentials which model the
behavior of the multielectron potential from the perspective of one of the electrons, thus reducing
the problem to a one-electron TDSE like Eq. (3.1). SAE potentials are commonly obtained throughthe use of density functional theory, and are often fit to analytical forms for ease of use and potential
physical interpretation [96]. The SAE potentials used in this thesis are those demonstrated in thework of Tong and Lin [97], which are expressed as
V (r) = −Zc + a1 e−a2 r + a3 re−a4 r + a5 e−a6 r.
r(3.48)

46
The first term in this potential, −Zc /r is the Coulomb tail seen at large r, the two Yukawa terms andthe exponential term describe the short-range interaction, and the ai ’s are obtained by fitting to thenumerical potential calculated from self-interaction free density functional theory (for details, see
Ref. [96]). We will see in Chapter 4 that such a form for the potential is convenient in determining
streaking time delays analytically.

3.1.5

The scattering phase and the WS time delay

The Wigner-Smith delay is most popularly displayed as the energy derivative of the asymptotic scattering phase for a given partial wave l (c.f., Eq. (2.1)). In the analytical method presented
in Chapter 4, we will need to obtain the scattering phase shifts in order to take their derivatives
directly. For this reason, we briefly review the origin of the scattering phase shifts. In general, ascattering process may be simply described as the interaction between an incident scatterer anda target. As the scatterer approaches the scattering center its wavefunction is modified, and theusual task is determining how and why it is modified by the particular potential of the target.
Photoionization can be considered as a half-scattering process [22], and thus observables in formal(full-)scattering theory become useful in describing the propagation of the electron as it leaves its
parent ion, the primary observable being the cross-section σ (Ω). The cross-section represents theflux of the outgoing scattered wavefunction as a function of the solid angle Ω, and as we will see,is closely related to the scattering phase.
First, we limit ourselves to the discussion of elastic scattering of electrons by short-range,
central potentials V (r) which approach zero asymptotically faster than 1/r, and we will thendiscuss the special case of the Coulomb potential afterwards. In a typical scattering process it isrequired that the observation method is made sufficiently far away such that the wave propagation
detected is unaffected by the presence of the scattering center. At such distances, the stationary
scattering state solution to the Schrödinger equation consists of three components: a plane wave ofdefinite linear momentum propagating towards the target, a wave that corresponds to transmission
through the target without scattering, and lastly a scattered wave [98]. The first term can be47
simply expressed asymptotically as a plane wave with definite linear momentum. This is also thesame asymptotic form of the second term, with the same momentum, as it is merely a continuationof the incoming wave. At large distances, the scattered wavefunction has the form of an outgoingspherical wave with an amplitude that varies with solid angle. Therefore, if E is the energy andk is the initial momentum of the particle, we can relate the cross section to the solution of thetime-independent Schrödinger equation
∇2−+ V (r) ψk (r) = Eψk (r) ,2(3.49)

whose asymptotic behavior at r → ∞ is of the formψk (r)|r→∞ ' eik·r + f (Ω)

eikrr(3.50)

where f (Ω) is the scattering amplitude as a function of solid angle Ω. The square modulus of thescattering amplitude gives the cross section for the process, and as we will show next, contains theinformation essential in determining the phase shifts [72].
We are mainly considering central potentials in this thesis, and therefore the direction of theincident wave vector k is an axis of rotational symmetry such that if it is taken as the polar axis,
the wavefunction and the scattering amplitude are not dependent upon the azimuthal angle φ. We
therefore decompose the solution in terms of partial waves by expanding in Legendre polynomials;
ψk (r, θ) =X yl (r)lf (Ω) = f (θ) =XrPl (cos θ)fl Pl (cos θ)(3.51)(3.52)lwhere yl is a solution of the radial equationl (l + 1)
d2+ 2 (E − V (r)) −yl (r) = 0
dr2
r2(3.53)and has an asymptotic form oflπyl |r→∞ ∼ al sin kr −+ φl .
2(3.54)

48
In Eq. (3.54), al is a normalization constant, and φl is the scattering phase for the l-th partial wave,
which is the same for all regular solutions of Eq. (3.53). It can be shown [72] that the scattering
amplitude can be represented in terms of the phase shifts as∞

1X
f (θ) =
(2l + 1) Pl (cos θ) eiφl sin φl ,k

(3.55)

l=0

and therefore the scattering cross-section σ (Ω) = |f (Ω)|2 may also be expressed in this manner.
The important point is that all of the information regarding the effect of the scattering potential
is contained in the phase shifts. We note that the only difference in a half-scattering process suchas ionization is that the phase shift obtained from a full-scattering process is divided by two.
Now, although associating time with a quantum operator faces fundamental conceptual difficulties [99], Eisenbud [68], Wigner [69], and Smith [70] pointed out that when restricted to thedomain of scattering states, an expectation value of the time operator can be formed using the∂, which for the case of scattering off a spherical potential [22]correspondence identity t → −i ∂Egives a group delay for each partial wave as∆tW S (E, l) = 2∂φl (E),∂E(3.56)and therefore in a half-scattering process, by dividing by two we end up with Eq. (2.1). This
expression then represents the time delay of the outgoing wavepacket relative to that of a freeparticle wavepacket traversing the same scattering region. It is implied from the use of asymptotic
solutions in the derivation that the scattering region extends to infinity, which justifies writing the
WS delay as Eq. (2.1).
In the above discussion, we have assumed that the binding potential decays to zero faster than
1/r, a condition which the Coulomb potential, which is of central importance to this thesis, does
not satisfy. The scattering problem may be adjusted accordingly by observing that the asymptotic
form of the wavefunction should now be a Coulomb wave [72] as opposed to a plane wave. For
the scattering of an electron by the Coulomb potential V (r) = −Z/r with nuclear charge Z, the49asymptotic form of the outgoing wavefunction is proportional tolπZψCoulomb |r→∞ ∼ sin kr −+ σl + ln (2kr)
2k(3.57)

Z
σl = arg Γ l + 1 − ik(3.58)

whereis the so-called Coulomb phase shift and the term Zk ln (2kr) is the well known logarithmic divergenceof the Coulomb potential, and is the reason that the scattering phase, and therefore the WS delay,
diverges for Coulomb scattering, as was discussed at the beginning of Chapter 2. We use thesephases and discuss the logarithmic divergence as it applies to attosecond streaking in further detailin Chapter 4.

3.2

Numerical Methods
In general, the above methods for solving Eq. (3.1) and its observables are not always

applicable, and so in such a case we need methods of obtaining the solutions numerically. For
example, our discussion of perturbation theory in section 3.1.3 depends on the condition that theinteraction of the laser with the electron is of significantly less strength than the interaction of theelectron with the binding Coulomb potential. Nevertheless, once the field is of sufficient strength, as
is common in strong-field laser physics, the two interactions are now of similar magnitude. In sucha case, perturbation theory can not be used; neither interaction can be considered as a perturbation
to the other.
In this section, we describe our methods for obtaining the solution of the relevant TDSE
numerically. For all cases, we discretize the TDSE on a spatial and temporal grid. Since thisgrid must be finite, it is possible that part of the wavefunction reaches the grid boundary during
propagation. Therefore we also describe our method of including effective absorbers along theboundaries of the grid in space, in order to suppress reflections at the boundaries. Such reflections
give non-physical effects in simulations, and therefore must be eliminated to a sufficient degree.

50
We conclude this section with a discussion of how to obtain time delays from streaking simulations
using the solutions of the corresponding TDSE.

3.2.1

Discretization and propagation of the TDSE

We are primarily interested in the interaction of matter when exposed to an ultrashort laser
pulse. That is, the operator V̂I (r, t) in Eq. (3.1) includes this time-dependent interaction. We describe Ĥ0 (r) using Eq. (3.21), which is time-independent and its eigenfunctions correspond to thefield-free bound states of the system. We mentioned in section 3.1.2 that the observables obtained
from the TDSE are gauge independent with respect to how we represent the interaction Hamiltonian. That is, the field-matter interaction term V̂I can be represented in different gauges, and it
was mentioned that there were advantages in using one over the other in numerical simulations,depending on the physics being studied. Despite the invariance of the solutions with respect to thegauge chosen, in practice the choice of gauge affects various aspects of the numerical calculations.
While the velocity gauge has been found to converge more rapidly for simulating the interaction of
atomic hydrogen with intense, low-frequency laser pulses [100], the length gauge has been shown tobe more accurate for calculating photomomentum spectra [101–104], such as is necessary for streaking simulations. We therefore use length gauge for our calculations, unless explicitly mentioned
otherwise.
A common method of solving second-order partial differential equations like Eq. (3.1) is
to discretize the wavefunction on a spatial-temporal grid with spatial step ∆xj along the j-thdimension (with number of points for M dimensions {Nx1 , Nx2 , ..., NxM }), and propagating thesolution forward in time by discrete intervals ∆t. In order to propagate the solution by ∆t, weneed to discretize the time-evolution operator discussed in section 3.1.3.
The goal is to obtain an accurate representation of the evolution operator over the interval[ti , ti+1 ] for i ∈ [0, N − 1] where N is the number of time intervals. We recall from the discussion
in section 3.1.3 that the time evolution operator is completely determined by the Hamiltonian.

51
Therefore, we begin by discretizing Eq. (3.25) as [94]:Z ti+1Û (ti+1 , ti ) = 1̂ − i

Ĥ (t1 ) Û (t1 , ti ) dt1 ,(3.59)

ti

and by iteration, we get the result asÛ (ti+1 , ti ) = 1̂ −∞X

Ûn (ti+1 , ti ) ,(3.60)

n=1

where
Ûn (ti+1 , ti ) = (−i)

nZ ti+1

Z t1
dt1

ti

Z tn−1
dt2 ...

tidtn Ĥ (t1 ) Ĥ (t2 ) ...Ĥ (tn ) .

(3.61)

ti

Now, given that the number of time intervals N is sufficiently large, we can make the approximationthat the Hamiltonian Ĥ (t) stays constant during each interval [ti , ti+1 ] and thusĤ (t) = Ĥ (ti ) for ti ≤ t ≤ ti+1 .
With this assumption, the integrals in (Eq. 3.61) are now simplified tohinĤ (ti ) ∆tÛn (ti+1 , ti ) = (−i)n.
n!

(3.62)(3.63)We are then able to formally sum Eq. (3.60) to yield our approximation for the evolution operatorasÛ (ti+1 , ti ) = e−iĤ(ti )∆t .

(3.64)

This means that for each small time step taken by the wavefunction, we have
Ψ (r, t + ∆t) ' e−iĤ(t)∆t Ψ (r, t) .

3.2.1.1(3.65)

Split-operator scheme

A further simplification to solving the TDSE in multiple dimensions can be made by decomposing the Hamiltonian along each of the NM spatial dimensions of the problem as (using thelength gauge as an example)
Ĥtot = Ĥx1 + Ĥx2 + ... + ĤxM ,(3.66)

52
where
Ĥx1
Ĥx2
ĤxM

∇2x1
V (r)
+ E · x1 +
,
2
M
∇2
V (r)
= − x2 + E · x 2 +
, ...
2
M
∇2x
V (r)
= − M + E · xM +2M
= −(3.67)(3.68)(3.69)

With this decomposition, we may rewrite Eq. (3.65) as
∆t

∆t

∆t

∆t

Ψ (r, t + ∆t) ' e−iĤx1 (t) 2 e−iĤx2 (t) 2 ... e−iĤxM (t)∆t ... e−iĤx2 (t) 2 e−iĤx1 (t) 2 ,(3.70)

using the Baker-Campbell Hausdorff formula, which can be shown to have accuracy O ∆t3 [105].
In general, we will not consider M > 3 in this thesis, however the strategy extends to any number ofdimensions and is not restricted by the coordinate geometry chosen. Using this method, known as
the split-operator scheme, the propagation of the wavefunction in time is now reduced to subsequent
propagation along a single dimension at a time. In the next section, we demonstrate the propagation
procedure, but using only a single dimension for clarity.

3.2.1.2

Crank-Nicolson Method

The Crank-Nicolson method [106] is a standard method for numerically obtaining the solutionof partial differential equations. It is a unitary, stable, and energy conserving method which can
describe the solution of the TDSE to second order accuracy in space and time. For small time steps∆t, the solution of the Schrödinger equation can be approximated as−iH(t)∆tΨ(t + ∆t) ≈ e

Ψ(t) ≈

1 − i ∆t
2 H
1 + i ∆t
2 H

Ψ(t).

(3.71)

where the Cayley unitary transformation [107] is used on the right hand side, which preserves thenorm of the wavefunction. It then follows that





∆t
∆t
1 + i H Ψ(t + ∆t) = 1 − i H Ψ(t).
22(3.72)

53
Now, the Hamiltonian Ĥ (t) can be decomposed into different coordinates, where each component
is a one-dimensional (1D) Hamiltonian given by
H = A(x, y, t)∂2∂+ B(x, y, t)+ V (x, y, t),2∂x∂x(3.73)

where x stands for the coordinate of interest and y represents the remaining spatial coordinates.
A (x, y) and B (x, y) are functions which are chosen to be appropriate for the geometry of thesystem. Then using the second-order discretization scheme, Eq. (3.72) reveals
Ψn+1 (t + ∆t) − 2Ψn (t + ∆t) + Ψn−1 (t + ∆t)∆t
An (y)
Ψn (t + ∆t) + i
2
(∆x)2

Ψn+1 (t + ∆t) − Ψn−1 (t + ∆t)+Bn (y)+ Vn (y)Ψn (t + ∆t)2∆x∆tΨn+1 (t) − 2Ψn (t) + Ψn−1 (t)= Ψn (t) − i
An (y)
2(∆x)2Ψn+1 (t) − Ψn−1 (t)+Bn (y)+ Vn (y)Ψn (t) ,2∆x(3.74)

where n represents the respective grid point in the x-direction on the discretized spatial grid.

54
Equation (3.74) leads to the following system of linear equations: 
 2 X1 1 X2



 0

 .
 ..



 0


0

···

3 X1

0

0

0

0

2 X2

3 X2

0

···

0

0

1 X3

2 X3

3 X3

···

..
.

..
.

..
.

..

.

0
..
.

0
..
.

0

0

0

···

1 XN −1

2 XN −1

0

0

0

···

0

1 XN

0




0



0



..

.



X
3 N −1 


X
2 N



 Ψ1 (t + ∆t)  Ψ2 (t + ∆t) 






Ψ
(t
+
∆t)


3




.
..








 ΨN −1 (t + ∆t) ΨN (t + ∆t)(3.75)=

 2 Y1 1 Y2



 0

 .
 ..



 0


0


···

3 Y1

0

0

0

0

2 Y2

3 Y2

0

···

0

0

1 Y3

2 Y3

3 Y3

···

..
.

..
.

..
.

..

.

0
..
.

0
..
.

0

0

0

···

1 YN −1

2 YN −1

0

0

0

···

0

1 YN

0




0



0



..

.



Y
3 N −1 


Y
2 N





 Ψ1 (t)  Ψ2 (t) 






Ψ
(t)


3

,


.
..








 ΨN −1 (t) ΨN (t)

where now N is the total number of grid points in the x-direction and
1 Xn

=

2 Xn

=

3 Xn

=

1 Yn

=

2 Yn

=

3 Yn

=

i∆t
i∆t
An (y) −Bn (y),
2
2(∆x)
4∆x
i∆t
i∆t
1−
An (y) +Vn (y),
2
(∆x)
2
i∆t
i∆t
An (y) +Bn (y),
2(∆x)2
4∆x
i∆t
i∆t
−
An (y) +Bn (y),
2
2(∆x)
4∆x
i∆t
i∆t
1+
An (y) −Vn (y),
2
(∆x)
2
i∆t
i∆t
−
An (y) −Bn (y).
2
2(∆x)
4∆x

(3.76)(3.77)(3.78)(3.79)(3.80)(3.81)The matrix on the left side of Eq. (3.75) is of tridiagonal form, and thus the linear equations can besolved by LU decomposition [108]. We note that the symmetry of the Hamiltonian is adapted by

55
modifying the boundaries of the grid in the relevant spatial dimension. As an example, in cylindricalcoordinates the radial coordinate ρ lies in the interval [0, ρmax ], which requires a modification of
the boundary condition at ρ = 0.
Due to recent developments in our code, we are able to do a Crank-Nicolson propagation
directly (as opposed to using the split-operator scheme, c.f., Eq. (3.70)) withΨ (r, t + ∆t) = eiĤ(t)∆t Ψ (r, t) ,(3.82)

which scales as O(N logN ), but has advantages in parallelization, specifically with regards to distributed memory. Therefore, given a large number (' 1000) CPU cores and with the use of PETSc
and SLEPc [109, 110], the direct Crank-Nicolson propagation with the total Hamiltonian becomesless costly than the split operator technique. Once implemented in our code, the direct CrankNicolson propagation was used as an alternative, in particular for the simulations in Chapter 6.

