Theory and Application of Strong Field Ionization in
Atomic Systems
by
S. R. Walker
B.S., University of Delaware, 2018

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2023

Committee Members:
Andreas Becker, Chair
Michael Litos
David Jonas
Thomas Schibli
Andy Lucas

ii
Walker, S. R. (Ph.D., Physics)
Theory and Application of Strong Field Ionization in Atomic Systems
Thesis directed by Prof. Andreas Becker

Throughout this thesis we explore ionization processes ranging from the single photon x-ray
and ultraviolet limit to the opposite adiabatic infrared limit where large numbers of photons are
required for ionization. We will summarize criteria responsible for the breakdown of perturbation
theory and outline key aspects of the non-perturbative theory. Our numerical calculations bridge theshort wavelength limit of single photon perturbative ionization to the long wavelength limit of nonperturbative multiphoton ionization to demonstrate an optimal wavelength regime for generationof electron pulses for probing chiral systems in rare gas atoms.
Breakdown of the widely used asymptotic saddle point approximation in strong-field physics
is analyzed for ionization of electrons bound to p−states. We resolve this issue by evaluating
Keldysh’s ionization amplitude exactly. Finally, we present an analytic model for the perturbative
two-photon ionization of atoms by arbitrary fields. As an application we reconstruct isolated
attosecond pulses and pulse trains from a provided autocorrelation trace generated in a pumpprobe ionization measurement. The results are generalized to the non-perturbative limit.

Dedication

To my grandparents John and Lori as well as my mom Amy and siblings Cody and Kelsi.

iv

Acknowledgements

I would like to express gratitude to everyone who has helped me get to this point. I would
like to thank my advisor Andreas Becker, current group membrs: Yonas Gebre, Bejan Ghomashi,
Keegan Finger, Rex Planalp and Noah Welikson, former group memebers: Joel Venzke, Brynn
Reiff, and Lucas Kolanz and all of my friends and family for their support before, during and after
the pandemic.

vContents

Chapter
1 Introduction

1

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32 Perturbative Theory of Light-Matter Interaction
2.1

2.2

Green’s functions and the Lippmann-Schwinger equation . . . . . . . . . . . . . . . . .

7

2.1.1

Scattering off an external potential . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.2

Atomic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3

Single active electron approximation . . . . . . . . . . . . . . . . . . . . . . . . . 14

Dirac and Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1

2.3

6

Perturbative multi-photon ionization . . . . . . . . . . . . . . . . . . . . . . . . 20

Breakdown of perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1

Coulomb force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.2

Bound state intensity parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3

Continuum state intensity parameter . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Non-perturbative Theory of Light-Matter Interaction

25

3.1

Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2

Contour deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1

Application to Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

vi
3.3

3.4

3.5

Motion in a static field: Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1

Newton’s second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.2

Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Ionization by a static field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1

Oppenheimer’s rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.2

Landau and Lifshitz rate: Correcting the pre-factor . . . . . . . . . . . . . . . . 50

Motion in a dynamic field: Volkov states . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.1

3.6

Newton’s second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Ionization by a dynamic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6.1

Keldysh rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6.2

PPT rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6.3

Velocity-gauge approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Selectivity in electron emission induced by ultrashort circularly polarized laser pulses

74

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2

Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3

Selectivity in electron emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4

4.3.1

General trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.2

Ionization thresholds and two-photon ionization . . . . . . . . . . . . . . . . . . 86

4.3.3

Doorway states in few-photon ionization regime . . . . . . . . . . . . . . . . . . 92

4.3.4

Application to Argon atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Extending the theory of strong-field ionization into the UV limit
5.1

100

Circular polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1.1

Short-wavelength breakdown of the saddle point approximation . . . . . . . . 101

5.1.2

Ionization amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.1.3

Photon Absorption/Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

vii
5.1.4
5.2

5.3

5.4

Long Pulse Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Numerical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.1

Ionization Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2.2

Photoelectron Energy and Angular Distributions . . . . . . . . . . . . . . . . . 114

Intensity parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.1

Weak-Field Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.3.2

Strong-Field Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Linear and elliptical polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.1

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.4.2

Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Strong-field processes involving polychromatic fields

131

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2

Characterization of vacuum and deep ultraviolet pulses via two-photon autocorrelation signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.3

6.2.1

Multi-color Gaussian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2.2

Derivation of ionization amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2.3

Application and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Non-perturbative limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.3.1

Time-slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.3.2

Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7 Summary

148

Bibliography

150

viii

TablesTable
5.1

Table of bound-state energy levels and Yukawa parameters. . . . . . . . . . . . . . . . 113

ix

FiguresFigure2.1

Angular distribution (with Γ ≡ w) in arbitrary units for a plane-wave (electron)
scattering of a Yukawa potential with Z = 1, µ = 1/5 a.u. and k = 1/8 a.u.

2.2

. . . . . . 10

Comparison of the total ionization yield between Fermi and Dirac’s golden ruleand the time-dependent Schrödinger equation. The linearly polarized field has anintensity of I = 1 × 1014 W/cm2 , duration of 16 cycles and wavelengths between 10
nm and 91.928 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3

Shaded regions correspond to regions where E < Ea.u. (blue), z1 < 1 (green), z < 1(red) and z < 0.1 (dark red). An estimate for the applicability of perturbation
theory is given by the dark red region. The vertical grey line corresponds to a laser
wavelength of 800 nm, at which an intensity of 2.6 × 1012 W/cm2 (horizontal line)
marks the limit where perturbation theory would approximately break down for a
Ti-sapphire laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1

Contour of integration CR,ϵ (left). Taking the limit as R → ∞ and ϵ → 0+ defines thecontour C from which I(s), I± (s) and IR (s) are defined (right). Taking the limit
R → ∞ forces IR (s) → 0 and taking the limit ϵ → 0+ allows C to avoid the branch
points z = ±1 of (1 − z 2 )ν−1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2

Comparison of J3 (s) with its asymptotic approximation, Eq. (3.18). . . . . . . . . . . 32

x
3.3

Contours of integration for the integrand ext−x /3 . The dark triangular regions cor3

respond to zones where the contour may begin or end (e−t /3 → 0). CA corresponds3

to a convergent solution as x → ∞, while CB± correspond to divergent solutions as
x → ∞.
3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Plot of the real part of f (t) where t = tr + iti . Integration contours connecting thebottom-left and top-left regions are chosen such that they pass through the saddle
points in a direction that the imaginary part of the exponent is constant.

3.5. . . . . . 38

Plot of the real part of f (t) where t = tr + iti . Integration contours connecting thebottom-left and top-left regions are chosen such that they pass through the saddle
points in a direction that the imaginary part of the exponent is constant. Here only√√
the saddle point at t = − ∣x∣ contributes to the integral since t = ∣x∣ connects thebottom left region to the right-most region describing divergent solutions. . . . . . . 40

3.6

Airy function (blue), compared with the asymptotic approximations for x → ∞ (Eq.
(3.53), orange x > 0) and x → −∞ (Eq. (3.50), orange x < 0). . . . . . . . . . . . . . . . 41

3.7

Sketch of components required to compute the Oppenheimer rate.

Blue repre-

sents the initial 1s state of hydrogen atom, orange shows the final continuum stateΦk⊥ ,zcl (r) (with k⊥ = 0), green represents the field free Coulomb potential and red
depicts the DC Stark potential. The classical turning point zcl = −10 a.u. is noted
for a field-strength of F = 0.05 a.u.. For F → 0 ionization is determined by the tailof the initial bound-state alone (Eq. (3.92)). . . . . . . . . . . . . . . . . . . . . . . . . 45
3.8

Ionization rate Γ ≡ w of Oppenheimer as a function of 1/F in atomic units. . . . . . . 49

3.9

Comparison of results for the total ionization yield obtained by the time-dependent
Schrödinger equation and the s−state PPT formulas [20]. The circularly polarized
field has an intensity of I = 1 × 1014 W/cm2 , duration of 16 cycles and wavelengths
between 10 nm and 800 nm. Properties of the Yukawa potential are discussed in themain text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

xi
3.10 Comparison of the total ionization yield for hydrogen atom between results of thetime-dependent Schrödinger equation (+ symbols) and the s−state PPT formulas(solid line). The circularly polarized field has an intensity of I = 1 × 1014 W/cm2 ,duration of 16 cycles and wavelengths between 10 nm and 800 nm. The black curve
represents application of the Coulomb correction [23, 102] to the circular s-state
PPT rate [20,136]. The approximation only agrees in the limit of γ ≪ 1 and outlines
difficulties associated with Coulomb corrections when γ ≳ 1.

. . . . . . . . . . . . . . 69

3.11 Comparison of the total ionization yield between the time-dependent Schrödinger
equation (data from Fig. 3.9) and s−state Reiss’ formulas [146]. The circularly
polarized field has an intensity of I = 1 × 1014 W/cm2 , duration of 16 cycles andwavelengths between 10 nm and 800 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1

Photon absorption channels for interaction of a left-handed circularly polarized laser
field (∆l = ±1, ∆m = −1) with an atom in initial (a) p (red: m = −1, green: m = 0,
blue: m = 1) and (b) d state (red: m = −2, −1, green: m = 0, 1, blue: m = 2) [167]. . . 80

4.2

Ionization probabilities P+ (stars with line), P0 (squares with line), and P− (circles
with line) as function of wavelength for neon-like atom at 5×1014 W/cm2 interactingwith 10-cycle left-handed circularly polarized laser pulses [167].

4.3. . . . . . . . . . . . 82

Comparison of numerical results (circles with line) for the ratios (a) P0 /P− and (b)P+ (lf = Nmin + 1)/P− with the scaling laws from lowest order perturbation theoryfor ∆l = 1 pathway (solid lines, Eqs. (4.8,4.9)) and those from nonadiabatic tunneling predictions (dotted lines, Eqs. (4.10,4.12)) as function of wavelength. Laserparameters are the same as in Fig. 4.2 [167]. . . . . . . . . . . . . . . . . . . . . . . . . 85

xii
4.4

Upper row: (a) Channels for single-photon ionization. Lower row: Comparison ofratios (b) P+ /P− (stars with line), P+ (l = 2)/P− (circles with line), and P+ (l = 0)/P−(squares with line), and (c) P+ /P0 (stars with line), P+ (l = 2)/P0 (circles with line)
and P+ (l = 0)/P0 (squares with line) as function of wavelength. Also shown are thepredictions based on lowest order perturbation theory for the ∆l = 1 channels (solidlines without symbols, Eqs. (4.8,4.9)) [167].

4.5. . . . . . . . . . . . . . . . . . . . . . . . 87

Upper row: (a) Channels for two-photon ionization. Lower row: Comparison ofratios (b) P+ /P− (stars with line), P+ (l = 3)/P− (circles with line), and P+ (l = 1)/P−(squares with line), as well as (c) P+ /P0 (stars with line), P+ (l = 3)/P0 (circles withline) and P+ (l = 1)/P0 (squares with line) as function of wavelength. Also shown are
the predictions in lowest order perturbation for the ∆l = 1 channels (lines without
symbols) [167]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.6

Comparison of photo-electron angular distributions for ionization from initial statem = 1 (a,d), m = 0 (b,e) and m = −1 (c,f) near one-photon (a-c: 53.8 nm) andtwo-photon (d-f: 104.12 nm) threshold [167]. . . . . . . . . . . . . . . . . . . . . . . . . 90

4.7

Excitation probabilities in 3s (orange squares), 3d (red circles) and all states (bluestars) for ionization from initial states (a) 2p1 , (b) 2p0 and (c) 2p−1 for neon-likeatom at 5 × 1014 W/cm2 ) interacting with 10-cycle laser pulses [167]. . . . . . . . . . 92

4.8(a) Two- and (b) three-photon transitions into doorway states in the n = 3 shell frominitial 2p1 state. Other non-resonant transitions are also shown using same arrowstyles as in Fig. 4.1 [167].

4.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Comparison of excitation probabilities in 3s (orange squares), 3p (green triangles), 3d(red circles) with total excitation probability (blue stars) from initial states (a) 2p1 ,(b) 2p0 and (c) 2p−1 for neon-like atom with 10-cycle laser pulses at peak intensity
5 × 1014 W/cm2 ) [167]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

xiii
4.10 Same as Fig. 4.9 but for ns (orange squares), np (green triangles), nd (red circles),nf (purple crosses), ng (brown diamonds), nh (pink plus signs), and ni (grey 3
pointed stars) states with n ≥ 4. Total excitation probabilities are represented byblue 5 pointed stars [167]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.11 Ratio P+ /P− (stars with solid lines) and P0 /P− (open circles with dotted lines) as afunction of wavelength for (a) neon-like atom (5 × 1014 W/cm2 ) and (b) argon-likeatom (5 × 1013 W/cm2 ) interacting with 10-cycle laser pulses [166].
5.1. . . . . . . . . . 97

Total ionization yield for the length gauge amplitude (sections 3.6.1 and 3.6.2) corresponding to a 16 cycle laser pulse with an intensity of 1014 W/cm2 . Results arecompared between the saddle-point approximation (red dashed curve) and the exact
time integration (solid black curve) for s-states (left panel) and p-states (right-panel).101

5.2

Total ionization yield for the length gauge amplitude (sections 3.6.1 and 3.6.2) corresponding to a 16 cycle laser pulse with an intensity of 1014 W/cm2 . The yield iscompared between the saddle-point approximation (red dashed curve) and exact time
integration (solid black curve). Top-left, bottom and top-right panels correspond toinitial magnetic quantum number mi = −1, 0 and 1, respectively. . . . . . . . . . . . . 102

5.3

Comparison of predictions for the s-state model with the TDSE results for exponential parameters a = a0 /5, a0 /3, a0 and ∞. The red solid (dashed) line corresponds tocalculations using zero-range initial and final states without (with) the saddle-point
approximation. The green curve in the calculations the asymptotic initial states
are replaced by numerical states of the atomic Hamiltonian. The black curve corresponds to calculations with the same numerical initial state, but now with Volkov
states where the plane-wave scattering state has been replaced with scattering statesof the atom. a = ∞ is included for the initial Hydrogen ground-state and final
Coulomb-corrected WKB Volkov state (blue curve) [23, 102]. . . . . . . . . . . . . . . 112

xiv
5.4

Comparison of the results of the p-state model with the TDSE results for exponential
parameter a = a0 /2, where the left panel corresponds to the sum of all p-orbitals andthe right panel corresponds to 2s ionization in addition to ionization by all p−states.
The blue line corresponds to calculations using zero-range initial and final states. For
the results presented by the orange curve the asymptotic initial states is replaced bynumerical states of the atomic Hamiltonian. The results shown by the green curve
correspond to calculations with the same numerical initial state, but now with Volkov
states where the plane-wave scattering state has been replaced with scattering statesof the atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.5

Photo-electron energy (left panels) and angular distributions (right panels) for the
Neon-like anion data given in Fig. 5.4. Both atomic initial and final states are chosen
and wavelengths of 10 nm (top panels), 100 nm (middle panels) and 800 nm (bottom
panels) are selected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.6

Comparison of the weak field approximations. The black (+) symbol represents
the total ionization for a = a0 /5 obtained from the TDSE calculations, as in Fig.
5.3a. The cyan curve represents results for the dominant pathways described in thissection. The green and red curves represent data obtained from Eq. (5.72) and Eq.
(5.74), respectively. The purple curve shows data using the application of Eq. (5.74)
where only the lowest order n = nth has been included. . . . . . . . . . . . . . . . . . . 121

5.7

Dominant ionization pathway yield from Fig.(5.6) where jlS (kn ξ) is treated exactly
and jlA (kA r) is expanded as in Eq.(5.76) to zeroth, first, second, third and fourthorder αmax (solid lines). Crosses correspond to the a = a0 /5 s-state TDSE data from
Fig.(5.3).

5.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Dominant ionization pathway yield from Fig.(5.6) where jlA (kA r) is treated exactly
and jlS (kn ξ) is expanded as in Eq.(5.77) from zeroth to seventh order βmax (solidlines). Crosses correspond to the a = a0 /5 s-state TDSE data from Fig.(5.3).

. . . . 123

xv
5.9

Comparison of the predictions of the p-state model with the TDSE results for exponential parameter a = a0 /2. The top (bottom) panel corresponds to interaction
with a laser pulses with an ellipticity of ϵ = 0 (ϵ = 0.5). The blue line shows thepredictions for zero-range initial and final states. The orange curve represents data
for the asymptotic initial states with numerical states of the atomic Hamiltonian.
The green curve corresponds to the application of the same numerical initial state,
but now with Volkov states where the plane-wave scattering state has been replacedwith scattering states of the atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.1(a) Macroscopic high harmonic spectrum in hydrogen atom driven by 20-cycle, 800
nm laser pulse at 4 × 1013 W/cm2 : full spectrum (black line) and spectrum after
application of a Gaussian filter about 9th harmonic (red line). (b) Temporal profile offiltered 9th harmonic and (c) autocorrelation signal using filtered 9th harmonic [168]. 139

6.2

Comparison of time-dependent amplitude ∣f˜(t)∣ (blue line) and phase variation ω(t)(red line) of the filtered 9th harmonic (dashed line) and reconstructed pulse with (a)1, (b) 2, (c) 4 and (d) 6 Gaussians (solid line). The temporal profile is also provided
as insets within each subplot [168].

6.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Error analysis of Gaussian pulse characterization method for filtered 9th (blue) and11th harmonics (red): (a) ErrorNg [Pion ] (Eq. 6.20), and (b) ErrorNg [f˜] (Eq. 6.21)[168].

6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141(a) Temporal profile of the filtered 9th to 11th harmonic spectral range, (b) autocorrelation signal, and comparison of time-dependent amplitude (blue) and phasevariation (red) of the original pulse (dashed lines) and reconstructions (solid lines)
obtained with one (c) and three (d) Gaussians per central frequencies at 9th and11th harmonic [168].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Chapter 1

Introduction

1.1

Background
The interaction of atoms with radiation fields is qualitatively different depending on theintensity of the applied radiation and energy of constituent photons. Electrons with binding energy
̵ > Ip behave differently than the sameIp exposed to very weak radiation with photon energies hω̵ ≪ Ip (ω is the angularelectrons interacting with intense fields comprised of photons with energy hω̵ is the reduced Plank constant). When weak fields comprised offrequency of each photon and hhighly energetic photons strike an atom, there is some small, but finite probability that the initiallybound electron will absorb a single quanta and ionize, where all additional energy is translated into
the kinetic motion of the ejected electron:
̵ − Ip .
Ek = hω(1.1)

This single-photon ionization process was first explored by Einstein [54] in 1905.
̵ < Ip , then multiple quanta
If the weak field is now comprised of photons containing energy hωmust be absorbed simultaneously for ionization, which was first discussed by Dirac in 1927 [46,153].
In this early work it was stated that “These (two-photon) terms correspond to processes in which
two light-quanta are emitted or absorbed simultaneously, and cannot arise in a light-quantum theory
in which there are no forces between the light quanta. The effects of these terms will be foundto be negligible, so that this disagreement with the light-quantum theory is not serious.” Dirac
claimed that multi-photon processes were an artifact of his theory and not a true physical process.

2Over time this has been proven untrue. Four years later the full theory of two-photon absorption
was developed by Göppert-Mayer in 1931 [67] with experimental verification three decades later by
Kaiser [91] in 1961.
In subsequent decades Dirac’s perspective changed until in 1958 [48] he stated that “Similarly
for a more general radiative process in which two or more photons are simultaneously emitted orabsorbed, the probability is proportional to (an intensity) factor I for each absorbed photon andI + hν 3 /c2 for each emitted photon (where h is the Plank constant, c is the speed of light, hν is thephoton energy and I is the field intensity). Thus, the process is stimulated by incident radiation in
the same direction and with the same frequency and polarization as any of the emitted photons.”
These ideas outline the most fundamental principles of perturbative multi-photon processes. When
weak radiation interacts with an atom the probability of absorbing n quanta is proportional to(I/Ia.u. )n ≪ 1, where Ia.u. is a characteristic value defined as the intensity associated with theelectric field binding an electron within the Hydrogen atom (see section 2.3). Ionization in this̵
regime is dominated by the lowest-order (nth = ⌈Ip /hω⌉)photon process.
The advent of the first Ruby laser by Maiman in 1960 [120] allowed experiments to utilize
stronger coherent radiation than was possible before and allowed for serious investigations into
processes involving a large number of field quanta. This advancement allowed Agostini et al. in 1979
[3] to irradiate xenon atoms with high enough intensity to not only observe the dominant lowestorder (nth ) ionization process, but an additional perturbative (nth +1) photon process corresponding
to a single free-free transition in the continuum. This process is known as above threshold ionization(ATI). As technology improved over the following years, higher order ATI peaks were observed
where the largest probability for emission was surprisingly found for the absorption of n > nthphotons and not the lowest order process predicted from perturbation theory [2, 138]. These weresome of the earliest explorations of non-perturbative multi-photon processes. Transitions are still
described by the absorption and emission of field quanta, but the framework of perturbation theory
breaks down.
Theoretical models for these processes have been available decades before these experimental

3
observations. These models originate from Oppenheimer’s model of ionization of Hydrogen by astatic field in 1928 [132]. Here, the initial Hydrogen ground-state perturbatively interacts withthe applied field once before transitioning into a final state described by an otherwise free electron
propagating in a static electric field. The framework of perturbation theory is still used, but the
final-state includes all orders of the field-interaction. Subsequently in 1965 Keldysh [98] applied
Oppenheimer’s model to ionization of atoms by linearly-polarized alternating fields. The final state
describes the motion of an otherwise free electron in a dynamical field. Both of these models areable to determine largely the correct ionization behavior in terms of physical parameters, but are
missing a proportionality factor due to the fact that the final state in this first-order amplitude
does not include the long-range Coulomb potential during and after ionization.
In subsequent years Keldysh’s model was extended by Perelomov, Popov and Terent’ev[135–137] to include arbitrary initial states and both circular and elliptical polarization. These earliest Keldysh-like theories employed asymptotic approximations most accurate for long-wavelength
radiation. Faisal and Reiss [57, 146] extended the model into the short-wavelength limit by evaluating the amplitude exactly in the vector-potential (velocity) gauge opposed to the electric field(length) gauge used in earlier works. This method pushes the theory of strong field ionization into
the short-wavelength limit, but fails for initial states with orbital angular momentum l > 0 due tothe use of the velocity-gauge [20, 146].

1.2

Structure of the thesis
The thesis is structured as follows. Starting in Chapter 2, we will outline fundamental

principles of the perturbative theory of light-matter interaction. Here, electrons propagate as they
do in atomic systems and applied weak radiation fields kick the electron a finite number of times
between atomic states before finally transitioning in the continuum. We will finish the Chapter bydiscussing necessary conditions for the applicability of the theory.
Subsequently in Chapter 3, we will summarize key aspects of the non-perturbative theoryof light-matter interaction, where both atomic reference-states and reference-states of an elec-

4tron propagating in the field are required to determine a low-order expansion which describes thenon-perturbative ionization process. We will start by describing accurate numerical methods formodeling the ionization of atoms. Afterwards we will describe the analytic models of Oppenheimer
and Keldysh [98, 132] and briefly summarize the influential models of Perelomov, Popov and Terent’ev as well as Reiss. As an application we will outline difficulties associated with the long-range
Coulomb interaction and applicability to ”real” atoms.
In Chapter 4 we present results of numerical calculations describing ionization where we
explore the transition from single photon to perturbative and non-perturbative multi-photon ionization. We explore optimal wavelength regimes for how atto- or femtosecond laser pulses may
potentially provide the opportunity to generate ultrashort spin-polarized electron pulses for probing chiral systems and magnetic properties of materials. A key element in the generation of spinpolarized electrons is a selectivity in ionization to the sense of the electron’s rotation in the initialstate with respect to the rotation direction of the laser field. We will therefore identify mechanisms
which increase the ionization from a specific magnetic sublevel over the others.
In Chapter 5 we demonstrate the breakdown of the asymptotic approximations used in theionization amplitude of Keldysh as well as Perelomov, Popov and Terent’ev for initial p-states irradiated by circularly polarized fields. To overcome this challenge we expand the ionization amplitude
for an arbitrary initial angular momentum state exactly in terms of partial wave components forboth circularly and elliptically polarized fields. Our spherical expansion has been motivated by
the cylindrical expansion employed by Reiss [146] in velocity-gauge and Bauer [23] in length-gauge,
however we believe our complementary method is more convenient for initial states consisting of
higher angular momentum components.
In Chapter 6 we will derive semi-analytic perturbative two-photon amplitudes for the ionization of atoms by arbitrary polychromatic fields. These formulas are applied to reconstruct isolated
attosecond pulses and pulse trains from a provided autocorrelation trace. A feedback loop is used
and ionization is determined from a model laser parameterized by a set of variational parameters.
The field is recovered when variational parameters are determined such that the exact autocorre-5
lation trace agrees with the trace generated by the parameterized field. The amplitude is extended
to the non-perturbative limit where time is partitioned into slices (tm−1 , tm ] (m ∈ Z) [63] narrow
enough such that perturbative formulas may be applied over each interval. We believe that this approach may be a useful alternative to the earlier non-perturbative methods in applications requiring
short pulses and long-range interactions.
We will complete the thesis with a brief summary.

Chapter 2

Perturbative Theory of Light-Matter Interaction

The theory section of this thesis is intended to systematically introduce concepts such asGreen’s functions, perturbation theory and non-perturbative approaches to ionization in a way thatbridges standard upper-division undergraduate quantum mechanics, first year graduate quantum
mechanics [125, 151, 152], more advanced Green’s function approaches to strong field ionization[5,57,98,107,132,146] and finally our own non-perturbative approach that generalizes the amplitudeof [98,132] by expanding the final scattering-state in partial waves [27,29,107] which will be discussed
in Chapter 5. We will try our best to honor the original texts, but may find our own way to theirresults. The present Chapter will focus on the perturbative theory while the subsequent Chapter
will discuss the nonperturbative approaches.
The first section will focus on Green’s functions and the Lippmann-Schwinger equation. We
will use reference states of a free particle scattering off of a Yukawa potential to motivate the idea ofa reference Green’s function and perturbation theory. In the second section we will describe multiphoton ionization of an atom interacting with a weak laser field. Here, field-free atomic states willtake the place of the plane-wave reference Green’s function and the (dipole) laser interaction willtake the place of the Yukawa potential described in the earlier section. We will focus on one- andtwo-photon ionization. In the third section we will discuss the breakdown of perturbation theory
and motivate the subsequent Chapter on nonperturbative approaches. Throughout this thesis we̵ = 1, whenwill use cgs units and switch to the system of atomic units (a.u.), where m = ∣e∣ = hconvenient.

7

2.1

Green’s functions and the Lippmann-Schwinger equation
The time-dependent Schrödinger equation̵[ih∂− H(x)] Ψj (x) = 0
∂t

(2.1)

describes the time-evolution of a physical state Ψj (x) under the influence of the Hamiltonian
H(x) =p2+ V (x)2m(2.2)

̵ r determines the momentum of a state and V (x) represents anfor a single particle. Here, p = −ih∇external potential with x ≡ {r, t}. The index j is included to emphasize that a system of orthogonalstates satisfies Eq. (2.1).
One may relate some known initial state Ψ(x) of the system to the state of the same system
at a later time through̵∫Ψ(x) = ih

dr0 G(x; r0 , t0 )Ψ(r0 , t0 ) ,

(2.3)

i
G(x; x′ ) = − ̵ θ(t − t′ ) ⨋ Ψj (x)Ψj (x′ )
h
j

(2.4)

R3

where the Green’s function

is determined from all states Ψj satisfying Eq. (2.1). Since we do not know the set of Ψj (x) it isoften convenient to suppose we have a complete set of eigenstates {ϕj (x)}j=1,2,⋯ corresponding toa simpler closely related Schrödinger equation with Hamiltonian Hi/f (x) where∂− Hi/f (x)] ϕj (x) = 0.
∂t

(2.5)

∂
− H(x)] G(x; x′ ) = δ (4) (x − x′ )
∂t

(2.6)

∂
− Hi/f (x)] Gi/f (x; x′ ) = δ (4) (x − x′ )
∂t

(2.7)

Ψ(x) = ϕi (x) + ∫ d4 x1 G(x; x1 )V (x1 )ϕi (t1 ).

(2.8)

̵
[ih
Using
̵
[ih
from Eq. (2.1) and̵
[ih
it is clear that

8
with
4
∫ d x1 ≡ ∫tti

dt1 ∫ dr1

(2.9)

and
δ (4) (x − x′ ) ≡ δ(t − t′ )δ (3) (r − r′ )(2.10)

where t1 describes the moment where the initial state ϕi (x) is set. This is called the LippmannSchwinger equation [112]. Continuing along these lines the full Green’s function can be expandedasG(x; x0 ) = Gi (x; x0 ) + ∫ d4 x1 G(x; x1 )Vi (x1 )Gi (x1 ; x0 )= Gf (x; x0 ) + ∫ d4 x1 Gf (x; x1 )Vf (x1 )G(x1 ; x0 ) (2.11)
with respect to the reference states where i/f are referred to as an initial- and final-state expansion.

2.1.1

Scattering off an external potential

As a concrete example, consider the Hamiltonian
H0 (t) =p2
2m(2.12)

corresponding to a free particle with states̵̵ 2 k2hϕk (x) = e−(i/h) 2m t ϕk (r) and ϕk (r) =eik⋅r.
(2π)3/2(2.13)The reference Green’s function becomes
3/2m(∆r)2i
dk ik⋅∆r−(i/h)
im̵ h̵ 2 k2 ∆t− 2ih∆t̵
2m
Gi (x; x′ ) = − ̵ θ(∆t) ∫
e
=
−
θ(∆t)
(
)
e
.
̵
̵
h
(2π)3
h
2πih∆t(2.14)

where in the Fourier basis
Φ̃(k) = ∫ dr ϕk (r)Φ(r) with

Φ(r) = ∫ dk Φ̃(k)ϕk (r).

(2.15)

one may find a simpler representation of the Green’s function by transforming r and r′ to give therepresentation in the momentum spaceiG̃i (k, t; k′ , t′ ) = − ̵ θ(∆t) ∫ dq ϕ̃q (k, t)ϕ̃q (k′ , t′ ).
h(2.16)

9
Transforming the states we haveϕ̃q (k) = δ (3) (k − q)(2.17)i̵ h̵ 2G̃0 (k, t; k′ , t′ ) = − ̵ θ(∆t)δ (3) (∆k)e−(i/h) 2m ∆th(2.18)

which gives

and the trivial equation of motionF=∆p ̵ ∆k
=h
= 0.
∆t
∆t

(2.19)

Individual momentum states ϕk (x) exactly follow the classical equation of motion and quantum
mechanics describes the interferences between momentum components k and k′ within a single
state.
Considering now Eq. (2.1) with potential
V (x) = −Z∣e∣2 −µrer(2.20)and truncating at first orderΨ(x) = ϕki (x) + ∫ d4 x1 Gi (x; x1 )V (x1 )ϕki (x1 ) = Ψ(in) (x) + Ψ(sc) (x)(2.21)

where the superscript (in) denotes the incoming plane-wave and (sc) the first-order scattered state.
The contribution to the scattered wave with momentum k is given byt22i̵ h̵ 2
M(sc) (k, t) = ∫ d4 x ϕk (x)Ψ(sc) (x) = − ̵ ∫ dt1 e(i/h) 2m (k −ki )t1 Ṽ (k; ki )h 0

(2.22)

where
t

∫

0

̵

̵

̵
̵
dt1 e(i/h)∆εt = e(i/h)∆εt/2 sinc((1/h)∆εt/2)t≡ δt ((1/h)∆ε/2)2mt→∞̵
ÐÐ→ δ (1) ((1/h)∆ε/2)= ̵ δ (1) (k − ki ) (2.23)
hk

enforces k = ki in the t → ∞ limit andṼ (k; ki ) = ∫ dr1 ϕk (r1 ) [−Z∣e∣2 −µr1−Z∣e∣2e] ϕki (r1 ) = 2 2
r1
2π (q + µ2 )(2.24)

10
with q = ∥k − ki ∥ = 2k sin(θ/2) yieldingM(sc) (k, t) =̵iZ∣e∣2 δt ((1/h)∆ε/2)̵
2π 2 h(q 2 + µ2 )(2.25)

with the scattered rate
Z 2 ∣e∣4 m δ (1) (k − ki )
̵ 3 k (q 2 + µ2 )24π 4 h(2.26)∞dλ(sc)
Z 2 ∣e∣4 mk
1= ∫ dk k 2 w(k k̂) = 4 ̵ 3dΩ
4π h (4k 2 sin2 (θ/2) + µ2 )20(2.27)

w(sc) (k) = lim t−1 ∣M(sc) (k, t)∣2 =t→∞

and angular distribution

which is the Rutherford scattering angular distribution as µ → 0.
In Fig. 2.1 an example of an angular distribution is shown for a plane-wave (electron) scattering of a Yukawa potential with Z = 1, µ = 1/5 a.u. and k = 1/8 a.u..
90
120

60
0.03

150

30

0.02
0.01

180

0

0

210

330

240

300
270

Figure 2.1: Angular distribution (with Γ ≡ w) in arbitrary units for a plane-wave (electron) scattering of a Yukawa potential with Z = 1, µ = 1/5 a.u. and k = 1/8 a.u.

The Lippmann-Schwinger equation recursively relates the solution of a simple problem to amore complicated problem of interest. In the next section we describe atomic reference states and11subsequently apply the ideas from this section to discuss ionization of an atom by the perturbationof a weak electric field.

