Atoms and Molecules in Strong Laser Fields: Double ...users.physik.fu-berlin.de/~ag-gross/theses/lein_phd.pdf · The numerical treatment of atoms or molecules in strong laser pulses

Atoms and Molecules in Strong LaserFields: Double Ionization, DissociativeIonization, and Harmonic Generation

Dissertation

zur Erlangung des

naturwissenschaftlichen Doktorgrades

der Bayerischen Julius-Maximilians-Universitat Wurzburg

vorgelegt von

Manfred Leinaus

Munchberg

Wurzburg 2001

Eingereicht am:

bei der Fakultat fur Chemie und Pharmazie

1. Gutachter:

2. Gutachter:

der Dissertation

1. Prufer:

2. Prufer:

der mundlichen Prufung

Tag der mundlichen Prufung:

Doktorurkunde ausgehandigt am:

Contents

Introduction 5

1 Theoretical background 91.1 Description of atoms and molecules in strong fields . . . . . . . . . . . . 91.2 Reduced dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Classical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Time-dependent multicomponent density functional theory . . . . . . . . 151.5 The Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . 171.6 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6.1 Split-operator method . . . . . . . . . . . . . . . . . . . . . . . 191.6.2 Obtaining the ground state . . . . . . . . . . . . . . . . . . . . . 201.6.3 Absorbing boundaries . . . . . . . . . . . . . . . . . . . . . . . 211.6.4 Choosing the grid parameters and time step . . . . . . . . . . . . 21

2 Double ionization of atoms 232.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 The mechanism of He double ionization in long-wavelength lasers . . . . 26

2.3.1 Time evolution in momentum-space . . . . . . . . . . . . . . . . 262.3.2 Coordinate expectation values . . . . . . . . . . . . . . . . . . . 312.3.3 Test for rescattering . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.4 Phase-space analysis . . . . . . . . . . . . . . . . . . . . . . . . 342.3.5 Comparison with the Hartree-Fock approach . . . . . . . . . . . 36

2.4 Photoelectron and recoil-ion spectra . . . . . . . . . . . . . . . . . . . . 412.4.1 Extension of the model . . . . . . . . . . . . . . . . . . . . . . . 432.4.2 Long-wavelength spectra . . . . . . . . . . . . . . . . . . . . . . 462.4.3 Short wavelengths: Above-threshold double ionization . . . . . . 502.4.4 Reconstruction of electron spectra from recoil-ion spectra . . . . 55

3

4 Contents

3 Strong-field dynamics of small molecules 603.1 Pulse-width and isotope effects in dissociative ionization . . . . . . . . . 60

3.1.1 Two-step model . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 653.1.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2 The strong-field dynamics of a model H2 molecule . . . . . . . . . . . . 733.2.1 The model H2 molecule . . . . . . . . . . . . . . . . . . . . . . 753.2.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 76

3.3 Even-harmonic generation due to beyond-Born-Oppenheimer dynamics . 813.3.1 The model HD molecule . . . . . . . . . . . . . . . . . . . . . . 833.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.3.3 Interpretation in terms of non-adiabatic couplings . . . . . . . . . 84

4 Outlook: Field propagation effects 884.1 The Maxwell-Schrodinger equations . . . . . . . . . . . . . . . . . . . . 884.2 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Summary 92

Zusammenfassung 96

Bibliography 100

Dank 107

Lebenslauf 109

Introduction

More than forty years have passed since researchers have observed laser radiation for thefirst time [1]. Today, the laser is an object of our daily life as it is part of CD players,printers, laser pointers, etc. Using the word “laser” has become so natural that we tend toforget its origin: Light Amplification through Stimulated Emission of Radiation. Thisterm points to the essential property of laser light, namely its coherence: Photons that areemitted through stimulation by incident light are always in phase with the incoming pho-tons. One of the reasons why lasers have become so important in technology is that theircoherent light can be focused with extreme precision. The wavelength is the lower limitfor the diameter of the beam waist, and this is – for visible light – below one micrometer.The focusability of lasers implies that it is possible to concentrate the radiation energyinto a small cross section and to achieve increased light intensities in this way. Such“compression” can be done not only in space but also in time: The generation of shortlaser pulses has been one of the most important parts of laser physics. Today, it is possibleto produce laser pulses as short as a few femtoseconds. The impact of this technology onfundamental research has been twofold. First, the short duration of femtosecond pulsesopens up the possibility to study the dynamics of chemical reactions in real time [2]. Inthis context, we can view the experimental setup as a camera with an extremely fast shut-ter. Ahmed Zewail was awarded the Nobel prize in chemistry for his pioneering work inthis field. Second, by concentrating the available radiation energy into short pulses, onecan easily generate intensities of 1015 W/cm2 and above. A whole new area of research –physics with intense laser fields – has thereby developed in the last twenty years.

With strong fields, a variety of new phenomena was found. Mostly, they result fromthe fact that atoms or molecules may absorb many photons if the intensity is large enough.The most prominent multiphoton effects are above-threshold ionization (ATI) [3] andhigh-harmonic generation (HHG) [4]. In ATI, an atom absorbs more photons than neededto overcome the ionization potential. ATI manifests itself in the kinetic-energy spectrumof the photoelectrons, where one finds series of peaks separated by the photon energy.HHG means the conversion of n photons with energies hω into one photon of energy nhω.HHG is the most promising method to produce coherent x rays and attosecond pulses.So far, the highest harmonics that have been observed in experiment are of the order 300[5, 6]. It was found that both ATI and HHG exhibit a strongly non-perturbative character,i.e. a theoretical description based on finite-order perturbation theory fails.

5

6 Introduction

More recently, strong non-sequential double ionization was observed in intense laserpulses [7, 8, 9, 10, 11]. Here, the term “non-sequential” indicates that the double-ion-ization yield is not correctly predicted by a model where two steps of ionization takeplace independently of each other. The measured yields are in fact several orders ofmagnitude larger. To understand this effect, the electron-electron interaction must betaken into account, and a major part of this thesis will be devoted to the investigation ofthe double-ionization mechanism.

When molecules are driven by intense laser pulses, other effects are observed in ad-dition to those mentioned above. For example, a molecule can dissociate by multiphotonabsorption. In analogy to atoms, we encounter above-threshold dissociation [12, 13].Even if the laser irradiation does not directly initiate dissociation, it changes the elec-tronic structure of the molecule – we say that the electronic states are “dressed.” Thismay result in a lowering of the dissociation threshold which is sometimes referred to asbond-softening [12, 13].

It was theoretically found [14, 15] that the ionization probability of molecules at largeinternuclear separations is much higher than at the ground-state equilibrium distance. Thedifference can be orders of magnitude. Presumably, this so-called charge resonance en-hanced ionization is the reason that the fragment kinetic energies following Coulombexplosion of molecular ions [16] are typically much smaller than what we would expect ifCoulomb explosion was initiated at the equilibrium distance. Obviously, the nuclear mo-tion decides upon whether the molecule probes the region of enhanced ionization or not.Consequently, the theoretical treatment of these processes should include both electronicand nuclear motion.

Harmonic generation depends on the nuclear motion in a more subtle way. Usually,the generation of a high-energy photon is explained such that after initial ionization theejected electron recombines with the core under emission of one photon. Now, the nucleimay change their positions during the time interval between ionization and recombination.One of the most striking results presented in this thesis will be that this can lead to theemission of even-order harmonics in cases where they would otherwise be forbidden,namely in isotopically asymmetric molecules such as HD.

The numerical treatment of atoms or molecules in strong laser pulses is very demand-ing. An exact non-relativistic quantum-mechanical calculation would require the solutionof the many-body Schrodinger equation for all nuclei and electrons. So far, this has beenachieved only for atoms, and only for atoms with one electron [17, 18] or two electrons[19, 20, 21]. The traditional approach in molecular physics is the Born-Oppenheimerapproximation which relies on the different time scales for electronic and nuclear dy-namics: In many cases – for example in the molecular ground state – the electrons movemuch faster than the nuclei. The Born-Oppenheimer approach assumes that the electronsadjust immediately to the nuclear configuration and thus adiabatically create the “Born-

Introduction 7

Oppenheimer surfaces” in which the nuclear dynamics takes place like in a given externalpotential. Yet, it is easy to see that these assumptions are not valid in cases where ion-ization and harmonic generation occur: Here, electrons are (permanently or temporarily)promoted into highly excited states. The energy differences between the highly excitedstates are small (infinitesimally small for continuum states). This means that the timescale for the electronic motion is much slower than in the ground state and that the adi-abatic approximation breaks down. Furthermore, the Born-Oppenheimer approximationis not easy to employ from a technical point of view: Too many electronic states wouldhave to be taken into account.

A well-known method to handle systems with many interacting particles is densityfunctional theory (DFT) [22, 23, 24]. However, the application of DFT to strong-fieldphysics requires that we use the non-perturbative time-dependent version of DFT [25, 26].Furthermore, dealing with molecules, we should treat nuclei and electrons on the samefooting within the framework of multicomponent time-dependent DFT [26, 27]. Althoughthe foundations of this theory are well defined, there have been few applications so far.The main problem here is that we need a good approximation for the exchange-correlationfunctionals involved in the theory, but few multicomponent functionals have been testedup to now. In this thesis, density functional calculations play a minor role, and will becarried out at the so-called “exchange-only” level. Nevertheless, we keep DFT in the backof our minds as it is one of the main motivations of this work to give reference solutionsin systems that allow for an exact numerical integration. These will then be useful in theconstruction of approximations within the framework of DFT.

Looking for a feasible approach to calculate the intense-field behavior of atoms andmolecules, but avoiding the Born-Oppenheimer approximation leads us to the concept ofreduced dimensionality. We are going to use models where the motion of the particlesis essentially aligned with the direction of the laser polarization. This approach can beformally justified to a certain degree and appears reasonable in the presence of a strongfield, where alignment with the polarization axis is enforced.

Chapter 1 gives a short summary of the theoretical background. We explain how wedescribe the interaction of the laser with atoms or molecules and discuss some aspectsof the electron dynamics on the basis of classical mechanics. We give an introduction totime-dependent DFT, and we also include a discussion of the Born-Oppenheimer approx-imation. Finally, the numerical methods are briefly outlined.

Chapter 2 is devoted to atoms. This part focuses on laser-induced double ionizationand develops a very useful concept: the Wigner transformation with respect to the elec-tronic center-of-mass coordinate as a tool to study the two-electron dynamics in phasespace. Further we discover the two-electron analogue of ATI, above-threshold doubleionization, and show how it should manifest itself in experiment.

In Chapter 3, we concentrate on molecules. We discuss the effect of the nuclear mo-

8 Introduction

tion on ionization and relate to the question under what circumstances charge resonanceenhanced ionization occurs. We study the dissociative ionization of molecules and explainthe experimentally observed pulse-width and isotope effects on the fragment kinetic ener-gies. Finally, we discuss even-harmonic generation by HD molecules as we have alreadymentioned above. We show that this effect cannot occur within the Born-Oppenheimerapproximation.

On the whole, this work serves a threefold purpose:

� The explanation of experimental observations which have not been fully understoodpreviously, such as non-sequential double ionization.

� The theoretical discovery of new strong-field phenomena. We find above-thresholddouble ionization and even-harmonic generation due to beyond-Born-Oppenheimerdynamics.

� The provision of reference solutions for the construction of approximate techniquesin theoretical strong-field physics and chemistry.

Chapter 1

Theoretical background

1.1 Description of atoms and molecules in strong fields

Let us first ask: What is an “exact” calculation of the atomic or molecular response to anexternal field? To answer this question, we must assume that the problem is governed bya physical theory for which the underlying equations are known. Although we can neverknow with final certainty what the correct equations are, it seems that all microscopicaspects of physics are perfectly describable by relativistic quantum field theory. Of coursewe do not want to break a fly on the wheel: there is a number of approximations that wecan employ safely as far as the topics of this thesis are concerned:

� Since we do not consider processes with creation or annihilation of massive parti-cles, we can treat the particles within ordinary quantum mechanics, not quantumfield theory. The nuclei are assumed to be elementary particles without substruc-ture.

� We assume non-relativistic motion of the particles , i.e., we use the non-relativisticSchrodinger equation. This approach is valid if the laser intensity is not too large.For an 800 nm laser of intensity I � 1016 W/cm2, the velocity amplitude of theelectronic quiver motion is only 0.06 c. In all applications presented in this thesis,the speed of the electron is even well below this value.

� We treat the laser field classically. This means that absorption and stimulated emis-sion of light is included, but fluorescence is not. The latter typically occurs on atime scale much longer than femtoseconds and is therefore not important for ourpurposes.

� For the interaction between particles and field, we employ the dipole approxima-tion, i.e., the electric field is assumed to be constant over the extension of one atom

9

10 1 Theoretical background

or molecule. The wavelength of visible light is much larger than small molecules,so this approximation is well justified. The interaction with the magnetic-field com-ponent of the light wave is neglected.

� Except for Chapter 4, we neglect field-propagation effects, i.e., the field is not al-tered by the interaction with the particles. This approximation breaks down if alaser pulse interacts with a dense sample of particles such as a solid or liquid. Then,part of the light is reflected or absorbed, and in the interior regions of the sample,the pulse will be different from the incident pulse.

Adopting these approximations, we would – in the present context – still speak of an“exact” calculation. The equation of motion is the time-dependent Schrodinger equation

i∂∂t

Ψ�t � � H

�t � Ψ � t �� (1.1)

where Ψ�t � is the many-particle wave function. Here and throughout this thesis, atomic

units are used, i.e., e � me� h � 4πε0

� 1. The Hamiltonian H�t � can be written as

H�t � �

Nn

∑α � 1

P2α

2Mα

� Ne

∑j � 1

p2j

2�

Wnn�

Wen�

Wee�� Ne

∑j � 1

r j � Nn

∑α � 1

ZαRα E�t �� (1.2)

The interaction with the electric field E�t � is included in the form of dipole terms de-

pending on the nuclear coordinates Rα and the electronic coordinates r j. For this reason,Eq. (1.2) is called the length-gauge form. Nn and Ne are the numbers of nuclei and elec-trons, respectively. Pα and p j are the canonical momentum operators

Pα� � i∇Rα � p j

� � i∇r j (1.3)

Mα and Zα are the nuclear masses and charges, respectively. All particle-particle interac-tions are of the Coulomb type:

Wnn�

12 ∑

α �� β

ZαZβ Rα � Rβ

� (1.4)

Wen� � ∑

α � j Zα r j � Rα

� (1.5)

Wee�

12 ∑

i �� j

1 ri � r j

(1.6)

The most appropriate way to deal with Eq. (1.1) is to separate into center-of-mass andinternal coordinates, and to transform the electronic coordinates to a body-fixed frame.Since we are not going to need the three-dimensional version of these transformations,we refer to Kreibich’s thesis for details [28].

1.2 Reduced dimensionality 11

1.2 Reduced dimensionality

The numerical integration of the full many-particle Schrodinger equation (1.1) is not asimple task, even for the hydrogen atom. With the current computing capacities, threedegrees of freedom is the maximum that can be treated on a personal workstation. Thecomputational effort increases exponentially with the number of particles. This meansthat we cannot hope for an exact treatment of such a simple system as the H2 moleculein the next years. To date, the most advanced calculations have been for the He atom[21] and for the H

�2 molecular ion [29]. In both cases, the electron(s) were treated in full

dimensionality, but in the case of H�2 , the molecular axis was assumed to be aligned with

the laser polarization.In many applications, the problem has been simplified by reducing the number of spa-

tial dimensions. For the nuclei, one can argue in the following way if we are primarilyinterested in the dynamics on the femtosecond time scale: The molecular rotations occuron the picosecond scale, so we can assume that the rotational motion is almost frozen.This might not be true for small molecules such as H2, but in this case, experiments in-dicate that the molecules become aligned with the laser polarization axis [30]. Generally,the molecules respond in such a way as if they were oriented. This is either due to dynam-ical alignment or because mainly those molecules which have parallel orientation respondsignificantly to the laser. Hence, it seems that a one-dimensional (1D) treatment of thenuclei, including vibration but no rotation, should be able to qualitatively account for theexperimental situation.

For electrons, we argue differently. As we are interested in strong fields, the electronsare heavily accelerated. If we employ linearly polarized light only, we can expect that theelectrons will mainly move along the direction of polarization. We consider now the bareCoulomb potential created by one nucleus with charge Z:

VClb�r � � � Z

r � � Z�

z2 � ρ2 (1.7)

Assuming that the important part of electron dynamics is in the longitudinal z coordinate,we average Eq. (1.7) over the transverse coordinate ρ to obtain a 1D potential:

Vav�z � � � Z

�z2 � ε

(1.8)

The parameter ε is interpreted as the squared transverse extension of the electronic wavefunction. This potential (1.8) has the computational advantage that it is not singular atz � 0. For this reason, it is called a soft-core Coulomb potential. It was first proposedto study the strong-field dynamics of the hydrogen atom [31]. Note that the electroncoordinate z can be negative or positive, so that it is possible that the electron passes


by the nucleus. This is desirable in the context of intense laser pulses but may be lessappropriate in other cases. For example, the so-called frozen planet configuration [32], adoubly excited state of helium where both electrons are localized on the same side of thenucleus, does not exist without the Coulomb singularity [33, 34]. Such states, however,are not stable in a strong field [35].

Similar to the interaction between electrons and nuclei we set up a one-dimensionalform of the electron-electron interaction. We could now adjust the smoothing parameterε such that some properties of the real 3D systems are reproduced, for instance ionizationpotentials. However, since we should generally not expect quantitative agreement of 1Dresults with experiment, we do not spend effort on that. Instead, we use ε � 1 everywhere.The resulting 1D Hamiltonian for linearly polarized light reads

H�t � �

Nn

∑α � 1

P2α

2Mα

� Ne

∑j � 1

p2j

2�

Wnn�

Wen�

Wee� � Ne

∑j � 1

z j � Nn

∑α

ZαRα E�t �� (1.9)

Here, Rα and z j are the laboratory-frame coordinates of nuclei and electrons along thelaser polarization axis, and the interactions are

Wnn�

12 ∑

α �� β

ZαZβ Rα � Rβ

� (1.10)

Wen� � ∑

α � j Zα� �z j � Rα � 2 � 1

� (1.11)

Wee�

12 ∑

i �� j

1� �zi � z j � 2 � 1

(1.12)

The transformation of Eq. (1.9) to the center-of-mass frame will be given for each systemof interest in the corresponding sections of Chapters 2 and 3.

1.3 Classical considerations

In this section, we consider the classical dynamics of a free electron in a linearly polarizedlaser field E0 sin

�ωt� φ � . This will be useful when we estimate adequate grid parameters

for simulations. Also it will be helpful for purposes of interpretation, particularly in thecontext of the recoil-momentum spectra of doubly charged ions.

For the coordinate z along the polarization axis, we have

z�t � � � E0 sin

�ωt� φ �� (1.13)

1.3 Classical considerations 13

This is the classical equation of motion for an electron after ionization at t � 0, under theassumption that the electron does not interact anymore with the core for t � 0. IntegratingEq. (1.13) with respect to time gives

z�t � � z0 � E0

ωcosφ � E0

ωcos

�ωt� φ � (1.14)

with the initial velocity z0� z

�t � 0 � . Apparently, the velocity oscillates with frequency

ω and amplitude E0�ω around the drift velocity vD

� z0 � � E0�ω � cosφ. The oscillation

covers the range

E0

ω� � 1 � cosφ �� z � z0 � E0

ω�1 � cosφ �� (1.15)

φ is the phase of the laser field at the time of ionization. It can generally assume anyarbitrary value so that for z0

� 0 the relevant range is

� 2E0

ω� z � 2E0

ω (1.16)

The time-dependent kinetic energy is

12

z�t � 2 �

12

v2D�

vDE0

ωcos

�ωt� φ � � 1

2E2

0

ω2 cos2 � ωt� φ �� (1.17)

Averaging this equation over one optical cycle T � 2π�ω, we obtain the average kinetic

energy

Eavkin

�12

v2D�

Up � (1.18)

where the so-called ponderomotive potential,

Up�

E20

4ω2 � (1.19)

is the energy contribution that arises from the oscillatory electronic motion.Integrating Eq. (1.14) once more with respect to time leads to

z�t � � z0 � E0

ω2 sinφ � �z0 � E0

ωcosφ � t

� E0

ω2 sin�ωt� φ � (1.20)

with the initial position z0� z

�t � 0 � . The drift motion is modulated by an oscillation

with the amplitude

α �E0

ω2 (1.21)


We like to know under what circumstances an electron can return to the origin whenit starts out with the initial conditions z0

� z0� 0. We first calculate the local minima and

maxima of z�t � by setting z � 0:

cosφ � cos�ωt� φ �� (1.22)

This condition is fulfilled for

ωtm� 2πm or ωt

�m

� 2πm � 2φ (1.23)

with an integer number m. Obviously, for φ�

� 2πn, there are two extrema in each opticalcycle. In particular, there is one an at t � 0. (In the special case φ � 2πn, there is onesaddle point per cycle and no local extrema.) The extremal values of z

�t � are

z�tm � � � E0 tm

ωcosφ (1.24)

z�t

�m � � � 2

E0

ω2 sinφ � E0 t�m

ωcosφ (1.25)

0 π 2ππ/2 3π/2φ

−1

0

1

sinφ

ωt

0z(t

)

One recollision

More than onerecollision

Figure 1.1: The phase φ of the laser field at the time of ionization determines whether theelectron recollides with the core. White: no recollision; shaded regions: as indicated. Theinset shows the time-dependent electron coordinate for φ � 1.72.

1.4 Time-dependent multicomponent density functional theory 15

Assume now that the electron returns to the core at a time tret, i.e., z�tret � � 0 for t � 0,

and tret does not coincide with a local extremum. Then the two extrema to the left and tothe right of tret must have different signs. We therefore require z

�t

�m� � z � tm � � 0 and t � 0,

and find the condition

tanφ � φ � πm� � 0 (1.26)

It is easy to see that Eq. (1.26) does not have any solutions for m�

if φ � �0 � π � 2 � or if

φ � �π � 3π

�2�, i.e., no recollisions occur for such phases. For φ � �

π�2 � π � or φ � �

3π�2 � 2π

�,

the electron reaches the origin at least once. The range of phases where the electroncrosses the origin more than once has to be determined by solution of a transcendentequation. In principle, phases can be chosen such that an arbitrarily large number ofzeros occurs. Fig. 1.1 shows the ranges of phases with no recollision, one recollision andmore than one recollision. Of course, this simplified way of calculating the number ofrecollisions from Eq. (1.20) implies that the electron does not change its velocity when itcrosses the position of the core.