3.2.2

Calculation of bound states

In order to determine the initial conditions for solving Eq. (3.1), which typically is an eigenstate of the system of interest, we need to solve the time-independent Schrödinger equationĤ0 Ψ (r) = EΨ (r) ,(3.83)

with certain boundary conditions (e.g., Ψ (r = 0) = Ψ (r → ∞) = 0 for a 3D Coulomb potential).
In order to solve this equation for one or many dimensions, it has been shown that the imaginary
time propagation (ITP) method as well as the Krylov-Schur method implemented in SLEPc are two
dependable tools in obtaining eigenvalues and eigenvectors for the system being studied [109, 110].
ITP is based upon the propagation of a random guess for a wavefunction (usually a Gaussian
wavepacket for atomic states) in the field-free Hamiltonian, i.e.,i∂Ψ (r, t)= Ĥ0 Ψ (r, t) .
∂t

(3.84)

56
One may write the time-dependent wavefunction Ψ (r, t) as an expansion in the Hilbert space ofĤ0 :Ψ (r, t) =X

cj φj (r) e−iEj t ,(3.85)jwhere φj (r) represents the jth eigenstate with energy eigenvalue Ej , and cj is the amplitude
coefficient, which is the projection of Ψ (r, t = 0) onto φj (r). If we propagate in imaginary time,
0i.e., t → −it starting from our initial guess, the wavefunction after N propagation steps of time
step ∆t becomesΨ (r, t = N ∆t) =X

cj φj (r) e−iEj N ∆t ,(3.86)j

If N is sufficiently large, the first term is vastly larger than the others, and so all other terms
become negligible, i.e.,cj e−iEj N ∆tcj
= e−i(E0 −Ej )N ∆t → 0.
−iE
N
∆t
0
c0 e
c0

(3.87)

Therefore, the ground state is solved for as0

φ0 (r) = Ψ (r, t = N ∆t)(3.88)

0where Ψ (r, t = N ∆t) denotes the normalized wavefunction in this instance. The excited states are
obtained with the same idea, however the states that are lower in energy than the calculated stateneed to be projected out after each propagation step.
ITP works well for most of our simulations, and is easy to implement. However, the accuracy
and efficiency of the method can at times be less ideal. In these cases we use the highly parallel
Krylov-Schur method using SLEPc [110]. In particular, when more than 4 or 5 eigenstates areneeded for a simulation of a process in more than one dimension, ITP converges at a relatively
slow rate. Using SLEPc, we are able to solve for many tens of eigenstates on a similar time scale.
Furthermore, in two or more dimensions, the split operator technique is used when implementing
ITP, which is limited in its parallel capabilities as compared to using SLEPc and Krylov-Schur (seeprevious section).

57
3.2.3

Exterior complex scaling

In order to simulate ionization processes, we need to propagate the ionized outgoing wavepacket
on the finite-grid. If no boundary conditions are implemented however, eventually this wavepacket
will be reflected by the boundaries of our grid. Such a reflection causes artificial processes to occur
in our simulations which are not physical, such as interference with a slower outgoing wavepacket,
and therefore must be eliminated. We use the exterior complex scaling (ECS) method to avoid
this complication [111–113]. In ECS, an exponentially decaying term in the propagator absorbs theoutgoing flux, which may happen if a complex coordinate axis is artificially set in the absorbing
region as (assuming 1D for simplification):x = x0 + (x − x0 ) eiη ,(3.89)

where η is the scaling angle with 0 < η < π/2 and x0 is the starting point of the absorbing boundary.
In the length-gauge, the solution of the reduced 1D TDSE from Eq. (3.1) in the absorption regionis2Ψ (t + ∆t) = e−i(p̂ /2+V0 (x)+E(t)x)∆t Ψ (t)iη= e−icos(2η)p̂ ∆t/2 e−sin(2η)p̂ ∆t/2 e−iV0 (x)∆t e−iE(t)[x0 +(x−x0 )e ]∆t Ψ (t) ,22(3.90)

with the field-free potential term V0 (x) untransformed in this region andiηe−iE(t)[x0 +(x−x0 )e ]∆t = e−iE(t)[x0 +(x−x0 )cosη]∆t eE(t)[x0 +(x−x0 )sinη]∆t .

(3.91)

It is clear that in the second row of Eq. (3.90), the second exponential decays exponentially andhence can absorb the outgoing probability that has entered the boundary region. It is important
to note however that the second exponential term on the right-hand side of Eq. (3.91) can become
an undesired source when the sign of the electric field E (t) is positive. Under certain extreme
conditions, this term causes the wavefunction to blow up, inducing numerical issues. It has therefore
been recommended by He et al. [111] and McCurdy and coworkers [112], amongst others, that the
ECS method be implemented without rotating the electric field interaction into the complex plane.

58
That is, one drops the eiη term in Eq. (3.91), and instead propagates using22

Ψ (t + ∆t) = e−icos(2η)p̂ ∆t/2 e−sin(2η)p̂ ∆t/2 e−iV0 (x)∆t e−iE(t)x∆t Ψ (t) ,(3.92)

meaning that the non-transformed standard field coupling is used over the entirety of the grid,
which includes the absorbing region. Further work on this method was done by McCurdy andcoworkers [112] and Scrinzi [113], amongst others. The work of [112] showed that numerical error
can arise for long-time propagations, but can be controlled by extending the finite grid. This
work also showed that the procedure is formally exact when applied in the appropriate gauge. The
method described in [113] is called infinite-range ECS. It is given this name because in principle, themethod also keeps a record of the dynamics in the region outside the absorber. In conclusion, thisrelatively simple modification of the traditional ECS method has been shown to be a dependable
absorber of the outgoing flux without any undesirable divergences, and also effectively suppresses
reflections as compared to the use of a masking function [111–113], and we therefore use it in our
simulations throughout this thesis.

3.3

Extraction of time delays from simulations of attosecond streaking
In Chapter 1, we introduced the streaking time delay as the temporal shift of the observed

streaking trace with respect to the trace predicted by the original streaking formula, kf = k0 −As (ti ). In order to obtain this delay in numerical simulations, we need to calculate the streaking
trace as accurately as possible without making excessive approximations [16]. We therefore simulate
the streaking experiment by numerically solving the following TDSE: 2
∂Ψ(r, t)pi=
+ V (r) + (EXUV (t) + Es (t)) · r Ψ(r, t),∂t
2(3.93)

where we assume that the ionizing and streaking fields EXU V and Es (t) are polarized along thesame axis, which we define as the ẑ-direction without loss of generality.
We must solve Eq. (3.93) for each delay between the XUV and streaking (IR) pulses to geta streaking trace. We do so on a grid in space and time by using the common Crank-Nicolson

59method described above. In each simulation we propagate the wavefunction on the grid for a
sufficient time such that both pulses have ceased and the ionized wavepacket is far from the ioniccore. We confirm in every simulation that the outgoing wavepackets stay on the grid and do notreach the boundaries. It is required to have a small enough time step to capture the interaction of
the attosecond XUV pulse with the atom or molecule, whilst also having a large grid to allow for the
separation of the photoelectron wavepacket from the remaining core. For this reason, separating
the ionized wavepacket in this manner can become costly in certain limits.
The streaking trace itself is characterized by a momentum distribution as a function of
XUV-IR delay. To this end, we spatially separate the ionized wavepacket from the rest of the
wavefunction, which is possible due to the long propagation times used in the simulations, and thenperform a Fourier transform of the remaining wavepacket. This therefore requires propagating theionizing wavepacket sufficiently far away from the nucleus so that the small error due to projection
onto plane waves instead of Coulomb waves is negligible [16]. After varying the XUV-IR delayτ a sufficient number of times, we obtain the streaking trace, in which the momentum of thephotoelectron along the polarization direction of the field kf,z is given as a function of τ . For thepresent case we normally present the streaking trace in the form of the expectation value of the
final momentum as a function of the relative delay in order to compare with other calculations. For
model systems with one spatial dimension this is trivial, however for systems with more than one
dimension, we calculate the expectation value of the momentum in the laser polarization direction
with an opening angle of 5 to 10 degrees [16]. As mentioned prior, the streaking time delay relates
to a momentum shift ∆k and can be understood as the temporal shift of the TDSE trace withrespect to the trace predicted by the original streaking formula. In practice, we do not compare
these two traces directly in order to obtain the streaking delay (see section 1.3.1.1). Instead we
extract the delay by fitting the TDSE streaking trace tokf,z (τ ) = k0,z − αAs (τ + ∆ts )(3.94)

where k0,z , α, and ∆ts are three fitting parameters, which can be retrieved using a least-square

60
method.
In Chapter 2, we discussed the existence of an absorption delay in the attosecond streaking ofresonant two-photon ionization of atoms. For extracting the absorption delay in such processes inour calculations, we obtain two streaking traces for each simulation. As discussed in section 2.3, we
may determine the continuum delay by calculating the streaking trace for a one-photon ionization
process of the same final electron kinetic energy [20]. Alternatively, we can get the continuum
delay via classical calculations of the streaking trace (c.f., section 2.2.2). Both methods are usedin certain cases throughout this thesis. It is then straightforward to calculate the absorption delay∆ta by subtracting the continuum delay ∆tc from the observed streaking delay ∆ts in the resonant
two-photon process, as is shown in Fig. 3.1.

61

1.3

quantumclassical
k 0 - A( )k f (a.u.)

1.25
1.2
1.15
1.1
-300

(a)
-200

-100

0

100

200

300

(a.u.)
1.2

k f (a.u.)

1.19

quantumclassical
k 0 - A( )

1.18

ts

1.17

tc

1.16
(b)
1.15
70

75

80

85

90

(a.u.)

Figure 3.1: Three streaking traces for a resonant two-photon ionization from numerical streaking
simulations (blue solid curve), classical electron trajectory calculations (green dashed line), and thereference free-particle streaking formula kff ree (τ ) = k0 −As (τ ) (red dashed-dotted curve). Panel (b)is an enlargement of the traces in panel (a) to show the streaking (∆ts ) and continuum (∆tc ) timedelay. Laser parameters used in the simulations are: IXUV = 1.0 × 1013 W/cm2 , ωXUV = 81.81 eV,
TXUV = 550 as, and φXUV = −π/2; Is = 1.0 × 1011 W/cm2 , λs = 2400 nm, Ns = 3, and φs = −π/2.

Chapter 4

Analytical estimates of time delays in streaked photoionization of atoms

With the measurement of a photoemission delay in an attosecond streaking experiment [17],it became crucial for theorists to explain the experimental observation with results of high levelcalculations. Although agreement with experiment has not been achieved in each case, much insightwas gained during the analysis of the problem. The streaking time delay ∆ts is usually studied
via the sum of the Wigner-Smith time delay ∆tW S [69, 70], related to the short-range part of theresidual ion potential, and a term ∆tCLC , related to the coupling between the laser field and thelong-range part of the Coulomb potential [17, 22, 24–26, 64, 73, 85–91, 114].
In this Chapter, we set out with an alternative estimate of the streaking time delay, which
consists of a sum of streaking field weighted field-free delays [92], in order to derive in section 4.1
an analytical formula describing the streaking delay. With the analytical formula, we are able toobtain predictions for the streaking time delay in a few seconds, while the results are in very good
agreement with those of numerical simulations, which typically take hours or days of computation
time. We then show that by separating long- and short-range terms in the formula, we can recover
the two aforementioned terms previously used to describe the observed delay, and hence provide
an alternative justification of the widely used sum rule. Finally, we proceed by showing that theanalytical formula offers the opportunity to effectively analyze other theoretical interpretations aswell as potential effects, such as the effect of an attochirp in the ionizing XUV pulse.We then show
an application of our analytical formula in determining the effect of an attochirp on the streaking
delay in section 4.2.2.

63

Short- and long-range contributions to the streaking delay1

4.1

In this section, we develop an analytical estimation of the streaking delay. Using our analyticalform, we are able to study the connection between the two interpretations of the streaking delay
discussed in Chapter 2. That is, under certain approximations the streaking field weighted field-freedelay sum may be further approximated by the sum of the Wigner-Smith delay and Coulomb laser
coupling term.

4.1.1

Field-free time delays and the cutoff Coulomb potential

The separation of short- and long-range behavior of the photoelectron phase in a streaking
process has previously been performed by attributing the short-range behavior to a Wigner-Smith(WS) delay, and the long range behavior to a Coulomb-laser-coupling (CLC) term. The WS delay,
as discussed before, refers to the energy derivative of the phase shift of the electronic wavepacket
induced by the potential as compared to a free particle. The phase shift due to a Coulomb potential can be evaluated using asymptotic forms (see, e.g., [71], section 3.1.5), where the long range
characteristics of the Coulomb potential show up in a logarithmic divergence term, and the phaseshift for an electron traversing across such a potential isφtotal = φCoulomb + φlog
= arg [Γ (1 + l + iη)] +(4.1)
Zln (2kr) .
k

(4.2)

The logarithmic divergence term makes it difficult to define a Wigner-Smith delay for a photoelectron escaping a Coulomb potential, and it is the tail of the Coulomb potential, coupled to the laser
field, which is deemed relevant for the CLC term in (e.g., [17, 22, 24–26, 64, 73, 85–91, 114]):∆ts = ∆tW S + ∆tCLC .

(4.3)

In this section we will show how the finite-range behavior of the streaking delay can recover thisseparation of short- and long-range effects.
1

The results of this section are also presented in C. Goldsmith, J. Su, A. Becker, and A. Jaroń-Becker, Phys.
Rev. A 96, 053410 (2017).

64
In section 2.2.2, we discussed that using a classical analysis it was found that the streaking
time delay can be approximated as1∆ts 'Es (ti )

Z Ts
ti

#
(0)
kz (t)dt.
Es (t) 1 −
k0,z
"

(4.4)

When put in the form of a discrete sum, we had
∆ts '

N
X
Es (tj )j=1

Es (tion )(j)∆tf f ,

(4.5)

(j)

where ∆tfield-free is a piecewise field-free time delay that the electron accumulates during its propagation in the time interval [tj , tj + δt] and over the related finite region [zj , zj + δz] of the potential
V (r) as compared to the propagation of a free particle over the same distance in space. This formula
states that the streaking delay may be considered as a sum of (streaking) field weighted field-freedelays over a finite range and duration. The finite-range behavior is dictated by the duration of
the streaking pulse. That is, the streaking delay is only accumulated until the end of the IR pulse,and hence probes the (long range) Coulomb potential over a finite distance only. It is therefore
useful to consider a cutoff Coulomb potential, which has a finite phase-shift, and therefore does notdiverge like the unmodified Coulomb potential.
In order to obtain an analytical expression, we write the field-free time delay as a difference:(j)∆tf f = ∆tf f (kj+1 , rj+1 ) − ∆tf f (kj , rj ).

(4.6)

Here, ∆tf f (k, r) is the field-free time delay accumulated during the propagation of the photoelectron, from the origin of the transition into the continuum by an XUV photon to location r, withpconstant momentum k = k (r) = 2(Easym − V (r)), where Easym is the asymptotic kinetic energyof the photoelectron. Considering the classical calculations for the streaking delay showing good
agreement with full quantum mechanical simulations [73,92], we assume a quantum-classical correspondence for the field-free time delay. Thus, we further decompose the (classical) time delay into
three well-known terms:∆tf f (k, r) = ∆tshort (k, r) + ∆tphase (k, r) + ∆tlog (k, r).