2.1.2

Atomic states

Atomic states are often represented in spherical-polar coordinatesx = r sin θ cos ϕ,

y = r sin θ sin ϕand z = r cos θ(2.28)

with some spherically symmetric potential Va (r) = Va (r). Using the Laplacian
∆=

1 ∂
L2
2 ∂
(r
)
+
r2 ∂r
∂r
r2

(2.29)

the time-independent Schrödinger equation (TISE) in atomic units is written as
1 ∂L2 ψ(r)2 ∂ψ(r)(r)++ 2(E − Va (r))ψ(r) = 0
r2 ∂r
∂r
r2

(2.30)

where the eigenspace of the angular momentum operator L2 is given by
L2 Ylm (r̂) = l(l + 1)Ylm (r̂)(2.31)

with spherical harmonics Ylm (r̂). Applying the separation
ψ = R(r)Ylm (r̂) with

R(r) =

χ(r)r(2.32)

we are left with the radial equationd2 χ(r)l(l + 1)
+ [2(E − Va (r)) −] χ(r) = 0.
2
dr
r2
r→0

Since ψ is finite everywhere in space we have χ(r) ÐÐ→ 0 and(2.33)

χ(r)r bounded for r → ∞ [27, 107].

For the special case of Hydrogen the analytic solution may be determined. Using Va (r) = −1/r
and defining√n = 1/ −2Eand ρ = 2r/n

(2.34)

the radial Schrödinger equation can be written as
2
1 n l(l + 1)
R′′ + R′ + [− + −]R = 0ρ
4 ρ
ρ2(2.35)

12
where primes indicate derivatives with respect to ρ. The solution R is found by taking the productof its solution for ρ → 0 and for ρ → ∞ with some other function which must be determined. Taking
ρ → 0 and setting R ∝ ρν one seesν(ν − 1)rν−2 + 2νρν−2 − l(l + 1)rν−2 → 0 ⇒ ν = l.

(2.36)

Similarly for ρ → ∞ the TISE becomes1
R′′ − R = 0 ⇒ R ∝ e−ρ/24(2.37)

R = ρl e−ρ/2 w(ρ).

(2.38)

yielding

Inserting R back into the Schrödinger equation one obtains the hypergeometric differential equation
ρw′′ (ρ) + (2l + 2 − ρ)w′ (ρ) + (n − l − 1)w(ρ) = 0(2.39)

which we will describe in the next subsection.

2.1.2.1

Hypergeometric functions

Eq. (2.39) is a particular case of the confluent hypergeometric differential equationzu′′ + (γ − z)u′ − αu = 0 ,(2.40)

which is satisfied by the hypergeometric function
1 F1 (α; γ; z) = 1 +∞α z α(α + 1) z 2(α)n z n++⋯= ∑,γ 1! γ(γ + 1) 2!
n=0 (γ)n n!

(2.41)

where(a)n =

Γ(a + n)Γ(a)(2.42)

is the Pochhammer symbol. Here, the sum converges for all finite z and α while γ cannot be zeroor a negative integer.

13
Since the confluent hypergeometric differential equation is second order there must be an
additional linearly independent solution which is determined by the substitution [107]u = z 1−γ u1 ,(2.43)

zu′′1 + (2 − γ − z)u′1 − (α − γ + 1)u1 = 0 ,(2.44)

where

which immediately yields the independent solutionz 1−γ 1 F1 (α − γ + 1; 2 − γ; z) ,(2.45)

where γ cannot be an integer greater than 1. The general solution is thereforeu = a 1 F1 (α; γ; z) + b z 1−γ 1 F1 (α − γ + 1; 2 − γ; z)(2.46)

where a and b are arbitrary constants.
Returning now to the determination of electronic states for atomic hydrogen we havew = a 1 F1 (−n + l + 1; 2l + 2; ρ)(2.47)

since solutions must be finite at the origin. Using the boundedness of R and properties of 1 F1 inthe limit where ρ → ∞ from Ref. [107] we have n to be an integer with n − l − 1 ≥ 0. The bound-statesolutions are therefore given by¿Á (n + 1)!
2ÁÀ
Rkl (r) = l+2(2r)l e−r/n 1 F1 (−n + l + 1; 2l + 2; 2r/n).
n (2l + 1)!
(n − l − 1)!

(2.48)

Following the same steps for continuum states we have parameters√n = −i/ 2E = −i/k,ρ = 2ikr(2.49)

and radial functions
Rkl (r) =Ck(2kr)l e−ikr 1 F1 (i/k + l + 1; 2l + 2; 2ikr)(2l + 1)!

with normalization√
Ck =on the k-scale.

2 π/2kke∣Γ(l + 1 − i/k)∣π(2.50)(2.51)

14
2.1.3

Single active electron approximation

For atoms other than Hydrogen analytic solutions to Eq. (2.33) are in general not known.
We assume each individual electron orbitals in an atom given by an averaged (fixed in time) atomic
potential
Va (r) = Vlong (r) + Vshort (r) + Vshell (r)(2.52)

generated from density functional theory (see Ref. [143]). The numerical potential Va (r) is fit to
the functions
Vlong (r) = −

Z∣e∣2
r

(2.53)

which corresponds to the long-range interaction between an ionized electron and the residual chargeZ∣e∣, while
Zc ∣e∣2 −crer(2.54)

Vshell (r) = − ∑ aj e−bj r(2.55)

Vshort (r) = −andjapproximate the average interaction of all other electrons at short distances.
To find the states in this case one defines a maximum radius rmax with boundary condition
χ(rmax ) = 0 and solves Eq. (2.33) numerically. The first approach we use involves expanding Eq.
(2.33) in some basis and solving the eigenvalue problem
Hl ∣χl ⟩ = E∣χl ⟩(2.56)

for each l to determine a truncated set of both bound-states and discrete continuum states thatspan the finite box. The second approach we follow involves integrating the radial function χ fromdr = 0 to rmax using Eq. (2.33) where a boundary condition drχ∣r=0 is set to some small constant

ϵ. For bound states a particular energy E < 0 is chosen as a lower bound. If χ(rmax ) = 0 (upto some tolerance) then the associated state is a bound-state. If the boundary condition at rmax
is not met, then E is increased by some increment and the boundary is checked yet again. For

15continuum states the procedure is simpler. All E > 0 satisfy Eq. (2.33) and therefore one only needs
to integrate the radial equation and normalize.
On the k−scale the parameter ϵ is determined by fitting each radial function to the asymptotic
solution for r → ∞ [27]√χk (r) ∼2
sin [kr + log(2kr)/k − lπ/2 + ηl (k)] ,π(2.57)

where the additional fit parameter ηl (k) describes the so-called scattering phase shifts relates radialfunctions Rkl (r) to the scattering states(−)ϕk (r) =1 ∞ l l −iηl (k)Rk,l (r) Ylm (k̂)Ylm (r̂) ,∑ ∑ i ek l=0 m=−l(2.58)

which describe the momentum of an electron in the continuum. For hydrogen atom we haveηl (k) = arg [Γ(l + 1 − i/k)] .
For short-range potentials the same procedure is applied to the asymptotic solution√
2
sin [kr − lπ/2 + ηl (k)] ,χk (r) ∼π(2.59)(2.60)whereηl (k) = 0(2.61)

for a zero-range potential. We almost always use the first method to determine a set of boundstates and the second method to describe the continuum.
Diagonalization ensures orthogonality of the bound states, but produces a set of discretized
continuum states. Assuming continuum wave-packets do not approach r = rmax , the discrete basis
is able to describe physical observables and represent the state well. Unfortunately, in the limitof t → ∞, corresponding to states of ejected electrons (r → ∞), determination and interpretationof the photo-electron distribution becomes difficult, but possible given the binning procedure ofRef. [84] as an example.
The shooting method proves to be an inexpensive straight-forward procedure for determining final scattering states with clear physical interpretation. When applied to bound states an16additional orthogonalization procedure is required and accuracy is not guaranteed. In practicethe discretized continuum proves to be an efficient computational basis where the true scattering
states determined through the shooting method may be projected onto the computational basis todescribe properties of the photo-electron distribution at the end of a calculation. It is importantto separate states corresponding to physical observables and convenient basis sets for numericalcalculations.

2.2

Dirac and Fermi’s golden rule
We will now consider the ionization of an atom initially occupying the atomic statei
ϕi (x) = Ri (r) Ylm(r̂)
i(2.62)

by a single photon from the linearly-polarized monochromatic electric field
E(t) =

E −iωt[e+ eiωt ] .
2(2.63)

Using the first-order expansion [47, 62]
Ψ(x) ≈ ϕi (x) + ∫ d4 x1 Gi (x; x1 )VL (x1 )ϕi (x1 )(2.64)VL (x) = ∣e∣E(t) ⋅ r(2.65)and laser interaction

the distribution of ejected photo-electrons can be described byM(k, t) = ∫i(−)
d x ϕk (x)Ψ(x) = − ̵ ∫
2h 0
4

t

dt1 e

̵ h̵ 2 k2 +Ip −hω)t̵(i/h)(
12m

Vi (k) ,(2.66)

where
Ha (x)ϕi (x) = −Ip ϕi (x),(2.67)

̵ 2 κ2 /2m andIp ≡ h(−)(k,l,m)Vi (k) ≡ ∫ dr ϕk (r) [∣e∣Ez] ϕi (r) = (∣e∣E/k) ∑ (−i)l eiηl (k) Ylm (k̂)(z)il,m(2.68)

17
with√(k,l,m)
(z)i
=

∞
4π
mi
2
0
m
∫ dr r Rk,l (r)rRi (r) ∫ dΩk Yl (r̂)Y1 (r̂)Yli (r̂).
3 0(2.69)

Since the energy of a final ionized state must be larger than the initial ground state we have
only included the single photon absorption contribution1∣e∣E(t) ⋅ r ↦ ∣e∣Ee−iωt z
2(2.70)

and assumed polarization in the ẑ-direction. Additionally the integral over three spherical harmonics
mi0m∫ dΩk Yl (r̂)Y1 (r̂)Yli (r̂)(2.71)

enforces the angular momentum selection rule mi = m and ∣li − 1∣ ≤ l ≤ li + 1 for linearly polarizedlasers in the ẑ-direction. For elliptically polarized lasers (in the x − y plane) the selection rules
remain relating l and li while m = mi ± 1 due to the harmonic contributions Y1±1 (r̂). For higherorder photon processes corresponding to Eq. (2.11) these rules hold between photon orders.
For the simplest case of an s-state interacting with a zero-range potential one has [136]Ri (r) ≈ Cκ,0 κ3/2
and
eik⋅r
=
ϕk (r) =(2π)3/2

√

e−κr
κr

2
l
m
∑ i jl (kr)Ylm (k̂)Yl (r̂)π l,m(2.72)(2.73)

which immediately yields
Vi (k) = [∣e∣E ⋅ (i∇k )] ϕ̃i (k) ,(2.74)e−ik⋅r1
ϕi (r1 )(2π)3/2(2.75)whereϕ̃i (k) = ∫ dr1

is the Fourier transform of the initial state and Cκ,0 is an asymptotic constant which describes thethe ground-state for κr ≫ 1.
The matrix element becomes
Vi (k) = [∣e∣E ⋅ (i∇k )] ∫ dr1−iCκ,0 (2κ)1/2
eik⋅r
ϕ
(r
)
=
[∣e∣Ekz ]i 1
π(k 2 + κ2 )2(2π)3/2(2.76)

18
after insertingϕ̃i (k) =

√

Cκ,0
κ
.
2 π(k 2 + κ2 )(2.77)

The amplitude is completed by using
t

∫

0

̵
̵
̵
̵
dt1 e(i/h)∆εt = e(i/h)∆εt/2 sinc((1/h)∆εt/2)t≡ δt ((1/h)∆ε/2)

̵2 2
2m
t→∞
̵ δ (1) ( h k + Ip − hω)̵ÐÐ→ 2h= ̵ δ (1) (k − kf ) (2.78)
hk
2m

̵ f=
where hk√̵ − Ip ] is the final momentum of the ejected electron.
2m [hω

The resulting amplitude is−iCκ,0 (2κ)1/2i̵
[∣e∣Ekz ]M(k, t) = − ̵ δt ((1/h)∆ε/2)2hπ(k 2 + κ2 )2(2.79)

w(k) = lim t−1 ∣M(k, t)∣2 .

(2.80)

2π (1)
2̵
lim t−1 δt ((1/h)∆ε/2)=δ (k − kf )t→∞vf

(2.81)

with partial rate
t→∞

Using

with
̵ f=
mvf = hk√̵ − Ip )2m(hω(2.82)

yields
2
(2κ)
Cκ,0
m
w(k) = ̵ 3 δ (1) (k − kf )[∣e∣2 E 2 kf cos2 θk ] .
22hπ(kf + κ2 )4(2.83)

Hence, angular distributions are given by2∞dλm Cκ,0 (2κ)= ∫ dk k 2 w(k) = ̵ 3[∣e∣2 E 2 kf3 cos2 θk ]dΩ
2h π(kf2 + κ2 )4
0(2.84)2πdλdλ m Cκ,0 (2κ)[∣e∣2 E 2 kf3 cos2 θk ]= ∫ dφk= ̵3 2d(cos θk )dΩ h−π
(kf + κ2 )4(2.85)

andwhere the ionization rate is determined by
2
2
4mCκ,04m Cκ,0 κ
2l +1
2 2 3 kf →0
λ = ̵3 2[∣e∣ E kf ] ÐÐÐ→ ̵ 3 7 ∣e∣2 E 2 kf3 ∝ kf f
2
4
3h (kf + κ )3h κ(2.86)

19
2l +1

which follows the expected kf f

(lf = 1) near-threshold scaling [27, 29, 172]. Exactly the same(−)

steps are followed in the case of more complicated atomic potentials where ϕi (r) and ϕk (r) arecalculated numerically. This approach is taken in Chapter 6.
TDSE
golden ruleion yield

10 -1

10 -2

10 -3

10

20

30

40

50

60

70

80

90

wavelength (nm)Figure 2.2: Comparison of the total ionization yield between Fermi and Dirac’s golden rule and thetime-dependent Schrödinger equation. The linearly polarized field has an intensity of I = 1 × 1014
W/cm2 , duration of 16 cycles and wavelengths between 10 nm and 91.928 nm.

In Fig. 2.2 we plot the zero-range total ionization yieldNion = 1 − exp [−λ (Ncycles

2π
)]
ω

(2.87)(solid line) for a 16 cycle laser at I = 1 × 1014 W/cm2 for wavelengths between 10 nm and 91.928
̵ = Ip = 13.6 eV. For comparison the blue dots
nm, with the longest wavelength corresponding to hω20
in Fig. 2.2 correspond to the numerical solution of the TDSE using 8th order B-splines (which will
be discussed in Chapter 3) where the ground state is described by the Yukawa potential
Va (r) = −Z∣e∣2 −r/aer(2.88)

with a range of one fifth of a Bohr radii (a = a0 /5), Z chosen such that Ip = 13.6 eV and Cκ,0 is
fit to the asymptotic tail of the exact ground state. The TDSE calculations use a monochromatic
16 cycle flat-top pulse which smoothly turns both on an off using a 2 cycle sin2 ramp function.
Differences in the two plots correspond to errors introduced by the asymptotic ground state andthe fact that we are not including the shape of the envelope in the analytic calculations.

2.2.1

Perturbative multi-photon ionization

Expanding Eq. (2.11) to second orderΨ(2) (x) = ∫ d4 x2 G0 (x; x2 ) [∣e∣E(t2 ) ⋅ r2 ] ∫ d4 x1 G0 (x2 ; x1 )[∣e∣E(t1 ) ⋅ r1 ]ϕi (x1 )= ∫ d4 x2 G0 (x; x2 ) [∣e∣E(t2 ) ⋅ r2 ] Ψ(1) (x2 ) (2.89)
describes the lowest-order two-photon correction whereΨ(1) (x) = ∫ d4 x1 G0 (x; x1 )[∣e∣E(t1 ) ⋅ r1 ]ϕi (x1 )(2.90)

corresponds to the first order correction as before. In Chapter 6 we will apply these formulas
to Helium in the case of a poly-chromatic field generated from high-order harmonic generation.
Assuming conditions described in the next section are met, Eq. (2.11) can be applied iteratively togenerate higher photon amplitudes where factors containing e−iωtj (eiωtj ) describe the absorption(emission) of a single photon during the j−th process.

2.3

Breakdown of perturbation theory
Several criteria exists which must all be simultaneously satisfied for the validity of perturba-

tion theory. Following Ref. [101] we will outline some of the simplest ones.

21
2.3.1

Coulomb force

The first criterium for the applicability of perturbation theory corresponds to the fact thatthe force associated with the external (laser) field must be significantly weaker than the Coulomb
force binding a free electron to the atomic core. Here, we approximate the Coulomb force at anelectronic distance of one Bohr radius a0 in Hydrogen as∣FCoul (r)∣ =∣e∣2 ∣e∣2
∼ 2
r2
a0

(2.91)

where the force applied by the electric field is∣FL ∣ = ∣e∣E.

(2.92)

Setting these two quantities equal to one another, we obtain the characteristic electric field
and intensity
Ea.u. ≡

c 2
∣e∣
= 5.14 × 109 V/cm ⇒ Ia.u. =Ea.u. = 3.51 × 1016 W/cm228π
a0(2.93)

binding the electron to the atomic core. For perturbation theory to be valid the external field
amplitude E must satisfy
E ≪ Ea.u.

(2.94)and therefore not significantly distort the ground state orbital motion.

2.3.2

Bound state intensity parameter̵ to the power
The second parameter we will describe relates the energy of a single photon hωbroadening of a level due to the application of a strong field ∣e∣Ea0 where the characteristic scale
̵ 2 /2ma2 and E is thea0 is the Bohr radius of the atom related to the ionization potential by Ip = h0field amplitude. The ratioγ=̵hω∣e∣Ea0(2.95)

is referred to as the Keldysh parameter [98]. When this value is significantly greater than one(γ ≫ 1) the level broadening is small compared to the energy contained in a single quanta. When22(γ ≪ 1) the level broadening is large compared to the photon energy and the number of photons
needed to transition between levels is indefinite.
In the literature Reiss [146] has introduced a more convenient ”bound-state” intensity parameter
z1 =

2Up
1
= 2 ,
Ip
γ

(2.96)

which re-expresses the Keldysh parameter in a way that it relates the ionization potential of theinitial bound state Ip to the time averaged quiver energy∣e∣2 E 2
1
Up ≡ mω⟨x(t)2 ⟩T =24mω 2

(2.97)

of a free electron in an external field.
One typically refers to γ ≫ 1 as the (perturbative) multi-photon regime since associating
features with a particular photon process is possible. Here, processes involving fewer photons aremost probable. The γ ≪ 1 limit is referred to as tunneling ionization where many different photonorders (possibly infinite) contribute to processes. In this limit the field-free atomic system is nolonger a good reference. It is important to recognize that multi-photon processes are still responsiblefor ionization, but perturbation theory fails. Processes involving a large number of photons cancontribute more than lower order processes.
In the static limit (γ = 0), ionization becomes tunneling through a time-independent barrier[107]. The electronic ground state of an atom like Hydrogen [124] is shifted and broadened by theelectric field F (in atomic units) as it increases. The energy follows the formiE(F ) = ε(F ) − Γ(F )2(2.98)

̵ describes the tunneling rate out of the Coulomb potential [132]. Perturbatively
where Γ(F )/hcorrecting the Hydrogen ground state as in Ref. [142] one obtains the expansion
1 ∞F 2n
E(F ) ∼ − ∑ en ( )2 n=04(2.99)

where en are real coefficients that grow factorially. This expansion is real and therefore cannot
describe Γ(F ) and the ionization process.

23
2.3.3

Continuum state intensity parameter

The third parameter we will discuss relates the quiver energy of a free electron to the energy
contained in a single field quanta
Up
z= ̵
hω

(2.100)which Reiss has called the ”continuum-state” intensity parameter [146]. A classical electron interacting with a time-dependent electric field accumulates a time-averaged kinetic energy Up . If a large
̵ are required to describe the classical motion, then the process
number of photons with energy hω
is inherently non-perturbative. In Chapter 5 we will expand the non-perturbative amplitude forionization in a strong circularly polarized field in a series with respect to z1 and z, and show thatas the laser wavelength increases so do z1 and z necessitating increasingly higher order terms.
In Fig. 2.3 we plot shaded regions where E < Ea.u. , z1 < 1 and z < 1. For given laser parameters
perturbation theory is limited by the continuum state intensity parameter z. The dark red shaded
region corresponds to z < 0.1 corresponding to a region where perturbation theory should be valid.
An intensity estimate of 2.6 × 1012 W/cm2 for ionization by a Ti-sapphire laser (800 nm) is notedby a horizontal and vertical line within the Figure.

24

Breakdown of perturbation theory
0.100

I (a.u.)0.001
Force(ω)
Bound(ω)

10-5Free(ω)0.1 Free(ω)

10-7

0.00

0.05

0.10

0.15

0.20

0.25

ω (a.u.)Figure 2.3: Shaded regions correspond to regions where E < Ea.u. (blue), z1 < 1 (green), z < 1 (red)and z < 0.1 (dark red). An estimate for the applicability of perturbation theory is given by thedark red region. The vertical grey line corresponds to a laser wavelength of 800 nm, at which an
intensity of 2.6 × 1012 W/cm2 (horizontal line) marks the limit where perturbation theory would
approximately break down for a Ti-sapphire laser.

Chapter 3

Non-perturbative Theory of Light-Matter Interaction

Attempts made at modeling (non-perturbative) strong field ionization dates back to at least
Oppenheimer in 1928 [132] where an expansion similar to the golden rule [47, 62] was generated
for the ionization of Hydrogen by a strong static field where the final scattering states of Sec.
2.2 have been replaced by states of an otherwise free electron in a static field (Airy functions).
Later in 1964 Keldysh applied essentially the same method to time-dependent fields and derived
the ionization amplitude [98]. In Keldysh’s work the quasi-classical method of steepest descent
was applied to approximate time-integration and the final continuum states have been replaced
by states of a free electron propagating in a dynamic electric field (Volkov states [98, 165]). The
quasi-classical approximation allows for a clear classical interpretation, but fails in the ultraviolet
limit where exact (fully quantum) time integration must be performed. In Chapter 5 we discard
the quasi-classical approximation in our own work where we have further extended Oppenheimer’s
and Keldysh’s amplitude into the ultraviolet limit. Our first-order amplitude is valid for arbitrary
initial and final states, but just as in Refs. [98, 132] higher order approximations are required todescribe long-range potentials. We will focus on the first-order expansion for this thesis, but theexact same method can be applied recursively to generate higher order corrections.
In the first section of this Chapter we will describe numerical solutions to the time-dependentproblem of ionization by a finite pulse which will be used in subsequent Chapters. Afterwards we will
discuss the main ingredients required to build the strong field approximation [57,98,107,132,136,146]
and key models that have motivated us to form our own approach. The goal here is to summarize

26
the points that have influenced our own work and not to determine which models are ”best” orprovide a complete review. We will try our best to respect the original works and emphasize key
points, but our derivation may deviate at times.
Our starting point is the idea of contour deformation and asymptotic approximations ofoscillatory integrals. Direct integration of integrals encountered in strong-field ionization are either
cumbersome or not available. Asymptotic approximations are able to extract simplified propertiesof a system in some reduced parameter space. These ideas are immediately applied to the quantum
propagation of a free electron in a static external field where we will recover the classical equationsof motion and describe quantum turning point behavior. Next, we will apply Oppenheimer’s
model [132] to describe the transition of a bound electron into the before mentioned states ofan electron propagating in an external field. For hydrogen atom (and all long-range potentials)
the final state must somehow include the influence of the atomic potential. Although mostly
correct, Oppenheimer’s rate [132] is missing a constant proportionality factor recovered by thes-state Landau and Lifshitz model [107] which includes the influence of both the Coulomb andlaser interaction in the final state. Afterwards, we will average the static ionization rate of a longwavelength field over a laser cycle to generate the so-called adiabatic ionization rate [5, 136]. At
shorter wavelengths the adiabatic approximation breaks down and the ionized electron must be
described by a final (Volkov) state which is determined by the full time-dependent electric field as
we will discuss. We will finally discuss the non-adiabatic rate given in Ref. [98] on which our own
method is based on and the complementary methods of Perelomov, Popov and Terent’ev [136] aswell as of Reiss [146] which have also influenced our approach.

3.1

Numerical solution
The most straightforward way of modeling strong-field problems for which perturbation the-ory fails is the direct numerical solution of the time-dependent Schrödinger equation. Ψ(x) is27
represented in spherical-polar coordinates where the state is expanded as
χl,m (r, t) m
Yl (r̂)rl,m

Ψ(x) = ∑(3.1)

in the basis of spherical harmonics over θ and ϕ and the radial function χl,m (r, t) is either represented
on a grid or expanded in some basis.
Finite-difference methods correspond to approaches where χl,m (r, t) is represented on a gridwith points at rj = jδr where j = 1, 2, . . . , Nr and rmax = rNr represents the edge of a finite ”box”.
r and functions f (r) are represented by a diagonal matrix evaluated at each rj . To cope withoutgoing ionized wave-packets that would otherwise reflect off the edge of the ”box” at r = rmax ,exterior complex scaling [78] is used where δr ↦ eiη δr at points rECS ≤ r ≤ rmax where, the derivative2d2−2iη d
↦
e
dr2
dr2

(3.2)

and η = −π/4 has been used. Oscillatory functions eikr corresponding to the kinetic energy k 2 /2
are replaced by −ik 2 /2 and exponentially suppressed. Assuming the region rECS ≤ r ≤ rmax is wide
enough and η is chosen appropriately the wave-packet is exponentially diminished as a function oftime.
For derivatives multiple different Taylor series are taken about some point rj which are added
together with appropriate weights such that all terms lower than the desired derivative vanish andhigher order terms correspond to some truncation error. The general procedure is referred to as
Fornberg’s method [64]. Using a second-order approximation in the case of the second derivative
centered about rj one hasf (rj−1 ) − 2f (rj ) + f (rj+1 )d2
f (r) ↦
2
dr
(δr)2

(3.3)

where f (0) and f (rNr +1 ) are equal to zero and omitted. At fourth order the central difference
becomes−(1/12)f (rj−2 ) + (4/3)f (rj−1 ) − (5/2)f (rj ) + (4/3)f (rj+1 ) − (1/12)f (rj+2 )d2
f (r) ↦
2
dr
(δr)2

(3.4)

28
for points corresponding to j = 2, ⋯, Nr − 1 and the forward difference
d2−20f (r1 ) + 6f (r2 ) + 4f (r3 ) − f (r4 )f (r)∣↦2dr12(δr)2
r=r1

(3.5)

at r1 and a similar backward difference at rNr to deal with the end of the box.
For the radial functions we have used both approximate states of the field-free atomic Hamiltonian [36] or B-splines. For the basis of atomic states,[−

1 d2
l(l + 1)+
+ Va (r) − iW (r)] χn,l (r) = En,l χn,l (r)
2
2 dr
2r2

(3.6)

is diagonalized (as a finite-difference representation) within a finite box where the complex absorbing
potential W (r) [178] suppresses wave-packets in the absorbing region at the end of the box.
Since the Hamiltonian is no longer Hermitian the energyiEn,l = εn,l − wn,l
2

(3.7)

is now complex where wn,l corresponds to the rate with which each state leaves the box and states
are normalized as
∫

rmax
0

dr χn,l (r)2 = 1(3.8)

with χn,l (r) ∈ C [129]. For deeply bound states wn,l ≈ 0 and χn,l (r) is approximately a realfunction. For the B-spline method we expand the radial coordinate similarly as in Refs. [1, 10, 15]
where exterior complex scaling has been implemented as in Ref. [123].
For time-integration of all methods described in this section, we have applied the CrankNicolson method where the known state ∣Ψ(t)⟩ at time t is used to determine the state ∣Ψ(t + δt)⟩
at a later time from the solution of[S + (iδt/2)H(t)] ∣Ψ(t + δt)⟩ = [S − (iδt/2)H(t)] ∣Ψ(t)⟩,(3.9)

where ∣Ψ(t)⟩ is a numerical representation of the state (column vector), H(t) is the discritized
Hamiltonian (matrix) and S is an overlap matrix between states of the numerical expansion. S

29
is exactly the identity matrix I for basis sets corresponding to orthonormal atomic states or gridpoints. In contrast, off-diagonal elements of S are nonzero for non-orthogonal basis-sets such as
B-splines.
Although methods discussed in this section are nearly exact in the limit of converged parameters difficulties arise from both the amount of time and computing resources needed for converged
calculations. In view of this as well as the need for approximate analytic models to physically interpret numerical results, in the following sections as well as in Chapter 5 we will discuss perturbationlike approximations to strong-field problems which are able to help interpret numerical results andgive a clear physical picture.

3.2

Contour deformation
Often when computing amplitudes, integrals of the form
I(s) = ∫ dz e−sf (z)C(3.10)

are encountered where, in general, the function f is complex valued and C is some arbitrary
contour. In the limit of large s the imaginary part of f causes the integrand to oscillate rapidly
where the resulting cancellations obscure the behavior of I(s). When f is analytic one can deformthe contour of integration such that the function f is almost entirely real-valued over the new
deformed contour C ′ . On this new deformed contour the now slowly oscillating integrand allows
for a clearer interpretation. As an example, we now apply this idea to extract the asymptotic
behavior of a Bessel function Jν (s) (ν > −1/2) in the large s limit.

3.2.1

Application to Bessel functions

Before applying this method to derive amplitudes we will sketch the simplest example presented in Appendix A of Ref. [155]. Intuition gained in this part is important to define the rate [98]from which our own method is derived. For index ν > −1/2 we may define the Bessel function asJν (s) =1(s/2)ν
isx
2 ν−1/2∫ dx e (1 − x )Γ(ν + 1/2)Γ(1/2) −1(3.11)

30
which only provides a simple representation in terms of elementary functions for half-integer ν [107]where√
Jn+1/2 (s) =2s
jn (s)π(3.12)

jn (s) = (−s)n (1 d n sin s
)
s ds
s

(3.13)

withfor integer n.
For arbitrary ν > −1/2 we may extract the s → ∞ behavior by defining
I(s) = ∫1−1

dx eisx (1 − x2 )ν−1/2 ,(3.14)

where we immediately recognize that the eisx product in the integrand rapidly oscillates obscuring
the asymptotic behavior of I(s). Realizing that f (z) = eisz (1 − z 2 )ν−1/2 is an analytic function
we may further define the closed loop contour CR,ϵ which contains no singularities for all R, ϵ > 0(c.f., Fig. 3.1). Taking the limit R → ∞, ϵ → 0+ we may define the closed loop contour C =limR→∞, ϵ→0+ CR,ϵ where the contribution from the line segment z = −1 to z = 1 describes thedesired integral I(s). I± (s) describes the integrand from z = 1 to z = 1 + i∞ (I+ (s)) and theintegrand from z = −1 + i∞ to z = −1 (I− (s)). Finally, the contribution IR (s) is defined fromz = 1 + i∞ to z = −1 + i∞, which closes the loop. Since f (z) is an analytic function we immediately
see that I(s) + I+ (s) + I− (s) + IR (s) = 0. Furthermore, there exists some A, c > 0 such that ∣f (z)∣ =e−sR ∣(1−(x+iR)2 )ν−1/2 ∣ < Ae−cR , and we immediately see that the contribution IR (s) tends towards
zero as R → 0 yielding I(s) = −I+ (s) − I− (s). Here, I± (s) has a simpler asymptotic behavior since
the integrand oscillates slowly with an exponential decay factor ∣eisz ∣ = e−sy .

31𝑧 = −1 + 𝑖𝑅𝑧 = 1 + 𝑖𝑅𝑧 = −1 + 𝑖∞𝑧 = 1 + 𝑖∞
𝐼𝑅 (𝑠)

𝐶

𝐶𝑅,𝜖
𝑅→∞
𝜖→0
𝐶𝑅,𝜖 → 𝐶𝜖𝑧 = −1𝐼− (𝑠)

𝐼+ (𝑠)

𝜖
𝑧=1

𝐼(𝑠)

𝑧 = −1𝑧=1

Figure 3.1: Contour of integration CR,ϵ (left). Taking the limit as R → ∞ and ϵ → 0+ defines thecontour C from which I(s), I± (s) and IR (s) are defined (right). Taking the limit R → ∞ forces
IR (s) → 0 and taking the limit ϵ → 0+ allows C to avoid the branch points z = ±1 of (1 − z 2 )ν−1/2 .

Writing
I± (s) = ie±is ∫0∞dy e−sy (1 − (±1 + iy)2 )ν−1/2(3.15)

we may now approximate the integrand f (z) to extract the asymptotic behavior of I± (s) yielding
the behavior of I(s). To this end, we may expand the polynomial factor as⎧⎪⎪⎪(∓2iy)ν−1/2 + O(y ν+1/2 )2 ν−1/2 ⎪(1 − (±1 + iy) )
=⎨
⎪
⎪
2ν−1
⎪
+ y ν−1/2 )
⎪
⎩ O(y

0≤y≤1

(3.16)

1 < y.

Since as s → ∞, e−sR falls off faster than all polynomial orders we may replace the polynomial
factor with the 0 ≤ y ≤ 1 contribution alone yielding the Laplace transform
I± (s) = ∫0∞

dy e−sy [(∓2iy)nu−1/2 + O(y ν+1/2 )] =π 3π2ν−1/2Γ(ν + 1/2)e±i(s−ν 2 + 4 ) + O(s−ν−3/2 ). (3.17)
ν+1/2
s

32
Combining these contributions we immediately see that√2π πJν (s) =cos (s − ν − ) + O(s−3/2 ).
πs
2 4(3.18)

Although determining the exact behavior of Jν (x) is difficult, we were able to extract the behavior
in the limit s → ∞ for arbitrary ν > −1/2. This idea is essential when approximating Green’s
functions in the long-wavelength limit. As an example in Fig. 3.2 both the approximate and exact
form of J3 (s) are plotted.
Bessel J comparison ν=3
0.4
0.2
0.0

J3 (s)2πs-0.2

coss -

3π
2

-

π

4



-0.4

0

10

20

30

40

50

60

70

s

Figure 3.2: Comparison of J3 (s) with its asymptotic approximation, Eq. (3.18).