1.4 Time-dependent multicomponent density functionaltheory

As it is one of the purposes of this work to provide exact reference solutions for the con-struction of approximations within density functional theory, we give now an introductioninto this theory. The basic foundation of DFT is the one-to-one correspondence betweendensities and external potentials. This correspondence was first shown by Hohenbergand Kohn [22] for a system of identical particles in a static external potential V

�r � . In

conventional applications of DFT, this is usually a many-electron system in the potentialcreated by a set of fixed nuclei. The Hohenberg-Kohn theorem states that the externalpotential is (up to an irrelevant constant) determined by the ground-state density n

�r � . For

time-dependent systems it was shown by Runge and Gross [25] that the density n�r � t �

determines the time-dependent external potential V�r � t � up to a purely time-dependent

function. This is true for fixed initial state Ψ�t � t0 � � Ψ0. Li and Tong have general-

ized [27] the Runge-Gross theorem to multicomponent systems, i.e., systems with variouskinds of particles: There is a one-to-one mapping of the form

�V1�r � t �� V2

�r � t ��

n1�r � t �� n2

�r � t �� (1.27)

where V j�r � t � are the external potentials for each component, and n j

�r � t � are the time-

dependent densities. Again, this is true for fixed initial many-particle state Ψ0. Conse-quently, all observables can be written as functionals depending on the set


�n1�r � t �� n2

�r � t �� and on Ψ0. Usually, one studies the case where an external pertur-

bation is turned on at a time t0 and the initial state at t0 is the ground state correspondingto the time-independent potentials for t � t0. Then, by virtue of the stationary Hohenberg-Kohn theorem, Ψ0 itself is a density functional so that all observables become pure densityfunctionals.

If we apply the multicomponent version of DFT to molecules, the Coulomb attractionbetween nuclei and electrons is no longer treated as external potential but as interaction.Hence, for a free molecule, it does not make sense to apply this theory in the laboratoryframe because the absence of a confining external potential would lead to spatially con-stant densities. Rather, one has to separate off the center-of-mass motion, and apply thedensity-functional approach to the internal degrees of freedom. For a diatomic molecule,we have to consider the density N

�R � t � of the “relative nuclear particle” (R is the differ-

ence vector between the two nuclei) and the electronic density in the body-fixed frame,n�r � t � .

In practice, DFT is employed in the form of the Kohn-Sham (KS) scheme [23]. Thismethod relies on the fact that the mapping between potentials and densities holds true in-dependently of the particle-particle interactions. In particular, a given set of densities canbe reproduced by a system of non-interacting particles with uniquely determined externalpotentials. These KS potentials are functionals of the densities and, conversely, we cancalculate the densities from the KS single-particle orbitals. For a diatomic molecule, wehave

N�R � t � �

χ�R � t � 2 � (1.28)

n�r � t � �

Ne

∑j � 1

ϕ j�r � t � 2 (1.29)

The time evolution of the nuclear orbital χ and the electronic orbitals ϕ j is given by theKS equations

i∂∂t

χ�R � t � �

� � ∇2R

2µn

�VS � n �N � n � �

R � t � � χ�R � t �� (1.30)

i∂∂t

ϕ j�r � t � �

� � ∇2r

2�

VS � e �N � n � �r � t � � ϕ j

�r � t �� (1.31)

Here, µn� M1M2

� �M1

�M2 � is the reduced nuclear mass. The KS potentials are usually

split into several terms,

VS � n �N � n � �R � t � � Vn

�R � t � � VH � n � n � �

R � t � � Vxc � n �N � n � �R � t �� (1.32)

VS � e �N � n � �r � t � � Ve

�r � t � � VH � e �N � n � �

r � t � � Vxc � e �N � n � �r � t �� (1.33)

1.5 The Born-Oppenheimer approximation 17

where Vn and Ve are the external potentials in the interacting system, and VH � n and VH � e arethe Hartree potentials:

VH � n � n � �R � t � �

�n�r � t � wen

�R � r � d3r (1.34)

VH � e �N � n � �r � t � �

�n�r

� � t � wee�r

� � r � d3r� � �

N�R � t � wen

�R � r � d3R (1.35)

The interactions are denoted as wen and wee. Equations (1.32), (1.33) are essentially thedefinition of the exchange-correlation (xc) potentials Vxc � n and Vxc � e. Of course, in thecase of atoms, we only need to consider the electronic KS equations, and the functionalsdepend on the electronic density only.

The so-called Hartree approximation amounts to neglecting the xc terms in Eqs. (1.32),(1.33). Going beyond the Hartree approach requires approximate functionals for the xcpotentials, and the central issue in DFT is the construction of such functionals.

1.5 The Born-Oppenheimer approximation

The traditional approach to the theoretical description of molecules is the Born-Oppen-heimer (BO) approximation [36] where one exploits the fact that the electronic motionis often much faster than the nuclear motion. The approximation can be applied both tothe stationary and to the time-dependent Schrodinger equation. Only the latter case isdiscussed here.

The total molecular wave function is first expanded in BO states:

Ψ��

Rα � � � r j � � t � � ∑k

χk��

Rα � � t � Φk� �

Rα � � � r j � � (1.36)

�Φk � is a complete and orthonormal set of stationary electronic eigenfunctions of the

operator Te�

Wnn�

Wen�

Wee, where Te� ∑ j � p2

j�2 � is the electronic kinetic-energy

operator. These eigenfunctions depend parametrically upon the nuclear positions Rα.The nuclear wave functions χk play the role of expansion coefficients and carry the fulltime dependence. In order to obtain the equation of motion for χk, we insert Eq. (1.36)into the time-dependent Schrodinger equation (1.1), multiply with Φl

��Rα � � � r j � �� and


integrate over the electronic coordinates:

i∂∂t

χl��

Rα � � t � �

� � ∑β

∇2Rβ

2Mβ

�V BO

l� �

Rα � �� χl��

Rα � � t � (1.37)

� E�t � ∑

k

Dlk��

Rα � � χk��

Rα � � t �� ∑

k

�Alk� �

Rα � � � 2∑β

Blkβ��

Rα � � ∇Rβ � χk��

Rα � � t ��Here,

V BOl��

Rα � � �� Φl Te�

Wnn�

Wen�

Wee Φl � (1.38)

are the Born-Oppenheimer potentials,

Dlk� �

Rα � � �� Φl �� ∑β ZβRβ � ∑j

r j ��Φk (1.39)

are the dipole matrix elements, and Alk, Blkβ are the non-adiabatic couplings:

Alk� � ∑

β

12Mβ

� Φl ∇2

Rβ

Φk � � (1.40)

Blkβ� � 1

2Mβ

� Φl ∇Rβ

Φk � (1.41)

They lead to a mixing of the different electronic BO states even in the absence of anexternal field. The BO approximation amounts to neglecting these terms so that onlythe first and second line of Eq. (1.37) remain. Within the BO approximation, transitionsbetween different electronic states can occur only if the system is perturbed by an externalfield.

Usually, the center-of-mass motion is separated off before the BO approximation isemployed. The structure of the resulting equation is the same as in Eq. (1.37). Only,the nuclear coordinates are then internal coordinates, the masses are replaced by the re-duced masses, and the expressions for the matrix elements involve the internal nuclearand electronic coordinates instead of the laboratory-frame coordinates.

The BO approximation provides an intuitive picture of the molecular motion: Thenuclear dynamics takes place on the BO potential energy surfaces V BO

l

��Rα � � as if these

were given external potentials. The laser field induces transitions from one surface toanother. This is quantified by the transition dipole matrix elements.

1.6 Numerical methods 19

1.6 Numerical methods

1.6.1 Split-operator method

Solving the time-dependent Schrodinger equation means that we find the time-dependentwave function Ψ

�t � for a given Hamiltonian and for given initial wave function Ψ

�t � 0 � .

The Hamiltonian can often be written as a sum of two contributions, one depending onlyon the momenta, the other depending only on the coordinates. Using the length-gaugeformulation, Eq. (1.2), the momentum-dependent contribution is the kinetic energy, whilethe other contains all interactions and external potentials:

H�t � � T

�V�t �� (1.42)

It follows from the Schrodinger equation that the evolution of the wave function withinan infinitesimal time increment dt is described by the unitary operator exp

� � iHdt � :Ψ�t�

dt � � exp� � iH

�t � dt � Ψ

�t �� (1.43)

For the numerical evaluation of Eq. (1.43) with a small but finite dt we use the split-operator method [37] where the evolution operator is split into a product of exponentials:

exp� � iH

�t � dt � � exp

� � iV�t � dt

�2 � exp

� � iT dt � exp� � iV

�t � dt

�2 � � O

�dt3 � (1.44)

The operator product on the right-hand side has the advantages that it is unitary, sym-metric under time-reversal, and easy to implement for the following reason: The operatorexp

� � iV�t � dt

�2 � is simply applied through multiplication in configuration space. Simi-

larly, the operator exp� � iT dt � is applied through multiplication in momentum space. The

momentum-space representation of the wave function is obtained from the configuration-space representation by a Fourier transformation. E.g., for a one-dimensional single-particle function, we have

ψ�p � �

�ψ�x � exp

� � ipx � dx (1.45)

(Using this notation, the momentum-space wave function is normalized to 2π.) On thecomputer, the wave function is represented on a grid with a finite number of points. Then,the efficient Fast Fourier Transformation (FFT) algorithm [38] can be used. Employingthe FFT, the computational time for one Fourier transform is proportional to N log

�N �

where N is the number of grid points. (Using the conventional evaluation of the discreteFourier transform, the computational effort scales with N2 because for each of the Ngrid points in momentum space, a sum over N positions in coordinate space has to becalculated.) The application of the operators exp

� � iV�t � dt

�2 � and exp

� � iT dt � requires


N multiplications, each. Thus, for large N, the Fourier transformation is the element thatmainly determines the computational time. Note, however, that the favorable FFT scalingapplies only when the number of grid points is a product of small prime numbers; ideal isthe case N � 2n.

1.6.2 Obtaining the ground state

The previous subsection explained how the numerical propagation in time is achieved.Before the propagation can be started we need to specify the initial state. Typically, weuse the ground state of the unperturbed system as initial state. The most convenient way toobtain the ground state is the “relaxation method” [39]: We begin with an arbitrary1 wavefunction Ψ

�t � 0 � and propagate in imaginary time t � � iτ. In terms of the eigenstates

ψk with energies Ek, the evolution over an interval dτ is given by

Ψ�τ � dτ � � ∑

k

� ψk Ψ�τ � � e � Ekdτ ψk (1.46)

In complete analogy to the propagation in real time we use the split-operator method asdescribed in Sec. 1.6.1. It is apparent from Eq. (1.46) that the state with the lowest energysuffers less damping than the excited states. In practice, the wave function is renormalizedin each time step so that after a sufficient number of steps, the normalized ground-statewave function ψ0 is obtained. The corresponding energy is

E0� lim

τ � ∞� 12dτ

ln� � Ψ � τ � dτ � Ψ � τ � dτ � �� Ψ � τ � Ψ � τ � � � (1.47)

Here, the numerator is meant to involve the wave function propagated for one time step,but before renormalization, just as it is given by Eq. (1.46). The energy calculated fromEq. (1.47) can be used as convergence criterion because it converges simultaneously withthe wave function.

The relaxation method may be employed to determine excited states as well. Toachieve this, all eigenstates with lower energies have to be determined first. Then, thesestates must be projected out in each time step.2

1In principle, the initial state is not completely arbitrary: It must not be orthogonal to the ground statewhich is to be obtained. However, small numerical errors, which can never be completely avoided, alwaysproduce small contributions which are not orthogonal to the ground state. Therefore, the correct groundstate is indeed obtained for an arbitrary initial state.

2This is again due to numerical errors. Without such errors it would suffice to perform the projectiononly once.

1.6 Numerical methods 21

1.6.3 Absorbing boundaries

Frequently, the numerical grid cannot be made large enough to contain the whole wavefunction. In particular, this is so when we deal with ionization in strong fields: Withina few optical cycles, the extension of the electronic wave function easily reaches severalhundred atomic units. Numerically, wave packets reaching the grid boundary must beabsorbed. Otherwise, a “ghost” wave packet reenters on the opposite side of the grid sincethe numerical Fourier transformation effectively imposes periodic boundary conditions.In propagation techniques other than the split-operator method, one usually has to specifythe boundary conditions explicitly. In that case, the absence of an absorber typically leadsto reflection.

Adding an imaginary (“optical”) potential to the external potential is a convenientway to achieve absorbing boundary conditions. For example, on a one-dimensional gridextending from � z0 to

�z0, one can use an optical potential of the form

Vopt�z � �

�� iλ�z�

z1 � 2 � � z0 � z � � z1

0 � � z1 � z � � z1� iλ�z � z1 � 2 � �

z1� z � � z0

(1.48)

with a parameter λ describing the strength of absorption. Using Eq. (1.48), the wave func-tion in the regions

� � z0 � � z1�

and�z1 � z0

�is effectively multiplied by exp

� � λdt�z � z1 � 2 �

in each time step. Therefore, instead of employing an optical potential, one can equiv-alently multiply in every time step with an absorbing mask function “by hand.” This iswhat one must do if a splitting technique is used where the outgoing wave packets aretransferred to an outer grid and propagated further in time. We will discuss this point inmore detail in Chapter 2.

The parameter λ has to be chosen with care: If λ is too small, the wave packets arenot entirely absorbed before they run into the boundary. On the other hand, if λ is toolarge, a part of the wave function is reflected. This is similar to the behavior of a wavepropagating along a rope with fixed ends.

1.6.4 Choosing the grid parameters and time step

For the numerical propagation one has to choose values for the grid spacings, the gridsize, and the size of the time step. Results become more precise with smaller steps andlarger grids at the cost of increasing computing time and increasing use of memory. Gen-erally, all results should be checked whether they remain stable if these parameters areimproved. In addition, we can give some helpful guidelines for estimating the minimumrequirements, based on the classical electronic motion described in Sec. 1.3:


� With an electronic grid spacing ∆z, the extension of the momentum-space grid is� � pmax � pmax�with pmax

� π�∆z. In a strong laser, this value must be larger than the

maximum momentum 2E0�ω that an electron can receive classically, see Eq. (1.16).

� The size of the grid along the electronic coordinates must be large enough to containthe oscillatory motion of the electrons in the laser field, at least during the timebetween ionization and recollision. Thus, the minimum range is from � 2α to 2α,where α is the classical oscillation amplitude, see Eq. (1.21).

� Using a time step ∆t, the maximum representable energy difference is ∆E � 2π�∆t.

This value should be larger than the difference between the maximum energy Emax

and the ground-state energy. By inserting the maximum drift velocity E0�ω of an

electron in a laser field into Eq. (1.18), we estimate Emax� 3Up per electron. Note,

however, that the time step derived from this argument is often much larger than thetime step required for an accurate propagation with the split-operator method.

Typical parameters for the calculations in this thesis are:

� Grid sizeElectronic coordinates: 100 to 400 a.u.Nuclear separation: 10 to 80 a.u.

� Spatial step sizeElectronic coordinates: 0.2 to 0.4 a.u.Nuclear separation: 0.1 a.u.

� Time step: 0.05 a.u.

Chapter 2

Double ionization of atoms

2.1 History

Double ionization of atoms has developed into one of the hottest topics in the last decadeafter Fittinghoff et al. [7] and others [8, 9, 10, 11] found anomalously large double-ionization rates of rare-gas atoms. The experimental single-ionization yields were alwaysin excellent agreement with single-active-electron (SAE) calculations, where only oneelectron is treated dynamically while all others form a “frozen core” screening the nuclearcharge. One could even go further and assume that single ionization in a long-wavelengthlaser is purely due to tunneling and apply the tunneling theory by Ammosov, Delone, andKrainov (ADK) [40] to achieve reasonable agreement with experiment. The release oftwo electrons in a linearly polarized laser, however, did not follow such simple rules. Asequential model of double ionization where the first and the second ionization step occurindependently of each other failed: The double-ionization probabilities predicted by thismodel were several orders of magnitude too small, at least in the low-intensity regime.On the other hand, for very high intensities, roughly I � 1015 W/cm2, good agreementwas found. The transition from the low-intensity to the high-intensity behavior gives riseto a “knee structure” in the intensity-dependence of the double-ionization yield. All ofthese findings can nicely be seen in Fig. 2.1 which is copied from the original article byWalker et al. [10].

Soon, a variety of theoretical approaches, such as diagrammatic S-matrix techniques[41, 42], 1D time-dependent simulations [43, 44, 45, 46, 47], and 3D time-dependentsimulations [21] were able to produce non-sequential ionization. Within the S-matrixapproach, it was possible to achieve quantitative agreement with the experimental He2

�

yield. Yet, although there has been general agreement that the electron-electron inter-action plays an important role, the physical picture of the ionization mechanism has re-mained controversial. Over the years, essentially three models have been proposed:

23

24 2 Double ionization of atoms

Figure 2.1: Measured He ion yield for linearly polarized 780 nm laser pulses with 100 fspulse width. Solid: SAE calculation; dashed: ADK yields; solid curve on right: calculatedsequential He2

�yield. The measured intensities are multiplied by 1.15. Taken from Ref.

[10].

1. A shake-off process [7]: One electron is removed very quickly so that the secondelectron effectively feels a rapid change in potential. This leads to excitation of thesecond electron, and ionization from excited states is much easier than ionizationfrom the ground state. In this model, the electron-electron interaction enters in theform of initial-state correlation.

2. A rescattering process [48]: After initial single ionization, the outer electron is ac-celerated back towards the core when the electric field reverses its sign. Recollisionof this electron with the core leads to e-2e scattering. This model relies on dynamicelectron-electron correlation rather than initial-state correlation. Rescattering ap-

2.2 Model 25

pears realistic because it is able to explain another phenomenon, too, namely thecut-off law in high-harmonic generation [48].

3. Collective tunneling [49]: Both electrons tunnel out simultaneously. In contrast tothe other two mechanisms, it was soon found that this process cannot quantitativelyaccount for the large double-ionization yields.

Until recently, the double-ionization experiments focused on the measurement of to-tal ion yields, giving little insight into the mechanism. The only hints were the findingsthat the “knee” was not observed for small wavelengths like 250nm [8], and that it wassuppressed when circular polarization was used [9]. In theoretical studies, one mostlytried to extract information from the time evolution in configuration space, without reach-ing unambiguous conclusions. The first direct experimental evidence of electron-electroncorrelation was given with the measurement of the He2

�[50] and Ne2

�recoil momenta

[51]. For intermediate intensities, the distribution of the recoil-momentum componentalong the polarization axis showed a double-hump structure with a local minimum atzero, implying that the configuration where both electrons move with equal momenta intoopposite directions is suppressed. Later, the full two-electron momentum correlation wasmeasured and confirmed the guess that the electrons are preferentially ejected into thesame direction [52]. This points towards the validity of the rescattering picture, as wewill see later in this chapter.

2.2 Model

We employ the linear model as described in Sec. 1.2 for a two-electron atom with nuclearmass Mn and nuclear charge Zn. We write the Hamiltonian in terms of the total center-of-mass coordinate RCM and electron coordinates z1, z2 measured relative to the nucleus.The Hamiltonian then separates into a center-of-mass term and an internal term:

H � HCM�

Hint (2.1)

with

HCM� � 1

2Mtot

∂2

∂R2CM

� � Zn � 2 � RCM E�t �� (2.2)

Hint� � 1

2µe

∂2

∂z21� 1

2µe

∂2

∂z22� 1

Mn

∂2

∂z1∂z2� Zn�z2

1�

1� Zn�

z22�

1

� 1� �z1 � z2 � 2 � 1�

qe�z1�

z2 � E � t �� (2.3)


Here, µe� Mn

� �Mn

�1 � is the electronic reduced mass, Mtot

� Mn�

2 is the total mass,and the field prefactor

qe�

Mn�

Zn

Mtot(2.4)

deviates slightly from unity if the number of electrons differs from the nuclear charge.In the case of the He atom, we have Zn

� 2 and thus qe� 1. In that case, the electric

field does not accelerate the center of mass, see Eq. (2.2). In the following, we consideronly the dynamics of the internal degrees of freedom. Further, we neglect the small mass-polarization term � � 1 � Mn � ∂2 � � ∂z1∂z2 � . The electric field is

E�t � � E0 f

�t � sin

�ω t � � (2.5)

where E0 is the peak amplitude, f�t � is an envelope function, and ω is the laser frequency.

We use the following numerical parameters: Grid size 400 � 400 a.u., spatial step size 0.38a.u., time step 0.05 a.u.

Since the Hamiltonian does not depend on the electron spins, the total spin-dependentwave function can be written as a product of a coordinate-dependent function and a spinfunction:

Ψσ1σ2

�z1 � z2 � t � � Ψ

�z1 � z2 � t � χσ1 � σ2 (2.6)

The Pauli principle requires that the total wave function must be antisymmetric with re-spect to the interchange of electrons. If χ describes a symmetric triplet, then Ψ

�z1 � z2 � t �

is antisymmetric under exchange of coordinates. Likewise, if χ stands for an antisym-metric singlet, then Ψ

�z1 � z2 � t � is symmetric. In this thesis, we concentrate on the case

that the system is initially in the singlet ground state. The time evolution does not alterthe symmetry, so Ψ

�z1 � z2 � t � remains symmetric for all times. Numerically, we find the

ground-state energy of the He model atom to be � 2.238 a.u. The ground-state energy ofthe real He atom is � 2.90 a.u.

2.3 The mechanism of He double ionization in long-wave-length lasers

2.3.1 Time evolution in momentum-space

The configuration-space evolution has been studied extensively for the 1D model [43,46, 47]. Since the conclusions drawn from those studies have been rather vague, we aregoing to find out what we can learn from a momentum-space analysis. We are confident

2.3 The mechanism of He double ionization in long-wavelength lasers 27

that the model describes non-sequential double ionization qualitatively correctly becauseit reproduces the characteristic knee structure in the He2

�yield [44]. We employ a laser

field with 780 nm wavelength and intensity 1015 W/cm2, and compare two cases:

� A field of constant amplitude, suddenly switched on at t � 0:

f�t � �

�0 � t � 01 � t � 0

(2.7)

� A sin2-shaped envelope function:

f�t � �

�� 0 � t � 0sin2 � tπ

6T � � 0 � t � 6T0 � t � 6T (2.8)

In the latter case, the total pulse length is 6T where T denotes the duration of one opticalcycle. Absorbing boundary conditions are applied. Therefore the norm is decaying withtime. This behavior is identified with ionization. At this stage, we do not yet distinguishbetween single and double ionization. Figure 2.2 shows the time-dependent electric fieldtogether with the norm � Ψ � t � Ψ � t � � . In the case of constant intensity, the norm startsto decay substantially before one optical cycle is completed. On the scale of the figure,the sin2 pulse does not yield noticeable ionization during the first two cycles. In thethird cycle, however, the field amplitude becomes almost as large as that of the field withconstant intensity, and strong ionization sets in. For both fields, the norm decreases in astepwise fashion with one step per half cycle. Apparently, the reason is that the electronsare ejected with highest probability at times of maximum electric field strength. Thisoccurs twice per optical cycle.