(4.7)

65
Each of these terms represents the derivative of a contribution to the (quantum) phase shift withrespect to the energy of the photoelectron. ∆tshort is the short-range contribution, while∆tphase (k, r) =1 ∂arg [Γ(1 + l + iη)]
k ∂k

(4.8)



1 ∂ Z
ln(2kr)k ∂k k

(4.9)

and∆tlog (k, r) =where l is the angular momentum of the photoelectron, η = −Z/k and Z is the charge of theresidual ion. Since the sum is terminated at the end of the streaking pulse, we see that the sum
over field-free delays will not diverge, despite that it contains a contribution from the logarithmic
divergence term.
We furthermore recall that ∆tf f (k, r) corresponds to the field-free time delay accumulated in
the atomic potential, which is cut-off at r, and the phase-shift can be evaluated as explained above.
The assumed quantum-classical correspondence in Eq. (4.7) is therefore expected to hold for cut-off
distances, at which the short-range part of the potential has vanished. Conversely, for distances
close to the core of the residual ion, at which multi-electron dynamics play a dominant role, theabove assumption is no longer valid. Thus, for small energies, at which the photoelectron explores
the short-range part of the potential for a significant time of the interaction with the pulse, thismay lead to discrepancies between the analytical estimates, to be derived in the next subsections,and the results of advanced ab-initio quantum calculations.

4.1.2
4.1.2.1

Analytical formulation
Hydrogen-like systems: Coulomb-phase and logarithmic terms

We first consider the case of hydrogen-like atoms and ions with a bare Coulomb potential
V (r) = −Z/r without short-range contributions. The spectral derivative in Eq. (4.9) is readily
determined as∆tlog (k, r) =Z
[1 − ln (2kr)] .
k3(4.10)

66
Again, it is important to emphasize that since the streaking time delay, Eq. (4.5) depends on finite
distances only, this term will not diverge. The Coulomb-phase term, Eq. (4.8), can be expressedin terms of the real part of the digamma function [26]. Here however, we use an expansion of thecomplex argument of the gamma function [115]:arg [Γ (x + iy)] = yψ (x) +

∞ 
X
n=0

y
− arctan
x+n



y
x+n


,

(4.11)

where ψ (x) is the polygamma function, and we use x = l + 1 and y = − Zk . Since l + 1 > 0
for any possible angular momentum quantum number l, the polygamma function ψ (l + 1) in thisexpansion is well defined, and can be written as:ψ (l + 1) = −γ +

l
X
1
j=1

j

,

(4.12)

where γ ' 0.5772 is the Euler constant. Using this expansion and the result for ∆tlog , we get theanalytical expression for the field-free delay as
#
"
l
X
Z
1
∆tf f (k, r) = 3 F (k, l) +− γ + 1 − ln (2kr)km(4.13)

m=1

where
F (k, l) =

∞
X
n=0

Z2
.
(n + l + 1)([k(n + l + 1)]2 + Z 2 )(4.14)In Fig. 4.1, the accuracy of the predictions of the analytical expression for ∆tf f (solidline) can be seen from the comparison with previously published data extrapolated from ab-initio
calculations (circles, [22]) as a function of the propagation time t after liberation of the electronpinto the continuum. In these calculations we have used r = kasym t, where kasym = 2Easym , inaccordance with the analysis in Ref. [22].
In order to calculate the streaking delay using Eq. (4.5), we must evaluate Eq. (4.13) at N
finite distances rj . These distances are estimated via the recursive relation
rj+1 = rj + k(rj )δtwhere r1 = rion and k (r1 ) =(4.15)p
2 (Easym − V (r1 )) are the initial position and momentum, respec-tively, of the photoelectron once it has arrived in the continuum. We note that the distances can

670

t ff (as)

-5
-10
-15
-20

0

10
20
propagation time (fs)30

Figure 4.1: Comparison of predictions of analytical formula for the field-free time delay ∆tf f , Eq.
(4.13), with results extrapolated from ab-initio calculations for hydrogen atom (Z = 1) [22] as afunction of propagation time (Taken from [23]).

also be determined by using a variety of other standard approaches to solve the differential equation
governing the classical trajectory r (t). For such a calculation it is necessary to find δt such thatconvergence of the streaking delay is achieved. In Fig. 4.2, we show the percent error for threedifferent final momenta of the electron as a function of the time-step used (log scale). We see thatthe calculation is very stable with respect to time step, with less than 1% error in the streaking
delay for all time steps shown. We therefore choose δt = 0.1 as it allows for quick and accurate
calculations.
Fig. 4.3 shows our analytical calculation of the streaking time delay for a hydrogen atomand helium ion compared to those of ab-initio calculations [22]. We find agreement within one
attosecond over the entire range of asymptotic electron kinetic energies studied.

4.1.2.2

Short-range term for multielectron atoms

There do not exist analytical solutions for the ionization (half-scattering) of an electron froman atom with more than one electron, and because the exact numerical solution for such an atom
requires excessively large calculations, the process is often modeled using single-active-electron(SAE) potentials. In such a potential, all electrons except the active photoelectron are assumed

68

Figure 4.2: Convergence of the time delay shown as percent error of ∆ts as a function of the logof the timestep δt used for the case of the ionization of the hydrogen ground state where the final
kinetic energy of the photoelectron is kf = 1.5 a.u. (solid blue line), kf = 2.0 a.u. (dashed orange),and kf = 2.5 a.u. (dotted green). Streaking field parameters were N = 3 cycles, λ = 800 nm,
Is = 1 × 1012 W/cm2 .

0

t s (as)

-10
-20
-30

H
He +

-40
-50
20

40

60

80

100

120

Easym (eV)Figure 4.3: Comparison of analytical estimates (solid lines) with results of ab-initio calculations(circles, [22]) of the streaking delay in hydrogen atom and helium ion. Parameters of the streaking
field: peak intensity Is = 1 × 1012 W/cm2 , wavelength λs = 800 nm and pulse duration of threecycles. In the analytical calculations, ionization of the photoelectron at the peak of the pulse andthe most probable location of the electron in the initial state has been assumed (Taken from [23]).

69
to be frozen during the interaction with the external fields. The corresponding SAE potentials are
given by
1
VSAE (r) = − + Vshort (r) ,r(4.16)

where the short-range part, Vshort , is typically modeled via a linear combination of Yukawa terms,
VYukawa (r) = a exp (−µr) /r, and exponential decay terms, Vshort (r) = a exp (−µr), with a and µconstant.
In determining the field-free time delay ∆tf f obtained by the photoelectron in a SAE potential, the sum of ∆tphase and ∆tlog is given by Eq. (4.13) for Z = 1. Analogous to the otherterms, the short-range contribution is given by the spectral derivative of the corresponding phaseshift δshort :∆tshort (k, r) =1 ∂δshort (k, r).
k ∂k(4.17)In order to obtain analytical estimates of this delay, we make use of the distorted-wave Born
approximation (DWBA), which accounts for scattering effects in a reference potential Vshort , andprovides good estimates as long as the difference to the real potential is sufficiently small, and the
kinetic energy of the electron is not too low. Within the DWBA, we may calculate the phase shift
due to the short-range part of the potential as [72]:1δshort (k, r) ' − 2
k

Z ∞

2
d%Fl,η
(k, r)Vshort (r),(4.18)

0with the bare Coulomb wave function
Fl,η (k, r) = Cl ei% %l+1 1 F1 (l + 1 + iη; 2l + 2; −2i%)(4.19)

where % = kr andπη

Cl = 2l e− 2|Γ(l + 1 + iη)|.
Γ(2l + 2)(4.20)For Yukawa and exponential potential terms, the integrals in Eq. (4.18) and the related
derivatives in Eq. (4.17) have the following closed form solutions, which are derived in the next
subsections:∆tY ukawa =δY ukawa[f (k) + g(k) − 1]
k2

(4.21)

700t s (as)

-5
-10
-15
-20
(a)
-25

50

100

150

0

t s (as)

-2
-4
-6
-8
(b)
-10
60

80

100

120

140

160

180

XUV Photon Energy (eV)Figure 4.4: Comparison of analytical predictions (solid lines) with results of ab-initio calculations(symbols) for the streaking time delay in the case of photoionization of (a) helium (ab-initio datafrom [22]) and (b) neon atom as a function of the XUV photon energy. Ab-initio results for Ne
are from calculations using the present SAE potential (circles, [24]) and multi-electron B-spline
R-matrix calculations (diamonds, [25]) are shown. Streaking laser parameters as in Fig. 4.3 (Taken
from [23]).

71
where the analytical forms of δY ukawa , f and g are given in Eqs. (4.25), (4.31) and (4.33) and
∆texp




δexp 2i(l + 1 − iη) iη
2i=− log 1 − kkµ − 2ik
k
µ
!
∞
X
πη η
1
2
l+
−
+
k
2
k
(l + 1 + n)2 + η 2
n=0c1 (k) − c2 (k) − c3 (k) + c4 (k)+,c0 (k)(4.22)

the analytical forms of δexp and ci (i = 0, . . . , 4) are given in Eq. (4.35), and Eq. (4.40) to Eq.
(4.44), respectively.
We tested the analytical streaking formula by making use of the following form of the SAE
potentials for electrons in the outermost shell of atoms:
VSAE (r) = −Z
a1 e−µ1 r + a2 e−µ2 r−− a3 e−µ3 r ,rr(4.23)

where the parameters for various noble gas neutrals and ions are given in Ref. [97]. The streaking
time delay can then be obtained in the same way as for hydrogen-like systems, using the analyticalformulas for ∆tshort , ∆tphase and ∆tlog . As can be seen in Fig. 4.4, the analytical predictions are ingood overall agreement with results of ab-initio calculations for photoionization of helium and neon
atom. We note that in the case of ionization of Ne we have considered that the 2p → d transition
is strongly dominant over the 2p → s one [86].

4.1.2.3

Short-range time delays for Yukawa Potentials

In this subsection we derive an analytical formula for the field-free time delay corresponding
to the emission of an electron from a short range Yukawa potential of the form
VY ukawa =aY exp[−µr],r(4.24)

72
where aY and µ are constants. Using the Distorted Wave Born Approximation the phase shift
induced on a l-partial wave in the potential is given by [116]:
δY ukawa 'aY|Γ (l + 1 + iη) |2 exp[−πη]
4kΓ(2l + 2)
−(l+1)  
µ2
2k1+ 2
× exp 2η arctan
µ
4k


4k 2
×2 F1 l + 1 + iη, l + 1 − iη; 2l + 2; 2
µ + 4k 2(4.25)

where
2 F1 (a, b; c; z)=∞X(a) (b) z n
n

n=0

=

n

(c)n n!

Γ(c)Γ(b)Γ(c − b)

Z 1

dt tb−1 (1 − t)c−b−1 (1 − zt)−a(4.26)

0is the convergent Gaussian hypergeometric function, given both as series and Euler integral representation, with(a)i =

Γ(a + i).
Γ(a)(4.27)

is the Pochhammer symbol.
Calculation of the corresponding time delay∆tY ukawa =∂δY ukawa1 ∂δY ukawa
=
∂E
k
∂k

(4.28)

requires to take the derivative of the convergent Gaussian hypergeometric function, in which threeof the four arguments depend on k. To this end, we make use of both representations of the function
as follows:∂
2 F1 (a (k) , b (k) ; c; z (k))∂k  

∂z 2 F1 (a, b; c; z)
2 F1 (a, b − 1; c; z)=(c − b)+ (b − c + az)∂kz(1 − z)2 F1 (a, b; c; z)
 
Z 1
∂a
Γ(c)
+
tb−1 (1 − t)c−b−1 (1 − tz)−a∂k Γ(b)Γ(c − b) 0
 


1−t
(0)
(0)
× log
+ ψ (b) − ψ (c − b) dt
t (1 − tz)∂a∂bwhere we used that ∂k= − ∂kand ψ (0) is the polygamma function.

(4.29)

73
Using this result the time delay is given as:∆tY ukawa =δY ukawa[f (k) + g(k) − 1]
k2

(4.30)

with
"

π
2kµ
f (k) = 2η+ 2
− arctan
2 4k + µ2



2k
µ



#
∞
1X
1
(2l + 2)µ2−+k(l + 1 + n)2 + η 2
4k 2 + µ2(4.31)

n=0

and



4k2
F
l
+
1
+
iη,
l
−
iη;
2l
+
2;
2
2 1
µ
µ2 +4k2− 2g(k) = 2 (l + 1 + iη) 
4k2
4k + µ22 F1 l + 1 + iη, l + 1 − iη; 2l + 2; µ2 +4k2
Z 1−iη
tl−iη (1 − t)l+iη (1 − tz)−(l+1+iη)
0
"
#
!
∞
X
(1 − t) 4k 2 + µ2
2iη
× log
−
dt
(4k 2 (1 − t) + µ2 ) t(l + 1 + n)2 + η 2
n=0


(4.32)

where we have replaced the polygamma function by its respective series representation. We notethat the full derivative of the hypergeometric function can alternatively be written as a series (asopposed to the integral), however the integral converges at a much higher rate than the series.

4.1.2.4

Short-range time delays for potentials with exponential decay

In this subsection we derive an analytical formula for the field-free time delay in a short range
potential with exponential decay
Vexp = aexp exp[−µr],(4.33)

where aexp and µ are constants. The corresponding phase shift δexp has a closed form expression
which can be represented in terms of the Appell F2 function as [117]aexp (2l + 2)|Γ(l + 1 + iη)|2 exp[−πη]k 2l(4.34)
Γ(2l + 2)µ2l+3−2ik 2ik
×F2 2l + 3; l + 1 + iη, l + 1 − iη; 2l + 2, 2l + 2;;µµ−(l+1−iη)22laexp (2l + 2)|Γ(l + 1 + iη)| exp[−πη]k
2ik
=
1−
2l+3
µ
Γ(2l + 2)µ
 

−2ik
2ik
× F1 l + 1 + iη; l + 2 + iη, l + 1 − iη; 2l + 2;;
µ 2ik − µ l + 1 − iη
2ik
−2ik
2ik
−
F1 l + 1 + iη; l + 2 + iη, l + 2 − iη; 2l + 2;;,
2l + 22ik − µµ 2ik − µ

δexp =

74
where in the second equation we expanded the F2 function in the more common Appell F1 series
as [117]
F2

k k0
c + s; a, a ; c, c − p; ;
h h
0




 0 s+p
k 0 −a X (a0 )m (−s − p)m
h −m
=
1−
1− 0
(4.35)
h(c − p)m m!
k
m=0


k
k
0
0
,
×F1 a; c + s − a , m + a ; c; ;h h − k0

where∞X(α)m+n (β1 )m (β2 )nxm y n(4.36)
(γ)
m!n!
m+n
m,n
Z 1
Γ(γ)
=
tα−1 (1 − t)γ−α−1 (1 − xt)−β1 (1 − yt)−β2 dt,
Γ(α)Γ(γ − α) 0F1 (α; β1 , β2 ; γ; x, y) =given both as series and Euler integral representations.
Calculation of the corresponding time delay∆texp =

∂δexp
1 ∂δexp=∂Ek ∂k(4.37)

requires to take the derivative of the Appell series, in which all of the arguments except γ depend
on k. To this end, we make use of both representations of F1 to get:∂
F1 (α(k); β1 (k), β2 (k); γ, x(k); y(k)) =∂k ∂x αβ1F1 (α + 1; β1 + 1, β2 ; γ + 1; x, y)∂kγ
 ∂y αβ2+F1 (α + 1; β1 , β2 + 1; γ + 1; x, y)
∂k
γ
Z 1
∂α
Γ(γ)
+
tα−1 (1 − t)γ−α−1 (1 − tx)−β1 (1 − ty)−β2∂k Γ(α)Γ(γ − α) 0 t(1 − ty)
(0)
(0)
× log
+ ψ (γ − α) − ψ (α) dt,(1 − t)(1 − tx)(4.38)∂β1∂β2
where we used that ∂α∂k = ∂k = − ∂k . Using this result the time delay is given by:∆texp =

δexp
k
+

2
k





2i(l + 1 − iη) iη
2i− log 1 − k(4.39)µ − 2ikkµ!
∞1πη η X
c1 (k) − c2 (k) − c3 (k) + c4 (k)
l+
−
+
,
2
k
(l + 1 + n)2 + η 2
c0 (k)
n=0