3.3

Motion in a static field: Airy functions
In a static field the motion of an otherwise free electron is described by the time-independent

Schrödinger equation[p2+ ∣e∣E ⋅ r] Φ(r) = εΦ(r)2m(3.19)

with energy eigenvalue ε and field E = Eẑ in the case of linear polarization. States are determinedby separation of variables whereΦ(r) = Φ⊥ (r⊥ )Φ∥ (z),ε = ε⊥ + εz

(3.20)

33
describes partitioning of the wave function into a part parallel to the field (∥) along the z−axis andperpendicular (⊥) part within the x-y plane. Inserting the state into Eq. (3.19) we have
⎡ 2⎤
p2op,⊥⎥
1 ⎢⎢ pop,∥1⎥+ ∣e∣Ez ⎥ Φ∥ (z) =[ε −] Φ⊥ (r⊥ )Φ∥ (z) ⎢⎢ 2m
Φ⊥ (r⊥ )2m⎥⎣⎦(3.21)

which immediately yields the perpendicular (⊥) plane-wave solutionΦ⊥ (r⊥ ) = Φk⊥ (r⊥ ) =eik⊥ ⋅r⊥
(2π)

ε⊥ =

̵ 2 k2h⊥2m(3.22)

that describes the expected free propagation.
The left-hand side of Eq. (3.21) describes the motion in the parallel direction and can be
written as Airy’s differential equation [4][(z − zcl )∂2−] Φ∥ (z) = 02∂z
L3(3.23)

withzcl ≡ ε∥ /(∣e∣E),

̵ 2 /(2m∣e∣E)]1/3 .
and L ≡ [h(3.24)

Here it is clear that for values of z to the left of the classical turning point zcl (z < zcl ) solutions
are locally oscillatory and to the right of the turning point, solutions either tend towards zero orare exponentially divergent. Here, we will only concern ourselves with the solutions which tend
towards zero.
Since Airy’s differential equation takes the formn

∑ (am + bm x)
m=0

dm y
=0
dxm

(3.25)

Laplace’s method may be applied [68]. The solution [107]y(x) = ∫ dt Z(t)ext(3.26)

C

is formed from the polynomialsnn

P (t) = ∑ am tm ,Q(t) = ∑ bm tm

m=0

m=0

(3.27)

34
and the function
Z(t) =

1 ∫ dt P (t)/Q(t)
e
,
Q(t)

(3.28)

where the contour C is chosen such that the integral is nonzero and finite. Note that
V (t) = ext Q(t)Z(t)(3.29)

returns to its original value where the contour can either be open or closed.
Starting with[∂2− x] y(x) = 0∂x2

Q(t) = −1,

Z(t) = −e−t /3(3.30)and identifying
P (t) = t2 ,and V (t) = ext−t /333(3.31)

the solution becomesy(x) = A ∫ dt ext−t /3
3C(3.32)

where the integrand goes to zero (Re t3 > 0) at each of its endpoints corresponding to the shaded
region in Fig. 3.3 and A is an arbitrary constant. In Fig. 3.3 the three most important contributions
correspond to contours CA , CB+ and CB− where CA results in a solution y(x) ≡ Ai(x) that goes tozero at x → ±∞ and CB± results in a divergent solution for x → ∞.

35

𝐶𝐴

𝑡

𝐶𝐵+

𝐶𝐵−

Figure 3.3: Contours of integration for the integrand ext−x /3 . The dark triangular regions corre3

spond to zones where the contour may begin or end (e−t /3 → 0). CA corresponds to a convergent
3

solution as x → ∞, while CB± correspond to divergent solutions as x → ∞.

Here we choose the bounded solutions along the imaginary axis (contour CA in Fig. 3.3,
where t = iφ) corresponding toy(x) = Ai(x) =1iφ3 /3+iφx∫ dφ e2π R(3.33)

where we have used the normalization convention(1)∫ dz Ai(z − x)Ai(z − y) = δ (x − y)(3.34)

1
Ai[(z − zcl )/L].
L(3.35)

1 ik⊥ ⋅r⊥e
Ai [(z − zcl )/L] .
2πL(3.36)

R

yieldingΦz (z) = Φzcl (z) =and finally [28]Φk⊥ zcl (r) =In the next subsection we will use solutions to Eq. (3.19) to describe the motion of a free particle
in a static field.

36
3.3.1

Newton’s second law

In this subsection we will use the momentum-space Green’s function G̃(k, t; k′ , t′ ) of Eq.
(3.19) to recover the classical equation of motion
ma = −∣e∣E.

(3.37)Starting with the states in the Schrödinger picture̵Φk⊥ zcl (x) = e−(i/h)ε(k⊥ zcl )t Φk⊥ zcl (r)(3.38)

we may write the position-basis Green’s function asiG(x; x′ ) = − ̵ θ(t − t′ ) ∫ dq⊥ ∫ dzcl Φq⊥ zcl (x)Φq⊥ zcl (x′ )hR2R(3.39)and Fourier transform with respect to each coordinate r and r′ to determine the momentum basis
representationiG̃(k, t; k′ , t′ ) = − ̵ θ(t − t′ ) ∫ dq⊥ ∫ dzcl Φ̃q⊥ zcl (k, t)Φ̃q⊥ zcl (k′ , t′ ).
h
R2R(3.40)

Evaluatingei(k∥ L) /3 −(i/h)ε(qe−ik⋅r̵(2)⊥ zcl )t
Φ
(x)
=
δ
(k
−
q
)
e
Φ̃q⊥ zcl (k, t) ≡ ∫ dr
q
z
⊥
⊥
⊥ cl
3/2
1/2
3
R
(2π)
(2π)
3

(3.41)

yieldsi̵G̃(k, t; k′ , t′ ) = − ̵ θ(∆t) δ (1) (∆k∥ + (∣e∣/h)E∆t)δ (2) (∆k⊥ )h⎡⎤̵ 2 ⎛ k∥2 + k∥ k∥′ + k∥′2
⎢
⎥
h
2⎞
⎢
̵
× exp ⎢−(i/h)+ k⊥ ∆t⎥⎥ (3.42)
2m ⎝
3
⎠ ⎥
⎢
⎣
⎦
̵ ′ describes the momentum of a free electron at time t′ in the past and hk̵after integration. hk
describes the resulting momentum at time t after the influence of E. The displacement in time is̵̵ − k′ ).
written as ∆t ≡ t − t′ and momentum h∆k= h(kUsing properties of the δ-distribution one immediately recognizes the expected classical equations of motionF⊥ =∆p⊥ ̵ ∆k⊥
=h
=0
∆t
∆t

and F∥ =∆p∥ ̵ ∆k∥=h= −∣e∣E.
∆t∆t(3.43)

37
The δ-functions in Eq. (3.42) describe classical evolution of a single momentum state, while thephase factors correspond to quantum interferences due to initial states composed of multiple plane
wave contributions (for example in an isolated wave-packet).

3.3.2

Asymptotic behavior

Determining the properties of an electron under the influence of a constant field far from theturning point is an important pre-requisite for describing strong field ionization. Since the Airyfunction (as we will soon see) tends towards zero as e− 3 ∣x∣
2

3/2, where x = (z − zcl )/L, and therefore

faster than the bound states which fall off as e−κr one immediately recognizes that the asymptotic
tail of the ground-state wave-function is most important for tunneling ionization. Steps taken here
to describe the Airy function will later be applied to determine ionization rates in an alternating
field [98, 136]. We may note that we intend to get results and gain intuition and not to provide arigorous proof.
Defining
f (t) = tx − t3 /3(3.44)

we may apply the saddle point approximation to determine the behavior of Ai(x) in the limit of x →±∞. In section 3.2 we extracted the asymptotic behavior of an integral with an oscillatory integrandby finding a new contour where the integrand was slowly changing and decayed exponentially. Herewe will do the same by deforming the contour of integration such that it passes through each valuet = ts where f ′ (ts ) = 0. The exponent is approximated as f (t) ≈ f (ts ) + 21 f ′′ (ts )(δt)2 whereδt = t − ts = ηeiϕs and ϕs is chosen such that the coefficient of η 2 becomes real. This corresponds toselecting contours such that Imf (t) is essentially constant (removing oscillations) and the integrandis built based on sharply-peaked Gaussian contributions corresponding to the approximation′′y(z) = ∑ ef (ts )+iϕs ∫ dη e(η /2)f (ts )e
2

s

2iϕs

(3.45)

Rfor complex z. It is assumed that the integrand is sharply peaked at each ts , where the precise
shape of the contour is unimportant as long as it passes through the saddle points at an angle ϕs .

38This procedure causes difficulties for the Airy function since√f ′ (ts ) = x − t2s = 0 → ts± = ± x,√
f ′′ (ts± ) = ∓2 x,

2and f (ts± ) = ± x3/2
3

(3.46)

corresponds to a multi-valued integrand with branch cuts, where the original integrand was entire.
Qualitatively, this imposes restrictions on which contours are allowed.
Starting with the relatively simple case of x → −∞ with√
ts± = ±i ∣x∣,√f ′′ (ts± ) = ±2i ∣x∣,2
and f (ts± ) = ∓i ∣x∣3/2
3(3.47)

we want the contour to start in the bottom-left shaded region and end up in the top-left shaded
region of Fig. 3.4 while traveling along steepest descent curves through ts defined by ϕs . Letting
x → −∞ we have ϕs− = π/4 and ϕs+ = 3π/4 where we may easily draw a contour from the bottom-left√dark region through the saddle point ts− at an angle of π/4 to ∣x∣ and then through the saddle
point ts+ at an angle of 3π/4 described in Fig. 3.4.

𝑡𝑡 = 𝑖 |𝑥|𝑡 = −𝑖 |𝑥|

Figure 3.4: Plot of the real part of f (t) where t = tr + iti . Integration contours connecting thebottom-left and top-left regions are chosen such that they pass through the saddle points in a
direction that the imaginary part of the exponent is constant.

39
Near the origin of the contour, the real part of f (t) is large and negative resulting in a small
contribution to the total integral compared to contributions from the points ts± meaning that the
precise orientation of the contour is unimportant far from the saddle points as long as they locallypass through the saddle points at an angle ϕs with respect to the real-axis. Along this first segment
the integral therefore accumulates a partial contribution1π2√ ∣x∣−1/4 exp [i ( ∣x∣3/2 + )]
3
4
2 πi

(3.48)

where we have assumed that the Gaussian is so sharply peaked that the bounds of integration forη may extend from minus infinity to infinity. The contour of integration lies between the large
positive valued ”ridges” of Ref (t).
After passing through ts− the contour now resides in the right-most dark region where again
the integrand is small and contributions can be neglected. To arrive at the desired top left shaded
region we must pass through ts+ along a contour described by the angle ϕs+ between the ”ridges”
of Ref (t) yielding the contribution12π− √ ∣x∣−1/4 exp [−i ( ∣x∣3/2 + )] .
3
4
2 πi

(3.49)

12πx→−∞
Ai(x) ÐÐÐ→ √ ∣x∣−1/4 sin ( ∣x∣3/2 + ) .
34
π(3.50)and the overall approximation

For x → ∞ the situation becomes more complicated. From√f ′ (ts ) = ∣x∣ − t2s = 0 → ts± = ± ∣x∣,√f ′′ (ts± ) = ∓2 ∣x∣,2
and f (ts± ) = ± ∣x∣3/2
3(3.51)

we immediately have that ϕs+ = 0 and ϕs− = π/2 corresponding to the yellow and purple curves inFig. 3.5 where we again want to determine a suitable contour connecting the top left and bottom left
triangular regions of Fig. 3.3. To achieve this goal we will determine contours where the imaginary
part of the exponent is zero and which connect the two regions.

40𝑡𝑡 = − |𝑥|𝑡=|𝑥|

Figure 3.5: Plot of the real part of f (t) where t = tr + iti . Integration contours connecting thebottom-left and top-left regions are chosen such that they pass through the saddle points in a
direction that the imaginary part of the exponent is constant. Here only the saddle point at√√
t = − ∣x∣ contributes to the integral since t = ∣x∣ connects the bottom left region to the rightmost region describing divergent solutions.

Writing t = tr + iti (tr , ti ∈ R) we have√
Imf (t) = 0 ⇒ tr = ± x + t2i /3or ti = 0

(3.52)√
where only the solution tr = − x + t2i /3 connects the desired regions. Defining a contour which√accumulates a contribution from t = − ∣x∣ connecting the bottom-left and top-left regions is straight√forward. We may use the contour mentioned above to pass through the saddle point t = − ∣x∣.
Since the integrand is essentially zero everywhere except through the saddle point we may replace
the contour with a vertical line passing through ts− and the job is done.
√
Deforming the contour such that it passes through t = ∣x∣ we are stuck in the right-mostshaded region in Fig. 3.3, which describes exponentially divergent solutions to Airy’s differential
equation. The only way to return the contour back to the top-left region is by crossing ”ridges”

41√∣x∣ which is ”forbidden”. The approximation therefore only includes the√solution corresponding to the saddle point t = − ∣x∣ along the path of ”steepest descent” forming
perpendicular to t =the approximation
2
3/2
1
x−>∞
Ai(x) ÐÐÐ→ √ ∣x∣−1/4 e− 3 ∣x∣
2 π(3.53)

Plotting Ai(x) with each of its asymptotic solutions, we see acceptable agreement as early as ∣x∣ ≳ 2
in Fig. 3.6.

Airy comparison

1.0

0.5

Ai(x)
AsymAi(x)

0.0

-0.5
-20

-15

-10

-5

0

5

x
Figure 3.6: Airy function (blue), compared with the asymptotic approximations for x → ∞ (Eq.
(3.53), orange x > 0) and x → −∞ (Eq. (3.50), orange x < 0).

3.4

Ionization by a static field
In section 2.3 we have shown that modeling the ionization of Hydrogen by a static electricfield is not possible at any order of (the standard) perturbation theory. In this section we will
describe two different approaches towards ionization of an atom by a static electric field.
The first approach given by Oppenheimer essentially follows the golden rule expansion of

42
section 2.2 where the final scattering state has been replaced with states of an electron in a static
field, as discussed in the prior section. Methods we will discuss later for ionization by dynamic
fields follow this idea. The Airy functions in section 3.3 are replaced by the time-dependent statesof an electron in a time-dependent field (section 3.5). For long-range potentials these methods
require higher order corrections or additional modifications since the final state does not see theatomic potential.
The second approach which is able to predict the correct ionization rate for Hydrogen is given
by Landau and Lifshitz. The final ionized state is described by the atomic ground-state whenever
the atomic potential is significantly stronger than the field-interaction. The Wentzel–Kramers–Brillouin(WKB) approximation is used to include both the field-interaction and atomic potential whereverboth contribute.

3.4.1

Oppenheimer’s rate

In this section we will use states of the electron in a static field discussed in the prior section toapproximate the static field rate. The original work [132] contained errors in the derivation. Laterin Ref. [177] the corrected version was reported with no derivation. We will therefore systematicallyderive the corrected rate and find our own way to the results of Ref. [177]. Keldysh [98] (section
3.6.1) uses the exact same method to describe ionization of an atom in an alternating field where
the Airy functions of section 3.3 are replaced by the Volkov states of section 3.5 which describe
motion of an electron in a dynamic field. In Ref. [98] and similarly in Ref. [136] the ionization rate
was derived using a similar contour deformation approach as discussed in section 3.2 and the saddle
point approximation (see section 3.3.2) which is valid for sufficiently low laser frequencies. In ourown work (chapter 5) we will generalize the work in Refs. [98, 132, 136] to arbitrary frequencies.
We start with the Hamiltonian
H(x) = Ha (x) + VDC (r)(3.54)

43
which is partitioned into an atomic contribution and a field interaction
Ha (x) =

p2
∣e∣2
−Z
,
2m
r

VDC (r) = ∣e∣E ⋅ r

and E = Eẑ

(3.55)

as before where the initial 1s state is given as
κ3/2̵ϕi (x) = e(i/h)Ip t ϕi (r) and ϕi (r) = √ e−κr .
π(3.56)

with ionization potential and bound-state κ-vectorIp =̵ 2 κ2h2m

̵ 2.
and κ ≡ mZ∣e∣2 /h(3.57)The rate is determined by expressing the exact wave function Ψ(x) with the initial-state Lippmann
Schwinger equation whereΨ(x) = ϕi (x) + ∫ d4 x1 G(x; x1 )[∣e∣E ⋅ r1 ]ϕi (x1 ).

(3.58)In the standard perturbation theory one would continue by expanding the exact Green’s function
G(x; x1 ) up to higher orders with respect to the atomic Green’s function Ga (x; x′ ) of Ha (x) andrest interaction [∣e∣E ⋅ r]. Unfortunately, such a perturbation series breaks down as described in theprior section and therefore an alternative expansion is required.
Oppenheimer’s approach involved now expressing G(x; x1 ) by the final-state Lippmann
Schwinger equation
G(x; x1 ) = GDC (x; x1 ) + ∫ d4 x2 GDC (x; x2 ) [−Z∣e∣2] G(x2 ; x1 )r2

(3.59)

where GDC (x; x′ ) describes the otherwise free propagation of an electron in a static field corresponding to
HDC (x) =p2+ ∣e∣E ⋅ r
2m(3.60)as discussed in a prior section.
Setting G(x2 ; x1 ) ≈ GDC (x2 ; x1 ) we immediately arrive at the first and second order correction
Ψ(1) (x) = ∫ d4 x1 GDC (x; x1 )[∣e∣E ⋅ r1 ]ϕi (x1 )(3.61)

44
andΨ(2) (x) = ∫ d4 x2 GDC (x; x2 ) [−Z∣e∣2] ∫ d4 x1 GDC (x2 ; x1 )[∣e∣E ⋅ r1 ]ϕi (x1 )
r2(3.62)

as given in Ref. [132]. Here, Ψ(1) (x) intuitively describes the transition of the initial ground state
ϕi (x) into a free particle state accelerating down the ∣e∣E⋅r potential surface, while Ψ(2) (x) describes
the same exact process where the ejected electron may scatter once of the atomic potential −Z∣e∣2 /r
before again propagating as a free particle down the surface ∣e∣E ⋅ r. G(x; x′ ) can be expanded up
to arbitrary order in −Z∣e∣2 /r although convergence is not guaranteed (just as in any perturbation
series). Although the first and second order terms were introduced in Ref. [132] only the first ordercontribution was considered.
The rate is determined by consideringM(k⊥ , zcl , t) ≡ ∫R3

dr Φk⊥ zcl (x) [Ψ(x) − ϕi (x)] ≈ M(1) (k⊥ , zcl , t)

(3.63)

which corresponds toiM(1) (k⊥ , zcl , t) = (− ̵ ) ∫ d4 x1 Φk⊥ zcl (x1 ) [∣e∣E ⋅ r1 ] ϕi (x1 )hti̵= (− ̵ ) ∫ dt1 Vi (k⊥ , zcl ) e(i/h)∆ε(k⊥ ,zcl )t1
h 0(3.64)

where∆ε(k⊥ , ztp ) ≡ ε⊥ + ε∥ + Ip

(3.65)Vi (k⊥ , zcl ) ≡ ∫ dr Φk⊥ zcl (r) [∣e∣E ⋅ r] ϕi (r).

(3.66)

andTime-integration may be immediately performed as
t

∫

0

̵
̵
̵
̵
dt1 e(i/h)∆εt = e(i/h)∆εt/2 sinc((1/h)∆εt/2)t≡ δt ((1/h)∆ε/2),(3.67)

where in the t → ∞ limit the energy conservation law
zcl = −

1(ε⊥ + Ip ) < 0∣e∣E(3.68)

45
is satisfied. Electrons have zero parallel velocity at the tunnel exit and as the perpendicular
momentum at ionization, k2⊥ , increases, states corresponding to the classical turning points zcl arepushed further away from the Coulomb potential. For ε⊥ = 0, Fig. 3.7 shows the atomic potential
and the laser interaction as well as the initial- and final-state wave-functions for F = 0.05 a.u..

Oppenhimer model for the tunnel ionization of Hydrogen
0.0-0.2
exp(-z)

-0.4
V, ϕ

π

Ai

L

L

classical turning
point zcl

-0.6

z-zcl

-



-

-

1
2

1
2

1
z 

Fz

-0.8
-1.0
-50

-40

-30

-20

-10

0

10

z
Figure 3.7: Sketch of components required to compute the Oppenheimer rate. Blue represents theinitial 1s state of hydrogen atom, orange shows the final continuum state Φk⊥ ,zcl (r) (with k⊥ = 0),
green represents the field free Coulomb potential and red depicts the DC Stark potential. The
classical turning point zcl = −10 a.u. is noted for a field-strength of F = 0.05 a.u.. For F → 0
ionization is determined by the tail of the initial bound-state alone (Eq. (3.92)).

Completing the amplitude involves computing the overlap matrix element
Vi (k⊥ , zcl ) = ∫

R2

dr⊥

e−ik⊥ ⋅r⊥1
κ3/2 −κ√r⊥2 +z 2
√ e
dz
Ai
[(z
−
z
)/L]
[∣e∣Ez].
cl∫(2π) R L
π(3.69)For the 1s-state of Hydrogen, the ionization amplitude in the perpendicular r⊥ direction has cylin-

46
drical symmetry around the z-axis allowing for the simplification∫ 2 dr⊥
R

∞
∞
∞
π dφ
e−ik⊥ ⋅r⊥r im(φk −φr )= ∑ (−i)m ∫ dr⊥ r⊥ Jm (k⊥ r⊥ ) ∫e= ∫ dr⊥ r⊥ J0 (k⊥ r⊥ ).
(2π)
00−π (2π)m=−∞(3.70)

Utilizing Ref. [69]
∫

∞

1

√
√
1
2
2
dx J(α x2 − 1)e−βx = √e− α +β ,22
α +β(3.71)

the perpendicular contribution
∫

∞
0

√
∞
∂
) ∫ dr J0 (k⊥ r2 − z 2 )e−κr∂κ ∣z∣⎤ √ 2 2
√
∞
κ ⎡⎢
∂
1
⎥ − k⊥ +κ ∣z∣⎥e⎢√= ∣z∣ (− ) ∫ du J0 (k⊥ ∣z∣ u2 − 1)e−κ∣z∣u = 2∣z∣+(3.72)22∂κ 1k⊥ + κ2 ⎢⎣k⊥ + κ ⎥⎦√dr⊥ r⊥ J0 (k⊥ r⊥ )e−κr⊥2 +z 2= (−is complete.
Defining K ≡√k⊥2 + κ2 , the final parallel contribution∣e∣E κ5/2 1∂
1
Vi (k⊥ , zcl ) = √( −) ∫ dz z Ai [(z − zcl )/L] e−K∣z∣ ∣ √
2
K ∂K
L
R
π K
K= k2 +κ2

(3.73)⊥

must now be computed. The integral over z is evaluated by using the variable substitution u =(z − zcl )/L:∫ dz zR

1
Ai [(z − zcl )/L] e−K∣z∣ = ∫ du (Lu − ∣zcl ∣)Ai(u)e−K∣Lu+zcl ∣LR(3.74)and partitioning the integral into two parts:−K∣Lu+zcl ∣∫ du (Lu − ∣zcl ∣)Ai(u)eR=∣zcl ∣/L∞∂[e−K∣zcl ∣ ∫du Ai(u)eKLu − eK∣zcl ∣ ∫du Ai(u)e−KLu ] (3.75)∂K−∞∣zcl ∣/L

corresponding to each piecewise portion of ∣z∣. The second integral from ∣zcl ∣/L to infinity may
be neglected in the weak field E → 0 limit where the bound of integration ∣zcl ∣/L goes to positive
infinity and the magnitude of the Airy function falls off faster than Ae−αu for all A, α > 0 as u → ∞
since
2
3/2
1
u→−∞
Ai(u) ÐÐÐ→ √ ∣u∣−1/4 e− 3 ∣u∣ .
2 π(3.76)

47
The first integral is evaluated with the help of the Heaviside step functionθ(∣zcl ∣ − Lu) = lim+
η→01dτ i(∣zcl ∣−Lu)τ
e∫
2πi R τ − iη

(3.77)

to givepu
p /3∫ du Ai(u)e = e3(3.78)

R

where∫∣zcl ∣/L−∞dτ i∣zcl ∣τ [(K−iτ )L]3 /3 1 (KL)3 /3
1
e
.
e
= e
∫
η→0 2πi R τ − iη2

du Ai(u)eKLu = lim+

(3.79)In Eq. (3.79) the integral over τ was determined using the Sokhotski–Plemelj theorem [125,139,154]
lim−i

η→0+ x − iη

1= πδ(x) − i p.v. .
x(3.80)

where we will show that the term corresponding to the Cauchy principal value (p.v.) is negligible
in the E → 0 limit.
Herep.v. ∫

3dτ i∣zcl ∣τ [(K−iτ )L]3 /3dτ i(∣zcl ∣−K 2 L3 )τ i(τ L)3 /3 −L3 Kτ 2ee= e(KL) /3 p.v. ∫
e
e
e
.
R τ
R τ

(3.81)To show that this term is negligible one must consider that the p.v. integral after the e(KL) /3 factor
3

on the right hand side is significantly smaller than 1/2. Since the overall integral is real we should
expect that the real part of the p.v. integral is identically zero. Explicitly writing it out we havedτ i(∣zcl ∣−K 2 L3 )τ i(τ L)3 /3 −L3 Kτ 2
e
e
e
]
R τ

Re [p.v. ∫= p.v. ∫32dτcos[(∣zcl ∣ − K 2 L3 )τ + (τ L)3 /3]e−L Kτ
R τ(3.82)

which is zero since we have an odd integrand over an even interval.
Investigating the imaginary part we have
Im [p.v. ∫dτ i(∣zcl ∣−K 2 L3 )τ i(τ L)3 /3 −L3 Kτ 2
e
e
e
]
R τ
3
2
dτ
sin[(∣zcl ∣ − K 2 L3 )τ + (τ L)3 /3]e−L Kτ . (3.83)
R τ

=∫

48
Since the sin-function has a zero at τ = 0, the first order pole exactly cancels out.
To the best of our knowledge, there is no analytic solution to this integral, therefore we willbound it from above and show that as E → 0 the integral goes to zero. Starting with the triangle
inequality
3
2
3
2
dτ
dτ
∣sin[(∣zcl ∣ − K 2 L3 )τ + (τ L)3 /3]∣ e−L Kτsin[(∣zcl ∣ − K 2 L3 )τ + (τ L)3 /3]e−L Kτ ∣ ≤ ∫R τR ∣τ ∣∣∫(3.84)
and replacing the ∣ sin(⋯)/τ ∣ factor by its maximum value M we have
3
2
3
2
dτ
∣sin[(∣zcl ∣ − K 2 L3 )τ + (τ L)3 /3]∣ e−L Kτ ≤ M ∫ dτ e−L Kτ = M∫
R ∣τ ∣
R

√

π
.
KL3

(3.85)Taking E → 0 we have L → ∞ and therefore the principal value contribution is negligble.
Applying Eq. (3.68) the matrix element is given byκ1/2 Ip − 2 K∣zcl ∣Vi (k⊥ , zcl ) = − √e 3≡ Vi (k⊥ ).
πK(3.86)

Computing then the ionization rate
w = lim ∫t→∞

R2dk⊥ ∫ dzcl
R∂
∣M(1) (k⊥ , zcl , t)∣2
∂t(3.87)

we have∞
2π(2π)2w = ̵ ∫ dk⊥ ∫ dzcl ∣Vi (k⊥ , zcl )∣2 δ(ε⊥ + ε∥ + Ip ) =dk⊥ k⊥ ∣Vi (k⊥ )∣2∫
̵h R2∣e∣E h 0R2∞∞4πm Ip4π 2 m2− 43 K∣zcl ∣
dε
∣V
(k
)∣
≈
dε
e
=
(3.88)
⊥
i
⊥
⊥
∫
∫
3
3
̵ 0
̵ κ 0
∣e∣E h∣e∣E hwhere in the prefactor we set K ≈ κ since the exponential factor is sharply peaked at ε⊥ = 0.
Integration is now performed by writing
4
4 Ip
ε⊥
K∣zcl ∣ ≈+23
3 ∣e∣Ea∣e∣Ea(3.89)

Ip − 4 Ipπ
w = π ( ̵ ) e 3 ∣e∣Ea ”=” e−2/3Fh2(3.90)

which yields

49
where the ”=” symbol denotes that the right-hand side is expressed in atomic units. Here the total
ionization rate, as shown in Fig. 3.8, is exponentially proportional to 1/F .
Oppenheimer rate
1.6

log(Γ) a.u.

1.4

1.2
1
2

π exp-

2
3F



1.0

0.8
0.0

0.2

0.4

0.6

0.8

1.0

1/F a.u.

Figure 3.8: Ionization rate Γ ≡ w of Oppenheimer as a function of 1/F in atomic units.

This first approximation of at an ionization rate in a static field predicts the correct exponential dependence in the F → 0 limit, but contains an incorrect pre-factor since the Coulomb
potential is neglected in the final continuum states, which will be explored further in section 3.4.2.

3.4.1.1

Final state form

In section 3.6.2 we will discuss a so-called final state ionization amplitude which uses −Z∣e∣2 /r
as a perturbation instead of ∣e∣E ⋅ r in the case of an alternating field. Here,i∣e∣2M(1) (k⊥ , zcl , t) = (− ̵ ) ∫ d4 x1 Φk⊥ zcl (x1 ) [−Z] ϕi (x1 )
h
r1
t
i
̵
= (− ̵ ) ∫ dt1 Vi (k⊥ , zcl ) e(i/h)∆ε(k⊥ ,zcl )t1 . (3.91)
h 050
represents the equivalent final state amplitude for the case of a static electric field. Setting Z∣e∣2 =2Ip /κ we have
Vi (k⊥ , zcl ) ≡ ∫ dr Φk⊥ zcl (r) [−2Ip] ϕi (r)κr(3.92)andVi (k⊥ , zcl ) = ∫

R2

dr⊥

√
⎡
⎤ κ3/2Ip
1e−ik⊥ ⋅r⊥⎥⎢
−κ r⊥2 +z 2
⎢
⎥
√
√
dz
Ai
[(z
−
z
)/L]
e
−2
cl
∫
⎢ κ r2 + z 2 ⎥ π(2π) R L⎣⎦⊥

Ip κ3/2 −K∣z∣1
κ1/2 Ip − 2 K∣zcl ∣= ∫ dz Ai [(z − zcl )/L] [−2
]√ e
e 3
=−√
(3.93)
L
κK
R
ππK
exactly as before leading to the same rateIπ
−4 p
w = ̵ Ip e 3 ∣e∣Ea .
h(3.94)In section 3.6.2.1 we show that initial state and final state amplitudes will also be identical
for the case of an arbitrary alternating electric field. If the final state is modified to account for
the atomic potential, this property does not hold in general.

3.4.2

Landau and Lifshitz rate: Correcting the pre-factor

The rate derived by Oppenheimer predicted the correct weak-field exponential factor with anincorrect proportionality constant. Landau and Lifshitz improved the pre-factor by including boththe atomic Coulomb potential and the applied field within the final state through the quasi-classicalWKB approximation [34,72,89,103,111,171]. The method requires determining the ground state ofhydrogen atom in parabolic coordinates and then determining regions of the wave function modified
with the quasi-classical approximation by the applied field.

3.4.2.1

Parabolic coordinates

The transformation to parabolic coordinates from the Cartesian system [107] involves
ξ = r + z,

η = r − z,

φ = tan−1 (y/x).

(3.95)

51
with backwards transformationx=√ξη cos φ,y=√ξη sin φ,1
z = (ξ − η)
2

(3.96)

and
r=

√

1
x2 + y 2 + z 2 = (ξ + η).
2(3.97)In this system the length element is determined by(dl)2 =

ξ+ηξ+η(dξ)2 +(dη)2 + ξη(dφ)24ξ
4η

(3.98)and the volume element is given by1
dr = (ξ + η)dξdηdφ.
4(3.99)

Determining the atomic states requires the Laplacian
∇2 =

4
∂
∂
∂
∂
1 ∂2
[ (ξ ) +(η )] +
ξ + η ∂ξ ∂ξ∂η ∂ηξη ∂φ2(3.100)which gives for the Schrödinger equation
∂
∂ϕ(r)
∂
∂ϕ(r)
1 ∂ 2 ϕ(r)
4
2
[ (ξ
)+
(η
)] +
) ϕ(r) = 0.
+ 2 (E +2
ξ + η ∂ξ
∂ξ
∂η
∂η
ξη ∂φ
ξ+η

(3.101)

Writingϕ(r) = f1 (ξ)f2 (η)eimφ(3.102)

separation of variables may be applied giving
df1 (ξ)11d(ξ
) + [ Eξ − m2 /ξ + β1 ] f1 (ξ) = 0
dξ
dξ
2
4

(3.103)d
df2 (η)11(η
) + [ Eη − m2 /η + β2 ] f2 (η) = 0
dη
dη
2
4

(3.104)

andwith separation parameters described by
β1 + β2 = 1.