We consider now the time-dependent momentum-space wave function

Ψ�p1 � p2 � t � �

�Ψ�z1 � z2 � t � e

� ip1z1� ip2z2 dz1dz2 (2.9)

We first discuss the case of constant intensity: Figure 2.3 shows the modulus squared ofthe momentum-space wave function at selected times during the first four periods T . Inthe distribution at t � 0 (the ground-state distribution) we find a slight preference in favorof configurations with parallel instead of antiparallel momenta. This can be understoodintuitively: The electrons avoid each other due to the electron-electron repulsion. Inparallel motion this is better achieved than in antiparallel motion. As dictated by the Pauliprinciple, the momentum-space distribution is always symmetric under reflection at thediagonal p1

� p2. Immediately after the laser is switched on with positive electric field,the electrons are accelerated into the negative z direction. Thus at t � T

�2 we observe


0 100 200 300 400 500 600 700t (a.u.)

0.0

0.2

0.4

0.6

0.8

1.0

Nor

m

−0.2

−0.1

0.0

0.1

0.2

E(t

) (a

.u.)

(a)

(b)

Figure 2.2: (a) Laser field with constant intensity (broken curve) and sin2-shaped pulse(solid curve). (b) The corresponding time-dependent norm.

two strong single-ionization wave packets: one where p1 is negative and another wherep2 is negative. Yet, we do not observe any density in the region where both momenta havelarge absolute values. This expresses the fact that the probability for double ionization isvery small at this time.

However, when the first cycle is finished, two small satellites (“Zipfel”)1 appear atabout (p1, p2) = (5, 3) a.u. and (3, 5) a.u. After another half cycle, similar structures arevisible at negative values of the momenta. For later times, the system keeps evolving in avery similar fashion, but becomes more structured because more and more wave packetsare superimposed and interfere with each other.

We conclude from Figure 2.3 that double ionization is effective every half period. Atthese times the two electrons are ejected with different momenta into the same direction.This is clear evidence for electron-electron correlation.

In Fig. 2.4, the momentum distribution is shown for the sin2 pulse. Note that the firstpanel is not for t � 0 but for t � 2T . We have chosen this time because for t � 2T , the

1Acronym derived from German “Zweifachionisationspeaks aus Femtosekunden-Laseranregung”(double-ionization peaks from femtosecond laser excitation).


-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p 2 (

a.u.

)

p1 (a.u.)

(d) t=3T/2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p 2 (

a.u.

)

p1 (a.u.)

(g) t=3T

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p 2 (

a.u.

)

p1 (a.u.)

(b) t=T/2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(a) t = 0

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p 2 (

a.u.

)

p1 (a.u.)

(c) t=T

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p 2 (

a.u.

)

p1 (a.u.)

(i) t=4T

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p 2 (

a.u.

)

p1 (a.u.)

(e) t=2T-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p 2 (

a.u.

)

p1 (a.u.)

(f) t=5T/2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p 2 (

a.u.

)

p1 (a.u.)

(h) t=7T/2

Figure 2.3: Momentum-space distribution Ψ�p1 � p2 � t � 2 at selected times t during the

action of a constant-intensity laser field with period T . Contour-line levels: Thick curve,4 � 10 � 2; thin curve, 2 � 10 � 3; broken curve, 2 � 10 � 5 .

intensity is well below its peak value whereas in the interval 2T � t � 4T , the electric fieldis very similar to the constant-intensity field (see Fig. 2.2). And indeed, the snapshots inFig. 2.4 are amazingly similar to those of Fig. 2.3. The double-ionization “Zipfel” arelocated at the same positions as before. On the falling edge of the pulse (t � 4T ), no morestructures appear at large momenta, indicating that ionization has become much weaker.

The comparison between Figs. 2.3 and 2.4 shows that pulse-shape effects on theionization mechanism are small. As soon as the intensity is large enough to cause single-


-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(b) t=5T/2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(c) t = 3T

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(d) t=7T/2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(a) t = 2T

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(e) t = 4T

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(i) t = 6T

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(f) t=9T/2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(g) t = 5T-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

p2 (

a.u

.)

p1 (a.u.)

(h) t=11T/2

Figure 2.4: Momentum-space distribution Ψ�p1 � p2 � t � 2 at selected times t during the

action of a sin2-shaped pulse with period T . Contour-line levels: Thick curve, 4 � 10 � 2;thin curve, 2 � 10 � 3 .

or double-ionization, these processes take place without being influenced much by theprevious evolution. This implies that we do not need to use an envelope function if weare interested in the ionization mechanism, saving us some computing time.


200 300 400 500 t (a.u.)−150

−50

50

150

<z 1>

, <z 2>

(a.

u.)

0 100 200 3000 t (a.u.)−200

−100

0

100

200

<z 1>

, <z 2>

(a.

u.)

(a)

(b)

laser field

laser field

Figure 2.5: Position expectation values � z1 � (circles ) and � z2 � (squares) calculated forparts of the wave function describing double ionization (see text). (a) Field with constantintensity; (b) sin2-shaped pulse. The laser field is plotted in arbitrary units.

2.3.2 Coordinate expectation values

To confirm the conclusions drawn from the momentum-space analysis, we plot in Fig. 2.5the position expectation values � z1 � , � z2 � calculated by integration over the triangles de-fined by

�z1 � z2

� 6a u � z2� z1 � and

�z1 � z2

� � 6a u � z2� z1 � . The probability located

in these regions corresponds to double ionization with both electrons on the same side ofthe nucleus. The restriction z2

� z1 selects one of the electrons as the outer one. Oth-erwise, the symmetry of the wave function would always lead to � z1 � � � z2 � . The twotime axes in Fig. 2.5 have been arranged such that those times that are expected to exhibitsimilar ionization dynamics are placed at the same horizontal position. I.e., t � 0 of panel(a) is directly above t � 2T of panel (b). As before, we find that the results for constantintensity are very similar to those for the sin2 pulse. We observe that both electron coor-dinates become large simultaneously, but increase with a different rate, i.e., the electronshave different velocities. The precise value of the velocities should not be extracted fromFig. 2.5 because some electron flux may always move into or out of the triangular regions.


2.3.3 Test for rescattering

The total ionization probability is simply calculated from the norm that remains on thenumerical grid:

pionization�t � � 1 � �

grid

Ψ � � z1 � z2 � t � Ψ � z1 � z2 � t � dz1dz2 (2.10)

In the present approach, the discrimination of double ionization against single ionizationis not straightforward if a lot of density is lost at the absorbing boundary. We assumethat the distinction can be made just before the absorption takes place: We calculate thetwo-electron current density

jk�z1 � z2 � t � �

12µei

� Ψ � � z1 � z2 � t � ∂∂zk

Ψ�z1 � z2 � t � � Ψ

�z1 � z2 � t � ∂

∂zkΨ � � z1 � z2 � t � � (2.11)

at coordinates close to the edge but still outside the absorbing region. The He�

and He2�

yields are then obtained by integrating over lines which are sections of the grid boundarycorresponding to single (I) and double ionization (II), and over time:

pI � t � �

t�0

dt� �

I

j�z1 � z2 � t � � nds � (2.12)

pII � t � �

t�0

dt� �II

j�z1 � z2 � t � � nds (2.13)

Here, j �

�j1 � j2 � and n is the unit vector perpendicular to the boundary. Part I contains

all boundary sections where one electron coordinate is smaller than 6 a.u. (The electronis assumed to be bound for z � 6 a.u.) Part II contains the remainder of the boundary.

We describe now a preliminary numerical test to find out whether the rescattering pic-ture is compatible with our quantum mechanical calculation. We consider the interactionwith two different “one-cycle pulses” as displayed in Fig. 2.6. In the first case, the fieldconsists of a single sine cycle whereas in the second case, a time delay is introduced be-tween the first and second half cycle. Panel (c) of the figure shows the time-dependentdouble-ionization yield obtained in either case. Apparently, double ionization is sup-pressed by about 60% when the time delay is employed. The interpretation within therescattering model is as follows. The first half-cycle creates an outgoing electron. Duringthe second half-cycle, the electron is accelerated back towards the core. If the emissionof the first electron is at a phase of the field that allows for a recollision (see Sec. 1.3)then e-2e scattering can lead to double ionization. In the time-delayed case, however, the


0 100 200 300t (a.u.)

0

0.002

0.004

0.006

0.008

pII

0 100 200 300−0.2

0

0.2E

(t)

(a.u

.)0 100 200 300

−0.2

0

0.2

E(t

) (a

.u.)

(c)

(b)

(a)

Figure 2.6: The upper panels show the time-dependent electric field for (a) a one-cycle“pulse” and (b) another where the second half cycle is time-delayed with respect to thefirst one. The resulting double-ionization yields are shown in panel (c). The line styledistinguishes case (a) from case (b).

first electron has time to travel far away from the core before the reverse acceleration setsin. Then, the probability for a recollision is much smaller, and so is the double-ionizationyield due to rescattering. The effect cannot be explained in terms of the shake-off model.There, the ejection of the second electron should not depend on how far the first electronis away. Introducing the time delay does not completely eliminate double-ionization. Thismay be explained by contributions from other mechanisms.

Consistently with the momentum-space snapshots, double ionization is negligible dur-ing the first half cycle. The delay of the second half cycle is the same as the delay in theonset of double ionization. This entirely excludes collective tunneling as the dominantmechanism since tunneling should recur every half period.

Note that this numerical test makes sense only if the probability density correspondingto the outgoing electron is still present on the grid at the time when the second ionizationstep takes place. Otherwise we would also observe a suppression as in Fig. 2.6(c), but thiswould merely be an effect due to the absorbing boundary.


2.3.4 Phase-space analysis

The preceding subsection provided strong indications that rescattering is the main mech-anism responsible for strong-field non-sequential ionization. In essence, this is a classicalprocess. Only the initial tunneling of one electron seems to be a quantum process whilethe free electron motion in the laser field and the recollision with the second electroncan be understood classically. Classical dynamics is often visualized in phase space: thespace spanned by coordinates and momenta. In quantum mechanics, the phase-spacedistribution is obtained via the Wigner transformation [53, 54]. The Wigner functioncorresponding to a one-particle wave function ϕ

�x � is given by

w�1 � � x � p � �

�ϕ � � x � y

2� ϕ � x

� y2� e

� ipy dy (2.14)

This quantity is always real, but can be positive or negative. The integration of w�1 � over

the momentum p correctly yields the coordinate probability distribution ϕ�x � 2 (except

for a prefactor 2π). Likewise, the integration over x yields the momentum distribution.Therefore, and despite the fact that the Wigner transform may assume negative values, it isusually interpreted as the probability distribution in the one-particle phase space spannedby the coordinate x and the momentum p.

In our 1D helium model, we have two electronic coordinates. Carrying out the Wignertransformation for each coordinate doubles the number of dimensions and thereby leadsto a quantity which is not easily analyzed. We therefore introduce the Wigner transforma-tion with respect to the electronic center-of-mass coordinate Z �

�z1�

z2 � � 2. We do nottransform with respect to the relative coordinate z � z2 � z1:

w�Z � P� z � �

�Ψ � � Z � y

2 � z2� Z � y

2� z

2� Ψ � Z

� y2 � z

2� Z � y

2� z

2� e

� iPy dy (2.15)

Here, P is the center-of-mass momentum. Integrating this quantity over the relative coor-dinate z yields the phase-space distribution for the center of mass:

w�Z � P � �

�w�Z � P� z � dz (2.16)

As far as the numerical evaluation of w�Z � P � is concerned, it is more efficient (yet equiv-

alent) to first integrate over z and then carry out the Fourier transformation. In this way,we do not need to handle the complicated integrand in Eq. (2.15), which depends on fourvariables.

The center-of-mass phase-space distribution has been calculated for selected timesduring the action of a laser field with constant intensity, switched on instantaneously att � 0. As in the preceding subsections, the wavelength is 780nm and the intensity is1015 W/cm2. If this thesis comes with a CD, you can watch the time evolution from t � 0


Figure 2.7: The time evolution of the center-of-mass phase-space distribution in a laserfield with wavelength 780 nm and intensity 1015 W/cm2. T is the duration of one period.

to t � 3T�2 by opening the video+audio clip “movie1.mov”. Several snapshots are also

shown in Fig. 2.7. The system starts out from the ground state which, on the scale ofthe figure, has a very narrow phase-space density around Z � 0 extending from aboutP � � 2a.u. to P � 2a.u. In the first half cycle, the electrons are accelerated towardsnegative coordinates and momenta. Thus, at t � T

�2, a very broad wave packet has

appeared in the region�Z � 0, P � 0 � . We have seen before that double ionization is

negligibly small at this time, so we can ascribe this density to pure single ionization.This is confirmed by the value of

P at the lower end of the wave packet: It agrees well

with the momentum 2E0�ω � 5 8a.u. that one free electron receives classically during

a complete half cycle of acceleration in the field, see Sec. 1.3. Thus the wave packetcorresponds to one electron at z � 2Z with momentum p � P while the second electronremains close to z � 0, p � 0. During the second half cycle, the field is negative andaccelerates the electrons into the direction of positive coordinates. Part of the single-ionization wave packet then returns to the core. The phase-space density crosses the


vertical line Z � 0 from left to right when the electron recollides with the core. Then,a substructure in the single-ionization wave packet appears as can be seen in the regionZ � 0 of the snapshot taken at t � T . Additionally, a new single-ionization wave packet isejected from the core region. For Z � 20 a.u., the overlap of the two wave packets givesrise to an interference pattern. However, the wave packets are still largely separated.This demonstrates an advantage of the phase-space analysis as compared to an analysisin configuration space: Wave packets ejected at different times are separated so that theirindividual time evolution can be studied. In configuration space (Imagine for examplethe projection of the phase-space distribution onto the Z axis), they would overlap in theirmost interesting parts.

Between t � T and t � 3T�2 we find that tails evolve out of the scattered portion of

the first single-ionization wave packet. These tails are accelerated more strongly towardsnegative momenta than the broad remainder of the wave packet. By close inspection, onefinds that the acceleration is in fact about twice as large, indicating that two electronsare now freely accelerated by the field. At t � 3T

�2 a double-ionization wave packet

is clearly visible at about�Z � P � �

� � 80 � � 7 5 � a.u. It is essentially isolated from therest. In the discussion of the recoil-ion momenta we will see that the value P � � 7 5 a.u.relates to a recoil momentum of 1.7 a.u., which is in good agreement with experiment. Inthe animation “movie1.mov”, the double-ionization regions are highlighted for the timest � 5T

�4 and t � 3T

�2.

The following conclusions can be drawn from the phase-space analysis:

� Double-ionization is negligible at times before the first recollision.

� The recollision of an electron with the core is clearly observed.

� The double-ionization density evolves out of the scattered parts of the single-ion-ization density.

2.3.5 Comparison with the Hartree-Fock approach

The exact numerical treatment of the He atom in long-wavelength lasers will presumablybe achieved in the near future [21]. However, this will not be feasible for heavier atomssince the number of grid points grows exponentially with the number of electrons. Upto now there is no reliable standard technique that one could possibly apply to calculatethe response of an arbitrary atom to strong laser fields. To devise such an approximatemethod it is essential to know which ingredients are necessary and which are not. Tobe more specific, the question is: To what level must the electron-electron interaction beincluded in order to reproduce the rescattering effects described before.


The single-active electron (SAE) approach may be viewed as the one with the “small-est” degree of electron-electron interaction. Here, only one valence electron is allowed tomove while all other electrons are frozen. The valence electron then feels a static potentialin which the nuclear Coulomb potential is partially screened. The SAE single-ionizationyields are in good agreement with experiment. However, the double-ionization yields –calculated by applying the SAE approximation first to the neutral atom and then to thesingly charged ion – are several orders of magnitude too small, see Sec. 2.1.

In the time-dependent Hartree-Fock (TDHF) method (see, e.g., [55]), the many-elec-tron wave function is approximated by a Slater determinant. In the case of two electrons,the TDHF approach is equivalent to the exchange-only density-functional approach [26].2

In the 1D model of He, the TDHF wave function reads

ΨHF � z1 � z2 � t � � ϕ�z1 � t � ϕ � z2 � t �� (2.17)

The time evolution of the single-particle orbital ϕ is governed by the one-electron Schrodin-ger equation

i∂∂t

ϕ�z � t � �

� � 12µe

∂2

∂z2�

Vs�z � t �� ϕ

�z � t �� (2.18)

where the single-particle potential Vs is given by

Vs�z � t � � � 2

�z2 � 1

� � ϕ�z

� � t � 2� �

z � z� � 2 � 1

dz� �

E�t � z (2.19)

This is a mean-field potential, i.e., each electron sees a potential that is averaged over allpossible positions of the other electron. In this respect, TDHF resembles the SAE method.However, both electrons are now allowed to move, in contrast to the SAE approach. Wecan therefore expect that the results for double ionization are improved. Indeed, the He2

�

yield calculated for the 1D model [44] is much larger than in the sequential SAE model.However, the knee structure is not reproduced. The TDHF double-ionization yield un-derestimates the exact yield at low intensities, and overestimates at high intensities. Thecrossing point is at about 9 � 1014 W/cm2.

Extensions of the TDHF method, based on the use of non-orthonormal sets of or-bitals, have been suggested. See the so-called time-dependent unrestricted Hartree-Focktheory [56, 57], and the time-dependent extended Hartree-Fock theory [58]. While theseapproaches generally provide some improvement, the results are still unsatisfactory.

We are going to investigate the TDHF center-of-mass phase-space dynamics in orderto find out why the Hartree-Fock approach is not capable of describing non-sequential

2The philosophy of DFT differs from HF in the sense that the Slater determinant is not interpreted as anapproximation for the real many-particle wave function.


double ionization. First, we consider the more general case of a symmetrized/antisym-metrized product of two single-particle wave functions ϕ1 and ϕ2 (not normalized, forsimplicity),

Ψs�a � z1 � z2 � � ϕ1

�z1 � ϕ2

�z2 � � ϕ1

�z2 � ϕ2

�z1 �� (2.20)

We insert into Eq. (2.15) and calculate w�Z � P � , Eq. (2.16). The result is:

ws�a � Z � P � � ��

�ϕ �2 � Z

� y2� ϕ1 � Z � y

2� eiPy

�2 dy ��

2

� wϕ1

�Z � P

2� wϕ2

�Z � P

2� � (2.21)

where wϕ1 and wϕ2 are the one-particle Wigner functions for ϕ1 and ϕ2. The second termarises from the symmetrization of the wave function and may be viewed as an exchangeterm. In the case of the Hartree-Fock wave function (2.17), we obtain the simpler form

wHF � Z � P � �12

�w

�1 � � Z � P

2� � 2 � (2.22)

where w�1 � is the one-particle Wigner function as defined in Eq. (2.14). Immediately,

we recognize that wHF � Z � P � � 0 always, contrary to a general Wigner function. Thissimplifies the interpretation as a probability distribution.

Consider now the case that ϕ�x � is a superposition of two wave packets, one centered

at x � � a and one centered at x ��

a:

ϕ�x � � ϕ � a

�x � � ϕa

�x �� (2.23)

The Wigner function w�1 � then contains maxima at x � � a. Therefore, by Eq. (2.22), the

two-particle quantity wHF � Z � P � contains maxima at Z � � a as well, corresponding to thecase where both electrons are localized at � a or where both electrons are localized at

�a.

However, the HF wave function

ΨHF � z1 � z2 � � ϕa�z1 � ϕa

�z2 � � ϕa

�z1 � ϕ � a

�z2 �� ϕ � a

�z1 � ϕa

�z2 � � ϕ � a

�z1 � ϕ � a

�z2 � (2.24)

also includes the possibility that the electrons are localized on opposite sides, correspond-ing to a phase-space density around Z � 0. By Eq. (2.22), the one-particle distributionthen contains density at x � 0. Thinking classically, this should not be the case. The solu-tion to the paradox is that the one-particle distribution exhibits rapid oscillations betweenpositive and negative values in the region at x � 0. These oscillations would average to


-3 0x

-20

2p

-4

0

4

8

w(1) (x,p)

-3 0 3 Z-2

02

P

-4

0

4

8

wHF (Z,P)

Figure 2.8: Left: One-particle Wigner function for a superposition of two wave packets,see Eq. (2.23). Right: Center-of-mass distribution, Eq. (2.22), for the Hartree-Fock wavefunction Eq. (2.17) using the superposition Eq. (2.23). In both cases, γ � 1 and a � 3.

zero if the Wigner function was convoluted with some smoothing window function. Byapplying the square in Eq. (2.22), we obtain a positive function which cannot average tozero.

We demonstrate this peculiar behavior of the HF center-of-mass Wigner function foran explicit example. We employ

ϕ �a�x � � e

� γ�x � a � 2

(2.25)

in Eq. (2.23) and find

w�1 � � x � p � �

2πγ

�e

� 2γ�x � a � 2 �

e� 2γ

�x

�a � 2 �

2e� 2γx2

cos�2pa � � e

� p2 � �2γ � (2.26)

Near x � 0, the term involving cos�2pa � gives rise to strong oscillations when p is varied.

This is evident in the left panel of Fig. 2.8, where w�1 � � x � p � is plotted for γ � 1 and

a � 3. The HF center-of-mass distribution wHF � Z � P � is shown on the right-hand side. Theoscillations near Z � 0 are the most dominant structure in this distribution. Integrationalong the P axis would lead to large values for Z � 0, while integration of w

�1 � � x � 0 � p �

along the p axis gives small values.We compute now the time-dependent center-of-mass phase-space distribution in the

TDHF approach. The same laser parameters as earlier in this section are employed. Onthe CD, these results can be found in the animation “movie2.mov”. Fig. 2.9 shows severalsnapshots, using the same grey scale as in Fig. 2.7. From the beginning, we recognizethe oscillatory substructure that has been explained above. Apart from this effect, the


Figure 2.9: Same as Fig. 2.7, but in the Hartree-Fock approximation.

observed wave packets are similar to the ones of the exact calculation. The first differencethat one observes is the long and thin double-ionization wave packet at t � T

�2. This

uncorrelated double ionization is absent in Fig. 2.7. Counterintuitively, at this time thereis more double ionization in TDHF than in the full propagation. However, this finding isconsistent with Ref. [44]. There, it was found that in the 1D model the TDHF double-ionization yield at 1015 W/cm2 is 1.4 times larger than the exact yield. If we chose anintensity below the “knee,” the TDHF yield would be orders of magnitude smaller thanthe exact yield. The most important difference between TDHF and exact propagationappears around t � T . The single-ionization wave packet does not suffer any noticeablechange when it crosses the vertical line Z � 0. This means that rescattering does notaffect the wave packet. Consistently, no tails corresponding to recollision-induced doubleionization appear between t � T and t � 3T

�2. Yet, there is some density arising from

uncorrelated double ionization, similar to what was already found in the first half cycle.