75
with
c0 (k) =c1 (k) =c2 (k) =c3 (k) =×c4 (k) =



−2ik
2ik
F1 l + 1 + iη; l + 2 + iη, l + 1 − iη; 2l + 2;;
µ 2ik − µ2ikl + 1 − iηF1 (l + 1 + iη;−
2l + 22ik − µ−2ik
2ik
l + 2 + iη, l + 2 − iη; 2l + 2;;
µ 2ik − µ2iµ(l + 1 − iη)− iη
(2l + 2)(2ik − µ)2ik − µ


2ik
−2ik
;
×F1 l + 1 + iη; l + 2 + iη, l + 2 − iη; 2l + 2;
µ 2ik − µ2i(l + 1 + iη) l + 2 + iηF1 (l + 2 + iη; l + 3 + iη,
2l + 2µ
−2ik
2ikµ(l + 1 + iη)l + 1 − iη; 2l + 3;;+
F1 (l + 2 + iη;µ 2ik − µ(2ik − µ)2

−2ik
2ik
l + 2 + iη, l + 2 − iη; 2l + 3;;
µ 2ik − µ2
l + 2 + iη
4k|l + 1 + iη|F1 (l + 2 + iη; l + 3 + iη,2(2l + 2) (2ik − µ)µ
2ikµ(l + 2 − iη)−2ik;+
l + 2 − iη; 2l + 3;
µ 2ik − µ(2ik − µ)2


−2ik
2ik
F1 l + 2 + iη; l + 2 + iη, l + 3 − iη; 2l + 3;;µ2ik − µ


−2ik
2ik
δF1 l + 1 + iη; l + 2 + iη, l + 1 − iη; 2l + 2;;
µ 2ik − µl + 1 − iη2ik−
δF1 (l + 1 + iη;
2l + 22ik − µ2ik
−2ik
l + 2 + iη, l + 2 − iη; 2l + 2;;
µ 2ik − µ(4.40)(4.41)(4.42)(4.43)(4.44)

where
∂α
Γ(γ)
δF1 (α; β1 , β2 ; γ; x; y) =∂k Γ(α)Γ(γ − α)
Z 1
×tα−1 (1 − t)γ−α−1 (1 − tx)−β1 (1 − ty)−β20 t(1 − ty)
(0)
(0)
× log
+ ψ (γ − α) − ψ (α) dt.
(1 − t)(1 − tx)(4.45)

76
0
-0.05

(a)
-0.15

short

(as)

-0.1

0
-0.1
-0.2

(b)

-0.3
20

40

60

80

100

120

140

XUV photon energy (eV)Figure 4.5: Absolute difference ∆short as a function of the XUV photon energy for the contribution
due to the short-range potential for (a) neon atom and (b) helium atom. Streaking laser parameters
as in Fig. 4.3 (Taken from [23]).

0
-0.1
-0.2
-0.3

(as)

(a)

phase

0
-0.2

(b)

-0.4
0
-1

(c)

-2
20

40

60

80

100

120

XUV photon energy (eV)Figure 4.6: Absolute difference ∆phase for photoionization of (a) neon, (b) helium and (c) hydrogen.
Streaking laser parameters as in Fig. 4.3 (Taken from [23]).

(a)

100

0

log

(a.s.)

50

(b)

20

10

0
40

60

80

100

120

XUV photon energy (eV)Figure 4.7: Absolute difference ∆log for photoionization of (a) neon, and (b) helium. Streaking
laser parameters as in Fig. 4.3 (Taken from [23]).

77
4.1.3

Relation to the Wigner-Smith Delay

Since we have established the usefulness of our analytical approximation, it is desirable tounderstand why it agrees with the streaking delay calculated in ab-initio simulations as a sum ofWigner-Smith and Coulomb-laser-coupling delays. We note that the field-free delay in Eq. (4.5) is
split into three contributions, and may therefore study the effect of the streaking field on each ofthese terms. To this end, we have determined
∆i =

N 
X
Es (tj )j=1

Es (tion )(j)− 1 ∆ti(4.46)

for each of the three contributions (i = short, phase, log). The corresponding results are shown in
Figs. 4.5, 4.6 and 4.7, respectively. The comparison clearly shows that for the short-range and the
Coulomb phase terms, the effect of the streaking field is significant for XUV photon energies near
the threshold of photoionization only. At photoelectron energies of 10 eV or more the respective
differences ∆i are below 1 as. On the other hand the effect of the streaking field on the logarithmic
term is much larger, even for photoelectron energies of several tens of eV.
Therefore, neglecting the photoelectron energies close to threshold, we can further approximate Eq. (4.5) as:∆ts =X Es (tj ) h

(j)

(j)

(j)

∆tshort + ∆tphase + ∆tlogi

Es (ti )jX
Es (tj ) (j)
(j)
(j)
'
∆tshort + ∆tphase +∆t
Es (ti ) log(4.47)(4.48)j= ∆tshort + ∆tphase +X Es (tj )jEs (ti )

(j)

∆tlog

= ∆tW S + ∆tCLC(4.49)(4.50)

where ∆tW S = ∆tshort + ∆tphase represents the well-known Wigner-Smith time delay for short
P E (t ) (j)range potentials, while ∆tCLC = j Ess (tji ) ∆tlog can be interpreted as the Coulomb-laser-coupling
term often used in the analysis of streaking time delays. Indeed, as shown in Fig. 4.8, the present
estimates for ∆tCLC are in good agreement with another analytical estimate ∆tCLC = Z[2 −ln(Easym TIR )]/(2Easym )3/2 [26], where TIR is the streaking field period.

78

0
-10

Z=1

t CLC (as)

-20
Z=2

-30
-40
-50
-60
-70
20

40

60

80

100

120

140

160

180

Easym (eV)Figure 4.8: Comparison of present analytical estimates (solid lines) for ∆tCLC with those of analternative analytical approximation (dashed lines, [26]). Streaking laser parameters as in Fig. 4.3(Taken from [23]).

79

4.2

Application of analytical streaking formula
In the previous section we derived an analytical formula for the streaking time delay. As

shown, the formula is useful to obtain accurate estimates of the delay in a short period of time, andfurthermore provides new insights into the separation of the time delay into the WS time delay anda Coulomb-laser-coupling term. However, in consideration of the good agreement with numericalresults and the efficiency to obtain predictions, the formula can also be easily applied to different
situations, and hence help to analyze other theoretical proposals and/or potential effects on thestreaking time delay. In this section we illustrate these opportunities for two examples.

4.2.1

Application to and analysis of recent theoretical proposal

Earlier this year, Kiesewetter et al. [27] performed a RABBITT experiment (c.f., section
1.3.1.2) using a novel analysis method, which they deemed RABBITT+, to extract the coarse
characteristics of the binding potential of helium, neon, and argon atoms. They assume a simple
pseudo-potential in their method, which allows for the modeling of the energy transfer occuring in
an XUV-pump NIR-probe attosecond experiment, given that the electron has propagated over alarge portion of the potential within the first half-cycle of the streaking field after ionization.
The pseudo-potential model from Ref. [27] is shown in Fig. 4.9. In the model, the electron
appears in the continuum at time t = 0 in the presence of the streaking field with period T . In thefirst quarter cycle of the field after liberation into the continuum (0 ≤ t ≤ T /4), the photoelectron
has an average momentum P1 and is assumed to experience a linear change in momentum along thelaser polarization direction. The momentum is then constant at P2 until after next field maximum
at t = T /2 + t0 , where the momentum assumes its asymptotic value p0 . The photoemission timedelay is defined as the energy derivative of the phase of the wave packet in this model potential
as compared to the phase of the corresponding free-particle (c.f., Eq. (2.1)). The wavepacket for
the free particle by definition experiences no net energy transfer from a potential, and according
to this model always has momentum p0 throughout the propagation. It follows that the amount80
of time delay accumulated during propagation should be directly related to the amount of energy
transfer occuring within the first half cycle of streaking field after the liberation of the electronin the continuum. Since it has been shown that RABBITT and streaking measurements provide
equivalent information concerning the time delay, the model by Kiesewetter et al. implies that theobserved (streaking or RABBITT) time delay is essentially determined within one half cycle of thestreaking field.
This model thus goes further then the interpretation provided by Su et al. [92] (c.f., section
2.2.2). While Su et al. showed that the streaking delay is accumulated until the end of the streaking
pulse, Kiesewetter et al. suggest that the streaking delay is already within a good approximationof its final value after only a quarter or half cycle of the streaking pulse. We cannot test thepseudo-potential model of Ref. [27] directly, however our analytical formula allows us to test thislast statement. To this end, we analyze the sum in Eq. (4.5) as a function of time tj during theelectron’s propagation, until the sum achieves its final value at the end of the streaking pulse. In
order to see to what end the assumption in Ref. [27] holds, concerning the time range over which
the streaking delay is effectively obtained, we find at what times the sum has converged within 10%
and 1% of its final value (denoted as T10% and T1% respectively), and compare these times to T /4
and T /2. For all calculations, the time of ionization tion is chosen to be the peak of the streaking
pulse:
2

Es (t) = ẑE0 sin



πt
τ




sin


2πc
t + φCEP .
λs(4.51)In Figure 4.10 we present the results of such a calculation for four different streaking field
wavelengths and for ionization of hydrogen atom. For each streaking field wavelength, we plot thesum (4.5) as a function of time for seven different XUV central frequencies. In addition, we show
the times T /4 and T /2 (dashed vertical lines) in order to compare with the assumptions of themodel by Kiesewetter et al. For each XUV central frequency considered, the times T1% (circles)
and T10% (stars) are indicated.
We see that the general behavior of Eq. (4.5) as a function of tj can be decomposed into

81

Figure 4.9: Pseudopotential as suggested in the work by Kiesewetter et al. [27]. (a) An electron isejected at time t = 0 in the presence of a weak NIR field F (t) with period T . For 0 ≤ t ≤ T /4, theelectron has an average momentum of P1 and experiences a constant restoring force along the laser
polarization direction. For t > T /4, its momentum is modeled as constant at P2 until after the next
field maximum at T /2 + t0 . After the net energy transfer has been established, the photoelectron
assumes its asymptotic momentum p0 . (b) Illustration of the corresponding spatial dependence ofthe pseudo-potential along the field polarization. (Taken from Ref. [27])82

Figure 4.10: Eq. (4.5) shown as a function of time tj for streaking wavelengths λs = 400 nm(a), 800 nm (b), 1600 nm (c), and 2400 nm (d). For each panel, the results are shown for final
electron kinetic energies corresponding to XUV central frequencies of ω = 44.22 eV (blue), 55.27
eV (orange), 68.03 eV (green), 82.48 eV (red), 98.64 (purple) eV, 116.49 eV (brown), and 136.05
eV (pink). The times T10% and T1% (as described in the text) are shown with stars and circles
respectively. For all simulations shown, a 3 cycle streaking pulse of intensity Is = 1 × 1012 W/cm2
is used.

83

Figure 4.11: The times T10% (left) and T1% (right) shown as a function of XUV central frequency for
different pulse durations; Ns = 3 cycles (circles), Ns = 4 cycles (stars), Ns = 5 cycles (diamonds),
and Ns = 6 cycles (squares). For the case of T10% , we further show T /2 (dashed light blue) andT /4 (solid green). The same streaking pulse intensity as for the results in Fig. 4.10 is used for all
calculations.

84
two regions. As also assumed by the pseudo-potential model, in the first region 0 ≤ t ≤ T /4 theelectron feels the influence of the Coulomb potential most strongly, and accordingly we see a sharp
increase in the magnitude of the time delay. Beyond t = T /4 the field changes and then oscillates
until the end of the pulse. Consequently, the accumulated streaking time delay as a function oftime oscillates as well. The magnitude of the oscillation decreases due to two reasons: (i) the
amplitude of the electric field decreases (reducing the weight of the field-free time delays) and (ii)the Coulomb potential decreases in strength (reducing the magnitude of the field-free time delays).
Thus, overall we observe a damped oscillation of the streaking time delay towards its final value atthe end of the pulse.
As we can see from the positioning of the stars and circles vs. the dashed vertical lines,
T10% < T /2 and T1% > T /2 for all of the streaking wavelengths and ionizing photon energies
considered. Furthermore, for almost every single case considered, T /4 < T10% < T /2. The cases
where this does not hold (appears as a “jump” in the results) are for the longer streaking wavelengths
considered, specifically for the highest XUV photon energy in the case of 1600 nm, and the threelargest XUV photon energies considered in the case of 2400 nm. For all four cases, the values of
T10% are much smaller than T /4, and we can understand the reasoning for these outliers as follows.
In each of these cases, the kinetic energy of the photoelectron is relatively large, and therefore theelectron propagates away from the core of the potential quickly. Therefore, by the time the sign ofthe field changes, due to the large decrease in the magnitude of the Coulomb potential, the terms(j)∆tf f are sufficiently small such that they do not contribute significantly to the total sum. Therefore∆ts approaches within 10% of the final value much faster as compared to the cases of less energetic
photoelectrons. In addition, we see this effect appear only for longer streaking wavelengths bysimilar reasoning. That is, the streaking field changes sign after a longer period of time, during
which the influence of the Coulomb potential has decreased significantly. This argument is further
supported by T10% and T1% both decreasing as a function of XUV photon energy for all streaking
pulse wavelengths considered. In further regards to T1% , we note that for no case considered does
this time fall between T /4 and T /2, and generally is reached towards the end of the streaking pulse.

85
Additionally, we present in Fig. 4.11 the results for T10% and T1% as a function of XUVcentral frequency using an 800 nm streaking field, now for four different streaking pulse durations.
We furthermore show T /2 and T /4 as horizontal lines in the Figure for T10% . We see that T10%
does not vary by more than a few atomic units with respect to pulse duration, which implies that
our observation T10% < T /2 is robust with respect to pulse duration. In contrast, T1% varies withpulse duration by at least 40 a.u. for every XUV central frequency shown. This demonstrates
that T1% is reached before the end of the pulse, but only once the oscillations of the field have asufficiently negligible influence on the propagation of the electron, which in general happens longafter the electron has propagated over half a period of the field. Therefore, from our analysis we canconclude that the prediction of Kiesewetter et al. regarding the streaking delay being accumulated
within the first half cycle of the streaking field after liberation of the electron into the continuumis accurate to within about 10% of the total streaking delay. In certain cases where the streaking
field has a longer wavelength or when the photoelectron has a relatively high kinetic energy, thislevel of accuracy can be achieved at times earlier than T /2.

4.2.2

Influence of the attochirp on attosecond streaking delays2

As mentioned in Chapter 1, only a fraction of the relative delay of 21 as, measured in [17],has been accounted for by theory [19, 25, 86, 118], which has generated much of the interest in theinterpretation of attosecond time delay measurements. It was recently suggested by Keller andco-workers [119] that the streaking time delay and other time measurements such as RABBITT[120, 121] may depend on the chirp of the ionizing attosecond XUV pulse. In the case of streaking
this was related to the presumption that different times for the transition of the photoelectron into
the continuum during the interaction with the XUV pulse would correspond to different asymptotic
kinetic energies of the electron. Previous theoretical studies focused on the analysis of the methodof measurement itself, namely the retrieval of the Wigner-Smith time delay from streaked energy
2

The results of this section are also presented in C. Goldsmith, A. Jaroń-Becker, and A. Becker, J. Phys. B 51,
025601 (2018).

86
spectra using chirped pulses [91, 120]. It was demonstrated that the spectrogram itself is sensitive
to the attochirp of the XUV pulse and for chirped pulses the retrieved time delays may dependon the accuracy of the retrieval algorithm and/or the central momentum approximation, in which
the momentum of the photoelectron is approximated by the center momentum of the unstreaked
spectrum. Here, we show an application of the analytical formula derived in the previous sectionin order to provide a complimentary study of the attochirp effect on the attosecond streaking timedelay. Instead of focusing on the experimental methods of extracting the WS time delays, we study
how the streaking time delay itself, intrinsic to the photoionization process in the superposition of
the two pulses, depends on the chirp of the attosecond XUV pulse.