(3.105)

52
Defining parameters√n = 1/ −2E,

ρ1 = ξ/n,ρ2 = η/n(3.106)

we haved2 f1 (ξ) 1 df1 (ξ)
1 1 ∣m∣ + 1
m2
(
+
+
[−
+
+
n
)
−
] f1 (ξ) = 0
1
dξ 2
ρ1 dξ
4 ρ1
2
4ρ21

(3.107)and a similar expression for f2 (η).
Solving exactly as in the spherical-polar case before we get√ξη eimφ
f
(
)
f
(
)√
n
,m
n
,m
2
n2 1
n
n
2π

(3.108)(p + ∣m∣)!
−ρ/2 ∣m∣/2
ρ
1 F1 (−p; ∣m∣ + 1; ρ)ep!

(3.109)11
ϕ1s (r) = √ e− 2 (ξ+η)π(3.110)1
1
ψ(r) = √ e− 2 (ξ+η)π(3.111)

ϕn1 ,n2 ,m (r) =with
1
fp,m (ρ) =
∣m∣!

√

2

and the ground state given by

3.4.2.2

Hydrogen rate

The initial statefrom the earlier subsection is the starting point of the approximation. Far from the atom the
Schrödinger equation
1 1
1
1
d2 χ
+ [− +++ F η] χ = 0(3.112)dη 2
4 2η 4η 2 4√
now includes the laser interaction where χ = ηψ. Choosing some point 1 ≪ η0 ≪ 1/F within thebarrier and applying the WKB approximation one has
√
χ=

η1
1
η0 ∣p0 ∣χ + η0pdη + iπ)
exp (
+ i∫
πp
2
4
η0

(3.113)

with momentum p(η1 ) = 0 at the tunnel exit and√p(η) =

1 1
1
1
1√
1
− +
+ 2 + Fη ≈Fη − 1 − √4 2η 4η
4
2
2η F η − 1(3.114)

53
within the barrier.
Taking the square∣χ∣2 =

η1
η0 −ξ ∣p0 ∣e
exp [−2 ∫∣p(η ′ )∣dη ′ − η0 ]πp
η0(3.115)and using ∣p0 ∣ ≈ 12 one has
η1
η1 √
η0
dη ′
√
∣χ∣2 = √1 − F η ′ dη ′ + ∫e−ξ exp [− ∫− η0 ]
η0 η ′ 1 − F η ′η0π Fη − 1(3.116)

where the first term of Eq. 3.114 has been used to approximate p in the prefactor and both terms
have been retained within the exponent.
Integrating and using F η0 ≪ 1 whenever possible∣χ∣2 =

4 −2/3F e−ξ
√eπFFη − 1(3.117)may be used to give the total rate∞

w = 2π ∫

0

where
1
dr ≈
2

√

η
dξ,
ξ

dr r vz ∣ψ∣2

vz =

√

(3.118)Fη − 1(3.119)andw=

4 −2/3Fe.
F(3.120)

Restoring units the ionization becomes (a = 1/κ)
Ip
Ip
4 Ip
) exp (−)
w = 16 ( ̵ ) (h∣e∣Ea
3 ∣e∣Ea(3.121)

which has the same exponential factor as the ionization rate of section 3.4.1 with the correct weakfield prefactor 4/F [177].

3.4.2.3

Short-range rate

The same procedure may now be followed for short-range potentials. The short-range asymptotic ground state1ψ(r) = Cκ,02√

κ e−κr= Cκ,0π r√

κ e− 2 (ξ+η)≈ Cκ,0π ξ+ηκ√κ e− 2 (ξ+η)πηκ(3.122)

54
is used and gives
1 2
1 2 ∣e∣E
4 Ip
w = Cκ,0F e−2/3F = Cκ,0
exp (−)
̵44hκ
3 ∣e∣Ea(3.123)where Cκ,0 is fit to the asymptotic tail of the field-free ground-state. The exponential factor is a
general property of ionization of particles bound by both long- and short-range potentials, while
the prefactor is determined by properties of the binding potential.

3.5

Motion in a dynamic field: Volkov states
In section 3.3 we solved Airy’s differential equation to determine the dynamics of an otherwise

free electron in an external electric field. These states were applied by Oppenheimer [132] (section
3.4.1) to determine an estimate for the ionization rate of Hydrogen. We now will follow the same
steps as in section 3.3 to determine the behavior of a free electron experiencing an arbitrary timedependent field. In the next subsection 3.6.1 we will follow the derivation by Keldysh [98] to apply
exactly the same method as in Ref. [132] and determine an estimate for the ionization of Hydrogenby a linearly polarized field. The so-called Volkov states form the basis for many standard nonperturbative approaches [57, 98, 136, 146] including our own generalization of Ref. [98] which will
be discussed in Chapter 5.
Motion of an otherwise free electron interacting with a time-dependent electric field is described by the Schrödinger equation2̵ ∂ Φ(r) = [ p + ∣e∣E(t) ⋅ r] Φ(r).
ih
∂t
2m

(3.124)

Recognizing that
E(t) = −

i∣e∣
i∣e∣
1∂
̵ A(t)⋅r̵ A(t)⋅r̵ ∂ e hc
A(t) and ih= [∣e∣E(t) ⋅ r] e hc
c ∂t∂t

(3.125)

we immediately see that the set of solutions must follow the formeik(t)⋅r(2π)3/2(3.126)∣e∣k(t) = k + ̵ A(t)hc

(3.127)

Φk (x) = T (t)
where

55
is the kinetic momentum of an electron in the field.

Inserting this solution one obtains the

Schrödinger equation
̵ih̵ 2 k(t)2∂h
T (t) =
T (t)
∂t
2m

(3.128)and the states̵tΦk (x) = e−(i/h) ∫0 dτ

3.5.1

̵ 2 k2 (τ )h2meik(t)⋅r.
(2π)3/2(3.129)

Newton’s second law

In this subsection we will use the momentum-space Green’s function G̃(k, t; k′ , t′ ) of Eq. 3.19
to recover the classical equation of motion
ma(t) = −∣e∣E(t).

(3.130)Starting with the Volkov states Φk (x) we may write the position-basis Green’s function asiG(x; x′ ) = − ̵ θ(t − t′ ) ∫ dq Φq (x)Φq (x′ )hR3(3.131)and Fourier transform with respect to each coordinate r and r′ to determine the momentum basis
representationiG̃(k, t; k′ , t′ ) = − ̵ θ(t − t′ ) ∫ dq Φ̃q (k, t)Φ̃q (k′ , t′ ).
hR3(3.132)

EvaluatingΦ̃q (k, t) ≡ ∫

R3

dr

2 2
e−ik⋅r̵ ∫ t dτ h̵ q (τ )(3)−(i/h)
0
2m
Φ
(x)
=
δ
(k
−
q(t))
e
q
(2π)3/2(3.133)yields̵2 2ti̵̵ ∫ dτ h q (τ ) ]G̃(k, t; k′ , t′ ) = − ̵ θ(∆t) δ (3) (∆k − (∣e∣/hc)∆A)
exp [−(i/h)
h2mt′(3.134)

̵ ′ describes the momentum of a free electron at time t′ in the past and hk̵after integration. hk
describes the resulting momentum at time t after the influence of E(t). Displacements are defined̵̵ − k′ ) in momentum and ∆A = A(t) − A(t′ ) in vector potential.
as ∆t ≡ t − t′ in time, h∆k
= h(k

56
Using properties of the δ-distribution one immediately recognizes the expected classical equations of motion
Favg =∆p ̵ ∆k ∣e∣ ∆A=h== −∣e∣Eavg
∆t
∆t
c ∆t

where

Eavg ≡ −

1 ∆A
.
c ∆t

(3.135)The δ-functions in Eq. (3.134) describe classical evolution of a single momentum state, while thephase factors given by∣e∣∣e∣q(τ ) = k + ̵ [A(τ ) − A(t)] = k′ + ̵ [A(τ ) − A(t′ )]hc
hc(3.136)

relate to quantum interferences due to initial states composed of multiple plane wave contributions(for example, in an isolated wave-packet).

3.6

Ionization by a dynamic field
We will now describe some of the standard golden-rule-like approaches to strong field ioniza-tion in a dynamic field. The earliest one is by Keldysh [98] to describe ionization of a hydrogen-likeatom in which the saddle point approximation for ionization is introduced. Additionally, we will
use the parameter γ described in section 2.3 to connect the γ ≫ 1 regime where perturbation theoryIpis valid to the static-field case (γ ≪ 1) where the exponential factor exp (− 43 ∣e∣Ea) is recovered. Af-

terwards we will show the equivalence of Keldysh’s amplitude with the complementary amplitudeof Perelomov, Popov and Terent’ev [136] where the results of Ref. [98] were generalized to arbitrary
initial states and additional laser configurations. We will focus on the case of short-range potentials
and circular polarization to show agreement with the TDSE in the long-wavelength limit. Finally,
we will discuss the velocity-gauge amplitude given by Reiss [146], where the saddle point method
is abandoned and the amplitude is integrated exactly.

3.6.1

Keldysh rate

Ionization is modeled for a hydrogen atom interacting with the linearly polarized field
E(t) = E cos(ωt)(3.137)

57
and prepared in the initial state
Ri (r) ≈ Cκ,0 κ3/2 e−κr .

(3.138)

Following steps given in Refs. [30, 98] we will use the saddle point approximation to determine a
rate accurate for a large range of γ-values. In the limit of tunnel and multi-photon ionization wewill simplify the rate.
The approximation starts by expanding the exact Green’s function in the initial state LippmannSchwinger equation
Ψ(x) = ϕi (x) + ∫ d4 x1 G(x; x1 )VL (x1 )ϕi (x1 )(3.139)to zeroth order in Volkov states̵ ∫t(−)−(i/h)t0
Φk (x) = e

∣e∣
2
̵
(hk+
c A(τ )) dτ + i∣e∣ A(t)⋅r
̵
hc
2m

(−)

ϕk (r).

(3.140)(−)

Here, we will choose ϕk (r) to be plane waves as it was done in Keldysh’s original paper [98],which has been recently generalized to exact atomic scattering states [58]. When the (plane-wave)
Volkov Green’s functioni
G(V ) (x; x′ ) = − ̵ θ(t − t′ ) ∫ dk Φk (x) Φk (x′ )h(3.141)is inserted into Eq. (3.139) we get the approximation
Ψ(x) ≈ ϕi (x) + ∫ d4 x1 G(V ) (x; x1 )VL (x1 )ϕi (x1 ).

(3.142)Emission of electrons is described by the outgoing partition(−)

Ψ(out) (x) = ∫ dk M(k, t)Φk (x)(3.143)

with coefficients(−)

M(k, t) = ∫ dr Φk (x)Ψ(x).

(3.144)The zeroth order contribution(−)

M(0) (k, t) = ∫ dr Φk (x)ϕi (x)(3.145)

58
is identically zero for finite-range Volkov states and vanishes for plane-wave Volkov states in thelong pulse limit [136]. The lowest order contribution therefore arises fromi(−)M(1) (k, t) = − ( ̵ ) ∫ d4 x1 Φk (x1 )[∣e∣E(t1 ) ⋅ r1 ]ϕi (x1 ),h(3.146)

where the ionization rate is determined by
w = lim ∫ dkt→∞∂
∣M(1) (k, t)∣2∂tt
2
= lim ( ̵ 2 ) Re ∫ dk cos(ωt)L(k, t) ∫ dt1 cos(ωt1 )L(k, t1 ) (3.147)t→∞ h
0in the long pulse limit (ti → 0, t → ∞) with∣e∣E
̵L(k, t) ≡ Vi (k − ̵ sin(ωt)) e(i/h)S(k,t) ,hω

(3.148)the matrix element(−)

Vi (k) ≡ ∫ dr1 ϕk (r1 )[∣e∣E(t) ⋅ r]ϕi (r)(3.149)and the action⎡ ̵⎤2⎢ (hk − ∣e∣⎥c A(τ ))⎢
S(k, t) ≡ ∫ dτ ⎢+ Ip ⎥⎥ .
(3.150)
2m
ti
⎢
⎥
⎣
⎦
The rate, Eq. (3.147), is evaluated by expanding L(k, t) in the Fourier componentst∞̵ (Ek + I˜p − nhω)̵ t] Ln (k)L(k, t) = ∑ exp [(i/h)(3.151)

n=−∞

̵

2

where I˜p = Ip + Up is the shifted ionization threshold, Ek = hk
2m is the ionized kinetic energy,
Up =

∣e∣2
⟨A± (t)2 ⟩T
2mc2(3.152)is the ponderomotive or cycle-averaged quiver energy and the Fourier component corresponding to̵ can be interpreted as the contribution determined by the absorption of n photons. Insertingnhω
this expansion we have∞π̵
w = lim ̵ ∫ dk ∑ ∣Ln+1 (k) + Ln−1 (k)∣2 δ(Ek + I˜p − nhω)
t→∞ h
n=−∞

(3.153)π1̵̵˜
k + Ip − nhω)φ] L(k, φ/ω)∫ dφ exp [−(i/hω)(E2π −π(3.154)withLn (k) =59
̵ n=
and hk√̵ − I˜p ).
2m(nhω

Using properties of the δ-distribution the rate simplifies to∞2π̵w = lim ̵ ∫ dk ∑ ∣L(k)∣2 δ(Ek + I˜p − nhω)
t→∞ h
n=−∞

(3.155)withL(k) =

π
1
∣e∣E
1
̵
(i/hω)S(u)∫ dφ cos φ L(k, φ/ω) =∮ du Vi (k − ̵ u) e
hω
2π −π
2π C

(3.156)

where the Fourier integral is expressed as a contour integral around the dog-bone contour C surrounding branch points u = ±1 and cut connecting the two [30]. The effective action is given
by

2
⎡ ̵
⎤
⎢ (hk − ∣e∣
⎥
ω Ev)
⎢
⎥
⎢
S(u) = ∫ √+ Ip ⎥ .
(3.157)
⎢
⎥
2
2m
0
1−v ⎢
⎥
⎣
⎦
̵ ≪ IpConcepts from sections 3.2 and 3.3 are applied to determine the Fourier coefficients when hωudv̵ ≪ Up . The action S(u) is expanded to second order and the contour is deformed such thatand hω
it passes through saddle points us (determined from S ′ (us ) = 0) along the path of steepest descent.
∣e∣

The slowly changing pre-exponential factor Vi (k − ω Eu) may be evaluated at u = us and Gaussian
integration is performed to evaluate L(k).
In Ref. [98] the rate is evaluated for a Hydrogen-like initial state
Ri (r) ≈ 2Cκ,0 κ3/2 e−κr(3.158)with plane-wave Volkov states yielding
Vi (k) = [∣e∣E ⋅ (i∇k )] ∫ dr1−iCκ,0 (2κ)5/2
eik⋅r
ϕ
(r
)
=
[∣e∣Ek∣∣ ]i
1
π(k 2 + κ2 )3(2π)3/2(3.159)where k∣∣ is the momentum component parallel to the field E. The matrix element contains a poleat k = ±iκ corresponding to the initial bound state [27, 107]. Setting S ′ (u)∣u=us = 0 and defining̵γ ≡ (hωκ/∣e∣E)
the two saddle points¿⎡⎤⎢ k∣∣ Ák⊥ 2 ⎥⎥⎢
ÁÀ
us± = γ ⎢ ± i 1 + ( ) ⎥
⎢κ
κ ⎥⎥
⎢
⎣
⎦

(3.160)∣e∣Eare obtained which correspond exactly to the poles of Vi (k) after the substitution k → k − hω
̵ u.

60
Expanding S(u) to second order and assuming k 2 ≪ κ2 the integral is evaluated with deformed contour passing through the saddle points us± (without crossing them) yielding
√
√
3/2
⎡ ˜
⎤
2
Cκ,0
⎢ 2Ip ⎛⎞
2Ip ⎛I˜p
1 + γ 2 ⎞⎥⎥
γ
−1
⎢
w=
ω ̵ √
sinh γ − γ
S (γ, ̵ ) exp ⎢− ̵2
h ⎝ 1 + γ2 ⎠hω
1 + 2γ 2 ⎠⎥⎥⎢ hω ⎝⎣⎦
after integration [30, 98] with

(3.161)

⎤
⎡
∞
⎢
⎞⎥⎥
⎛
γ
S(γ, x) ≡ ∑ exp ⎢⎢−2(⟨x + 1⟩ − x + a) sinh−1 γ − √⎥⎝⎢1 + γ 2 ⎠⎥⎦a=0⎣⎧⎤1/2 ⎫⎪⎡⎢ 2γ⎪⎪⎥ ⎪(⟨x + 1⟩ − x + a)⎥ ⎬ . (3.162)
× D+ ⎨⎢ √⎥ ⎪⎪⎢⎣ 1 + γ 2
⎪
⎦ ⎪
⎩
⎭
˜
a describes the number of quanta absorbed above the ionization threshold Ip , ⟨x⟩ corresponds tothe integer part of x and D+ (z) ≡ ∫0 ey −z dy is the Dawson function. When performing the finalz

2

2

integral over k interference effects between the two saddle points u = us± are neglected. Errors
introduced by this approximation are small for the total rate w, but must be included to describe
the angular emission rate dw
dΩ [30]. Cκ,0 is numerically fit to the asymptotic tail of the initial state.
Here we follow Ref. [98] and set Cκ,0 = 2 for the 1s state of Hydrogen.
3.6.1.1

Multiphoton limit

Keldysh’s amplitude has been approximated in the weak-field multiphoton limit. Here thelowest near-threshold order a = 0 dominates and the rate
¿
√
⎡Á
⎤
˜ ̵
⎢Á
2Ip 1 ⟨Ip /hω+1⟩
I˜p
I˜p
I˜p
I˜p ⎥
exp (2 ⟨ ̵ + 1⟩ − ̵ ) D+ ⎢⎢À2 (⟨ ̵ + 1⟩ − ̵ )⎥⎥w = 2 ω ̵ ( 2)
h 4γ
hω
hω
hω
hω ⎥
⎢
⎣
⎦

(3.163)is determined by taking γ ≫ 1 in the rate, truncating the sum and approximating sinh−1 γ ≈ log(2γ).
̵ + 1⟩ describe the influence of ionization thresholds and quanta absorbed
Portions containing ⟨I˜p /hω
in the perturbative limit. Care must be taken since the saddle point approximation may not bevalid. We will discuss this further in Chapter 5.

3.6.1.2

Tunnel Limit

For very long wavelengths and/or very strong fields where γ ≪ 1 the sum in Eq. (3.162) is̵approximated as an integral since large a are the dominant contributions yielding S(γ, I˜p /hω)
≈

61
√

3π/4γ 2 . Expanding the exponent to 3rd order in γ the rate of tunnel ionization is given by
√
w=

6π
2

√

4Ip γ
ωIp
γ3
exp
[−
(1
−
)] .
̵
̵
hγ
3hω
10

(3.164)

Taking ω → 0 the exponent in Eq. (3.164) corresponds to the ionization rate of atoms in a quasistatic field. Here, one has
√
w=

6π
2

√

∣e∣EIp4 Ip κ
̵h2 κ exp [− 3 ∣e∣E ] =

√ √
Ip
4 Ip
6π
∣e∣Ea ̵ 2 exp [−].
h23 ∣e∣Ea(3.165)

Averaging the static Landau and Lifshitz rate wstat from section 3.4.2 over a laser cycle [5, 136] anadiabatic rate¿√
Á 3 ∣e∣Ea
I
3 Ip
4 Ip
p
À
exp [−]
wstat = 16 ( ̵ )wadiabatic = Á
2π Ip
2π ∣e∣Ea
3 ∣e∣Eah(3.166)is determined.
The Keldysh rate [98] is unable to recover the correct adiabatic pre-factor since the final(ionized) Volkov state does not include the Coulomb potential as in Ref. [132]. In later sections
as well as in Chapter 5 we will show that the approximation is accurate for short-range potentials.
Often semi-classical long-range Coulomb corrections are applied in the limit of long wavelength
radiation. We will discuss one of the simplest Coulomb corrections [102] to give an idea of how one
might better approximate the long-range rate.

3.6.2

PPT rate

In this section we will show that the first order Keldysh ionization amplitude is equivalent
to the PPT amplitude [136] up to boundary terms for zero-range final states and all applied fields
E(t). After establishing equivalence with the Keldysh rate we will quickly summarize the rate ofs− and p−state ionization in circularly polarized fields [20]. Numerical solutions of the TDSE will
be used to establish accuracy of the rate for s-states and in Chapter 5 we will use the p−state rate
to discuss breakdown of the saddle point approximation and when asymptotic approximations to
the initial state become inappropriate.

62
3.6.2.1

Equivalence with the Keldysh amplitude for zero-range Volkov states

We may write Keldysh’s amplitude as̵ ∫ d4 x1 Φk (x) [∣e∣E(t1 ) ⋅ r1 ] ϕi (x1 )M(1) (k, t) = −(i/h)
=∫

t
t0

̵

dt1 e(i/h)S(k,t1 ) (−∂) ϕ̃i (k(t1 )) (3.167)
∂t1

where
ϕ̃i (k, A(t)) = ∫ dr1 ϕk+ ∣e∣ A(t) (r1 )ϕi (r1 ) ≡ ϕ̃i (k(t))(3.168)̵̵ + ∣e∣ A(t) ≡ p(t).
hk(t)
≡ hk
c

(3.169)̵hcand̵ ∂ we see that
Acting on both sides with ih
∂t
̵
ih

∂
∂S
̵
̵
[M(1) (k, t) + e(i/h)S(k,t) ϕ̃i (k(t))] = − ( ) e(i/h)S(k,t) ϕ̃i (k(t))∂t∂t(3.170)and solving for the amplitude we get
M(1) (k, t) = − [M(0) (k, t1 )∣ti∂S (i/h)S(k,t̵1)+ ( ̵ ) ∫ dt1 ()eϕ̃i (k(t1 ))t1 =t0∂t1
h t0
t(3.171)

where we recognize that the boundary term̵M(0) (k, t) = e(i/h)S(k,t) ϕ̃i (k(t)) ,(3.172)p2 (t)
∂S
)=
+ Ip
∂t
2m

(3.173)p2 (t1 )i
M(1) (k, t) = ( ̵ ) ∫ d4 x1 Φk (x1 ) [+ Ip ] ϕi (x1 ).
h2m(3.174)can be neglected. Using(
the remaining term is written as

To arrive at the plane-wave PPT formulas we recognize−Ip ϕi (x) = [p̂2+ Va (r)] ϕi (x)2m(3.175)

63
which gives usp̂2 − p2 (t1 )iM(1) (k, t) = − ( ̵ ) ∫ d4 x1 Φk (x1 ) [ 1+ Va (r1 )] ϕi (x1 ).
h2m(3.176)Integrating by parts we see thatp̂2
p2 (t)
Φk (x1 ) =Φk (x1 )2m
2m(3.177)i
M(1) (k, t) = − ( ̵ ) ∫ d4 x1 Φk (x1 )Va (r1 )ϕi (x1 ).
h(3.178)yields the desired result

For finite-range final states the situation is different. Writing down the PPT amplitude
and substituting in finite-range states gives a different rate than using the equivalent Keldysh
amplitude with the same states. Starting with Eq. (3.176) we separate the amplitude into a PPTlike contribution and a perturbation
M(1) (k, t) = M(1,PPT) (k, t) + δM(1) (k, t)(3.179)i(−)
M(1,PPT) (k, t) = − ( ̵ ) ∫ d4 x1 Φk (x1 )Va (r1 )ϕi (x1 )h(3.180)p̂2 − p2 (t1 )i(−)δM(1) (k, t) = − ( ̵ ) ∫ d4 x1 Φk (x1 ) [ 1] ϕi (x1 ).
h2m(3.181)

where

and

Using the Fourier representation′

ϕi (x) = (2π)−3/2 ∫ dk′ eik ⋅r ϕ̃i (k′ , t)

(3.182)

and
(−)

′′

(−)

Φk (x) = (2π)−3/2 ∫ dk′′ eik ⋅r Φ̃k (k′′ , t)

(3.183)

we see that the perturbation becomesδM(1) (k, t) =⎡ ̵ 2
̵ ⋅ (hq̵ − ∣e∣ A(t1 )) − ( ∣e∣ A(t1 ))2 ⎤⎥t⎢ (hq) + 2(hk)
i
(−)
c
c
⎢
⎥ ϕ̃i (k + q, t1 ).
− ( ̵ ) ∫ dt1 ∫ dq Φ̃k (k + q, t1 ) ⎢
⎥
h t0
2m
⎢
⎥
⎣
⎦
(3.184)

64
(−)

If ϕk

is a plane wave then the corresponding Volkov state contains a single Fourier com-

̵ = ∣e∣ A(t) and gives δM(1) = 0 as before. Atomic scattering states
ponent corresponding to hqcare described by a distribution of vectors q which leads to δM(1) ≠ 0 in general and therefore adifferent result.

3.6.2.2

Ionization by circularly polarized fields

In Refs. [135–137] the amplitude was generalized to the case of arbitrary initial states and extended to both circular and elliptical polarizations. Additional quasi-classical Coulomb corrections
have been applied to the PPT formulas. We will instead describe the simpler Coulomb corrections [23, 102] which are closely related to the WKB corrections used by Landau and Lifshitz intheir static rate (Sec.3.4.2).
Starting with the right-/left- handed (±) circularly polarized field
A± (t) = −A [sin(ωt)ex ∓ cos(ωt)ey ](3.185)the final state Lippmann-Schwinger equation is applied to determine the first order expansioniM(1) (k, t) = − ( ̵ ) ∫ d4 x1 Φk (x1 )Va (r1 )ϕi (x1 ).
h(3.186)Using̵Va (r)ϕi (x) = (ihp̂2∂−) ϕi (x)∂t 2m(3.187)and integrating by parts the amplitude becomes̵ 2 k(t1 )2tih̵M(1) (k, t) = ( ̵ ) ∫ dt1 [+ Ip ] e(i/h)S(k,t1 ) ϕ̃i (k(t1 )).
h 02m(3.188)∣e∣with k(t) ≡ k + hc
̵ A(t) andϕ̃i (k) ≡ ∫ dre−ik⋅rϕi (r).
(2π)3/2(3.189)The rate can then be determined as
w = lim ∫ dk
t→∞

t
∂
2
∣M(1) (k, t)∣2 = lim ( ̵ 2 ) Re ∫ dk L(k, t) ∫ dt1 L(k, t1 )
t→∞ h
∂t
0

(3.190)65
in the long pulse limit (ti → 0, t → ∞) withL(k, t) ≡ [̵ 2 k(t1 )2h̵+ Ip ] e(i/h)S(k,t1 ) ϕ̃i (k(t1 )).
2m(3.191)

Expanding L(k, t) again as a Fourier series the rate is represented as∞2π̵w = ̵ ∫ dk ∑ ∣Ln (k)∣2 δ(Ek + I˜p − nhω)hn=−∞(3.192)withLn (k) ≡̵ 2 k(t)2
π/ω
h
ω
̵
dt [
+ Ip ] e(i/h)S(k,t) ϕ̃i (k(t))∣.
∫
2π −π/ω
2m
k=kn(3.193)

Switching now to atomic units∣Ln± (k)∣2 = ∣2π
1iS± (k,φ/ω)∣∫ dφ Ξi (k± (φ/ω))e
2π −π
k=kn

(3.194)k± (t)2+ Ip ] ϕ̃i (k± (t))
2

(3.195)withΞi (k± (t)) ≡ [where the integral is approximated using the saddle point approximation to obtain [20]∣Ln± (k)∣2k=kn =√∣Ξi (k(ti ))∣2k=kn
2√e−2n[arcoshχn (kz )− 1−1/χn (kx ) ]2
2πn 1 − 1/χn (kz )(3.196)from the zero-range initial state Eq. (2.72) with saddle points
ωti =π± ϕk + 2πN + i arcoshχn (kz )2which gives(3.197)√kn
1e−2n[arcoshχn (kz )− 1−1/χn (kz ) ]
√
dkz
∑
∫
2π n=nth n 0
1 − 1/χn (kz )2

2
Cκ,0
κ
(l =0)
w± i =∞

2(3.198)for an s-state,√kn
1
kz2 e−2n[arcoshχn (kz )− 1−1/χn (kz ) ]
√
dk
∑
z
∫
2πκ n=nth n 0
1 − 1/χn (kz )2

2
3Cκ,1(l =1,mi =0)
w± i
=

∞

2

(3.199)for a p0 -state and√kn1e−2n[arcoshχn (kz )− 1−1/χn (kz ) ]
√
dk
∑
z
∫
16πκ n=nth n 0
χn (kz )2 1 − 1/χn (kz )2√2
× [2χn (kz )2 (1 ∓ sgn(mi ) 1 − 1/χn (kz )2 ) − (1 + (κ/A0 )2 )n/n0 ] (3.200)

2
3Cκ,1
A20
(l =1,∣mi ∣=1)
w± i
=

∞

2

66
for a p±1 -state whereχn (kz ) =nω
√,
A0 kn2 − kz2(3.201)√nth = ⌈n0 ⌉, n0 = I˜p /ω and A0 ≡ (1/ω) I/2 with I is the laser intensity in atomic units.
Results for the s-state yield:
Nion = 1 − exp [−w (Ncycles

2π
)]
ω

(3.202)are compared with those of numerical solutions of the TDSE (8’th-order B-splines) in Fig. 3.9 forthe Yukawa potentialZ
Va (r) = − e−r/ar(3.203)

with a = 1/5 a.u. and Z are chosen such that Ip = 1/2 a.u.. TDSE calculations are performed for a
16 cycle flat-top pulse with a 2 cycle sin2 ramp on and off. The laser intensity is fixed at I = 1 × 1014
W/cm2 and wavelengths are chosen in the interval from 10 nm to 800 nm. Jumps at 91.928 nm
and longer wavelengths correspond to the single photon and higher order ionization thresholds.

67

S-State PPT Comparison
10−1

Nion

10−2

10−3

10−4

10−5

10

PPT
TDSE

20

40

100
λ (nm)

200

400

800

Figure 3.9: Comparison of results for the total ionization yield obtained by the time-dependent
Schrödinger equation and the s−state PPT formulas [20]. The circularly polarized field has anintensity of I = 1 × 1014 W/cm2 , duration of 16 cycles and wavelengths between 10 nm and 800 nm.
Properties of the Yukawa potential are discussed in the main text.

At the first threshold (91.928 nm) disagreement between the results of TDSE calculations
and those for the PPT formulas are due to the use of a short pulse in the TDSE calculations andthe approximate long-pulse rate in the PPT formulas. For longer pulse durations errors of this form
diminish. At the short wavelength (10 nm) limit the PPT rate starts to deviate from the TDSE
data set due to breakdown of the saddle point approximation. For zero-range s− states and current
laser parameters the approximation works well even for single-photon ionization. In Chapter 5 weshow that this disagreement greatly increases in the case of p-states where we will derive our own
complementary rate.

68
3.6.2.3

Coulomb corrections

In the long-wavelength limit the Landau and Lifshitz ionization rate [107] may be recoveredby applying a similar quasi-classical correction to the final ionized Volkov state. We will stick to
one of the simplest [23, 102] whereΦk (x) ↦ exp [i ∫

r

1/2F

dr′ δp(r′ )] Φk (x),(3.204)√
with δp(r) = rp01(r) and p0 (r) ≈ i 1 − 2F r. We assume sufficiently weak fields giving:exp [i ∫r

1/2F

dr′ δp(r′ )] ≈2
Fr

(3.205)where the long-range Hydrogen rate may be approximated as
wHydrogen ∼ Qwzero-range(3.206)

with Q = (2/F )2 and Cκ,0 = 2. The WKB-like Coulomb correction relates the Hydrogen rate toa closely related zero-range rate. In Ref. [140] improvements were made to the correction withacceptable, but not perfect agreement with results of ab-initio numerical calculations. The lack offully quantum Coulomb corrections for ionization of atoms where γ ≳ 1 proves to be one of thelargest difficulties associated with the theory of strong field ionization.
In Fig. 3.10 we show a comparison of the PPT yield [20, 136] with the simplified WKB-like
Coulomb correction given in Refs. [23, 102] and the TDSE results for ionization of hydrogen atom
by a 16-cycle flat top laser pulse with a two cycle sin2 ramp up and down with an intensity of
I = 1 × 1014 W/cm2 and wavelengths between 10 and 800 nm. In the long wavelength (∼ 800 nm)
limit where γ is smallest the approximation performs well, while for the shortest wavelengths (∼ 350
nm and shorter) the approximation fails completely.

69

Zero-Range + WKB Coulomb Correction

101

100

10−1

Nion

10−2

10−3

10−4

10−5

10−6

final = zero-range, initial = Constant x zero-range
TDSE
10

20

40

100

200

400

800

λ (nm)

Figure 3.10: Comparison of the total ionization yield for hydrogen atom between results of thetime-dependent Schrödinger equation (+ symbols) and the s−state PPT formulas (solid line). The
circularly polarized field has an intensity of I = 1 × 1014 W/cm2 , duration of 16 cycles and wavelengths between 10 nm and 800 nm. The black curve represents application of the Coulomb
correction [23,102] to the circular s-state PPT rate [20,136]. The approximation only agrees in thelimit of γ ≪ 1 and outlines difficulties associated with Coulomb corrections when γ ≳ 1.

For the remainder of the thesis we will apply the amplitude in the short-range case andconsider only this simplest correction. In Chapter 4 we will compare p-state ionization ratios
N+1 /N−1 , and N0 /N−1 between its three magnetic projections m = −1, 0 and 1. Approximating
Nion = 1 − exp [−w (Ncycles

2π
2π)] ≈ w (Ncycles )ωω(3.207)

the Coulomb pre-factor Q vanishes in ionization ratios and it is clear that zero-range estimates oflong-range properties are valid in the long-wavelength limit where Nion ≲ 0.1.