The conclusion from these results is that double ionization in the TDHF approach isof a very different nature than the recollision-induced double ionization that we found in

2.4 Photoelectron and recoil-ion spectra 41

the exact calculation. Hence, the mean-field level of electron-electron interaction as it isincluded in TDHF is not adequate for the description of intense-field double ionization.

2.4 Photoelectron and recoil-ion spectra

So far, we have always employed absorbing boundary conditions, and the informationabout the outgoing electronic wave packets has been discarded. Within this approach, it isnot possible to calculate the energy or momentum spectrum of the ejected electrons. Also,the momentum distribution of the singly and doubly charged recoil ions is not accessible.

In experiment, total electron spectra were routinely determined. Because multiple-ionization events are outnumbered by single-ionization events, one was essentially ob-serving electrons from single ionization.3 Recently, however, double-ionization distribu-tions have been measured by several groups. The first experiments presented the mea-surement of He2

�momenta [50] and Ne2

�momenta [51]. Later, coincidence techniques

were used to determine the correlated two-electron momentum distribution [52]. Energyspectra for electrons coincident with single ionization or double ionization have also beenpublished [61].

The most important experimental observation was the double-hump structure in therecoil-ion spectra: The momentum component p2

�z along the polarization axis shows

two broad maxima (one at positive p2�

z , one at negative p2�

z ), and a shallow minimumat zero. Let us briefly show why this finding is not compatible with the shake-off andthe collective-tunneling models. Assume that an ion with charge q is created at t � t0

with initial momentum pion�t0 � � 0 and initial position zion

�t0 � � 0. If the electric field is

monochromatic,

E�t � � E0 sin

�ω t � � (2.27)

then, by integration of the classical equation of motion, the ion momentum is

pion�t � �

qE0

ω�cos

�ω t0 � � cos

�ω t �� (2.28)

The drift momentum is

pdrift�

qE0

ωcos

�ω t0 �� (2.29)

When the electric field is turned off, the final ion momentum will approximately equalthis value if the switch-off process is slow enough (“adiabatic”). We demonstrate this by

3This is true for multiphoton ionization in intense lasers. For direct one-photon double ionization,coincidence measurements of both electrons have been performed some years ago [59, 60].


inspection of the example

E�t � � E0 e

� γ�t � t0 � sin

�ω t � (2.30)

In this case,

pion�t � �

qE0�γ2 � ω2 � �

γsin�ω t0 � � ω

�cos t0 � � � qE0 e

� γ�t � t0 ��

γ2 � ω2 � �γsin

�ω t � � ωcos

�ω t � � �

(2.31)

and the final ion momentum is

limt � ∞

pion�t � �

qE0

γ2 � ω2

�γsin

�ω t0 � � ωcos

�ω t0 � � (2.32)

In the adiabatic limit γ � ω, this asymptotic value equals the drift momentum Eq. (2.29).As another example we consider a pulse shape with linear ramps:

E�t � �

��

�E0

tt1

sin�ω t �� 0 � t � t1

E0 sin�ω t �� t1

� t � t2

E0 � 1 � t � t2t3 � t2

� sin�ω t �� t2

� t � t3

0 � t � t3

(2.33)

If the time of ionization t0 is within the interval of constant amplitude, t1� t0

� t2, wefind that the ion momentum for t2 � t � t3 is

pion�t � �

qE0

ω� t � t2

t3 � t2� 1 � cos

�ω t � � qE0

ωcos

�ω t0 �� qE0

ω2�t3 � t2 � �

sin�ω t2 � � sin

�ω t � � (2.34)

The final ion momentum is obtained by inserting the time t3:

pion�t3 � �

qE0

ωcos

�ω t0 � � qE0

ω2�t3 � t2 � �


�ω t3 � � (2.35)

As before, the final momentum equals the drift momentum Eq. (2.29) for sufficiently longswitch-off times t3 � t2. Numerical calculations are often restricted to short pulse dura-tions, so we cannot employ pulses in which the field is turned off adiabatically. However,Eq. (2.35) shows that the same effect is achieved by choosing


�ω t3 �� (2.36)


We will take advantage of this behavior in order to eliminate unrealistic pulse-shape ef-fects on the final ion momentum.

The probability for single ionization is largest at times of maximum electric fieldstrength, i.e., for t � π

�2 � t � 3π

�2 � Insertion into Eq. (2.29) shows that the ions re-

ceive zero final momentum in that case. This agrees with the experimental He�

momen-tum distributions, which consist of a single peak at zero momentum [50, 51]. If doubleionization was caused by shake-off or collective tunneling, the same argument would ap-ply because both ionization steps would occur almost simultaneously at maximum fieldstrength. In the rescattering model, up to one cycle may pass between the first and secondionization step, see Sec. 1.3. At the recollision time, the electric field need not be at itsmaximum, so ions with momenta different from zero can be ejected. Details of the kine-matics are given in Refs. [51, 62], showing that the experimental recoil-ion distributionsare compatible with the rescattering model.

To continue our investigation of double ionization, it seems advantageous to extendthe theoretical description in such a way that photoelectron and recoil-ion spectra can becalculated. In the following, we first describe our extended model. Then, the model willbe employed for a variety of laser parameters.

2.4.1 Extension of the model

We subdivide the full configuration space into several parts as shown in Fig. 2.10. The in-

��

��

��

��

��

��

��

��

��

��

��

� � � � � � � � �

��

��

��

2��

��

1

400

a.u

.

z

z

CBA

Figure 2.10: Partitioning of the two-electron configuration space. Regions A, B, and Ccorrespond to the neutral atom, single ionization, and double ionization, respectively.


ner region A corresponds to the case where both electrons are less than 200 a.u. away fromthe nucleus. This is exactly the region that has been used for the calculations described inthe previous section. Additionally, we include now the region B (one electron farther than200 a.u. from the nucleus) and the region C (both electrons farther than 200 a.u. from thenucleus).

Without the mass-polarization terms, the two-electron Hamiltonian reads [cf. Eq. (2.3)]

Hint�

p21

2µe

� p22

2µe� Zn�

z21�

1� Zn�

z22�

1� 1� �z1 � z2 � 2 � 1

�qe�z1�

z2 � E � t �� (2.37)

We apply a gauge transformation Ψ�z1 � z2 � t � � eiΛ

�z1 � z2 � t � Ψ

� �z1 � z2 � t � with

Λ�z1 � z2 � t � � qe

�z1�

z2 � A � t � � q2e

µe

t�0

A�t

� � 2 dt� � (2.38)

where

A�t � � � t�

0

E�t

� � dt� (2.39)

By substitution into the time-dependent Schrodinger equation, we find that Ψ� �

z1 � z2 � t �obeys the transformed equation

i∂∂t

Ψ� �

z1 � z2 � t � � Hvelint Ψ

� �z1 � z2 � t � (2.40)

with

Hvelint

�

p21

2µe

� p22

2µe

� qe

µe

�p1�

p2 � A � t �� Zn�

z21�

1� Zn�

z22�

1

� 1� �z1 � z2 � 2 � 1

(2.41)

This is the velocity-gauge version of the Hamiltonian: The interaction with the field entersin the form of terms proportional to the particle momenta. Usually, a term proportionalto A

�t � 2 appears in the Hamiltonian. Here, it has been transformed away by introducing

the second term in the gauge function Λ, Eq. (2.38). To strictly identify the quantity


A�t � with the z-component of a vector potential, we would need to multiply it with an

additional prefactor c (using atomic units).Now, in the regions B and C, we neglect the interaction between the nucleus and the

far-away electron(s). We also neglect the electron-electron interaction. This is plausiblefor B where one electron remains near the origin. As for region C, we can assume thatthe electron-electron repulsion has driven the electrons far enough away from each otherbefore they enter C. Within these approximations the electrons are decoupled in B and C.

For example, if z1 � 200 a.u., the Hamiltonian for electron 1 is just

Hvel1

�p2

1

2µe

� qe

µep1 A

�t �� (2.42)

Then, the evolution operator exp� � iH dt � does not depend on z1, only on p1. Using the

split-operator method, we need not perform the Fourier transformations with respect toz1. The propagation scheme for one time step can be illustrated in this way (we drop theprime that denotes the gauge-transformed wave function):

A

Ψ�z1 � z2 � t �

�

e� iV dt

�2 Ψ

�z1 � z2 � t �

�FT

�

Ψ�p1 � p2 � t �

�

e� iTdt Ψ

�p1 � p2 � t �

�FT

�

¯Ψ�z1 � z2 � t �

�

e� iV dt

�2 ¯Ψ

�z1 � z2 � t �

Ψ�z1 � z2 � t � dt �

B � part�

z1 � 200 a u �

Ψ�p1 � z2 � t �

�

e� iV

�z2 � dt

�2 Ψ

�p1 � z2 � t �

�FT�variable 2 only � �

Ψ�p1 � p2 � t �

�

e� iT dt Ψ

�p1 � p2 � t �

�FT�variable 2 only � �

¯Ψ�p1 � z2 � t �

�

e� iV

�z2 � dt

�2 ¯Ψ

�p1 � z2 � t �

Ψ�p1 � z2 � t � dt �

C

Ψ�p1 � p2 � t �

�

e� iTdtΨ

�p1 � p2 � t �

Ψ�p1 � p2 � t � dt �

Here, FT stands for Fourier transformation, and the part of B where z2 � 200 a.u. is

treated in analogy to what is written down for the part where z1 � 200 a.u.

Finally, we have to specify how the probability density is transferred from one region


to another. We assume that electrons can only move from A to B or from B to C. Thismeans that the electrons cannot reenter the inner regions. We do not use an optical po-tential added to the external potential. Rather, the wave function is multiplied “by hand”with a mask function in regular time intervals, so that the area close to the boundary isdepleted. Instead of discarding the portion of the wave function which is absorbed at theboundary of region A, it is Fourier transformed and coherently added to the wave functionin region B. In the same way, the transfer from B to C is achieved. This procedure maybe applied in each time step. To save computing time, however, the transfer is done onlyevery 10 to 25 time steps. The absorption strength and the frequency of transfer must beadjusted for each set of parameters to avoid unphysical reflections at the boundaries.

The propagation is continued for a while after the electric field has been switched off.In this way we ensure that practically all density corresponding to ionization reaches partsB/C so that the calculated distributions are converged. At the end of the propagation, thespectra of single-ionization electrons are calculated from the wave function in region B,while double-ionization spectra are obtained from the wave function in region C.

With the knowledge of the electron spectra, it is straightforward to calculate the recoil-ion spectra. The reason is that the laser photons carry negligible momenta: One 780 nmphoton has a momentum of 0.00043 a.u. Therefore the total momentum of the system ispractically conserved. The recoil momentum p2

�of a doubly charged ion is then related

to the electron momenta by

p2�

� � � p1�

p2 � � � P (2.43)

We employ laser pulses with various intensities and wavelengths. The laser is turnedon and off using linear ramps. Around mid-pulse, the amplitude is held constant. At firstsight, this kind of pulse shape does not appear realistic. However, by setting the durationof the ramps to an integer multiple of one optical cycle, we can fulfill condition (2.36).Thereby, unrealistic pulse-shape effects are avoided as explained before.

2.4.2 Long-wavelength spectra

We first study the wavelength 780 nm since most experiments have been done for wave-lengths around this value. The duration of the pulses is 8 optical cycles, which equals20.8 fs. The leading linear ramp lasts for two cycles, and so does the falling ramp. Thelaser intensity is varied from 1 � 1014 W/cm2 to 2 � 1015 W/cm2. Figure 2.11 shows theresulting double-ionization two-electron momentum distributions. The electron-electronrepulsion causes the electrons to avoid each other, leading to the tendency that p1 differsfrom p2. For this reason, the density along the diagonal p1

� p2 is generally small. Nev-ertheless, for intensities between 6.6 � 1014 W/cm2 and 1.3 � 1015 W/cm2, the major partof the density is in the regions where both momenta have the same sign: The electrons


Figure 2.11: Two-electron momentum distributions for double ionization at (a)1 � 1014 W/cm2, (b) 3 � 1014 W/cm2, (c) 6.6 � 1014 W/cm2, (d) 1 � 1015 W/cm2, (e)1.3 � 1015 W/cm2, and (f) 2 � 1015 W/cm2. Grey scales from 0 to (a) 1.5 � 10 � 6, (b)4 � 10

� 3, (c)–(d) 1.5 � 10� 2, (e)–(f) 2.5 � 10

� 2.

are preferentially ejected into the same direction. This is in agreement with experiment[52]. Yet, in the measured distribution of Ref. [52], no suppression along the diagonal isobserved. Note that this is probably because the experimental distribution is integratedover the coordinates perpendicular to the polarization direction. Our calculations shouldrather be compared to a cut of the full-dimensional momentum distributions along thepolarization axis. Such spectra have not yet been published although the experimentaltechniques used in Ref. [52] do in principle allow such measurements. At the intensity


−5.0 −2.5 0 2.5 5.0

p2+

(a.u.)

0.000.050.100.000.020.040.00

0.02

−5.0 −2.5 0 2.5 5.0

p2+

(a.u.)

0.00

0.02

0.000

0.005

diffe

rent

ial p

roba

bilit

y 0

2x10−6

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2.12: Calculated momentum distributions of He2�

recoil ions for intensities as inFig. 2.11.

2 � 1015 W/cm2, the field is sufficiently strong to make correlation effects irrelevant: Thedensity along the diagonal is no longer suppressed, and the preference for ejection in thesame direction has disappeared. Furthermore, vertical and horizontal lines indicate thatthe distribution resembles a product of two single-particle momentum distributions.

By integration of the two-electron spectra we arrive at the He2�

momentum distribu-tions in Fig. 2.12. The ion momenta extend to about � 2 a.u. for the lowest intensity andup to � 5 a.u. for the highest intensity. In the intermediate intensity regime, the distri-butions exhibit two maxima, located symmetrically with respect to the origin. At zero,there is a minimum. This general structure is predicted by the rescattering model [51].The non-zero positions where the maxima are located move towards smaller momenta asthe intensity is decreased. Additionally, a peak at zero appears, most clearly visible atthe lowest intensity. In the high-intensity regime, where sequential double ionization isdominant, a single broad maximum is observed. Here, the electrons are most likely andindependently of each other ejected at times of maximum electric field, giving rise to zerofinal ion momentum.

In Fig. 2.13, experimental spectra4 are shown for comparison. The experimentally de-termined intensities have an uncertainty of up to 50%, and it seems that we must compareFig. 2.12(d) to Fig. 2.13(b). Here, we find excellent agreement, not only in the “camel”

4The data has been provided by R. Dorner. It is identical to the data shown in Fig. 3 of Ref. [50].


−5.0 −2.5 0 2.5 5.0

pz2+

(a.u.)

050

100

yiel

d (a

rb. u

.)−5.0 −2.5 0 2.5 5.0

pz2+

(a.u.)

0100200(a) (b)

Figure 2.13: Experimental distribution of He2�

momentum components along the polar-ization axis for the intensities (a) 2.9 � 1014 W/cm2 and (b) 6.6 � 1014 W/cm2. Cf. Fig. 3of Ref. [50].

structure but also in the total width of the base and in the shallow mininum at zero mo-mentum. For lower intensity, theory and experiment remain consistent: The double-humpstructure disappears and the width of the spectrum decreases.

The phase-space analysis in Sec. 2.3 yielded an electronic center-of-mass momentumof about -7.5 a.u. at t � 3T

�2. This corresponds to p2

� �3T�2 � � 7 5 a.u. At this time,

the electric field is zero and cos�ω t � � � 1. Using Eqs. (2.28) and (2.29), we find that the

final drift momentum of the ion is then 1.7 a.u. Figures 2.12(d) and 2.13(b) agree wellwith this value.

The large probability for non-zero recoil-ion momenta at intermediate laser inten-sities is another strong indication in favor of the rescattering model. The structures inthe low-intensity spectra, panels (a) and (b) of Fig. 2.12, have not yet been confirmedexperimentally. S-matrix calculations for zero-range potentials, however, yield similarstructures [63] if – not quite consistently with the form of the potential – the energy levelsof the real He atom and He

�ion are employed. It seems that these structures are due to

rescattering processes where the second electron is promoted into an excited state fromwhich it tunnels out afterwards [63].

Finally, we show in Fig. 2.14 an example of an electron-kinetic-energy spectrum forsingle ionization. The energy is given in units of the ponderomotive potential Up definedin Eq. (1.19). The ATI peaks are not clearly resolved with the chosen grid parameters.However, it is obvious that the spectrum consists of two plateaus. The first extends upto 2.5 Up, the second ranges up to 10 Up. This is in reasonable agreement with expectedcut-off laws. Classically, the maximum drift velocity vmax for direct ionization is E0

�ω

(see Sec. 1.3), so the maximum expected energy is v2max�2 � 2Up. The second cut-off is

understood within a detailed analysis of the classical electron trajectories: When an elec-tron is scattered from the core, the maximum final energy occurs for elastic backscatteringby 180

�

. This energy is numerically determined to be 10.007 Up [64, 65]. Apparently, the


0 2 4 6 8 10 12 14Ekin / Up

10−12

10−10

10−8

10−6

10−4

10−2

100

diffe

rent

ial p

roba

bilit

y

Figure 2.14: Spectrum of electron kinetic energies for single ionization at the intensity6.6 � 1014 W/cm2. The dashed lines indicate the expected cut-offs for direct electrons andfor scattered electrons.

second plateau comprises the scattered electrons while the first comprises direct electrons.The value of 10 Up has been confirmed in one-electron quantum-mechanical calculations[65]. It is interesting to note that the cut-off law apparently remains unchanged when thedynamics of the second electron is included.

2.4.3 Short wavelengths: Above-threshold double ionization

Very close examination of panels (a) and (b) in Fig. 2.11 reveals densely spaced concentriccircles with their centers at the origin. These structures are not artifacts as we demonstratein the following by employing shorter wavelengths. On the left-hand side of Fig. 2.15, thetwo-electron momentum distributions are displayed for 250 nm and 400 nm wavelength.As indicated, two different intensities have been used. The plots differ a lot from thelong-wavelength spectra. The most striking feature is the clear appearance of the ringstructure. It is most pronounced for short wavelength and low intensity.

Neglecting the kinetic energy of the nucleus and approximating the electronic reducedmass (µe

� 0 99986) by one, the total kinetic energy is given by

Etot�

p21

2� p2

2

2 (2.44)

Therefore, the circles are lines of constant energy: Obviously, the total energy assumes


Figure 2.15: Left: Two-electron momentum distributions for double ionization by laserpulses with wavelengths and intensities as indicated in the panels on the right-hand side.Right: Total-kinetic-energy spectra for the various wavelengths and intensities.

only quantized values. The energy spectra on the right-hand side of Fig. 2.15 are ob-tained by integration of the momentum distribution. The simplest way to perform theintegration numerically is to calculate the total energy for each grid point in momentumspace and then count the number of entries in adequately chosen energy intervals. We findthat the energy spectra consist of well-defined peaks. Each peak corresponds to a circlein the two-electron momentum distribution. The separation between two adjacent max-ima is constant throughout the energy spectrum and equals the photon energy hω. Here,the well-known one-electron effect above-threshold ionization has found its two-electronanalogue: above-threshold double ionization (ATDI).

Figure 2.16 shows the energy spectra for 250 nm and three different intensities. One


0 0.2 0.4 0.6 0.8 1 1.2Etot (a.u.)

02468

100

1

dif

fere

nti

al p

rob

abili

ty0.0

0.1

0.2 I=2x1014

W/cm2

I=3x1014

W/cm2

I=4x1014

W/cm2

13

1415

16 17

14 15

16

17 18

14

15 16

17

18 19

Figure 2.16: Total-kinetic-energy spectra for double ionization in 250nm pulses at variousintensities. The arrows point to the peak energies expected from Eq. (2.45) for the givennumber of absorbed photons.

recognizes that the peaks shift towards lower energies with increasing intensity. Thisis due to the fact that the applied field raises the energies of the continuum states andthereby increases the ionization barrier. The energy increase for a one-electron continuumstate equals the ponderomotive potential Up while the shift of the ground-state can beneglected in a first approximation [66]. The double-ionization threshold is thus increasedby 2Up. Then, analogous to ordinary short-pulse ATI, the position of the ATDI peaks isapproximately

En� nhω � � I

�2 �

p�

2Up � � (2.45)

where n is the number of absorbed photons and I�2 �

p is the double-ionization potential.The energies predicted by this equation are indicated by arrows in Fig. 2.16. They give agood estimate for the peak energies. The number of absorbed photons is specified for eacharrow. While the simple Eq. (2.45) is useful for quick predictions, we can – at least in ourmodel system – go one step further and calculate the small ac Stark shift of the groundstate. For this purpose we employ Floquet theory [67] for our model atom subject to amonochromatic laser field: We calculate the Floquet spectrum from the time-dependent


Table 2.1: For the intensities and wavelength of Fig. 2.16, the table lists the ac Stark shifts ∆Stark

of the ground state. Also given are the differences ∆num between the numerical

ATDI peak positions and the values calculated from Eq. (2.45).

Intensity (W/cm2) 2 � 1014 3 � 1014 4 � 1014 ∆Stark

(a.u.) 0.0096 0.014 0.025

∆num (a.u.) 0.009 0.015 0.020

wave function by taking the modulus squared of the Fourier transformed autocorrelationfunction [37]. From these spectra, we extract the ac Stark shifts

∆Stark

given in Table 2.1.

In the same table, we list the differences ∆num

between the numerical ATDI energies and

the values expected from Eq. (2.45). For each intensity, ∆num

is averaged over the 5

to 6 peaks visible in the ATDI spectrum. The small discrepancy between the numericalresults and Eq. (2.45) agrees indeed well with the Stark shift. The remaining difference isextremely small and is comparable to the numerical uncertainties.

−3 −2 −1 0 1 2 3p2+ (a.u.)

0.00

0.05

0.00

0.01

0.02

dif

fere

nti

al p

rob

abili

ty

0.0

0.1

0.2

0.00

0.05 λ=250 nmI=3x10

14 W/cm

2

λ=250 nm

I=3x1014

W/cm2

I=1015

W/cm2

I=1015

W/cm2

λ=400 nm

λ=400 nm

Figure 2.17: Recoil-ion momentum distributions of doubly charged ions for the laserparameters of Fig. 2.15.


We continue with the calculation of the recoil-ion momenta because these are mea-sured with much less effort than the correlated two-electron distributions. For the samelaser parameters as in Fig. 2.15, the recoil-ion spectra are shown in Fig. 2.17. Again, apronounced peak structure is found, at least in the case of small wavelength and intensity.However, only part of these peaks are due to the ring structure of Fig. 2.15 and the rela-tion is somewhat indirect. One verifies easily that in the outer regions of Fig. 2.17, i.e.,in the region of large absolute values of p2

�, the separation between two adjacent max-

ima corresponds to the photon energy hω if we calculate the energy as Etot� � p2

� � 2 �4.