4.2.2.1

Accounting for the attochirp in the model

In the previous section we showed that for asymptotic photoelectron energies Easym not too
close to threshold the effect of the streaking field on the short-range and Coulomb phase parts ofthe field-free delay are negligible and we can approximate the streaking delay via∆ts (ω, ti ) = ∆tW S (ω) +NX
Es (tj )j=1

Es (ti )(j)∆tlog (ω),(4.52)

where ∆tW S = ∆tshort + ∆tphase represents the Wigner-Smith time delay [68–70] and the second
term can be interpreted as the Coulomb-laser-coupling term often part of the commonly used
analysis of the streaking time delay [22, 24–26, 64, 73, 75, 76, 84–91, 114]. The Wigner-Smith timedelay can be either estimated analytically (see previous section) or determined using literature
values based on advanced calculations (e.g., [29, 79]). In the calculations presented below we haveused the data for the Wigner-Smith time delay obtained by Kheifets [29]. The logarithmic term
can be calculated analytically using"(j)∆tlog (ω) = Z

1 − ln (2kj+1 rj+1 ) 1 − ln (2kj rj )
−
3
kj+1
kj3

#
(4.53)

along the method outlined in Section 4.1.
In Figure 4.12 we present results of the analytical estimates for the streaking time delay forphotoemission from a neon atom ((a) 2p and (b) 2s) and argon atom ((c) 3p and (d) 3s) as a87function of XUV central frequency. In the calculations we assumed a Gaussian XUV laser pulse
with a FWHM pulse duration of τF W HM = 40 as and used a streaking pulse with sin2 envelope
consisting of 3 cycles. The maxima of the two fields were set to coincide in time.
We furthermore compare estimates for the streaking delays for three different times of transition of the photoelectron into the continuum during the XUV laser pulse, i.e., we show delays forthe cases ti = −20 as, ti = 0 as, and ti = 20 as. This comparison shows that the streaking timedelay does not have a significant dependence upon the time of transition. This is due to the fact thestreaking pulse does not vary much over the interval of 40 as. In addition, the estimates of the delay
for the neon atom are in good agreement with results from ab-initio simulations (diamonds, [25]).
It is important to note however that the current model formula does not provide an improvement
concerning the discrepancies between the experimental data of [17] and theoretical predictions.
Since we discussed in Section 4.1 that the predictions of the model formula are expected [23, 93] tobe less reliable for XUV central frequencies near the ionization threshold, we restrict ourselves in
the frequency range of the attosecond pulse in the analysis below.
While the time delays for Ne and Ar(3s) increase quite smoothly as a function of the XUV
frequency, while a pronounced minimum appears around the Cooper minimum for Ar(3p). The
minimum arises from the WS time delay and is related to the change of sign in the dipole matrix
element [29, 79, 122–124].
In order to study the effect of the attochirp using Eq. (4.52), we determine the expectation
value of the streaking time delay in a chirped laser pulse asZh∆ts i (ω0 , b) =Zdω

dti P (ω, ω0 , ti , b) ∆ts (ω, ti ) ,(4.54)

where P (ω, ω0 , ti , b) is the probability distribution for a transition at a certain frequency ω at agiven time ti ; ω0 is the central frequency of the pulse, and b is a parameter which describes the chirpof the pulse (see Eq. (4.58) below). Since the dependence of ∆ts (ω, ti ) on ti during the interaction
with the XUV pulse is negligible (see Fig. 4.12), we can further approximateZh∆ts i (ω0 , b) 'dω∆ts (ω) Pavg (ω, ω0 , b) ,(4.55)

88

Figure 4.12: Estimates of the streaking delay for Neon ((a) 2p, (b) 2s) and Argon ((c) 3p, (d) 3s)
given by Eq. (4.52), as a function of the central XUV frequency for times of the transition in thecontinuum, ti = −τF W HM /2 (dash-dotted line), ti = 0 (solid line), and ti = τF W HM /2 (dashedline) are compared. Also shown are results from ab-initio calculations (diamonds), [25]. In all
calculations a streaking pulse of wavelength λs = 800 nm, sin2 -envelope consisting of 3 cycles, andpeak intensity Is = 1 × 1012 W/cm2 has been considered (Taken from [28]).

89

0.04

0.2

Pavg (arb. u)

Ne, 2p

(Mb)

0.35
0.3
0.25

0.15
0.1

0.03

0.02

0.01

0.05

(a)
40

60

80

100

(eV)

120

140

0

(b)
40

50

60

70

80

(eV)

Figure 4.13: (a) Neon 2p photoionization cross section [29] and time averaged probability for (b)ω0 = 60 eV. The chirp is taken to be b = 0 eV/fs (solid line), b = 15 eV/fs (dashed line), andb = 25 eV/fs (dash-dotted) and τ = 450 as (Taken from [28]).

90

Figure 4.14: Same as Fig. 4.13 but for (a) Argon 3p photoionization cross-section [29]; time
averaged probability for (b) ω0 = 45 eV, (c) ω0 = 60 eV, and (d) ω0 = 80 eV (Taken from [28]).

91
where Pavg (ω, ω0 , b) is the time averaged probability distribution. In our studies below we consider
a linear chirped laser Gaussian pulse for which [125]Z ∞g (ω, ω0 , t, b) U (t)dt,

Pavg (ω, ω0 , b) = N σ(ω)(4.56)−∞

where σ (ω) is the photoionization cross-section and"#− (16ln2)2 t2
U (t) = exp
τ2(4.57)

pis the temporal intensity distribution with τ /8 ln (2) being the FWHM pulse duration of theGaussian pulse. The instantaneous spectral distribution of the pulse is"#(ω − ω0 − bt)2,
g (ω, ω0 , t, b) = exp −γ 2 /4ln2(4.58)hi1/2
2 2
with γ 2 = Γ2 − τ 4b , where Γ = τ1 64 (ln2)2 + τ 4 b2
and N is a normalization factor.
For an unchirped pulse (i.e., b = 0) the instantaneous spectral distribution g is timeindependent and Pavg is given by the photoionization cross section weighted by a Gaussian distribution centered about the central frequency (ω0 ) (solid lines in Figs 4.13 (b) and 4.14(b)-(d)).
For photoemission from the 2p state of Neon (and the s-states, not shown), the cross section varies
rather smoothly as a function of XUV frequency (Fig. 4.13(a)). Consequently, a temporal chirp inthe pulse causes a broader and less intense time averaged probability distribution while the positionof the maximum in the distribution does not shift significantly, independent of the central frequency
ω0 (Fig. 4.13(b)).
In contrast in the case of Ar(3p) (Fig. 4.14) the variation in the photoionization cross section
near the frequency ωC for the Cooper minimum leads to a shift of the maximum of Pavg . For
central frequencies which are slightly below ωC (Fig. 4.14(b)) the frequency at which the maximum
appears is less than ω0 , and decreases as the chirp of the pulse increases due to the rapid decreaseof the photoionization cross section. We note that the effect of Fano resonances below 30 eV (seeFig. 4.14(a)) on the present results cannot be further analyzed, since the resonances appear outside
the frequency range which is applicable for our model formula. At frequencies just above ωC (Fig.

t s (b = 0) (as)0.5

0t s (b) -t s (b) -t s (b = 0) (as)

92

-0.5

(a)
-1

50

55

60

65

70

0.2
0
-0.2
-0.4
-0.6
-0.8
-1

75

30
20

t s (b = 0) (as)

(c)

80

90

100

110

XUV central frequency (eV)
3

(d)

2.5
2
1.5
1

10

t s (b) -t s (b) -t s (b = 0) (as)XUV central frequency (eV)

(b)

0
50

60

70

XUV central frequency (eV)

0.5
0
-0.5

60

70

80

90

100

XUV central frequency (eV)Figure 4.15: Change in streaking time delay for photoionization of Ne from 2p (a) and 2s (b)shells and of Ar from 3p (c) and 3s (d) in chirped laser pulses (with respect to an unchirped pulse)
for b = 5 eV/fs (line with plus signs), b = 10 eV/fs (dashed), b = 15 eV/fs (dash-dot), b = 20 eV/fs(dotted), and b = 25 eV/fs (solid) (Taken from [28]).

93
4.14(c)), we observe a small effect of the chirp due to the increase of the photoionization crosssection. On the other hand for frequencies significantly above ωC (Fig. 4.14(d)), the distributionof Pavg for Ar(3p) changes similar to the case of Ne(2p) as well as the s-states.
We also note that the analytical formula allows us to predict that for very large chirp"  #−16ln (2) t 2
lim g (ω, ω0 , ti , b) = exp,(4.59)b→∞3
τwhile the time averaged probability distribution is given byτlim Pavg (ω, ω0 , b) = N σ(ω)
b→∞
8

r

3π
ln2

(4.60)and the expectation value of the streaking delay, Eq. (4.55), becomes independent of the central
frequency ω0 . As test calculations have shown however, this limit is only reached approximately
for unrealistic values of the chirp parameter b of a few hundred eV/fs.

4.2.2.2

Effect of the Cooper minimum in Argon

As can be seen from Eq. (4.55) the effect of the attochirp on the streaking time delay arises
from the convolution of the time averaged probability distribution Pavg (ω, ω0 , b) and the variation of
the delay with XUV photon energy ω. In Fig. 4.15 we present predictions for the difference between
the streaking time delays obtained in a chirped pulse and an unchirped pulse for (a) Ne(2p), (b)Ne(2s) as well as for (c) Ar(3p) and (d) Ar(3s). In each case, the difference increases as the chirpparameter b increases, and we observe variations as a function of XUV central frequency. For Neonthe effect is smaller than 1 as over the whole range of frequencies, even for the largest attochirp
considered (b = 25 eV/fs). This is due to the rather smooth variation as a function of frequency ofboth the streaking time delay and the photoionization cross section for photoemission from the 2p
and 2s subshells.
According to our results in the previous section, the influence of the chirp is much larger
when the time delay and the cross section change rapidly as a function of the laser frequency.
Consequently, we observe significant changes of up to a couple of tens of attoseconds in the streaking

94time delay at small XUV frequencies for photoemission from the 3p shell in Ar, with a pronounced
maximum centered near the Cooper minimum (Fig. 4.15(c)). This confirms previous expectations[79] that time delays and related effects are most pronounced near a Cooper minimum. The effectis larger at frequencies lower than ωC than for ω > ωC , in agreement with the results for Pavg in
Fig. 4.14. In contrast, the changes in the streaking time delay for ionization of Ar(3s) as well as ofAr(3p) at frequencies significantly above the Cooper minimum are as small as for photoionizationof Ne atom. We have further analyzed the impact of the two contributions to the streaking delay,
namely Wigner-Smith term and Coulomb-laser coupling term, on the change observed in Fig. 4.15.
The data in Fig. 4.16 show that the significant change for the Ar(3p) photoemission primarily resultsfrom the Wigner-Smith contribution (Fig. 4.16(a)). This is due to the fact that the Wigner-Smith

30

20

10

0
45

t CLC
(b = 0) (as)
s

(a)

t CLC
(b) s

t WS
(b) s

t WS
(b = 0) (as)sterm varies more rapidly than the Coulomb-laser coupling term around the Cooper minimum.

50

55

60

65

70

XUV central frequency (eV)

75

1

(b)

0
-1
-2
-3
-4
-5
45

50

55

60

65

70

75

XUV central frequency (eV)Figure 4.16: Comparison of the contributions from the (a) Wigner-Smith and (b) Coulomb-laser
coupling terms for the change in the streaking delay for photoionization of Ar from 3p shell. Linesand symbols as in Fig. 4.15(c) (Taken from [28]).

Finally, we present in Fig. 4.17 the relative time delays for photoemission from the different
subshells in (a) Ne and (b) Ar as a function of the chirp parameter b. As expected, in the case of
Ne it is found that an attochirp has a negligible effect on the intrinsic value of the relative streaking
time delay. In particular, the relative time delay at 100 eV remains at about 12-13 as independentof the value of the chirp parameter, and thus cannot account for the discrepancy between theoretical

95

24

(a)

(b)
0

t 3p-3s (as)t 2p-2s (as)

22
20
18
16

-20

-40

14
12

0

10

20

temporal chrip b (eV/fs)

30

-60

0

5

10

15

temporal chrip b (eV/fs)

20

Figure 4.17: Difference of streaking time delays for photoemission (a) from 2p and 2s in Neon at
XUV central frequencies of 80 eV (solid), 90 eV (dashed), and 100 eV (dash-dotted) and (b) from
3p and 3s in Argon at XUV central frequencies of 55 eV (solid), 60 eV (dashed), 70 eV (dash-dot),
and 80 eV (dotted) as a function of the chirp parameter b (Taken from [28]).

96
predictions and the experiment of [17]. On the other hand, for argon the relative delay can varyby several attoseconds for XUV laser frequencies near the Cooper minimum.

Chapter 5

Absorption time delays in resonant and non-resonant two-photon ionization

In recent studies, it has been suggested to apply the attosecond streaking principle to temporally resolve electron dynamics in two-photon resonant ionization [20, 21, 67, 80] and double ionization [90]. These studies showed that the time delay between the absorption of two photons may
be retrieved from the streaking or RABBITT time delays, and that this delay can be controlled byparameters of the applied XUV laser, such as pulse duration in the case of Ref. [20], and central
frequency as shown in Ref. [67]. Further analysis is necessary to understand whether or not such
results can be used for advanced time-resolved measurements.
As discussed in section 2.3, Su and coworkers [20] found that for a two-photon ionization
process, an additional time-delay, called the absorption delay ∆tabs , may occur that is not accounted
for by the streaking time delay obtained in the continuum (as in the case of one-photon ionization,
see also Chapter 4). Furthermore, ∆tabs is observed to be non-zero in the case of resonant twophoton ionization only [16]. As the absorption delay depends on the energy level structure of thetarget, it may open a route towards probing attosecond dynamics in atoms and molecules. In this
Chapter, we show under which conditions the absorption delay can be approximated using timedependent perturbation theory. We then demonstrate how our perturbative model can be used toilluminate dynamics.
In section 5.1, we derive an analytical formula that captures the absorption delay through an
isolated resonance, and compare to the predictions of TDSE simulations. Furthermore, we show
how the inclusion of an additional resonance improves agreement with TDSE calculations. Then, in

98section 5.2 we analyze how the behavior of the absorption delay is modified in cases where multiple
intermediate resonances are within the bandwidth of the XUV pulse. In addition, we discuss how
the streaking results are related to observables in attosecond transient absorption spectroscopy [14],which is an alternative method for time-resolved measurements. Finally, in section 5.3 we show
results of the absorption delay in the simplest molecule H2+ using a one-dimensional model.

5.1

Two-photon ionization through an isolated resonance1
The work by Su et al. [20] proposed that the time delay in streaking of two-photon ionization

can be parametrized by:∆t(2ω)= ∆tabs + ∆tc ,
s

(5.1)

where ∆tc is the time delay accumulated by the photoelectron during propagation in the continuum
after absorbing two photons, and ∆tabs is the delay which contains temporal information about thetransition of the electron from the ground state into the continuum (see section 2.3). We note that(2ω)

in the context of this Chapter, the two-photon streaking delay ∆ts

in Eq. (5.1) corresponds tothe two-photon process, and therefore we refer to it as simply ∆ts for the rest of this thesis.
In the previous Chapter, we have considered analysis of the second term in the sum (5.1),
i.e., the continuum delay. We saw that the continuum delay could be accounted for via classicalsimulations or the sum (4.5). Indeed, in one photon ionization processes being probed by attosecond
streaking, the continuum delay accounts for the entirety of the observed streaking delay. In thisrespect, the liberation of the electron into the continuum is instantaneous within the resolution of
the simulation.
Through our analysis below, we derive an analytical formula for the absorption delay whichin the case of an isolated resonance shows the strong linear dependence of the absorption delay
upon the XUV pulse duration previously seen in the TDSE simulations of Ref. [20] (c.f., section
2.3). Furthermore, our analysis reveals a dependence on the detuning of the central frequency of the1
The results of this section are also presented in C. Goldsmith, J. Su, A. Jaroń-Becker, and A. Becker, J. Phys.
B: At. Mol. Opt. Phys. 51, 155602 (2018).

99
XUV pulse from the resonance. In addition, we estimate the effect of the inclusion of an additional
intermediate state.