70
3.6.3

Velocity-gauge approach

When solving the TDSE exactly, all observables are gauge-independent. In other words exact
solutions to the TDSE within the dipole approximation should yield the same observables wheneitherVL (x) = ∣e∣E(t) ⋅ r(3.208)orVL (x) =∣e∣
A(t)2A(t) ⋅ p +mc
2mc2(3.209)is used to describe interactions between an electron and an external field [151, 152]. However, both
perturbative approximations [33] and nonperturbative approaches [23, 146] are, in general, gauge
dependent unless expanded to infinite order.
We will outline the most common velocity-gauge amplitude given by Reiss [146] and note that
Faisal [57] has produced a similar rate which allows for more general final states. The amplitude
starts with the velocity-gauge version of the same initial state Lippmann-Schwinger expansion as
used in Keldysh’s amplitude. The first order contribution is(1,VG)

M±

i
(VG)
(VG)
(k, t) = − ( ̵ ) ∫ d4 x1 Φk± (x1 )VL± (x1 )ϕi (x1 ),h(3.210)

where
(VG)

VL±

(x) =

∣e∣
∣e∣2
A± (t) ⋅ p̂ +A± (t)2
mc
2mc2

and

eik⋅r
.
(2π)3/2(3.212)

̵ + A± (t))2(hk∂ (VG)(VG)c
Φk± (x) =Φk± (x)∂t
2m(3.213)(VG)̵ ∫t−(i/h)tΦk± (x) = e
Using
̵
ih

∣e∣
2
̵
(hk+
c A± (τ )) dτ
2m(3.211)

0

∣e∣

with
(VG)
̵ Φ(VG) (x)p̂ Φk± (x) = (hk)k±̵and ih∂ϕi (x) = −Ip ϕi (x)∂t(3.214)

we have(1,VG)M±i(VG)(k, t) = ( ̵ ) (Ek + Ip ) ∫ d4 x1 Φk± (x1 )ϕi (x1 )h(3.215)

71
where vanishing boundary terms have been omitted.
Proceeding as before

t

(VG)

4
∫ d x1 Φk± (x1 )ϕi (x1 ) = ϕ̃i (k) ∫0̵dt1 e(i/h)S± (k,t1 )(3.216)where
ϕ̃i (k) = ∫ dr1e−ik⋅r1
ϕi (r1 )(2π)3/2(3.217)is the Fourier transform of the initial state and using the asymptotic stateiϕi (r) = Cκ,li κ3/2 (κr)ν−1 e−κr Ylm(r̂),
i(3.218)the momentum representation is given asϕ̃i (k) = (−i)li

Cκ,li κν+1/2 miκ
Γ(li + ν + 2)−(li +1/2)
i√Pν+1/2(√) ≡ R̃i (k)Ylm(k̂),
Yli (k̂) √ik( k 2 + κ2 )ν+3/2
k 2 + κ2

which simplifies toϕ̃i (k) =

√

Cκ,0
κ
2 π(k 2 + κ2 )(3.219)(3.220)in the case of a zero-range (ν = 0) s-state.
̵

To complete the amplitude the k ⋅ ξ ± (t) term in e(i/h)S± (k,t1 ) is expanded using the JacobiAnger expansion∞

eik⋅ξ± (t) = eikρ ξ cos(±φk −ωt) = ∑ in e−inωt Jn (kρ ξ)e±inφk(3.221)n=−∞which gives
t

∫

0

̵

∞

dt1 e(i/h)S± (k,t1 ) = e−ik⋅ξ± (0) ∑ in Jn (kρ ξ)e±inφk δt ([N (k) − n]ω/2)(3.222)n=−∞and(1,VG)

M±∞i(k, t) = ( ̵ ) (Ek + Ip ) ϕ̃i (k)e−ik⋅ξ± (0) ∑ in Jn (kρ ξ)e±inφk δt ([N (k) − n]ω/2)hn=−∞(3.223)after time integration. Taking the long pulse limit as before we havedwn (2π)mkn ̵=(nhω − Up )2 ∣ϕ̃i (kn )∣2 Jn2 (kn ξ sin θk )̵3dΩh(3.224)

72
and
wn =

π
(2π)2 mkn ̵
2
(n
hω
−
U
)
dθk sin θk ∣ϕ̃i (kn )∣2 Jn2 (kn ξ sin θk )
p
∫
̵3
h
0

(3.225)where we assumed that ϕ̃i (kn ) contains a single spherical harmonic component which removes theφk dependence from the angular integral (∣eimi φk ∣2 = 1).
Additionally, as given the velocity-gauge formulas incorrectly predict an identical ionizationiirate for co- and counter-rotating electrons [166, 167] since ∣Ylm(k̂)∣2 = ∣Yl−m(k̂)∣2 in the ∣ϕ̃i (kn )∣2ii∣e∣

product. The inclusion of e−i c A(t)⋅r in the length-gauge Volkov states resolves this issue, as shown
in Refs. [20, 136]. In Fig. 3.9 we compare the yield from Reiss’ rate∞w = ∑ wn

(3.226)n=nth(with nth = ⌈I˜p /ω⌉) to the numerical data produced by simulations of the TDSE, as discussed in
the PPT section 3.6.2. Again differences persist near ionization thresholds since we have modeled
the total yield using a long pulse rate.

73

Velocity-gauge S-State Comparison
10−1

Nion

10−2

10−3

10−4

10−5

10

Velocity-Gauge
TDSE

20

40

100
λ (nm)

200

400

800

Figure 3.11: Comparison of the total ionization yield between the time-dependent Schrödinger
equation (data from Fig. 3.9) and s−state Reiss’ formulas [146]. The circularly polarized field has
an intensity of I = 1 × 1014 W/cm2 , duration of 16 cycles and wavelengths between 10 nm and 800
nm.

For zero-range s-states and laser parameters considered, both the velocity-gauge expansion
discussed here as well as the PPT expansions of section 3.6.2 are able to accurately describe theionization yield. In Chapter 5 we will show that for ionization of an initial p−state, the velocitygauge formulas discussed here fail at all wavelengths and the saddle point approximation applied
to the PPT formulas is unable to accurately predict ionization across a large range of wavelengths.
We will resolve this issue by integrating the length gauge rate of Keldysh [98] exactly using a similar
expansion as used in this section.

Chapter 4

Selectivity in electron emission induced by ultrashort circularly polarized laser
pulses4.1

Introduction1
Spin polarization of atomic photo-electrons is an important aspect in exploring the structureof atoms, molecules and solids [99]. The generation of ultrashort spin-polarized electron pulses canbe achieved by ionizing a gaseous atomic target with ultrashort circularly polarized laser pulses.
These electron pulses are useful tools to characterize and probe chiral systems and magnetic properties of materials on an ultrafast time scale. Towards an understanding of the mechanisms behind
the generation of spin-polarized electrons it is essential to study the sensitivity of the ionization
probability on the sense of the electron’s rotation in the initial state of the atom with respect tothe helicity of the applied circularly polarized laser field [18, 77]. Thus, it is interesting to identify
laser parameter regimes in which the ionization rate strongly depends on the number and/or thesign of the magnetic quantum number in the initial state.
Spin polarization of photo-electrons and the related dependence of atomic photoionizationon the relative rotation between electrons in the initial state and the applied field has first beenstudied for single-photon ionization. While it was first believed that spin polarization of photoelectrons is a relativistic effect [141], Fano showed that electrons can be spin polarized in thenonrelativistic energy region as well [59]. In general, for single-photon ionization the emission
1

Part of the material presented in this Chapter has been previously published in S. Walker, L. Kolanz, J. Venzke,
and A. Becker, Physical Review A 103, L061101 (2021) and in S. Walker, L. Kolanz, J. Venzke, and A. Becker,
Physical Review Research 3, 043051 (2021).

75
of electrons co-rotating with respect to the rotation of the electric field of the applied field ismore likely than that of counter-rotating electrons. Stronger spin polarization was predicted forenergy regions near Cooper minima. Experimental [26,79] and further theoretical [38] investigations
supported these predictions by Fano. Preference of the emission of co-rotating over counter-rotating
electrons has also been found in the ionization of atoms prepared in Rydberg states with microwave
radiation [150, 180]. In the seventies and eighties of last century the quest for sources generating
spin-polarized electrons continued along with the development of laser technology. Significant
electron spin polarization has been predicted and observed in two- and three-photon ionizationof atomic targets [8, 16, 49, 50, 60, 61, 70, 71, 104–106, 157]. At the corresponding laser frequencies
the mechanism for spin polarization is related to intermediate resonances. The cross sections fornear-resonant two- or three-photon ionization can be large if high-power lasers with narrow spectral
bandwidth are available.
Using ultrashort laser pulses fine-tuning of the laser frequency to specific resonances is generally not possible and (near-)resonant effects, previously observed with lasers having a long pulse
duration, are often less pronounced in the ultrashort pulse regime. However, in the highly nonperturbative tunneling regime of strong-field ionization a different selectivity in the emission ofelectrons on the sense of the electron rotation in the initial state has been predicted recently [18].
In contrast to the general observations in the single-photon ionization regime, strong-field tunnel ionization yields larger ionization probabilities for counter-rotating electrons than for thoseco-rotating with respect to the helicity of the applied field. The effect is due to nonadiabatic tunneling in which the two kind of electrons have, contrary to the standard assumption in strong-fieldtunneling theory, nonzero velocities at the exit of the tunnel barrier. This prediction sparked significant experimental [45, 52, 53, 77, 80, 85, 118, 158, 176] and theoretical [11–14, 17, 19–22, 24, 25, 35,
40, 44, 45, 74, 76, 82, 88, 90, 92–97, 110, 113–118, 122, 128, 166, 169, 170, 174–176, 179, 181, 182] interest
over the last decade, including the observation of spin polarization of electrons of about 30%-50%
for the interaction of atoms with laser pulses at 800 nm [21, 77, 158] and even larger one at about
400 nm [176].

76
The role of excited states on the selectivity in the emission of counter- versus co-rotating
electrons has been investigated in a few studies [13, 25, 44, 82, 90, 166, 176, 182]. It has been shownthat resonant excitation can lead to peaks in the helicity dependent enhancement of ionization at
wavelengths in the deep multiphoton regime [182]. On- and off-resonance two-photon ionizationof an inner p-sub-shell electron has been discussed in the context of fluorescence polarization [82].
More recently, experimental and theoretical work has demonstrated that at wavelengths of about
400 nm Freeman resonances [65] provide a scheme to separate counter-rotating electrons fromco-rotating electrons in the photo-electron energy spectrum of xenon atoms [176]. In this caseresonant ionization via an intermediate state close to the threshold, which is accessible to thecounter-rotating electrons only, leads to a selection in the electron emission at a specific kinetic
energy. Therefore, the scheme provides an opportunity to generate spin-polarized electrons byenergy gating. Furthermore, strong asymmetries in the ionization with left- and right-handed
circularly polarized light for the lithium atom prepared in the polarized 2p(m = +1) state has been
observed [44]. In that combined experimental-theoretical study the emission of counter-rotating
electrons is found to be favored when the wavelength is tuned to the 2p−3s transition. We have moregenerally analyzed the impact of resonant transitions via excited states, that – due to the selection
rules – can only be accessed by counter-rotating electrons. As we will show below, these states act
as doorway states in the few-photon ionization regime and are responsible for a dependence of thetotal ionization probability on the magnetic quantum number of the initial state which is stronger
than those in both the single-photon and the tunneling regimes. The maximum ratios of total
ionization of counter- over co-rotating electrons, found in the numerical calculations, are about ten
to one, indicating the possibility to achieve a large selectivity in the overall electron emission.
More specifically, based on numerical solutions of the time-dependent Schrödinger equation
we predict a surprisingly large enhancement in the emission of electrons, that are initially counterrotating with respect to rotation of the applied field, during the interaction of rare-gas atoms withultrashort circular polarized laser pulses in a previously unexplored wavelength regime [166, 167].
In order to determine the mechanism behind this surprising enhancement we performed additional

77
numerical calculations where specific emphasis was given to the intermediate-wavelength regime
where the trend in the ionization probability of counter-rotating electrons strongly differs from thosefor electrons in the other two initial magnetic sub-levels. Theoretical analysis and numerical results
indicate that the enhanced emission of counter-rotating electrons occurs via photon absorption
channels, which are only accessible for electrons rotating opposite to the rotation direction of
the external electric field. Physical mechanisms behind the enhancement are identified as threshold
effects, in which the emission into continuum states with low angular momentum quantum numbers
is favored, and resonant enhanced ionization involving transitions via specific excited states in theatom [166, 167].

4.2

Numerical methods
To study the dependence of strong-field ionization on the value and the sign of the magnetic

quantum number, we have solved the TDSE for the interaction of an atom with an intense laser
pulse in atomic units, wherei∂∇2
Ψ(r, t) = [−+ E(t) ⋅ r + V (r)] Ψ(r, t) .
∂t
2

(4.1)

Here, V (r) is a single-active electron (SAE) model potential for an electron in the 2p-shell in Neonand in the 3p shell in Argon given by:
1 Zc e−cr
V (r) = − −− ∑ aj e−bj r ,
r
r
j

(4.2)

where for Neon, ZC = 9, c = 2.0872, a1 = −5.4072, a2 = 1.0374, b1 = 4.1537 and b2 = 67.1114.
Similarly for Argon ZC = 17, c = 0.8472, a1 = −15.3755, a2 = −27.7431, a3 = 2.1705, b1 = 1.2484,
b2 = 4.3924 and b3 = 87.9345. The potential has been constructed using a method for benchmarking
tests between TDSE and time-dependent density functional theory calculations [143]. We will
discuss general features in detail for Neon [167] and note that we have observed the exact same
trends in Argon [166].

78
For the circularly polarized laser pulse we set the vector potential of the applied field as:
A(t) = A0 sin2 (where A0 =√πt
) [sin (ωA t) x̂ − ϵ cos (ωA t)ŷ]
τ

(4.3)

2 , τ = 2πN , c is the speed of light, I is the peak intensity, N is thec2 I/(1 + ϵ2 )ωAωA

number of cycles in the pulse, and ωA is the central frequency of the vector potential. This ensuresthat the electric field integrates to zero. ωA is determined such that the spectral distribution of theE-field matches the given physical central frequency ωE [164]. The electric field is obtained byE(t) = −1∂
A(t) .
c ∂t(4.4)

We have considered a wavelength regime from 10 nm, which corresponds to ionization bya single-photon, to 1000 nm, corresponding to tunneling ionization. At all wavelengths we haveapplied intense laser pulses with a duration of 10-cycles and a peak intensity of 5 × 1014 W/cm2 .
Without loss of generality we have chosen left-handed circularly polarized pulses (i.e. ϵ = −1). In
each calculation we have prepared the atomic system in one of the three magnetic sub-levels of
the p-shell. Eq. (4.1) has been solved in three spatial dimensions by using previously discussed
grid-based [163] and numerical basis-state methods [37, 168]. At the end of each simulation we
have determined the population in the excited states, characterized by the quantum numbers n,
l and m, via projection on the corresponding bound states of the atomic potential, which havebeen determined numerically. The ionization probability has then been evaluated as the difference
between unity and the total probability left in the bound states.
In the grid-based method we have expanded Ψ(r, t) in spherical harmonics up to lmax =∣mmax ∣ = 40 and discretized the radius using fourth order finite difference. The wave function has
been propagated in time with a time step of 0.05 a.u., on a grid with spacing of 0.05 a.u., maximum
radius of 500 a.u., and exterior complex scaling on the outer 25 a.u. of the grid using the CrankNicolson method to propagate the wave function in time. This method was mainly used for thesimulations at long wavelengths.
In the numerical basis state method we have also expanded Ψ(r, t) in spherical harmonics
with lmax = ∣mmax ∣ = 40 in a box with maximum radius of 500 a.u. The basis was generated by

79
diagonalizing the field-free Hamiltonian including a double-exponential complex absorbing potential[78] on the outer 50 a.u. of the domain using a second order finite difference method with a grid
spacing of 0.03 a.u. In the calculations we have chosen a maximum principal quantum number
nmax = 500 corresponding to a maximum energy per angular momentum block of about 4.9 a.u.
The wave function is propagated in time using the Crank-Nicolson method with a time step of 0.05
a.u. We account for outgoing boundary conditions and the results have been checked against thoseof the grid-based approach. Transitions to occupied states other than the initial state have beenprohibited in the basis code calculations.

4.3

Selectivity in electron emission
We first discuss concepts to enhance the emission probability of electrons from initial atomic

states with different magnetic quantum numbers via photon absorption. The absorption of aphoton from a left-handed circularly polarized field leads to changes of orbital angular and magnetic
quantum numbers of ∆l = ±1 and ∆m = −1. The schematic representations in Fig. 4.1 show theallowed pathways for the absorption of the first three photons from initial (a) p and (b) d states.

80

Figure 4.1: Photon absorption channels for interaction of a left-handed circularly polarized laser
field (∆l = ±1, ∆m = −1) with an atom in initial (a) p (red: m = −1, green: m = 0, blue: m = 1) and(b) d state (red: m = −2, −1, green: m = 0, 1, blue: m = 2) [167].

In the case of an initial p state (Fig. 4.1(a)) for electrons in each of the three magnetic sublevels transitions with ∆l = 1 are allowed leading to a successive increase of the orbital angularquantum number with the absorption of each photon, where additional photon absorption channels
are accessible for the counter-rotating electrons (here, electrons in the initial m = 1 level, blue
arrows). For example, for a counter-rotating electron the absorption of a (left-handed circularly
polarized) photon induces a change in magnetic quantum number to m = 0 and therefore transitions
into both l = 2- and l = 0-levels are allowed. In contrast, for the other two initial sub-levels only
the transition to l = 2 is possible. Similar considerations hold for the absorption of additional
photons. If the probability via these additional channels can be enhanced, we should therefore
expect a strong selectivity of the emission of counter-rotating electrons in the interaction with acircularly polarized laser field. As our results below indicate this can be achieved either near thethresholds of photon absorption (section 4.3.2) or via intermediate resonances which act as doorway
states accessible for the counter-rotating electrons only (section 4.3.3). These additional pathways

81
to ionization have been also mentioned in Refs. [13, 25, 44, 82, 90, 166, 176]. Resonant transitions
via different pathways for counter-rotating electrons in two-photon ionization has been discussed
in Refs. [82, 90]. Since selective emission via these additional pathways does not rely on a splittingof energy levels, the enhancement can be achieved independent of the duration of the applied laser
pulse, which is interesting for the generation of ultrashort spin-polarized electron pulses.
With our numerical results presented below we study the applicability of these concepts forthe initial 2p state in the neon atom, similar results have been found for argon atom as well [166].
Reasoning along similar lines as above can be however done for initial states with a larger orbital
angular momentum number as well, as it is illustrated for an initial d state in Fig. 4.1(b). Besides
the ∆l = 1 pathway, which is open for electrons from each of the five initial m-levels, additional
pathways are available for electrons from the initial levels with m = 0, 1 (red arrows) and evenmore for those from the m = 2 state (blue arrows). Thus, by identifying specific frequency regionsin which the transition probability via one or some of these additional pathways is large one can
expect an enhancement in the emission of certain groups of electrons.

4.3.1

General trends

In Fig. 4.2 we present the numerical results for the ionization probability from the three 2p
orbitals of neon-like atom as a function of wavelength in a 10-cycle left-handed circularly polarizedlaser pulse at peak intensity of 5×1014 W/cm2 . P+ , where the index denotes the magnetic quantum
number of the initial orbital (m = 1, 2p+ , stars with line), corresponds to the ionization probability
for counter-rotating electrons, while P− (circles with line) is the one for co-rotating electrons andP0 (squares with line) denotes the ionization probability for the initial state with m = 0.

82

ionizat ion pr obablilit y

10 − 110 − 210 − 310 − 410 − 5
10 1

Figure 4.2:

10 2
w avelengt h (nm )10 3

Ionization probabilities P+ (stars with line), P0 (squares with line), and P− (circles

with line) as function of wavelength for neon-like atom at 5 × 1014 W/cm2 interacting with 10-cycle
left-handed circularly polarized laser pulses [167].

There are a few general trends which we will discuss before we further analyze more specificfeatures in the next subsections. In the short wavelength regime the single-photon ionization
probability for each of the three sub-shells increases approximately proportional to λ7/2 , as it isexpected at large photon energies based on the Born approximation [32]. In agreement with earlier
work [59, 104] for single-photon transitions co-rotating electrons are easier to ionize than counterrotating electrons, while the ionization probability for the initial orbital with m = 0 remains in
between those for the other two sub-shells (section 4.3.2). The differences in the trends of theionization probabilities near the one-photon threshold will be discussed in section 4.3.2.
In the two-, few- and multiphoton ionization regime the dependencies of P− and P0 on the
wavelength remain rather similar. As discussed out the outset of section 4.3 (c.f., Fig. 4.1(a)), forelectrons in initial p− and p0 states absorption of photons is allowed through the pathway with83
∆l = 1 only. Therefore, the absorption of N photons results in the final state with lf = N + l andmf = m−N , where l is the angular momentum quantum number of the initial state. In lowest order
perturbation theory the transition amplitudes for both initial states share identical intermediate
reduced matrix elements and differ from each other only by a 3j symbol. A perturbative scaling
law for the ratio of ionization probabilities via the ∆l = +1 pathway for electrons in initial states(l, m) and (l, m′ ) is therefore given by2⎛ l+k
1 l+k−1 ⎞⎜⎟⎜⎟Nmin ⎝k − m −1 m + 1 − k ⎠
Pm
= ∏
,
2
Pm′
k=1 ⎛
⎞
l+k
1
l+k−1
⎜
⎟
⎜
⎟
′
′
⎝k − m −1 m + 1 − k ⎠(4.5)

where Nmin = ⌈Ip /ω⌉ is the minimum number of photons absorbed and Ip is the field free ionization
potential. Using¿⎛ l+k
1 l+k−1 ⎞Á (2k + l − m − 1)(2k + l − m)l−mÀ⎜
⎟ = (−1) Á,⎜⎟(2k + 2l − 1)(2k + 2l)(2k + 2l + 1)
⎝k − m −1 m + 1 − k ⎠(4.6)

Pm Nmin (2k + l − m − 1)(2k + l − m)(l − m′ )!(2Nmin + l − m)!
=
.
= ∏
′
′
Pm′
(l − m)!(2Nmin + l − m′ )!
k=1 (2k + l − m − 1)(2k + l − m )(4.7)

we getThus, for an initial p state with l = 1 we haveP0
1=
P− Nmin + 1(4.8)

P+ (lf = Nmin + 1)
1=,
P−
(Nmin + 1)(2Nmin + 1)

(4.9)

and

where P+ (lf = Nmin + 1) is the perturbative ionization probability for counter-rotating electrons
along the ∆l = +1 pathway only.
The comparison in Fig. 4.11(a) shows that the numerical results for P0 /P− (circles with line)
overall agree well with the perturbative scaling law (solid line, Eq. (4.8)) at the shorter wavelengths.
The differences near the photon absorption thresholds are due to the bandwidth of the short pulse

84
used in the calculations. The deviation at large wavelengths indicates the transition to the tunnelingregime. In this regime the numerical results for P0 /P− agree rather well with predictions based on
nonadiabatic tunneling theory (dotted line in Fig. 4.11(a), [18, 20]):¿2P0 Áγ 3 (ζ0 − 1)2 (ζ0 + 1)
À I 1+γ=Á,√P−
16Ip3 ζ02 + γ 2 ( ζ 2 + γ 2 − ζ √1 + γ 2 )200
where I is the laser intensity, γ is the Keldysh parameter and ζ0 is determined by¿¿⎛ 1 Á ζ 2 + γ2 ⎞Á ζ 2 + γ2
Á
Á
À 0
À 0
⎟.
= tanh ⎜1 + γ21−ζ
1 + γ2 ⎠0⎝(4.10)(4.11)

In contrast, the results for P+ /P− (circles with line) in Fig. 4.11(b) show a deviation fromboth the perturbative scaling law (solid line, Eq. (4.9)) and the prediction of the nonadiabatic
tunneling theory (dotted line, [18, 20]):√√
2
P+ ⎛ ζ02 + γ 2 + ζ0 1 + γ 2 ⎞√= √P − ⎝ ζ 2 + γ 2 − ζ0 1 + γ 2 ⎠(4.12)

0in the intermediate two- and few-photon ionization regime. Here, P+ /P− changes by two ordersof magnitude, reversing from 1:10 to 10:1. An enhancement of the emission of counter-rotating
electrons in the few-photon ionization regime has also been observed in numerical results for strongfield detachment of F− [13]. As we will discuss in the next subsections, for the present results thestrong variation in the ratio can be explained as due to the impact of the additional pathways
available for the counter-rotating electrons (c.f., Fig. 4.1(a)). From the results in Fig. 4.11(b) we
can identify that these pathways should have a strong impact at the one- and two-photon ionizationthresholds, the region at about 70 nm and the few-photon ionization regime.

P0 /P −85

10 − 1(a)
10 1

P + /P −

10 1

10 2
w avelengt h (nm )

10 3

10 2
w avelengt h (nm )

10 3

(b)

10 0

10 − 1

10 1

Figure 4.3:

Comparison of numerical results (circles with line) for the ratios (a) P0 /P− and(b) P+ (lf = Nmin + 1)/P− with the scaling laws from lowest order perturbation theory for ∆l = 1
pathway (solid lines, Eqs. (4.8,4.9)) and those from nonadiabatic tunneling predictions (dotted lines,
Eqs. (4.10,4.12)) as function of wavelength. Laser parameters are the same as in Fig. 4.2 [167].

86
Before proceeding, we note that the present numerical results agree well with the predictionsof the nonadiabatic tunneling theory at long wavelengths. A similar degree of agreement has beenreported before for results from an analytical R-matrix theory [94], 2D-TDSE calculations [17]and the R-matrix with time dependence method [13]. Furthermore, the results of the analyticalR-matrix theory [94] indicate a maximum in the P+ /P− ratio as well, which occurs in a wavelength
regime close to that one where the maximum is found in the present results. Finally, we note thatthe trend of the results for P+ and P− in Figs. 4.2 and 4.11(b) near 1000 nm may indicate thatthe relative emission probability of counter- versus co-rotating electrons reverses at even longer
wavelengths. Such a reversal has been predicted in past theoretical studies [174, 182].

4.3.2

Ionization thresholds and two-photon ionization

At wavelengths near the one-photon ionization threshold the relative probability for emissionof counter-rotating electrons increases as compared to those from the initial levels with m = −1 andm = 0. This can be seen from the ratios P+ /P− (Fig. 4.4(b), stars with line) and P+ /P0 (Fig. 4.4(c),stars with line). The enhancement is due to the fact that for small photo-electron energies thetransition in the continuum is favored for final states with low angular momentum quantum numbers[32]. Since transitions into continuum states with lf = 0 are only allowed for counter-rotating
electrons (Fig. 4.4(a)) the emission of those electrons is enhanced near the threshold, which raises
the two ratios. This interpretation is verified by the comparison with reduced ratios, in which wedeliberately neglected the probability for the channel to one of the final states in the P+ results.
To this end, we have computed the continuum populations for a certain angular momentum by
projecting the wave function onto the corresponding continuum functions used in the basis method
and adding these coefficients squared. The reduced ratios show the dominant emission in the lf = 2channel (squares with line) and the lf = 0-channel (circles with line) at short wavelengths and near
the threshold, respectively.

87

0.5

1.0

0.4

0.8

0.3

0.4

0.1

0.2

25
50
w avelengt h (nm )

(c)

0.6

0.2

0.0

Figure 4.4:

1.2

(b)

P + /P0

P + /P −

0.6

0.0

25
50
w avelengt h (nm )Upper row: (a) Channels for single-photon ionization. Lower row: Comparison ofratios (b) P+ /P− (stars with line), P+ (l = 2)/P− (circles with line), and P+ (l = 0)/P− (squares withline), and (c) P+ /P0 (stars with line), P+ (l = 2)/P0 (circles with line) and P+ (l = 0)/P0 (squares withline) as function of wavelength. Also shown are the predictions based on lowest order perturbation
theory for the ∆l = 1 channels (solid lines without symbols, Eqs. (4.8,4.9)) [167].

The same effect explains the enhancement in the emission of counter-rotating electrons forwavelengths near and just above 100 nm, at which the two-photon ionization channel closes. Since
the pathway into the lf = 1-channel is allowed for the counter-rotating electrons only (Fig. 4.5(a)),

88
predominant emission of counter-rotating electrons into the final channel with the lowest possible
angular momentum quantum number enhances the ratio of P+ /P− and P+ /P0 near the threshold(Fig. 4.5(b,c)).

(c)

3.0

0.8

2.5

0.6

2.0

P + /P0

P + /P −

1.0

3.5

(b)

0.4

1.5
1.0

0.2
0.0

0.5
50
75
100
w avelengt h (nm )

0.0

50
75
100
w avelengt h (nm )Figure 4.5: Upper row: (a) Channels for two-photon ionization. Lower row: Comparison of ratios(b) P+ /P− (stars with line), P+ (l = 3)/P− (circles with line), and P+ (l = 1)/P− (squares with line),
as well as (c) P+ /P0 (stars with line), P+ (l = 3)/P0 (circles with line) and P+ (l = 1)/P0 (squares
with line) as function of wavelength. Also shown are the predictions in lowest order perturbation
for the ∆l = 1 channels (lines without symbols) [167].

89
The impact of the channels with lowest angular momentum quantum number in the final
state can be also investigated via photo-electron angular distributions. For ionization along the∆l = 1 pathway, available for all electrons, we expect an angular distribution proportional to thespherical harmonics ∣Y2m−1 ∣2 (for single photon ionization) and ∣Y3m−2 ∣2 (for two-photon ionization).
As can be seen, from the comparison of the numerical results in Fig. 4.6, obtained at the one-photon(left column, 53.8 nm) and the two-photon threshold (right column, 104.12 nm), this is the casefor ionization from both 2p0 (panels (b) and (e)) and 2p−1 (panels (c) and (f)). In contrast, theangular distributions for the counter-rotating electrons at the one-photon threshold (Fig. 4.6(a))
show signatures of an interference. Specifically, in the projections it is seen that the node that would
occur for a pure ∣Y20 ∣2 distribution is not present. This indicates the presence of an interferencewith a spherical harmonic having a different, here lower, angular momentum, as expected. At thetwo-photon threshold the contribution from l = 3 is significantly smaller than that from l = 1 sointerference is barely visible in the photo-electron angular distributions (Fig. 4.6(d)).

10p z (a rb . un it s)

10p z (a rb . un it s)

−1
)
1
(b)
it s
n
−1
0 .u
px ( a 0
rbrb. u
n it s 1 − 1 (a)
py−1
1 t s)(c)i−1
0 . unpx ( a 0
rbrb. u
n it s 1 − 1 (a
)
py

0.0

Figure 4.6:

0.2

0.4

p z (a rb . un it s)

−1
)
1
(a)
it s
n
−1
0 .u
px ( a 0
rbrb. u
n it s 1 − 1 (a
)
py

0

−1
)
1
(d)
it s
n
−1
0 .u
px ( a 0
rbrb. u
n it s 1 − 1 (a
)
py

1
0

p z (a rb . un it s)

0

1

−1
)
1
(e)
it s
n
−1
0 .u
px ( a 0
rbrb. u
n it s 1 − 1 (a
)
py

1
0

p z (a rb . un it s)

1

p z (a rb . un it s)

90−1
1 t s)(f)i−1
0 . unpx ( a 0
rbrb. u
n it s 1 − 1 (a
)
py

0.6

0.8

1.0

Comparison of photo-electron angular distributions for ionization from initial statem = 1 (a,d), m = 0 (b,e) and m = −1 (c,f) near one-photon (a-c: 53.8 nm) and two-photon (d-f:104.12 nm) threshold [167].

91
The enhancement in the P+ -yield and the corresponding ratios at a wavelength of about 70 nm
in the two-photon ionization regime is due to resonant ionization via the intermediate 3s doorway
state. While transition into the 3d state is allowed from all three initial states via the absorptionof one left-handed circularly polarized photon, the 3s state is only accessible for electrons in theinitial 2p−1 state (c.f., Fig. 4.1(a)). Indications for the impact of the doorway state are given by theratios P+ (lf = 1)/P− and P+ (lf = 1)/P0 in Fig. 4.5(b,c) and the large population in the 3s state forionization from the initial 2p+ state at the end of the pulse (Fig. 4.7(a)). The relative importance ofresonant transition via s and d states in the two-photon ionization regime has also been discussed
for interaction of counter-rotating electrons prepared in excited 2p state in hydrogen atom [90].
Similarly, an enhancement in the emission of counter-rotating electrons has been observed for aresonant 2p−3s transition in the three-photon ionization of lithium atoms, prepared in the polarized
2p(m = +1) state [44]. Together, the data clearly support the interpretation that the resonant
ionization pathway, which is accessible for the counter-rotating electrons only, is the origin for theenhancement in the P+ -yield. The comparison with the final populations in the states of the n = 3level for the other two initial states in Fig. 4.7(b,c) show that in those cases only the 3d state ispopulated and the population decreases as a function of wavelength in the two-photon ionization
regime.

92
10 − 1(a)10 − 3

excit at ion pr obablilit y

10 − 510 − 710 − 1(b)

10 − 310 − 510 − 710 − 1(c)10 − 310 − 510 − 7
50

55

60

65

70

75

80

w avelengt h (nm )Figure 4.7:

Excitation probabilities in 3s (orange squares), 3d (red circles) and all states (bluestars) for ionization from initial states (a) 2p1 , (b) 2p0 and (c) 2p−1 for neon-like atom at 5 × 1014
W/cm2 ) interacting with 10-cycle laser pulses [167].

4.3.3

Doorway states in few-photon ionization regime

In contrast to the two-photon ionization regime, for few-photon ionization states in the n = 3level can only be accessed by the counter-rotating electrons, as shown in the two schemes in Fig. 4.8.
Corresponding to each of these additional pathways one can expect resonant enhanced ionizationin the P+ -yield. In contrast, there are no pathways for resonant enhanced ionization via the n = 3

93states for electrons in the other initial states.