The reason for this behavior is obvious if only one circle of Fig. 2.15 with energy Etot

is considered, as is schematically displayed in Fig. 2.18. The integration that leads fromthe two-electron momentum distribution (shown as the upper part of Fig. 2.18) to the dis-tribution of p2

�(shown as the lower part of Fig. 2.18) is along the lines where p1

�p2

is constant, i.e., along vertical lines in the figure. A single ring with energy Etot in the�p1 � p2 � plane gives rise to a distribution of p2

�with two peaks and sharp cut-offs at the

lower and upper ends because the value of the vertical line integral is largest when the in-tegration is along a tangent to the circle. In this case, we have p1

� p2� � �

Etot. Hence,by p2

�� p1

�p2 � , the peaks are located at � pmax

� � 2�

Etot, in agreement with theabove result.

In the region of recoil momenta around zero, additional maxima are visible. Thesecannot be explained by the above argument. They are due to structures within the ATDI

−(2Etot)1/2

0 (2Etot)1/2

p 2

p1

−p2+

/ 2

p12+p2

2=2Etot

Figure 2.18: The appearance of peaks in the recoil-ion momentum distribution is illus-trated. See text for details.


rings, mainly within the innermost ring.ATDI has not yet been observed in experiment, neither in electron spectra nor in

recoil-ion spectra. With the widely used red laser pulses, it is hard to resolve the ATDIpeaks. However, since the experimental groups continue their efforts to study double ion-ization of atoms, we can expect that our predictions will be confirmed by measurementsin the near future. We mention that in a very recent 3D calculation [68], ATDI has beenpredicted for He under the influence of XUV lasers with wavelengths around 20 nm.

2.4.4 Reconstruction of electron spectra from recoil-ion spectra

In general, the total-kinetic-energy spectrum cannot be retrieved from the recoil-ion spec-trum because information is lost in the integration procedure. Under reasonable assump-tions, however, such a reconstruction should be possible. To that end, we introduce thefunction h

�E � φ � which is the distribution of the total kinetic energy E and the “angle”

φ which is defined by tanφ � p2�

p1. Varying φ from 0 to 2π means nothing more thangoing once around an ATDI ring in Fig. 2.15. Our main assumption is that we can writeh�E � φ � as a product of an energy distribution and an angular distribution:

h�E � φ � � hE

�E � hφ

�φ �� (2.46)

A minor restriction is the assumption of forward/backward symmetry:

hφ�φ � � hφ

� 3π2 � φ � (2.47)

Equation (2.47) means that the spectrum is invariant if the direction of the polarizationaxis is reversed. This symmetry is only broken for very short pulses where the phase ofthe field relative to the envelope becomes important. Inspection of the calculated spectrashows that the assumption of forward/backward symmetry is generally justified.

The Pauli principle guarantees that the distribution is exactly invariant under exchangeof p1 and p2. This implies

hφ�φ � � hφ � π

2 � φ � (2.48)

Equations (2.46)–(2.48) allow us to write the distribution M�P� p � of the total momentum

P � p1�

p2 and the relative momentum p � p2 � p1 as

M�P� p � �

12

h � E � P� p �� φ � P� p � ��

12

hE� 1

4� P2 � p2 � � hφ

�arctan

P�

pP � p

� (2.49)


without the need to distinguish several cases for the angular argument of h�E � φ � . The

prefactor 1�2 arises from the variable transformation

�P� p � � �

E � φ � . Inserting Eq. (2.49)into the recoil-ion momentum distribution

R � p2�

� �

� ∞�� ∞

M � � p2� � p � dp (2.50)

yields after the substitution E � 14

� �p2

� � 2 � p2 � :

R � p2� � �

� ∞��p2 � � 2

�4

hE�E � hφ

�arctan

p2� � 2 f � E � p2

� �p2

� �2 f�E � p2

� � dEf�E � p2

� � � (2.51)

where

f�E � u � �

�E � 1

4u2 (2.52)

Finally, we replace the upper limit of the integral by a finite value Emax. This is a harmlessapproximation since we can choose Emax arbitrarily large. Then, for given functionsR�p2

� � and hφ�φ � , Eq. (2.51) is a Volterra equation for hE

�E � [38].

The singularity in 1�

f�E � p2

� � complicates the solution of the integral equation. Wecalculate the value R

�u � δu � at the slightly shifted argument u � δu by splitting the integral

into two terms:

R�u � δu � � R1

�R2 (2.53)

with

R1�

u2 �4�

�u � δu � 2

�4

hE�E � hφ

�arctan

u � δu � 2 f�E � u � δu �

u � δu�

2 f�E � u � δu � � dE

f�E � u � δu � � (2.54)

R2�

Emax�u2

�4

hE�E � hφ

�arctan

u � δu � 2 f�E � u � δu �

u � δu�

2 f�E � u � δu � � dE

f�E � u � δu � (2.55)

The first integral is over a small interval and the integrand is strongly peaked when E isclose to the lower limit

�u � δu � 2 � 4 because of the singularity in 1

�f�E � u � . Therefore,

we approximate

R1 � hE� � u � δu � 2

4� hφ � π

4� u2 �

4��u � δu � 2

�4

dEf�E � u � δu �

� hE� � u � δu � 2

4� hφ � π

4� �

2uδu (2.56)


Assume now that hE�E � is known for E � E1 with an arbitrary value of E1, and set u �

�4E1. Then, R2 can be calculated from Eq. (2.55) and, since the function R

�u � is given,

R1 follows from Eq. (2.53). We can then solve Eq. (2.56) for hE� � u � δu � 2 � 4 � . Hence,

the numerical scheme for the determination of hE�E � starts with setting hE

�E � � 0 for

E � Emax and then builds the function downwards from Emax by repeating the procedureas outlined above.

The function hφ has to be provided as an input to the reconstruction scheme. Ap-parently, it is not easy to find such a function for long wavelengths because the angularstructure is rather complicated, see Fig. 2.11. The short-wavelength spectra, Fig. 2.15, onthe other hand, are not very anisotropic in the

�p1 � p2 � plane. Here, we are going to apply

the scheme with the simplest possible choice,

hφ�φ � �

12π (2.57)

The rapidly varying substructures within the ADTI rings should in principle enter theangular function. However, the effect of these substructures can be greatly reduced bysmoothing the resulting energy spectrum, which is much more efficient than trying to

0 0.5 1 1.5 2Etot (a.u.)

0.00

0.05

−0.1

0.0

0.1

0.2

hE (

a.u

.)

0.0

0.1

0.2(a)

(b)

(c)

Figure 2.19: Reconstruction of the total-kinetic-energy spectrum from the recoil-ion mo-mentum distribution for a 400nm pulse with intensity 3 � 1014 W/cm2: (a) exact spectrum,(b) spectrum from solution of Eq. (2.51), (c) smoothed version of spectrum (b).


reproduce the intricate φ-dependence.The result of the method is presented in Fig. 2.19 for the 400 nm pulse with inten-

sity 3 � 1014 W/cm2. The upper panel shows the exact calculated spectrum (same as inFig. 2.15). Panel (b) contains the spectrum reconstructed from the recoil-ion spectrumby solution of the integral equation. Convolution with a Gaussian of 0.04 a.u. width(full width at half maximum) gives the smoothed spectrum in panel (c). Considering thecrudeness of the approximation hφ

�φ � � 1

�2π, the agreement with the exact spectrum is

excellent. Not only the peak positions are correct, but also the envelope is well reproducedexcept for the range of energies lower than 0.5 a.u.

Another example is given in Fig. 2.20. For this particular set of parameters (wave-length 190 nm, intensity 6.6 � 1014 W/cm2), we find not only one, but two series of ATDIpeaks in the energy spectrum. In each of the series, we find the usual separation hω be-tween neighboring maxima. The appearance of more than one series is familiar fromordinary ATI and is due to laser-induced resonances: During the switch-on process, ex-cited states of the system are temporarily brought into multiphoton resonance with theground state and are thus populated by multiphoton absorption. Around mid-pulse, theseintermediate states are no longer in resonance with the ground state. Then, ionizationfrom these states gives rise to shifted peaks. In this more complex case, the reconstruc-

0 0.4 0.8 1.2Etot (a.u.)

0.00.20.40.6−1

0

1

0

hE (

a.u

.)

0

0.5

1

1.5

2(a)

(b)

(c)

Figure 2.20: Same as Fig. 2.19 for a 190 nm pulse with intensity 6.6 � 1014 W/cm2.


tion also works, but not as nicely as before. The smoothing is an obstacle if the two seriesare to be resolved. Either, one has to be content with the rapid oscillations in panel (b), orone has to relinquish some resolution.

The integral equation Eq. (2.51) has been derived for the 1D model, i.e., assumingthat the electrons are ejected strictly along the polarization axis. Since the reconstructionmethod is intended to facilitate the experimental observation of ATDI peaks without hav-ing to use coincidence techniques, we should discuss the relation between the spectra inone and three dimensions. In the 1D model, the ATDI peaks in the recoil-ion spectra cor-respond to the case where two electrons are ejected with identical moduli p1

� p2 into thesame direction. In 3D, the two electrons are ejected not exactly parallel, but at some anglewith respect to the polarization axis. 3D calculations of double ionization for wavelengthsaround 200 nm [69] indicate that the angle between the two electrons deviates not morethan π

�6 from zero or π. Furthermore, the recoil-ion momenta are strongly aligned with

the polarization axis [50, 51, 70]. For equal moduli p1 , p2 of the electron momenta (see

Fig. 2.21), it follows from the approximate momentum conservation p2�

� � � p1�

p2 �that the angle of electron emission θ is less than π

�12. The absolute value of the recoil-ion

momentum, p2

� � �� p1

cosθ � p2

cosθ �� 2

p1

cosθ

(2.58)

is then in the range

2 p1 �

�

p2

� �

� 0 97 � 2 p1 � (2.59)

This means that in 3D, the peak positions should differ by only 3% from the 1D peakpositions. Consequently, we expect that the proposed reconstruction procedure can beapplied to experimental spectra and will be helpful in the detection of the ATDI peaks.

θθ

2

p2+p

p

1

Figure 2.21: Connection between recoil-ion momentum p2�

and electron momenta inthree dimensions.

Chapter 3

Strong-field dynamics of smallmolecules

The physics of laser-driven molecules is a vast subject. Naturally, this chapter can onlycover a small section of the field. Nevertheless, we are going to describe a respectablevariety of effects, including dissociation, single and double ionization, and harmonic gen-eration. The calculations are performed without adopting the Born-Oppenheimer approx-imation. The BO framework will only be invoked for purposes of interpretation.

3.1 Pulse-width and isotope effects in dissociative ioniza-tion

Our conception of the term “dissociative ionization” is that a molecule is first ionizedby laser irradiation. The molecular ion then dissociates into charged and/or neutral frag-ments. From the kinetic energies of these fragments, we hope to learn about the strong-field dynamics of the molecule.

As motivation for the following work, we show in Fig. 3.1 four experimental kinetic-energy spectra for the H

�/D

�ions detected in the dissociative ionization of H2 and D2.1

In general, three maxima appear; the small low-energy peak at about 0.2 eV, however,is not observed for the short 28 fs pulses. The origin of the various maxima has beenexplained previously [13, 73, 74]:

� The small peak at 0.2 eV and the large peak at 0.5 eV result from the fragmentationinto a neutral atom and a proton/deuteron. The smaller energy is detected when the

1The spectra have been measured by C. Trump et al. at the Max-Born Institute in Berlin, see Refs.[71, 72].

60

3.1 Pulse-width and isotope effects in dissociative ionization 61

0 1 2 3 4 50,0

0,2

0,4

0,6

0,8

1,0

(d) 80 fs(c) 28 fs

(a) 28 fs (b) 80 fs

D+ Y

ield

[arb

. un

its]

H+ Y

ield

[arb

. un

its]

Kinetic Energy [eV]

0 1 2 3 4 50,0

0,2

0,4

0,6

0,8

1,0

0 1 2 3 4 50,0

0,2

0,4

0,6

0,8

1,0

0 1 2 3 4 50,0

0,2

0,4

0,6

0,8

1,0

Figure 3.1: Measured kinetic energies of H�

/D�

ions for dissociative ionization of H2/D2

in 790 nm pulses with intensity 5 � 1014 W/cm2. Pulse widths as indicated. Cf. Refs.[71, 72].

molecular ion dissociates by absorption of one photon. The larger energy is foundwhen two photons are absorbed. Actually, the two-photon dissociation proceeds byfirst absorbing three photons and then emitting one photon. This is shown schemat-ically in Fig. 3.2, where the Born-Oppenheimer picture is used for visualization.

� The broad maximum at higher energies is due to Coulomb explosion: The molecu-lar ion is ionized a second time, and fragments into two protons/deuterons.

There is an unexpected feature in Fig. 3.1. In the case of D�

ions, the 28 fs pulse producesa two-photon line at a slightly larger energy than the 80 fs pulse. Also, the line widthincreases for the shorter pulse duration. In the case of H

�ions, the two-photon peak does

not shift. See Tab. 3.1 for the detailed data. These pulse-width and isotope effects arenot expected because one naively thinks that the final energy should depend only on themolecular energy levels and on the photon energy. It seems that dynamic effects duringthe dissociation process have to be taken into account for a full understanding of the two-photon dissociation.

62 3 Strong-field dynamics of small molecules

R

V B

O (

R)

hω

hω

hω

hω

ungerade

gerade

R

V B

O (

R)

hω2

hω

ungerade

gerade

Figure 3.2: Mechanism of the effective two-photon absorption in H�2 /D

�2 . Left: Inter-

pretation in terms of the unperturbed BO potentials. Right: Interpretation in terms ofthe diabatic dressed potentials which are nothing but the BO potentials shifted by integermultiples of the photon energy hω.

Table 3.1: Position and width (FWHM) of the effective two–photon dissociation line inFig. 3.1.

Isotope pulse width Position FWHM

H�

28 fs 0.47 eV 230 meV

D�

28 fs 0.57 eV 310 meV

H�

80 fs 0.47 eV 160 meV

D�

80 fs 0.47 eV 140 meV

3.1.1 Two-step model

The realistic description of dissociative ionization is not simple because one has to takeinto account the dynamics of both the neutral molecule and the molecular ion. In prin-ciple, after single ionization of the molecule, the ejected electron should be treated sim-ilarly as in Sec. 2.4. In the case of the H2 molecule, however, the computational effortfor the splitting into inner and outer regions becomes too large. We introduce a simplifiedtwo-step model consisting of (i) ionization of the molecule and (ii) dissociation of themolecular ion where in stage (ii), the ejected electron is not included in the calculation.

The single-ionization probability of H2/D2 is calculated with the internuclear distance


fixed at the ground-state equilibrium distance R0. The assumption of fixed nuclei is adhoc, but will be shown to be justified later. The corresponding 1D Hamiltonian for theinternal degrees of freedom reads

HR0�

2

∑k � 1

� � 12

∂2

µe∂z2k� 1� �

zk � R0�2 � 2 � 1

� 1� �zk�

R0�2 � 2 � 1 �� 1� �

z1 � z2 � 2 � 1

�E0 f

�t � sin

�ω t � � z1

�z2 � � (3.1)

where µe� 2M

� �2M

�1 � is the electronic reduced mass, M is the mass of one nucleus.

We start the propagation from the ground state. The single-ionization probability is ob-tained in a way which is in principle equivalent to Eq. (2.12), but simpler to implement.We employ the absorbing boundary in the form of multiplication with a mask function.This function is split into a single-ionization and a double-ionization mask: One-electronejection is assumed when the coordinate of the second electron satisfies z � 7 a.u. Whenthe wave function is multiplied with the single-ionization mask, the norm decreases by acertain amount. This decrease is identified with an increase in the single-ionization prob-ability. We use a grid size of 164 � 164 a.u., a grid-point separation of 0.32 a.u., and 1500time steps per optical cycle.

After ionization, the dynamics of the 1D molecular ion is governed by the Hamiltonian

Hion� � 1

2µe

∂2

∂z2 � 1M

∂2

∂R2� 1

R � 1� �

z � R�2 � 2 � 1

� 1� �

z�

R�2 � 2 � 1�

qe z E0 f�t � sin

�ω t � (3.2)

with

qe�

2M�

22M

�1 (3.3)

Here, the two degrees of freedom are the internuclear distance R and the electron coordi-nate z measured relative to the nuclear center of mass. The prefactor qe differs from onebecause of the non-vanishing total charge of the molecular ion: The interaction with thefield in terms of the three laboratory-frame coordinates R1, R2, and z

� �R1�

R2 � � 2 reads

�z� �

R1�

R2 � � 2 � R1 � R2�E�t � �

�z � � R1

�R2 � � 2 �

E�t � �

�qez � RCMN

�E�t �� (3.4)

where we have used RCM�

�M�R1�

R2 � � z� �

R1�

R2 � � 2 � � �2M

�1 � . The crucial question

is, how to choose the initial state for the propagation of the ionic wave function. The


ground state of the ion is not an adequate choice because we have assumed that the initialionization occurs at the equilibrium distance of H2/D2. We use

Ψ�z � R � t � t0 � � χH2

�D2

0

�R � Φ0

�R � z �� (3.5)

where χH2�D2

0

�R � denotes the vibrational ground state of H2/D2, and Φ0

�R � z � is the elec-

tronic ground-state wave function of the molecular ion at fixed internuclear distance R(electronic Born-Oppenheimer wave function). With Eq. (3.5) we assume that the nuclearposition is unchanged by the ionization process, and that the evolution of the molecularion begins in the electronic ground state. Since the initial ionization can occur at anytime t0, we should perform the propagation of the ionic wave function for a large num-ber of starting times. In practice, we restrict ourselves to initial times when the electricfield passes through zero to prevent unphysical switch-on effects. The grid size is 77 a.u.for the nuclear coordinate and 164 a.u. for the electronic coordinate. The correspondingspatial step sizes are 0.075 a.u. and 0.32 a.u., and the time step is 0.1 a.u.

The experimental pulses are approximately Gaussian shaped. On the computer, weprefer to use a sin2 envelope function f

�t � because of its finite duration. Denoting the

experimental pulse width τp, we set the total duration of the sin2 pulse to�

8τp. In thisway, the full width at half maximum of the amplitude E0 f

�t � is the same as in experiment.

It is known [75, 76] that for large enough intensities, the two-photon dissociation doesnot happen in the center of the laser focus, but only in the spatial region with intensitiesin the range

�1 � 3 � � 1014 W/cm2. This is confirmed by our calculations where the two-

photon peak is mainly produced at intensities around 1.5 � 1014 W/cm2. The results givenbelow are for this value. The wavelength is 790 nm as in the experiment by Trump et al.[71, 72].

The grid is chosen large enough in the nuclear degree of freedom to ensure that no lossoccurs due to dissociation within the propagation time. We subdivide the wave functioninto a portion Ψb where the molecular ion is bound (R � 5 a.u.) and a portion Ψf wherethe molecular ion is fragmented (R � 5 a.u.) The nuclear-momentum distribution of thefragments is obtained by integrating the modulus squared of the momentum representa-tion Ψf

�PR � pz � over the electronic coordinate. Transformation to the energy scale yields

the fragment kinetic-energy spectrum. The total kinetic-energy release is shared equallyby the two fragments if the center of mass is assumed to rest. Thus, the ion kinetic energyis one half of the calculated energy release.

By its construction, this model does not include the Coulomb explosion which followsionization of the molecular ion. This is due to the fact that the information about the partsof the wave function corresponding to ionization is lost whenever absorbing boundaryconditions are employed. If one is interested in the fragment kinetic energies followingColoumb explosion, a splitting technique similar to the one described in Sec. 2.4 can beused [77].


3.1.2 Numerical results

Calculations have been performed for H2 and D2, and for three different pulse widths,τp

� 30, 50, and 80 fs. Let us first discuss the ionization of H2/D2. Since we have fixedthe internuclear distance at R � R0

� 2 2 a.u., there is no dependence on the isotope. Thesingle-ionization probability per half cycle is shown in Fig. 3.3(b) for τp

� 80 fs, in Fig.3.4(b) for τp

� 50 fs, and Fig. 3.5(b) for τp� 30 fs. We choose to integrate the ionization

rate over half cycles because the norm typically decreases in a stepwise fashion with onestep per half cycle. At each time t when the electric field is zero, a vertical bar indicatesthe ionization probability during the half cycle immediately before t. The most strikingfeature in these plots is that, for 80 fs and 50 fs pulses, ionization is most effective attimes before mid-pulse (denoted as t3 in the figures). A second smaller maximum ofionization is found on the falling edge of the pulse. For τp

� 30 fs, the time-dependentshape is similar, except that the first maximum is located at mid-pulse. These effectsare probably related to stabilization phenomena that were already found (although forhigher laser frequencies) in previous simulations (see [78, 79] and references therein):The comparatively large ionization rates on the leading and falling edges of the pulse canbe either due to a dynamical effect, induced by the temporal change of the intensity, or dueto “adiabatic stabilization” making the molecule more stable against ionization at largerintensities (independent of the pulse shape). Here, the term “adiabatic” indicates that themolecule is adiabatically transferred into the stabilized field-dressed state while the laseris switched on.

For various times of ionization (denoted as tn, n � 1 � 5 in the Figs. 3.3, 3.4, and3.5), we calculate the kinetic-energy spectra of the H

�/D

�ions. Panels (c) and (d) of

each figure show the results. To assess the contribution of a particular time of ionization,each kinetic-energy spectrum is weighted with the corresponding ionization probabilityof panels (b). Here and in the following, we assume that the probability that dissociationstarts at a time tn is given by the ionization probability during the half cycle immediatelybefore tn. Note that the energy scale in panels (c) and (d) is very much stretched ascompared to Fig. 3.1. The figures demonstrate that the kinetic-energy release dependsstrongly on the time of ionization. In general, ionization at a later time leads to a broaderpeak located at higher energy. A comparison of the two isotopes at equal ionization timesshows that D

�usually acquires a larger energy than H

�. For very late ionization times

[see t5 in Figs. 3.4(d) and 3.5(c), and t3 � t5 in Fig. 3.5(d)], additional contributions atlow energy appear. In all cases shown in Fig. 3.3 (80 fs pulses) and 3.4 (50 fs pulses), thefinal wave function has gerade symmetry, i.e., the same symmetry as the electronic groundstate of the molecular ion. This means that the fragmentation results from an effectivetwo-photon absorption (compare Fig. 3.2). For 30 fs pulse width, on the other hand, thelow-energy maxima belonging to t5 in Fig. 3.5(c) and t4, t5 in Fig. 3.5(d) are mainly due


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Ekin (eV)

0

0.02

0.04

dp/d

Eki

n (e

V−

1 )

60 80 100 120 140 t (fs)0

0.01

0.02

dp(H

2+)

per

h.c.