5.1.1

Time-dependent perturbation theory analysis

We wish to model a two-photon ionization process, starting from an initial bound state|ii into a final continuum state |f i with respect to parameters of the ionizing attosecond pulse.
Assuming the ionizing pulse is sufficiently weak, the amplitude cf i can be written within second
order perturbation theory (c.f., section 3.1.3) as [126]:
cf i (ω) =

Z
Z ∞
X
m

iδf t

µf m e

Z t
E(t)

−∞

0

µmi eiδm t E(t0 )dt0 dt ,

(5.2)

−∞

where µjk is the transition dipole moment between states |ji and |ki, E (t) is the field envelopeand ω is the central frequency of the XUV pulse, δf = Ef − Em − ω, and δm = Em − Ei − ω with
Ef = Ei + 2ω. The sum with integral represents the expansion of the intermediate state over all
field-free states of the target atom or molecule. Eq. (5.2) can be solved analytically for a Gaussian
2

2

field envelope E (t) = E0 e−t /2T [126]cf i = πE02 T 2
where F (x) = e−x

2

Z
X


m µf m µmi

2 T2
−δm

e


2
√
−i
F (δm T ) ,
π

(5.3)

R x −t2
dt is Dawson’s integral. Assuming that the bandwidth of the attosecond
0 epulse predominantly covers only one intermediate (resonant) state |ri we further approximate [126]cf i = πE02 T 2−δr2 T 2µf r µri e


Z
2 X
− i√
m µf m µmi F (δm T ) ,
π

(5.4)

since the Gaussian term decays quickly as a function of detuning and therefore contributes for the
resonant state only.
The absorption time delay is related to the phase of the two-photon transition amplitude as
Im (cf i )φf = arg cf i = arctan.
Re (cf i )(5.5)

Before proceeding, we note that it is only the absolute value of the dipoles which are of concern.
The dipole matrix element µf m = |µf m | eiθf m , involving the transition into the continuum state, has

100a phase θf m whose derivative with respect to energy is merely the widely discussed Wigner-Smith
delay. This delay, and therefore the corresponding phase, is accounted for in the continuum delay∆tc , or equivalently does not contribute to the absorption delay. On the other hand, the dipole
matrix elements µmi between bound states are, in general, real. Therefore, we proceed to get thephase related to the absorption time delay as follows:
φabs



Z
2 X |µf m µmi | F (δm T )= arctan − √mπ|µf r µri | e−δr2 T 2



Z
X
|µf m µmi | F (δm T )2
δr2 T 2= −arctan √eF (δr T ) +,m6=r
π|µf r µri | e−δr2 T 2

(5.6)
(5.7)

where we have separated the contribution by the resonant state from those by the other states.
Assuming that the latter contributions are small, we make a Taylor expansion by considering theclosest (dipole allowed) non-resonant state |bi in the sum over m only and obtain:(res)(non−res)φabs = φabs + φabs

(5.8)

with
(res)
φabs




2 δr2 T 2
= −arctan √ e
F (δr T )
π

(5.9)
2

(non−res)φabs

2

2 a2b (δb , δr , T )eδr T F (δr T )ab (δb , δr , T )
√
+
= −
2
π
1 + erfi2 (δr T )1 + erfi2 (δr T )(5.10)|µf b µbi |2 2F (δb T ) eδr T ,|µf r µri |(5.11)

whereab (δb , δr , T ) =22

erfi(δr T ) = √2π F (δr T )eδr T , and the above expansion was made about ab = 0. The absorption timedelay is given by the spectral derivative of φabs :(res)(non−res)∆tabs (δr , δb , T ) = ∆tabs (δr , T ) + ∆tabs(res)(non−res)(δb , δr , T )
(res)

=

∂φabs
∂φabs
+
∂Ef
∂Ef

=

∂φabs
T
e(δr T )√+∂Ef
π 1 + erfi2 (δr T )2=(non−res)∂φabs∂δr ∂φabs+∂Ef ∂δr∂Ef(non−res).

(5.12)

where we used δr = Er − Ei − ω = Er − 21 (Ei + Ef ). The first term is the contribution from theresonant intermediate state and the second term is the approximate correction due to the closest

101(dipole allowed) non-resonant state. In the present analysis we obtained the latter correction by
calculating the corresponding phase and performing the derivative numerically. As we will show
below, the correction term is typically small and we neglected it in the main part of our analysis(unless mentioned otherwise).
(res)Eq. (5.12) shows that the absorption delay ∆tabs (δr , T ) depends on the detuning from theresonance δr and the duration of the ionizing laser pulse T , while it is independent of other laser(res)or target parameters. In Fig. 5.1, we present the predictions for ∆tabs (δr , T ) as a function of δrand T . As one would expect, for large durations (i.e., small energy bandwidth of the pulse) theabsorption delay is non-negligible for small detunings from the resonance only, while the detuning(res)range over which ∆tabs (δr , T ) is significant increases with a decrease of the pulse duration.
5.1.2

Comparison with numerical simulations

In order to test if the predictions of the model formula, Eq. (5.12), are reliable we have
performed numerical simulations of a streaking measurement (see section 3.3 for details). Sinceaccording to our analytical model the general trends for the absorption time delay through a single(non−res)intermediate resonance (i.e. when ∆tabs' 0) are independent of the specific form of theatomic potential, we have chosen a model system for the main part of our analysis.
To this end, we have considered a one-dimensional (1D) potential:
V (x) = − √Z,x2 + a(5.13)

where Z is the effective charge and a is a soft-core parameter. Choosing Z = 3.0 and a = 0.15,
the energies of the three lowest states are E1 = −5.32, E2 = −2.31, and E3 = −1.30 a.u. Theseeigenstates are well separated such that one can selectively study the two-photon absorption into
the continuum via a single intermediate resonance. In order to keep the duration of the XUV pulse
shorter than the oscillation period of the streaking pulse, we have used a 3-cycle streaking field of
2400 nm in wavelength with Is = 1.0×1011 W/cm2 and φs = −π/2. In order to test the predictionsof the perturbative analysis in case of nearby additional states, we have also made use of previously

102

(res)

Figure 5.1: Resonant contribution to the absorption delay ∆tabs (in attoseconds) as a function ofdetuning δr and pulse duration T (Taken from Ref. [1]).

103
obtained data for the absorption delay for resonant two-photon ionization of the helium atom [20].
To obtain the absorption time delay from the numerical results for the streaking delay weneed to retrieve the contribution due to the continuum time delay. As discussed in Chapters 2 and4, several techniques are known to obtain ∆tc from the well-studied streaking of photoionization.
One of these techniques makes use of the fact that the propagation in the continuum can be often
well described by classical analysis [73]. We have therefore used a classical trajectory method todetermine the continuum time delay in this particular instance. To this end, we consider the Newton
equation for the dynamics of the electron in the field generated by the Coulomb potential of theresidual ion and the streaking field of the linearly polarized laser. The corresponding differential
equation has been solved using the fourth-order Runge–Kutta method. As initial conditions, wehave chosen the most probable position of the electron in the ground state of the potential andthe initial momentum of the electron due to the absorption of a photon in the Coulomb potential.
Alternatively, one can sample the initial conditions in Monte-Carlo calculations [73]. The delay
between the XUV pulse and streaking pulse corresponds to the instant of the appearance of thephotoelectron in the continuum during the streaking pulse. We have therefore varied this initialtime instant to obtain a classical streaking trace in terms of the classical final momentum. The
continuum time delay ∆tc can be then retrieved from this classical streaking trace in the same
way as ∆ts from the quantum streaking simulations. For further details of this classical trajectory
calculations we refer to [92]. An example of the procedure is shown in Fig. 5.2 for a resonant twophoton ionization of the model system. The absorption time delay is then given as ∆tabs = ∆ts −∆tc
(c.f., Eq. (5.1)).
Considering resonant two-photon ionization at the central frequency of the pulse, i.e. detuningδr = 0, we have performed streaking simulations using the model potential and an ionizing pulse
with ω = 3.01 = E2 − E1 . In Fig. 5.3 (a), we present the numerically determined absorption timedelay ∆tabs = ∆ts − ∆tc (circles) as a function of the pulse duration T . The numerical results arecompared with the theoretical model predictions for the absorption time delay ∆tabs (Eq. (5.12),
solid line), including both resonant and non-resonant contributions, and the resonant contribution

104

quantumclassical
k 0 - A( )final momentum (a.u.)

1.3

1.25

1.2

1.15

1.1

(a)

-300

-200

-100

0

100

200

300

delay between XUV and streaking pulses (a.u.)

final momentum (a.u.)

1.2

1.19

quantumclassical
k 0 - A( )

1.18

ts

1.17

tc

1.16

(b)
1.15
70

75

80

85

90

delay between XUV and streaking pulses (a.u.)

Figure 5.2: Three streaking traces for a resonant two-photon ionization from numerical streaking
simulations (blue solid curve), classical electron trajectory calculations (green dashed line), and the(free)
reference free-particle streaking formula kf(ti ) = k0 − As (ti ) (red dashed-dotted curve). Panel(b) is an enlargement of the traces in panel (a) to show the streaking (∆ts ) and continuum (∆tc )time delay. Laser parameters used in the simulations are: IXUV = 1.0 × 1013 W/cm2 , ωXUV = 81.81
eV, T = 550 as, and φXUV = −π/2; Is = 1.0 × 1011 W/cm2 , λs = 2400 nm, Ns = 3, and φs = −π/2
(Taken from Ref. [1]).

105

700
600

t abs (as)

500
400
300
200
100
0

(b)
0

1000

2000

3000

4000

5000

6000

T (as)

Figure 5.3: Absorption time delay from numerical simulations (blue circles) and model predictions(res)(non−res)(res)for ∆tabs + ∆tabs(red solid line) and ∆tabs (black dashed line) as a function of pulse
duration for a Gaussian XUV pulse tuned into resonance with the first excited state of (a) the model
system and (b) helium atom (data for numerical simulations taken from [20]). Laser parameters:
IXUV = 1.0 × 1013 W/cm2 (resonant), φXUV = −π/2; Is = 1.0 × 1011 W/cm2 , λs = 2400 nm,
Ns = 3, and φs = −π/2 (Taken from Ref. [1]).

106
(res)

(res)

∆tabs (dashed line) only. Note that in the case of resonant central frequency limδr →0 ∆tabs = √Tπ .
The absorption time delays extracted from the numerical simulations indeed reproduce the linear
dependence on the pulse duration as predicted by perturbation theory. Although the correction
from the non-resonant contribution is small (c.f., solid line vs. dashed line), its inclusion further
improves the agreement between predictions and numerical data.
In Fig. 5.3 (b), we compare predictions from perturbation theory to TDSE simulations of
the absorption delay in 3D helium atom taken from [20]. In the calculations of the dipole matrix
elements for the helium atom, we used bound state wavefunctions in the Coulomb field of aneffective charge Ze = 27/16 and the plane wave approximation for the continuum states. The
comparison reveals that while the numerical results qualitatively reproduce the linear dependenceof the absorption time delay on the pulse duration, there is an almost constant offset. This offset
was interpreted before as the influence of the streaking field on the resonant 2p state [20], which is
not taken into account in the present perturbation model. In the present comparison we howeveralso observe that in the case of the helium atom consideration of the non-resonant correction term(c.f., solid line vs. dashed line) has a much larger effect than for the model system (Fig. 5.3 (a)),
where the states are more separated in energy. Thus, it is likely that in the case of helium atom
further improvement may require to consider more than one non-resonant state in the analysis,
which is beyond the scope of the present section.
As mentioned above, in our perturbative model analysis of the absorption time delay wehave neglected the influence of the streaking pulse. In case of the model system with the transition
through a strongly bound resonant state, this assumption is confirmed by the results of our numerical simulations. For example, in Table 5.1 it is shown that, although the streaking and continuum
time delays (not shown) vary with a change of either the wavelength or the intensity of the streaking
pulse, the variation of the extracted absorption time delays is less than 5 as. This implies thatby accurate determination of the continuum time delay one is able to obtain the absorption timedelay, independent of the parameters of the applied streaking pulse, if the intermediate resonant
state is well isolated from other states and the effect of the streaking field on the intermediate state

T
250.0
350.0
450.0

λs = 2400
Is = 1011
137.79193.47247.66λs = 3200
Is = 1011
134.34190.52244.87λs = 4800
Is = 1010
140.08195.96250.49

(res)

(res)

107

(non−res)∆tabs∆tabs + ∆tabs

141.05
197.47253.89137.40194.86251.86

Table 5.1: Results of simulations for the absorption delay (in as) in the model system as a function of
the pulse length T (in as) of a two-photon resonantly ionizing pulse and streaking pulses at different√(res)wavelengths λ (in nm), intensities I (in W/cm2 ). Also shown are the predictions ∆tabs = T / π(res)(non−res)(in as) and ∆tabs + ∆tabs(in as) from perturbation theory (Taken from Ref. [1]).

is negligible. In the following section, we shall see that this last assumption in particular is notalways justified.
For the case of a well-isolated intermediate state, as in the case of the 1D model system,(res)

we have also considered how ∆tabs

changes for small detunings, i.e. δr 6= 0. The predictions of(res)

the perturbative analysis for the absorption delay ∆tabs

as a function of pulse duration T forvarious detunings δr are shown in Fig. 5.4. We note that for the 1D model system, the contribution(non−res)∆tabs

is less than 10 as even for the largest δr considered, as can be seen in Fig. 5.5.

One can see that for small T , or more generally small product δr T , the absorption delay
approximately depends linearly on the pulse duration, as in the on-resonant case. This can beunderstood by expanding the model formula in a Taylor series for x = δr T about x = 0:





1
4
(res)
3
5
∆tabs (x) ' √x +O x
.
x+ 1−
π
πδr

(5.14)For a given detuning δr and sufficiently small T , the first term of the expansion is dominant and√(res)the absorption delay is approximately determined by ∆tabs = T / π. Physically, this corresponds
to the situation that the frequency for the resonant transition lies well within the bandwidth of theionizing laser pulse.
(res)

On the other hand, we see that the model prediction for the absorption delay ∆tabs

goes

to zero for large pulse durations, independent of the detuning δr . Physically, this corresponds tothe situation that the energy bandwidth of the pulse is sufficiently small and the contributions at
the resonant frequency become negligible. In this case the streaking delay is determined by the108t (res)
(as)
abs

400

300

200

100

0

0

1000

2000

3000

T (as)(res)

Figure 5.4: Absorption delay ∆tabs as a function of pulse duration for detunings δr = 0.68 eV(solid line), 0.82 eV (dashed line), 0.95 eV (dash-dot line), 1.09 eV (stars), 1.36 eV (diamonds),
2.04 eV (boxes), and 2.72 eV (circles) (Taken from Ref. [1]).

2(non res) (as)
tabs

0
2
4
6
8
0

500 1000 1500 2000 2500 3000 3500
T (as)(non−res)

Figure 5.5: Absorption delay ∆tabs
as a function of pulse duration for detunings δr = 0.68
eV (solid line), 0.82 eV (dashed line), 0.95 eV (dash-dot line), 1.09 eV (stars), 1.36 eV (diamonds),
2.04 eV (boxes), and 2.72 eV (circles).

109
continuum time delay only, as discussed before [20]. This can be also shown by taking the limit of(res)∆tabs for T → ∞ as:(res)lim ∆tabs (δr , T ) =

T →∞

d
dT



2

T eδr T21
√ lim d
π T →∞ dT 1 + erfi2 (δr T )(5.15)

1 + 2δr2 T 2
T →∞ 2δr erfi (δr T )δr T 2=0= lim
T →∞ erfi (δr T )=lim(5.16)(5.17)

where we have repeatedly applied L’Hospital’s rule.
(res)

Besides these limits, further analysis of ∆tabs

reveals another interesting dependence: If

the detuning δr is inversely proportional to the pulse duration T , i.e. δr T = const., the absorption
delay depends linearly on T . Since the half-width of a pulse with Gaussian envelope is given by√∆ω1/2 =

(res)
2ln2
T , we therefore expect ∆tabs ∝ Tfor δr = ∆ω1/2 . In Fig. 5.6, we present resultsof TDSE simulations for the streaking, continuum and absorption delay as a function of the pulse(res)duration for this case using the 1D model system. Indeed, the linear relationship of ∆tabs on thepulse duration is captured by the ab-initio calculations of ∆tabs for pulse durations T > 200 as.
For shorter pulse durations we observe a deviation from the linear trend, since the corresponding
broad bandwidth of the pulse covers more than one resonance. In the next section, we take thenext steps towards analyzing the role of multiple resonances.