Figure 4.8:(a) Two- and (b) three-photon transitions into doorway states in the n = 3 shellfrom initial 2p1 state. Other non-resonant transitions are also shown using same arrow styles as inFig. 4.1 [167].

The importance of the resonant transitions is supported by the results of our numerical calculations in Fig. 4.9. The comparison of the total and 3l-excitation probabilities for the three initialstates at the end of the pulse shows the stronger role of the overall excitation for counter-rotating
electrons (Fig. 4.9(a)) as compared to the other initial states (Fig. 4.9(b,c)) in the wavelength
regime between 100 nm and 200 nm. It is also seen that a major part of the total excitation arises
from populations in either the 3p-level (two photon resonant transition) or the 3d-level (three photon resonant absorption). In contrast, there is only very small final population in the 3d state forthe other two initial states. This population arises from one-photon transitions due to the broad
bandwidth of the ultrashort pulse. Similarly, we see some small population in the 3s state due to
one-photon absorption for the initial 2p1 state.

94

10 0

(a)
10 − 210 − 4

excit at ion pr obablilit y

10 − 610 − 810 0(b)
10 − 210 − 410 − 610 − 810 0(c)10 − 210 − 410 − 610 − 810 2

2 × 10 2

3 × 10 2

w avelengt h (nm )Figure 4.9: Comparison of excitation probabilities in 3s (orange squares), 3p (green triangles), 3d(red circles) with total excitation probability (blue stars) from initial states (a) 2p1 , (b) 2p0 and(c) 2p−1 for neon-like atom with 10-cycle laser pulses at peak intensity 5 × 1014 W/cm2 ) [167].

The remaining part of the total excitation probabilities for the counter-rotating electrons aswell as the contributions for the other initial states arise from higher excited states. This is shown in
Fig. 4.10, where we present the corresponding populations in states with n ≥ 4, separately for eachl-quantum number up to l = 6 (i states). The excitation for the initial states with (b) m = 0 and (c)m = −1 both solely arise from the f - (at about 110 nm), g- (∼ 170 nm), h- (∼ 200 nm), and i−levels

95(∼ 240 nm), confirming that mechanism and pathway to excitation and ionization from these twostates are similar. The population of levels with successively larger l-value in separate regimes, as
the wavelength increases, is in agreement with the interpretation of a resonant enhancement alongthe ∆l = 1 pathway. In contrast, the results for the counter-rotating electrons (Fig. 4.10(a)) again
indicate the strong impact of the additional pathways. In this case the populations in the excitedstates along the ∆l = 1 pathway do not provide significant contributions to the total excitation
probability. For example, the population in the nf states (purple crosses) at about 110 nm for
initial state m = 1 is smaller than the populations in the corresponding states for the interaction
with the other two initial states. However, its contribution to the total excitation probability is
negligible. Instead, the populations in the excited states with lower l are, in general, by about
two orders of magnitude stronger than those with the higher l’s. This observation agrees withtheoretical results for on-resonance two-photon ionization of an inner atomic np-sub-shell by acircularly polarized light field [82].

96

10 0

(a)
10 − 210 − 4

excit at ion pr obablilit y

10 − 610 − 810 0(b)
10 − 210 − 410 − 610 − 810 0(c)10 − 210 − 410 − 610 − 810 2

2 × 10 2

3 × 10 2

w avelengt h (nm )Figure 4.10: Same as Fig. 4.9 but for ns (orange squares), np (green triangles), nd (red circles), nf(purple crosses), ng (brown diamonds), nh (pink plus signs), and ni (grey 3 pointed stars) stateswith n ≥ 4. Total excitation probabilities are represented by blue 5 pointed stars [167].

97
4.3.4

Application to Argon atoms

Figure 4.11: Ratio P+ /P− (stars with solid lines) and P0 /P− (open circles with dotted lines) as afunction of wavelength for (a) neon-like atom (5 × 1014 W/cm2 ) and (b) argon-like atom (5 × 1013
W/cm2 ) interacting with 10-cycle laser pulses [166].

In general, the results for an argon-like atom can be interpreted in the same way as those forthe neon-like atom. A few additional features arise due to the more complex level structure among
the doorway states closest in energy to the initial 3p-levels.
In Fig. 4.11, we plot the P+ /P− ratio (solid blue line) and P0 /P− ionization ratio (solid blackline) now for both neon atoms (panel a) and argon atoms (panel b), where all laser parameters in
the case for argon have been chosen identically to the neon-like case except the use of a reduced
laser intensity of 5 × 1013 W/cm2 . The large maximum in the single-photon regime for the argonlike atom at about 30 nm and several of the small maxima in the two-photon regime for both
atoms are due to the Cooper minimum and resonant transitions, respectively. These signals are
therefore short-pulse features of the effects studied for continuous lasers or long pulses in earlier
studies [59, 104]. Here, the vertical solid lines correspond to ionization thresholds where a maxima
in the co- vs. counter-rotating ratio for both neon and argon atoms again lies approximately at the
five-photon ionization threshold corresponding to the transition between tunneling and multiphoton
ionization.

98
When analyzing the P0 /P− ratio we again see the perturbative scaling law, Eq.(4.8), where
the effects of resonances and Cooper minima are not observed. We must note that the upward
slope of the ratio in Fig.(4.11 b) above 600 nm is likely a convergence error in the numerical results
but does not alter our results and analysis.

4.4

Summary
To summarize, we have performed ab-initio numerical calculations of the time-dependent

Schrödinger equation to study the dependence of the interaction of argon- and neon-like atoms[166, 168] with an intense ultrashort circularly polarized laser pulse on the magnetic quantum
number of the initial state. In our analysis we have considered a broad wavelength regime, ranging
from single-photon to tunneling ionization. The numerical results for ionization of electrons corotating with respect to the rotation direction of the field and those from the initial level with m = 0
agree well with the predictions from lowest order perturbation theory at short and nonadiabatic
tunneling theory [18,20] at long wavelengths. In contrast, the data for the counter-rotating electrons
deviate significantly from both these predictions in the intermediate wavelength regime.
At the one- and two-photon ionization thresholds the enhancement in the emission of counterrotating electrons can be explained via the transition into channels with low angular momentum
quantum number in the continuum, which are accessible for these type of electrons only. Further
increase in the ionization yield for counter-rotating electrons (Fig. 4.2) and in the ionization ratioof counter- over co-rotating electrons (Fig. 4.11(b)) in the two- and few-photon ionization regime
comes along with much stronger excitation probabilities found for the interaction of counter-rotating
electrons with a circularly polarized short laser pulse. Together with the finding that the population
is mainly in excited states with lower angular momentum quantum number our results indicate that
resonant enhanced ionization via specific pathways and doorway states, which are accessible for thecounter-rotating electrons only, is the physical mechanism behind the enhancement. The overallsimilar trends in ionization and excitation probabilities for initial states with m = 0 and m = 1,
from which none of the doorway states can be excited, further support this interpretation.

99
The additional pathways for counter-rotating electrons exist in all atoms with ground states
having l ≥ 1 and may therefore provide an alternative route towards the generation of ultrashortspin-polarized electron pulses. Energy gating of the photo-electrons, as proposed and demonstrated
in Ref. [176], can even further enhance the selectivity in the ionization process. Increasing the wavelength beyond the few-photon ionization regime, the excitation probability becomes less relevant
due to the large number of photons needed for resonant transitions into excited states close to
the threshold. Correspondingly, the relative strength of the emission of counter-rotating over corotating electrons gets significantly smaller, decreasing from a ratio of ten to one at 250 nm to lessthan three to one in the tunneling regime at 1000 nm (Fig. 4.11(b)).

Chapter 5

Extending the theory of strong-field ionization into the UV limit

Our recent work [166,167] discussed in the previous Chapter motivated us to seek out accurate
models of strong field ionization from a general initial state at arbitrary photon energies. The
current approach involves expanding the Keldysh ionization amplitude [98] in Fourier components(in time) and partial waves (in space). We are able to describe the strong field process in termsof photon absorption pathways and angular momentum selection rules. Almost exact results forthe ionization of electrons bound to short range potentials in the presence of a strong circularly
polarized field at an intensity of I = 1 × 1014 W/cm2 and with wavelengths between 10 and 800 nm
are found in comparison with ab-initio numerical TDSE simulations. We will, in particular, discuss
the correct choice of states in the 10-200 nm wavelength regime where the asymptotic tail of thewave-function alone cannot describe ionization.

5.1

Circular polarization
In section 3.6.2, we summarized the results of Refs. [20, 136], for the ionization of electrons

bound by short-range potentials for the case of a strong circularly polarized field. The success ofthese formulas was then demonstrated in Fig. 3.9, where satisfactory agreement was found withnumerical solutions of the TDSE applied to a 16-cycle laser pulse with an intensity of I = 1 × 1014
W/cm2 and wavelengths between 10 nm and 800 nm. Discrepancies were observed near ionizationthresholds since we did not include the pulse envelope, and at the short (10 nm) limit due to the useof the quasi-classical saddle point approximation. Errors of the first kind are well understood and101
may be resolved using rate equations and numerical time integration including the pulse envelope.
In this section we will discuss errors of the second kind introduced by the use of saddle points.

5.1.1

Short-wavelength breakdown of the saddle point approximation

In Fig. 5.1 we integrate the length gauge amplitude exactly in time (Eq. (5.42)) to demonstrate short-wavelength errors introduced by the saddle-point approximation. The exact results
will be derived in the subsequent section. In the left panel of Fig. 5.1 we compare the saddle point
PPT result of section 3.6.2, as also shown in Fig. 3.9, to the exact same amplitude without theapproximation for an initial s-state. It is clear that for nearly all wavelengths longer than the
single photon ionization threshold (∼ 92 nm), both the exact (black curve) and the approximate
amplitude (red dashed curve) predict essentially the same results.
S-State Saddle-Point Comparison

P-State Saddle-Point Comparison

100

10−1

10−1
10−2

10−3

Nion

Nion

10−2

10−3
10−4

10−4
10−5
10−5

10−6

exact
saddle-point

10

20

exact

40

100
λ (nm)

200

400

800

10−7
10

20

saddle-point40100
λ (nm)

200

400

800

Figure 5.1: Total ionization yield for the length gauge amplitude (sections 3.6.1 and 3.6.2) corresponding to a 16 cycle laser pulse with an intensity of 1014 W/cm2 . Results are compared between
the saddle-point approximation (red dashed curve) and the exact time integration (solid black
curve) for s-states (left panel) and p-states (right-panel).

In contrast, in the right panel of Fig. 5.1 the same comparison is done for initial p-states,
where it is clear that the exact (black curve) and the approximate amplitude (red dashed curve)

102
disagree over almost the entire wavelength regime considered. To further explore this behavior we
decompose the ionization yield into contributions corresponding to each of the three initial statesm = −1, 0 and 1 where the field is assumed to be right handed circularly polarized.
P-1-State Saddle-Point Comparison

10−1

10−1

10−2

10−2

10−3

10−3

10−4

10−4

10−5

10−5

10−6

10−6
exact

10−7
10

20

saddle-point

40

P1-State Saddle-Point Comparison

100

Nion

Nion

100

exact

100
λ (nm)

200

400

800

10−7
10

saddle-point

20

40

100
λ (nm)

200

400

800

P0-State Saddle-Point Comparison

100
10−1

Nion

10−2
10−3
10−4
10−5
10−6
exact

10−7
10

20

saddle-point40100
λ (nm)

200

400

800

Figure 5.2: Total ionization yield for the length gauge amplitude (sections 3.6.1 and 3.6.2) corresponding to a 16 cycle laser pulse with an intensity of 1014 W/cm2 . The yield is compared between
the saddle-point approximation (red dashed curve) and exact time integration (solid black curve).
Top-left, bottom and top-right panels correspond to initial magnetic quantum number mi = −1, 0
and 1, respectively.

From this comparison it is obvious that the saddle-point approximation overestimates theyield for each magnetic sub-level, where the worst agreement is observed near the short wavelength103
10 nm limit, while the best agreement is found at the 800 nm long wavelength limit. To verify thatthe correct pre-factor has been used, the p0 −yield was calculated exactly from the s−yield usedin the left panel of Fig. 5.1, where the s−state factor κ/2π was replaced with 3kz2 /2πκ with theappropriate term for Cκ,li .

5.1.2

Ionization amplitude

We will now discuss the accurate ionization amplitude used in Figs. 5.1 and 5.2 for the case of
circular polarization with application to elliptically and linearly polarized pulses in a later section.
The exact solution to the TDSE can be expressed as a Lippmann-Schwinger-type integral equation
Ψ± (x) = ϕi (x) + ∫ d4 x1 G± (x; x1 )VL± (x1 )ϕi (x1 ),(5.1)

where we use the notation
4∫ d x1 ≡ ∫tt0

dt1 ∫ dr1

(5.2)

for integration over intermediate coordinates and + and − in ± correspond to right- and left-handed
polarized fields. Here, t0 is the moment when the field is turned on and t is the time instant when
the field is turned off and measurements are made.
The initial atomic state is either chosen to be the exact numerical eigenstates for a SAE
potential̵̵i
ϕi (x) ≡ e(i/h)Ip t ϕi (r) = e(i/h)Ip t Ri (r)Ylm
(r)
i

(5.3)

p̂2
+ Va (r)] ϕi (r) = −Ip ϕi (r)
2m

(5.4)

with Ri given byĤa (r)ϕi (r) = [or the approximate asymptotic states from Ref. [136], whereRi (r) ≈ Cκli κ3/2 (κr)ν−1 e−κr

(5.5)

which is accurate for calculations at long wavelengths, where ionization is dominated by the tail√
̵ ≡ 2mIp is the bound(κr ≫ 1) of the ground state. As before, Ip is the ionization potential, hκ104state momentum and we obtain Cκli by fitting the asymptotic state to the κr ≫ 1 part of the exact
state.
We will investigate accurate solutions for the case of short range potentials where the solution
to the atomic Schrödinger equation for κr ≫ 1 gives ν = 0. For long range potentials (which wewill treat in a later work) the power law becomes ν ≡ (κC /κ) with the Coulomb momentum
̵ C = mZ∣e∣2 /h̵ and residual ionic charge Z∣e∣. To derive the first order amplitude we expand thehκfull Green’s function in Eq. (5.1) to zeroth order in Volkov states̵ ∫t(−)−(i/h)t0
Φk± (x) = e

∣e∣
2
̵
(hk+
c A± (τ )) dτ + i∣e∣ A (t)⋅r
±
̵
2m
hc

(−)

ϕk (r).

(5.6)

(−)

Here, ϕk (r) is chosen to either be plane-waves or atomic continuum states obtained throughthe shooting method. The earlier correspond to the usual zero-range Volkov states and the lattercorrespond to the finite-range atom Volkov states with properties given in Ref. [58].
When the Volkov Green’s function
i
(V )
(−)
(−)
G± (x; x′ ) = − ̵ θ(t − t′ ) ⨋ Φk± (x) Φk± (x′ )
h
k

(5.7)

is inserted in Eq. (5.1) we arrive at the approximation(V )Ψ± (x) ≈ ϕi (x) + ∫ d4 x1 G± (x; x1 )VL± (x1 )ϕi (x1 ).

(5.8)

To determine ionization properties we restrict ourselves to the outgoing part of the wave-function
spanned by continuum Volkov states
(out)

Ψ±

(−)

(x) = ∫ dk M± (k, t)Φk± (x)(5.9)

with the coefficient given by(−)

M± (k, t) = ∫ dr Φk± (x)Ψ± (x).

(5.10)The zeroth order contribution
(0)

(−)

M± (k, t) = ∫ dr Φk± (x)ϕi (x)(5.11)

105
is identically zero for finite-range Volkov states and dies off for plane-wave Volkov states in the longpulse limit [136]. The lowest order contribution therefore comes from
i
(1)
(−)
M± (k, t) = − ( ̵ ) ∫ d4 x1 Φk± (x1 )[∣e∣E± (t1 ) ⋅ r1 ]ϕi (x1 )h(5.12)and is the well known Keldysh amplitude [98].

5.1.3

Photon Absorption/Emission

We select t0 = 0 and write the amplitude asM(1) (k, t) = ∫

t

0

̵

dt1 e(i/h)S(k,t1 ) (−∂) ϕ̃i (k, A(t1 ))∂t1(5.13)

with
(−)

i∣e∣

̵ A(t)⋅r1
ϕi (r1 )ϕ̃i (k, A(t)) = ∫ dr1 ϕk (r1 )e− hc(5.14)We then expand the terms in Eq. (5.13) in partial waves to determine the ionization amplitude.
To evaluate time integrals via Fourier components we replace the sum over saddle points by a sum
over discrete energy levels. The discrete states describe the total quanta absorbed (n) and the final
energy after ionization.
Starting with the ϕ̃i term we express the vector potential contribution as∞i∣e∣̵ A± (t)⋅re− hc= 4π ∑lA±n±n−in ωtl∑ (−i) A jlA (kA r)YlA A (Â± (0))YlA A (r̂)e A(5.15)

lA =0 nA =−lA

where kA ≡

∣e∣A
̵
hc

A
is the vector potential momentum. Yl±n(r̂) represents the angular momentumA̵transfer and e−inA ωt represents the total energy transfer nA hω.
Since lA ≥ ∣nA ∣, we note that, ingeneral, nA describes the net absorption or emission of energy and not necessarily an order of
perturbation theory. This factor is completed by expanding the continuum state in the term as(−)

ϕk (r) =

lk
1 ∞
m
m
l −iη (k)∑ ∑ i k e lk Rk,lk (r)Ylk k (k̂)Ylk k (r̂)
k lk =0 mk =−lk(−)

where Rk,lk is the radial continuum state with phase shift ηlk (k). Here, ϕk(5.16)
describes scattering

states which are asymptotically given at r → ∞ by a plane wave plus an ingoing spherical wave [107].

106
Putting it all together we have(−∂±nA −inA ωt̵̵ ∑ (nA hω)A
) ϕ̃i (k, A(t)) = (i/h)
lA e∂tlA ,nA
⎡
⎤
⎢
⎥
l
l
l
⎢
⎥ mi ±nAiAk⎥Y(k̂) (5.17)
× ∑ Klk (k)Illik ,lA (k) ⎢⎢
⎥ lk
⎢
⎥
lk
⎢−(mi ± nA ) ±nA mi ⎥⎣⎦with coefficientslA ±nAA
A±nlA ≡ 4π(−i) YlA (Â± (0)),(5.18)

Klk (k) ≡ (−i)lk eiηlk (k) .

(5.19)

andThe Fourier component e−inA ωt describes the transfer of nA quanta of energy to the ionizedelectron and the angular integrals are given by
⎡
⎤
⎢ l
l2
l1 ⎥⎥
⎢ 3
⎢
⎥ ≡ ∫ dΩr Y m3 (r̂1 )Y m2 (r̂1 )Y m1 (r̂1 )
⎢
⎥
1 l3
l2
l1
⎢
⎥
⎢−m3 m2 m1 ⎥⎣⎦(5.20)

which provides the corresponding angular momentum selection rules which are evaluated as Wigner3j symbols. The radial integral
Illik ,lA (k) ≡∞12∫ dr1 r1 Rk,lk (r1 )jlA (kA r1 )Ri (r1 ),k 0(5.21)

determines how the initial state influences the photo-electron distribution. We numerically evaluate
this integral for all cases in this section and the subsequent section on numerical applications.
Using the expansionjl (x) =√√ππ (x/2)l2
Jl+1/2 (x) =0 F1 (l + 3/2; −x /4),2x
2 Γ(l + 3/2)

(5.22)

for asymptotic initial states and zero-range Volkov states the integral can be evaluated exactly as
Illik ,lA (k) = Cκ,li2ν−1/2 Γ([lk + lA + ν + 2]/2)Γ([lk + lA + ν + 3]/2)
Γ(lk + 3/2)Γ(lA + 3/2)
κ3/2

k lk kA lA
lk + lA + ν + 2 lk + lA + ν + 3
3
3
k 2
kA 2
× ( ) ( ) F4 (,; lk + , lA + ; − ( ) , − ( ) ) . (5.23)
κ
κ
2
2
2
2
κ
κ

107
The Appell F4 function [9] is defined as∞(a)m+n (b)m+n xm y nm,n=0 (c1 )m (c2 )n m! n!

F4 (a, b; c1 , c2 ; x, y) = ∑with domain of convergence

√

∣x∣+

(5.24)√∣y∣ < 1 (or k +kA < κ) and can be extended to the entire domain

via various analytic continuations [6, 7, 56, 83]. In section 5.3 we will find a simpler expansion interms of orders of the intensity parameters z1 and z, where lowest-order contributions correspondto the weak-field limit and progressively higher-order terms are required when transitioning into
the strong-field limit.
To finish the derivation we expand the action term in partial waves and perform time integralssuch that⎡ ̵⎤
2⎢ (hk + ∣e∣⎥c A± (τ ))⎢̵ t + hk̵ ⋅ (ξ (t) − ξ (0)) ,S± (k, t) = ∫ dτ ⎢+ Ip ⎥⎥ = N (k)hω
±
±
2m
0
⎢
⎥
⎣
⎦
t

(5.25)

̵ 2 k2hA
2m(5.26)t∣e∣∫ A± (τ )dτ = ξ[cos(ωt)x̂ ± sin(ωt)ŷ]mc(5.27)

1
N (k) ≡ ̵ (Ek + I˜p ) ,
hω

Ek =

̵ 2 k2
h
,
2m

Up ≡

and
ξ ± (t) ≡∣e∣Awith ξ ≡ ωmc . Using the same partial wave expansion as before we have∞eik⋅ξ± (t) = 4π ∑

lS

±n
±n
−in ωtl∑ i S jlS (k ξ)YlS S (ξ̂ ± (0))YlS S (k̂)e S .

(5.28)

lS =0 nS =−lS

For a given number of quanta nA from the vector potential term we have
t

∫

0

̵

dt1 e(i/h)S± (k,t1 )−inA ωt1 =∞e−ik⋅ξ± (0) ∑

lS

±n
±n
∑ XlS S (k)YlS S (k̂)δt ([N (k) − (nA + nS )]ω/2) (5.29)

lS =0 nS =−lS

with coefficientSS
Xl±n(k) ≡ 4πilS jlS (kξ)Yl±n(ξ̂ ± (0)).
SS(5.30)

δt (x) ≡ eixt sinc(xt)t(5.31)The shape term

108
describes the distribution of final energy states after the absorption of nA + nS photons by a finite
flat-top pulse.
Combining all contributions we get for the ionization amplitude
∞

(1)

lA

̵ −ik⋅ξ± (0) ∑M± (k, t) = (i/h)e±nA
̵∑ (nA hω)AlA

lA =0 nA =−lA×

li +lA∑lk =max(∣li −lA ∣,∣mi ±nA ∣)⎡⎢⎢Klk (k) Illik ,lA (k) ⎢⎢

lk

lA

⎢
⎢−(mi ± nA ) ±nA
⎣

∞

× ∑

⎤
li ⎥⎥ m ±n
⎥ Y i A (k̂)
⎥ lk
⎥
mi ⎥
⎦

lS

±n
±n
∑ XlS S (k)YlS S (k̂)δt ([N (k) − (nA + nS )]ω/2) (5.32)

lS =0 nS =−lS

with angular momentum components determined by
i ±nA
S
Yl±n
(k̂)Ylm(k̂) =
S
k

⎡
⎤
⎢
l
lS
lk ⎥⎥ m ±(n +n )⎢⎢⎥ Y i A S (k̂) (5.33)
∑
⎢
⎥ l
⎢
⎥
l=max(∣lk −lS ∣,∣mi ±[nA +nS ]∣) ⎢−(m ± [n + n ]) ±ni
ASS m i ± nA ⎥⎣⎦lk +lSand the yield given by
(ion)

P±
5.1.4

(t) = ∫ dk ∣M± (k, t)∣2 .

(5.34)Long Pulse Limit

To determine the time-averaged rate for a long pulse we use(ion)w± = lim t−1 P±
t→∞

(t)

(5.35)

where it is useful to introduce the distribution(1)

W± (k) = lim t−1 ∣M± (k, t)∣2 ≈ lim t−1 ∣M± (k, t)∣2 .
t→∞

t→∞

(5.36)Taking the limit we see that the only time-dependence comes from cross terms involving δt (x).
Using the sinc representation of the Dirac delta function we see that
lim t−1 δt ([N (k) − n]ω/2)δt ([N (k) − n′ ]ω/2)t→∞̵ δ([N (k) − n]hω)̵ δn,n′ = 2π δ(k − kn ) δn,n′= 2π hvn(5.37)

109
which results in the quantized final state momenta
̵ n≡
hk

√

̵ − I˜p ) ≡ mvn
2m (nhω(5.38)and the distribution∞(1)

W± (k) = ∑ ∣M± (kn )∣2 δ(k − kn )(5.39)

n=nth̵
after the absorption of n photons with threshold value nth = ⌈I˜p /hω⌉and effective ionization potential I˜p = Ip + Up .
The amplitude for each photon process becomes∞(1)

M± (kn ) = e−ikn ⋅ξ± (0)∑l=∣mi ±n∣

Clmi ±n (kn )Ylmi ±n (k̂).

(5.40)

with components
̵Clmi ±n (kn ) ≡ (i/h)

√

lA
2π ∞
±nA
̵
∑ ∑ (nA hω)AlA
vn lA =0 nA =−lA

⎡
⎤
⎢
⎥
l
l
l
⎢
⎥
i
A
k
lk ,lA⎢⎥×
Klk (kn ) Ili (kn ) ⎢∑⎥⎢⎥lk =max(∣li −lA ∣,∣mi ±nA ∣)⎢−(mi ± nA ) ±nA mi ⎥
⎣
⎦
⎡
⎤
⎢
⎥
∞
l
l
l
⎢
⎥
S
k
±(n−nA )⎢⎥ . (5.41)
× ∑ XlS(kn ) ⎢⎥⎢⎥lS =∣n−nA ∣⎢−(mi ± n) ±(n − nA ) mi ± nA ⎥⎣⎦li +lA

Performing the sum of the contributions for each photon process we obtain the rate as∞

w± = ∫ dk W± (k) = ∑ ∫ dΩk
n=nth

∞
∞
∞
dwn±
= ∑
∑ wln± = ∑ wn±
dΩk n=nth l=∣mi ±n∣n=nth(5.42)

where we have introduced the partial rates
wln± = kn2 ∣Clmi ±n (kn )∣2

and wn± =

∞

∑

wln±

(5.43)

∞
dw±
dwn±
= ∑
.
dΩk n=nth dΩk(5.44)

l=∣mi ±n∣as well as the angular differential rates
dwn±= kn2 ∣M± (kn )∣2
dΩk

and

110

5.2

Numerical Applications

5.2.1

Ionization Yield

As an initial test we compare predictions for the ionization of spinless Hydrogen-like andNeon-like anions using Eq. (5.42) with results of the time dependent Schrödinger equation. Our
application involves using a Yukawa potential
Va (r) = −Za ∣e∣2 −r/aer(5.45)

with valence binding energy of Ip ≈ 13.6 eV and the electron interacting with a 16 cycle circularflat-top pulse with intensity 1 × 1014 W/cm2 at wavelengths between 10 nm and 800 nm. The
exponential factor a describes the range of the Yukawa potential while the prefactor Za is chosensuch that the binding energy remains the same for all ranges considered.
Numerical comparisons with the TDSE results are performed for three selections of initial- andfinal-states. In the first set we use zero-range initial and final states to determine errors introducedby the saddle-point approximation. In the next set of calculations we replace the asymptotic initialstates with numerical states of the atomic Hamiltonian to determine errors introduced by the useof asymptotic zero-range initial state. Finally, we report another set of calculations in which for
the Volkov states the plane-wave scattering state has been replaced with scattering states of theatom [58].
In Fig. 5.3 we compare the predictions of the model rate to the TDSE results for the case ofan s-state and exponential parameters a = a0 /5, a0 /3, a0 and ∞ (a0 is the Bohr radius) to showthat the model can provide accurate results for short-range potentials at all wavelengths. For the
TDSE calculations we use velocity gauge and expand the wave function in a basis of sphericalharmonics for the angular dimensions (lmax = ∣mmax ∣ = 30) and a basis of 8th order B-splines in
the radial dimension. The 600 B-spline nodes are placed such that the spacing between nodes isquadratic near the origin then becomes constant at a chosen radius (30 a.u.). The maximum radiusof the box is 500 a.u., where exterior complex scaling has been applied to the last 50 a.u.. The

111
Crank-Nicolson method has been used to propagate the wave function in time with a step size of
0.1 a.u.. Additional two cycle sin2 ramp on and ramp off are included to the 16 cycle flat-top pulse
to ensure that the vector potential smoothly goes to zero at t → ±∞ . The zero-range yield fors-states predicts essentially the same yield for all but the shortest wavelengths and agrees well with
the TDSE results for the case a = a0 /5. Expanding the atomic range to a = a0 /3 and a0 makes it
clear that atomic initial and final states are required to obtain reliable results. The case of a = ∞is included in the bottom-right panel of Fig. 5.3, where the quasi-classical Coulomb corrections
from section 3.6.2.3 are applied (blue curve), but a break down is observed in the short wavelength
limit. In contrast, Coulomb-Volkov states (solid black curve) predict the correct ionization yield forwavelengths shorter than the single photon ionization threshold. An additional green curve which
disagrees for all wavelengths has been included showing the results for using an atomic initial state
and a plane-wave Volkov state.

112
S-State Ionization Probablility for a = a0 /5

Ionization Probablility for a = a0 /3

10−1

10−1

10−2

10−3

Nion

Nion

10−2

10−4

10−4

final/initial = zero-range

10−5

10

final/initial = zero-range saddle pt
final = zero-range, initial = short-range
final/initial = short-range
TDSE

20

40

100
λ (nm)

10−5
200

400

800

10

S-State Ionization Probablility for a = a0

final/initial = zero-range
final = zero-range, initial = short-range
final/initial = short-range
TDSE

20

40

100
λ (nm)

200

400

800

Ionization Probablility for a = ∞

100

10−1

10−1

10−2

Pion

10−2
Nion

10−3

10−3

10−4

final/initial = zero-range

10−4

final/initial = zero-range saddle pt
final = zero-range, initial = short-range
final/initial = short-range
TDSE

10−5
10

20

40

10−3

100
λ (nm)

final = zero-range, initial = Constant x zero-range
final = zero-range, initial = long-range
final/initial = long-range
TDSE

10−5

200

400

800

10−6

10

20

40

100

200

400

800

λ (nm)

Figure 5.3: Comparison of predictions for the s-state model with the TDSE results for exponential
parameters a = a0 /5, a0 /3, a0 and ∞. The red solid (dashed) line corresponds to calculations using
zero-range initial and final states without (with) the saddle-point approximation. The green curvein the calculations the asymptotic initial states are replaced by numerical states of the atomic
Hamiltonian. The black curve corresponds to calculations with the same numerical initial state,
but now with Volkov states where the plane-wave scattering state has been replaced with scattering
states of the atom. a = ∞ is included for the initial Hydrogen ground-state and final Coulombcorrected WKB Volkov state (blue curve) [23, 102].

For the case of a Neon-like anion we chose a valence ionization potential Ip (2p) = 13.6 eV toenable direct comparison with the Hydrogen-like data. The Yukawa range parameter of a = a0 /2
was chosen to obtain 1s, 2s and 2p bound states with parameters given in Table 5.1. We will assume

113
all orbitals are occupied and neglect spin. Calculating the total ionization includes calculating theionization for each occupied orbital and adding the results. This corresponds to the single active
electron approximation. Although we work with Yukawa potentials, application to other potentials
should be straightforward.
State
Ip (Hartree)

1s
33.9

2s
1.32

2p
0.50

a = a0 /2
Zc = 10.15

Table 5.1: Table of bound-state energy levels and Yukawa parameters.

To obtain the total ionization yield for the Neon-like anion we calculate the single orbital
yield
(j)

Nion = 1 − e−w(j) T(5.46)

as before and add those up
(1s)

(2s)

(2p )

(2p )

(2p

Nion = Nion + Nion + Nion 1 + Nion 0 + Nion −1)(5.47)

to get the total yield. The results are shown in Fig. 5.4 for the case of a Yukawa potential withrange parameter a = a0 /2.
As one can anticipate, the occupied core 1s orbital can be neglected since the yield is muchsmaller than all the other yields for all wavelengths considered. The TDSE results show reduced
ionization in the co-rotating (2p1 ) orbital due to transitions into the occupied 2s state. This ismarked by the dashed E2p − E2s line in the left panel of Fig. 5.4. At the same time, the 2s orbital
experiences resonance enhanced ionization due to the occupied co-rotating state. The combinationof these effects are negated and the model produces accurate predictions for the total ionization
yield (right panel of Fig. 5.4). The effect of resonances can be included explicitly as shown in
Ref. [98]. We have again plotted results for each initial/final state combination, as in Fig. 5.3.
Ionization is best described by the use of initial and final atomic states, where some disagreement
is observed near ionization thresholds since we have not included properties of the pulse envelope.

114
P-State Ionization Yield (Circular)

100

Total Ionization Yield (Circular)

100

10−2

10−2

10−3

10−3

Nion

10−1

Nion

10−1

10−4

10−4

10−5

10−5

final/initial = zero-range
final = zero-range, initial = short-range
final/initial = short-range
TDSE

10−6

10−7

10

20

40

100

final/initial = zero-range
final = zero-range, initial = short-range
final/initial = short-range
TDSE

10−6

200

400

800

10−7

10

20

40

λ (nm)

100

200

400

800

λ (nm)

Figure 5.4: Comparison of the results of the p-state model with the TDSE results for exponential
parameter a = a0 /2, where the left panel corresponds to the sum of all p-orbitals and the right panel
corresponds to 2s ionization in addition to ionization by all p−states. The blue line corresponds tocalculations using zero-range initial and final states. For the results presented by the orange curve
the asymptotic initial states is replaced by numerical states of the atomic Hamiltonian. The results
shown by the green curve correspond to calculations with the same numerical initial state, but now
with Volkov states where the plane-wave scattering state has been replaced with scattering statesof the atom.