60 80 100 120 140 t (fs)

−0.05

0

0.05

00

E(t

) (a

.u.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.02

0.04

dp/d

Eki

n (e

V−

1 )

t1 t2 t3 t4 t5

t1

t2

t3

t4

t5

t1

t2 t3

t4

t5

(c) H+

(d) D+

(a)

(b)

Figure 3.3: Calculated fragmentation of H2/D2 in an 80 fs pulse with a peak intensity of1.5 � 1014 W/cm2. (a) Time-dependent laser field. (b) Single-ionization probability perhalf cycle. (c), (d) Kinetic-energy spectra of H

�/D

�ions for different times of ionization,

weighted by the corresponding ionization probabilities.

to one-photon absorption (ungerade symmetry of the final wave function). It seems thatthe low-energy ions (originating both from one-photon and two-photon absorption) areresponsible for the shoulder that is experimentally observed on the low-energy side of thetwo-photon peak for short pulses.

From the figures we conclude that a wide range of ionization times contributes, and


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Ekin (eV)

0

0.02

0.04

0.06

dp/d

Eki

n (e

V−

1 )

40 60 80 100 t (fs)0

0.01

0.02

dp(H

2+)

per

h.c.

40 60 80 100 t (fs)

−0.05

0

0.05

00

E(t

) (a

.u.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.02

0.04

dp/d

Eki

n (e

V−

1 )

t1 t2 t3 t4 t5

t1

t2t3

t4

t5

t1

t2

t3

t4t5

(c) H+

(d) D+

(a)

(b)

Figure 3.4: Same as Fig. 3.3, but for a 50 fs pulse.

that it is not sufficient to restrict the analysis to a particular one as it was done in Ref.[77]. Instead, one should add up all contributions, properly weighted with the ionizationprobabilities at each time. Since we do not have access to the wave function of the entiresystem (molecular ion plus free electron), we are restricted to an incoherent summation ofthe different contributions. This means that we add up the various distributions shown inpanels (c)/(d) instead of adding up the corresponding wave functions. This approximationwould be exact if the state φt1 of the ejected electron for ionization at time t1 was orthog-onal to state φt2 for any t1

�� t2. We identify φtn as the electronic wave packet produced


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.01

0.02

0.03

0.04

dp/d

Eki

n (e

V−

1 )

30 40 50 60 t (fs)0

0.005

0.01

0.015

0.02

dp(H

2+)

per

h.c.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Ekin (eV)

0

0.005

0.01

0.015

0.02

dp/d

Eki

n (e

V−

1 )30 40 50 60 t (fs)

−0.1

−0.05

0

0.05

0.1

E(t

) (a

.u.)

t1 t2 t3 t4 t5

t1

t2

t3

t4

t5

t5

t4

t3

t2t1

(a)

(b)

(c) H+

(d) D+

Figure 3.5: Same as Fig. 3.3, but for a 30 fs pulse.

by one half cycle immediately before tn. We thus expect that the condition for incoherentsummation is well fulfilled since the stepwise increase in the ionization probability indi-cates that the wave packets produced by different half cycles are well separated from eachother in space, and therefore are almost orthogonal.

For each pulse width, we sample 15 equally spaced ionization times around mid-pulse,with spacings of two optical cycles for τp

� 80 fs, one optical cycle for τp� 50 fs, and

a half cycle for τp� 30 fs. The summation leads to the total spectra plotted in Fig. 3.6.


0 0.2 0.4 0.6 0.8 1Ekin (eV)

0

0.4

0.8

1.2

H+, 30 fs

D+, 30 fs

0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2Y

ield

(ar

b. u

nits

)

H+, 50 fs

D+, 50 fs

0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2

H+, 80 fs

D+, 80 fs

(a)

(b)

(c)

Figure 3.6: Calculated kinetic-energy spectra of H�

/D�

ions for laser pulses with a peakintensity of 1.5 � 1014 W/cm2 and different pulse widths. The spectra are normalized suchthat the peaks beyond 0.2 eV have a maximum value of 1.

For 80 fs pulse width, the H�

and the D�

peak are located at the same energy. This isconsistent with the experimental result. Two competing phenomena are effective here:On one hand, D

�ions receive larger energy than H

�ions for equal ionization times; on

the other hand, early ionization times give larger yields of D�

ions than H�

ions [seepanels (c) and (d) of Fig. 3.3]. For the shorter 50 fs pulse, the maximum of the H

�

peak remains at the same energy while the D�

ions receive a larger energy. If we shortenthe pulse width to 30 fs, the H

�peak also shifts toward higher energies. Interestingly,

the low-energy D�

ions become so dominant that the peak maximum is now located at alower energy than the H

�peak. Note, however, that D

�ions with larger energy are also

present in the spectrum.The fact that the calculated peaks are all located at lower energies than in experiment

is due to the 1D model, in which the binding energies and the equilibrium distances differfrom the 3D values. The ground-state energy of 1D H

�2 , its binding energy, and its equi-

librium distance are � 0.776 a.u., 0.106 a.u., and 2.6 a.u., respectively. The 3D values are� 0.597 a.u., 0.097 a.u., and 2.0 a.u.The situation may be characterized in the following way: If we perform our calcu-

lations with 80 fs and 50 fs, then we can well reproduce the trends found in experiment


for 80 fs and 28 fs. In the calculation with 30 fs, these effects become so strong that thekinetic-energy spectra differ qualitatively from the experimental ones. An explanation forthis behavior will be given below.

It was found experimentally that the one-photon line exhibits a similar behavior [80]:The kinetic energy also becomes larger with decreasing pulse duration and finally mergeswith the two-photon peak, giving rise to a shoulder structure. This is compatible withthe present calculations. We saw that parts of the low-energy fragments stem from one-photon dissociation, at least in the case of the 30 fs pulse.

3.1.3 Interpretation

The experimental trends are reproduced by the calculation, but we wish to understandtwo points: Why do D

�ions tend to receive larger energies, and why does the energy

release depend on the time of ionization? Previous work has shown that the two-photondissociation essentially involves only the two lowest electronic states, see Fig. 3.2. Itshould be possible to give an interpretation in terms of potential curves, within the Born-Oppenheimer picture. In the presence of a strong field, however, the BO potentials aredistorted: We are dealing with “field-dressed” states. For a laser field with constant in-tensity, the dressed potential surfaces are obtained from Floquet theory [67] or from thediagonalization of the R-dependent field-quantized electron-plus-field Hamiltonian [81].The relation between the two methods is discussed in Refs. [82, 83]. Here, we approachthe problem in the spirit of the second method. We consider the two lowest electronicstates coupled to the field mode with frequency ω. At fixed internuclear distance R theenergies of the electronic states are Vg

�R � for the gerade ground state and Vu

�R � for the

ungerade excited state. The dipole matrix element is denoted Dug�R � . The entire system

(electron plus photon field) has an infinite number of energy levels. Without couplingbetween electron and field, the levels are � Vu � 2ω � Vg � 2ω � Vu � ω � Vg � ω � Vu � Vg � Vu

� ω � Vg� ω � Vu

�2ω � Vg

�2ω �

Now, we introduce the coupling. There is direct coupling only between states with differ-ent symmetry (gerade/ungerade), and only between states, the photon numbers of whichdiffer by one [81]. Therefore, the total set of states separates into two parts, which aredecoupled from each other. We consider only one of them (the other contains essentiallythe same states, just with their energies shifted by ω): � Vg � 2ω � Vu � ω � Vg � Vu

� ω � Vg�

2ω � For a large number of photons in the mode with frequency ω, the coupling matrix elementis [81]

c�R � � � � Dug

�R � qe E0 � � 2 (3.6)


The coupling to other photon modes corresponds to spontaneous processes, and is ne-glected here. Then, the matrix form of the Hamiltonian reads

H �

��

. . .Vg�R � � 2ω c

�R �

c�R � Vu

�R � � ωc�R � Vg

�R � c

�R �

c�R � Vu

�R � � ω c

�R �

c�R � Vg

�R � � 2ω

. . .

�� (3.7)

This matrix is real, symmetric, and tridiagonal. It can be easily diagonalized numerically,if a finite number of photon states is chosen. The eigenvalues of the matrix are the energiesof the field-dressed states. As a result of the truncation of the matrix, the spectrum ofeigenvalues is unphysical at the lower and upper end and gives energies that diverge forR ∞. The physical dressed-state energies are found in the central part of the spectrum.Here, the eigenvalues are grouped in sets of two: � V1

�R � � 2ω � V2

�R � � 2ω � V1

�R �� V2

�R �� V1

�R � � 2ω � V2

�R � � 2ω �

V1�R � and V2

�R � are the desired potential curves. The corresponding eigenstates are the

stationary states of the electron-plus-field system. If the nuclear motion is sufficientlyslow, then we can assume that the system adiabatically remains in the same dressed state,i.e., no transitions from one adiabatic surface to another occur. However, if the two adi-abatic curves exhibit an avoided crossing with a small gap, then the system usually mi-grates from one into the other adiabatic state. This is called diabatic motion. The avoidedcrossings occur at internuclear distances where the energy difference between the twounperturbed electronic levels equals on odd multiple of the photon energy.

We wish to construct a potential curve that describes the effective two-photon dissoci-ation which proceeds by first absorbing three photons and then emitting one photon. Wetherefore have to assume diabatic motion at the internuclear distance where five-photonabsorption is possible, i.e., we assume that no five-photon absorption takes place. Indeed,the gap between the two adiabatic curves is numerically found to be so small at the five-photon crossing that it can hardly be distinguished visually. Adiabatic motion is assumedat the three-photon and the one-photon crossings, i.e., the molecular ion follows the pathwhich is schematically shown on the right-hand side of Fig. 3.2.

The most important point is that we are dealing with femtosecond pulses, i.e., the pulseduration is comparable to the time scale for dissociation. Therefore, it is not sufficient tocalculate the adiabatic potentials for only one intensity. Rather, we compute the potentials


for each instant in time, using the field amplitude at that time. Consequently, the nucleimove in a time-dependent potential.

In order to arrive at a simple picture, we treat the nuclear motion classically using theinitial conditions

�R � pR � �

�R0 � 0 � , where R0 is the equilibrium distance of H2. We diag-

onalize a 40 � 40 matrix, i.e., 40 photon states are used. Figure 3.7 shows the result for apulse width of 80 fs. The time-dependent potential is plotted as a surface, starting at thetime t2 of Fig. 3.3 (two optical cycles before mid-pulse), and ending after the dissociationprocess is essentially finished. The energy scale is chosen such that the asymptotic poten-tial level for R ∞ equals the ground-state energy of the 1D hydrogen atom, � 18.2 eV( � 0.67 a.u.). Around mid-pulse, the potential exhibits two minima. The outer minimumat R � 6 a.u. gradually becomes flatter with increasing time and eventually disappears. (Itreappears near the end of the pulse, but without influencing the dissociation, which is al-ready completed by that time.) For two different times of ionization (t2 and t4 of Fig. 3.3),the classically calculated nuclear coordinate R

�t � of H

�2 and D

�2 is plotted onto the sur-

face. Along these paths, the nuclei experience a sequence of acceleration and decelerationprocesses.

First, compare the dynamics of the two isotopes. Since the dissociation of D�2 is

-0.05 0 0.05E (a.u.)

H2+

D2+

110

120

130

140

150

160

170

t(fs)

2 4 6 8 10 12R (a.u.)

-18.4

-18.2

-18

-17.8

-17.6

V (eV)

(a)(b)

Figure 3.7: (a) Time-dependent adiabatic potential for an 80 fs pulse with a peak inten-sity of 1.5 � 1014 W/cm2. Classical trajectories for H

�2 and D

�2 are plotted on top of the

potential for two different starting times. (b) Time-dependent laser field.

3.2 The strong-field dynamics of a model H2 molecule 73

slower, it needs a longer time to pass the outer minimum in the potential. Consequently,the slow-down on the slope between R � 6 a.u. and R � 9 a.u. is less effective, leading toa larger kinetic energy for D

�2 . Now, consider the case of two different ionization times.

Clearly, the potential experienced by the ion will be different in either case. As one cansee from the figure, the accelerating slope between R � 4 a.u. and R � 6 a.u. is similarlysteep in both cases, but the decelerating region R � 6–9 a.u. is flatter at later times. As aresult, later times of ionization produce faster fragments. Indeed, a close look at the figurereveals that two trajectories belonging to the same isotope but different ionization timeshave different asymptotic velocities.

We are now able to explain why there is good agreement of the 28 fs experiment withthe 50 fs calculation but less agreement with the 30 fs calculation. Since the fragments aregenerally slower in the 1D model than in the experiment, the complete dissociation pro-cess necessarily takes a longer time in the 1D model. We have learned from the classicaltrajectories that the fragments receive a shift towards larger kinetic energies if the disso-ciation is slow. Thus the effect should be stronger in the 1D model and should alreadyappear for intermediate pulse widths like 50 fs.

The two-state model can explain the isotope and pulse-width effects on the fragmentspectra. Yet, we cannot expect that the spectra can be accurately calculated with onlytwo electronic states, because the dissociation is accompanied by ionization which is notincluded in the two-state model. Ionization diminishes the two-photon dissociation yieldso that the correct weights for each time of ionization cannot be obtained from the two-state model. Furthermore, the energies of the dressed states are only approximate if thecoupling to higher electronic states is neglected.

3.2 The strong-field dynamics of a model H2 molecule

In this section we are going to investigate the dynamics of a linear H2 molecule withone nuclear degree of freedom (the internuclear separation) and two electronic degrees offreedom. So far, this system has only been studied in Kreibich’s thesis [28]. A similarmodel, but with clamped nuclei, has been employed in Ref. [84].

There are two motivations to study this system. First, the non-sequential double ion-ization of molecules currently attracts growing interest [85, 86, 87, 88]. Until a few yearsago, laser-induced double ionization was investigated almost exclusively for atoms. The-oretical studies of molecular double ionization are rare, so it is an open question if thedouble-ionization mechanism is the same as in the case of atoms. Second, it is worthwhileto find out, if the phenomenon termed charge resonance enhanced ionization (CREI) playsa role in H2 or not. CREI means that the ionization probability is strongly enhanced at in-ternuclear distances much larger than the equilibrium distance [14, 15]. This phenomenon


can be visualized in the following way. The valence electrons move in a double-well po-tential created by the two nuclei. In a laser pulse, the potential is periodically distortedby the electric field, and an electron can temporarily get trapped in the upper well [15].For certain internuclear distances, the internal barrier between the two wells is low andnarrow, and tunneling of an electron through the barrier leads directly into the continuum.The presence of the lower well effectively reduces the potential barrier that the electronhas to overcome. This situation – which is schematically shown in Fig. 3.8 – gives rise toan enhanced ionization rate. Experiments indicate that molecular ions are, for this reason,preferentially ionized at internuclear distances significantly larger than the equilibriumdistance: The ionization of the molecular ion initiates Coulomb explosion, and the re-sulting kinetic energies of the fragments are much lower than the energy calculated forCoulomb explosion from the equilibrium distance [16, 89, 90, 91]. Typically, this energydeficit is about 50%, indicating that Coulomb explosion is initiated at about twice theequilibrium separation.

Using fixed nuclei, it was shown that H2 exhibits CREI (see Ref. [84] for one-dimensional H2 and Ref. [92] for three-dimensional H2). Yet, only a non-BO calculationcan clarify whether the molecule actually probes the internuclear distances of enhancedionization, after starting out from the molecular ground state.

z

v

v(z)

−zE(t)

Figure 3.8: Mechanism of charge resonance enhanced ionization.


3.2.1 The model H2 molecule

In the 1D H2 model, the dynamics of the internal degrees of freedom is determined by theHamiltonian

H � � 1M

∂2

∂R2� 1

R� 1� �

z1 � z2 � 2 � 1� 2

∑k � 1

� � 12µe

∂2

∂z2k� 1� �

zk � R�2 � 2 � 1

� 1� �zk�

R�2 � 2 � 1�

zk E0 f�t � sin

�ω t � � (3.8)

As usual, R is the internuclear distance, z1 and z2 are the electron coordinates relative tothe nuclear center of mass, and mass-polarization terms are neglected.

The ground-state energy of the model molecule is E0� � 1 385 a.u., its ground-state

equilibrium separation is R0� 2 2 a.u., and the dissociation energy is D0

� 0 0454 a.u.(1.24 eV). For the three-dimensional H2 molecule, these values are E0

� � 1 165 a.u.,R0

� 1 4 a.u., and D0� 0 165 a.u. (4.48 eV). The dissociation potential is obviously

much too low in the 1D model. One may thus expect that the model predicts too largedissociation yields or too strong expansion of the molecule. Nevertheless, the numer-ical results show that the dissociation probability is negligible and that the strong-fielddynamics takes places near the ground-state geometry.

The typical numerical grid size is 10 � 90 � 90 a.u., where the first number is the exten-sion in the nuclear coordinate. The step sizes are 0.1 a.u. for the nuclear coordinate and0.36 a.u. for the electrons. 2000 time steps per optical cycle are used. For laser pulseswith 790 nm wavelength and a sin2 envelope function f

�t � , the total pulse duration is

chosen to be 28 optical cycles. This corresponds to a full width at half maximum of 27 fsin intensity.

The single- and double-ionization probabilities are calculated in the same way as de-scribed in the preceding section. The additional nuclear degree of freedom offers thepossibility to calculate the ionization probabilities for each internuclear separation R. Thetotal probabilities follow by integration over all R.

Single ionization is often followed by dissociation and double ionization is alwaysfollowed by Coulomb explosion. In these cases, one electron is far away from the nu-cleus. As the corresponding parts of the many-body wave function are absorbed at thegrid boundary, the present model does not include these processes. Furthermore, sincethe dynamics of the molecular ion created by single ionization is not accessible in thisapproach, the calculated double ionization does not include the process that an interme-diate molecular ion first expands towards larger internuclear distances and then becomes


ionized.

3.2.2 Results and discussion

Figure 3.9 shows the single- and double-ionization yields pI and pII in the intensity rangefrom 3 � 1013 W/cm2 to 2 � 1014 W/cm2. Also shown is the ratio pII � pI and the ionizationprobability of the 1D H

�2 molecular ion from its ground state. We observe that the ratio

pII � pI is typically several orders of magnitude larger than the ionization probability of H�2 .

This indicates clearly that double ionization is non-sequential in the intensity range underconsideration. Again, “non-sequential” means that it is not possible to obtain the doubleionization yield as the product of the single-ionization probability and the H

�2 ionization

probability. Since the model does not include the ionization of long-lived intermediate H�2

ions, the strong double-ionization signal must stem from an almost simultaneous ejectionof both electrons.

At the highest intensity, pI is 56%, i.e., the onset of saturation in the single-ionizationcurve will be just above 2 � 1014 W/cm2. The knee structure in the intensity dependence

3x1013

1014

2x1014

5x1013

Intensity (W/cm2)

10−10

10−8

10−6

10−4

10−2

100

Ioni

zatio

n pr

obab

ility

Figure 3.9: Probabilities of single ionization (squares) and double ionization (circles)versus laser intensity for the model H2 molecule in 790 nm laser pulses. Triangles: Ratioof double to single ionization. Diamonds: Ionization probability for the 1D H

�2 molecular

ion.


of the double-ionization yield typically occurs at about the same intensity where singleionization saturates, see Chapter 2 and, for molecules, Refs. [85, 86, 87]. Consequently, ifthere is such a knee structure in the present case, then it must be outside the intensity rangeof Fig. 3.9. In view of the computational effort, however, we do not explore intensitieshigher than 2 � 1014 W/cm2. We note that the ratio double to single ionization is much lessdependent on the intensity than the ionization probabilities themselves, and varies around0.2.

To gain further insight into the double-ionization mechanism, we show in panels (a)–(e) of Figs. 3.10 and 3.11 snapshots of the two-electron configuration-space distribution

Γee�z1 � z2 � t � �

�dR

Ψ�R � z1 � z2 � t � 2 (3.9)

for the intensity 2 � 1014 W/cm2. The snapshots are taken at times close to mid-pulse andare separated by time intervals of 0.05T where T is one optical cycle. On the scale ofthe figures, the ground-state resides in a very small region around the origin. The densityalong the axes corresponds to single ionization, and the density in the region where bothelectron coordinates have large absolute values corresponds to double ionization. Thepictures are quite similar to those published previously for the 1D helium atom, see, e.g.,Ref. [47]. In the double-ionization process, either both electrons emerge on oppositesides in patterns parallel to the axes (second and fourth quadrant) or they are ejected inthe form of jets on the same side of the nuclei (first and third quadrant). The direct doubleionization of H2 seems to proceed in a similar manner as for He. Figs. 3.10(f)–(j) and3.11(f)–(j) show momentum-distribution snapshots of the outer regions which we defineby

z1 � z2

� 8 a.u. We find that the double ionization in Fig. 3.10 evolves to a stage[panels (e), (j)] where one electron is almost at rest, but nevertheless with a tendency thatboth electrons move into the same direction. This configuration will effectively receiveno additional ponderomotive momentum shift during the remainder of the pulse since theelectric field is at a local minimum at t � 13 75T . I.e., the momentum distribution inFig. 3.10(j) can be viewed as the final distribution for this process. In almost all doubleionization of Fig. 3.11, the electrons are initially ejected with momenta pointing in thesame direction. However, the momenta will suffer an additional shift between zero and� E0

�ω � � 1 3 a.u. per electron, where the value � 1 3 a.u. is for electrons ejected at

t � 14T (zero field). Therefore, the final distribution will partly contain electrons movingin opposite directions.

To address the question, whether CREI plays a role in the ionization of H2, we analyzethe numerical single-ionization probability pI into contributions from different values ofR, i.e., we calculate the differential probability dpI � dR, see Fig. 3.12 The distributionturns out to be almost independent of the laser intensity. (This is also true for intensitiesnot shown in the figure.) It is a single peak with its maximum at R � 2 5 a.u. The peak


Figure 3.10: (a)–(e): Snapshots of the two-electron density Γee�z1 � z2 � t � , Eq. (3.9), for the

model H2 molecule in a 790 nm pulse with intensity 2 � 1014 W/cm2, taken at the times(a) t � 13.55T , (b) t � 13.6T , (c) t � 13.65T , (d) t � 13.7T , and (e) t � 13.75T . (f)–(j):Snapshots of the two-electron momentum distribution for the doubly ionized part of thewave function, taken at the same times. A logarithmic grey scale is used.

Figure 3.11: Same as Fig. 3.10 for the times (a) t � 13.8T , (b) t � 13.85T , (c) t � 13.9T ,(d) t � 13.95T , and (e) t � 14T .


Figure 3.12: R-dependent single-ionization yield for 790 nm pulses withintensities (a) 6 � 1013 W/cm2, (b)1 � 1014 W/cm2, and (c) 2 � 1014 W/cm2

(solid curves). The dot-dashed curve in(c) is the ground-state nuclear densityin arbitrary units.