5.2

Absorption delay in the case of multiple resonances
As seen in the previous section, when more than one resonance is within the bandwidth of

the ionizing pulse, the analytical solution presented is no longer well applicable. While Eq. (5.3)
is still valid, the result will depend on the accurate calculation of more than one dipole moment.
Thus, the sum has to be calculated numerically and the dipole moments must be calculated togood approximation.

110

200

t

abs

tc

150

time delay (as)

ts
100
50
0
-50
-100

0

100

200

300

400

T (as)

Figure 5.6: Absorption delay as a function of XUV pulse duration, for a detuning chosen suchthat the resonance is at FWHM of the XUV pulse (Taken from Ref. [1]).

111
5.2.1

Analysis using model potentials

In order to be able to clearly distinguish between the regimes of isolated resonances andoverlapping resonances, we use the same 1D model potential as in the previous section, Eq. (5.13),
but with Z = 3 and α = 1.0. The lowest six states for this potential are E0 = −2.336, E1 =−1.360, E2 = −0.8697, E3 = −0.5946, E4 = −0.4303, and E5 = −0.3244 a.u. Thus, the first two
allowed transitions from the ground state are well separated as compared to a typical bandwidth(' 0.5 a.u. in the case of Ref. [17]) of the ionizing pulse. This is desirable as it allows us to stillobserve the effects of individual resonances, while analyzing additional effects in the regions where
the resonances overlap.
Using this potential, we solved for the absorption delay across a central frequency range
1.5–2.1 a.u., which spans frequencies resonant with the 3rd excited state to the 5th excited state.
In order to calculate the absorption delay in TDSE simulations, we again use ∆tabs = ∆ts − ∆tc .
The streaking delay ∆ts is obtained from the streaking trace for two-photon ionization. For thedetermination of the continuum delay ∆tc , instead of using classical simulations as in the previous
section, we use the other method described in section 3.3, which is to solve the TDSE for thecorresponding one-photon ionization process. The results of the TDSE simulations for the streaking
time delay, the continuum time delay, and the obtained absorption time delay are shown in Fig
5.7 (a), where we see three peaks in the absorption delay. The first peak is relatively broad and iscentered around the central frequency resonant with the third excited state (ω30 = 1.7414 a.u., seesection 3.1.3.2 for notation), and the peak corresponding to the highest energy is centered aroundthe fifth excited state resonant frequency (ω50 = 2.0116 a.u.). Another sharp peak appears in thevicinity of the fourth excited state resonance (ω40 = 1.9057 a.u.).
To gain insights into these features, we first use the model calculations of the absorption
delay based on time-dependent perturbation theory which was presented in the previous section.
Here, we are required by Eq. (5.3) to know the transition dipole moments corresponding to every
contributing state. As a further approximation however, we assume that the dipole moments

112
between the bound states and the continuum are approximately constant in this energy regime,
and use the dipole moment products as fitting parameters for the absorption delay. In this sense,
we fit the result of second order perturbation theory to the TDSE calculations, with the dipole
moment product µf j µji for different intermediate states |ji as constant complex numbers. The
results of this fit are shown in Fig. 5.7 (b), for the cases where states |3i and |5i are included asintermediates (dashed line), and when the first excited state |1i (solid line) is included as well. We
note that the even parity states |2i and |4i do not contribute, as single photon transitions between
these states and the (even parity) ground state are dipole forbidden, i.e., the dipole moments µ02
and µ04 are identically zero, and therefore are not considered as fitting parameters.
The two peaks which appear about the third and fifth excited state resonances can be well
reproduced in the cases where these excited states are included, and further improvement (at low
energies) is achieved when the first excited state is additionally included. From the previous section,(res)the absorption delay ∆tabs , as a function of detuning and therefore central frequency, peaks with aroughly Gaussian shape about the resonant frequency for an isolated resonance (c.f., Fig. 5.1). Dueto the model potential having well isolated resonances, the model should capture these features,
which is indeed what we observe. However, the sharp peak at approximately 1.9 a.u. is not capturedby the perturbation theory model, and therefore cannot be attributed to an overlapping of the thirdand fifth excited state resonances.
The effect of overlapping resonances is observed, however, in the frequency regime between
the first and third excited state resonances. This can be seen in Fig. 5.7 (b) in the result of themodel calculation when the first excited state is included. What we observe is that the decreaseof the absorption delay at frequencies ω < ω30 is captured only once the first excited state isincluded in the fit. This is interesting since for a frequency resonant with the first excited state,at ω = ω10 = 0.976 a.u., there is not sufficient energy to ionize the system from the ground state(E0 = −2.336 a.u.) with two photons of the central frequency. Therefore, the influence of the firstexcited state is due to the broad bandwidth of the attosecond pulse and its strong coupling to theground state.

113

Figure 5.7: (a) TDSE calculations of the absorption delay ∆tabs (diamonds), the two-photon
streaking delay ∆ts (stars), and the continuum delay ∆tc (circles) for the model system as a functionof XUV central frequency. The pulse parameters used are λs = 2400 nm, Ns = 3 cycles, Is =FWHM = 350 as. For the simulation of the resonant two1 × 1011 W/cm2 , φs = φXU V = −π/2, τXUV
photon ionization, an XUV intensity of IXUV = 1 × 1013 W/cm2 is used, whereas for the continuum
delay ∆tc , an intensity of IXUV = 1 × 1014 W/cm2 is used to initiate the one-photon ionization.
In (b), the absorption delay from TDSE calculations (diamonds) are compared with perturbation
theory predictions, when including the third and fifth excited states as intermediates (dashed line)
as well as when the first excited state is also included (solid line). The dipole moment productsµf j µji for intermediate state |ji have been used as (constant) fitting parameters.

114
The feature which is not captured by the model is the sharpest and tallest peak in the absorption delay profile. This peak appears in the region around ω = 1.9 a.u, which is very close
to the fourth excited state resonance (ω40 = 1.9057 a.u.), and it was mentioned above that thistransition is dipole forbidden. Our model however, beyond the assumptions of second order perturbation theory, only considers interaction with the XUV pulse. Therefore, in order to understandthe occurence of the sharp peak, we need to consider other effects beyond our model, e.g., theinfluence of the streaking field. In section 5.1, we showed that the streaking pulse has no effect on
the absorption delay for the system and pulses considered in that study where the resonant state
was strongly bound. In general, it has usually been assumed that the streaking pulse plays norole on the transition of the electron into the continuum itself. However, it was suggested in theprevious section as well as in Ref. [20] that the discrepancies seen between our model calculationsfor He atom and TDSE simulations in Fig. 5.3 (b) may be due to the effect of the infrared pulse
upon the intermediate states.
The influence of the IR field on electronic transitions is known from other methods involving
the simultaneous application of XUV and IR pulses [14, 15]. For example, in attosecond transientabsorption spectroscopy (ATAS) (c.f., section 1.3.2), sidebands are observed around energies corresponding to so-called “dark states” [15], as shown in Fig. 5.8. Dark states are states that cannot
be populated via transition from the ground state due to dipole selection rules. These sidebands
are referred to in the literature as light-induced states (LISs) [14] and occur in the region of an
ATAS spectrum where the XUV and IR pulses overlap. In a simple picture of this effect, each LIS
may be associated with a two-photon process that involves the absorption of an XUV photon andthen either absorption or stimulated emission of an IR photon, such that the atom may end up
with population in a state that cannot be populated by a single XUV photon, as illustrated in theschematic shown in Fig. 5.8. In the current context of attosecond streaking, such effects would bepresent in the absorption delay if an electron populating a dark state, having undergone the process
shown in Fig. 5.8, transitions into the continuum via absorption of a second XUV photon.
In order to determine whether the same physics is responsible for the occurence of the sharp

115

Figure 5.8: The “dark” 2s state in He may be populated via absorption or stimulated emission of
an IR photon in addition to an XUV photon for a transition from the initial 1s state (Taken from
Ref. [14]).

peak in Fig. 5.7, we performed a few test calculations. First, contrary to what we saw in thecase of the isolated resonance (c.f., Table 5.1) the peak should depend on the intensity of the IR
field, but not the XUV intensity. Second, the other peaks that we explained via our perturbativemodel should not depend on the IR intensity. Lastly, the position of the peak should not be at theforbidden transition frequency ω40 , but should be offset by an IR photon, i.e., ω = ω40 −ωIR ' 1.886
a.u.
The first and last expectations are satisfied, as demonstrated by the results in Fig. 5.9 (a),where the absorption delay for various frequencies in the region around the sharpest peak of Fig. 5.7
is plotted as a function of streaking pulse intensity. We see that there is a significant dependence
upon the streaking field intensity when the XUV central frequency is tuned to ω40 − ωIR , yet nosuch dependence is present at the other central frequencies considered. In addition, we see in Fig.
5.9 (b) that the absorption delay in the vicinity of the fourth excited state does not depend on the
XUV intensity. The second condition is shown to be satisfied in Fig. 5.9 (c), where we see that theabsorption delay about the dipole-allowed third excited state transition has minimal dependence
on the streaking field intensity, which implies that the absorption delay at this frequency can be

116

Figure 5.9: (a) Absorption delay as a function of streaking pulse intensity for three different
XUV central frequencies. All other parameters remain fixed to the same values as in Fig. 5.7. (b)Absorption delay as a function of ionizing pulse intensity for the case of the dipole-forbidden fourth
excited state minus a streaking photon (blue line plus diamonds), and the case where resonant withthe dipole-allowed third excited state (red line plus asterisks). All other pulse parameters remain
fixed. (c) The streaking (blue dashed with circles), continuum (red dashed with diamonds), andabsorption (solid green line with asterisks) delay as a function of streaking pulse intensity when
the ionizing pulse is tuned to resonance with the dipole-allowed third excited state.

117
well described by the method of section 5.1. Our results indicate that it is primarily the streaking
pulse which enables the accumulation of an absorption delay in the vicinity of one-photon forbidden
transitions.
We further note that the peak in the absorption delay only appears at ω = ω40 − ωIR = 1.886
a.u., and not at the frequency ω = ω40 + ωIR = 1.9254 a.u. Remembering that Fig. 5.7 is plotted
as a function of XUV central frequency, this asymmetry implies that the dark state is populated
via absorption of one XUV photon and then subsequent absorption of an IR photon, but is not
populated via absorption of one XUV photon and the emission of an IR photon. This is contrary towhat is observed in ATAS experiments and simulations (see Fig. 5.8), where peaks corresponding toboth absorption and stimulated emission of an infrared photon are observed. Currently the reason
for this is unknown, and further investigation is needed.
We have shown in this section that overlapping resonances can give rise to absorption delays ofdifferent behavior to that of the previous section. In addition, we have shown that in contrast to thestrongly bound case studied in Chapter 4, the streaking pulse may play a role in the accumulationof an absorption delay, specifically it can contribute an infrared photon to populate an intermediate
state which would otherwise be forbidden for a transition initiated by a single XUV photon.

5.3

Absorption delay in H+
2
In the quest to control interactions of light and matter on the attosecond timescale, it isdesirable that the methods used to obtain information for atoms may also be extended to learn
about molecules. In particular, we wish to see whether the absorption delay is a useful observablefor dynamics in molecules. As a first step, we will consider only electronic resonances in a model
for a simple one-electron diatomic molecule, while neglecting rovibrational transitions. To this end,in this section we study the behavior of the absorption delay in resonant two-photon ionization of a
1D model of the simplest molecule H+
2 under the Born-Oppenheimer (fixed-nuclei) approximation.

118

Figure 5.10: Potential energy surface for the 1D model of H2+ . The ground state, and the first
four excited states are shown as a function of internuclear distance. Also shown is the equilibrium
internuclear distance of Req = 2.64 a.u. (dashed line).

5.3.1

Simulations of the absorption delay

For our TDSE simulations of the streaking of our model H+
2 molecule, we take a onedimensional Born-Oppenheimer potential of the form
V (r) = − q

1

R 2

r+ 2

+α

−q

1

R 2

r− 2

+
+α

1
,
R

(5.18)

with the nuclei located at ± R2 where R is the internuclear distance. On the right hand side of Eq.
(5.18), the first two terms represent the attractive potential felt by the electron due to each of thenuclei, and the last term represents the energy due to nuclear repulsion. We solved for the BornOppenheimer potential energy surfaces of the molecular model by solving the time-independent
Schrödinger equation for varying internuclear distances R. We found the internuclear distance atequilibrium at Req = 2.64 a.u. by solving for the minimum of the ground state potential energy
surface, as shown in Fig. 5.10 (dashed line). The lowest electronic energies at the equilibrium
internuclear distance En (Req ) are E0 = −1.1591, E1 = −0.8502, E2 = −0.4905, E3 = −0.3278,
E4 = −0.2251, and E5 = −0.1665 a.u.
As mentioned, we wish to see if the concept of the absorption delay may be extended toa molecular system. Therefore, we calculated the absorption delay as a function of XUV central

119Figure 5.11: Absorption delay as a function of XUV central frequency for the resonant two-photon
ionization of H+
2 . Field parameters for the streaking pulse used are the same as those in Fig. 5.7.
For the XUV pulse, a sine squared envelope is used with a pulse duration of TXU V = 3 fs, and thesame intensity and CEP is used as in Fig. 5.7.

120
frequency in exactly the same manner as in the previous section. The range of XUV central
frequencies considered for the two-photon process was from 0.65 a.u. to 1.1591 a.u., where theupper bound is simply the ionization potential. Within this range, the electron is most likely totransition into the continuum via the absorption of two XUV photons. The results for the absorption
delay are shown in Fig. 5.11. We no longer see the pronounced features of individual resonances asin the previous two sections. Instead, the absorption delay appears to contain contributions fromseveral resonances, which gives a less clear picture. In order to further analyze the result, we again
turn to second order time dependent perturbation theory to help shine light on the physics. In
particular, as in the previous section we wish to know which excited states play a role over thecentral frequency range considered.

5.3.2

Role of the excited states and interfering pathways

Our analysis is made in a similar manner as in the previous section. We use Eq. (5.3), andallow more excited states to participate in the sum over intermediate states. The most distinctive
feature in Fig. 5.11 is the asymmetric, Fano-like lineshape centered around the |0i → |3i resonanttransition at ω30 = 0.8313 a.u. As seen in the previous examples of the current Chapter, theabsorption delay for resonant two-photon transitions as a function of XUV central frequency is
generally at a maximum when tuned exactly to resonance. This is in contrast to what is shown in
Fig. 5.11, where at ω = ω30 , the absorption delay is close to zero, which is perhaps indicative of aninterference effect. In order to reveal which states are responsible for this feature, we systematicallyallow for more intermediate states |mi to contribute to the sum in Eq. (5.3) and analyze the resulting
absorption delay for each number of contributing intermediate states.
In order to accurately determine the dipole moment contributions in Eq. (5.3), we havenumerically calculated the transition dipole moments, using the plane wave approximation for thebound-free transitions giving way to µf m . We then calculated cf i and φf using Eqs. (5.3) and(5.5), and obtained ∆tabs = ∂φf /∂E through the use of a numerical derivative. The absorption
delay calculated in this manner, when considering only the 3rd excited state as an intermediate is121

Figure 5.12: Resonant two-photon ionization of the 1D model of H+
2 calculated via time-dependent
perturbation theory (blue solid line) where (a) m = 1, (b) m = 1, 3, and (c) m = 1, 3, and 5 areconsidered in the sum in Eq. (5.3) as compared to TDSE simulations (red stars). The vertical lines
mark the third (black) and fifth (purple) excited state resonances.