5.2.2

Photoelectron Energy and Angular Distributions

After we have verified the accuracy of the formulas we will describe how to extract energy
and angular photo-electron distributions with application to the before mentioned Neon-like anion.
The population of the ground state and each energy level in the continuum is described by
⎡
⎢
⎢
⎢
⎢
⎢
d ⎢⎢
dt ⎢⎢
⎢
⎢
⎢
⎢
⎢
⎣

⎤ ⎡
⎤
⎥ ⎢ −w ⎥
⎥
⎥ ⎢
⎥
⎥ ⎢
⎥
⎥ ⎢
⎢
⎥
Nnth ⎥ ⎢ wnth ⎥⎥⎥ Ni .
⎥=⎢
⎥
⎥ ⎢
⎥ ⎢
⎥
Nnth +1 ⎥ ⎢ wnth +1 ⎥
⎥ ⎢
⎥
⎥
⎥ ⎢
⎥ ⎢
⎥
⎥
⎥ ⎢
⋮
⋮
⎦ ⎣
⎦
Ni

(5.48)

115
Inserting the solution into the rate equations we get the yield for each ATI peak nwnNion .
w(5.49)dwn
dΩk(5.50)

dNion 1 dw
=
Nion
dΩk
w dΩk

(5.51)

dNn
1 dwn
=
Nn .
dΩk wn dΩk(5.52)

Nn =Using

wn = ∫ dΩk
we obtain the total angular distribution

and that for each photon process

Angular distributions for ionization by a circularly polarized field and initial states that include aisingle Ylm(r̂)-term have cylindrical symmetry in the φk direction. Integrating over this degree ofifreedom, one obtains the distributions
2π
dNion
dNion
dNion
≡∫
dφk
= 2π
∣
d(cos θk )dΩkdΩk φk =00(5.53)

2π
dNn
dNn
dNn
≡∫
dφk
= 2π
∣
.
d(cos θk )dΩkdΩk φk =00(5.54)

andIn Fig. 5.5 we plot the photo-electron energy and angular distributions corresponding to the
Neon-like anion data with both atomic initial and final states as for the results presented in Fig. 5.4.
Wavelengths of 10 nm, 100 nm and 800 nm are selected. The 10 nm and 100 nm data correspond to
the perturbative multiphoton limit since the energy distribution is linear on a log scale. In contrast,
the kinetic energy distribution at 800 nm is peaked at energies larger than the threshold value nthdemonstrating the expected behavior in the non-adiabatic limit [20]. The angular distributions
become more localized at the equator as wavelengths increase since ionized electrons belong tohigher-order spherical harmonic contributions with l ≈ m [167].

116

10nm Energy Distribution

10−1

10nm Angular Distribution
0°

Total
2s
2p-1
2p1
2p0

10−4
10−7

0.00300.00250.00200.00150.0010NndNiond(cos θk )

10−10
10

Total
2s
2p-1
2p1
2p0

45°

−13

0.000590°0.0000

10−16
10−19
10−22

135°
0

100

200
300
Kinetic Energy eV.

400

180°

500

θk

100nm Energy Distribution

100nm Angular Distribution
0°

Total
2p-1
2p1
2p0

10−2
10−4

10−8

dNion
d(cos θk )

Nn

10−6

10−10

0.00350.00300.00250.00200.00150.00100.00050.0000

Total
2p-1
2p1
2p0

45°

90°

10−12
10−14
10

−16

135°
0

10

20

30
40
50
Kinetic Energy eV.

60

70

180°

80

θk

800nm Energy Distribution×10−6

800nm Angular Distribution
×10−5

Total
2p-1
2p1
2p0

2.5

dNion
d(cos θk )

Nn

2.0

1.5

0°
3.0
2.5
2.0
1.5
1.0
0.5
0.0

Total
2p-1
2p1
2p0

45°

90°

1.0

0.5
135°
0.0

0

5

10
15
Kinetic Energy eV.

20

25

180°
θk

Figure 5.5: Photo-electron energy (left panels) and angular distributions (right panels) for the Neonlike anion data given in Fig. 5.4. Both atomic initial and final states are chosen and wavelengthsof 10 nm (top panels), 100 nm (middle panels) and 800 nm (bottom panels) are selected.

117

5.3

Intensity parameters
The general ionization formula can be accurately approximated as a two-step process for laserparameters used in the current work. The initial state is promoted to a small set of intermediate
momentum states via the absorption of nA photons before an additional n−nA photons is absorbed
to transfer in a final state. We will consider an initial s-state and use these ideas to determine dominant pathways for ionization. Furthermore, we will determine a simplified weak-field description
and discuss successive corrections required when transitioning into the strong-field limit.
Starting with an initial s-state (li = mi = 0) we immediately recognize the rule lk = lA from⎡⎤⎢lk
lA 0⎥⎥ δlk ,lA
⎢
⎢
⎥= √
⎢
⎥
⎢
⎥
4π
⎢−(mi ± nA ) ±nA 0⎥⎣⎦(5.55)

which yields the coefficient
∞

∞

±n
l
̵ √1
̵
Cl±n (kn ) = (i/h)∑ (nA hω)
∑ AlA A KlA (kn ) I0A (kn )2vn nA =−∞lA =∣nA ∣

⎡
⎤
⎢ l
⎥
l
l
⎢
⎥
S
A
±(n−nA )⎥ . (5.56)
× ∑ XlS(kn ) ⎢⎢⎥⎢⎥lS =∣n−nA ∣⎢∓n ±(n − nA ) ±nA ⎥⎣⎦∞As lA increases the integral IlliA (kn ) ≡ IlliA ,lA (kn ) decreases due to decreased overlaps of both thecontinuum state Rk,lA (r) and the Bessel function jlA (kA r) with the initial bound state Ri (r). The
coefficient should therefore be dominated by the lowest order contribution lA = ∣nA ∣. Similarly,
we should expect that for the intensity and wavelengths considered the sum over lS should bedominated by the lowest order contribution (∣n − nA ∣) yielding
∞

∣n ∣
±nA
̵
̵ √1
K∣nA ∣ (kn ) I0 A (kn )Cl±n (kn ) = (i/h)∑ (nA hω)A∣n∣
A
2vn nA =−∞

⎡
⎤
⎢ l
⎥
∣n
−
n
∣
∣n
∣
⎢
⎥
A
A
±(n−n )⎥ . (5.57)
× X∣n−n ∣A (kn ) ⎢⎢⎥A⎢⎥⎢∓n ±(n − nA ) ±nA ⎥⎣⎦Continuing along the same lines, the sum over nA may be reduced by restricting ourselvesto the largest contributions. This is done by choosing the set of nA which minimizes ∣n − nA ∣ + ∣nA ∣118
for a particular n. Geometrically this amounts to identifying the set of nA which minimizes thetotal distance (∣n − nA ∣ + ∣nA ∣) over which the magnetic quantum number changes from the initialstate to the intermediate state (∣nA ∣) and then to the final state (∣n − nA ∣). This is achieved for all
0 ≤ nA ≤ n for l = n. Thus, the l > n contributions may be neglected for fixed n since the distance∣nA ∣ + ∣n − nA ∣ is larger than for n = l yielding the approximation∞w± ≈ ∑ wnn± .

(5.58)n=nth

In Fig. 5.6 we show that approximating the rate as described above accurately determines theionization yield for the case of a = a0 /5. For later comparison we use zero-range Volkov states andan asymptotic initial state, but the conclusion holds in general.

5.3.1

Weak-Field Limit

We will now apply the above mentioned selection rules to the special case of a zero-range
potential for kA ≪ κ. The radial integral is approximated byjlA (kA r) ≈lA !(2kA r)lA
(2lA + 1)!

yielding
I0nA (kn ) =Cκ,li nA !
Γ(nA + 3/2)

Using
1
Yl±l (θ, φ) =
l!

√

√

(5.59)

κ (kn kA )nA.
2 (kn2 + κ2 )nA +1l(2l + 1)! 1(∓ sin θe±φ )4π2(5.60)(5.61)

the final-state angular integral can be evaluated as⎡⎤¿⎢n(n − nA )nA ⎥⎥Á (2(n − nA ) + 1)!(2nA + 1)!
⎢
n!
Á
À
⎢
⎥=
.
⎢
⎥ (n − n )!n !
4π(2n + 1)!
⎢⎥
AA⎢∓n ±(n − nA ) ±nA ⎥⎣⎦(5.62)

The coefficient now requires evaluating±(n−n )±(n−n )AA

2 (n+nA )AAA
A±nj(n−nA ) (kξ)Y(n−n )A (ξ̂ ± (0))Yl±n(Â± (0)).
nA KnA (kn )X(n−n ) (kn ) = (4π) iA(5.63)

119
Using the same spherical harmonic identity as earlier(−i)nA
1 n±(n−n )A
Y(n−n )A (ξ̂ ± (0))Yn±n(Â± (0)) =(∓ )AA(n − nA )!nA ! 2√(2(n − nA ) + 1)!(2nA + 1)!
4π(5.64)and the weak field approximationjlS (kn ξ) ≈lS !(2kn ξ)lS
(2lS + 1)!

(5.65)

we have¿i n (2kn ξ)(n−nA ) Á±(n−nA )ÁÀ (2nA + 1)! .
A
(k
)X
(k
)
=
4π
(∓
A±n
K
)
n
n
n
nA
A
(n−nA )2nA !
(2(n − nA ) + 1)!

(5.66)

Applying
nA ! Γ(nA + 3/2) =√π(2nA + 1)!/22nA +1(5.67)

we finally have
̵ (∓i)√
Cn±n (kn ) = 2(i/h)

nn! Cκ,li√n̵κ (kn ξ)n n (nA hω)
2kA /ξ A() .
∑vn (kn2 + κ2 ) nA =0 (n − nA )! kn2 + κ2

(2n + 1)!

(5.68)

The coefficient can be simplified using the exponential integral function
n

1
1
1
nA
z nA =[1 + e1/z (n − ) E−n ( )]z n!
zznA =0 (n − nA )!
∑yielding
Cκ,l (∓ikn ξ)
Cn±n (kn ) = i √in(2n + 1)!

√

̵
Up
hκ
[1 + ( ̵ ) en E−n (n)]mknhω

(5.69)(5.70)

after the weak-field substitution
2m
̵ − Up ) ≈ ( 2m ) nhω.
̵
kn2 + κ2 = ( ̵ 2 ) (nhω̵2hh

(5.71)

Using the hypergeometric representation of E−n (1/z)
Cκ,l (∓ikn ξ)
Cn±n (kn ) = i √in(2n + 1)!

√

̵
Up
hκ
1
[1 + ( ̵ ) 2 F0 (−n, 1; ; − )]
mkn
nhω
n

(5.72)

̵ we see that the 2 F0 term is negligible leading to the simplified coefficientand Up ≪ Ip < nhωCκ,l (∓ikn ξ)
Cn±n (kn ) = i √in(2n + 1)!

√

̵
hκ
mkn

(5.73)

120
and the partial rate(s)2wn,n± = Cκ,li(κ vn ) (kn ξ)2n
(2n + 1)!

(5.74)

which gives the expected near-threshold (l = n) scaling of ∼ kn2l+1 [27, 29, 172] shown in section 2.2.
In Fig. 5.6 we demonstrate the accuracy of the above mentioned weak field approximations.
The black (+) symbol represents the total ionization yield for a = a0 /5 obtained from the TDSE
calculations, as in Fig. 5.3a). The cyan curve represents data when we keep only the dominant
pathways described in this section. The green curve represents the results obtained by applicationof the first weak field approximation (Eq. (5.72)). Finally, the red curve represents predictions
we get by applying the second simplified weak field approximation (Eq. (5.74)). We observe thatthe dominant pathways accurately describe ionization for all wavelengths considered. Additionally,both weak field approximations accurately reproduce the yield for one, two and three photon
ionization while application of Eq. (5.74) holds well up to five photon ionization. This result issurprising since Eq. (5.74) is a further approximation of Eq. (5.72).
Since kA ≪ κ we see that jlA (kA r) is approximated accurately by the lowest (kA r)lA term atshort wavelengths. Similarly, since dominant contributions to ionization are described by the lowest
energy near-threshold photo-electrons we have kn ξ ≪ 1 yielding that jlS (kn ξ) is approximated wellby the term with order (kn ξ)lS . The additional purple curve in Fig. 5.6 represents results obtainedusing the application of Eq. (5.74) where only the lowest order n = nth has been included to show
accuracy of the near-threshold approximation for photon orders above the threshold value (n > nth ).

121
Weak-Field Approximation
10−1

Nion

10−2

10−3

10−4

10−5

10−6
10

dominant pathways only
weak field approximation 1
weak field approximation 2
weak field approximation 2 (knth only)
TDSE

20

40

100
λ (nm)

200

400

800

Figure 5.6: Comparison of the weak field approximations. The black (+) symbol represents thetotal ionization for a = a0 /5 obtained from the TDSE calculations, as in Fig. 5.3a. The cyan curve
represents results for the dominant pathways described in this section. The green and red curves
represent data obtained from Eq. (5.72) and Eq. (5.74), respectively. The purple curve shows datausing the application of Eq. (5.74) where only the lowest order n = nth has been included.

5.3.2

Strong-Field Limit

The light blue curve in Fig. 5.6 demonstrates that the reduced set of pathways described in
the prior weak-field subsection describe ionization up to the largest 800 nm wavelength considered.
Other curves in Fig. 5.6 further approximate the rate from the lowest-order polynomial terms ofjlA (kA r) and jlS (kn ξ). One would therefore expect that the weak-field approximations will tend
towards TDSE predictions as higher contributions are used to approximate jlA (kA r) and jlS (kn ξ).
To demonstrate this trend we will expand the radial integrand as(−1)α (lA + α)!
(kA r)2α
α!(2l
+
2α
+
1)!
A
α=0

αmax

jlA (kA r) ≈ (2kA r)lA ∑(5.75)

122
to express the radial integral as a series expansion
αmax

l (α)

I0lk (kn ) ≈ ∑ z1α I0k(kn )(5.76)

α=0

with respect to the bound-state intensity parameter z1 ≡ 2Up /Ip [146, 148] (see section 2.3) whichis related to the Keldysh parameter γ [98] by z1 = 1/γ 2 . In Fig. 5.7 we have compared predictions
using Eq. (5.76) with orders αmax = 0 − 4 to the TDSE results which show that significantly higher
order expansions are required to approximate the rate as the wavelength increases.
z1α = [2Up /Ip ]α expansion (βmax = ∞)
10−1

Nion

10−2

10−3

10−4

10−5

αmax = 0αmax = 1
αmax = 2
αmax = 3
αmax = 4
TDSE

10−6
10

20

40

100
λ (nm)

200

400

800

Figure 5.7: Dominant ionization pathway yield from Fig.(5.6) where jlS (kn ξ) is treated exactly andjlA (kA r) is expanded as in Eq.(5.76) to zeroth, first, second, third and fourth order αmax (solidlines). Crosses correspond to the a = a0 /5 s-state TDSE data from Fig.(5.3).

We will express jlS (kn ξ) in a similar series expansion with respect to the non-perturbative
̵ yielding
intensity parameter z = Up /hω,
βmax

(β)

jlS (kn ξ) ≈ ∑ z β BlS (kn ) with
β=0

(β)

BlS (kn ) ≡β(2kn ξ)lS (lS + β)!
Ek[−4 ( ̵ n )] .
β!(2lS + 2β + 1)!
hω(5.77)

123
In Fig. 5.8 we have compare results using Eq. (5.77) with βmax = 0 − 7 with the TDSE results. It isagain clear that significantly higher order expansions are required to approximate the rate as the
wavelength becomes large.
z β = [Up /h̄ω]β expansion (αmax = ∞)
10−1

Nion

10−2
10−3
10−4
10−5
10−6
10

βmax = 0βmax = 1βmax = 2βmax = 3βmax = 4βmax = 5βmax = 6βmax = 7
TDSE

20

40

100
λ (nm)

200

400

800

Figure 5.8: Dominant ionization pathway yield from Fig.(5.6) where jlA (kA r) is treated exactly
and jlS (kn ξ) is expanded as in Eq.(5.77) from zeroth to seventh order βmax (solid lines). Crosses
correspond to the a = a0 /5 s-state TDSE data from Fig.(5.3).

Combining these expansions it is obvious that probability amplitudes may be written as
αmax βmax

Cn±n (kn ) ≈ ∑ ∑ z1α z β Cn±n(α,β) (kn ),(5.78)

α=0 β=0

where an expansion is performed over both z1 and z in disagreement with Reiss [147] who claimed
that the length-gauge rate corresponds to an expansion over z1 = 1/γ 2 only. To summarize, inthis section we were able to evaluate the first-order Keldysh amplitude exactly for the case ofarbitrary initial and final states. We described the importance of using correct atomic initial andfinal states at the shortest wavelengths and identified issues with the application of the saddle-point

124
approximation to p−states. We finished the section by finding simplified rates in the UV limit andappropriate strong-field corrections while transitioning into the strong-field IR limit.

5.4

Linear and elliptical polarization

5.4.1

Results

Expanding linearly and elliptically polarized laser pulses as a linear combination of both righthanded and left-handed circularly polarized fields exactly the same steps can be used to determine
the ionization amplitude. The TDSE for ionization by an elliptically polarized field can be written
as
̵
ih

∂
Ψϵ (r, t) = [Ha + ∣e∣Eϵ (t) ⋅ r] Ψϵ (r, t)∂t(5.79)Eϵ (t) = E [cos(ωt)x̂ + ϵ sin(ωt)ŷ],(5.80)withEϵ (t) = −

1 dAϵ
(t),
c dt

Aϵ (t) = −A [sin(ωt)x̂ − ϵ cos(ωt)ŷ] ,

and

A=

cF
.
ω

(5.81)The vector potential is now represented as
Aϵ (t) = (1+ϵ1−ϵ) A+ (t) + () A− (t)22(5.82)

in terms of the right-handed (+) and left-handed (−) circularly polarized fields A± (t). Expanding
the exponential terms as in Eq. (5.28) and Eq. (5.15) a rate is determined exactly as in the lastsection dedicated to circular polarization. We will now give numerical comparisons of the predictions
with the TDSE results before deriving the amplitude in the next subsection.
In Fig. 5.9 we reproduce the right panel of Fig. 5.4 where all laser parameters are the same
except for the laser ellipticity. In the top panel (bottom panel) we show the good agreement with
the TDSE results for the case of linear (elliptical ϵ = 0.5) polarization at all wavelengths, where thebest agreement is again found for the case when both atomic initial and final states are used. It isclear that the formulas yield a similar level of agreement for these cases as we have found earlier
in the case of circular polarization.

125
Total Ionization Yield (Linear)

100

10−1

10−2

Nion

10−3

10−4

10−5

final/initial = zero-range
final = zero-range, initial = short-range
final/initial = short-range
TDSE

10−6

10−7

10

20

40

100

200

400

800

400

800

λ (nm)

Total Ionization Yield (Elliptical)

100

10−1

10−2

Nion

10−3

10−4

10−5

final/initial = zero-range
final = zero-range, initial = short-range
final/initial = short-range
TDSE

10−6

10−7

10

20

40

100

200

λ (nm)

Figure 5.9: Comparison of the predictions of the p-state model with the TDSE results for exponential parameter a = a0 /2. The top (bottom) panel corresponds to interaction with a laser pulses with
an ellipticity of ϵ = 0 (ϵ = 0.5). The blue line shows the predictions for zero-range initial and final
states. The orange curve represents data for the asymptotic initial states with numerical states of
the atomic Hamiltonian. The green curve corresponds to the application of the same numericalinitial state, but now with Volkov states where the plane-wave scattering state has been replacedwith scattering states of the atom.

126
5.4.2

Derivation

Using the partial wave expansion, Eq. (5.15), the amplitudeM(1)ϵ (k, t) = ∫

t

0

̵

dt1 e(i/h)Sϵ (k,t1 ) (−

i∣e∣
∂
(−)
̵ Aϵ (t1 )⋅r1
) ∫ dr1 ϕk (r1 ) e− hc
ϕi (r1 )∂t1(5.83)

is evaluated using
A+
∞
i∣e∣
∂
nA
(−)
̵ Aϵ (t1 )⋅r1(−
) ∫ dr1 ϕk (r1 ) e− hc
ϕi (r1 ) = ω ∑∑ YlA+ + (Â+ (0))∂t1lA =0 nA =−lA

l

+

∞

lA−

× ∑

∑

+

∞

−n

AlAA+,lAA− YlA A− (Â− (0))e−i(nA+ +nA− )ωt1 ∑nlA− =0 nA− =−lA−

+

,n

+

−

−

lk

m
∑ Klk (k)Ylk k (k̂)

lk =0 mk =−lk

⎡
⎢ l
lA−
⎢ k
lk ,lA− ,lA+× Ini ,li(k, kA , ϵ) ⎢⎢⎢⎢−mk −nA−
⎣

lA+
nA+

⎤
li ⎥⎥
⎥ , (5.84)
⎥
⎥
mi ⎥
⎦

where
AlAA+,lAA− = 16π 2 i(nA+ + nA− )(−i)lA+ +lA− ,
n

,n

+

−

and Klk (k) = (−i)lk eiσlk (k) .

(5.85)

Other contributions correspond to the angular integral
⎡
⎤
⎢ l
l3
l2
l1 ⎥⎥
⎢ 4
⎢
⎥ ≡ ∫ der Y m4 (r̂)Y m3 (r̂)Y m2 (r̂)Y m1 (r̂)
⎢
⎥
l4
l3
l2
l1
⎢
⎥
⎢−m4 m3 m2 m1 ⎥⎣⎦(5.86)

expanded in terms of 3j-symbols and the radial integral
l ,l

,l

Inki ,lAi − A+ (k, kA , ϵ) ≡

∞
1−ϵ
1
1+ϵ
2
) kA r1 ) jlA+ (() kA r1 ) Rni ,li (r1 ) (5.87)∫ dr1 r1 Rk,lk (r1 )jlA− ((k 0
22which is evaluated numerically.
For asymptotic initial states and plane-wave Volkov states the radial integral becomes
l ,l ,lInki ,lAi − A+ (k, kA , ϵ) =√ Γ ( lk +lA− +lA+ +ν+1 ) Γ ( lk +lA− +lA+ +ν+2 )
Cκli
π
2
2
3
3
3/2−ν
3/2
2
κ
Γ (lk + 2 ) Γ (lA− + 2 ) Γ (lA+ + 23 )k lk 1 − ϵ kA lA− 1 + ϵ kA lA+ (3) lk + lA− + lA+ + ν + 1 lk + lA− + lA+ + ν + 2
× ( ) [()] [()] FC (
,
κ
2
κ
2
κ
2
2
3
3
3
k 2
1 − ϵ kA 2
1 + ϵ kA 2
; lk + , lA− + , lA+ + ; − ( ) , − [()] , − [()] ) (5.88)
2
2
2
κ
2
κ
2
κ

127
where
(3)

FC (a, b; c1 , c2 , c3 ; x1 , x2 , x3 ) ≡∞(a)i1 +i2 +i3 (b)i1 +i2 +i3 xi11 xi22 xi33
i1 ,i2 ,i3 =0 (c1 )i1 (c2 )i2 (c3 )i3 i1 ! i2 ! i3 !
∑(5.89)

is the Lauricella hypergeometric series [108] which may be evaluated past its radius of convergence√√√∣x1 ∣ + ∣x2 ∣ + ∣x3 ∣ < 1 (or k + kA < κ) through various analytic continuations. This exact solution
allows for systematic approximations in the weak field limit (kA ≪ κ) just as we have shown it forcircular polarization in section 5.3.
The amplitude is completed from the action
⎤
⎡ ̵
2
2
t
⎥
⎢ (hk + ∣e∣c Aϵ (τ ))̵ ⋅ (ξ (t) − ξ (0)) + ∣e∣ ∫ dτ Aϵ (τ )2 ,+ Ip ⎥⎥ = (Ek + Ip ) t + hk
dτ ⎢⎢
ϵ
ϵ
2
2m
2mc 0
0
⎥
⎢
⎦
⎣
(5.90)Sϵ (k, t) = ∫twhere the ponderomotive termt∣e∣2 A2 (1 − ϵ2 )∣e∣22sin(−2ωt)
dτ
A
(τ
)
=
U
t
+
ϵ
p
∫
2mc2 0
8ωmc2

with

Up ≡

∣e∣2 A2 (1 + ϵ2 )4mc2

(5.91)

has additional time-dependent oscillations due to the presence of both right- and left-handed fields.
The quiver motion
ξ ϵ (t) = (1+ϵ1−ϵ) ξ + (t) + () ξ − (t)2
2

(5.92)is separated into left- and right-handed contributions giving the exponential partitions
̵e(i/h)Sϵ (k,t) = e−ik⋅ξϵ (0) eiN (k)ωt ei∣e∣2 A2 (1−ϵ2 )sin(−2ωt)
2
̵
8hωmc

ei( 2 )k⋅ξ− (t) ei( 2 )k⋅ξ+ (t)
1−ϵ

1+ϵ

(5.93)

with1
N (k) ≡ ̵ (Ek + I˜p ) .
hω

(5.94)

The factors∞ei( 2 )k⋅ξ± (t) = ∑
1±ϵ

lS±

∑

lS± =0 nS± =−lS±

±n

±n

±

±

XlS± (k, ξ, ϵ)YlS S± (ξ̂ ± (0))YlS S± (k̂)e−inS± ωt(5.95)

are evaluated as before, whereei∣e∣2 A2 (1−ϵ2 )sin(−2ωt)
2
̵
8hωmc

∞

= ∑ Ba (A, ω, ϵ)e−2iaωta=−∞(5.96)

128
completes the exponential withX± (k, ξ, ϵ) ≡ 4πilS+ /L jlS+ /L ((

1±ϵ
) kξ)
2

and Ba (A, ω, ϵ) = Ja (∣e∣2 A2 (1 − ϵ2 )
).
2
̵
8hωmc

(5.97)

Putting it all together the exponential action factor becomes
∞

̵

lS+

e(i/h)Sϵ (k,t) = e−ik⋅ξϵ (0) ∑∑lS+ =0 nS+ =−lS+

∞

lS−

× ∑

∑

lS− =0 nS− =−lS−

n

n

+

+

XlS+ (k, ξ, ϵ)YlS S+ (ξ̂ + (0))YlS S+ (k̂)

−n

−n

−

−

∞

XlS− (k, ξ, ϵ)YlS S− (ξ̂ − (0))YlS S− (k̂) ∑ Ba (A, ω, ϵ)ei[N (k)−(nS+ +nS− +2a)]ωt . (5.98)
a=−∞

Using
t

∫

0

dt1 ei[N (k)−(nA+ +nA− +nS+ +nS− +2a)]ωt1 = δt ([N (k) − (nA+ + nA− + nS+ + nS− + 2a)]ω/2)(5.99)

the amplitude for a finite pulse becomes∞lA+−ik⋅ξϵ (0)M(1)∑ϵ (k, t) = ωe∑lA+ =0 nA+ =−lA+

∞

n

lA−

YlAA+ (Â+ (0)) ∑+∑lA− =0 nA− =−lA−

n

−n

,n

AlAA+,lAA− YlA A− (Â− (0))
+

⎡
⎢
lk
lA−
⎢
l ,l ,l× ∑ Klk (k)Inki ,lAi − A+ (k, kA , ϵ) ⎢⎢⎢lk =0⎢−(mi + nA+ − nA− ) −nA−
⎣
∞

∞

× ∑
⎡

∞ ⎢
⎢

lS+

∑

lS+ =0 nS+ =−lS+n
XlS+ (k, ξ, ϵ)YlS S+ (ξ̂ + (0))
+

l

lS−

× ∑ ⎢⎢
l=0 ⎢⎢−(mi + nA+ − nA− + nS+ − nS− ) −nS−
⎣
∞

∞

∑

lS−

nS+

lA+
nA+

⎤
li ⎥⎥
⎥
⎥
⎥
mi ⎥
⎦

−n

∑

lS− =0 nS− =−lS−

lS+

−

−

XlS− (k, ξ, ϵ)YlS S− (ξ̂ − (0))−⎤⎥⎥ mi +nA+ −nA− +nS+ −nS−
⎥Y
(k̂)
⎥ l
⎥
(mi + nA+ − nA− )⎥⎦lk

× ∑ Ba (A, ω, ϵ)δt ([N (k) − (nA+ + nA− + nS+ + nS− + 2a)]ω/2). (5.100)a=−∞5.4.2.1

Long pulse limit

Next, we will compute the rate for a long pulse
Wϵ (k) = lim t−1 ∣Mϵ (k, t)∣2 ≈ lim t−1 ∣Mϵ(1) (k, t)∣2 .
t→∞

t→∞

(5.101)

129
To carry out the t → ∞ limit we only need to consider properties of δt ([N (k) − n]ω/2) and known
properties of the Dirac delta distribution
lim t−1 δt ([N (k) − n]ω/2)δt ([N (k) − n′ ]ω/2) =t→∞′lim ei(n −n)ωt/2t→∞sin([N (k) − n]ωt/2) sin([N (k) − n′ ]ωt/2)
t[N (k) − n][N (k) − n′ ]ω 2 /4̵ δ([N (k) − n]hω)̵ δn,n′ . (5.102)= π δ([N (k) − n]ω/2) δn,n′ = 2π h
To finish the derivation of the rate we will replace the delta function in energy by a delta functionin k-space where√̵ δ([N (k) − n]hω)̵ δn,n′ = 2πm δ(k − kn ) δn,n′ ,
2π h
̵ n
hk

with

kn ≡

2m ̵
˜
̵ 2 (nhω − Ip )h(5.103)

and
∞

2
2
(1)
Wϵ (k) = lim t−1 ∣M(1)ϵ (k, t)∣ = ∑ ∣Mnϵ (k̂)∣ δ(k − kn ).
t→∞(5.104)n=n0Using the selection rulesn = nA+ + nA− + nS+ + nS− + 2a⇒nS− = n − nA+ − nA− − nS+ − 2a(5.105)the sum over nS− is eliminated and we get the amplitude as√M(1)nϵ (k̂) = ωA+A−∞
2π −ikn ⋅ξϵ (0) ∞nA−nAnA ,nAe∑∑ YlA+ + (Â+ (0)) ∑∑ AlA++,lA− − YlA− − (Â− (0))
vnlA =0 nA =−lA
lA =0 nA =−lA

l

+

+

l

+

−

−

−

⎡
⎢
∞
lk
lA−
⎢
l ,l ,l× ∑ Klk (k)Inki ,lAi − A+ (k, kA , ϵ) ⎢⎢⎢lk =0⎢−(mi + nA+ − nA− ) −nA−
⎣
∞

× ∑

lS+

∞

n

∑

lS+ =0 nS+ =−lS+n

XlS+ (k, ξ, ϵ)YlS S+ (ξ̂ + (0)) ∑× YlS A+−++nA− +nS+ +2a−n

lA+
nA+

⎤
li ⎥⎥
⎥
⎥
⎥
mi ⎥
⎦

∞

∑ Ba (A, ω, ϵ)XlS− (k, ξ, ϵ)lS− =0 a=−∞∞m +2(nA+ +nS+ +a)−n(ξ̂ − (0)) ∑ Yl i
l=0

⎡
⎢
l
lS−
lS+
⎢
× ⎢⎢
⎢
⎢−[mi + 2(nA+ + nS+ + a) − n] (nA+ + nS+ + nA− + 2a − n) nS+
⎣

(k̂)

⎤
⎥
⎥
⎥ . (5.106)⎥⎥(mi + nA+ − nA− )⎥⎦lk

The partial wave expansions used when expanding the ionization amplitude for circularly polarized fields apply directly to the case of linearly and elliptically polarized fields as well. Although130
tedious, we believe that these formulas will prove to be an inexpensive alternative to numerical calculations by the TDSE. The amplitude can be greatly simplified in specific limits such as ϵ ≈ ±1 justas in section 5.3 where the right- (left-)handed circularly polarized fields may be non-perturbativewith corrections by a perturbative field of opposite left- (right-)handed polarization.

Chapter 6

Strong-field processes involving polychromatic fields

6.1

Introduction1
The generation of ultrashort vacuum ultraviolet (VUV) and deep ultraviolet (DUV) laser

pulses [41, 66, 145] is important since excitation and ionization energies of many atoms, molecules,
nanoparticles, and materials lie in this spectral region. Such laser sources are used to trigger,
steer, probe, and image physical processes and chemical reactions on the ultrafast time scale, down
to the attosecond regime of electron dynamics [31, 55, 100, 109, 119, 127, 131, 156, 162]. Temporalpulse characterization is often required to enable the analysis of the spectroscopic data. Many
ultrashort pulse characterization methods rely on the measurement of ions or photoelectrons via
the autocorrelation or the cross correlation approach (for an overview, see [133]). In cross correlation
methods the pulse to be characterized is used in superposition with a well-characterized infrared
pulse [39,73,87,121,134,159], while for autocorrelation measurements, two replicas of the unknown
pulse are used [126, 130, 149, 160, 161].
In this Chapter we will derive semi-analytic perturbative two-photon amplitudes for theionization of atoms by arbitrary polychromatic fields (section 6.2). These formulas are applied
to reconstruct isolated attosecond pulses and pulse trains from a provided autocorrelation trace(ionization as a function of relative delay). A feedback loop is used and ionization is determinedfrom a model laser parameterized by a set of variational parameters. The field is recovered by
1

Part of the material presented in this Chapter has been previously published in S. Walker, R. Reiff, A. JaronBecker, and A. Becker, Optics Letters 46, 3083–3086 (2021) [168].