1 2 3 4 5 6R (a.u.)

0.0

0.4

0.00

0.02

dpI /d

R

0.000

0.002 (a)

(b)

(c)

1 2 3 4 5 6R (a.u.)

0

0.04

0

0.004

dpII /d

R

0

4x10−4

(a)

(b)

(c)

Figure 3.13: Same as in Fig. 3.12 fordouble ionization.

is located only slightly above the equilibrium distance R0� 2 2 a.u. For comparison, the

nuclear density of the unperturbed ground state is shown in panel (c).For double ionization, the R-dependent yield is given in Fig. 3.13. These distributions

extend to somewhat larger R, but the location of the maxima is almost unchanged ascompared to single ionization.

For the highest intensity, we plot snapshots of the two-body density

Γne�R � z � t � � 2

�dz

� Ψ�R � z � z � � t � 2 (3.10)

in Fig. 3.14. For each internuclear separation R, this quantity gives the electron densityas a function of z. The snapshots are taken during a half cycle with negative electric fieldso that we can trace the emission of an electron in positive z direction. Obviously, theelectron is ejected at nuclear configurations close to the equilibrium geometry. It is alsoapparent that there is only a small change in R during the time between the ejection of anelectron from the core region and its detection at the grid boundary. E.g., in the right-handside of panel (e), the lower end of the distribution is always at R � 1 8 a.u., independentlyof the electron coordinate z. In addition, we find density at larger R, see, e.g., in panel(b) around z � 15 a.u. This is the probability for an electron which was ejected during an


Figure 3.14: Snapshots of the two-body density Γne�R � z � t � , Eq. (3.10), for the intensity

2 � 1014 W/cm2 taken at (a) t � 13.6T , (b) t � 13.7T , (c) t � 13.8T , (d) t � 13.9T , and(e) t � 14T . Logarithmic grey scale.

earlier half cycle and is now oscillating in the field. In this case, the nuclei have had timeto separate from each other so that the density is found at slightly increased R.

The ionization of H�2 from its ground state is very different, see Fig. 3.15 for a direct

comparison. The molecular ion is ionized at distances much larger than the equilibriumdistance 2.6 a.u. Except for the highest intensity, the probability distributions have sev-eral maxima. For all intensities, the distributions are located between R � 4 a.u. andR � 10 a.u. which is the range where the critical distances for enhanced ionization areexpected [84, 92]. This means that in H

�2 , the ionization probability at the ground-state

geometry is so small that the molecule first expands to distances of enhanced ionizationand is then ionized.

3.3 Even-harmonic generation due to beyond-Born-Oppenheimer dynamics 81

2 4 6 8 10 12 14R (a.u.)

0

0.04

0

3x10−6

dpI /d

R

0

2x10−8 (a)

(b)

(c)

Figure 3.15: R-dependent ionization yield for the 1D H�2 molecule in 790 nm pulses with

intensities as in Fig. 3.12 (solid curves). The dot-dashed curve in panel (c) is the ground-state nuclear density of 1D H

�2 in arbitrary units.

The difference between H2 and H�2 is possibly due to a screening effect: The double-

well potential seen by one electron is shallower if a second electron screens the nuclearcharges. Hence, the localization of an electron in the upper well (see Fig. 3.8) is less likely.Furthermore, if an electron sits in the upper well, then, as a consequence of electron-electron correlation, the second electron is most likely located in the lower well. Thesuppression of the ionization barrier by the presence of the lower well is therefore lesseffective.

To summarize, single- and double-ionization of the model H2 molecule occur nearthe ground-state geometry. Apparently, CREI does not have any effect on the ionizationdynamics of the neutral molecule. The assumption made in the preceding section thatthe ionization probability of H2 can be calculated at the equilibrium internuclear distanceis thus justified. A non-sequential double-ionization process similar to atomic doubleionization was found.

3.3 Even-harmonic generation due to beyond-Born-Op-penheimer dynamics

Atoms and molecules driven by intense laser pulses radiate at frequencies that are integermultiples of the laser frequency. This so-called harmonic generation [4, 93, 94, 95] has


been the subject of numerous experimental and theoretical studies, mainly because theemission of high-order harmonics is a promising method to produce coherent x-rays andattosecond pulses: Experimentally, orders up to about 300 have been reported [5, 6].

We first discuss what frequencies are expected to appear in the harmonic spectrum.Frequencies other than the usual integer multiples may be emitted by a single atom ormolecule as a consequence of resonances. This is quite analogous to resonance-inducedATI peak series, see Sec. 2.4. However, if a large number of atoms or molecules is in-volved in the process, then the coherent part of the resulting spectrum consists of pureinteger multiples of the fundamental frequency since only these can survive the propa-gation through the medium [96], while the others undergo destructive interference. Ad-ditionally, an incoherent fluorescence background may be present [93]. Typically, onlyodd harmonics are observed in gas-phase experiments. One routinely invokes the conceptof inversion symmetry to explain the absence of even harmonics. Intuitively speaking,when an electron in a symmetric potential V

�r � � V

� � r � is driven by a laser, it performsa “symmetric” oscillation of the form z

�t � � � z

�t�

T�2 � . Such an oscillation consists of

odd frequency components only. More precisely, parity is a good quantum number in asystem with inversion symmetry, i.e., the eigenstates of the system have a defined parity,�

1 or � 1. In the picture of perturbation theory, the generation of harmonic radiationproceeds by first absorbing n laser photons of frequency ω and then emitting one photonof frequency nω. The latter process must be a transition between two states of differentparity. Therefore, the initial absorption cannot involve an even number of photons. Thisleads to the fact that in general neither atoms nor molecules produce even harmonics. Onecould try to break the symmetry by choosing heteropolar molecules: the potential seenby the electrons in such a molecule is not inversion symmetric if we think of the nucleias fixed. However, the full field-free molecular Hamiltonian (including all nuclear andelectronic coordinates) is always inversion symmetric so that even-harmonic generationis still forbidden.

Violations of the above selection rule have been found in numerical calculations [97];they are caused by accidental degeneracies of Floquet states. It is demonstrated in Ref.[97] that the even harmonics generated in this way are a special case of radiation whichgenerally occurs at non-integer multiples of the fundamental frequency. (As mentionedabove, such non-integer harmonics can be produced at the single-molecule level but arenot phase matched in a gas of molecules). A simple example for such a degeneracy isan excited state that has the same parity as the ground state and the energy of which isan odd multiple of the photon energy above the ground-state level. Consider now theradiative transition into this excited state from a highly excited state which was populatedby absorption of an odd number of laser photons. In contrast to a transition into the groundstate, the energy of the emitted photon is an even multiple of the laser photon energy.

In an ensemble of oriented diatomic molecules, the inversion symmetry is broken


except in the case of homonuclear molecules. We consider now an interesting kind ofsymmetry breaking which is realized in the HD molecule: the nuclear masses M1 � M2

are different while the nuclear charges are equal. The HD molecule possesses a perma-nent electric dipole moment which has been calculated previously [98]. As we will showbelow, the asymmetry also gives rise to the generation of even harmonics. Note that thepotential seen by the electrons is still symmetric if we use the Born-Oppenheimer approx-imation. I.e., we are dealing with even-harmonic generation by non-Born-Oppenheimerdynamics.

3.3.1 The model HD molecule

In the HD molecule, the nuclear center of mass does not coincide with the nuclear centerof charge. Then, keeping the definition of the electron coordinates relative to the nuclearcenter of mass, the dipole operator contains not only the electron coordinates z1, z2 butalso the internuclear distance R � R1 � R2:

D � � � z1�

z2 � � λR (3.11)

Here, λ �

�M2 � M1 � � Mn is the mass-asymmetry parameter. Mn

� M1�

M2 denotes thetotal mass of the nuclei. The 1D Hamiltonian for the internal degrees of freedom (onenuclear and two electronic degrees of freedom) is

H � � 12µn

∂2

∂R2 � 12µe

� ∂2

∂z21

� ∂2

∂z22� � 1

R�

w�z1 � z2 �

� 2

∑j � 1

w � z j � M2

MnR � � 2

∑j � 1

w � z j � � M1

MnR � � E

�t � D � (3.12)

with the reduced masses

µn�

M1M2

Mn� µe

�Mn

Mn�

1(3.13)

and the interaction

w�x � y � �

1� �x � y � 2 � 1

(3.14)

The grid parameters are the same as in the preceding section. We employ a laser withpeak intensity 1014 W/cm2 and wavelength 770 nm. The field is switched on with a linearramp during the first 10 optical cycles. Afterwards, the amplitude is held constant. Thetotal time of propagation is 30 optical cycles.


For an ensemble of molecules, the spectrum of emitted radiation is approximately(neglecting incoherent fluorescence) proportional to the squared modulus of the Fouriertransformed dipole acceleration expectation value [99],

S�ω � � ��

�eiωt d2

dt2� Ψ � t � D Ψ � t � � dt ��

2 (3.15)

The dipole acceleration expectation value follows from the Ehrenfest theorem:

d2

dt2� Ψ � t � D Ψ � t � � �

1µe

� Ψ�t � ��

∂H∂z1

� ∂H∂z2��Ψ�t � � λ

µn

� Ψ�t � ��

∂H∂R ��Ψ

�t � (3.16)

3.3.2 Results

The harmonic spectrum generated by the model HD molecule is shown in Fig. 3.16(a).Figure 3.16(b) is the spectrum for the H2 model under exactly the same conditions. Inboth cases, we observe the familiar rapid decrease of magnitude in the low-frequencyrange (harmonic order lower than 17). Between the 19th and the 25th order, the envelopeexhibits a local maximum. At higher frequencies, we find a plateau extending up to the45th order. H2 generates only odd harmonics, whereas HD produces a series of even har-monics throughout the entire range of observed frequencies. In the low-order range, theeven harmonics are much weaker than the odd harmonics, but the shape of the envelopeis similar. In the plateau region, the relative contribution of the even harmonics becomeslarger. For the highest frequencies shown in the spectrum, even and odd harmonics are ofthe same order of magnitude. This is in contrast to other mechanisms of symmetry break-ing, where even and odd harmonics usually exhibit similar envelope shapes throughoutthe whole spectrum. See, e.g., Ref. [100] for static magnetic fields or Ref. [101] for staticelectric fields.

3.3.3 Interpretation in terms of non-adiabatic couplings

To show that the even-harmonic generation by the HD molecule occurs only beyond theBorn-Oppenheimer approximation, we expand the wave function in Born-Oppenheimerstates:

Ψ�R � z1 � z2 � t � � ∑

jχ j�R � t � Φ j

�R � z1 � z2 �� (3.17)


where�Φk�R � z1 � z2 � � is an orthonormal set of stationary electronic eigenfunctions, cf.

Sec. 1.5. Then, the time-dependent Schrodinger equation reads

i∂∂t

χk�R � t � �

� � 12µn

∂2

∂R2�

Vk�R �� χk

�R � t � � E

�t � ∑

jDk j

�R � χ j

�R � t �� ∑

j

�Ak j

�R � � 2Bk j

�R � ∂

∂R� χ j

�R � t �� (3.18)

where Vk�R � are the BO potentials and Dk j

�R � � � Φk

D Φ j � are the dipole matrix ele-

ments. Ak j and Bk j are the non-adiabatic couplings,

Ak j�R � � � 1

2µn

�Φk�R � z1 � z2 � � ∂2

∂R2 Φ j�R � z1 � z2 � dz1 dz2 (3.19)

0 10 20 30 40 50 60harmonic order

10−10

10−8

10−6

10−4

10−2

100

S(ω

)/ω

(ar

b. u

nits

)

0 10 20 30 40 50 6010

−10

10−8

10−6

10−4

10−2

100

(a)

(b)

Figure 3.16: Harmonic spectrum generated by (a) the model HD molecule and (b) themodel H2 molecule, driven by a laser with peak intensity 1014 W/cm2 and wavelength770 nm. The plotted quantity S

�ω � � ω is proportional to the number of emitted photons.


and

Bk j�R � � � 1

2µn

�Φk�R � z1 � z2 � � ∂

∂RΦ j�R � z1 � z2 � dz1 dz2 (3.20)

For HD, the origin of the electronic coordinate system is not at the geometric centerbetween the nuclei. Therefore, the electronic states Φ j are not eigenstates of the electronicinversion operator Pe which acts as

�z1 � z1 � z2 � z2 � . We can, however, shift the

origin to the geometric center

RG�

R1�

R2

2 (3.21)

The relation to the nuclear center of mass

RCMN�

M1R1�

M2R2

M1�

M2(3.22)

is

RCMN� RG � λ

2R (3.23)

Therefore, the shift of the origin to the geometric center is achieved by the replacement

Φ j�R � z1 � z2 � � Φ j

�R � z1 � λ

2R � z2 � λ

2R � (3.24)

where the shifted states Φ j are eigenstates of Pe (gerade/ungerade) because they are theeigenstates of an electronic Hamiltonian with a symmetric potential. (The Φ j would beidentical with the electronic Born-Oppenheimer states of the H2 system, if the reducedelectronic mass was not slightly different.) We immediately find that

Dk j�R � � Dk j

�R � � � Φk

D Φ j � � (3.25)

where D is the shifted dipole operator D � � � z1�

z2 � . The states Φ j are eigenstates of Pe,and the dipole operator D changes its sign under inversion. Therefore, the dipole operatorcouples only states with different eigenvalues of Pe. Adopting the Born-Oppenheimerapproximation means that the non-adiabatic couplings in Eq. (3.18) are neglected. Then,as a consequence of Eq. (3.25), HD obeys the same selection rules as H2. Beyond theBorn-Oppenheimer approximation, the situation is different: In the case of H2, the non-adiabatic couplings Ak j and Bk j are zero if the states k and j have different symmetry,leaving the selection rules unchanged. In the case of HD, the non-adiabatic couplingsmix all states. The replacement Φ j Φ j in Eqs. (3.19), (3.20) leads to additional terms


because the derivative ∂�∂R acts on the R-dependence of the coordinate shift λ

2 R. Forexample, Eq. (3.20) becomes

Bk j�R � � Bk j

�R � � λ

4µn

�Φk�R � z1 � z2 � � � ∂

∂z1

� ∂∂z2� Φ j

�R � z1 � z2 � dz1 dz2 (3.26)

with

Bk j�R � � � 1

2µn

�Φk�R � z1 � z2 � � ∂

∂RΦ j�R � z1 � z2 � dz1 dz2 (3.27)

The second term in Eq. (3.26) enables transitions between states of different symmetry.The prefactor λ indicates that these transitions increase with the mass asymmetry. Moreprecisely, the relevant factor is

λµn

�M2 � M1

M1M2 (3.28)

The selection rules that prevent even-harmonic generation are thus violated in the case oforiented HD. While the non-adiabatic matrix elements couple only states of equal sym-metry in H2, they couple all states in HD.

The numerical results show that the generation of even harmonics is strongest at highharmonic frequencies. We conclude that non-adiabatic effects are particularly importantin the high-order range. The explanation of this behavior lies in the mechanism of high-harmonic generation: An electron is temporarily promoted into highly excited states, i.e.,continuum states. The energy differences between these states are infinitesimally small.The Born-Oppenheimer approximation then breaks down since the electronic motion isnot fast compared to the nuclear motion.

Chapter 4

Outlook: Field propagation effects

The description of atoms and molecules in strong fields essentially requires the consider-ation of two points:

� The response of the single atom/molecule to laser irradiation.

� The propagation of the electromagnetic field.

The preceding chapters exclusively dealt with the first point. The effect of the laser-induced radiation on the incoming field was ignored. This approach is valid if the laserinteracts with a dilute gas of particles. If we are dealing with a dense sample of particles,however, field propagation effects become important. Reflection or absorption of light arepossible, and the field seen by the particles will depend on the position in space. Further-more, the most consistent way to calculate harmonic spectra is via field propagation sincethe superposition of the contributions from all particles is automatically included. Thischapter gives a brief description how to treat the combined dynamics of the field and theparticles, see, e.g., [102, 103].

4.1 The Maxwell-Schrodinger equations

We number the atoms/molecules with indices; for example, Ψk�t � denotes the time-

dependent wave function of the kth atom/molecule. The non-relativistic response of eachsystem to the electric field E

�r � t � follows from the time-dependent Schrodinger equation

i∂∂t

Ψk�t � �

�Hk � 0 � Dk E

�r � t � � Ψk

�t � (4.1)

88

4.1 The Maxwell-Schrodinger equations 89

where Hk � 0 denotes the Hamiltonians of the unperturbed systems, and

Dk� � � Ne

∑j � 1

r j � Nn

∑α � 1

ZαRα (4.2)

is the dipole operator of the kth atom/molecule, expressed in terms of the electronic andnuclear coordinates. In Eq. (4.1), the coupling of the particles to the magnetic field isneglected.

The time evolution of the electromagnetic field is governed by Maxwell’s equations.The two relevant equations are

1c

∂∂t

�E�r � t � � 4πP

�r � t � �

� ∇ � B�r � t � � � (4.3)

1c

∂B�r � t �

∂t� � ∇ � E

�r � t �� (4.4)

B�r � t � is the magnetic field and P

�r � t � is the polarization. In Eq. (4.4), magnetization

effects are neglected, i.e., B � H. Taking the time derivative of Eq. (4.3) and insertingEq. (4.4) yields

1c2

∂2

∂t2

�E�r � t � � 4πP

�r � t � �

� 4π∇�∇P � � ∇2E

�r � t �� (4.5)

Here, we have used ∇E � � 4π∇P. The first term on the right-hand side vanishes for“transverse” fields satisfying ∇E � 0.

The effect of the particle dynamics on the field is introduced by relating the polar-ization to the microscopic dipole moments. To that end we fix the center of mass of thekth atom/molecule at the position RCM � k and specify the density ρ

�r � of atoms/molecules.

For simplicity, we assume that only one species of atoms or molecules is present. Then,

P�RCM � k � t � � ρ

�RCM � k � � Ψk

�t � Dk

Ψk�t � � (4.6)

We consider now the case that a linearly polarized laser pulse propagates along the xdirection with the electric field vector parallel to the z axis. We assume that the density ρdepends only on x. In this case, the electric field and the polarization are of the form

E�r � t � � � 0 � 0 � E � x � t � � � (4.7)

P�r � t � � � 0 � 0 � P � x � t � � (4.8)

Then, the wave equation (4.5) takes the simpler form

1c2

∂2

∂t2

�E�x � t � � 4πP

�x � t � �

�∂2

∂x2 E�x � t �� (4.9)

The combination of Eq. (4.9) and Eqs. (4.1) can be used to calculate the propagation of apulse through a dense gas. This amounts to the simultaneous solution of a large numberof Schrodinger equations, which are coupled to the wave equation (4.9) via Eq. (4.6).

90 4 Outlook: Field propagation effects

4.2 Numerical example

As an example, we calculate the propagation of a 500 nm laser pulse with 2 � 1014 W/cm2

peak intensity through a layer of one-dimensional hydrogen atoms. For an atom at theposition x, the Hamiltonian reads,

Hx� � 1

2µe

∂2

∂z2 � 1�z2 � 1

�zE�x � t � (4.10)

with µe� Mn

� �Mn

�1 � . In order to observe absorption and reflection of light, we choose

a high density of atoms:

ρ�x � � ρ0 e

��x � x0 � 2 �

2d2(4.11)

with ρ0� 0 02 (a.u.)

� 3, d � 1200 a.u, and x0� 108000 a.u. The Schrodinger equation is

solved at 220 different positions, located symmetrically around x0 and covering a spatialrange of 6600 a.u.

The initial pulse shape is chosen to be Gaussian,

E�x � t � 0 � � E0 e

��x � x1 � 2 �

2σ2sin � ω0

c

�x � x1 � � � (4.12)

with σ � 14000 a.u. This corresponds to an initial temporal full width at half maximumof 2

�ln2σ

�c � 170 a.u. As the differential equation (4.9) is of second order in time, we

must specify the initial time-derivative of E�x � t � . Numerically, a pulse propagating in the

direction of positive x is employed by not only fixing E�x � t � 0 � but also

E�x � t � 0 � dt � � E

�x�

cdt � t � 0 �� (4.13)

The propagation is performed with 800 time steps per optical cycle, and spatial stepsof 30 a.u. for the field and 0.38 a.u. for the electron coordinate. The double derivatives inEq. (4.9) are approximated by simple three-point formulas.

In Fig. 4.1, the electric field of the incoming pulse is shown in the upper panel. Inthe lower panel, the field is plotted at t � 965 a.u. At this time, the pulse has passed thelayer of atoms which is located at x0

� 108000 a.u. Since the pulse is partially reflected,the amplitude of the transmitted field is lower than the initial amplitude. The pulse shapeof the transmitted part is very similar to the incoming pulse. The reflected part, however,appears with a distorted envelope. Although the layer of atoms is smaller than the laserwavelength, it seems that phase mismatching is responsible for this effect. Integer mul-tiples of the laser frequency are only phase matched in the forward direction, not in thebackward direction. This applies to the fundamental frequency as well. Figure 4.2 showsthe frequency spectrum of the reflected and the transmitted pulse. In the latter, the typi-cal odd harmonics are observed. They are not very clearly resolved because the pulse isextremely short. The spectrum of the reflected pulse, on the other hand, does not exhibitany regular structure.

4.2 Numerical example 91

0 100000 200000x (a.u.)

−0.08

−0.04

0

0.04

0.08

E(x

) (a

.u.)

−0.08

−0.04

0

0.04

0.08

E(x

) (a

.u.)

(a)

(b)

atoms

Figure 4.1: Propagation of a 500 nm laser pulse with intensity 2 � 1014 W/cm2 through alayer of model H atoms. (a) Initial electric field. (b) Final electric field. The density ofatoms is plotted in arbitrary units.

0 5 10 15 20ω/ω0

10−12

10−10

10−8

10−6

10−4

10−2

|E(ω

)|2 (

arb.

uni

ts)

Figure 4.2: Frequency spectrum of the transmitted (solid) and reflected (dashed) pulse inFig. 4.1.

Chapter 5

Summary

The dynamics of atoms and molecules in strong laser pulses displays a huge variety of ef-fects. The subject of this work is the theoretical investigation and explanation of selectedstrong-field phenomena. As a consequence of the highly non-linear response to intenselight, the field cannot be treated as a small perturbation. In the case of molecules, multi-photon absorption can easily populate highly excited states so that the Born-Oppenheimerapproach breaks down. Therefore, non-perturbative methods beyond the Born-Oppenhei-mer approximation must be applied. However, even for the smallest atoms and molecules,the full numerical solution of the time-dependent Schrodinger equation is extremely de-manding. We approach the problem by considering only the most important degreesof freedom. In the case of linearly polarized intense femtosecond pulses, these are theelectronic coordinates along the polarization axis and – in molecules – the internuclearseparation. This leads us to model systems that can be treated numerically with reason-able effort. Of course, the theoretician’s long-term goal is to find approximate methodsthat give reliable quantitative results for arbitrary systems in strong fields. To find ade-quate approximations, it is essential to consider systems where exact reference solutionsare available.