122
shown in Fig. 5.12 (a) (solid line) as compared to the TDSE simulations (stars). We clearly seethat a single intermediate state cannot explain the feature seen around ω = ω30 . The absorption
delay, when including both the first and third excited states as intermediates, is presented in Fig.
5.12 (b). We now see the appearance of an asymmetric feature in the model calculation of theabsorption delay centered around the third excited state resonance, however it is of a significantly
larger magnitude than that found in the TDSE simulations. Finally, we consider including the fifth
excited state as a possible intermediate in Fig. 5.12 (c), and we see that a similar feature appears
about the fifth excited state resonance ω50 . However, due to a high density of states in this region
and the large bandwidth of the attosecond pulse, a larger number of intermediates are likely to becontributing, causing a different lineshape than is seen in the TDSE result. Therefore, in order toobtain better quantitative agreement in future work, more intermediate states need to be included
to predict this lineshape.
We note that in panels (b) and (c) of Fig. 5.12, we see a large, divergent peak around thecentral frequency corresponding to ω20 = 0.6686 a.u., in contrast to the TDSE result (stars), whichshows a smooth minimum in the absorption delay around this region. It is therefore anotherindication that our model calculation, when considering the 1st, 3rd, and 5th excited states as intermediates, is not sufficient in describing the dynamics in this region. It is unlikely that continuing
to add dipole-allowed intermediate states in our second-order model will account for this particularfeature, and therefore future work will need to go beyond the additional approximations made in
our model. For example, going beyond second order perturbation theory, which was discussed in
the previous section, is such a consideration. We saw that a third order process in the form of theIR pulse contributing a photon during transition into the continuum could lead to the occupationof an otherwise forbidden intermediate state. The same could be true in the current case for thesecond excited state of the model, but further investigation is needed.

Chapter 6

Single photon ionization with isolated elliptically polarized attosecond pulses

As discussed in Chapter 1, isolated circularly and elliptically polarized attosecond pulses
have recently been demonstrated experimentally [9]. Since this is a new development, the use of
these pulses for applications in attosecond spectroscopy has not been thoroughly explored. In this
Chapter we study photoionization of atoms with isolated elliptically polarized attosecond pulses,
and take a few first steps towards exploring the possibility of gaining information about both thepulse and the system being probed with the use of an isolated attosecond pulse only, i.e., without afemtosecond pulse. In section 6.1, we motivate the method and describe our numerical simulations.
Then, in section 6.2, we demonstrate the ability to control the resulting photoelectron angular
distribution through changing the ellipticity of the applied pulse, and postulate the possible connection of the rotation of the distribution to a time. Finally, in section 6.3, we discuss opportunities
for future study regarding so-called “ring current” states.

6.1

Unassisted XUV ionization
Up until now in this thesis, we have only considered methods in which an atom or moleculeis exposed to both an XUV attosecond pulse and a femtosecond NIR pulse. The goal of this section
is to analyze the photoelectron angular distribution (PAD) resulting from the application of an
isolated elliptically polarized attosecond pulse to an atom or molecule, and to explore if dynamical
information can be obtained. To this end, we show that the PAD depends upon the ellipticity of
the applied attosecond pulse, and postulate the use of such a dependency to study dynamics on a124
finer temporal resolution scale than the attoclock, which is a similar method.
The attoclock technique (see Refs. [13, 66], section 1.3.1.3) exposes an atom or molecule toa femtosecond circularly polarized laser pulse with a duration of two (or less) cycles. Normally, acircularly polarized pulse of many cycles will result in an isotropic PAD with respect to emission
angle. In the attoclock technique however, the femtosecond pulse has few enough cycles such thatthe emission of an electron is predicted to be most probable in the positive and negative directions
along the axis where the field vector points at its maximum intensity. This axis is typically chosen
in numerical attoclock simulations as the y−axis in the plane of polarization. In such a case,the positive ŷ−direction is defined as the “time zero” of the clock. The obtained photoelectron
spectrum in the attoclock setup is then measured in the plane of polarization. The attoclock timedelays are determined by the angular offset of the maximum of the PAD from the direction oftime zero (90◦ for the positive ŷ−direction). This angular offset is then converted into a time byconsidering that a full rotation of 360◦ corresponds to one period of the field (2.67 fs for an 800 nm
pulse).
It is desirable to see if we can extend the concept of the attoclock to the case of an isolated
XUV pulse which has a period on the scale of attoseconds. In order to achieve a similar symmetryof the PAD to the attoclock with an attosecond pulse of several cycle duration, we use an ellipticallypolarized pulse, constructed such that the field reaches its maximum when its polarization vector
lies along the positive ŷ−direction, i.e., the major axis of the polarization ellipse. Thus in thefollowing, we simulate the interaction of an atom with such a pulse, and determine the offset angle
between the maximum of the calculated PAD and the positive y−axis. We will then briefly consider
the consequences of converting this angle to a time, as is done for the attoclock.

6.1.1

TDSE simulations

We simulate the photoemission on a Cartesian 2D grid, corresponding to the polarization
plane of the pulse, and use the SLEPc method described in Chapter 3 to solve the TDSE. Although
we use length gauge for the interaction Hamiltonian, we set the vector potential as opposed to the125electric field, primarily due to how the code implementing the SLEPc method is set up. To thisend, we use
A0 (t)
A (t) = p[cos (ωt + φCEP ) ŷ − ξ sin (ωt + φCEP ) x̂] ,1 + ξ2(6.1)

where −1 ≤ ξ ≤ 1 is the ellipticity of the pulse, ω describes the central frequency, φCEP is the carrier
envelope phase, and A0 (t) is the envelope of the pulse, for which we use a Gaussian function:"#(t − TFWHM /2)2
A0 (t) = A0 exp −(6.2)
2(2 ln 2) TFWHM
where TFWHM is the full-width-half-maximum (FWHM) duration of the pulse envelope, and A0
is the strength of the vector potential, which is related to the strength of the electric field as
A0 = ωc E0 . Note that we have chosen the x − y plane as the plane of polarization with the y-axis
defining the major axis of the ellipse. Furthermore, ξ = ±1 corresponds to left (right) circularly
polarized light, and ξ = 0 corresponds to linearly polarized light along the y-axis. Throughout thissection we use an attosecond pulse with intensity IXU V = |E0 |2 = 1013 W/cm2 , central frequency
ω = 1.0 a.u., φCEP = 0, and a pulse duration of 5 cycles FWHM. The ellipticity of the pulse will
be varied in the calculations. In our simulations, we make use of a model hydrogen atom in 2D
Cartesian space (x, y). The potential is chosen as1
V (x, y) = − p
x2 + y 2 + α 2(6.3)

with α = 0.8. Using a grid spacing of ∆x = ∆y = 0.2 and a grid size of [-750, 750] a.u. for bothx− and y−axes, the use of the Krylov-Schur method gives a ground state energy of Eg = −0.4996
a.u., and the first and second excited states are degenerate with an energy of E1,2 = −0.1842 a.u.,which are reasonably close to the n = 1 and n = 2 energy levels of the 3D hydrogen atom.
We obtain the photoelectron momentum distributions in a similar way as outlined in thecase of linear streaking (see section 3.3). That is, we propagate the ionized wavepacket sufficiently
far on the grid such that it is well separated from the remaining ion core, while confirming that it
does not reach the boundaries. Assuming that we have propagated sufficiently far such that theerror from projecting onto a plane wave is negligible, we cut out the remaining core, and performa Fourier transform to obtain the photoemission spectrum.

126

6.2

Dependence of angular distributions on ellipticity
Using the method described above, we obtain the PAD for the ionization of our model system

initially in the ground state |gi with isolated elliptically polarized attosecond pulses of various
ellipticities. The results are shown in Fig. 6.1, where we show the PAD in momentum space
for a few selected ellipticities. In the case of ξ = 0.0 (a), we see the typical p-wave behavior ofphotoemission pumped by a linearly polarized pulse from an s−like ground state. As the ellipticity
is increased, the PAD extends over a broader angular range until for ξ = 1.0 ((e), right handed
circular polarization) we observe the expected isotropic emission pattern. Besides covering a broader
range in angular emission, the angle of maximum emission does not remain at 90◦ as the ellipticity
is increased from 0 to 1.0. This variation is not resolved in the presentation of the PADs in Fig.
6.1, but it becomes obvious in Fig. 6.2, where we have replotted the angular distributions (for fixed
k = 1.0 a.u.) for the same value of ellipticity as in Fig. 6.1. We can now clearly see that as theellipticity is increased, the angle for maximum emission is offset further from the y−axis (denoted
as 0◦ in the Figure). In Fig. 6.3, we summarize these results by showing this offset angle as afunction of ellipticity, spanning across the entire range of ellipticity from left circularly polarized
to right circularly polarized. We see that the maximum of the PAD rotates in the direction of thepolarization of the field such that the angular offset is an odd function with respect to ellipticity,i.e. representing the offset as θmax (ξ), we have θmax (ξ) = −θmax (−ξ). As the limit of circularly
polarized light is approached, the offset angle diverges since we reach an isotropic angular emission
pattern in this case. Therefore, we can conclude that the ellipticity and helicity (left vs. right)of the ionizing attosecond elliptical pulse can be determined from the PAD. Furthermore, such adependence of the preferred emission direction on the ellipticity may prove to be useful in potential
control applications.
Having in mind that PADs generated by the interaction of atoms with circularly polarized
femtosecond pulses were used to obtain information into temporal dynamics, it now becomes tempting to associate the observed rotation of the PAD with a time. The elliptically polarized pulses used

127

Figure 6.1: PADs for ionization from the ground state of the 2D hydrogen model for ellipticitiesof ξ = (a) 0.0 (linearly polarized), (b) 0.3, (c) 0.6, (d) 0.8, (e) 0.9, and (f) 1.0 (right circularly
polarized). Other laser parameters are given in the text.

128

Figure 6.2: PADs along kmax as a function of angle. Ellipticities are the same as in Fig. 6.1. The
peak of the distribution (red vertical line) and the location corresponding to zero degrees rotationfrom the major axis (black vertical line) are shown.

129

Figure 6.3: Angle of maximum emission in the PADs from ionization of the 1s state of the 2Dhydrogen model, plotted as (a) an angle and (b) a time (see text for relation), as a function of
ellipticity ξ.

in the calculations have a period of 152 as, and therefore we can associate a time with the observed
angle by using the conversion 360◦ = 152 as. Accordingly, the obtained angle of maximum emission
is replotted as time vs. ellipticity in Fig. 6.3 (b). Assuming that this conversion is meaningful, we
see that the times measured using this technique are below one attosecond.

6.3

Future work
In considering potential applications of the method described in this Chapter, we pose thequestion of whether the technique can be used to temporally resolve the rotation of the electronprobability distribution in a so-called “ring current state”, which has been observed in both atoms
and conjugated ring molecules [127, 128]. Furthermore, it has been recently proposed that the
attoclock could be used to time resolve the rotation of such a state [129]. Ring current states aresuperposition states which are characterized by an electron distribution which rotates in time, subsequently creating a magnetic field. These states have a well-defined angular momentum projection
quantum number ±m, where +(-) corresponds to counterclockwise (clockwise) rotation, if viewing
from above the polarization plane. In this section, we demonstrate the preparation of a ring current
state, using our 2D hydrogen model potential, to visualize the temporal- and angular-distribution

130of the state as a function of time. The demonstration is provided in order to motivate future proofof principle studies with respect to using isolated elliptically polarized attosecond pulses to image
the oscillation of the ring current state in time.

Figure 6.4: The first three eigenstates of the 2D model hydrogen atom are shown, where |gi
denotes the ground state, and since we only consider three eigenstates in this section, we label thefirst two excited states as |px i and |py i without a principal quantum number.

The first three eigenstates for the 2D hydrogen model defined by the potential in Eq. (6.3)
are shown in Fig. 6.4, where we label the states as |gi for the ground state, and since the first two
excited states have angular nodes aligned with the x− or y− axis, we denote them as |px i and|py i according to the direction along which the maximum of the electron probability distribution
lies. It is important to note that because we are in 2D Cartesian coordinates, one cannot label the
eigenstates of the 2D model Hamiltonian with the common quantum numbers of 3D atoms, andthe use of the labels px and py is a choice made for convenience, as these orbitals are similar to the|2px i and |2py i states of hydrogen. Since we limit ourselves to these three eigenstates, no principal
quantum number n is needed in the current discussion to distinguish between the different states.
In our model potential, a ring current state can be prepared by superposing the ground state withthe states |p± i defined as|p± i = |px i ± i |py i(6.4)|Ψ± i = cg |gi + cp |p± i ,(6.5)

such that the ring current states are

131
where |gi denotes the ground state, and |px i and |py i refer to the states labeled as such in Fig.
q2
6.4. For the current demonstration, let cg = √13 and cp =3 , and without loss of generality,
assume we wish to prepare the state |p+ i. The probability distribution |Ψ+ (t) |2 rotates in the
counterclockwise direction with a period of rotation of τ = 2π/ |Ep − Es | ' 20 a.u., or 483.8 as.
Since this rotation occurs on the attosecond time scale, it provides a test case to see if the rotation
can be temporally resolved to a high degree by ionizing the state with an isolated ellipticallypolarized attosecond pulse. We demonstrate the rotation of the ring current state by allowing
the superposition state to oscillate freely in time, and calculate the x̂ and ŷ components of thetime-dependent dipole moment, which is defined as the negative expectation value of position as afunction of time:d (t) = − hΨ+ (t) |r| Ψ+ (t)i .

(6.6)

The results are shown in Fig. 6.5, where the x− (blue) and y− (orange) components of d (t)demonstrate that the oscillation period is 20 a.u. as expected.
The goal for a possible next step will be to resolve the oscillating phase of the ring current
as a function of time, such that imaging of the superposition state becomes possible. Furthermore,
future work could also focus on the resonant two-photon ionization using the ring current state as
an intermediate, and whether or not the concept of the absorption delay for resonant two-photon
ionization with an isolated linearly polarized attosecond pulses, as discussed in detail in Chapter
5, can be extended to the proposed study.

132

Figure 6.5: Vector components of the time-dependent dipole moment, Eq. (6.6), of the ring
current state, when allowed to freely evolve in time. We see that both dx (blue) and dy (orange)
oscillate with a period of about 20 a.u.

Chapter 7

Summary and perspective

This thesis involved theoretical efforts to resolve electron dynamics on attosecond timescales
through the study of photoemission time delays in one- and two-photon ionization, particularly
through the use of the attosecond streaking technique. We have developed analytical models forthe two contributions to the photoemission delay: the so-called continuum and absorption timedelays, which agree well with numerical simulations. We have further shown that these analyticalformulae can be used to provide a deeper understanding of the origin of the time delays themselves,
and also to analyze effects in attosecond streaking not considered up to this point.
To this end, we covered in the first three Chapters a description of fundamental concepts
regarding attosecond laser physics and the basic techniques of the field, previous studies regarding
the subject of photoemission time delays, as well as an overview of the analytical and numericalmethods used in this thesis to analyze laser-matter interactions.
We then continued in Chapter 4 by analyzing the continuum delay in one-photon ionization.
We were able to derive an analytical formula for the contribution, and showed that it can be used
to efficiently obtain accurate estimates of the streaking time delay. Furthermore, we provided new
insights in the separation of the time delay into the WS time delay and a Coulomb-laser couplingterm. We then showed how the analytical formula could also be easily applied to other scenarios,and hence to analyze a recent theoretical proposal as well as the potential effects of the attochirpon the continuum delay.
In Chapter 5, we used time-dependent perturbation theory to derive an analytical expression

134of the absorption delay, and obtained good agreement with numerical simulations in the case of
an isolated resonance. We then extended our studies to gain insight into cases where multiple
resonances contribute to the absorption delay. Furthermore, by comparing our analytical results
to numerical simulations for a model system, we were able to determine that the streaking pulse
can play a role in populating so-called dark states, similar to that of the near-infrared field in
attosecond transient absorption spectroscopy. In the final section, we showed an application of ouranalytical formula to the study of the absorption delay in a model H+
2 molecule, and showed thatnovel features in the absorption delay are obtained. We ended the Chapter by suggesting that
future studies should be made to gain further insights into these features in order to study electron
dynamics in molecules.
Lastly, in Chapter 6 we began to study the ionization of an atom interacting with an isolated
elliptically polarized attosecond pulse, since these pulses have now become available in experiment.
We studied the photoelectron angular distribution resulting from the application of pulses withvarying ellipticity. We were able to determine that there is an offset of the angular distribution
with respect to the major polarization axis of the elliptical pulse, and that the offset depends
on ellipticity, and is asymmetric with respect to helicity. Finally, we considered the attoclock
concept to demonstrate that the rotations correspond to a resolution on a sub-attosecond timescale, assuming that such a conversion is applicable. We concluded by suggesting that this techniquecould potentially be used in future studies to image dynamics involving ring current states in atoms.

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