132
determining variational parameters such that the exact autocorrelation trace agrees with the trace
generated by the parameterized field. In the last section (section 6.3) of this Chapter we will extend
the formulas into the non-perturbative limit. Time is partitioned into slices (tm−1 , tm ] (m ∈ Z) [63]narrow enough such that the formulas of section 6.2 may be applied. We see that time-slicing
extends analytic perturbative principles into the nonperturbative limit. These formulas may prove
to be useful for nonperturbative processes involving short pulses and electrons bound by long-range
potentials.

6.2

Characterization of vacuum and deep ultraviolet pulses via two-photon
autocorrelation signals
In an autocorrelation measurement, ion signals are recorded as a function of the time delay

between the two replicas of the pulse. Generalized two-photon ionization cross sections for theinteraction of a single Gaussian pulse with the target are then used to fit the autocorrelation
trace and determine the pulse duration. This powerful method has been applied to estimate theduration of attosecond extreme ultraviolet (XUV) pulses [160, 161], VUV and DUV pulses [66,149]. Overall, the application of current pulse characterization techniques in the important VUVand DUV spectral region is limited, since many methods, such as the widely used reconstructionof attosecond beating by interference of two-photon transitions (RABBITT) [134] or attosecond
streaking technique [87], rely on the ionization of the target by absorption of a single photon fromthe unknown pulse.
We consider an extension of the single-Gaussian autocorrelation technique, which enables
the characterization of the temporal pulse envelope of an isolated ultrashort VUV pulse or a pulse
train, i.e. the time-dependent amplitude and phase variation of the electric field. The extension
is based on the analytical solution of the time-dependent Schrödinger equation (TDSE) for the
perturbative two-photon ionization of an atom by a Gaussian laser pulse in the single-active-electron
approximation [86]. Since the solution includes both resonant and non-resonant pathways, it canbe used for the characterization of broadband pulses with photon energies in the regime of typical133
atomic excitation energies. The unknown pulse is approximated as a superposition of Gaussian
pulses, for which the amplitudes and temporal widths are determined via fitting to the two-photon
autocorrelation signal generated by the unknown laser pulse. We demonstrate this extension of
the single-Gaussian autocorrelation technique via applications based on results for ultrashort VUVpulses from numerical simulations of macroscopic high harmonic generation (HHG).

6.2.1

Multi-color Gaussian approach

For the characterization, we use that the electric field of any arbitrary pulse can be written
as the real part of an expansion in a basis of multi-color Gaussian functions (Hartree atomic units̵ = m = 1 a.u.):
are used: e = h⎤⎡ (t − τ )2
n,j
⎥
⎢
−
iω
t
f˜(t) = ∣f˜(t)∣e−iϕ(t) = ∑ f˜n,j exp ⎢−
j ⎥ ,
2
⎥
⎢
2T
n,j
n,j
⎦
⎣

(6.1)

where ∣f˜(t)∣ and ϕ(t) are the time-dependent amplitude and phase of the unknown pulse. f˜n,j isthe complex amplitude, τn,j is a translation in time, and Tn,j is the width of a Gaussian pulse
with central frequency ωj . In the examples below, we consider linearly polarized pulses, but the
expansion can be applied to other polarizations as well. Since each Gaussian has an independent
phase factor, in the multi-Gaussian approach the nonlinear phase accumulation ω(t) ≡ dϕ
dt can be
determined from the interferences in the two-photon autocorrelation trace. If a large frequency
variation is expected, an alternative basis of linearly chirped Gaussian functions can be utilized by−2−2
substituting Tn,j
→ Tn,j+ iαn,j , where αn,j is an additional parameter. For all pulses considered in

the present work, we have used αn,j = 0 to limit the number of fitting parameters.
Next, we utilize the analytic solution of the two-photon ionization amplitude for interactionof an atom in the single-active-electron approximation with two Gaussian laser pulses:
π(2)
af,i (τn2 ,j2 , τn1 ,j1 ) = − f˜n2 ,j2 f˜n1 ,j1 Tn2 ,j2 Tn1 ,j1 ∑ zf,m zm,i
4
Em <0
1
1
× exp (− Tn22 ,j2 ∆2f,m − Tn21 ,j1 ∆2m,i + i∆f,m τn2 ,j2 + i∆m,i τn1 ,j1 )2
2
1
× {1 + erf [ √ (τn2 ,j2 − τn1 ,j1 + i (Tn22 ,j2 ∆f,m − Tn21 ,j1 ∆m,i ))]} . (6.2)
2T̄

134
We note that Eq. (6.2) is a generalization of the solution for a single Gaussian pulse given in Ref. [86]and is derived in section 6.2.2. Here, T̄ 2 = Tn22 ,j2 + Tn21 ,j1 , ∆m,i = Em,i − ωj1 , and ∆f,m = Ef,m − ωj2 ,where Em,i and Ef,m are the energy differences between the field-free intermediate (m) states
and the initial (i) and the final (f ) states, respectively. zf,m and zm,i are the transition dipole
moments between the respective states which can be obtained via various theoretical techniques
for an atom. In the present applications, we restrict the sum to intermediate states with Em < 0.
To compute erf(x + iy), we have used a continued fraction approximation for the majority of thecomplex plane [43] and patched the (nonphysical) singularities near the origin with the global
Padé approximation [173] and other singularities near the imaginary axis with the standard Padé
approximation. Since we want to characterize ultrashort pulses with large bandwidths, we havealso taken into account the one-photon ionization amplitude for interaction with the n-th Gaussian
pulse which is given by
√
π˜
1 2 2
(1)
af,i (τn,j ) = −i∆f,i + i∆f,i τn,j ) .
fn,j Tn,j zf,i exp (− Tn,j
2
2

(6.3)

The autocorrelation signal for the interaction with two replicas of the arbitrary pulse delayed
by a time interval τ to each other can then be approximated by the multi-color Gaussian approachas:(1)

PNg (τ ) = ∫

Ef >0

(2)

2

∣Af,i (τ ) + Af,i (τ )∣ dEf ,(6.4)

where the integral over Ef includes all final levels f for the one- and two-photon processes, respectively,
(1)

(1)

Af,i (τ ) = ∑ ∑ af,i (τn,j ) (1 + eiEf,i τ ) ,
n

(2)

(6.5)

j

(2)

Af,i (τ ) = ∑ ∑ ∑ ∑ af,i (τn2 ,j2 , τn1 ,j1 ) (1 + eiEf,i τ )n 2 j2 n 1 j1

(2)

(2)

+ af,i (τn2 ,j2 + τ, τn1 ,j1 )eiωj2 τ + af,i (τn2 ,j2 , τn1 ,j1 + τ )eiωj1 τ(6.6)

In order to characterize the envelope of the unknown ultrashort pulse, we introduce an objectivefunction between the autocorrelation signal Pexact (τ ), measured with the unknown pulse, and the135approximate multi-Gaussian signal PNg (τ ) as:
objNg (τ ) =∣Pexact (τ ) − PNg (τ )∣,Pexact (0)(6.7)

where Pexact (0) is the maximum ionization yield for which the two replicas of the pulse perfectly
overlap. For a given number of Gaussian pulses, Ng , the L2 norm of the objective function isthen minimized with respect to variation of f˜n,j , τn,j and Tn,j using the standard non-linear leastsquares approach, where the objective function has a complexity that scales as the square of thenumber of basis functions used. Without loss of generality, for exactly one n, j we chose f˜n,j realand fix τn,j = 0.
6.2.2

Derivation of ionization amplitudes

Before applying the multi-color Gaussian approach we discuss the derivation of the one- andtwo-photon ionization amplitudes used in the approach. The perturbative ionization amplitude(Chapter 2) used in the previous section can be described by the Lippmann-Schwinger equation
Ψ(x) = ϕi (x) + ∫ d4 x1 G(x; x1 )VL (x1 )ϕi (x1 ),(6.8)

where ϕi is the initial atomic state calculated from the time-independent Schrödinger equation fora single-active electron potential and VL is the laser interaction. The exact Green’s function G is
now expanded as
G(x; x1 ) = G0 (x; x1 ) + ∫ d4 x2 G(x; x2 )VL (x2 )G0 (x2 ; x1 )(6.9)

in terms of the atomic Green’s function
G0 (x; x′ ) = −i θ (t − t′ ) ⨋ ϕj (x) ϕj (x′ ),j(6.10)

where ⨋j describes a sum over all bound and an integral over all continuum states ϕj (x).
The two photon ionization amplitude is then determined from the second order truncation
Ψ(x) ≈ ϕi (x) + ∫ d4 x1 G0 (x; x1 )VL (x1 )ϕi (x1 )+ ∫ d4 x2 ∫ d4 x1 G0 (x; x2 )VL (x2 )G0 (x2 ; x1 )VL (x1 )ϕi (x1 ) (6.11)

136
using the interaction potential
VL (x) = F(t) ⋅ r = Re [f (t)] ẑ ⋅ r,

(6.12)

where⎡ (t − τ )2
⎤
n,j
⎢
⎥
−iϕ(t)˜˜
˜⎥f (t) = ∣f (t)∣ e= ∑ fn,j exp ⎢−−iωt(6.13)
j
2
⎢
⎥
2Tn,j
n,j
⎣
⎦
is the laser field expanded in a basis of multi-color Gaussian functions. f˜n,j is the complex amplitude, τn,j is a translation in time, and Tn,j is the width of a Gaussian pulse with central frequency
ωj . A detailed description of the first- and second-order terms are presented in sections 6.2.2.1
and 6.2.2.2.

6.2.2.1

One-photon ionization amplitude

The first-order term in Eq. (6.11), corresponds to the lowest order one-photon process
Ψ(1) (x) = ∫ d4 x1 G0 (x; x1 )VL (x1 )ϕi (x1 ).

(6.14)

Inserting the field interaction and taking t0 → −∞ for a finite pulse the correction becomes
(1)

Ψ

t
⎡ (t − τ )2 ⎤−i
1n,j ⎥ i(Em −Ei )t1 ˜⎢⎥e(fn,j e−iωj t1 + f˜n,j eiωj t1 )(x) =⨋ ϕm (x)zm,i ∑ ∫ dt1 exp ⎢⎢−
2
⎥
2 m
2T
n,j
n,j
⎣
⎦
−∞

and simplifies toΨ(1,abs) (x) =t⎡ (t − τ )2
⎤
−i
1
n,j
⎢
⎥
˜
⎢−
ϕ
(x)z
+ i∆m,i,j t1 ⎥ .
fdt
exp
m,i ∑ n,j ∫
1
⨋ m
2
⎢
⎥
2 m
2T
n,j
n,j
⎣
⎦

(6.15)−∞

for photon absorption processes. Taking the limit as t → ∞ and projecting onto a continuum state
ϕf (x) gives the contribution for each Gaussian in the laser field as√∆2f,i ⎞⎛π(1)zf,i Tn,j f˜n,j exp iτn,j ∆f,i −.
af,i (τn,m ) = −i
2
2
2Tn,j
⎝
⎠
6.2.2.2(6.16)

Two-photon ionization amplitude

The second-order, two photon term in Eq. (6.11) is given byΨ(2) (x) = ∫ d4 x2 ∫ d4 x1 G0 (x; x2 )VL (x2 )G0 (x2 ; x1 )VL (x1 )ϕi (x1 ).

(6.17)

137
Inserting the field and following the same steps as the one-photon case before we obtain the ϕf (x)component
π(2)af,i (τn2 ,j2 , τn1 ,j1 ) = − ⨋ zf,m zm,i f˜n1 ,j1 Tn1 ,j1 f˜n2 ,j2 Tn2 ,j2
4 m∆2f,m ⎞
∆2m,i⎛−
× exp iτn1 ,j1 ∆m,i + iτn2 ,j2 ∆f,m −2Tn21 ,j1 2Tn22 ,j2 ⎠
⎝
⎡
⎤ ⎫
⎧
⎪
⎥ ⎪
⎪
1
⎪ ⎢⎢
⎪
⎥ ⎪
2
2
(τn2 ,j2 − τn1 ,j1 − i∆m,i Tn1 ,j1 + i∆f,m Tn2 ,j2 )⎥ + 1⎬
× ⎨erf ⎢ √ √
⎢
⎥
⎪
2
2
⎪
⎪
⎥ ⎪
⎪
⎩ ⎢⎣ 2 Tn1 ,j1 + Tn2 ,j2⎭⎦ ⎪after Gaussian integration where photon absorption processes alone are considered.
In general, erf(x+iy), is poorly conditioned and expensive to compute in the standard integral
form. We use a continued fraction approximation for the majority of the complex plane [43] andpatch the nonphysical singularities near the origin with the global Padé approximate and othersingularities near the imaginary axis with the standard Padé approximate. The evaluation involves
defining g(z, a) = ea erf(z), and approximating it as
⎧
⎪
⎪
⎪
⎪
ea [5/5]erf(±2i) (z)∣z ∓ 2i∣ < 1.1
⎪
⎪
⎪
⎪
√
⎪
⎪
2
⎪
2 4/π+bz⎪g(z, a) ≈ ⎨sign (Re [z]) e2a − e2a−z 1+bz2
z∈Ω
⎪
⎪
⎪
⎪
−1
⎪
⎪
⎛
⎞
−z 2
⎪
1/2
ze
a
2
⎪
√
⎪
sign
(Re
[z])
e
1
−
)
(z
+
else,
1
⎪
π
⎪
1+
⎪
⎝
⎠
z 2 +3/2⎩(6.18)

in the complex plane. Here b ≈ 0.14, Ω is the set of all z such that ∣z∣ < 1.4 and ∣z ± 2i∣ ≥ 1.1, and[m/n]f (z0 ) (z) is the Padé approximate of a function f centered around z0 with order m in the
numerator and order n in the denominator.

6.2.3

Application and results

To test the characterization with this multi-Gaussian approach, we have obtained ultrashortpulses using results of numerical macroscopic simulations of HHG spectra [144]. First, calculations of the microscopic single-atom response have been performed by solving the time-dependent
Schrödinger equation within the dipole approximation with Hamiltonian11H(t) = − ∇2 − + E(t) ⋅ y ,2
r(6.19)

138
where E(t) and y are the electric field and polarization direction of the driving laser pulse. The
wavefunction has been expanded in 30 spherical harmonics and the radial part of the wavefunction
and the potential have been discretized utilizing fourth order finite difference on a radial grid withspacing dr = 0.2 a.u. and grid sizes up to rmax = 100 a.u. [163]. As absorbing boundary, we
have used exterior complex scaling (ECS), where the edge of the grid (10%) is rotated into complex
space by an angle η = π/4. We have used the Crank-Nicolson method to propagate the wavefunction
starting from the initial state of the hydrogen atom with time step dt = 0.1. To obtain the HHG
spectra P (ω) = ∣ã(ω)∣2 , the dipole acceleration a(t) has been evaluated using the Ehrenfest theorem∂(− 1r )⟩ and the complex harmonic response ã(ω) is then obtained by taking the Fouriera(t) = ⟨− ∂ytransform (without windowing).
For the macroscopic radiation signal, we have considered the low gas density regime that is
free from longitudinal phase-matching effects. We then followed the approach used in Ref. [81],in which the macroscopic yield is obtained as the superposition of the fields generated at different
points in the medium. Application of this approach requires single-atom simulations for a large
number of intensities and phase factors. In the present simulations 5 × 105 single-atom results
have been used via an interpolation scheme based on 100 exact TDSE results. The success of theinterpolation method has been verified elsewhere [144].
In Fig. 6.1(a), we show the below- and near-threshold part of the numerical macroscopic
HHG signal obtained for a gas jet of hydrogen atoms interacting with a 20-cycle, 800 nm pulse
at peak intensity 4 × 1013 W/cm2 . From this spectrum, we have first extracted isolated ultrashortpulses by applying a Gaussian filter in the frequency domain centered about the harmonic H0 , i.e.
e−(H−H0 ) /2β where β = 0.35 in the present work. The temporal shape of the pulse filtered about the22

9th harmonic varies significantly from that of a Gaussian pulse (Fig. 6.1(b)). For the application of
the characterization method, we used replicas of the pulse with an intensity of 3.16×1011 W/cm2 todetermine the autocorrelation signal shown in Fig. 6.1(c). The signal has been obtained by solving
the time-dependent Schrödinger equation for the interaction of a helium atom with the two pulses;
a single-active-electron potential for the helium atom [143] has been used for these calculations.

139

intensity (arb. units)

100

(a)

10-5

10-10

10-15

1

3

5

13

ionization probability

0

-2
0

500

(c)
15
10
5

-500

time (a.u.)

Figure 6.1:

11

10-7

2

-500

9

harmonic order

10-3
(b)

field strength (a.u.)

7

0

500

delay (a.u.)(a) Macroscopic high harmonic spectrum in hydrogen atom driven by 20-cycle, 800

nm laser pulse at 4 × 1013 W/cm2 : full spectrum (black line) and spectrum after application of a
Gaussian filter about 9th harmonic (red line). (b) Temporal profile of filtered 9th harmonic and(c) autocorrelation signal using filtered 9th harmonic [168].

To characterize the filtered 9th harmonic, we have chosen a single-color approach in which all
Gaussian pulses have the central frequency of the 9th harmonic. The results of the approximations
for the time dependent amplitude ∣f˜(t)∣ (blue) and phase variation ω(t) (red) using up to six
Gaussian pulses (solid lines) are compared in Fig. 6.2 with the original pulse (dashed lines). The
reconstruction with a single Gaussian (panel (a)) provides an estimate of the pulse width, but
the double-hump structure in the amplitude and the phase variation is, of course, not reproduced.
Already inclusion of a second Gaussian pulse (panel (b)) provides a significant improvement in thisrespect, however the minimum in between the two humps, the small post-pulse structure and most140of the phase variation is still not well reproduced. Using four (c) and six (d) Gaussian pulses, firstthe main part of the amplitude and then the post-pulse structure and even the phase variation in
major parts of the pulse, is well reproduced. We however note that using autocorrelation signals,
time-independent phases cannot be determined and a certain pulse cannot be distinguished fromthe pulse with time-reversed envelope.
10-3

10-3

3 (a)
2
3

1
0

0.54

3 (b)

0.52

2

0.52

10-3

3

0.5

0
-3
-500

0

-500

0.54

1

0.48

500

0

0

500

-3
-500

-500

2
3

10

0.54

3 (d)

0

500

0.52

2

0.52
3

0.5

0
-3
-500

0.54

-3

0

1

0.48

500

0

0

500

time (a.u.)
Figure 6.2:

0.48

500

10-3

3 (c)

-500

0

time (a.u.)

10-3

0

0.5

0

time (a.u.)

1

10-3

10-3

0.5

0
-3
-500

-500

0

0.48

500

0

500

time (a.u.)Comparison of time-dependent amplitude ∣f˜(t)∣ (blue line) and phase variation ω(t)(red line) of the filtered 9th harmonic (dashed line) and reconstructed pulse with (a) 1, (b) 2, (c)4 and (d) 6 Gaussians (solid line). The temporal profile is also provided as insets within each
subplot [168].

While the results in Fig. 6.2 visualize the potential of the method, we quantify the convergenceby errors for the ionization signal using the objective function as:√
ErrorNg [Pion ] =∫ dτ objNg (τ )2 ,(6.20)

141
and the error in the field defined as:
ErrorNg [f˜] =√2∫ dt ∣f˜exact (t) − f˜Ng (t)∣(6.21)

in Fig. 6.3. Results for both the filtered 9th (blue) and the filtered 11th (red) harmonic arecompared with each other. In both cases, the error in the ionization signal (panel (a)) drops
smoothly with an increase of the number of Gaussians included in the method. The number ofGaussians necessary for similar degree of convergence is smaller for the 11th harmonic, since thepulse envelope (not shown) is less complex than that of the 9th harmonic. The convergence inthe error for the ionization signal corresponds to a decrease of the error in the field (panel (b))
that reflects the observation in Fig. 6.2 that the pulse envelope is well reconstructed with just a
few Gaussians. At the same time, we note that already two Gaussians are sufficient to provide anestimate of the FWHM pulse duration below 10 a.u. for for both harmonics.

10-1

9th
11th

2
1.5
1
0.5

10-2
1

2

3

4

Ng
Figure 6.3:

10-2
2.5 (b)9th
11th

field error

ion error

100

(a)

5

6

0

1

2

3

4

5

6

Ng

Error analysis of Gaussian pulse characterization method for filtered 9th (blue) and11th harmonics (red): (a) ErrorNg [Pion ] (Eq. 6.20), and (b) ErrorNg [f˜] (Eq. 6.21) [168].

After applying the method to isolated ultrashort pulses by filtering the spectrum about asingle harmonic, we consider the more complex temporal profile of a pulse train. To this end, we
have used a flat top filter to the macroscopic HHG spectrum, where the leading edge is the same
Gaussian filter used for the 9th harmonic and the trailing edge is a Gaussian filter applied to the11th harmonic. The temporal shape of the resulting pulse train and the autocorrelation signal

142

3

10-3

10-7

ionization probability

field strength (a.u.)

obtained with the pulse train are shown in Fig. 6.4(a) and (b), respectively.

(a)

0

-3
-500

0

500

3

(b)

2
1
0
-500

0

time (a.u.)
4

10

delay (a.u.)

-3

0.8

4

10

-3

0.8

(d)

(c)
2

0.7

2

0.7

0

0
0.6

-2
-4

500

-500

0

time (a.u.)

500

0.5

0.6

-2
-4

-500

0

500

0.5

time (a.u.)

Figure 6.4: (a) Temporal profile of the filtered 9th to 11th harmonic spectral range, (b) autocorrelation signal, and comparison of time-dependent amplitude (blue) and phase variation (red) of
the original pulse (dashed lines) and reconstructions (solid lines) obtained with one (c) and three(d) Gaussians per central frequencies at 9th and 11th harmonic [168].

Since the spectral filtering covers two harmonics, we have used a two-color approach withcentral frequencies equal to those of the 9th and 11th harmonics. Comparison of the original pulse
train with the reconstructions shows that one Gaussian pulse per central frequency (panel (c))
provides a good approximation, including the duration of the train as well as the duration of the143
individual pulses in the train. However, the rather flat overall envelope with several pulses in the
train having similar amplitudes as well as most of the phase variation is not well reproduced. Using
three Gaussian pulses per central frequency (panel (d)) improves the agreement with the original
pulse in these aspects significantly.
Thus, using an extension of the two-photon autocorrelation technique based on multiple
Gaussian pulses about one or several central frequencies enables a characterization of the amplitude
and phase variation of isolated ultrashort pulses and pulse trains. The potential of the method isdemonstrated using VUV pulses obtained from numerical simulations of macroscopic high harmonic
spectra. We note that the method can be further improved by using alternative basis sets with moreparameters or more sophisticated search algorithms in the fitting procedure. The same approach
can also be applied to two-photon cross-correlation signals of two isolated ultrashort VUV pulses,
e.g., if one of the two pulses is first characterized via autocorrelation. Furthermore, the analytic
solutions of two-photon absorption may be extended to other techniques such as RABBITT for
VUV and DUV pulse characterization. The accuracy of the method depends on the determinationof one- and two-photon transition dipoles, for which several theoretical methods are available.

6.3

Non-perturbative limit

6.3.1

Time-slicing

In the prior section we derived efficient semi-analytical models for perturbative multiphoton processes via arbitrary polychromatic pulses. In this limit difficulties such as the long-range
Coulomb potential are non-existent and there are no limitations on the properties of the incident
radiation as long as the intensity is sufficiently weak and perturbation theory is extended to a
sufficiently high order. Unfortunately, when transitioning into the strong field limit perturbation
theory will break down (section 2.3) and alternative methods are required.
In the previous Chapters 3, 4 and 5 we explored various non-perturbative models for strong
field processes, where each had unique benefits and difficulties. The first and most reliable method

144
is the direct numerical integration of the time dependent Schrödinger equation, which proves to be
exact in the limit of converged parameters, but may become prohibitively computational expensive
and requires additional models for analysis. The rest of the strong field approximations [57,98,136,
146] provide nearly exact analytic results for short-range potentials and long monochromatic pulses.
For short polychromatic pulses these formulas either become cumbersome or require additional
numerical time integration. Additionally, due to the lack of accurate Coulomb corrections acrossthe entire electromagnetic spectrum (Chapter 5) application to electrons bound by long-range forces
may be reliably approximated only in the limits γ ≫ 1 and γ ≪ 1. Higher order corrections suchas in Ref. [58] may yield correct long-range threshold behavior for arbitrary γ, but those have notbeen systematically investigated.
In this section we will combine favorable properties of both perturbative and non-perturbativemodels in order to determine a semi-analytic strong field amplitude for processes involving arbitrary polychromatic fields and electrons bound by arbitrary (possibly long range) forces. Here, we
decompose the interaction picture propagator UI (tf , ti ) [151] defined by∣ΨI (tf )⟩ = UI (tf , ti )∣ΨI (ti )⟩(6.22)

into M different time-levels tm between some initial time ti ≡ t0 and final time tf ≡ tM , wheret0 < t1 < ⋯ < tM [63, 75]. Taking final and initial matrix elements ∣EM ⟩ and ∣E0 ⟩ we haveM

⟨EM ∣UI (tM , t0 )∣E0 ⟩ = ⟨EM ∣T [ ∏ UI (m)] ∣E0 ⟩,(6.23)

m=1

where we introduce the shorthand notation
UI (m) ≡ UI (tm , tm−1 )(6.24)and T is the time-ordering operator.
Assuming each time-slice δtm ≡ tm − tm−1 is sufficiently small, then the propagator UI (m)may be approximated perturbatively over the interval as it is done in numerical time integration
schemes such as the Crank Nicolson method [42] which we have used for direct numerical solutions

145of the TDSE. Each interval is perturbative, but the combined contribution from all intervals is nonperturbative. Instead of evaluating the propagator over δtm numerically, we now choose UI as theinteraction picture propagator for a complete basis of atomic reference states ∣Em ⟩ interacting withan external laser field VL (x) = E(t) ⋅ r just as we have done for perturbative two-photon ionizationby short pulses (section 6.2.2). We expand the external field as⎡ (t − τ )2
⎤
n,j
⎢
⎥
⎥ ẑ.
f̃ (t) = ∑ ∑ f˜n,j (t)ẑ = ∑ ∑ f˜n,j exp ⎢−
−
iω
t
j
2
⎢
⎥
2Tn,j
n j
n j
⎣
⎦

(6.25)

E(t) = Re[f̃ (t)](6.26)

andas before. We will use E(t) = F (t)ẑ to avoid confusion with what we will later call the energy attime t, E(t).
The approximation involves finding sufficiently small time intervals δtm such that
∞

(k)

(1)

UI (m) = 1 + ∑ UI (m) ≈ 1 + UI (m)(6.27)

k=1

approximates the interval well. The last expression on the right is the first order Dyson approximation [51]. Here δtm may be chosen independently for each m.
In this approximation we decompose the transition amplitude into a linear combination ofGaussian contributions
(1)

(1)

UI (m) = ∑ Un,j (m)n,jwith approximation(Dyson)UI(1)(m) ≈ 1 + ∑ Un,j (m).
n,jNext, one partitions the first order coefficient into a part responsible for absorption and one foremission of a photon
(1)

(abs)

(emi)

Un,j (m) = Un,j (m) + Un,j

(m),

146
where
√

1 2
π˜
2
(abs)
(∆(abs)fn,j Tn,j zm exp (− Tn,jm ) + i∆m τn,j )
8
2
1
1
2
2
× {erf [ √(tm − τn,j − iTn,j∆(abs)(tm−1 − τn,j − iTn,j∆(abs)m )] − erf [ √m )]} (6.28)
2Tn,j
2Tn,j

(abs)
⟨Em ∣Un,j (m)∣Em−1 ⟩ = −i

and
√

1 2
π˜
fn,j Tn,j zm exp (− Tn,j(∆(emi))2 + i∆(emi)τn,j )
m
m
8
2
1
1
2
2
(tm − τn,j − iTn,j∆(emi))] − erf [ √(tm−1 − τn,j − iTn,j∆(emi))]} (6.29)
× {erf [ √
m
m
2Tn,j
2Tn,j

(emi)
⟨Em ∣Un,j (m)∣Em−1 ⟩ = −iare the dipole-transition matrix elements for single photon absorption and emission process, re(abs)spectively. The shorthand notations ∆m(emi)= Em − Em−1 − ωj and ∆m= Em − Em−1 + ωj are

used.

6.3.2

Perturbation theory

The approximate propagator becomes
M

M

m=1

m=1

(1)

M

(K,M )UI (T /2, −T /2) = T [ ∏ UI (m)] ≈ T [ ∏ (1 + UI (m))] ≡ ∑ UI

(6.30)

K=0

after each time-slice has been expanded in a first-order Dyson series. The sum over K on the righthand side corresponds to terms containing K first-order Dyson contributions, where the zeroth-orderterm(0,M )

UI(0)≡ 1 = UI

(6.31)

corresponds to the zeroth order of the standard Dyson series.
At first order(1,M )

UI

M

(1)

(1)

≡ ∑ ∑ Un,j (m) = UI

(6.32)

m=1 n,jsimilarly represents the first-order Dyson series, where all ”diagrams” containing a single photon
are summed over. At second order(2,M )UI⎤⎡ M⎤ ⎡ M −1
⎥
⎢
⎥⎢
(1)
(1)
≡ ⎢⎢ ∑∑ Un2 ,j2 (m2 )⎥⎥ ⎢⎢ ∑ ∑ Un1 ,j1 (m1 )⎥⎥⎥⎢m2 =m1 +1 n2 ,j2⎥ ⎢m1 =1 n1 ,j1⎦⎣⎦⎣(6.33)

147
deviates from the Dyson result unless δtm → 0 (M → ∞) where(2)

UI(2,M )= lim UI
M →∞.

(6.34)

Continuing now to the Kth order process(K,M )UI⎡⎤ ⎡ M +2−K
⎤ ⎡M +1−K
⎤
M
⎢
⎥ ⎢
⎥⎢
⎥
(1)
(1)
(1)
≡ ⎢⎢
∑
∑ UnK ,jK (mK )⎥⎥ ⋯ ⎢⎢ ∑∑ Un2 ,j2 (m2 )⎥⎥ ⎢⎢ ∑ ∑ Un1 ,j1 (m1 )⎥⎥⎢mK =mK−1 +1 nK ,jK⎥ ⎢m2 =m1 +1 n2 ,j2⎥ ⎢ m1 =1 n1 ,j1
⎥
⎣
⎦ ⎣
⎦⎣
⎦
(6.35)

where the Dyson prediction(K)

UI(K,M )= lim UI
M →∞(6.36)

is again recovered for δtm → 0 (M → ∞).
Such an expansion extends analytic perturbative principles into a new non-perturbative limit
where results agree with the time-dependent Schrödinger equation in the limit of converged parameters. The approximation is immediately applicable to problems involving short pulses andlong-range interactions. Integration over each finite time-interval (tm−1 , tm ] is performed analytically and may be interpreted in terms of photon absorption and emission processes. The propagatoraccounts for the temporal variation of the field over the interval in contrast to the Crank-Nicolson
method which assumes a piecewise constant over each δtm . Since the time-dependence of the field isapproximated over the interval it is expected that significantly larger δtm may be required yielding
an improvement over the Crank-Nicholson method. We plan on exploring these ideas more in the
future.

Chapter 7

Summary

Throughout this thesis we have explored ionization processes ranging from the x-ray andultraviolet limit where electrons propagate as they would in an atomic system and applied radiation induces electronic transitions between atomic states before eventually transferring into thecontinuum to the opposite adiabatic infrared limit where large numbers of photons are required forionization and electrons appear to tunnel through a static barrier. We summarized criteria responsible for the breakdown of perturbation theory and outlined key aspects of the non-perturbative
theory. We discussed key steps responsible to derive the ionization amplitude of Keldysh and recognized its origins in the tunneling model of Oppenheimer. Additionally, we proved equivalenceof Keldysh’s amplitude with the model of Perelomov, Popov and Terent’ev for the case of a final
plane-wave Volkov state and all possible combinations of initial states and applied radiation fields.
Numerical calculations were performed bridging the short wavelength limit of single photon
perturbative ionization to the long wavelength limit of non-perturbative multiphoton ionization.
These calculations found an optimal wavelength regime for generation of electron pulses for probing
chiral systems in the intermediate wavelength limit where neither adiabatic tunneling theory nor
perturbation theory are capable of describing the ionization process.
Breakdown was demonstrated of the asymptotic saddle point approximation for ionization ofelectrons bound to p−states. We overcame this issue by evaluating Keldysh’s amplitude exactly. Our
explorations recognized the intermediate wavelength limit where the asymptotic tail of the initialbound-state alone cannot determine ionization. Descriptions of ionization in this intermediate

149
wavelength regime prove difficult for electrons bound by long-range potentials both due to theinapplicability of perturbation theory and lack of sufficient Coulomb corrections to describe proper
long-range threshold behavior in the strong field theories.
In the final Chapter we developed an analytic model for the perturbative two-photon ionization of atoms by arbitrary polychromatic radiation fields. As an application these formulas wereapplied to reconstruct isolated attosecond pulses and pulse trains from a provided autocorrelation
trace generated in a pump probe ionization measurement. We generalized the amplitude to thenon-perturbative limit by partitioning time into sufficiently small slices such that our perturbative
formulas may be applied over each interval. We believe that this complementary approach may allow for an accurate semi-analytic physical description of strong-field processes involving long-range
Coulomb potentials and short laser pulses.

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