The first part of this work deals with non-sequential multiphoton double ionization ofatoms. After the experimental observation of unexpectedly large double-ionization ratesfor rare-gas atoms, various theoretical approaches tried to clarify the nature of this pro-cess. Although the experimental findings could be reproduced by theory, the physicalmechanism of the two-electron ejection remained controversial until recently. Last year,the recoil momenta of doubly charged ions were measured, revealing that the ions pref-erentially receive non-zero momenta. More precisely, the distributions exhibited a localminimum at zero. Among the proposed double-ionization models, the rescattering sce-nario was the only one that could explain this result. In this model, an electron is firstejected by tunnel ionization. Upon reversal of the electric-field direction, the electron can

92

Summary 93

return to the core and transfer part of its energy to another electron.In this thesis, we have investigated the two-electron dynamics in momentum space and

in the center-of-mass phase space. A model He atom with two electronic coordinates wasused. The phase-space dynamics was studied by calculating the time-dependent Wignerfunction describing the coordinate-momentum correlation for the electronic center-of-mass. The numerical results confirm the classical rescattering picture as much as a quan-tum mechanical calculation possibly can. In consistency with experiment, we find that thetwo electrons are preferentially ejected into the same direction so that the recoil-ion mo-mentum is non-zero. In a numerical experiment, we introduced a time delay between twohalf cycles to reduce the probability that the outer electrons recollides with the core. Thedouble-ionization was thereby suppressed. A single half cycle alone produces negligibledouble ionization. In the phase-space evolution, we were able to identify the scattering ofthe returning electron from the core and the following two-electron ejection. In a Hartree-Fock calculation, no scattering could be identified. Double-ionization in the Hartree-Fockapproach occurs due to uncorrelated ejection of two electrons. The signatures of this pro-cess in phase space are very different from the correlated recollision process. Some in-teresting properties of the Hartree-Fock center-of-mass Wigner function were found. Forexample, it is never smaller than zero, contrary to a general Wigner function. In orderto calculate electron and ion spectra, the numerical method had to be extended such thatthe information about electrons far away from the nucleus is not lost at the absorbing gridboundary. The numerical recoil-ion momentum spectra agree well with experiment. Inparticular, the double-hump structure at intermediate intensities is reproduced. This struc-ture disappears at higher intensities where double ionization is uncorrelated. Altogether,the body of presented results strongly supports the rescattering model at intermediate in-tensities. The ATI spectrum from single ionization was also discussed briefly. We foundthat the two-electron calculation reproduces the cutoff energies which are known fromsingle-active-electron calculations.

When the wavelength of the laser approaches the ultraviolet region, we find above-threshold double ionization. This is the two-electron analogue of ordinary ATI and man-ifests itself as peaks in the spectrum of the total electronic kinetic energy. These peaksare separated by the photon energy and result from the fact that only an integer number ofphotons can be absorbed by the system. This effect has not yet been observed experimen-tally. We have shown that the recoil-ion momentum distributions exhibit ATDI-inducedstructures as well. Furthermore, we have proposed a numerical method to reconstruct theelectron energy spectrum from the recoil-ion spectrum. Therefore, we suggest that theATDI peaks can be detected by measuring the recoil-ion momenta. A simple formulainvolving the double-ionization potential and the ponderomotive potential has been givento predict the energies of the ATDI peaks. Similar to one-electron ATI, more than oneseries of ATDI peaks may result from laser-induced resonances.

94 Summary

In the second part, we have described several aspects of the strong-field dynamicsof small molecules. Linear model molecules with one or two electrons were used. Weapplied a two-step model to calculate the fragment kinetic energies for dissociative ioniza-tion of H2 and D2. For several pulse widths, we have calculated first the time-dependentionization probability of the neutral molecule and afterwards – for various times of ioniza-tion – the fragmentation of the molecular ion. The simulation reproduced the experimentalobservation that the dissociation via effective two-photon absorption exhibits pulse-widthand isotope effects: In the case of D2, the fragment kinetic energies become larger upondecreasing the pulse duration. This shift is not found for H2. We have explained theseunexpected effects by exploring the classical nuclear dynamics in the time-dependent adi-abatic potential curves.

We have investigated the strong-field dynamics of a linear H2 model molecule withone nuclear and two electronic degrees of freedom. The single- and double-ionizationprobabilities were calculated for various laser intensities. We found strong non-sequentialdouble-ionization. The ratio of double to single ionization is almost independent of theintensity. The time-evolution in configuration space indicates that the mechanism of thetwo-electron ejection is similar as in atoms. Single and double ionization occur at in-ternuclear distances close to the ground-state equilibrium distance although for clampednuclei, the ionization probability is known to be strongly enhanced at critical internucleardistances larger than the equilibrium distance. Our results justify the frequently madeassumption that H2 is ionized at the equilibrium distance. The H

�2 molecular ion behaves

differently in that it first expands to large internuclear distances before ionization takesplace.

We have then compared the harmonic spectrum generated by an H2 model moleculewith the spectrum generated by an HD model molecule. The striking difference betweenboth cases is that the oriented HD molecule produces even and odd harmonic orders whileH2 produces odd harmonics only. The relative strength of the even harmonics is largestin the high-frequency region. In gas-phase experiments, one typically observes radiationfrom unoriented molecules. Then, even harmonics are forbidden to all orders in pertur-bation theory. In the calculation, we deal with oriented molecules so that the selectionrule is possibly violated. We have shown that even-harmonic generation by HD is dueto beyond-Born-Oppenheimer dynamics. The matrix elements describing non-adiabaticcoupling between different electronic states mix all states in HD while they couple onlystates of equal symmetry in H2.

Finally, we have briefly discussed how the propagation of a laser pulse through anensemble of atoms or molecules is treated by combining Maxwell’s equations with theSchrodinger equation. A numerical example shows that reflection and transmission oflight are included in such a framework and that the harmonic spectrum generated by theparticles can be conveniently calculated as the spectrum of the outgoing pulse.

Summary 95

To conclude, we were able to reproduce and explain a variety of experimental results.Moreover, we predicted two interesting phenomena which have not yet been found inexperiment: Above-threshold double ionization and even-harmonic generation by non-Born-Oppenheimer dynamics. The interaction of intense laser pulses with matter stillbares many open questions which deserve further research. A few examples are the re-sponse to elliptically polarized lasers, triple-ionization processes, or non-Born-Oppenhei-mer wave-packet dynamics.

Zusammenfassung

Atome und Molekule, die starken Laserpulsen ausgesetzt werden, weisen eine enormvielfaltige Dynamik auf. Gegenstand dieser Arbeit ist die theoretische Untersuchung undErklarung ausgewahlter, durch starke Felder hervorgerufener Phanomene. Als Folge derhochgradig nichtlinearen Reaktion auf intensives Licht kann das Feld nicht als kleineStorung behandelt werden. Im Fall von Molekulen gelangt man durch Absorption vielerPhotonen leicht in den Bereich hochangeregter Zustande, fur welche der Born-Oppenhei-mer-Zugang versagt. Deshalb mussen nichtperturbative Methoden jenseits der Born-Op-penheimer-Naherung angewandt werden. Die volle numerische Losung der zeitabhangi-gen Schrodingergleichung ist jedoch selbst fur die kleinsten Atome und Molekule extremaufwendig. Wir gehen das Problem dadurch an, dass wir nur die wichtigsten Freiheitsgra-de in Betracht ziehen. Fur linear polarisierte intensive Femtosekundenpulse sind dies dieElektronenkoordinaten entlang der Polarisationsachse und – in Molekulen – der Abstandzwischen den Kernen. Auf diese Weise gelangen wir zu Modellsystemen, die mit vernunf-tigem Aufwand numerisch behandelt werden konnen. Naturlich ist es das langfristige Zieldes Theoretikers, approximative Methoden zu finden, die verlassliche Ergebnisse fur be-liebige Systeme in starken Feldern liefern. Um aber angemessene Naherungen aufzustel-len, ist es notwendig, Systeme zu betrachten, die exakte Referenzlosungen ermoglichen.

Der erste Teil dieser Arbeit befasst sich mit nichtsequentieller Doppelionisation vonAtomen. Nach der experimentellen Beobachtung unerwartet hoher Doppelionisationsra-ten bei Edelgasatomen versuchte man mit verschiedenen theoretischen Zugangen, dieNatur dieses Prozesses aufzuklaren. Obwohl die experimentellen Befunde reproduziertwerden konnten, blieb der physikalische Mechanismus der Zweifachionisation zunachstumstritten. Im letzten Jahr gelang es, die Ruckstoßimpulse der doppelt geladenen Ionen zumessen, mit dem Ergebnis, dass diese Ionen mit großer Wahrscheinlichkeit von Null ver-schiedene Impulse tragen. Genauer ausgedruckt fand man Verteilungen mit einem lokalenMinimum bei Null. Nur eines der Modelle, die zur Erklarung der starken Doppelionisati-on vorgeschlagen wurden, konnte dieses Ergebnis erklaren: Im Ruckstoßmodell verlasstzunachst ein Elektron das Atom durch Tunnelionisation. Nachdem das elektrische Feldsein Vorzeichen umgekehrt hat, kann das Elektron zum Rumpf zuruckkehren und einenTeil seiner Energie auf ein anderes Elektron ubertragen.

Im Rahmen dieser Dissertation wurde die Dynamik der beiden Elektronen im Impuls-raum und im Phasenraum der Schwerpunktsbewegung untersucht. Dazu wurde ein Mo-

96

Zusammenfassung 97

dellatom mit zwei Elektronenkoordinaten benutzt. Zur Erforschung der Phasenraumdyna-mik berechneten wir die zeitabhangige Wignerfunktion. Diese Verteilungsfunktion ist dasquantenmechanische Analogon zur klassischen Phasenraumverteilung, gibt also die Kor-relation zwischen Impuls und Ort an. Im vorliegenden Fall handelt es sich dabei um denGesamtimpuls der beiden Elektronen und die Ortskoordinate des elektronischen Schwer-punkts. Die numerischen Ergebnisse bestatigen das klassische Ruckstreubild so detailliertwie dies in einer quantenmechanischen Rechnung moglich ist. Wir finden in Ubereinstim-mung mit dem Experiment, dass die beiden Elektronen mit großter Wahrscheinlichkeit indie gleiche Richtung emittiert werden, so dass der Ruckstoßimpuls ungleich Null ist. Ineinem numerischen Experiment fuhrten wir eine Zeitverzogerung zwischen zwei opti-schen Halbzyklen ein, um die Wahrscheinlichkeit fur die Ruckkehr des außeren Elektronszu vermindern. Die Zweifachionisation wurde dadurch unterdruckt. Ein Halbzyklus al-lein bewirkt eine vernachlassigbar kleine Wahrscheinlichkeit fur Doppelionisation. In derZeitentwicklung der Phasenraumverteilung konnten wir die Streuung des zuruckkehren-den Elektrons am Rumpf identifizieren. In einer Hartree-Fock-Rechnung wurde keineStreuung beobachtet. Im Hartree-Fock-Zugang findet Zweifachionisation aufgrund un-korrelierter Emission zweier Elektronen statt. Dieser Prozess hinterlasst wesentlich an-dere Signaturen im Phasenraum. Einige interessante Eigenschaften der Hartree-Fock-Wignerfunktion wurden gefunden. Beispielsweise wird diese im Gegensatz zum allge-meinen Fall niemals negativ. Um die Spektren von Elektronen und Ionen zu berechnen,mussten wir die numerische Methode derart erweitern, dass die Information uber Elek-tronen weit weg vom Kern nicht an absorbierenden Gitterrandern verloren geht. Die nu-merischen Ruckstoßionenimpulsverteilungen stimmen gut mit dem Experiment uberein.Insbesondere werden im mittleren Intensitatsbereich die beiden symmetrisch um Null lie-genden Maxima reproduziert. Diese Struktur verschwindet bei hoheren Intensitaten, weildie Doppelionisation in diesem Bereich unkorreliert ablauft. Die Gesamtheit der prasen-tierten Ergebnisse spricht klar fur die Richtigkeit des Ruckstoßmodells bei mittleren In-tensitaten. Wir haben auch das Elektronenspektrum bei Einfachionisation kurz diskutiert.Es zeigte sich, dass die Zweielektronenrechnung die Maximalenergien reproduziert, wel-che von Einelektronenrechnungen bekannt sind.

Wenn sich die Wellenlange des Lasers dem ultravioletten Bereich nahert, finden wir

”Zweifachionisation uber der Schwelle“. Dieser Effekt, den wir entsprechend dem engli-

schen Ausdruck”above-threshold double ionization“ meist als ATDI abkurzen, bedeu-

tet, dass die Gesamtenergieverteilung der beiden Elektronen scharfe Maxima im Ab-stand der Photonenenergie enthalt. Dieses Verhalten steht in Analogie zu dem bekanntenEinelektroneneffekt, welcher gewohnlich mit dem Kurzel ATI (

”above-threshold ioniza-

tion“) bezeichnet wird und sich in den Photoelektronenspektren ebenso durch Maxima imAbstand der Photonenenergie außert. Fur zwei Elektronen wurde dieses Phanomen bis-lang nicht experimentell beobachtet. Wir haben gezeigt, dass die Ionenimpulsverteilun-

98 Zusammenfassung

gen ebenfalls ATDI-Strukturen enthalten. Des weiteren wurde eine numerische Methodevorgestellt, mit deren Hilfe das Energiespektrum der Elektronen aus den Ionenimpulsenrekonstruiert werden kann. Wir schlagen deshalb vor, die ATDI-Maxima durch Messungder Ruckstoßionenimpulse zu entdecken. Die Energien der ATDI-Maxima konnen mit-tels einer einfachen Formel vorhergesagt werden, welche das Zweifachionisationspoten-tial und das sogenannte ponderomotive Potential enthalt. Das letztere gibt die Energie-erhohung der elektronischen Kontinuumszustande durch die Prasenz des Laserfeldes anund ist darum mit einer Zunahme des Ionisationspotentials gleichzusetzen. Als Folge vonlaserinduzierten Resonanzen kann ahnlich wie bei Einfachionisation mehr als eine Serievon Maxima in den Spektren auftreten.

Im zweiten Teil dieser Arbeit wurden einige Aspekte der Dynamik kleiner Molekulein starken Feldern besprochen. Dazu wurden lineare Modelle mit einem oder zwei Elek-tronen verwendet. Mit Hilfe eines Zweistufenmodells haben wir die kinetischen Energiender Fragmente berechnet, welche aus der dissoziativen Ionisation von H2 bzw. D2 resul-tieren. Fur mehrere Pulsbreiten haben wir zunachst die zeitabhangige Ionisationswahr-scheinlichkeit des neutralen Molekuls berechnet, anschließend wurde die Fragmentationdes Molekulions fur verschiedene Ionisationszeiten simuliert. Die Rechnung reproduzier-te den experimentellen Befund, dass die Zweiphotonendissoziation Abhangigkeiten vonder Pulsdauer und vom Isotop aufweist: Im Fall D2 vergroßern sich die Energien der Frag-mente mit abnehmender Pulsdauer. Bei H2 wird die Verschiebung nicht beobachtet. Wirhaben diese unerwarteten Effekte erklart, indem wir die klassische Bewegung der Kernein den zeitabhangigen adiabatischen Potentialen untersucht haben.

Weiterhin wurde die Dynamik eines linearen H2-Modellmolekuls mit einem Kernfrei-heitsgrad und zwei elektronischen Freiheitsgraden eingehend untersucht. Fur verschiede-ne Laserintensitaten haben wir die Wahrscheinlichkeiten fur Einfach- und Doppelioni-sation berechnet. Das Molekul zeigt starke nichtsequentielle Doppelionisation, und dasVerhaltnis zwischen Doppel- und Einfachionisation ist fast unabhangig von der Inten-sitat. Die Zeitentwicklung im Konfigurationsraum offenbart, dass die Dynamik der beidenElektronen sehr ahnlich wie bei Atomen verlauft. Sowohl Einfach- als auch Doppelionisa-tion finden bei Kern-Kern-Abstanden nahe dem Gleichgewichtsabstand im Grundzustandstatt. Dies steht im Gegensatz zu der Tatsache, dass die berechnete Ionisationswahrschein-lichkeit bei festgehaltenen Kernen bekanntermaßen stark erhoht ist, wenn deren Abstandin einem kritischen Bereich oberhalb des Gleichgewichtsabstands liegt. Unsere Ergebnis-se rechtfertigen die weitverbreitete Annahme, dass H2 beim Gleichgewichtsabstand ioni-siert wird. Das H

�2 -Molekulion verhalt sich grundsatzlich anders in der Hinsicht, dass es

sich zunachst hin zu hoheren Kern-Kern-Abstanden ausdehnt, bevor Ionisation stattfindet.Unter dem Einfluss starker Laserfelder geben Atome und Molekule koharente Strah-

lung ab, welche nicht nur die Frequenz des Lasers sondern auch hohere Frequenzenenthalt. Weil dies im Normalfall ganzzahlige Vielfache der Grundfrequenz sind, spricht

Zusammenfassung 99

man von der Erzeugung eines harmonischen Spektrums. Das von einem H2-Modellmole-kul erzeugte Spektrum wurde in dieser Arbeit mit demjenigen eines HD-Modellmolekulsverglichen. Der bedeutendste Unterschied ist der, dass ein ausgerichtetes HD-Molekulgerade und ungerade harmonische Ordnungen abstrahlt wahrend das H2-Molekul aus-schließlich ungerade Harmonische produziert. Die relative Starke der geraden Harmoni-schen ist am großten im Bereich hoher Frequenzen. In Gasphasenexperimenten beob-achtet man typischerweise die Strahlung unorientierter Molekule. In diesem Fall sindgerade Harmonische storungstheoretisch verboten. Unsere Rechnung bezieht sich auforientierte Molekule, so dass jene Auswahlregel verletzt sein kann. Wir haben gezeigt,dass die Erzeugung gerader Harmonischer bei HD ein Effekt außerhalb der Born-Oppen-heimer-Naherung ist. Die Matrixelemente, welche die nichtadiabatische Kopplung zwi-schen verschiedenen elektronischen Zustanden beschreiben, mischen im HD-Molekul alleZustande, wahrend sie in H2 nur Zustande gleicher Symmetrie koppeln.

Schließlich haben wir kurz diskutiert, wie die Ausbreitung eines Laserpulses in ei-nem Ensemble von Atomen oder Molekulen zu behandeln ist. Dies geschieht durch dieKombination der Maxwellgleichungen mit der Schrodingergleichung. Ein numerischesBeispiel zeigt, dass diese Methode Reflexion und Transmission von Licht beschreibt, unddass man das erzeugte harmonische Spektrum bequem durch Fourieranalyse des trans-mittierten Pulses erhalten kann.

Wir konnten eine Reihe experimenteller Ergebnisse reproduzieren und erklaren. Daru-berhinaus sagen wir zwei interessante Effekte vorher: Doppelionisation uber der Schwelleund die Erzeugung gerader Harmonischer durch Dynamik jenseits der Born-Oppenhei-mer-Naherung. Die Wechselwirkung intensiver Laserpulse mit Materie bietet noch immerviele offene Fragen, die zukunftige Forschungsprojekte motivieren werden. Wir nennennur einige Beispiele: Das Verhalten in elliptisch polarisierten Pulsen, Dreifachionisationoder Wellenpaketdynamik jenseits der Born-Oppenheimer-Naherung.

Bibliography

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Dank

Zunachst mochte ich mich bei meinen Betreuern Prof. Volker Engel und Prof. Hardy Großherzlich bedanken. Sie haben in einer vorbildlichen Kooperation zwischen den Institutenfur Physikalische Chemie und Theoretische Physik das Zustandekommen dieser Arbeitermoglicht. Ich habe enorm von der Gelegenheit profitiert, die Forschungsgebiete beiderArbeitsgruppen auskundschaften zu konnen. Die fachlichen Diskussionen fanden stets inherzlicher Atmosphare statt. Beiden Betreuuern habe ich es auch zu verdanken, dass ichzahlreiche Forschungsaufenthalte im Ausland absolvieren konnte.

Samtliche Mitglieder beider Gruppen haben mir stets mit ihrem Rat zur Seite gestan-den. Insbesondere bedanke ich mich bei Thomas Kreibich, mit dem ich gegen Ende derPromotion intensiv zusammengearbeitet habe. Oliver Rubner und Stefan Meyer habenmir die Eingewohnung im Institut fur Physikalische Chemie leicht gemacht.

Weiterhin bedanke ich mich (in zufalliger Reihenfolge) bei Klaus Capelle, Michael Braun,Miguel Marques, Zhenwen Shen, Martin Petersilka, Andreas Vetter, Vladimir Ermoshin,Holger Dietz, Martin Luders, Robert van Leeuwen, Achim Herwig, Ihsan Boustani, Nek-tarios Lathiotakis, Lars Fast, Nikitas Gidopoulos, Heiko Appel, Nicole Helbig und MarcoErdmann.

Lebenslauf

Personliche Daten:

Name: Manfred LeinGeburtsdatum: 23.6.1972Geburtsort: MunchbergStaatsangehorigkeit: deutsch

Schulbildung:

1978-1982 Grundschule Helmbrechts1982-1991 Math.-naturwissensch. Gymnasium Munchberg1991 Abitur

Grundwehrdienst:

Okt.-Dez. 1991 Grundausbildung in GoslarJan.-Sept.1992 Radarflugstellung Dobraberg

Studium und berufliche Tatigkeiten:

1992-1995 Studium der Physik an der Universitat Wurzburg1995-1996 Auslandsjahr an der University at Albany, USAMai 1996 Abschluss

”Master of Science“

Aug.-Sept. 1996 Werkstudententatigkeit am Max-Planck-Institutfur Quantenoptik in Garching

1997-1998 Diplomarbeit bei Prof. E.K.U. GroßApril 1998 Diplom in Physik

Promotion:

1998-2001 Promotion bei Prof. V. Engel und Prof. E.K.U. Groß

Auslandsaufenthalte wahrend der Promotion:

April 1999 bei Prof. J. Perdew an der Tulane University, New OrleansOkt. 1999 bei Prof. A. Beswick an der Universite Paul Sabatier, ToulouseDez. 1999 bei Prof. D. Neuhauser an der University of California

at Los AngelesJuli 2000 Sommerschule Femtochemie und Femtobiologie in El Escorial

Documents

Atoms and Molecules in Strong Laser Fields: Double ...users.physik.fu-berlin.de/~ag-gross/theses/lein_phd.pdf · The numerical treatment of atoms or molecules in strong laser pulses